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A configurable analog VLSI neural network with spiking neurons and self-regulating plastic synapses which classifies overlapping patterns M. Giulioni∗ Italian National Inst. of Health, Rome, Italy INFN-RM2, Rome, Italy giulioni@roma2.infn.it M. Pannunzi Italian National Inst. of Health, Rome, Italy INFN-RM1, Rome, Italy D. Badoni INFN-RM2, Rome, Italy V. Dante Italian National Inst. of Health, Rome, Italy INFN-RM1, Rome, Italy P. Del Giudice Italian National Inst. of Health, Rome, Italy INFN-RM1, Rome, Italy Abstract We summarize the implementation of an analog VLSI chip hosting a network of 32 integrate-and-fire (IF) neurons with spike-frequency adaptation and 2,048 Hebbian plastic bistable spike-driven stochastic synapses endowed with a selfregulating mechanism which stops unnecessary synaptic changes. The synaptic matrix can be flexibly configured and provides both recurrent and AER-based connectivity with external, AER compliant devices. We demonstrate the ability of the network to efficiently classify overlapping patterns, thanks to the self-regulating mechanism. 1 Introduction Neuromorphic analog, VLSI devices [12] try to derive organizational and computational principles from biologically plausible models of neural systems, aiming at providing in the long run an electronic substrate for innovative, bio-inspired computational paradigms. In line with standard assumptions in computational neuroscience, neuromorphic devices are endowed with adaptive capabilities through various forms of plasticity in the synapses which connect the neural elements. A widely adopted framework goes under the name of Hebbian learning, by which the efficacy of a synapse is potentiated (the post-synaptic effect of a spike is enhanced) if the pre- and post-synaptic neurons are simultaneously active on a suitable time scale. Different mechanisms have been proposed, some relying on the average firing rates of the pre- and post-synaptic neurons, (rate-based Hebbian learning), others based on tight constraints on the time lags between pre- and post-synaptic spikes (“Spike-Timing-Dependent-Plasticity”). The synaptic circuits described in what follows implement a stochastic version of rate-based Hebbian learning. In the last decade, it has been realized that general constraints plausibly met by any concrete implementation of a synaptic device in a neural network, bear profound consequences on ∗http://neural.iss.infn.it/ 1 the capacity of the network as a memory system. Specifically, once one accepts that a synaptic element can neither have an unlimited dynamic range (i.e. synaptic efficacy is bounded), nor can it undergo arbitrarily small changes (i.e. synaptic efficacy has a finite analog depth), it has been proven ([1], [7]) that a deterministic learning prescription implies an extremely low memory capacity, and a severe “palimpsest” property: new memories quickly erase the trace of older ones. It turns out that a stochastic mechanism provides a general, logically appealing and very efficient solution: given the pre- and post-synaptic neural activities, the synapse is still made eligible for changing its efficacy according to a Hebbian prescription, but it actually changes its state with a given probability. The stochastic element of the learning dynamics would imply ad hoc new elements, were it not for the fact that for a spike-driven implementation of the synapse, the noisy activity of the neurons in the network can provide the needed “noise generator” [7]. Therefore, for an efficient learning electronic network, the implementation of the neuron as a spiking element is not only a requirement of “biological plausibility”, but a compelling computational requirement. Learning in networks of spiking IF neurons with stochastic plastic synapses has been studied theoretically [7], [10], [2], and stochastic, bi-stable synaptic models have been implemented in silicon [8], [6]. One of the limitations so far, both at the theoretical and the implementation level, has been the artificially simple statistics of the stimuli to be learnt (e.g., no overlap between their neural representations). Very recently in [4] a modification of the above stochastic, bi-stable synaptic model has been proposed, endowed with a regulatory mechanism termed “stop learning” such that synaptic up or down-regulation depends on the average activity of the postsynaptic neuron in the recent past; a synapse pointing to a neuron that is found to be highly active, or poorly active, should not be further potentiated or depressed, respectively. The reason behind the prescription is essentially that for correlated patterns to be learnt by the network, a successful strategy should de-emphasize the coherent synaptic Hebbian potentiation that would result for the overlapping part of the synaptic matrix, and that would ultimately spoil the ability to distinguish the patterns. A detailed learning strategy along this line was proven in [13] to be appropriate for linearly separable patterns for a Perceptron-like network; the extension to spiking and recurrent networks is currently studied. In section 2 we give an overview of the chip architecture and of the implemented synaptic model. In section 3 we show an example of the measures effectuated on the chip useful to characterize the synaptic and neuronal parameters. In section 4 we report some characterization results compared with a theoretical prediction obtained from a chip-oriented simulation. The last paragraph describes chip performances in a simple classification task, and illustrate the improvement brought about by the stop-learning mechanism. 2 Chip architecture and main features The chip, already described in [3] implements a recurrent network of 32 integrate-and-fire neurons with spike-frequency adaptation and bi-stable, stochastic, Hebbian synapses. A completely reconfigurable synaptic matrix supports up to all-to-all recurrent connectivity, and AER-based external connectivity. Besides establishing an arbitrary synaptic connectivity, the excitatory/inhibitory nature of each synapse can also be set. The implemented neuron is the IF neuron with constant leakage term and a lower bound for the membrane potential V (t) introduced in [12] and studied theoretically in [9]. The circuit is borrowed from the low-power design described in [11], to which we refer the reader for details. Only 2 neurons can be directly probed (i.e., their “membrane potential” sampled), while for all of them the emitted spikes can be monitored via AER [5]. The dendritic tree of each neuron is composed of up to 31 activated recurrent synapses and up to 32 activated external, AER ones. For the recurrent synapses, each impinging spike triggers short-time (and possibly long-term) changes in the state of the synapse, as detailed below. Spikes from neurons outside the chip come in the form of AER events, and are targeted to the correct AER synapse by the X-Y Decoder. Synapses which are set to be excitatory, either AER or recurrent are plastic; inhibitory synapses are fixed. Spikes generated by the neurons in the chip are arbitrated for access to the AER bus for monitoring and/or mapping to external targets. The synaptic circuit described in [3] implements the model proposed in [4] and briefly motivated in the Introduction. The synapse possesses only two states of efficacy (a bi-stable device): the internal synaptic dynamics is associated with an internal variable X; when X > θX the efficacy is set to be 2 potentiated, otherwise is set to be depressed. X is subjected to short-term, spike-driven dynamics: upon the arrival of an impinging spike, X is candidate for an upward or downward jump, depending on the instantaneous value of the post-synaptic potential Vpost being above or below a threshold θV . The jump is actually performed or not depending on a further variable as explained below. In the absence of intervening spikes X is forced to drift towards a “high” or “low” value depending on whether the last jump left it above or below θX. This preserves the synaptic efficacy on long time scale. A further variable is associated with the post-synaptic neuron dynamics, which essentially measures the average firing activity. Following [4], by analogy with the role played by the intracellular concentration of calcium ions upon spike emission, we will call it a “calcium variable” C(t). C(t) undergoes an upward jump when the postsynaptic neuron emits a spike, and linearly decays between two spikes. It therefore integrates the spikes sequence and, when compared to suitable thresholds as detailed below, it determines which candidate synaptic jumps will be allowed to occur; for example, it can constrain the synapse to stop up-regulating because the post-synaptic neuron is already very active. C(t) acts as a regulatory element of the synaptic dynamics. The resulting short-term dynamics for the internal synaptic variable X is described by the following conditions: X(t) →X(t)+Jup if Vpost(t) > θV and VT H1 < C(t) < VT H3; X(t) →X(t)−Jdw if Vpost(t) ≤θV and VT H1 < C(t) < VT H2 where Jup and Jdw are positive constants. Detailed description of circuits implementing these conditions can be found in [3]. In figure 1 we illustrate the effect of the calcium dynamics on X. Increasing input forces the postsynaptic neuron to fire at increasing frequencies. As long as C(t) < VT H2 = VT H3 X undergoes both up and down jumps. When C(t) > VT H2 = VT H3 jumps are inhibited and X is forced to drift towards its lower bound. V (t) post C(t) X(t) V (t) pre 40 ms 2V 1V 1V 1V VTH2 Figure 1: Illustrative example of the stop-learning mechanism (see text). Top to bottom: postsynaptic neuron potential Vpost, calcium variable C, internal synaptic variable X, pre-synaptic neuron potential Vpre 3 LTP/LTD probabilities: measurement VS chip-oriented simulation We report synapse potentiation (LTP) / depression (LTD)from the chip and we compare experimental results to simulations. For each synapse in a subset of 31, we generate a pre-synaptic poisson spike train at 70 Hz. The post synaptic neuron is forced to fire a poisson spike train by applying an external DC current and a poisson train of inhibitory spikes through AER. Setting to zero both the potentiated and depressed efficacies, the activity of the post-synaptic neuron can be easily tuned by varying the amplitude of the DC current and the frequency of the inhibitory AER train. We initialize the 31 (AER) synapses to depressed (potentiated) and we monitor the post-synaptic neuron activity during a stimulation 3 trial lasting 0.5 seconds. At the end of the trial we read the synaptic state using an AER protocol developed to this purpose. For each chosen value of the post-synaptic firing rate, we evaluate the probability to find synapses in a potentiated (depressed) state repeating the test 50 times. The results reported in figure 2 (solid lines) represent the average LTP and LTD probabilities per trail over the 31 synapses. Tests were performed both with active and inactive Calcium mechanism. When calcium mechanism is inactive, the LTP is monotonically increasing with the post-synaptic firing rate while when the calcium circuit is activated the LTP probability has a max form Vpost around 80 Hz. Identical tests were also run in simulation (dashed curves in figure 2). For the purpose of a meaningful comparison with the chip behaviour relevant parameter affecting neural and synaptic dynamics and their distributions (due to inhomogenities and mismatches) are characterized. Simulated and measured data are in qualitative agreement. The parameters we chose for these tests are the same used for the classification task described in the next paragraph. 0 50 100 150 200 250 300 0 0.1 0.2 0.3 0.4 0.5 0.6 Fraction of potentiated synapses w+ νpost [Hz] Experiment: solid line Simulation: dashed line Figure 2: Transition probabilities. Red and blue lines are LTP probabilities with and without calcium stop-learning mechanism respectively. Gray lines are LTD probabilities without calcium stoplearning mechanism, the case LTD with Ca mechanism is not shown. Error bars are standard deviations over the 50 trials 4 Learning overlapping patterns We configured the synaptic matrix to have a perceptron like network with 1 output and 32 inputs (32 AER synapses). 31 synapses are set as plastic excitatory ones, the 32nd is set as inhibitory and used to modulate the post-synpatic neuron activity. Our aim is to teach the perceptron to classify two patterns through a semi-supervised learning strategy: “Up” and “Down”. We expect that after learning the perceptron will respond with high output frequency for pattern “Up” and with low output frequency for pattern “Down”. The self regulating Ca mechanism is exploited to improve performances when Up and Down patterns have a significant overlap. The learning is semi-supervised: for each pattern a “teacher” input is sent to the output neuron steering its activity to be high or low, as desired. At the end of the learning period the “teacher” is turned off and the perceptron output is driven only by the input stimuli: in this conditions its classification ability is tested. We present learning performances for input patterns with increasing overlap, and demonstrate the effect of the stop learning mechanism (overlap ranging from 6 to 14 synapses). Upon stimulation active pre-synaptic inputs are poisson spike trains at 70 Hz, while inactive inputs are poisson spike trains at 10 Hz. Each trial lasts half a second. Up and Down patterns are randomly presented with equal probability. The teaching signal, a combination of an excitatory constant cur4 rent and of an inhibitory AER spike train, forces the output firing rate to 50 or 0.5 Hz. One run lasts for 150 trials which is sufficient for the stabilization of the output frequencies. At the end of each trial we turn off the teaching signal, we freeze the synaptic dynamics and we read the state of each synapse using an AER protocol developed for this purpose. In these conditions we performed a 5 seconds test (“Checking Phase”) to measure the perceptron frequencies when pattern Up or pattern Down are presented. Each experiment includes 50 runs. For each run we change: a) the “definition” of patterns Up and Down: inputs activated by pattern Up and Down are chosen randomly at the beginning of each run; b) the initial synaptic state, with the constraint that only about 30 % of the synapses are potentiated; c) the stimulation sequence. For the first experiment we turned off the stop learning mechanism and we chose orthogonalpatterns. In this case the perceptron was able to correctly classify the stimuli: after about 50 trials, choosing a suitable threshold, one can discriminate the perceptron ouput to different patterns (lower left panel on figure 4). The output frequency separation slightly increases until trial number 100 remaining almost stable after that point. We then studied the case of overlapped patterns both with active and inactive Calcium mechanism. We repeated the experiment with an increasing overlap: 6, 10 and 14. (implying an increase in the coding level from 0.5 for the orthogonal case to 0.7 for the overlap equal to 14). Only the threshold Kup high is active (the threshold above which up jumps are inhibnited). The Calcium circuit parameters are tuned so that the Ca variable passes Kup high for the mean firing rate of the postsynaptic neuron around 80 Hz. We show in figure 3 the distributions of the potentiated fraction of the synapses over the 50 runs at different stages along the run for overlap 10 with inactive (upper panels) and active (lower panels) calcium mechanism. We divided synapses in three subgroups: Up (red) synapses with pre-synaptic input activated solely by Up pattern, Down (blue) synapses with pre-synaptic inputs activated only by Down pattern, and Overlap (green) synapses with pre-synpatic inputs activated by both pattern Up and Down. The state of the synapses is recorded after every learning step. Accumulating statistics over the 50 runs we obtain the distributions reported in figure 3. The fraction of potentiated synapses is calculated over the number of synapses belonging to each subgroup. When the stop learning mechanism is inactive, at the end of the experiment, the green Ca mechanism inactive 0 0.5 1 0 0.2 0.4 0.6 0.8 1 trial 2 w+ P(w+) 0 0.5 1 0 0.2 0.4 0.6 0.8 1 trial 50 w+ 0 0.5 1 0 0.2 0.4 0.6 0.8 1 trial 100 w+ 0 0.5 1 0 0.2 0.4 0.6 0.8 1 trial 150 w+ synapses Overlap synapses Up synapses Down Ca mechanism active 0 0.5 1 0 0.2 0.4 0.6 0.8 1 trial 2 w+ P(w+) 0 0.5 1 0 0.2 0.4 0.6 0.8 1 trial 50 w+ 0 0.5 1 0 0.2 0.4 0.6 0.8 1 trial 100 w+ 0 0.5 1 0 0.2 0.4 0.6 0.8 1 trial 150 w+ synapses Overlap synapses Up synapses Down Figure 3: Distribution of the fraction of potentiated synapses. The number of inputs belonging to both patterns is 10. distribution of overlap synapses is broad, when the Calcium mechanism is active, synapses overlap tend to be depotentiated. This result is the “microscopic” effect of the stop learning mechanism: once the number of potentiated synapses is sufficient to drive the perceptron output frequency above 80 Hz, the overlap synapses tend to be depotentiated. Overlap synapses would be pushed half of the 5 times to the potentiated state and half of the times to the depressed state, so that it is more likely for the Up synapses to reach earlier the potentiated state. When the stop learning mechanism is active, the potentiated synapses are enogh to drive the output neuron about 80 Hz, further potentiation is inhibited for all synapses so that overlap synapses get depressed on average. This happens under the condition that the transition probability are sufficiently small to avoid that at each trial the learning is completely disrupted. The distribution of the output frequencies for increasing overlap is illustrated in figure 4 (Ca mechanism inactive in the upper panels, active for the lower panels). The frequencies are recorded during the “checking phase”. In blue the histograms of the output frequency for the down pattern, in red those for up pattern. It is clear from the figure that the output frequency distribution remain well separated even for high overlap when the Calcium mechanism is active. A quantitative parameter to describe the distribution separation is δ = νup −νdn σ2νup + σ2νdn (1) δ values are summarized in table 1. Ca mechanism inactive 0 100 200 0 0.2 0.4 0.6 0.8 1 νck [Hz] P(νck) Overlap 0 0 100 200 0 0.2 0.4 0.6 0.8 1 νck [Hz] Overlap 6 0 100 200 0 0.2 0.4 0.6 0.8 1 νck [Hz] Overlap 10 0 100 200 0 0.2 0.4 0.6 0.8 1 νck [Hz] Overlap 14 Pattern Down Pattern Up Ca mechanism active 0 100 200 0 0.2 0.4 0.6 0.8 1 νck [Hz] Overlap 0 P(νck) 0 100 200 0 0.2 0.4 0.6 0.8 1 νck [Hz] Overlap 6 0 100 200 0 0.2 0.4 0.6 0.8 1 νck [Hz] Overlap 10 0 100 200 0 0.2 0.4 0.6 0.8 1 νck [Hz] Overlap 14 Pattern Down Pattern Up Figure 4: Distributions of perceptron frequencies after learning two overlapped patterns. Blue bars refer to pattern Down stimulation, red bars refers to pattern Up. Each panel refers to overlap. Table 1: Discrimination power [seconds] overlap 0 overlap 6 overlap 10 overlap 14 Ca OFF 4.39 1.87 1.59 0.99 Ca ON 5.29 2.20 1.88 1.66 For each run the number of potentiated synapses is different due to the random choices of Up, Down and Overlap synapses for each run and the mismatches affecting the behavior of different synapses. The failure of the discrimination for high overlap in the absence of this stop learning mechanism is due to the fact that the number of potentiated synapses can overcome the effect of the teaching signal for the down pattern. The Calcium mechanism, defining a maximum number of allowed potentiated synapses, limits this problem. This offer the possibility of establishing a priori threshold to discriminate the perceptron outputs on the basis of the frequency corresponding to the maximum value of the LTP probability curve. 6 5 Conclusions We briefly illustrate an analog VLSI chip implementing a network of 32 IF neurons and 2,048 reconfigurable, Hebbian, plastic, stop-learning synapses. Circuit parameters has been measured as well as their dispersion across the chip. Using these data a chip-oriented simulation was set up and its results, compared to experimental ones, demonstrate that circuits behavior follow the theoretical predictions. Once configured the network as a perceptron (31 AER synapses and one output neuron), a classification task has been performed. Stimuli with an increasing overlap have been used. The results show the ability of the network to efficiently classify the presented patterns as well as the improvement of the performances due to the calcium stop-learning mechanism. References [1] D.J. Amit and S. Fusi. Neural Computation, 6:957, 1994. [2] D.J. Amit and G. Mongillo. Neural Computation, 15:565, 2003. [3] D. Badoni, M. Giulioni, V. Dante, and P. Del Giudice. In Proc. IEEE International Symposium on Circuits and Systems ISCAS06, pages 1227–1230, 2006. [4] J.M. Brader, W. Senn, and S. Fusi. Neural Computation (in press), 2007. [5] V. Dante, P. Del Giudice, and A. M. Whatley. The neuromorphic engineer newsletter. 2005. [6] E. Chicca et al. IEEE Transactions on Neural Networks, 14(5):1297, 2003. [7] S. Fusi. Biological Cybernetics, 87:459, 2002. [8] S. Fusi, M. Annunziato, D. Badoni, A. Salamon, and D.J. Amit. Neural Computation, 12:2227, 2000. [9] S. Fusi and M. Mattia. Neural Computation, 11:633, 1999. [10] P. Del Giudice, S. Fusi, and M. Mattia. Journal of Physiology Paris, 97:659, 2003. [11] G. Indiveri. In Proc. IEEE International Symposium on Circuits and Systems, 2003. [12] C. Mead. Analog VLSI and neural systems. Addison-Wesley, 1989. [13] W. Senn and S. Fusi. 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The discriminant center-surround hypothesis for bottom-up saliency Dashan Gao Vijay Mahadevan Nuno Vasconcelos Department of Electrical and Computer Engineering University of California, San Diego {dgao, vmahadev, nuno}@ucsd.edu Abstract The classical hypothesis, that bottom-up saliency is a center-surround process, is combined with a more recent hypothesis that all saliency decisions are optimal in a decision-theoretic sense. The combined hypothesis is denoted as discriminant center-surround saliency, and the corresponding optimal saliency architecture is derived. This architecture equates the saliency of each image location to the discriminant power of a set of features with respect to the classification problem that opposes stimuli at center and surround, at that location. It is shown that the resulting saliency detector makes accurate quantitative predictions for various aspects of the psychophysics of human saliency, including non-linear properties beyond the reach of previous saliency models. Furthermore, it is shown that discriminant center-surround saliency can be easily generalized to various stimulus modalities (such as color, orientation and motion), and provides optimal solutions for many other saliency problems of interest for computer vision. Optimal solutions, under this hypothesis, are derived for a number of the former (including static natural images, dense motion fields, and even dynamic textures), and applied to a number of the latter (the prediction of human eye fixations, motion-based saliency in the presence of ego-motion, and motion-based saliency in the presence of highly dynamic backgrounds). In result, discriminant saliency is shown to predict eye fixations better than previous models, and produces background subtraction algorithms that outperform the state-of-the-art in computer vision. 1 Introduction The psychophysics of visual saliency and attention have been extensively studied during the last decades. As a result of these studies, it is now well known that saliency mechanisms exist for a number of classes of visual stimuli, including color, orientation, depth, and motion, among others. More recently, there has been an increasing effort to introduce computational models for saliency. One approach that has become quite popular, both in the biological and computer vision communities, is to equate saliency with center-surround differencing. It was initially proposed in [12], and has since been applied to saliency detection in both static imagery and motion analysis, as well as to computer vision problems such as robotics, or video compression. While difference-based modeling is successful at replicating many observations from psychophysics, it has three significant limitations. First, it does not explain those observations in terms of fundamental computational principles for neural organization. For example, it implies that visual perception relies on a linear measure of similarity (difference between feature responses in center and surround). This is at odds with well known properties of higher level human judgments of similarity, which tend not to be symmetric or even compliant with Euclidean geometry [20]. Second, the psychophysics of saliency offers strong evidence for the existence of both non-linearities and asymmetries which are not easily reconciled with this model. Third, although the center-surround hypothesis intrinsically poses 1 saliency as a classification problem (of distinguishing center from surround), there is little basis on which to justify difference-based measures as optimal in a classification sense. From an evolutionary perspective, this raises questions about the biological plausibility of the difference-based paradigm. An alternative hypothesis is that all saliency decisions are optimal in a decision-theoretic sense. This hypothesis has been denoted as discriminant saliency in [6], where it was somewhat narrowly proposed as the justification for a top-down saliency algorithm. While this algorithm is of interest only for object recognition, the hypothesis of decision theoretic optimality is much more general, and applicable to any form of center-surround saliency. This has motivated us to test its ability to explain the psychophysics of human saliency, which is better documented for the bottom-up neural pathway. We start from the combined hypothesis that 1) bottom-up saliency is based on centersurround processing, and 2) this processing is optimal in a decision theoretic sense. In particular, it is hypothesized that, in the absence of high-level goals, the most salient locations of the visual field are those that enable the discrimination between center and surround with smallest expected probability of error. This is referred to as the discriminant center-surround hypothesis and, by definition, produces saliency measures that are optimal in a classification sense. It is also clearly tied to a larger principle for neural organization: that all perceptual mechanisms are optimal in a decision-theoretic sense. In this work, we present the results of an experimental evaluation of the plausibility of the discriminant center-surround hypothesis. Our study evaluates the ability of saliency algorithms, that are optimal under this hypothesis, to both • reproduce subject behavior in classical psychophysics experiments, and • solve saliency problems of practical significance, with respect to a number of classes of visual stimuli. We derive decision-theoretic optimal center-surround algorithms for a number of saliency problems, ranging from static spatial saliency, to motion-based saliency in the presence of egomotion or even complex dynamic backgrounds. Regarding the ability to replicate psychophysics, the results of this study show that discriminant saliency not only replicates all anecdotal observations that can be explained by linear models, such as that of [12], but can also make (surprisingly accurate) quantitative predictions for non-linear aspects of human saliency, which are beyond the reach of the existing approaches. With respect to practical saliency algorithms, they show that discriminant saliency not only is more accurate than difference-based methods in predicting human eye fixations, but actually produces background subtraction algorithms that outperform the state-of-the-art in computer vision. In particular, it is shown that, by simply modifying the probabilistic models employed in the (decision-theoretic optimal) saliency measure - from well known models of natural image statistics, to the statistics of simple optical-flow motion features, to more sophisticated dynamic texture models - it is possible to produce saliency detectors for either static or dynamic stimuli, which are insensitive to background image variability due to texture, egomotion, or scene dynamics. 2 Discriminant center-surround saliency A common hypothesis for bottom-up saliency is that the saliency of each location is determined by how distinct the stimulus at the location is from the stimuli in its surround (e.g., [11]). This hypothesis is inspired by the ubiquity of “center-surround” mechanisms in the early stages of biological vision [10]. It can be combined with the hypothesis of decision-theoretic optimality, by defining a classification problem that equates • the class of interest, at location l, with the observed responses of a pre-defined set of features X within a neighborhood W1 l of l (the center), • the null hypothesis with the responses within a surrounding window W0 l (the surround ), The saliency of location l∗is then equated with the power of the feature set X to discriminate between center and surround. Mathematically, the feature responses within the two windows are interpreted as observations drawn from a random process X(l) = (X1(l), . . . , Xd(l)), of dimension d, conditioned on the state of a hidden random variable Y (l). The observed feature vector at any location j is denoted by x(j) = (x1(j), . . . , xd(j)), and feature vectors x(j) such that j ∈Wc l , c ∈ 2 {0, 1} are drawn from class c (i.e., Y (l) = c), according to conditional densities PX(l)|Y (l)(x|c). The saliency of location l, S(l), is quantified by the mutual information between features, X, and class label, Y , S(l) = Il(X; Y ) = X c Z pX(l),Y (l)(x, c) log pX(l),Y (l)(x, c) pX(l)(x)pY (l)(c)dx. (1) The l subscript emphasizes the fact that the mutual information is defined locally, within Wl. The function S(l) is referred to as the saliency map. 3 Discriminant saliency detection in static imagery Since human saliency has been most thoroughly studied in the domain of static stimuli, we first derive the optimal solution for discriminant saliency in this domain. We then study the ability of the discriminant center-surround saliency hypothesis to explain the fundamental properties of the psychophysics of pre-attentive vision. 3.1 Feature decomposition The building blocks of the static discriminant saliency detector are shown in Figure 1. The first stage, feature decomposition, follows the proposal of [11], which closely mimics the earliest stages of biological visual processing. The image to process is first subject to a feature decomposition into an intensity map and four broadly-tuned color channels, I = (r + g + b)/3, R = ⌊˜r −(˜g +˜b)/2⌋+, G = ⌊˜g −(˜r + ˜b)/2⌋+, B = ⌊˜b −˜(r + ˜g)/2⌋+, and Y = ⌊(˜r + ˜g)/2 −|˜r −˜g|/2⌋+, where ˜r = r/I, ˜g = g/I,˜b = b/I, and ⌊x⌋+ = max(x, 0). The four color channels are, in turn, combined into two color opponent channels, R −G for red/green and B −Y for blue/yellow opponency. These and the intensity map are convolved with three Mexican hat wavelet filters, centered at spatial frequencies 0.02, 0.04 and 0.08 cycle/pixel, to generate nine feature channels. The feature space X consists of these channels, plus a Gabor decomposition of the intensity map, implemented with a dictionary of zero-mean Gabor filters at 3 spatial scales (centered at frequencies of 0.08, 0.16, and 0.32 cycle/pixel) and 4 directions (evenly spread from 0 to π). 3.2 Leveraging natural image statistics In general, the computation of (1) is impractical, since it requires density estimates on a potentially high-dimensional feature space. This complexity can, however, be drastically reduced by exploiting a well known statistical property of band-pass natural image features, e.g. Gabor or wavelet coefficients: that features of this type exhibit strongly consistent patterns of dependence (bow-tie shaped conditional distributions) across a very wide range of classes of natural imagery [2, 9, 21]. The consistency of these feature dependencies suggests that they are, in general, not greatly informative about the image class [21, 2] and, in the particular case of saliency, about whether the observed feature vectors originate in the center or surround. Hence, (1) can usually be well approximated by the sum of marginal mutual informations [21]1, i.e., S(l) = d X i=1 Il(Xi; Y ). (2) Since (2) only requires estimates of marginal densities, it has significantly less complexity than (1). This complexity can, indeed, be further reduced by resorting to the well known fact that the marginal densities are accurately modeled by a generalized Gaussian distribution (GGD) [13]. In this case, all computations have a simple closed form [4] and can be mapped into a neural network that replicates the standard architecture of V1: a cascade of linear filtering, divisive normalization, quadratic nonlinearity and spatial pooling [7]. 1Note that this approximation does not assume that the features are independently distributed, but simply that their dependencies are not informative about the class. 3 Feature decomposition 6 Color (R/G, B/Y) Intensity Orientation Feature maps Feature saliency maps Saliency map Figure 1: Bottom-up discriminant saliency detector. (a) 5 10 20 30 40 50 60 70 80 90 0 0.5 1 1.5 2 2.5 3 Orientation contrast (deg) Saliency 0 10 20 30 40 50 60 70 80 90 1.7 1.75 1.8 1.85 1.9 Orientation contrast (deg) Saliency (b) (c) Figure 2: The nonlinearity of human saliency responses to orientation contrast [14] (a) is replicated by discriminant saliency (b), but not by the model of [11] (c). 3.3 Consistency with psychophysics To evaluate the consistency of discriminant saliency with psychophysics, we start by applying the discriminant saliency detector to a series of displays used in classical studies of visual attention [18, 19, 14]2. In [7], we have shown that discriminant saliency reproduces the anecdotal properties of saliency - percept of pop-out for single feature search, disregard of feature conjunctions, and search asymmetries for feature presence vs. absence - that have previously been shown possible to replicate with linear saliency models [11]. Here, we focus on quantitative predictions of human performance, and compare the output of discriminant saliency with both human data and that of the differencebased center-surround saliency model [11]3. The first experiment tests the ability of the saliency models to predict a well known nonlinearity of human saliency. Nothdurft [14] has characterized the saliency of pop-out targets due to orientation contrast, by comparing the conspicuousness of orientation defined targets and luminance defined ones, and using luminance as a reference for relative target salience. He showed that the saliency of a target increases with orientation contrast, but in a non-linear manner: 1) there exists a threshold below which the effect of pop-out vanishes, and 2) above this threshold saliency increases with contrast, saturating after some point. The results of this experiment are illustrated in Figure 2, which presents plots of saliency strength vs orientation contrast for human subjects [14] (in (a)), for discriminant saliency (in (b)), and for the difference-based model of [11]. Note that discriminant saliency closely predicts the strong threshold and saturation effects characteristic of subject performance, but the difference-based model shows no such compliance. The second experiment tests the ability of the models to make accurate quantitative predictions of search asymmetries. It replicates the experiment designed by Treisman [19] to show that the asymmetries of human saliency comply with Weber’s law. Figure 3 (a) shows one example of the displays used in the experiment, where the central target (vertical bar) differs from distractors (a set of identical vertical bars) only in length. Figure 3 (b) shows a scatter plot of the values of discriminant saliency obtained across the set of displays. Each point corresponds to the saliency at the target location in one display, and the dashed line shows that, like human perception, discriminant saliency follows Weber’s law: target saliency is approximately linear in the ratio between the difference of target/distractor length (∆x) and distractor length (x). For comparison, Figure 3 (c) presents the corresponding scatter plot for the model of [11], which clearly does not replicate human performance. 4 Applications of discriminant saliency We have, so far, presented quantitative evidence in support of the hypothesis that pre-attentive vision implements decision-theoretical center-surround saliency. This evidence is strengthened by the 2For the computation of the discriminant saliency maps, we followed the common practice of psychophysics and physiology [18, 10], to set the size of the center window to a value comparable to that of the display items, and the size of the surround window is 6 times of that of the center. Informal experimentation has shown that the saliency results are not substantively affected by variations around the parameter values adopted. 3Results obtained with the MATLAB implementation available in [22]. 4 (a) 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 ∆ x/x Saliency 0 0.2 0.4 0.6 0.8 1.65 1.7 1.75 1.8 1.85 1.9 1.95 ∆ x/x Saliency (b) (c) Figure 3: An example display (a) and performance of saliency detectors (discriminant saliency (b) and [11] (c)) on Weber’s law experiment. 0.8 0.85 0.9 0.95 0.98 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Inter−subject ROC area Saliency ROC area discriminant saliency Itti et al. Bruce et al. Figure 4: Average ROC area, as a function of inter-subject ROC area, for the saliency algorithms. Saliency model Discriminant Itti et al. [11] Bruce et al. [1] ROC area 0.7694 0.7287 0.7547 Table 1: ROC areas for different saliency models with respect to all human fixations. already mentioned one-to-one mapping between the discriminant saliency detector proposed above and the standard model for the neurophysiology of V1 [7]. Another interesting property of discriminant saliency is that its optimality is independent of the stimulus dimension under consideration, or of specific feature sets. In fact, (1) can be applied to any type of stimuli, and any type of features, as long as it is possible to estimate the required probability distributions from the center and surround neighborhoods. This encouraged us to derive discriminant saliency detectors for various computer vision applications, ranging from the prediction of human eye fixations, to the detection of salient moving objects, to background subtraction in the context of highly dynamic scenes. The outputs of these discriminant saliency detectors are next compared with either human performance, or the state-of-the-art in computer vision for each application. 4.1 Prediction of eye fixations on natural images We start by using the static discriminant saliency detector of the previous section to predict human eye fixations. For this, the saliency maps were compared to the eye fixations of human subjects in an image viewing task. The experimental protocol was that of [1], using fixation data collected from 20 subjects and 120 natural images. Under this protocol, all saliency maps are first quantized into a binary mask that classifies each image location as either a fixation or non-fixation [17]. Using the measured human fixations as ground truth, a receiver operator characteristic (ROC) curve is then generated by varying the quantization threshold. Perfect prediction corresponds to an ROC area (area under the ROC curve) of 1, while chance performance occurs at an area of 0.5. The predictions of discriminant saliency are compared to those of the methods of [11] and [1]. Table 1 presents average ROC areas for all detectors, across the entire image set. It is clear that discriminant saliency achieves the best performance among the three detectors. For a more detailed analysis, we also plot (in Figure 4) the ROC areas of the three detectors as a function of the “intersubject” ROC area (a measure of the consistency of eye movements among human subjects [8]), for the first two fixations - which are more likely to be driven by bottom-up mechanisms than the later ones [17]. Again, discriminant saliency exhibits the strongest correlation with human performance, this happens at all levels of inter-subject consistency, and the difference is largest when the latter is strong. In this region, the performance of discriminant saliency (.85) is close to 90% of that of humans (.95), while the other two detectors only achieve close to 85% (.81). 4.2 Discriminant saliency on motion fields Similarly to the static case, center-surround discriminant saliency can produce motion-based saliency maps if combined with motion features. We have implemented a simple motion-based detector by computing a dense motion vector map (optical flow) between pairs of consecutive images, and using the magnitude of the motion vector at each location as motion feature. The probability distributions of this feature, within center and surround, were estimated with histograms, and the motion saliency maps computed with (2). 5 Figure 5: Optical flow-based saliency in the presence of egomotion. Despite the simplicity of our motion representation, the discriminant saliency detector exhibits interesting performance. Figure 5 shows several frames (top row) from a video sequence, and their discriminant motion saliency maps (bottom row). The sequence depicts a leopard running in a grassland, which is tracked by a moving camera. This results in significant variability of the background, due to egomotion, making the detection of foreground motion (leopard), a non-trivial task. As shown in the saliency maps, discriminant saliency successfully disregards the egomotion component of the optical flow, detecting the leopard as most salient. 4.3 Discriminant Saliency with dynamic background While the results of Figure 5 are probably within the reach of previously proposed saliency models, they illustrate the flexibility of discriminant saliency. In this section we move to a domain where traditional saliency algorithms almost invariably fail. This consists of videos of scenes with complex and dynamic backgrounds (e.g. water waves, or tree leaves). In order to capture the motion patterns characteristic of these backgrounds it is necessary to rely on reasonably sophisticated probabilistic models, such as the dynamic texture model [5]. Such models are very difficult to fit in the conventional, e.g. difference-based, saliency frameworks but naturally compatible with the discriminant saliency hypothesis. We next combine discriminant center-surround saliency with the dynamic texture model, to produce a background-subtraction algorithm for scenes with complex background dynamics. While background subtraction is a classic problem in computer vision, there has been relatively little progress for these type of scenes (e.g. see [15] for a review). A dynamic texture (DT) [5, 3] is an autoregressive, generative model for video. It models the spatial component of the video and the underlying temporal dynamics as two stochastic processes. A video is represented as a time-evolving state process xt ∈Rn, and the appearance of a frame yt ∈Rm is a linear function of the current state vector with some observation noise. The system equations are xt = Axt−1 + vt yt = Cxt + wt (3) where A ∈Rn×n is the state transition matrix, C ∈Rm×n is the observation matrix. The state and observation noise are given by vt ∼iid N(0, Q,) and wt ∼iid N(0, R), respectively. Finally, the initial condition is distributed as x1 ∼N(µ, S). Given a sequence of images, the parameters of the dynamic texture can be learned for the center and surround regions at each image location, enabling a probabilistic description of the video, with which the mutual information of (2) can be evaluated. We applied the dynamic texture-based discriminant saliency (DTDS) detector to three video sequences containing objects moving in water. The first (Water-Bottle from [23]) depicts a bottle floating in water which is hit by rain drops, as shown in Figure 7(a). The second and third, Boat and Surfer, are composed of boats/surfers moving in water, and shown in Figure 8(a) and 9(a). These sequences are more challenging, since the micro-texture of the water surface is superimposed on a lower frequency sweeping wave (Surfer) and interspersed with high frequency components due to turbulent wakes (created by the boat, surfer, and crest of the sweeping wave). Figures 7(b), 8(b) and 9(b), show the saliency maps produced by discriminant saliency for the three sequences. The DTDS detector performs surprisingly well, in all cases, at detecting the foreground objects while ignoring the movements of the background. In fact, the DTDS detector is close to an ideal backgroundsubtraction algorithm for these scenes. 6 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Detection rate (DR) False positive rate (FPR) Discriminant Salliency GMM 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Detection rate (DR) False positive rate (FPR) Discriminant Saliency GMM 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Detection rate (DR) False positive rate (FPR) Discriminant Salliency GMM (a) (b) (c) Figure 6: Performance of background subtraction algorithms on: (a) Water-Bottle, (b) Boat, and (c) Surfer. (a) (b) (c) Figure 7: Results on Bottle: (a) original; b) discriminant saliency with DT; and c) GMM model of [16, 24]. For comparison, we present the output of a state-of-the-art background subtraction algorithm, a Gaussian mixture model (GMM) [16, 24]. As can be seen in Figures 7(c), 8(c) and 9(c), the resulting foreground detection is very noisy, and cannot adapt to the highly dynamic nature of the water surface. Note, in particular, that the waves produced by boat and surfer, as well as the sweeping wave crest, create serious difficulties for this algorithm. Unlike the saliency maps of DTDS, the resulting foreground maps would be difficult to analyze by subsequent vision (e.g. object tracking) modules. To produce a quantitative comparison of the saliency maps, these were thresholded at a large range of values. The results were compared with ground-truth foreground masks, and an ROC curve produced for each algorithm. The results are shown in Figure 6, where it is clear that while DTDS tends to do well on these videos, the GMM based background model does fairly poorly. References [1] N. D. Bruce and J. K. Tsotsos. Saliency based on information maximization. In Proc. NIPS, 2005. [2] R. Buccigrossi and E. Simoncelli. Image compression via joint statistical characterization in the wavelet domain. IEEE Transactions on Image Processing, 8:1688–1701, 1999. [3] A. B. Chan and N. Vasconcelos. Modeling, clustering, and segmenting video with mixtures of dynamic textures. IEEE Trans. PAMI, In Press. [4] M. N. Do and M. Vetterli. Wavelet-based texture retrieval using generalized gaussian density and kullback-leibler distance. IEEE Trans. Image Processing, 11(2):146–158, 2002. [5] G. Doretto, A. Chiuso, Y. N. Wu, and S. Soatto. Dynamic textures. Int. J. Comput. Vis., 51, 2003. [6] D. Gao and N. Vasconcelos. Discriminant saliency for visual recognition from cluttered scenes. In Proc. NIPS, pages 481–488, 2004. [7] D. Gao and N. Vasconcelos. Decision-theoretic saliency: computational principle, biological plausibility, and implications for neurophysiology and psychophysics. submitted to Neural Computation, 2007. [8] J. Harel, C. Koch, and P. Perona. Graph-based visual saliency. In Proc. NIPS, 2006. [9] J. Huang and D. Mumford. Statistics of Natural Images and Models. In Proc. IEEE Conf. CVPR, 1999. [10] D. H. Hubel and T. N. Wiesel. Receptive fields and functional architecture in two nonstriate visual areas (18 and 19) of the cat. J. Neurophysiol., 28:229–289, 1965. 7 (a) (b) (c) Figure 8: Results on Boats: (a) original; b) discriminant saliency with DT; and c) GMM model of [16, 24]. (a) (b) (c) Figure 9: Results on Surfer: (a) original; b) discriminant saliency with DT; and c) GMM model of [16, 24]. [11] L. Itti and C. Koch. A saliency-based search mechanism for overt and covert shifts of visual attention. Vision Research, 40:1489–1506, 2000. [12] L. Itti, C. Koch, and E. Niebur. A model of saliency-based visual attention for rapid scene analysis. IEEE Trans. PAMI, 20(11), 1998. [13] S. G. Mallat. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Trans. PAMI, 11(7):674–693, 1989. [14] H. C. Nothdurft. The conspicuousness of orientation and motion contrast. Spat. Vis., 7, 1993. [15] Y. Sheikh and M. Shah. Bayesian modeling of dynamic scenes for object detection. IEEE Trans. on PAMI, 27(11):1778–92, 2005. [16] C. Stauffer and W. Grimson. Adaptive background mixture models for real-time tracking. In CVPR, pages 246–52, 1999. [17] B. W. Tatler, R. J. Baddeley, and I. D. Gilchrist. Visual correlates of fixation selection: effects of scale and time. Vision Research, 45:643–659, 2005. [18] A. Treisman and G. Gelade. A feature-integratrion theory of attention. Cognit. Psych., 12, 1980. [19] A. Treisman and S. Gormican. Feature analysis in early vision: Evidence from search asymmetries. Psychological Review, 95:14–58, 1988. [20] A. Tversky. Features of similarity. Psychol. Rev., 84, 1977. [21] N. Vasconcelos. Scalable discriminant feature selection for image retrieval. In CVPR, 2004. [22] D. Walther and C. Koch. Modeling attention to salient proto-objects. Neural Networks, 19, 2006. [23] J. Zhong and S. Sclaroff. Segmenting foreground objects from a dynamic textured background via a robust Kalman filter. In ICCV, 2003. [24] Z. Zivkovic. Improved adaptive Gaussian mixture model for background subtraction. In ICVR, 2004. 8
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Statistical Analysis of Semi-Supervised Regression John Lafferty Computer Science Department Carnegie Mellon University Pittsburgh, PA 15213 lafferty@cs.cmu.edu Larry Wasserman Department of Statistics Carnegie Mellon University Pittsburgh, PA 15213 larry@stat.cmu.edu Abstract Semi-supervised methods use unlabeled data in addition to labeled data to construct predictors. While existing semi-supervised methods have shown some promising empirical performance, their development has been based largely based on heuristics. In this paper we study semi-supervised learning from the viewpoint of minimax theory. Our first result shows that some common methods based on regularization using graph Laplacians do not lead to faster minimax rates of convergence. Thus, the estimators that use the unlabeled data do not have smaller risk than the estimators that use only labeled data. We then develop several new approaches that provably lead to improved performance. The statistical tools of minimax analysis are thus used to offer some new perspective on the problem of semi-supervised learning. 1 Introduction Suppose that we have labeled data L = {(X1, Y1), . . . (Xn, Yn)} and unlabeled data U = {Xn+1, . . . X N} where N ≫n and Xi ∈RD. Ordinary regression and classification techniques use L to predict Y from X. Semi-supervised methods also use the unlabeled data U in an attempt to improve the predictions. To justify these procedures, it is common to invoke one or both of the following assumptions: Manifold Assumption (M): The distribution of X lives on a low dimensional manifold. Semi-Supervised Smoothness Assumption (SSS): The regression function m(x) = EY | X = x is very smooth where the density p(x) of X is large. In particular, if there is a path connecting Xi and X j on which p(x) is large, then Yi and Y j should be similar with high probability. While these assumptions are somewhat intuitive, and synthetic examples can easily be constructed to demonstrate good performance of various techniques, there has been very little theoretical analysis of semi-supervised learning that rigorously shows how the assumptions lead to improved performance of the estimators. In this paper we provide a statistical analysis of semi-supervised methods for regression, and propose some new techniques that provably lead to better inferences, under appropriate assumptions. In particular, we explore precise formulations of SSS, which is motivated by the intuition that high density level sets correspond to clusters of similar objects, but as stated above is quite vague. To the best of our knowledge, no papers have made the assumption precise and then explored its consequences in terms of rates of convergence, with the exception of one of the first papers on semi-supervised learning, by Castelli and Cover (1996), which evaluated a simple mixture model, and the recent paper of Rigollet (2006) in the context of classification. This situation is striking, given the level of activity in this area within the machine learning community; for example, the recent survey of semi-supervised learning by Zhu (2006) contains 163 references. 1 Among our findings are: 1. Under the manifold assumption M, the semi-supervised smoothness assumption SSS is superfluous. This point was made heuristically by Bickel and Li (2006), but we show that in fact ordinary regression methods are automatically adaptive if the distribution of X concentrates on a manifold. 2. Without the manifold assumption M, the semi-supervised smoothness assumption SSS as usually defined is too weak, and current methods don’t lead to improved inferences. In particular, methods that use regularization based on graph Laplacians do not achieve faster rates of convergence. 3. Assuming specific conditions that relate m and p, we develop new semi-supervised methods that lead to improved estimation. In particular, we propose estimators that reduce bias by estimating the Hessian of the regression function, improve the choice of bandwidths using unlabeled data, and estimate the regression function on level sets. The focus of the paper is on a theoretical analysis of semi-supervised regression techniques, rather than the development of practical new algorithms and techniques. While we emphasize regression, most of our results have analogues for classification. Our intent is to bring the statistical perspective of minimax analysis to bear on the problem, in order to study the interplay between the labeled sample size and the unlabeled sample size, and between the regression function and the data density. By studying simplified versions of the problem, our analysis suggests how precise formulations of assumptions M and SSS can be made and exploited to lead to improved estimators. 2 Preliminaries The data are (X1, Y1, R1), . . . , (X N, YN, RN) where Ri ∈{0, 1} and we observe Yi only if Ri = 1. The labeled data are L = {(Xi, Yi) Ri = 1} and the unlabeled data are U = {(Xi, Yi) Ri = 0}. For convenience, assume that data are labeled so that Ri = 1 for i = 1, . . . , n and Ri = 0 for i = n + 1, . . . , N. Thus, the labeled sample size is n, and the unlabeled sample size is u = N −n. Let p(x) be the density of X and let m(x) = E(Y | X = x) denote the regression function. Assume that R ⊥⊥Y | X (missing at random) and that Ri | Xi ∼Bernoulli(π(Xi)). Finally, let µ = P(Ri = 1) = R π(x)p(x)dx. For simplicity we assume that π(x) = µ for all x. The missing at random assumption R ⊥⊥Y | X is crucial, although this point is rarely emphasized in the machine learning literature. It is clear that without some further conditions, the unlabeled data are useless. The key assumption we need is that there is some correspondence between the shape of the regression function m and the shape of the data density p. We will use minimax theory to judge the quality of an estimator. Let R denote a class of regression functions and let F denote a class of density functions. In the classical setting, we observe labeled data (X1, Y2), . . . , (Xn, Yn). The pointwise minimax risk, or mean squared error (MSE), is defined by Rn(x) = inf b mn sup m∈R,p∈F E(b mn(x) −m(x))2 (1) where the infimum is over all estimators. The global minimax risk is defined by Rn = inf b mn sup m∈R,p∈F E Z (b mn(x) −m(x))2dx. (2) A typical assumption is that R is the Sobolev space of order two, meaning essentially that m has smooth second derivatives. In this case we have1 Rn ≍n−4/(4+D). The minimax rate is achieved by kernel estimators and local polynomial estimators. In particular, for kernel estimators if we use a product kernel with common bandwidth hn for each variable, choosing hn ∼n−1/(4+D) yields an 1We write an ≍bn to mean that an/bn is bounded away from 0 and infinity for large n. We have suppressed some technicalities such as moment assumptions on ϵ = Y −m(X). 2 estimator with the minimax rate. The difficulty is that the rate Rn ≍n−4/(4+D) is extremely slow when D is large. In more detail, let C > 0 and let B be a positive definite matrix, and define R = m m(x) −m(x0) −(x −x0)T ∇m(x0) ≤C 2 (x −x0)T B(x −x0) (3) F = p p(x) ≥b > 0, |p(x1) −p(x2)| ≤c∥x1 −x2∥α 2 . (4) Fan (1993) shows that the local linear estimator is asymptotically minimax for this class. This estimator is given by b mn(x) = a0 where (a0, a1) minimizes Pn i=1(Yi−a0−aT 1 (Xi−x))2K(H−1/2(Xi− x)), where K is a symmetric kernel and H is a matrix of bandwidths. The asymptotic MSE of the local linear estimator b m(x) using the labeled data is R(H) = 1 2µ2 2(K)tr(Hm(x)H) 2 + 1 n|H|1/2 ν0σ 2 p(x) + o( tr(H)) (5) where Hm(x) is the Hessian of m at x, µ2(K) = R K 2(u) du and ν0 is a constant. The optimal bandwidth matrix H∗is given by H∗= ν0σ 2|Hm|1/2 µ2 2(K)nDp(x) !2/(D+4) (Hm)−1 (6) and R(H∗) = O(n−4/(4+D)). This result is important to what follows, because it suggests that if the Hessian Hm of the regression function is related to the Hessian Hp of the data density, one may be able to estimate the optimal bandwidth matrix from unlabeled data in order to reduce the risk. 3 The Manifold Assumption It is common in the literature to invoke both M and SSS. But if M holds, SSS is not needed. This is argued by Bickel and Li (2006) who say, “We can unwittingly take advantage of low dimensional structure without knowing it.” Suppose X ∈RD has support on a manifold M with dimension d < D. Let b mh be the local linear estimator with diagonal bandwidth matrix H = h2I. Then Bickel and Li show that the bias and variance are b(x) = h2J1(x)(1 + oP(1)) and v(x) = J2(x) nhd (1 + oP(1)) (7) for some functions J1 and J2. Choosing h ≍n−1/(4+d) yields a risk of order n−4/(4+d), which is the optimal rate for data that to lie on a manifold of dimension d. To use the above result we would need to know d. Bickel and Li argue heuristically that the following procedure will lead to a reasonable bandwidth. First, estimate d using the procedure in Levina and Bickel (2005). Now let B = {λ1/n1/(bd+4), . . . , λB/n1/(bd+4)} be a set of bandwidths, scaling the asymptotic order n−1/(bd+4) by different constants. Finally, choose the bandwidth h ∈B that minimizes a local cross-validation score. We now show that, in fact, one can skip the step of estimating d. Let E1, . . . , En be independent Bernoulli (θ = 1 2) random variables. Split the data into two groups, so that I0 = {i Ei = 0} and I1 = {i Ei = 1}. Let H = {n−1/(4+d) 1 ≤d ≤D}. Construct b mh for h ∈H using the data in I0, and estimate the risk from I1 by setting bR(h) = |I1|−1 P i∈I1 (Yi −b mh(Xi))2. Finally, let bh minimize bR(h) and set b m = b mbh. For simplicity, let us assume that both Yi and Xi are bounded by a finite constant B. Theorem 1. Suppose that and |Yi| ≤B and |Xi j| ≤B for all i and j. Assume the conditions in Bickel and Li (2006). Suppose that the data density p(x) is supported on a manifold of dimension d ≥4. Then we have that E(b m(x) −m(x))2 = e O 1 n4/(4+d) . (8) 3 The notation e O allows for logarithmic factors in n. Proof. The risk is, up to a constant, R(h) = E(Y −b m(X))2, where (X, Y) is a new pair and Y = m(X) + ϵ. Note that (Y −b mh(X))2 = Y 2 −2Y b mh(X) + b m2 h(X), so R(h) = E(Y 2) − 2E(Y b mh(X)) + b m2 h(X). Let n1 = |I1|. Then, bR(h) = 1 n1 X i∈I1 Y 2 i −2 n1 X i∈I1 Yi b mh(Xi) + 1 n1 X i∈I1 b m2 h(Xi). (9) By conditioning on the data in I0 and applying Bernstein’s inequality, we have P max h∈H |bR(h) −R(h)| > ϵ ≤ X h∈H P |bR(h) −R(h)| > ϵ ≤De−ncϵ2 (10) for some c > 0. Setting ϵn = √C log n/n for suitably large C, we conclude that P max h∈H |bR(h) −R(h)| > r C log n n ! −→0. (11) Let h∗minimize R(h) over H. Then, except on a set of probability tending to 0, R(bh) ≤ bR(bh) + r C log n n ≤bR(h∗) + r C log n n (12) ≤ R(h∗) + 2 r C log n n = O 1 n4/(4+d) + 2 r C log n n = e O 1 n4/(4+d) (13) where we used the assumption d ≥4 in the last equality. If d = 4 then O(√log n/n) = e O(n−4/(4+d)); if d > 4 then O(√log n/n) = o n4/(4+d) . □ We conclude that ordinary regression methods are automatically adaptive, and achieve the lowdimensional minimax rate if the distribution of X concentrates on a manifold; there is no need for semi-supervised methods in this case. Similar results apply to classification. 4 Kernel Regression with Laplacian Regularization In practice, it is unlikely that the distribution of X would be supported exactly on a low-dimensional manifold. Nevertheless, the shape of the data density p(x) might provide information about the regression function m(x), in which case the unlabeled data are informative. Several recent methods for semi-supervised learning attempt to exploit the smoothness assumption SSS using regularization operators defined with respect to graph Laplacians (Zhu et al., 2003; Zhou et al., 2004; Belkin et al., 2005). The technique of Zhu et al. (2003) is based on Gaussian random fields and harmonic functions defined with respect to discrete Laplace operators. To express this method in statistical terms, recall that standard kernel regression corresponds to the locally constant estimator b mn(x) = arg min m(x) n X i=1 Kh(Xi, x)(Yi −m(x))2 = Pn i=1 Kh(Xi, x) Yi Pn i=1 Kh(Xi, x) (14) where Kh is a symmetric kernel depending on bandwidth parameters h. In the semi-supervised approach of Zhu et al. (2003), the locally constant estimate b m(x) is formed using not only the labeled data, but also using the estimates at the unlabeled points. Suppose that the first n data points (X1, Y1), . . . , (Xn, Yn) are labeled, and the next u = N −n points are unlabeled, Xn+1, . . . , Xn+u. The semi-supervised regression estimate is then (b m(X1), b m(X2), . . . , b m(X N)) given by b m = arg min m N X i=1 N X j=1 Kh(Xi, X j) (m(Xi) −m(X j))2 (15) 4 where the minimization is carried out subject to the constraint m(Xi) = Yi, i = 1, . . . , n. Thus, the estimates are coupled, unlike the standard kernel regression estimate (14) where the estimate at each point x can be formed independently, given the labeled data. The estimator can be written in closed form as a linear smoother b m = C−1 B Y = G Y where b m = (b m(Xn+1), . . . , m(Xn+u))T is the vector of estimates over the unlabeled test points, and Y = (Y1, . . . , Yn)T is vector of labeled values. The (N −n)×(N −n) matrix C and the (N −n)×n matrix B denote blocks of the combinatorial Laplacian on the data graph corresponding to the labeled and unlabeled data: 1 = A BT B C (16) where the Laplacian 1 = 1i j has entries 1i j = P k Kh(Xi, Xk) if i = j −Kh(Xi, X j) otherwise. (17) This expresses the effective kernel G in terms of geometric objects such as heat kernels for the discrete diffusion equations (Smola and Kondor, 2003). This estimator assumes the noise is zero, since b m(Xi) = Yi for i = 1, . . . , n. To work in the standard model Y = m(X) + ϵ, the natural extension of the harmonic function approach is manifold regularization (Belkin et al., 2005; Sindhwani et al., 2005; Tsang and Kwok, 2006). Here the estimator is chosen to minimize the regularized empirical risk functional Rγ (m)= N X i=1 n X j=1 K H(Xi, X j) Y j −m(Xi) 2+γ N X i=1 N X j=1 K H(Xi, X j) m(X j) −m(Xi) 2 (18) where H is a matrix of bandwidths and K H(Xi, X j) = K(H−1/2(Xi −X j)). When γ = 0 the standard kernel smoother is obtained. The regularization term is J (m) ≡ N X i=1 N X j=1 K H(Xi, X j) m(X j) −m(Xi) 2 = 2mT 1m (19) where 1 is the combinatorial Laplacian associated with K H. This regularization term is motivated by the semi-supervised smoothness assumption—it favors functions m for which m(Xi) is close to m(X j) when Xi and X j are similar, according to the kernel function. The name manifold regularization is justified by the fact that 1 2J (m) → R M ∥∇m(x)∥2 dMx, the energy of m over the manifold. While this regularizer has primarily been used for SVM classifiers (Belkin et al., 2005), it can be used much more generally. For an appropriate choice of γ , minimizing the functional (18) can be expected to give essentially the same results as the harmonic function approach that minimizes (15). Theorem 2. Suppose that D ≥2. Let e m H,γ minimize (18), and let 1p,H be the differential operator defined by 1p,H f (x) = 1 2trace(H f (x)H) + ∇p(x)T H∇f (x) p(x) . (20) Then the asymptotic MSE of e m H,γ (x) is e M = c1µσ 2 n(µ + γ )p(x)|H|1/2 + c2(µ + γ ) µ I −γ µ1p,H −1 1p,Hm(x) !2 + o( tr(H)) (21) where µ = P(Ri = 1). Note that the bias of the standard kernel estimator, in the notation of this theorem, is b(x) = c21p,Hm(x), and the variance is V (x) = c1/np(x)|H|1/2. Thus, this result agrees with the standard supervised MSE in the special case γ = 0. It follows from this theorem that e M = M + o( tr(H)) where M is the usual MSE for a kernel estimator. Therefore, the minimum of e M has the same leading order in H as the minimum of M. The proof is given in the full version of the paper. The implication of this theorem is that the estimator that uses Laplacian regularization has the same rate of convergence as the usual kernel estimator, and thus the unlabeled data have not improved the estimator asymptotically. 5 5 Semi-Supervised Methods With Improved Rates The previous result is negative, in the sense that it shows unlabeled data do not help to improve the rate of convergence. This is because the bias and variance of a manifold regularized kernel estimator are of the same order in H as the bias and variance of standard kernel regression. We now demonstrate how improved rates of convergence can be obtained by formulating and exploiting appropriate SSS assumptions. We describe three different approaches: semi-supervised bias reduction, improved bandwidth selection, and averaging over level sets. 5.1 Semi-Supervised Bias Reduction We first show a positive result by formulating an SSS assumption that links the shape of p to the shape of m by positing a relationship between the Hessian Hm of m and the Hessian Hp of p. Under this SSS assumption, we can improve the rate of convergence by reducing the bias. To illustrate the idea, take p(x) known (i.e., N = ∞) and suppose that Hm(x) = Hp(x). Define e mn(x) = b mn(x) −1 2µ2 2(K)tr(Hm(x)H) (22) where b mn(x) is the local linear estimator. Theorem 3. The risk of e mn(x) is O n−8/(8+D) . Proof. First note that the variance of the estimator e mn, conditional on X1, . . . , Xn, is Var(e mn(x)|X1, . . . , Xn) = Var(b mn(x)|X1, . . . , Xn). Now, the term 1 2µ2 2(K)tr(Hm(x)H) is precisely the bias of the local linear estimator, under the SSS assumption that Hp(x) = Hm(x). Thus, the first order bias term has been removed. The result now follows from the fact that the next term in the bias of the local linear estimator is of order O(tr(H)4). □ By assuming 2ℓderivatives are matched, we get the rate n−(4+4ℓ)/(4+4ℓ+D). When p is estimated from the data, the risk will be inflated by N −4/(4+D) assuming standard smoothness assumptions on p. This term will not dominate the improved rate n−(4+4ℓ)/(4+4ℓ+D) as long as N > nℓ. The assumption that Hm = Hp can be replaced by the more realistic assumption that Hm = g(p; β) for some parameterized family of functions g(·; β). Semiparametric methods can then be used to estimate β. This approach is taken in the following section. 5.2 Improved Bandwidth Selection Let b H be the estimated bandwidth using the labeled data. We will now show how a bandwidth b H∗can be estimated using the labeled and unlabeled data together, such that, under appropriate assumptions, lim sup n→∞ |R( b H∗) −R(H∗)| |R( b H) −R(H∗)| = 0, where H∗= arg min H R(H). (23) Therefore, the unlabeled data allow us to construct an estimator that gets closer to the oracle risk. The improvement is weaker than the bias adjustment method. But it has the virtue that the optimal local linear rate is maintained even if the proposed model linking Hm to p is incorrect. We begin in one dimension to make the ideas clear. Let b m H denote the local linear estimator with bandwidth H ∈R, H > 0. To use the unlabeled data, note that the optimal (global) bandwidth is H∗= (c2B/(4nc1 A))1/5 where A = R m′′(x)2dx and B = R dx/p(x). Let bp(x) be the kernel density estimator of p using X1, . . . , X N and bandwidth h = O(N −1/5). We assume (SSS) m′′(x) = Gθ(p) for some function G depending on finitely many parameters θ. Now let \ m′′(x) = Gbθ(bp), and define b H∗= c2bB 4nc1 bA 1/5 where bA = R (\ m′′(x))2 dx and bB = R dx/bp(x). 6 Theorem 4. Suppose that \ m′′(x) −m′′(x) = OP(N −β) where β > 2 5. Let N = N(n) →∞as n →∞. If N/n1/4 →∞, then lim sup n→∞ |R( b H∗) −R(H∗)| |R( b H) −R(H∗)| = 0. (24) Proof. The risk is R(H) = c1H4 Z (m′′(x))2dx + c2 nH Z dx p(x) + o 1 nH . (25) The oracle bandwidth is H∗= c3/n1/5 and then R(H∗) = O(n−4/5). Now let b H be the bandwidth estimated by cross-validation. Then, since R′(H∗) = 0 and H∗= O(n−1/5), we have R( b H) = ( b H −H∗)2 2 R′′(H∗) + O(| b H −H∗|3) (26) = ( b H −H∗)2 2 O(n−2/5) + O(| b H −H∗|3). (27) From Girard (1998), b H −H∗= OP(n−3/10). Hence, R( b H) −R(H∗) = OP(n−1). Also, bp(x) − p(x) = O(N −2/5). Since \ m′′(x) −m′′(x) = OP(N −β), b H∗−H∗= OP N −2/5 n1/5 + OP N −β n1/5 . (28) The first term is oP(n−3/10) since N > n1/4. The second term is oP(n−3/10) since β > 2/5. Thus R( b H∗) −R(H∗) = oP(1/n) and the result follows. □ The proof in the multidimensional case is essentially the same as in the one dimensional case, except that we use the multivariate version of Girard’s result, namely, H∗−b H = OP(n−(D+2)/(2(D+4))). This leads to the following result. Theorem 5. Let N = N(n). If N/nD/4 →∞, bθ −θ = OP(N −β) for some β > 2 4+D then lim sup n→∞ |R( b H∗) −R(H∗)| |R( b H) −R(H∗)| = 0. (29) 5.3 Averaging over Level Sets Recall that SSS is motivated by the intuition that high density level sets should correspond to clusters of similar objects. Another approach to quantifying SSS is to make this cluster assumption explicit. Rigollet (2006) shows one way to do this in classification. Here we focus on regression. Suppose that L = {x p(x) > λ} can be decomposed into a finite number of connected, compact, convex sets C1, . . . , Cg where λ is chosen so that Lc has negligible probability. For N large we can replace L with L = {x bp(x) > λ} with small loss in accuracy, where bp is an estimate of p using the unlabeled data; see Rigollet (2006) for details. Let k j = Pn i=1 I (Xi ∈C j) and for x ∈C j define b m(x) = Pn i=1 Yi I (Xi ∈C j) k j . (30) Thus, b m(x) simply averages the labels of the data that fall in the set to which x belongs. If the regression function is slowly varying in over this set, the risk should be small. A similar estimator is considered by Cortes and Mohri (2006), but they do not provide estimates of the risk. Theorem 6. The risk of b m(x) for x ∈L ∩C j is bounded by O 1 nπ j + O δ2 jξ2 j (31) where δ j = supx∈C j ∥∇m(x)∥, ξ j = diameter(C j) and π j = P(X ∈C j). 7 Proof. Since the k j are Binomial, k j = nπ j + o(1) almost surely. Thus, the variance of b m(x) is O(1/(nπ j)). The mean, given X1, . . . , Xn, is 1 k j X i Xi∈C j m(Xi) = m(x) + 1 k j X i Xi∈C j (m(Xi) −m(x)). (32) Now m(Xi)−m(x) = (X j −x)T ∇m(ui) for some ui between x and Xi. Hence, |m(Xi)−m(x)| ≤ ∥X j −x∥supx∈C j ∥∇m(x)∥and so the bias is bounded by δ jξ j. □ This result reveals an interesting bias-variance tradeoff. Making λ smaller decreases the variance and increases the bias. Suppose the two terms are balanced at λ = λ∗. Then we will beat the usual rate of convergence if π j(λ∗) > n−D/(4+D). 6 Conclusion Semi-supervised methods have been very successful in many problems. Our results suggest that the standard explanations for this success are not correct. We have indicated some new approaches to understanding and exploiting the relationship between the labeled and unlabeled data. Of course, we make no claim that these are the only ways of incorporating unlabeled data. But our results indicate that decoupling the manifold assumption and the semi-supervised smoothness assumption is crucial to clarifying the problem. 7 Acknowlegments We thank Partha Niyogi for several interesting discussions. This work was supported in part by NSF grant CCF-0625879. References BELKIN, M., NIYOGI, P. and SINDHWANI, V. (2005). On manifold regularization. In Proceedings of the Tenth International Workshop on Artificial Intelligence and Statistics (AISTAT 2005). BICKEL, P. and LI, B. (2006). Local polynomial regression on unknown manifolds. Tech. rep., Department of Statistics, UC Berkeley. CASTELLI, V. and COVER, T. (1996). The relative value of labeled and unlabeled samples in pattern recognition with an unknown mixing parameter. IEEE Trans. on Info. Theory 42 2101–2117. CORTES, C. and MOHRI, M. (2006). On transductive regression. In Advances in Neural Information Processing Systems (NIPS), vol. 19. FAN, J. (1993). Local linear regression smoothers and their minimax efficiencies. The Annals of Statistics 21 196–216. GIRARD, D. (1998). Asymptotic comparison of (partial) cross-validation, gcv and randomized gcv in nonparametric regression. Ann. Statist. 12 315–334. LEVINA, E. and BICKEL, P. (2005). Maximum likelihood estimation of intrinsic dimension. In Advances in Neural Information Processing Systems (NIPS), vol. 17. NIYOGI, P. (2007). Manifold regularization and semi-supervised learning: Some theoretical analyses. Tech. rep., Departments of Computer Science and Statistics, University of Chicago. RIGOLLET, P. (2006). Generalization error bounds in semi-supervised classification under the cluster assumption. arxiv.org/math/0604233 . SINDHWANI, V., NIYOGI, P., BELKIN, M. and KEERTHI, S. (2005). Linear manifold regularization for large scale semi-supervised learning. In Proc. of the 22nd ICML Workshop on Learning with Partially Classified Training Data. SMOLA, A. and KONDOR, R. (2003). Kernels and regularization on graphs. In Conference on Learning Theory, COLT/KW. TSANG, I. and KWOK, J. (2006). Large-scale sparsified manifold regularization. In Advances in Neural Information Processing Systems (NIPS), vol. 19. ZHOU, D., BOUSQUET, O., LAL, T., WESTON, J. and SCHÖLKOPF, B. (2004). Learning with local and global consistency. In Advances in Neural Information Processing Systems (NIPS), vol. 16. ZHU, X. (2006). Semi-supervised learning literature review. Tech. rep., University of Wisconsin. ZHU, X., GHAHRAMANI, Z. and LAFFERTY, J. (2003). Semi-supervised learning using Gaussian fields and harmonic functions. In ICML-03, 20th International Conference on Machine Learning. 8
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Hierarchical Apprenticeship Learning, with Application to Quadruped Locomotion J. Zico Kolter, Pieter Abbeel, Andrew Y. Ng Department of Computer Science Stanford University Stanford, CA 94305 {kolter, pabbeel, ang}@cs.stanford.edu Abstract We consider apprenticeship learning—learning from expert demonstrations—in the setting of large, complex domains. Past work in apprenticeship learning requires that the expert demonstrate complete trajectories through the domain. However, in many problems even an expert has difficulty controlling the system, which makes this approach infeasible. For example, consider the task of teaching a quadruped robot to navigate over extreme terrain; demonstrating an optimal policy (i.e., an optimal set of foot locations over the entire terrain) is a highly non-trivial task, even for an expert. In this paper we propose a method for hierarchical apprenticeship learning, which allows the algorithm to accept isolated advice at different hierarchical levels of the control task. This type of advice is often feasible for experts to give, even if the expert is unable to demonstrate complete trajectories. This allows us to extend the apprenticeship learning paradigm to much larger, more challenging domains. In particular, in this paper we apply the hierarchical apprenticeship learning algorithm to the task of quadruped locomotion over extreme terrain, and achieve, to the best of our knowledge, results superior to any previously published work. 1 Introduction In this paper we consider apprenticeship learning in the setting of large, complex domains. While most reinforcement learning algorithms operate under the Markov decision process (MDP) formalism (where the reward function is typically assumed to be given a priori), past work [1, 13, 11] has noted that often the reward function itself is difficult to specify by hand, since it must quantify the trade off between many features. Apprenticeship learning is based on the insight that often it is easier for an “expert” to demonstrate the desired behavior than it is to specify a reward function that induces this behavior. However, when attempting to apply apprenticeship learning to large domains, several challenges arise. First, past algorithms for apprenticeship learning require the expert to demonstrate complete trajectories in the domain, and we are specifically concerned with domains that are sufficiently complex so that even this task is not feasible. Second, these past algorithms require the ability to solve the “easier” problem of finding a nearly optimal policy given some candidate reward function, and even this is challenging in large domains. Indeed, such domains often necessitate hierarchical control in order to reduce the complexity of the control task [2, 4, 15, 12]. As a motivating application, consider the task of navigating a quadruped robot (shown in Figure 1(a)) over challenging, irregular terrain (shown in Figure 1(b,c)). In a naive approach, the dimensionality of the state space is prohibitively large: the robot has 12 independently actuated joints, and the state must also specify the current three-dimensional position and orientation of the robot, leading to an 18-dimensional state space that is well beyond the capabilities of standard RL algorithms. Fortunately, this control task succumbs very naturally to a hierarchical decomposition: we first plan a general path over the terrain, then plan footsteps along this path, and finally plan joint movements 1 Figure 1: (a) LittleDog robot, designed and built by Boston Dynamics, Inc. (b) Typical terrain. (c) Height map of the depicted terrain. (Black = 0cm altitude, white = 12cm altitude.) to achieve these footsteps. However, it is very challenging to specify a proper reward, specifically for the higher levels of control, as this requires quantifying the trade-off between many features, including progress toward a goal, the height differential between feet, the slope of the terrain underneath its feet, etc. Moreover, consider the apprenticeship learning task of specifying a complete set of foot locations, across an entire terrain, that properly captures all the trade-offs above; this itself is a highly non-trivial task. Motivated by these difficulties, we present a unified method for hierarchical apprenticeship learning. Our approach is based on the insight that, while it may be difficult for an expert to specify entire optimal trajectories in a large domain, it is much easier to “teach hierarchically”: that is, if we employ a hierarchical control scheme to solve our problem, it is much easier for the expert to give advice independently at each level of this hierarchy. At the lower levels of the control hierarchy, our method only requires that the expert be able to demonstrate good local behavior, rather than behavior that is optimal for the entire task. This type of advice is often feasible for the expert to give even when the expert is entirely unable to give full trajectory demonstrations. Thus the approach allows for apprenticeship learning in extremely complex, previously intractable domains. The contributions of this paper are twofold. First, we introduce the hierarchical apprenticeship learning algorithm. This algorithm extends the apprenticeship learning paradigm to complex, highdimensional control tasks by allowing an expert to demonstrate desired behavior at multiple levels of abstraction. Second, we apply the hierarchical apprenticeship approach to the quadruped locomotion problem discussed above. By applying this method, we achieve performance that is, to the best of our knowledge, well beyond any published results for quadruped locomotion.1 The remainder of this paper is organized as follows. In Section 2 we discuss preliminaries and notation. In Section 3 we present the general formulation of the hierarchical apprenticeship learning algorithm. In Section 4 we present experimental results, both on a hierarchical multi-room grid world, and on the real-world quadruped locomotion task. Finally, in Section 5 we discuss related work and conclude the paper. 2 Preliminaries and Notation A Markov decision process (MDP) is a tuple (S, A, T, H, D, R), where S is a set of states; A is a set of actions, T = {Psa} is a set of state transition probabilities (here, Psa is the state transition distribution upon taking action a in state s); H is the horizon which corresponds to the number of time-steps considered; D is a distribution over initial states; and R : S →R is a reward function. As we are often concerned with MDPs for which no reward function is given, we use the notation MDP\R to denote an MDP minus the reward function. A policy π is a mapping from states to a probability distribution over actions. The value of a policy π is given by V (π) = E hPH t=0 R(st)|π i , where the expectation is taken with respect to the random state sequence s0, s1, . . . , sH drawn by stating from the state s0 (drawn from distribution D) and picking actions according to π. 1There are several other institutions working with the LittleDog robot, and many have developed (unpublished) systems that are also very capable. As of the date of submission, we believe that the controller presented in this paper is on par with the very best controllers developed at other institutions. For instance, although direct comparison is difficult, the fastest running time that any team achieved during public evaluations was 39 seconds. In Section 4 we present results crossing terrain of comparable difficulty and distance in 30-35 seconds. 2 Often the reward function R can be represented more compactly as a function of the state. Let φ : S →Rn be a mapping from states to a set of features. We consider the case where the reward function R is a linear combination of the features: R(s) = wT φ(s) for parameters w ∈Rn. Then we have that the value of a policy φ is linear in the reward function weights V (π) = E[PH t=0 R(st)|π] = E[PH t=0 wT φ(st)|π] = wT E[PH t=0 φ(st)|π] = wT µφ(π) (1) where we used linearity of expectation to bring w outside of the expectation. The last quantity defines the vector of feature expectations µφ(π) = E[PH t=0 φ(st)|π]. 3 The Hierarchical Apprenticeship Learning Algorithm We now present our hierarchical apprenticeship learning algorithm (hereafter HAL). For simplicity, we present a two level hierarchical formulation of the control task, referred to generically as the low-level and high-level controllers. The extension to higher order hierarchies poses no difficulties. 3.1 Reward Decomposition in HAL At the heart of the HAL algorithm is a simple decomposition of the reward function that links the two levels of control. Suppose that we are given a hierarchical decomposition of a control task in the form of two MDP\Rs — a low-level and a high-level MDP\R, denoted Mℓ= (Sℓ, Aℓ, Tℓ, Hℓ, Dℓ) and Mh = (Sh, Ah, Th, Hh, Dh) respectively — and a partitioning function ψ : Sℓ→Sh that maps low level states to high-level states (the assumption here is that |Sh| ≪|Sℓ| so that this hierarchical decomposition actually provides a computational gain).2 For example, in the case of the quadruped locomotion problem the low-level MDP\R describes the state of all four feet, while the high-level MDP\R describes only the position of the robot’s center of mass. As is standard in apprenticeship learning, we suppose that the rewards in the low-level MDP\R can be represented as a linear function of state features, R(sℓ) = wT φ(sℓ). The HAL algorithm assumes that the reward of a high-level state is equal to the average reward over all its corresponding low-level states. Formally R(sh) = 1 N(sh) X sℓ∈ψ−1(sh) R(sℓ) = 1 N(sh) X sℓ∈ψ−1(sh) wT φ(sℓ) = 1 N(sh)wT X sℓ∈ψ−1(sh) φ(sℓ) (2) where ψ−1(sh) denotes the inverse image of the partitioning function and N(sh) = |ψ−1(sh)|. While this may not always be the most ideal decomposition of the reward function in many cases— for example, we may want to let the reward of a high-level state be the maximum of its low level state rewards to capture the fact that an ideal agent would always seek to maximize reward at the lower level, or alternatively the minimum of its low level state rewards to be robust to worst-case outcomes—it captures the idea that in the absence of other prior information, it seems reasonable to assume a uniform distribution over the low-level states corresponding to a high-level state. An important consequence of (2) is that the high level reward is now also linear in the low-level reward weights w. This will enable us in the subsequent sections to formulate a unified hierarchical apprenticeship learning algorithm that is able to incorporate expert advice at both the high level and the low level simultaneously. 3.2 Expert Advice at the High Level Similar to past apprenticeship learning methods, expert advice at the high level consists of full policies demonstrated by the expert. However, because the high-level MDP\R can be significantly simpler than the low-level MDP\R, this task can be substantially easier. If the expert suggests that π(i) h,E is an optimal policy for some given MDP\R M (i) h , then this corresponds to the following constraint, which states that the expert’s policy outperforms all other policies: V (i)(π(i) h,E) ≥V (i)(π(i) h ) ∀π(i) h . Equivalently, using (1), we can formulate this constraint as follows: wT µ(i) φ (π(i) h,E) ≥wT µφ(π(i) h ) ∀π(i) h . While we may not be able to obtain the exact feature expectations of the expert’s policy if the highlevel transitions are stochastic, observing a single expert demonstration corresponds to receiving 2As with much work in reinforcement learning, it is the assumption of this paper that the hierarchical decomposition of a control task is given by a system designer. While there has also been recent work on the automated discovery of state abstractions[5], we have found that there is often a very natural decomposition of control tasks into multiple levels (as we will discuss for the specific case of quadruped locomotion). 3 a sample from these feature expectations, so we simply use the observed expert features counts ˆµ(i) φ (π(i) h,E) in lieu of the true expectations. By standard sample complexity arguments [1], it can be shown that a sufficient number of observed feature counts will converge to the true expectation. To resolve the ambiguity in w, and to allow the expert to provide noisy advice, we use regularization and slack variables (similar to standard SVM formulations), which results in the following formulation: minw,η 1 2∥w∥2 2 + Ch Pn i=1 η(i) s.t. wT ˆµ(i) φ (π(i) h,E) ≥wT µφ(π(i) h ) + 1 −η(i) ∀π(i) h , i where π(i) h indexes over all high-level policies, i indexes over all MDPs, and Ch is a regularization constant.3 Despite the fact that there are an exponential number of possible policies there are wellknown algorithms that are able to solve this optimization problem; however, we defer this discussion until after presenting our complete formulation. 3.3 Expert Advice at the Low Level Our approach differs from standard apprenticeship learning when we consider advice at the low level. Unlike the apprenticeship learning paradigm where an expert specifies full trajectories in the target domain, we allow for an expert to specify single, greedy actions in the low-level domain. Specifically, if the agent is in state sℓand the expert suggests that the best greedy action would move to state s′ ℓ, this corresponds directly to a constraint on the reward function, namely that R(s′ ℓ) ≥R(s′′ ℓ) for all other states s′′ ℓthat can be reached from the current state (we say that s′′ ℓis “reachable” from the current state sℓif ∃a s.t.Psℓa(s′′ ℓ) > ϵ for some 0 < ϵ ≤1).4 This results in the following constraints on the reward function parameters w, wT φ(s′ ℓ) ≥wT φ(s′′ ℓ) for all s′′ ℓreachable from sℓ. As before, to resolve the ambiguity in w and to allow for the expert to provide noisy advice, we use regularization and slack variables. This gives: minw,ξ 1 2∥w∥2 2 + Cℓ Pm j=1 ξ(j) s.t. wT φ(s′ ℓ (j)) ≥wT φ(s′′ ℓ (j)) + 1 −ξ(j) ∀s′′ ℓ (j), j where s′′ ℓ (j) indexes over all states reachable from s′ ℓ (j) and j indexes over all low-level demonstrations provided by the expert. 3.4 The Unified HAL Algorithm From (2) we see the high level and low level rewards are a linear combination of the same set of reward weights w. This allows us to combine both types of expert advice presented above to obtain the following unified optimization problem minw,η,ξ 1 2∥w∥2 2 + Cℓ Pm j=1 ξ(j) + Ch Pn i=1 η(i) s.t. wT φ(s′ ℓ (j)) ≥wT φ(s′′ ℓ (j)) + 1 −ξ(j) ∀s′′ ℓ (j), j wT ˆµ(i) φ (π(i) h,E) ≥wT µφ(π(i) h ) + 1 −η(i) ∀π(i) h , i. (3) This optimization problem is convex, and can be solved efficiently. In particular, even though the optimization problem has an exponentially large number of constraints (one constraint per policy), the optimum can be found efficiently (i.e., in polynomial time) using, for example, the ellipsoid method, since we can efficiently identify a constraint that is violated.5 However, in practice we found the following constraint generation method more efficient: 3This formulation is not entirely correct by itself, due to the fact that it is impossible to separate a policy from all policies (including itself) by a margin of one, and so the exact solution to this problem will be w = 0. To deal with this, one typically scales the margin or slack by some loss function that quantifies how different two policies are [16, 17], and this is the approach taken by Ratliff, et al. [13] in their maximum margin planning algorithm. Alternatively, Abbeel & Ng [1], solve the optimization problem without any slack, and notice that as soon as the problem becomes infeasible, the expert’s policy lies in the convex hull of the generated policies. However, in our full formulation with low-level advice also taken into account, this becomes less of an issue, and so we present the above formulation for simplicity. In all experiments where we use only the high-level constraints, we employ margin scaling as in [13]. 4Alternatively, one interpret low-level advice at the level of actions, and interpret the expert picking action a as the constraint that P s′ Psa(s′)R(s′) ≥P s′ Psa′(s′)R(s′) ∀a′ ̸= a. However, in the domains we consider, where there is a clear set of “reachable” states from each state, the formalism above seems more natural. 5 Similar techniques are employed by [17] to solve structured prediction problems. Alternatively, Ratliff, et al. [13] take a different approach, and move the constraints into the objective by eliminating the slack variables, then employ a subgradient method. 4 0 100 200 300 400 500 600 700 800 900 1000 50 100 150 200 250 300 Number of Training Samples Suboptimality of Policy HAL Flat Apprenticeship Learning 0 2 4 6 8 10 12 14 16 18 20 50 100 150 200 250 300 350 400 # of Training MDPs Suboptimality of policy HAL High−Level Contraints Only Low−Level Constraints Only Figure 2: (a) Picture of the multi-room gridworld environment. (b) Performance versus number of training samples for HAL and flat apprenticeship learning. (c) Performance versus number of training MDPs for HAL versus using only low-level or only high-level constraints. 1. Begin with no expert path constraints. 2. Find the current reward weights by solving the current optimization problem. 3. Solve the reinforcement learning problem at the high level of the hierarchy to find the optimal (high-level) policies for the current reward for each MDP\R, i. If the optimal policy violates the current (high level) constraints, then add this constraint to the current optimization problem and goto Step (2). Otherwise, no constraints are violated and the current reward weights are the solution of the optimization problem. 4 Experimental Results 4.1 Gridworld In this section we present results on a multi-room gridworld domain with unknown cost. While this is not meant to be a challenging control task, it allows us to compare the performance of HAL to traditional “flat” (non-hierarchical) apprenticeship learning methods, as these algorithms are feasible in such domains. The grid world domain has a very natural hierarchical decomposition: if we average the cost over each room, we can form a “high-level” approximation of the grid world. Our hierarchical controller first plans in this domain to choose a path over the rooms. Then for each room along this path we plan a low-level path to the desired exit. Figure 2(b) shows the performance versus number of training examples provided to the algorithm (where one training example equals one action demonstrated by the expert).6 As expected, the flat apprenticeship learning algorithm eventually converges to a superior policy, since it employs full value iteration to find the optimal policy, while HAL uses the (non-optimal) hierarchical controller. However, for small amounts of training data, HAL outperforms the flat method, since it is able to leverage the small amount of data provided by the expert at both levels of the hierarchy. Figure 2(c) shows performance versus number of MDPs in the training set for HAL and well as for algorithms which receive the same training data as HAL (that is, both high level and low level expert demonstrations), but which make use of only one or the other. Here we see that HAL performs substantially better. This is not meant to be a direct comparison of the different methods, since HAL obtains more training data per MDP than the single-level approaches. Rather, this experiment illustrates that in situations where one has access to both high-level and low-level advice, it is advantageous to use 6Experimental details: We consider a 111x111 grid world, evenly divided into 100 rooms of size 10x10 each. There are walls around each room, except for a door of size 2 that connects a room to each of its neighbors (a picture of the domain is shown in figure 2(a)). Each state has 40 binary features, sampled from a distribution particular to that room, and the reward function is chosen randomly to have 10 “small” [-0.75, -0.25], negative rewards, 20 “medium” [-1.0 -2.0] negative rewards, and 10 “high” [-3.0 -5.0] negative rewards. In all cases we generated multiple training MDPs, which differ in which features are active at each state and we provided the algorithm with one expert demonstration for each sampled MDP. After training on each MDP we tested on 25 holdout MDPs generated by the same process. In all cases the results were averaged over 10 runs. For all our experiments, we fixed the ratio of Ch/Cℓso that the both constraints were equally weighted (i.e., if it typically took t low level actions to accomplish one high-level action, then we used a ratio of Ch/Cℓ= t). Given this fixed scaling, we found that the algorithm was generally insensitive (in terms of the resulting policy’s suboptimality) to scaling of the slack penalties. In the comparison of HAL with flat apprenticeship learning in Figure 2(b), one training example corresponds to one expert action. Concretely, for HAL the number of training examples for a given training MDP corresponds to the number of high level actions in the high level demonstration plus the (equal) number of low level expert actions provided. For flat apprenticeship learning the number of training examples for a given training MDP corresponds to the number of expert actions in the expert’s full trajectory demonstration. 5 Figure 3: (a) High-level (path) expert demonstration. (b) Low-level (footstep) expert demonstration. both. This will be especially important in domains such as the quadruped locomotion task, where we have access to very few training MDPs (i.e., different terrains). 4.2 Quadruped Robot In this section we present the primary experimental result of this paper, a successful application of hierarchical apprenticeship learning to the task of quadruped locomotion. Videos of the results in this section are available at http://cs.stanford.edu/˜kolter/nips07videos. 4.2.1 Hierarchical Control for Quadruped Locomotion The LittleDog robot, shown in Figure 1, is designed and built by Boston Dynamics, Inc. The robot consists of 12 independently actuated servo motors, three on each leg, with two at the hip and one at the knee. It is equipped with an internal IMU and foot force sensors. We estimate the robot’s state using a motion capture system that tracks reflective markers on the robot’s body. We perform all computation on a desktop computer, and send commands to the robot via a wireless connection. As mentioned in the introduction, we employ a hierarchical control scheme for navigating the quadruped over the terrain. Due to space constraints, we describe the complete control system briefly, but a much more detailed description can be found in [8]. The high level controller is a body path planner, that plans an approximate trajectory for the robot’s center of mass over the terrain; the low-level controller is a footstep planner that, given a path for the robot’s center, plans a set of footsteps that follow this path. The footstep planner uses a reward function that specifies the relative trade-off between several different features of the robot’s state, including (i) several features capturing the roughness and slope of the terrain at several different spatial scales around the robot’s feet, (ii) distance of the foot location from the robot’s desired center, (iii) the area and inradius of the support triangle formed by the three stationary feet, and other similar features. Kinematic feasibility is required for all candidate foot locations and collision of the legs with obstacles is forbidden. To form the high-level cost, we aggregate features from the footstep planner. In particular, for each foot we consider all the footstep features within a 3 cm radius of the foot’s “home” position (the desired position of the foot relative to the center of mass in the absence of all other discriminating features), and aggregate these features to form the features for the body path planner. While this is an approximation, we found that it performed very well in practice, possibly due to its ability to account for stochasticity of the domain. After forming the cost function for both levels, we used value iteration to find the optimal policy for the body path planner, and a five-step lookahead receding horizon search to find a good set of footsteps for the footstep planner. 4.2.2 Hierarchical Apprenticeship Learning for Quadruped Locomotion All experiments were carried out on two terrains: a relatively easy terrain for training, and a significantly more challenging terrain for testing. To give advice at the high level, we specified complete body trajectories for the robot’s center of mass, as shown in Figure 3(a). To give advice for the low level we looked for situations in which the robot stepped in a suboptimal location, and then indicated the correct greedy foot placement, as shown in Figure 3(b). The entire training set con6 Figure 4: Snapshots of quadruped while traversing the testing terrain. Figure 5: Body and footstep plans for different constraints on the training (left) and testing (right) terrains: (Red) No Learning, (Green) HAL, (Blue) Path Only, (Yellow) Footstep Only. sisted of a single high-level path demonstration across the training terrain, and 20 low-level footstep demonstrations on this terrain; it took about 10 minutes to collect the data. Even from this small amount of training data, the learned system achieved excellent performance, not only on the training board, but also on the much more difficult testing board. Figure 4 shows snapshots of the quadruped crossing the testing board. Figure 5 shows the resulting footsteps taken for each of the different types of constraints, which shows a very large qualitative difference between the footsteps chosen before and after training. Table 1 shows the crossing times for each of the different types of constraints. As shown, he HAL algorithm outperforms all the intermediate methods. Using only footstep constraints does quite well on the training board, but on the testing board the lack of high-level training leads the robot to take a very roundabout route, and it performs much worse. The quadruped fails at crossing the testing terrain when learning from the path-level demonstration only or when not learning at all. Finally, prior to undertaking our work on hierarchical apprenticeship learning, we invested several weeks attempting to hand-tune controller capable of picking good footsteps across challenging terrain. However, none of our previous efforts could significantly outperform the controller presented here, learned from about 10 minutes worth of data, and many of our previous efforts performed substantially worse. 5 Related Work and Discussion The work presented in this paper relates to many areas of reinforcement learning, including apprenticeship learning and hierarchical reinforcement learning, and to a large body of past work in quadruped locomotion. In the introduction and in the formulation of our algorithm we discussed the connection to the inverse reinforcement learning algorithm of [1] and the maximum margin planning algorithm of [13]. In addition, there has been subsequent work [14] that extends the maximum margin planning framework to allow for the automated addition of new features through a boosting procedure; There has also been much recent work in reinforcement learning on hierarchical reinforcement learning; a recent survey is [2]. However, all the work in this area that we are aware of deals with the more standard reinforcement learning formulation where known rewards are given to the agent as it acts in a (possibly unknown) environment. In contrast, our work follows the apprenticeship learning paradigm where the model, but not the rewards, are known to the agent. Prior work on legged locomotion has mostly focused on generating gaits for stably traversing fairly flat 7 HAL Feet Only Path Only No Learning Training Time (sec) 31.03 33.46 — 40.25 Testing Time (sec) 35.25 45.70 — — Table 1: Execution times for different constraints on training and testing terrains. Dashes indicate that the robot fell over and did not reach the goal. terrain (see, among many others, [10], [7]). Only very few learning algorithms, which attempt to generalize to previously unseen terrains, have been successfully applied before [6, 3, 9]. The terrains considered in this paper go well beyond the difficulty level considered in prior work. 6 Acknowledgements We gratefully acknowledge the anonymous reviewers for helpful suggestions. This work was supported by the DARPA Learning Locomotion program under contract number FA8650-05-C-7261. References [1] Pieter Abbeel and Andrew Y. Ng. Apprenticeship learning via inverse reinforcement learning. In Proceedings of the International Conference on Machine Learning, 2004. [2] Andrew G. Barto and Sridhar Mahadevan. Recent advances in hierarchical reinforcement learning. Discrete Event Dynamic Systems: Theory and Applications, 13:41–77, 2003. [3] Joel Chestnutt, James Kuffner, Koichi Nishiwaki, and Satoshi Kagami. Planning biped navigation strategies in complex environments. In Proceedings of the International Conference on Humanoid Robotics, 2003. [4] Thomas G. Dietterich. Hierarchical reinforcement learning with the MAXQ value function decomposition. Journal of Artificial Intelligence Research, 13:227–303, 2000. [5] Nicholas K. Jong and Peter Stone. State abstraction discovery from irrelevant state variables. In Proceedings of the International Joint Conference on Artificial Intelligence, 2005. [6] H. Kim, T. Kang, V. G. Loc, and H. R. Choi. Gait planning of quadruped walking and climbing robot for locomotion in 3D environment. In Proceedings of the International Conference on Robotics and Automation, 2005. [7] Nate Kohl and Peter Stone. Machine learning for fast quadrupedal locomotion. In Proceedings of AAAI, 2004. [8] J. Zico Kolter, Mike P. Rodgers, and Andrew Y. Ng. A complete control architecture for quadruped locomotion over rough terrain. In Proceedings of the International Conference on Robotics and Automation (to appear), 2008. [9] Honglak Lee, Yirong Shen, Chih-Han Yu, Gurjeet Singh, and Andrew Y. Ng. Quadruped robot obstacle negotiation via reinforcement learning. In Proceedings of the International Conference on Robotics and Automation, 2006. [10] Jun Morimoto and Christopher G. Atkeson. Minimax differential dynamic programming: An application to robust biped walking. In Neural Information Processing Systems 15, 2002. [11] Gergeley Neu and Csaba Szepesv´ari. Apprenticeship learning using inverse reinforcement learning and gradient methods. In Proceedings of Uncertainty in Artificial Intelligence, 2007. [12] Ronald Parr and Stuart Russell. Reinforcement learning with hierarchcies of machines. In Neural Information Processing Systems 10, 1998. [13] Nathan Ratliff, J. Andrew Bagnell, and Martin Zinkevich. Maximum margin planning. In Proceedings of the International Conference on Machine Learning, 2006. [14] Nathan Ratliff, David Bradley, J. Andrew Bagnell, and Joel Chestnutt. Boosting structured prediction for imitation learning. In Neural Information Processing Systems 19, 2007. [15] Richard S. Sutton, Doina Precup, and Satinder Singh. Between mdps and semi-mdps: A framework for temporal abstraction in reinforcement learning. Artificial Intelligence, 112:181–211, 1999. [16] Ben Taskar, Vassil Chatalbashev, Daphne Koller, and Carlos Guestrin. Learning structured prediction models: A large margin approach. In Proceedings of the International Conference on Machine Learning, 2005. [17] I. Tsochantaridis, T. Joachims, T. Hofmann, and Y. Altun. Large margin methods for structured and interdependent output variables. Journal of Machine Learning Research, 6:1453–1484, 2005. 8
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Colored Maximum Variance Unfolding Le Song†, Alex Smola†, Karsten Borgwardt‡ and Arthur Gretton∗ †National ICT Australia, Canberra, Australia ‡University of Cambridge, Cambridge, United Kingdom ∗MPI for Biological Cybernetics, T¨ubingen, Germany {le.song,alex.smola}@nicta.com.au kmb51@eng.cam.ac.uk,arthur.gretton@tuebingen.mpg.de Abstract Maximum variance unfolding (MVU) is an effective heuristic for dimensionality reduction. It produces a low-dimensional representation of the data by maximizing the variance of their embeddings while preserving the local distances of the original data. We show that MVU also optimizes a statistical dependence measure which aims to retain the identity of individual observations under the distancepreserving constraints. This general view allows us to design “colored” variants of MVU, which produce low-dimensional representations for a given task, e.g. subject to class labels or other side information. 1 Introduction In recent years maximum variance unfolding (MVU), introduced by Saul et al. [1], has gained popularity as a method for dimensionality reduction. This method is based on a simple heuristic: maximizing the overall variance of the embedding while preserving the local distances between neighboring observations. Sun et al. [2] show that there is a dual connection between MVU and the goal of finding a fast mixing Markov chain. This connection is intriguing. However, it offers limited insight as to why MVU can be used for data representation. This paper provides a statistical interpretation of MVU. We show that the algorithm attempts to extract features from the data which simultaneously preserve the identity of individual observations and their local distance structure. Our reasoning relies on a dependence measure between sets of observations, the Hilbert-Schmidt Independence Criterion (HSIC) [3]. Relaxing the requirement of retaining maximal information about individual observations, we are able to obtain “colored” MVU. Unlike traditional MVU which takes only one source of information into account, “colored” MVU allows us to integrate two sources of information into a single embedding. That is, we are able to find an embedding that leverages between two goals: • preserve the local distance structure according to the first source of information (the data); • and maximally align with the second sources of information (side information). Note that not all features inherent in the data are interesting for an ulterior objective. For instance, if we want to retain a reduced representation of the data for later classification, then only those discriminative features will be relevant. “Colored” MVU achieves the goal of elucidating primarily relevant features by aligning the embedding to the objective provided in the side information. Some examples illustrate this situation in more details: • Given a-bag-of-pixels representation of images (the data), such as USPS digits, find an embedding which reflects the categories of the images (side information). • Given a vector space representation of texts on the web (the data), such as newsgroups, find an embedding which reflects a hierarchy of the topics (side information). 1 • Given a TF/IDF representation of documents (the data), such as NIPS papers, find an embedding which reflects co-authorship relations between the documents (side information). There is a strong motivation for not simply merging the two sources of information into a single distance metric: Firstly, the data and the side information may be heterogenous. It is unclear how to combine them into a single distance metric; Secondly, the side information may appear in the form of similarity rather than distance. For instance, co-authorship relations is a similarity between documents (if two papers share more authors, they tends to be more similar), but it does not induce a distance between the documents (if two papers share no authors, we cannot assert they are far apart). Thirdly, at test time (i.e. when inserting a new observation into an existing embedding) only one source of information might be available, i.e. the side information is missing. 2 Maximum Variance Unfolding We begin by giving a brief overview of MVU and its projection variants, as proposed in [1]. Given a set of m observations Z = {z1, . . . , zm} ⊆Z and a distance metric d : Z × Z →[0, ∞) find an inner product matrix (kernel matrix) K ∈Rm×m with K ⪰0 such that 1. The distances are preserved, i.e. Kii + Kjj −2Kij = d2 ij for all (i, j) pairs which are sufficiently close to each other, such as the n nearest neighbors of each observation. We denote this set by N. We will also use N to denote the graph formed by having these (i, j) pairs as edges. 2. The embedded data is centered, i.e. K1 = 0 (where 1 = (1, . . . , 1)⊤and 0 = (0, . . . , 0)⊤). 3. The trace of K is maximized (the maximum variance unfolding part). Several variants of this algorithm, including a large scale variant [4] have been proposed. By and large the optimization problem looks as follows: maximize K⪰0 tr K subject to K1 = 0 and Kii + Kjj −2Kij = d2 ij for all (i, j) ∈N. (1) Numerous variants of (1) exist, e.g. where the distances are only allow to shrink, where slack variables are added to the objective function to allow approximate distance preserving, or where one uses low-rank expansions of K to cope with the computational complexity of semidefinite programming. A major drawback with MVU is that its results necessarily come as somewhat of a surprise. That is, it is never clear before invoking MVU what specific interesting results it might produce. While in hindsight it is easy to find an insightful interpretation of the outcome, it is not a-priori clear which aspect of the data the representation might emphasize. A second drawback is that while in general generating brilliant results, its statistical origins are somewhat more obscure. We aim to address these problems by means of the Hilbert-Schmidt Independence Criterion. 3 Hilbert-Schmidt Independence Criterion Let sets of observations X and Y be drawn jointly from some distribution Prxy. The HilbertSchmidt Independence Criterion (HSIC) [3] measures the dependence between two random variables, x and y, by computing the square of the norm of the cross-covariance operator over the domain X × Y in Hilbert Space. It can be shown, provided the Hilbert Space is universal, that this norm vanishes if and only if x and y are independent. A large value suggests strong dependence with respect to the choice of kernels. Let F and G be the reproducing kernel Hilbert Spaces (RKHS) on X and Y with associated kernels k : X × X →R and l : Y × Y →R respectively. The cross-covariance operator Cxy : G 7→F is defined as [5] Cxy = Exy [(k(x, ·) −µx)(l(y, ·) −µy)] , (2) where µx = E[k(x, ·)] and µy = E[l(y, ·)]. HSIC is then defined as the square of the HilbertSchmidt norm of Cxy, that is HSIC(F, G, Prxy) := ∥Cxy∥2 HS . In term of kernels HSIC is Exx′yy′[k(x, x′)l(y, y′)] + Exx′[k(x, x′)]Eyy′[l(y, y′)] −2Exy[Ex′[k(x, x′)]Ey′[l(y, y′)]]. (3) 2 Given the samples (X, Y ) = {(x1, y1), . . . , (xm, ym)} of size m drawn from the joint distribution, Prxy, an empirical estimate of HSIC is [3] HSIC(F, G, Z) = (m −1)−2 tr HKHL, (4) where K, L ∈Rm×m are the kernel matrices for the data and the labels respectively, and Hij = δij −m−1 centers the data and the labels in the feature space. (For convenience, we will drop the normalization and use tr HKHL as HSIC.) HSIC has been used to measure independence between random variables [3], to select features or to cluster data (see the Appendix for further details). Here we use it in a different way: We try to construct a kernel matrix K for the dimension-reduced data X which preserves the local distance structure of the original data Z, such that X is maximally dependent on the side information Y as seen from its kernel matrix L. HSIC has several advantages as a dependence criterion. First, it satisfies concentration of measure conditions [3]: for random draws of observation from Prxy, HSIC provides values which are very similar. This is desirable, as we want our metric embedding to be robust to small changes. Second, HSIC is easy to compute, since only the kernel matrices are required and no density estimation is needed. The freedom of choosing a kernel for L allows us to incorporate prior knowledge into the dependence estimation process. The consequence is that we are able to incorporate various side information by simply choosing an appropriate kernel for Y . 4 Colored Maximum Variance Unfolding We state the algorithmic modification first and subsequently we explain why this is reasonable: the key idea is to replace tr K in (1) by tr KL, where L is the covariance matrix of the domain (side information) with respect to which we would like to extract features. For instance, in the case of NIPS papers which happen to have author information, L would be the kernel matrix arising from coauthorship and d(z, z′) would be the Euclidean distance between the vector space representations of the documents. Key to our reasoning is the following lemma: Lemma 1 Denote by L a positive semidefinite matrix in Rm×m and let H ∈Rm×m be defined as Hij = δij −m−1. Then the following two optimization problems are equivalent: maximize K tr HKHL subject to K ⪰0 and constraints on Kii + Kjj −2Kij. (5a) maximize K tr KL subject to K ⪰0 and constraints on Kii + Kjj −2Kij and K1 = 0. (5b) Any solution of (5b) solves (5a) and any solution of (5a) solves (5b) after centering K ←HKH. Proof Denote by Ka and Kb the solutions of (5a) and (5b) respectively. Kb is feasible for (5a) and tr KbL = tr HKbHL. This implies that tr HKaHL ≥tr HKbHL. Vice versa HKaH is feasible for (5b). Moreover tr HKaHL ≤tr KbL by requirement on the optimality of Kb. Combining both inequalities shows that tr HKaHL = tr KbL, hence both solutions are equivalent. This means that the centering imposed in MVU via constraints is equivalent to the centering in HSIC by means of the dependence measure tr HKHL itself. In other words, MVU equivalently maximizes tr HKHI, i.e. the dependence between K and the identity matrix I which corresponds to retain maximal diversity between observations via Lij = δij. This suggests the following colored version of MVU: maximize K tr HKHL subject to K ⪰0 and Kii + Kjj −2Kij = d2 ij for all (i, j) ∈N. (6) Using (6) we see that we are now extracting a Euclidean embedding which maximally depends on the coloring matrix L (for the side information) while preserving local distance structure. A second advantage of (6) is that whenever we restrict K further, e.g. by only allowing for K be part of a linear subspace formed by the principal vectors in some space, (6) remains feasible, whereas the (constrained) MVU formulation may become infeasible (i.e. K1 = 0 may not be satisfied). 3 5 Dual Problem To gain further insight into the structure of the solution of (6) we derive its dual problem. Our approach uses the results from [2]. First we define matrices Eij ∈Rm×m for each edge (i, j) ∈N, such that it has only four nonzero entries Eij ii = Eij jj = 1 and Eij ij = Eij ji = −1. Then the distance preserving constraint can be written as tr KEij = d2 ij. Thus we have the following Lagrangian: L = tr KHLH + tr KZ − X (i,j)∈N wij(tr KEij −d2 ij) = tr K(HLH + Z − X (i,j)∈N wijEij) + X (i,j)∈N wijd2 ij where Z ⪰0 and wij ≥0. (7) Setting the derivative of L with respect to K to zero, yields HLH + Z −P (i,j)∈N wijEij = 0. Plugging this condition into (7) gives us the dual problem. minimize wij X (i,j)∈N wijd2 ij subject to G(w) ⪰HLH where G(w) = X (i,j)∈N wijEij. (8) Note that G(w) amounts to the Graph Laplacian of a weighted graph with adjacency matrix given by w. The dual constraint G(w) ⪰HLH effectively requires that the eigen-spectrum of the graph Laplacian is bounded from below by that of HLH. We are interested in the properties of the solution K of the primal problem, in particular the number of nonzero eigenvalues. Recall that at optimality the Karush-Kuhn-Tucker conditions imply tr KZ = 0, i.e. the row space of K lies in the null space of Z. Thus the rank of K is upper bounded by the dimension of the null space of Z. Recall that Z = G(w) −HLH ⪰0, and by design G(w) ⪰0 since it is the graph Laplacian of a weighted graph with edge weights wij. If G(w) corresponds to a connected graphs, only one eigenvalue of G(w) vanishes. Hence, the eigenvectors of Z with zero eigenvalues would correspond to those lying in the image of HLH. If L arises from a label kernel matrix, e.g. for an n-class classification problem, then we will only have up to n vanishing eigenvalues in Z. This translates into only up to n nonvanishing eigenvalues in K. Contrast this observation with plain MVU. In this case L = I, that is, only one eigenvalue of HLH vanishes. Hence it is likely that G(w) −HLH will have many vanishing eigenvalues which translates into many nonzero eigenvalues of K. This is corroborated by experiments (Section 7). 6 Implementation Details In practice, instead of requiring the distances to remain unchanged in the embedding we only require them to be preserved approximately [4]. We do so by penalizing the slackness between the original distance and the embedding distance, i.e. maximize K tr HKHL −ν X (i,j)∈N Kii + Kjj −2Kij −d2 ij 2 subject to K ⪰0 (9) Here ν controls the tradeoff between dependence maximization and distance preservation. The semidefinite program usually has a time complexity up to O(m6). This renders direct implementation of the above problem infeasible for anything but toy problems. To reduce the computation, we approximate K using an orthonormal set of vectors V (of size m×n) and a smaller positive definite matrix A (of size n × n), i.e. K = VAV⊤. Conveniently we choose the number of dimensions n to be much smaller than m (n ≪m) such that the resulting semidefinite program with respect to A becomes tractable (clearly this is an approximation). To obtain the matrix V we employ a regularization scheme as proposed in [4]. First, we construct a nearest neighbor graph according to N (we will also refer to this graph and its adjacency matrix as N). Then we form V by stacking together the bottom n eigenvectors of the graph Laplacian of the neighborhood graph via N. The key idea is that neighbors in the original space remain neighbors in 4 the embedding space. As we require them to have similar locations, the bottom eigenvectors of the graph Laplacian provide a set of good bases for functions smoothly varying across the graph. Subsequent to the semidefinite program we perform local refinement of the embedding via gradient descent. Here the objective is reformulated using an m × n dimensional vector X, i.e. K = XX⊤. The initial value X0 is obtained using the n leading eigenvectors of the solution of (9). 7 Experiments Ultimately the justification for an algorithm is practical applicability. We demonstrate this based on three datasets: embedding of digits of the USPS database, the Newsgroups 20 dataset containing Usenet articles in text form, and a collection of NIPS papers from 1987 to 1999.1 We compare “colored” MVU (also called MUHSIC, maximum unfolding via HSIC) to MVU [1] and PCA, highlighting places where MUHSIC produces more meaningful results by incorporating side information. Further details, such as effects of the adjacency matrices and a comparison to Neighborhood Component Analysis [6] are relegated to the appendix due to limitations of space. For images we use the Euclidean distance between pixel values as the base metric. For text documents, we perform four standard preprocessing steps: (i) the words are stemmed using the Porter stemmer; (ii) we filter out common but meaningless stopwords; (iii) we delete words that appear in less than 3 documents; (iv) we represent each document as a vector using the usual TF/IDF (term frequency / inverse document frequency) weighting scheme. As before, the Euclidean distance on those vectors is used to find the nearest neighbors. As in [4] we construct the nearest neighbor graph by considering the 1% nearest neighbors of each point. Subsequently the adjacency matrix of this graph is symmetrized. The regularization parameter ν as given in (9) is set to 1 as a default. Moreover, as in [4] we choose 10 dimensions (n = 10) to decompose the embedding matrix K. Final visualization is carried out using 2 dimensions. This makes our results very comparable to previous work. USPS Digits This dataset consists of images of hand written digits of a resolution of 16×16 pixels. We normalized the data to the range [−1, 1] and used the test set containing 2007 observations. Since it is a digit recognition task, we have Y ∈[0, . . . , 9]. Y is used to construct the matrix L by applying the kernel k(y, y′) = δy,y′. This kernel further promotes embedding where images from the same class are grouped tighter. Figure 1 shows the results produced by MUHSIC, MVU and PCA. The overall properties the embeddings are similar across the three methods (‘2’ on the left, ‘1’ on the right, ‘7’ on top, and ‘8’ at the bottom). Arguably MUHSIC produces a clearer visualization. For instance, images of ‘5’ are clustered tighter in this case than the other two methods. Furthermore, MUHSIC also results in much better separation between images from different classes. For instance, the overlap between ‘4’ and ‘6’ produce by MVU and PCA are largely reduced by MUHSIC. Similar results also hold for ‘0’ and ‘5’. Figure 1 also shows the eigenspectrum of K produced by different methods. The eigenvalues are sorted in descending order and normalized by the trace of K. Each patch in the color bar represents an eigenvalue. We see that MUHSIC results in 3 significant eigenvalues, MVU results in 10, while PCA produces a grading of many eigenvalues (as can be seen by an almost continuously changing spectrum in the spectral diagram). This confirms our reasoning of Section 5 that the spectrum generated by MUHSIC is likely to be considerably sparser than that of MVU. Newsgroups This dataset consists of Usenet articles collected from 20 different newsgroups. We use a subset of 2000 documents for our experiments (100 articles from each newsgroup). We remove the headers from the articles before the preprocessing while keeping the subject line. There is a clear hierarchy in the newsgroups. For instance, 5 topics are related to computer science, 3 are related to religion, and 4 are related to recreation. We will use these different topics as side information and apply a delta kernel k(y, y′) = δy,y′ on them. Similar to USPS digits we want to preserve the identity of individual newsgroups. While we did not encode hierarchical information for MVU we recover a meaningful hierarchy among topics, as can be seen in Figure 2. 1Preprocessed data are available at http://www.it.usyd.edu.au/∼lesong/muhsic datasets.html. 5 Figure 1: Embedding of 2007 USPS digits produced by MUHSIC, MVU and PCA respectively. Colors of the dots are used to denote digits from different classes. The color bar below each figure shows the eigenspectrum of the learned kernel matrix K. Figure 2: Embedding of 2000 newsgroup articles produced by MUHSIC, MVU and PCA respectively. Colors and shapes of the dots are used to denote articles from different newsgroups. The color bar below each figure shows the eigenspectrum of the learned kernel matrix K. A distinctive feature of the visualizations is that MUHSIC groups articles from individual topics more tightly than MVU and PCA. Furthermore, the semantic information is also well preserved by MUHSIC. For instance, on the left side of the embedding, all computer science topics are placed adjacent to each other; comp.sys.ibm.pc.hardware and comp.os.ms-windows.misc are adjacent and well separated from comp.sys.mac.hardware and comp.windows.x and comp.graphics. The latter is meaningful since Apple computers are more popular in graphics (so are X windows based systems for scientific visualization). Likewise we see that on the top we find all recreational topics (with rec.sport.baseball and rec.sport.hockey clearly distinguished from the rec.autos and rec.motorcycles groups). A similar adjacency between talk.politics.mideast and soc.religion.christian is quite interesting. The layout suggests that the content of talk.politics.guns and of sci.crypt is quite different from other Usenet discussions. NIPS Papers We used the 1735 regular NIPS papers from 1987 to 1999. They are scanned from the proceedings and transformed into text files via OCR. The table of contents (TOC) is also available. We parse the TOC and construct a coauthor network from it. Our goal is to embed the papers by taking the coauthor information into account. As kernel k(y, y′) we simply use the number of authors shared by two papers. To illustrate this we highlighted some known researchers. Furthermore, we also annotated some papers to show the semantics revealed by the embedding. Figure 3 shows the results produced by MUHSIC, MVU and PCA. All three methods correctly represent the two major topics of NIPS papers: artificial systems, i.e. machine learning (they are positioned on the left side of the visualization) and natural systems, 6 Figure 3: Embedding of 1735 NIPS papers produced by MUHSIC, MVU and PCA. Papers by some representative (combinations of) researchers are highlighted as indicated by the legend. The color bar below each figure shows the eigenspectrum of the learned kernel matrix K. The yellow diamond in the graph denotes the current paper as submitted to NIPS. This paper is placed in the location of its nearest neighbor; more details are in the appendix. 7 i.e. computational neuroscience (which lie on the right). This is be confirmed by examining the highlighted researchers. For instance, the papers by Smola, Sch¨olkopf and Jordan are embedded on the left, whereas the many papers by Sejnowski, Dayan and Bialek can be found on the right. Unique to the visualization of MUHSIC is that there is a clear grouping of the papers by researchers. For instance, papers on reinforcement learning (Barto, Singh and Sutton) are on the upper left corner; papers by Hinton (computational cognitive science) are near the lower left corner; and papers by Sejnowski and Dayan (computational neuroscientists) are clustered to the right side and adjacent to each other. Interestingly, papers by Jordan (at that time best-known for his work in graphical models) are grouped close to the papers on reinforcement learning. This is because Singh used to be a postdoc of Jordan. Another interesting trend is that papers on new fields of research are embedded on the edges. For instance, papers on reinforcement learning (Barto, Singh and Sutton), are along the left edge. This is consistent with the fact that they presented some interesting new results during this period (recall that the time period of the dataset is 1987 to 1999). Note that while MUHSIC groups papers according to authors, thereby preserving the macroscopic structure of the data it also reveals the microscopic semantics between the papers. For instance, the 4 papers (numbered from 6 to 9 in Figure 3) by Smola, Scholk¨opf, Hinton and Dayan are very close to each other. Although their titles do not convey strong similarity information, these papers all used handwritten digits for the experiments. A second example are papers by Dayan. Although most of his papers are on the neuroscience side, two of his papers (numbered 14 and 15) on reinforcement learning can be found on the machine learning side. A third example are papers by Bialek and Hinton on spiking neurons (numbered 20, 21 and 23). Although Hinton’s papers are mainly on the left, his paper on spiking Boltzmann machines is closer to Bialek’s two papers on spiking neurons. 8 Discussion In summary, MUHSIC provides an embedding of the data which preserves side information possibly available at training time. This way we have a means of controlling which representation of the data we obtain rather than having to rely on our luck that the representation found by MVU just happens to match what we want to obtain. It makes feature extraction robust to spurious interactions between observations and noise (see the appendix for an example of adjacency matrices and further discussion). A fortuitous side-effect is that if the matrix containing side information is of low rank, the reduced representation learned by MUHSIC can be lower rank than that obtained by MVU, too. Finally, we showed that MVU and MUHSIC can be formulated as feature extraction for obtaining maximally dependent features. This provides an information theoretic footing for the (brilliant) heuristic of maximizing the trace of a covariance matrix [1]. The notion of extracting features of the data which are maximally dependent on the original data is far more general than what we described in this paper. In particular, one may show that feature selection [7] and clustering [8] can also be seen as special cases of this framework. Acknowledgments NICTA is funded through the Australian Government’s Backing Australia’s Ability initiative, in part through the ARC.This research was supported by the Pascal Network. References [1] K. Q. Weinberger, F. Sha, and L. K. Saul. Learning a kernel matrix for nonlinear dimensionality reduction. In Proceedings of the 21st International Conference on Machine Learning, Banff, Canada, 2004. [2] J. Sun, S. Boyd, L. Xiao, and P. Diaconis. The fastest mixing markove process on a graph and a connection to a maximum variance unfolding problem. SIAM Review, 48(4):681–699, 2006. [3] A. Gretton, O. Bousquet, A.J. Smola, and B. Sch¨olkopf. Measuring statistical dependence with HilbertSchmidt norms. In S. Jain, H. U. Simon, and E. Tomita, editors, Proceedings Algorithmic Learning Theory, pages 63–77, Berlin, Germany, 2005. Springer-Verlag. [4] K. Weinberger, F. Sha, Q. Zhu, and L. Saul. Graph laplacian regularization for large-scale semidefinte programming. In Neural Information Processing Systems, 2006. [5] K. Fukumizu, F. R. Bach, and M. I. Jordan. Dimensionality reduction for supervised learning with reproducing kernel hilbert spaces. J. Mach. Learn. Res., 5:73–99, 2004. [6] J. Goldberger, S. Roweis, G. Hinton, and R. Salakhutdinov. Neighbourhood component analysis. In Advances in Neural Information Processing Systems 17, 2004. [7] L. Song, A. Smola, A. Gretton, K. Borgwardt, and J. Bedo. Supervised feature selection via dependence estimation. In Proc. Intl. Conf. Machine Learning, 2007. [8] L. Song, A. Smola, A. Gretton, and K. Borgwardt. A dependence maximization view of clustering. In Proc. Intl. Conf. Machine Learning, 2007. 8
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Adaptive Embedded Subgraph Algorithms using Walk-Sum Analysis Venkat Chandrasekaran, Jason K. Johnson, and Alan S. Willsky Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology venkatc@mit.edu, jasonj@mit.edu, willsky@mit.edu Abstract We consider the estimation problem in Gaussian graphical models with arbitrary structure. We analyze the Embedded Trees algorithm, which solves a sequence of problems on tractable subgraphs thereby leading to the solution of the estimation problem on an intractable graph. Our analysis is based on the recently developed walk-sum interpretation of Gaussian estimation. We show that non-stationary iterations of the Embedded Trees algorithm using any sequence of subgraphs converge in walk-summable models. Based on walk-sum calculations, we develop adaptive methods that optimize the choice of subgraphs used at each iteration with a view to achieving maximum reduction in error. These adaptive procedures provide a significant speedup in convergence over stationary iterative methods, and also appear to converge in a larger class of models. 1 Introduction Stochastic processes defined on graphs offer a compact representation for the Markov structure in a large collection of random variables. We consider the class of Gaussian processes defined on graphs, or Gaussian graphical models, which are used to model natural phenomena in many large-scale applications [1, 2]. In such models, the estimation problem can be solved by directly inverting the information matrix. However, the resulting complexity is cubic in the number of variables, thus being prohibitively complex in applications involving hundreds of thousands of variables. Algorithms such as Belief Propagation and the junction-tree method are effective for computing exact estimates in graphical models that are tree-structured or have low treewidth [3], but for graphs with high treewidth the junction-tree approach is intractable. We describe a rich class of iterative algorithms for estimation in Gaussian graphical models with arbitrary structure. Specifically, we discuss the Embedded Trees (ET) iteration [4] that solves a sequence of estimation problems on trees, or more generally tractable subgraphs, leading to the solution of the original problem on the intractable graph. We analyze non-stationary iterations of the ET algorithm that perform inference calculations on an arbitrary sequence of subgraphs. Our analysis is based on the recently developed walk-sum interpretation of inference in Gaussian graphical models [5]. We show that in the broad class of so-called walk-summable models, the ET algorithm converges for any arbitrary sequence of subgraphs used. The walk-summability of a model is easily tested [5, 6], thus providing a simple sufficient condition for the convergence of such non-stationary algorithms. Previous convergence results [6, 7] analyzed stationary or “cyclo-stationary” iterations that use the same subgraph at each iteration or cycle through a fixed sequence of subgraphs. The focus of this paper is on analyzing, and developing algorithms based on, arbitrary non-stationary iterations that use any (non-cyclic) sequence of subgraphs, and the recently developed concept of walk-sums appears to be critical to this analysis. 1 Given this great flexibility in choosing successive iterative steps, we develop algorithms that adaptively optimize the choice of subgraphs to achieve maximum reduction in error. These algorithms take advantage of walk-sum calculations, which are useful in showing that our methods minimize an upper-bound on the error at each iteration. We develop two procedures to adaptively choose subgraphs. The first method finds the best tree at each iteration by solving an appropriately formulated maximum-weight spanning tree problem, with the weight of each edge being a function of the partial correlation coefficient of the edge and the residual errors at the nodes that compose the edge. The second method, building on this first method, adds extra edges in a greedy manner to the tree resulting from the first method to form a thin hypertree. Simulation results demonstrate that these non-stationary algorithms provide a significant speedup in convergence over stationary and cyclic iterative methods. Since the class of walk-summable models is broad (including attractive models, diagonally dominant models, and so-called pairwise-normalizable models), our methods provide a convergent, computationally attractive method for inference. We also provide empirical evidence to show that our adaptive methods (with a minor modification) converge in many non-walk-summable models when stationary iterations diverge. The estimation problem in Gaussian graphical models involves solving a linear system with a sparse, symmetric, positive-definite matrix. Such systems are commonly encountered in other areas of machine learning and signal processing as well [8, 9]. Therefore, our methods are broadly applicable beyond estimation in Gaussian models. Some of the results presented here appear in more detail in a longer paper [10], which provides complete proofs as well as a detailed description of walk-sum diagrams that give a graphical interpretation of our algorithms (we show an example in this paper). The report also considers problems involving communication “failure” between nodes for distributed sensor network applications. 2 Background Let G = (V, E) be a graph with vertices V , and edges E ⊂ V 2 that link pairs of vertices together. Here, V 2 represents the set of all unordered pairs of vertices. Consider a Gaussian distribution in information form [5] p(x) ∝exp{−1 2xT Jx + hT x}, where J−1 is the covariance matrix and J−1h is the mean. The matrix J, also called the information matrix, is sparse according to graph G, i.e. Js,t = Jt,s = 0 if and only if {s, t} /∈E. Thus, G represents the graph with respect to which p(x) is Markov, i.e. p(x) satisfies the conditional independencies implied by the separators of G. The Gaussian mean estimation problem reduces to solving the following linear system of equations: Jx = h, (1) where x is the mean vector. Convergent iterations that compute the mean can also be used in turn to compute variances using a variety of methods [4, 11]. Thus, we focus on the problem of estimating the mean at each node. Throughout the rest of this paper, we assume that J is normalized to have 1’s along the diagonal.1 Such a re-scaling does not affect the convergence results in this paper, and our analysis and algorithms can be easily generalized to the un-normalized case [10]. 2.1 Walk-sums We give a brief overview of the walk-sum framework developed in [5]. Let J = I −R. The offdiagonal entries of the matrix R have the same sparsity structure as that of J, and consequently that of the graph G. For Gaussian processes defined on graphs, element Rs,t corresponds to the conditional correlation coefficient between the variables at vertices s and t conditioned on knowledge of all the other variables (also known as the partial correlation coefficient [5]). A walk is a sequence of vertices {wi}ℓ i=0 such that each step {wi, wi+1} ∈E, 0 ≤i ≤ℓ−1, with no restriction on crossing the same vertex or traversing the same edge multiple times. The weight of a walk is the product of the edge-wise partial correlation coefficients of the edges composing the walk: φ(w) ≜Qℓ−1 i=0 Rwi,wi+1. We then have that (Rℓ)s,t is the sum of the weights of all length-ℓwalks from s to t in G. With this point of view, we can interpret J−1 as follows: (J−1)s,t = ((I −R)−1)s,t = ∞ X ℓ=0 (Rℓ)s,t = ∞ X ℓ=0 φ(s ℓ→t), (2) 1This can be achieved by performing the transformation ˜J ←D−1 2 JD−1 2 , where D is a diagonal matrix containing the diagonal entries of J. 2 where φ(s ℓ→t) represents the sum of the weights of all the length-ℓwalks from s to t (the set of all such walks is finite). Thus, (J−1)s,t is the length-ordered sum over all walks in G from s to t. This, however, is a very specific way to compute the inverse that converges if the spectral radius ϱ(R) < 1. Other algorithms may compute walks according to different orders (rather than length-based orders). To analyze arbitrary algorithms that submit to a walk-sum interpretation, the following concept of walk-summability was developed in [5]. A model is said to be walk-summable if for each pair of vertices s, t ∈V , the absolute sum over all walks from s to t in G converges: ¯φ(s →t) ≜ X w∈W(s→t) |φ(w)| < ∞. (3) Here, W(s →t) represents the set of all walks from s to t, and ¯φ(s →t) denotes the absolute walk-sum2 over this set. Based on the absolute convergence condition, walk-summability implies that walk-sums over a countable set of walks in G can be computed in any order. As a result, we have the following interpretation in walk-summable models: (J−1)s,t = φ(s →t), (4) xt = (J−1h)t = X s∈V hsφ(s →t) ≜φ(h; ∗→t), (5) where the wildcard character ∗denotes a union over all vertices in V , and φ(h; W) denotes a reweighting of each walk in W by the corresponding h value at the starting node. Note that in (4) we relax the constraint that the sum is ordered by length, and do not explicitly specify an ordering on the walks (such as in (2)). In words, (J−1)s,t is the walk-sum over the set of all walks from s to t, and xt is the walk-sum over all walks ending at t, re-weighted by h. As shown in [5], the walk-summability of a model is equivalent to ϱ( ¯R) < 1, where ¯R denotes the matrix of the absolute values of the elements of R. Also, a broad class of models are walk-summable, including diagonally-dominant models, so-called pairwise normalizable models, and models for which the underlying graph G is non-frustrated, i.e. each cycle has an even number of negative partial correlation coefficients. Walk-summability implies that a model is valid, i.e. has positivedefinite information/covariance. Concatenation of walks We briefly describe the concatenation operation for walks and walk-sets, which plays a key role in walk-sum analysis. Let u = u0 · · · uend and v = vstartv1 · · · vℓ(v) be walks with uend = vstart. The concatenation of these walks is defined to be u·v ≜u0 · · · uendv1 · · · vℓ(v). Now consider a walk-set U with all walks ending at uend and another walk-set V with all walks beginning at vstart. If uend = vstart, then the concatenation of U and V is defined: U ⊗V ≜{u · v : u ∈U, v ∈V}. 2.2 Embedded Trees algorithm We describe the Embedded Trees iteration that performs a sequence of updates on trees, or more generally tractable subgraphs, leading to the solution of (1) on an intractable graph. Each iteration involves an inference calculation on a subgraph of all the variables V . Let (V, S) be some subgraph of G, i.e. S ⊂E (see examples in Figure 1). Let J be split according to S as J = JS −KS, so that the entries of J corresponding to edges in S are assigned to JS, and those corresponding to E\S are part of KS. The diagonal entries of J are all part of JS; thus, KS has zeroes along the diagonal.3 Based on this splitting, we can transform (1) to JSx = KSx+h, which suggests a natural recursion: JSbx(n) = KSbx(n−1) + h. If JS is invertible, and it is tractable to apply J−1 S to a vector, then ET offers an effective method to solve (1) (assuming ϱ(J−1 S KS) < 1). If the subgraph used changes with each iteration, then we obtain the following non-stationary ET iteration: bx(n) = J−1 Sn (KSnbx(n−1) + h), (6) where {Sn}∞ n=1 is any arbitrary sequence of subgraphs. An important degree of freedom is the choice of the subgraph Sn at iteration n, which forms the focus of Section 4 of this paper. In [10] we also consider a more general class of algorithms that update subsets of variables at each iteration. 2We generally denote the walk-sum of the set W(∼) by φ(∼). 3KS can have non-zero diagonal in general, but we only consider the zero diagonal case here. 3 Figure 1: (Left) G and three embedded trees S1, S2, S3; (Right) Corresponding walk-sum diagram. 3 Walk-Sum Analysis and Convergence of the Embedded Trees algorithm In this section, we provide a walk-sum interpretation for the ET algorithm. Using this analysis, we show that the non-stationary ET iteration (6) converges in walk-summable models for an arbitrary choice of subgraphs {Sn}∞ n=1. Before proceeding with the analysis, we point out that one potential complication with the ET algorithm is that the matrix JS corresponding to some subgraph S may be indefinite or singular, even if the original model J is positive-definite. Importantly, such a problem never arises in walk-summable models with JS being positive-definite for any subgraph S if J is walk-summable. This is easily seen because walks in the subgraph S are a subset of the walks in G, and thus if absolute walk-sums in G are well-defined, then so are absolute walk-sums in S. Therefore, JS is walk-summable, and hence, positive-definite. Consider the following recursively defined set of walks for s, t ∈V : Wn(s →t) = ∪u,v∈V Wn−1(s →u) ⊗W(u E\Sn(1) −→ v) ⊗W(v Sn −→t) [ W(s Sn −→t) = Wn−1(s →∗) ⊗W(∗ E\Sn(1) −→ •) ⊗W(• Sn −→t) [ W(s Sn −→t), (7) with W0(s →t) = ∅. Here, ∗and • are used as wildcard characters (a union over all elements in V ), and ⊗denotes concatenation of walk-sets as described previously. The set Wn−1(s →∗) denotes walks that start at node s computed at the previous iteration. The middle term W(∗ E\Sn(1) −→ •) denotes a length-1 walk (called a hop) across an edge in E\Sn. Finally, W(• Sn −→t) denotes walks in Sn that end at node t. Thus, the first term in (7) refers to previously computed walks starting at s, which hop across an edge in E\Sn, and then finally propagate only in Sn (ending at t). The second term W(s Sn −→t) denotes walks from s to t that only live within Sn. The following proposition (proved in [10]) shows that the walks contained in these walk-sets are precisely those computed by the ET algorithm at iteration n. For simplicity, we denote φ(Wn(s →t)) by φn(s →t). Proposition 1 Let bx(n) be the estimate at iteration n in the ET algorithm (6) with initial guess bx(0) = 0. Then, bx(n) t = φn(h; ∗→t) = P s∈V hsφn(s →t) in walk-summable models. We note that the classic Gauss-Jacobi algorithm [6], a stationary iteration with JS = I and KS = R, can be interpreted as a walk-sum algorithm: bx(n) t in this method computes all walks up to length n ending at t. Figure 1 gives an example of a walk-sum diagram, which provides a graphical representation of the walks accumulated by the walk-sets (7). The diagram is the three-level graph on the right, and corresponds to an ET iteration based on the subgraphs S1, S2, S3 of the 3 × 3 grid G (on the left). Each level n in the diagram consists of the subgraph Sn used at iteration n (solid edges), and information from the previous level (iteration) n −1 is transmitted through the dashed edges in E\Sn. The directed nature of these dashed edges is critical as they capture the one-directional flow of computations from iteration to iteration, while the undirected edges within each level capture the inference computation at each iteration. Consider a node v at level n of the diagram. Walks in the diagram that start at any node and end at v at level n, re-weighted by h, are exactly the walks computed by the ET algorithm in bx(n) v . For more examples of such diagrams, see [10]. Given this walk-sum interpretation of the ET algorithm, we can analyze the walk-sets (7) to prove the convergence of ET in walk-summable models by showing that the walk-sets eventually contain all the walks required for the computation of J−1h in (5). We have the following convergence theorem for which we only provide a brief sketch of the complete proof [10]. 4 Theroem 1 Let bx(n) be the estimate at iteration n in the ET algorithm (6) with initial guess bx(0) = 0. Then, bx(n) →J−1h element-wise as n →∞in walk-summable models. Proof outline: Proving this statement is done in the following stages. Validity: The walks in Wn are valid walks in G, i.e. Wn(s →t) ⊆W(s →t). Nesting: The walk-sets Wn(s →t) are nested, i.e. Wn−1(s →t) ⊆Wn(s →t), ∀n. Completeness: Let w ∈W(s →t). There exists an N > 0 such that w ∈WN(s →t). Using the nesting property, we conclude that for all n ≥N, w ∈Wn(s →t). These steps combined together allow us to conclude that φn(s →t) →φ(s →t) as n →∞. This conclusion relies on the fact that φ(Wn) →φ(∪nWn) as n →∞for a sequence of nested walk-sets Wn−1 ⊆Wn in walk-summable models, which is a consequence of the sum-partition theorem for absolutely summable series [5, 10, 12]. Given the walk-sum interpretation from Proposition 1, one can check that bx(n) →J−1h element-wise as n →∞. □ Thus, the ET algorithm converges to the correct solution of (1) in walk-summable models for any sequence of subgraphs with bx(0) = 0. It is then straightforward to show that convergence can be achieved for any initial guess [10]. Note that we have taken advantage of the absolute convergence property in walk-summable models (3) by not focusing on the order in which walks are computed, but only that they are eventually computed. In [10], we prove that walk-summability is also a necessary condition for the complete flexibility in the choice of subgraphs — there exists at least one sequence of subgraphs that results in a divergent ET iteration in non-walk-summable models. 4 Adaptive algorithms Let e(n) = x−bx(n) be the error at iteration n and let h(n) = Je(n) = h−Jbx(n) be the corresponding residual error (which is tractable to compute). We begin by describing an algorithm to choose the “next-best” tree Sn in the ET iteration (6). The error at iteration n can be re-written as follows: e(n) = (J−1 −J−1 Sn )h(n−1). Thus, we have the walk-sum interpretation e(n) t = φ(h(n−1); ∗ G\Sn −→t), where G\Sn denotes walks that do not live entirely within Sn. Using this expression for the error, we have the following bound that is tight for attractive models (Rs,t ≥0 for all s, t ∈V ) and non-negative h(n−1): ∥e(n)∥ℓ1 = X t∈V |φ(h(n−1); ∗ G\Sn −→t)| ≤ ¯φ(|h(n−1)|; G\Sn) = ¯φ(|h(n−1)|; G) −¯φ(|h(n−1)|; Sn). (8) Hence, minimizing the error at iteration n corresponds to finding the tree Sn that maximizes the second term ¯φ(|h(n−1)|; Sn). This leads us to the following maximum walk-sum tree problem: arg max Sn a tree ¯φ(|h(n−1)|; Sn) (9) Finding the optimal such tree is combinatorially complex. Therefore, we develop a relaxation that minimizes a looser upper bound than (8). Specifically, consider an edge {u, v} and all the walks that live on this single edge W({u, v}) = {uv, vu, uvu, vuv, uvuv, vuvu, . . . }. One can check that the contribution based on these single-edge walks can be computed as: σu,v = X w∈W({u,v}) ¯φ(|h(n−1)|; w) = |h(n−1) u | + |h(n−1) v | |Ru,v| 1 −|Ru,v|. (10) This weight provides a measure of the error-reduction capacity of edge {u, v} by itself at iteration n. These single-edge walks for edges in Sn are a subset of all the walks in Sn, and consequently provide a lower-bound on ¯φ(|h(n−1)|; Sn). Therefore, the maximization arg max Sn a tree X {u,v}∈Sn σu,v (11) 5 Figure 2: Grayscale images of residual errors in an 8 × 8 grid at successive iterations, and corresponding trees chosen by adaptive method. Figure 3: Grayscale images of residual errors in an 8 × 8 grid at successive iterations, and corresponding hypertrees chosen by adaptive method. is equivalent to minimizing a looser upper-bound than (8). This relaxed problem can be solved efficiently using a maximum-weight spanning tree algorithm that has complexity O(|E| log log |V |) for sparse graphs [13]. Given the maximum-weight spanning tree of the graph, a natural extension is to build a thin hypertree by adding extra “strong” edges to the tree, subject to the constraint that the resulting graph has low treewidth. Unfortunately, to do so optimally is an NP-hard optimization problem [14]. Hence, we settle on a simple greedy algorithm. For each edge not included in the tree, in order of decreasing edge weight, we add the edge to the graph if two conditions are met: first, we are able to easily verify that the treewidth stays less than M, and second, the length of the unique path in Sn between the endpoints is less than L. In order to bound the tree width, we maintain a counter at each node of the total number of added edges that result in a path through that node. Comparing to another method for constructing junction trees from spanning trees [15], one can check that the maximum node count is an upper-bound on the treewidth. We note that by using an appropriate directed representation of Sn relative to an arbitrary root, it is simple to identify the path between two nodes with complexity linear in path length (< L).4 Hence, the additional complexity of this greedy algorithm over that of the tree-selection procedure described previously is O(L|E|). In Figure 2 and Figure 3 we present a simple demonstration of the tree and hypertree selection procedures respectively, and the corresponding change in error achieved. The grayscale images represent the residual errors at the nodes of an 8 × 8 grid similar to G in Figure 1 (with white representing 1 and black representing 0), and the graphs beside them show the trees/hypertrees chosen based on these residual errors using the methods described above (the grid edge partial correlation coefficients are the same for all edges). Notice that the first tree in Figure 2 tries to include as many edges as possible that are incident on the nodes with high residual error. Such edges are useful for capturing walks ending at the high-error nodes, which contribute to the set of walks in (5). The first hypertree in Figure 3 actually includes all the edges incident on the higherror nodes. The residual errors after inference on these subgraphs are shown next in Figure 2 and Figure 3. As expected, the hypertree seems to achieve greater reduction in error compared to the spanning tree. Again, at this iteration, the subgraphs chosen by our methods adapt based on the errors at the various nodes. 5 Experimental illustration 5.1 Walk-summable models We test the adaptive algorithms on densely connected nearest-neighbor grid-structured models (similar to G in Figure 1). We generate random grid models — the grid edge partial correlation coef4One sets two pointers into the tree starting from any two nodes and then iteratively walks up the tree, always advancing from the point that is deeper in the tree, until the nearest ancestor of the two nodes is reached. 6 Figure 4: (Left) Average number of iterations required for the normalized residual to reduce by a factor of 10−6 over 100 randomly generated 75 × 75 grid models; (Center) Convergence plot for a randomly generated 511×511 grid model; (Right) Convergence range in terms of partial correlation for 16-node cyclic model with edges to neighbors two steps away. Figure 5: (Left) 16-node graphical model; (Right) two embedded spanning trees T1, T2. ficients are chosen uniformly from [−1, 1] and R is scaled so that ϱ( ¯R) = 0.99. The vector h is chosen to be the all-ones vector. The table on the left in Figure 4 shows the average number of iterations required by various algorithms to reduce the normalized residual error ∥h(n)∥2 ∥h(0)∥2 by a factor of 10−6. The average was computed based on 100 randomly generated 75 × 75 grid models. The plot in Figure 4 shows the decrease in the normalized residual error as a function of the number of iterations on a randomly generated 511 × 511 grid model. All these models are poorly conditioned because they are barely walk-summable (ϱ( ¯R) = 0.99). The stationary one-tree iteration uses a tree similar to S1 in Figure 1, and the two-tree iteration alternates between trees similar to S1 and S3 in Figure 1 [4]. The adaptive hypertree method uses M = 6 and L = 8. We also note that in practice the per-iteration costs of the adaptive tree and hypertree algorithms are roughly comparable. These results show that our adaptive algorithms demonstrate significantly superior convergence properties compared to stationary methods, thus providing a convergent, computationally attractive method for estimation in walk-summable models. Our methods are applicable beyond Gaussian estimation to other problems that require solution of linear systems based on sparse, symmetric, positive-definite matrices. Several recent papers that develop machine learning algorithms are based on solving such systems of equations [8, 9]; in fact, both of these papers involve linear systems based on diagonally-dominant matrices, which are walk-summable. 5.2 Non-walk-summable models Next, we give empirical evidence that our adaptive methods provide convergence over a broader range of models than stationary iterations. One potential complication in non-walk-summable models is that the subgraph models chosen by the stationary and adaptive algorithms may be indefinite or singular even though J is positive-definite. In order to avoid this problem in the adaptive ET algorithm, the trees Sn chosen at each iteration must be valid (i.e., have positive-definite JSn). A simple modification to the maximum-weight spanning tree algorithm achieves this goal — we add an extra condition to the algorithm to test for diagonal dominance of the chosen tree model (as all symmetric, diagonally-dominant models are positive definite [6]). That is, at each step of the maximum-weight spanning tree algorithm, we only add an edge if it does not create a cycle and maintains a diagonally-dominant tractable subgraph model. Consider the 16-node model on the left in Figure 5. Let all the edge partial correlation coefficients be r. The range of r for which J is positive-definite is roughly (−0.46, 0.25), and the range for which the model is walk-summable is (−0.25, 0.25) (in this range all the algorithms, both stationary and adaptive, converge). For the onetree iteration we use tree T1, and for the two-tree iteration we alternate between trees T1 and T2 (see 7 Figure 5). As the table on the right in Figure 4 demonstrates, the adaptive tree algorithm without the diagonal-dominance (DD) check provides convergence over a much broader range of models than the one-tree and two-tree iterations, but not for all valid models. However, the modified adaptive tree algorithm with the DD check appears to converge almost up to the validity threshold. We have also observed such behavior empirically in many other (though not all) non-walk-summable models where the adaptive ET algorithm with the DD condition converges while stationary methods diverge. Thus, our adaptive methods, compared to stationary iterations, not only provide faster convergence rates in walk-summable models but also converge for a broader class of models. 6 Discussion We analyze non-stationary iterations of the ET algorithm that use any sequence of subgraphs for estimation in Gaussian graphical models. Our analysis is based on the recently developed walk-sum interpretation of inference in Gaussian models, and we show that the ET algorithm converges for any sequence of subgraphs used in walk-summable models. These convergence results motivate the development of methods to choose subgraphs adaptively at each iteration to achieve maximum reduction in error. The adaptive procedures are based on walk-sum calculations, and minimize an upper bound on the error at each iteration. Our simulation results show that the adaptive algorithms provide a significant speedup in convergence over stationary methods. Moreover, these adaptive methods also seem to provide convergence over a broader class of models than stationary algorithms. Our adaptive algorithms are greedy in that they only choose the “next-best” subgraph. An interesting question is to develop tractable methods to compute the next K best subgraphs jointly to achieve maximum reduction in error after K iterations. The experiment with non-walk-summable models suggests that walk-sum analysis could be useful to provide convergent algorithms for non-walksummable models, perhaps with restrictions on the order in which walk-sums are computed. Finally, subgraph preconditioners have been shown to improve the convergence rate of the conjugategradient method; using walk-sum analysis to select such preconditioners is of clear interest. References [1] M. Luettgen, W. Carl, and A. Willsky. Efficient multiscale regularization with application to optical flow. IEEE Transactions on Image Processing, 3(1):41–64, Jan. 1994. [2] P. Rusmevichientong and B. Van Roy. An Analysis of Turbo Decoding with Gaussian densities. In Advances in Neural Information Processing Systems 12, 2000. [3] J. Pearl. Probabilistic Reasoning in Intelligent Systems. Morgan Kauffman, San Mateo, CA, 1988. [4] E. Sudderth, M. Wainwright, and A. Willsky. Embedded Trees: Estimation of Gaussian processes on graphs with cycles. IEEE Transactions on Signal Processing, 52(11):3136–3150, Nov. 2004. [5] D. Malioutov, J. Johnson, and A. Willsky. Walk-Sums and Belief Propagation in Gaussian Graphical Models. Journal of Machine Learning Research, 7:2031–2064, Oct. 2006. [6] R. Varga. Matrix Iterative Analysis. Springer-Verlag, New York, 2000. [7] R. Bru, F. Pedroche, and D. Szyld. Overlapping Additive and Multiplicative Schwarz iterations for Hmatrices. Linear Algebra and its Applications, 393:91–105, Dec. 2004. [8] D. Zhou, J. Huang, and B. Scholkopf. Learning from Labeled and Unlabeled data on a directed graph. In Proceedings of the 22nd International Conference on Machine Learning, 2005. [9] D. Zhou and C. Burges. Spectral Clustering and Transductive Learning with multiple views. In Proceedings of the 24th International Conference on Machine Learning, 2007. [10] V. Chandrasekaran, J. Johnson, and A. Willsky. Estimation in Gaussian Graphical Models using Tractable Subgraphs: A Walk-Sum Analysis. To appear in IEEE Transactions on Signal Processing. [11] D. Malioutov, J. Johnson, and A. Willsky. GMRF variance approximation using spliced wavelet bases. In IEEE International Conference on Acoustics, Speech and Signal Processing, 2007. [12] R. Godement. Analysis I: Convergence, Elementary Functions. Springer-Verlag, New York, 2004. [13] T. Cormen, C. Leiserson, R. Rivest, and C. Stein. Introduction to Algorithms. MIT Press, 2001. [14] N. Srebro. Maximum Likelihood Markov Networks: An Algorithmic Approach. Master’s thesis, Massachusetts Institute of Technology, 2000. [15] F. Kschischang, B. Frey, and H. Loeliger. Factor graphs and the sum-product algorithm. IEEE Transactions on Information Theory, 47(2):498–519, Feb. 2001. 8
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Ultrafast Monte Carlo for Kernel Estimators and Generalized Statistical Summations Michael P. Holmes, Alexander G. Gray, and Charles Lee Isbell, Jr. College Of Computing Georgia Institute of Technology Atlanta, GA 30327 {mph, agray, isbell}@cc.gatech.edu Abstract Machine learning contains many computational bottlenecks in the form of nested summations over datasets. Kernel estimators and other methods are burdened by these expensive computations. Exact evaluation is typically O(n2) or higher, which severely limits application to large datasets. We present a multi-stage stratified Monte Carlo method for approximating such summations with probabilistic relative error control. The essential idea is fast approximation by sampling in trees. This method differs from many previous scalability techniques (such as standard multi-tree methods) in that its error is stochastic, but we derive conditions for error control and demonstrate that they work. Further, we give a theoretical sample complexity for the method that is independent of dataset size, and show that this appears to hold in experiments, where speedups reach as high as 1014, many orders of magnitude beyond the previous state of the art. 1 Introduction Many machine learning methods have computational bottlenecks in the form of nested summations that become intractable for large datasets. We are particularly motivated by the nonparametric kernel estimators (e.g. kernel density estimation), but a variety of other methods require computations of similar form. In this work we formalize the general class of nested summations and present a new multi-stage Monte Carlo method for approximating any problem in the class with rigorous relative error control. Key to the efficiency of this method is the use of tree-based data stratification, i.e. sampling in trees. We derive error guarantees and sample complexity bounds, with the intriguing result that runtime depends not on dataset size but on statistical features such as variance and kurtosis, which can be controlled through stratification. We also present experiments that validate these theoretical results and demonstrate tremendous speedup over the prior state of the art. Previous approaches to algorithmic acceleration of this kind fall into roughly two groups: 1) methods that run non-accelerated algorithms on subsets of the data, typically without error bounds, and 2) multi-tree methods with deterministic error bounds. The former are of less interest due to the lack of error control, while the latter are good when exact error control is required, but have built-in overconservatism that limits speedup, and are difficult to extend to new problems. Our Monte Carlo approach offers much larger speedup and a generality that makes it simple to adapt to new problems, while retaining strong error control. While there are non-summative problems to which the standard multi-tree methodology is applicable and our Monte Carlo method is not, our method appears to give greater speedup by many orders of magnitude on problems where both methods can be used. In summary, this work makes the following contributions: formulation of the class of generalized nested data summations; derivation of recursive Monte Carlo algorithms with rigorous error guarantees for this class of computation; derivation of sample complexity bounds showing no explicit 1 dependence on dataset size; variance-driven tree-based stratified sampling of datasets, which allows Monte Carlo approximation to be effective with small sample sizes; application to kernel regression and kernel conditional density estimation; empirical demonstration of speedups as high as 1014 on datasets with points numbering in the millions. It is the combination of all these elements that enables our method to perform so far beyond the previous state of the art. 2 Problem definition and previous work We first illustrate the problem class by giving expressions for the least-squares cross-validation scores used to optimize bandwidths in kernel regression (KR), kernel density estimation (KDE), and kernel conditional density estimation (KCDE): SKR = 1 n X i yi − P j̸=i Kh(||xi −xj||)yj P j̸=i Kh(||xi −xj||) !2 SKDE = 1 n X i „ 1 (n −1)2 X j̸=i X k̸=i Z Kh(||x −xj||)Kh(||x −xk||)dx − 2 (n −1) X j̸=i Kh(||xi −xj||) « SKCDE = 1 n X i „P j̸=i P k̸=i Kh2(||xi −xj||)Kh2(||xi −xk||) R Kh1(y −yj)Kh1(y −yk)dy P j̸=i P k̸=i Kh2(||xi −xj||)Kh2(||xi −xk||) −2 P j̸=i Kh2(||xi −xj||)Kh1(yi −yj) P j̸=i Kh2(||xi −xj||) « . These nested sums have quadratic and cubic computation times that are intractable for large datasets. We would like a method for quickly approximating these and similar computations in a simple and general way. We begin by formulating an inductive generalization of the problem class: B (Xc) → X i∈I(Xc) f (Xc, Xi) (1) G (Xc) →B (Xc) | X i∈I(Xc) f (Xc, G1 (Xc, Xi) , G2 (Xc, Xi) , . . . ) . (2) B represents the base case, in which a tuple of constant arguments Xc may be specified and a tuple of variable arguments Xi is indexed by a set I, which may be a function of Xc. For instance, in the innermost leave-one-out summations of SKR, Xc is the single point xi while I(Xc) indexes all single points other than xi. Note that |I| is the number of terms in a summation of type B, and therefore represents the base time complexity. Whenever I consists of all k-tuples or leave-one-out k-tuples, the base complexity is O(nk), where n is the size of the dataset. The inductive case G is either: 1) the base case B, or 2) a sum where the arguments to the summand function are Xc and a series of nested instances of type G. In SKR the outermost summation is an example of this. The base complexity here is |I| multiplied by the maximum base complexity among the nested instances, e.g. if, as in SKR, I is all single points and the most expensive inner G is O(n), then the overall base complexity is O(n2). Previous work. Past efforts at scaling this class of computation have fallen into roughly two groups. First are methods where data is simply subsampled before running a non-accelerated algorithm. Stochastic gradient descent and its variants (e.g. [1]) are prototypical here. While these approaches can have asymptotic convergence, there are no error guarantees for finite sample sizes. This is not show-stopping in practice, but the lack of quality assurance is a critical shortcoming. Our approach also exploits the speedup that comes from sampling, but provides a rigorous relative error guarantee and is able to automatically determine the necessary sample size to provide that guarantee. The other main class of acceleration methods consists of those employing “higher order divide and conquer” or multi-tree techniques that give either exact answers or deterministic error bounds (e.g. [2, 3, 4]). These approaches apply to a broad class of “generalized n-body problems” (GNPs), and feature the use of multiple spatial partitioning structures such as kd-trees or ball trees to decompose and reuse portions of computational work. While the class of GNPs has yet to be formally defined, the generalized summations we address are clearly related and have at least partial overlap. 2 The standard multi-tree methodology has three significant drawbacks. First, although it gives deterministic error bounds, the bounds are usually quite loose, resulting in overconservatism that prevents aggressive approximation that could give greater speed. Second, creating a new multi-tree method to accelerate a given algorithm requires complex custom derivation of error bounds and pruning rules. Third, the standard multi-tree approach is conjectured to reduce O(np) computations at best to O(nlog p). This still leaves an intractable computation for p as small as 4. In [5], the first of these concerns began to be addressed by employing sample-based bounds within a multi-tree error propagation framework. The present work builds on that idea by moving to a fully Monte Carlo scheme where multiple trees are used for variance-reducing stratification. Error is rigorously controlled and driven by sample variance, allowing the Monte Carlo approach to make aggressive approximations and avoid the overconservatism of deterministic multi-tree methods. This yields greater speedups by many orders of magnitude. Further, our Monte Carlo approach handles the class of nested summations in full generality, making it easy to specialize to new problems. Lastly, the computational complexity of our method is not directly dependent on dataset size, which means it can address high degrees of nesting that would make the standard multi-tree approach intractable. The main tradeoff is that Monte Carlo error bounds are probabilistic, though the bound probability is a parameter to the algorithm. Thus, we believe the Monte Carlo approach is superior for all situations that can tolerate minor stochasticity in the approximated output. 3 Single-stage Monte Carlo We first derive a Monte Carlo approximation for the base case of a single-stage, flat summation, i.e. Equation 1. The basic results for this simple case (up to and including Algorithm 1 and Theorem 1) mirror the standard development of Monte Carlo as in [6] or [7], with some modification to accommodate our particular problem setup. We then move beyond to present novel sample complexity bounds and extend the single-stage results to the multi-stage and multi-stage stratified cases. These extensions allow us to efficiently bring Monte Carlo principles to bear on the entire class of generalized summations, while yielding insights into the dependence of computational complexity on sample statistics and how tree-based methods can improve those statistics. To begin, note that the summation B (Xc) can be written as nE[fi] = nµf, where n = |I| and the expectation is taken over a discrete distribution Pf that puts mass 1 n on each term fi = f(Xc, Xi). Our goal is to produce an estimate ˆB that has low relative error with high probability. More precisely, for a specified ϵ and α, we want | ˆB −B| ≤ϵ|B| with probability at least 1 −α. This is equivalent to estimating µf by ˆµf such that |ˆµf −µf| ≤ϵ|µf|. Let ˆµf be the sample mean of m samples taken from Pf. From the Central Limit Theorem, we have asymptotically ˆµf ⇝N(µf, ˆσ2 f/m), where ˆσ2 f is the sample variance, from which we can construct the standard confidence interval: |ˆµf −µf| ≤ zα/2ˆσf/√m with probability 1 −α. When ˆµf satisfies this bound, our relative error condition is implied by zα/2ˆσf/√m ≤ϵ|µf|, and we also have |µf| ≥|ˆµf| −zα/2ˆσf/√m. Combining these, we can ensure our target relative error by requiring that zα/2ˆσf/√m ≤ϵ(|ˆµf| −zα/2ˆσf/√m), which rearranges to: m ≥z2 α/2 (1 + ϵ)2 ϵ2 ˆσ2 f ˆµ2 f . (3) Equation 3 gives an empirically testable condition that guarantees the target relative error level with probability 1 −α, given that ˆµf has reached its asymptotic distribution N(µf, ˆσ2 f/m). This suggests an iterative sampling procedure in which m starts at a value mmin chosen to make the normal approximation valid, and then is increased until the condition of Equation 3 is met. This procedure is summarized in Algorithm 1, and we state its error guarantee as a theorem. Theorem 1. Given mmin large enough to put ˆµf in its asymptotic normal regime, with probability at least 1 −α Algorithm 1 approximates the summation S with relative error no greater than ϵ . Proof. We have already established that Equation 3 is a sufficient condition for ϵ relative error with probability 1 −α. Algorithm 1 simply increases the sample size until this condition is met. Sample Complexity. Because we are interested in fast approximations, Algorithm 1 is only useful if it terminates with m significantly smaller than the number of terms in the full summation. Equation 3 3 Algorithm 1 Iterative Monte Carlo approximation for flat summations. MC-Approx(S, Xc, ϵ, α, mmin) samples ←∅, mneeded ←mmin repeat addSamples(samples, mneeded, S, Xc) m, ˆµf, ˆσ2 f ←calcStats(samples) mthresh ←z2 α/2(1 + ϵ)2ˆσ2 f/ϵ2ˆµ2 f mneeded ←mthresh −m until m ≥mthresh return |S.I|ˆµf addSamples(samples, mneeded, S, Xc) for i = 1 to mneeded Xi ←rand(S.I) samples ←samples ∪S.f(Xc, Xi) end for calcStats(samples) m ←count(samples) ˆµf ←avg(samples) ˆσ2 f ←var(samples) return m, ˆµf, ˆσ2 f gives an empirical test indicating when m is large enough for sampling to terminate; we now provide an upper bound, in terms of the distributional properties of the full set of fi, for the value of m at which Equation 3 will be satisfied. Theorem 2. Given mmin large enough to put ˆµf and ˆσf in their asymptotic normal regimes, with probability at least 1 −2α Algorithm 1 terminates with m ≤O σ2 f µ2 f + σf |µf | q µ4f σ4 f −1 . Proof. The termination condition is driven by ˆσ2 f/ˆµ2 f, so we proceed by bounding this ratio. First, with probability 1 −α we have a lower bound on the absolute value of the sample mean: |ˆµf| ≥|µf| −zα/2ˆσf/√m. Next, because the sample variance is asymptotically distributed as N(σ2 f, (µ4f −σ4 f)/m), where µ4f is the fourth central moment, we can apply the delta method to infer that ˆσf converges in distribution to N(σf, (µ4f −σ4 f)/4σ2 fm). Using the normal-based confidence interval, this gives the following 1 −α upper bound for the sample standard deviation: ˆσf ≤σf + zα/2 q µ4f −σ4 f/(2σf √m). We now combine these bounds, but since we only know that each bound individually covers at least a 1 −α fraction of outcomes, we can only guarantee they will jointly hold with probability at least 1 −2α, giving the following 1 −2α bound: ˆσf |ˆµf| ≤ σf + zα/2 √ µ4f −σ4 f 2σf √m |µf| −zα/2 σf √m . Combining this with Equation 3 and solving for m shows that, with probability at least 1 −2α, the algorithm will terminate with m no larger than: z2 α/2 2 (1 + 2ϵ)2 ϵ2 σf |µf| " σf |µf| + ϵ(1 + ϵ) (1 + 2ϵ)2 rµ4f σ4 f −1 + r σf |µf| s σf |µf| + 2ϵ(1 + ϵ) (1 + 2ϵ)2 rµ4f σ4 f −1 # . (4) Three aspects of this bound are salient. First, computation time is liberated from dataset size. This is because the sample complexity depends only on the distributional features (σ2 f, µf, and µ4f) of the summation terms, and not on the number of terms. For i.i.d. datasets in particular, these distributional features are convergent, which means the sample or computational complexity converges to a constant while speedup becomes unbounded as the dataset size goes to infinity. Second, the bound has sensible dependence on σf/|µf| and µ4f/σ4 f. The former is a standard dispersion measure known as the coefficient of variation, and the latter is the kurtosis. Algorithm 1 therefore gives greatest speedup for summations whose terms have low dispersion and low kurtosis. The intuition is that sampling is most efficient when values are concentrated tightly in a few clusters, making it easy to get a representative sample set. This motivates the additional speedup we later gain by stratifying the dataset into low-variance regions. Finally, the sample complexity bound indicates whether Algorithm 1 will actually give speedup for any particular problem. For a given summation, let the speedup be defined as the total number of terms n divided by the number of terms evaluated by the approximation. For a desired speedup τ, we need n ≥τmbound, where mbound is the expression in Equation 4. This is the fundamental characterization of whether speedup will be attained. 4 Algorithm 2 Iterative Monte Carlo approximation for nested summations. MC-Approx: as in Algorithm 1 calcStats: as in Algorithm 1 addSamples(samples, mneeded, S, Xc) for i = 1 to mneeded Xi ←rand(S.I(Xc)) mcArgs ←map(MC-Approx(∗, Xc ◦Xi, . . .), ⟨S.Gj⟩) samples ←samples ∪S.f(Xc, mcArgs) end for 4 Multi-stage Monte Carlo We now turn to the inductive case of nested summations, i.e. Equation 2. The approach we take is to apply the single-stage Monte Carlo algorithm over the terms fi as before, but with recursive invocation to obtain approximations for the arguments Gj. Algorithm 2 specifies this procedure. Theorem 3. Given mmin large enough to put ˆµf in its asymptotic normal regime, with probability at least 1 −α Algorithm 2 approximates the summation S with relative error no greater than ϵ . Proof. We begin by noting that the proof of correctness for Algorithm 1 rests on 1) the ability to sample from a distribution Pf whose expectation is µf = 1 n P i fi, and 2) the ability to invoke the CLT on the sample mean ˆµf in terms of the sample variance ˆσ2 f. Given these properties, Equation 3 follows as a sufficient condition for relative error no greater than ϵ with probability at least 1−α. We therefore need only establish that Algorithm 2 samples from a distribution having these properties. For each sampled fi, let bGj be the recursive approximation for argument Gj. We assume bGj has been drawn from a CLT-type normal distribution. Because the bGj are recursively approximated, this is an inductive hypothesis, with the remainder of the proof showing that if the hypothesis holds for the recursive invocations, it also holds for the outer invocation. The base case, where all recursions must bottom out, is the type-B summation already shown to give CLT-governed answers (see proof of Theorem 1). Let bGm = ( bG1, bG2, . . .) be the vector of bGj values after each bGj has been estimated from mj samples (P mj = m), and let G be the vector of true Gj values. Since each component bGj converges in distribution to N(Gj, σ2 j /mj), bGm satisfies bGm ⇝N(G, Σm). We leave the detailed entries of the covariance Σm unspecified, except to note that its jjth element is σ2 j /mj, and that its off-diagonal elements may be non-zero if the bGj are generated in a correlated way (this can be used as a variance reduction technique). Given the asymptotic normality of bGm, the same arguments used to derive the multivariate delta method can be used, with some modification, to show that fi( bGm) ⇝N(fi(G), ▽f(G)Σm▽T f (G)). Thus, asymptotically, fi( bGm) is normally distributed around its true value with a variance that depends on both the gradient of f and the covariance matrix of the approximated arguments in bGm. This being the case, uniform sampling of the recursively estimated fi is equivalent to sampling from a distribution ˜Pf that gives weight 1 n to a normal distribution centered on each fi. The expectation over ˜Pf is µf, and since the algorithm uses a simple sample mean the CLT does apply. These are the properties we need for correctness, and the applicability of the CLT combined with the proven base case completes the inductive proof. Note that the variance over ˜Pf works out to ˜σ2 f = σ2 f + 1 n P i∈I σ2 i , where σ2 i = ▽f(G)Σm▽T f (G). In other words, the variance with recursive approximation is the exact variance σ2 f plus the average of the variances σ2 i of the approximated fi. Likewise one could write an expression for the kurtosis ˜µ4f. Because we are still dealing with a sample mean, Theorem 2 still holds in the nested case. Corollary 2.1. Given mmin large enough to put ˆµf and ˆσf in their asymptotic normal regimes, with probability at least 1 −2α Algorithm 2 terminates with m ≤O ˜σ2 f µ2 f + ˜σf |µf | q ˜µ4f ˜σ4 f −1 . It is important to point out that the 1 −α confidences and ϵ relative error bounds of the recursively approximated arguments do not pass through to or compound in the overall estimator ˆµf: their influence appears in the variance σ2 i of each sampled fi, which in turn contributes to the overall variance ˜σ2 f, and the error from ˜σ2 f is independently controlled by the outermost sampling procedure. 5 Algorithm 3 Iterative Monte Carlo approximation for nested summations with stratification. MC-Approx: as in Algorithm 1 calcStats(strata, samples) m ←count(samples) ˆµfs ←stratAvg(strata, samples) ˆσ2 fs ←stratV ar(strata, samples) return m, ˆµfs, ˆσ2 fs addSamples(strata, samples, mneeded, S, Xc) needPerStrat = optAlloc(samples, strata, mneeded) for s = 1 to strata.count ms = needPerStrat[s] for i = 1 to ms Xi ←rand(S.I(Xc), strata[s]) mcArgs ←map(MC-Approx(∗, Xc ◦Xi, . . .), ⟨S.Gj⟩) samples[s] ←samples[s] ∪S.f(Xc, mcArgs) end for end for 5 Variance Reduction With Algorithm 2 we have coverage of the entire generalized summation problem class, and our focus turns to maximizing efficiency. As noted above, Theorem 2 implies we need fewer samples when the summation terms are tightly concentrated in a few clusters. We formalize this by spatially partitioning the data to enable a stratified sampling scheme. Additionally, by use of correlated sampling we induce covariance between recursively estimated summations whenever the overall variance can be reduced by doing so. Adding these techniques to recursive Monte Carlo makes for an extremely fast, accurate, and general approximation scheme. Stratification. Stratification is a standard Monte Carlo principle whereby the values being sampled are partitioned into subsets (strata) whose contributions are separately estimated and then combined. The idea is that strata with higher variance can be sampled more heavily than those with lower variance, thereby making more efficient use of samples than in uniform sampling. Application of this principle requires the development of an effective partitioning scheme for each new domain of interest. In the case of generalized summations, the values being sampled are the fi, which are not known a priori and cannot be directly stratified. However, since f is generally a function with some degree of continuity, its output is similar for similar values of its arguments. We therefore stratify the argument space, i.e. the input datasets, by use of spatial partitioning structures. Though any spatial partitioning could be used, in this work we use modified kd-trees that recursively split the data along the dimension of highest variance. The approximation procedure runs as it did before, except that the sampling and sample statistics are modified to make use of the trees. Trees are expanded up to a user-specified number of nodes, prioritized by a heuristic of expanding nodes in order of largest size times average per-dimensional standard deviation. This heuristic will later be justified by the variance expression for the stratified sample mean. The approximation procedure is summarized in Algorithm 3, and we now establish its error guarantee. Theorem 4. Given mmin large enough to put ˆµf in its asymptotic normal regime, with probability at least 1 −α Algorithm 3 approximates the summation S with relative error no greater than ϵ . Proof. Identical to Theorem 3, but we need to establish that 1) the sample mean remains unbiased under stratification, and 2) the CLT still holds under stratification. These turn out to be standard properties of the stratified sample mean and its variance estimator (see [7]): ˆµfs = X j pj ˆµj (5) ˆσ2(ˆµfs) = ˆσ2 fs m ; ˆσ2 fs ≜m X j p2 j ˆσ2 j mj = X j p2 j qj ˆσ2 j , (6) where j indexes the strata, ˆµj and ˆσ2 j are the sample mean and variance of stratum j, pj is the fraction of summation terms in stratum j, and qj is the fraction of samples drawn from stratum j. Algorithm 3 modifies the addSamples subroutine to sample in stratified fashion, and computes the stratified ˆµfs and ˆσ2 fs instead of ˆµf and ˆσ2 f in calcStats. Since these estimators satisfy the two conditions necessary for the error guarantee, this establishes the theorem. The true variance σ2(ˆµfs) is identical to Equation 6 but with the exact σ2 j substituted for ˆσ2 j . In [7], it is shown that σ2 fs ≤σ2 f, i.e. stratification never increases variance, and that any refinement of a 6 stratification can only reduce σ2 fs. Although the sample allocation fractions qj can be chosen arbitrarily, σ2 fs is minimized when qj ∝pjσj. With this optimal allocation, σ2 fs reduces to (P j pjσj)2. This motivates our kd-tree expansion heuristic, as described above, which tries to first split the nodes with highest pjσj, i.e. the nodes with highest contribution to the variance under optimal allocation. While we never know the σj exactly, Algorithm 3 uses the sample estimates ˆσj at each stage to approximate the optimal allocation (this is the optAlloc routine). Finally, the Theorem 2 sample complexity still holds for the CLT-governed stratified sample mean. Corollary 2.2. Given mmin large enough to put ˆµfs and ˆσfs in their asymptotic normal regimes, with probability at least 1 −2α Algorithm 3 terminates with m ≤O σ2 fs µ2 f + σfs |µf | q µ4fs σ4 fs −1 . Correlated Sampling. The variance of recursively estimated fi, as expressed by ▽f(G)Σm▽T f (G), depends on the full covariance matrix of the estimated arguments. If the gradient of f is such that the variance of fi depends negatively (positively) on a covariance σjk, we can reduce the variance by inducing positive (negative) covariance between Gj and Gk. Covariance can be induced by sharing sampled points across the estimates of Gj and Gk, assuming they both use the same datasets. In some cases the expression for fi’s variance is such that the effect of correlated sampling is datadependent; when this happens, it is easy to test and check whether correlation helps. All experiments presented here were benefited by correlated sampling on top of stratification. 6 Experiments We present experimental results in two phases. First, we compare stratified multi-stage Monte Carlo approximations to exact evaluations on tractable datasets. We show that the error distributions conform closely to our asymptotic theory. Second, having verified accuracy to the extent possible, we run our method on datasets containing millions of points in order to show 1) validation of the theoretical prediction that runtime is roughly independent of dataset size, and 2) many orders of magnitude speedup (as high as 1014) relative to exact computation. These results are presented for two method-dataset pairs: kernel regression on a dataset containing 2 million 4-dimensional redshift measurements used for quasar identification, and kernel conditional density estimation on an n-body galaxy simulation dataset containing 3.5 million 3-dimensional locations. In the KR case, the fourth dimension is regressed against the other three, while in KCDE the distribution of the third dimension is predicted as a function of the first two. In both cases we are evaluating the cross-validated score functions used for bandwidth optimization, i.e. SKR and SKCDE as described in Section 2. Error Control. The objective of this first set of experiments is to validate the guarantee that relative error will be less than or equal to ϵ with probability 1 −α. We measured the distribution of error on a series of random data subsets up to the highest size for which the exact computation was tractable. For the O(n2) SKR, the limit was n = 10K, while for the O(n3) SKCDE it was n = 250. For each dataset we randomly chose and evaluated 100 bandwidths with 1 −α = 0.95 and ϵ = 0.1. Figure 1 shows the full quantile spreads of the relative errors. The most salient feature is the relationship of the 95% quantile line (dashed) to the threshold line at ϵ = 0.1 (solid). Full compliance with asymptotic theory would require the dashed line never to be above the solid. This is basically the case for KCDE,1 while the KR line never goes above 0.134. The approximation is therefore quite good, and could be improved if desired by increasing mmin or the number of strata, but in this case we chose to trade a slight increase in error for an increase in speed. Speedup. Given the validation of the error guarantees, we now turn to computational performance. As before, we ran on a series of random subsets of the data, this time with n ranging into the millions. At each value of n, we randomly chose and evaluated 100 bandwidths, measuring the time for each evaluation. Figure 2 presents the average evaluation time versus dataset size for both methods. The most striking feature of these graphs is their flatness as n increases by orders of magnitude. This is in accord with Theorem 2 and its corollaries, which predict sample and computational complexity independent of dataset size. Speedups2 for KR range from 1.8 thousand at n = 50K to 2.8 million at n = 2M. KCDE speedups range from 70 million at n = 50K to 1014 at n = 3.5M. This performance is many orders of magnitude better than that of previous methods. 1The spike in the max quantile is due to a single outlier point. 2All speedups are relative to extrapolated runtimes based on the O() order of the exact computation. 7 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 0.05 0.1 0.15 0.2 0.25 dataset size relative error 99%−max 90%−99% 75%−90% 50%−75% 25%−50% 10%−25% 1%−10% min−1% 95% error = 0.1 50 100 150 200 250 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 dataset size relative error 99%−max 90%−99% 75%−90% 50%−75% 25%−50% 10%−25% 1%−10% min−1% 95% error = 0.1 Figure 1: Error distribution vs. dataset size for KR (left), and KCDE (right). 0 500,000 1,000,000 1,500,000 2,000,000 0 1000 2000 3000 4000 dataset size avg. computation time (ms) 0 1,000,000 2,000,000 3,000,000 −1000 0 1000 2000 3000 4000 5000 6000 dataset size avg. computation time (ms) Figure 2: Runtime vs. dataset size for KR (left), and KCDE (right). Error bars are one standard deviation. 7 Conclusion We have presented a multi-stage stratified Monte Carlo method for efficiently approximating a broad class of generalized nested summations. Summations of this type lead to computational bottlenecks in kernel estimators and elsewhere in machine learning. The theory derived for this Monte Carlo approach predicts: 1) relative error no greater than ϵ with probability at least 1−α, for user-specified ϵ and α, and 2) sample and computational complexity independent of dataset size. Our experimental results validate these theoretical guarantees on real datasets, where we accelerate kernel crossvalidation scores by as much as 1014 on millions of points. This is many orders of magnitude faster than the previous state of the art. In addition to applications, future work will likely include automatic selection of stratification granularity, additional variance reduction techniques, and further generalization to other computational bottlenecks such as linear algebraic operations. References [1] Nicol N. Schraudolph and Thore Graepel. Combining conjugate direction methods with stochastic approximation of gradients. In Workshop on Artificial Intelligence and Statistics (AISTATS), 2003. [2] Alexander G. Gray and Andrew W. Moore. N-body problems in statistical learning. In Advances in Neural Information Processing Systems (NIPS) 13, 2000. [3] Mike Klaas, Mark Briers, Nando de Freitas, and Arnaud Doucet. Fast particle smoothing: If I had a million particles. In International Conference on Machine Learning (ICML), 2006. [4] Ping Wang, Dongryeol Lee, Alexander Gray, and James M. Rehg. Fast mean shift with accurate and stable convergence. In Workshop on Artificial Intelligence and Statistics (AISTATS), 2007. [5] Michael P. Holmes, Alexander G. Gray, and Charles Lee Isbell Jr. Fast nonparametric conditional density estimation. In Uncertainty in Artificial Intelligence (UAI), 2007. [6] Reuven Y. Rubinstein. Simulation and the Monte Carlo Method. John Wiley & Sons, 1981. [7] Paul Glasserman. Monte Carlo methods in financial engineering. Springer-Verlag, 2004. 8
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Inferring Neural Firing Rates from Spike Trains Using Gaussian Processes John P. Cunningham1, Byron M. Yu1,2,3, Krishna V. Shenoy1,2 1Department of Electrical Engineering, 2Neurosciences Program, Stanford University, Stanford, CA 94305 {jcunnin,byronyu,shenoy}@stanford.edu Maneesh Sahani3 3Gatsby Computational Neuroscience Unit, UCL Alexandra House, 17 Queen Square, London, WC1N 3AR, UK maneesh@gatsby.ucl.ac.uk Abstract Neural spike trains present challenges to analytical efforts due to their noisy, spiking nature. Many studies of neuroscientific and neural prosthetic importance rely on a smoothed, denoised estimate of the spike train’s underlying firing rate. Current techniques to find time-varying firing rates require ad hoc choices of parameters, offer no confidence intervals on their estimates, and can obscure potentially important single trial variability. We present a new method, based on a Gaussian Process prior, for inferring probabilistically optimal estimates of firing rate functions underlying single or multiple neural spike trains. We test the performance of the method on simulated data and experimentally gathered neural spike trains, and we demonstrate improvements over conventional estimators. 1 Introduction Neuronal activity, particularly in cerebral cortex, is highly variable. Even when experimental conditions are repeated closely, the same neuron may produce quite different spike trains from trial to trial. This variability may be due to both randomness in the spiking process and to differences in cognitive processing on different experimental trials. One common view is that a spike train is generated from a smooth underlying function of time (the firing rate) and that this function carries a significant portion of the neural information. If this is the case, questions of neuroscientific and neural prosthetic importance may require an accurate estimate of the firing rate. Unfortunately, these estimates are complicated by the fact that spike data gives only a sparse observation of its underlying rate. Typically, researchers average across many trials to find a smooth estimate (averaging out spiking noise). However, averaging across many roughly similar trials can obscure important temporal features [1]. Thus, estimating the underlying rate from only one spike train (or a small number of spike trains believed to be generated from the same underlying rate) is an important but challenging problem. The most common approach to the problem has been to collect spikes from multiple trials in a peristimulus-time histogram (PSTH), which is then sometimes smoothed by convolution or splines [2], [3]. Bin sizes and smoothness parameters are typically chosen ad hoc (but see [4], [5]) and the result is fundamentally a multi-trial analysis. An alternative is to convolve a single spike train with a kernel. Again, the kernel shape and time scale are frequently ad hoc. For multiple trials, researchers may average over multiple kernel-smoothed estimates. [2] gives a thorough review of classical methods. 1 More recently, point process likelihood methods have been adapted to spike data [6]–[8]. These methods optimize (implicitly or explicitly) the conditional intensity function λ(t|x(t), H(t)) — which gives the probability of a spike in [t, t + dt), given an underlying rate function x(t) and the history of previous spikes H(t) — with respect to x(t). In a regression setting, this rate x(t) may be learned as a function of an observed covariate, such as a sensory stimulus or limb movement. In the unsupervised setting of interest here, it is constrained only by prior expectations such as smoothness. Probabilistic methods enjoy two advantages over kernel smoothing. First, they allow explicit modelling of interactions between spikes through the history term H(t) (e.g., refractory periods). Second, as we will see, the probabilistic framework provides a principled way to share information between trials and to select smoothing parameters. In neuroscience, most applications of point process methods use maximum likelihood estimation. In the unsupervised setting, it has been most common to optimize x(t) within the span of an arbitrary basis (such as a spline basis [3]). In other fields, a theory of generalized Cox processes has been developed, where the point process is conditionally Poisson, and x(t) is obtained by applying a link function to a draw from a random process, often a Gaussian process (GP) (e.g. [9]). In this approach, parameters of the GP, which set the scale and smoothness of x(t) can be learned by optimizing the (approximate) marginal likelihood or evidence, as in GP classification or regression. However, the link function, which ensures a nonnegative intensity, introduces possibly undesirable artifacts. For instance, an exponential link leads to a process that grows less smooth as the intensity increases. Here, we make two advances. First, we adapt the theory of GP-driven point processes to incorporate a history-dependent conditional likelihood, suitable for spike trains. Second, we formulate the problem such that nonnegativity in x(t) is achieved without a distorting link function or sacrifice of tractability. We also demonstrate the power of numerical techniques that makes application of GP methods to this problem computationally tractable. We show that GP methods employing evidence optimization outperform both kernel smoothing and maximum-likelihood point process models. 2 Gaussian Process Model For Spike Trains Spike trains can often be well modelled by gamma-interval point processes [6], [10]. We assume the underlying nonnegative firing rate x(t) : t ∈[0, T] is a draw from a GP, and then we assume that our spike train is a conditionally inhomogeneous gamma-interval process (IGIP), given x(t). The spike train is represented by a list of spike times y = {y0, . . ., yN}. Since we will model this spike train as an IGIP1, y | x(t) is by definition a renewal process, so we can write: p(y | x(t)) = N Y i=1 p(yi | yi−1, x(t)) · p0(y0 | x(t)) · pT (T | yN, x(t)), (1) where p0(·) is the density of the first spike occuring at y0, and pT (·) is the density of no spikes being observed on (yN, T]; the density for IGIP intervals (of order γ ≥1) (see e.g. [6]) can be written as: p(yi | yi−1, x(t)) = γx(yi) Γ(γ) γ Z yi yi−1 x(u)du γ−1 exp −γ Z yi yi−1 x(u)du . (2) The true p0(·) and pT (·) under this gamma-interval spiking model are not closed form, so we simplify these distributions as intervals of an inhomogeneous Poisson process (IP). This step, which we find to sacrifice very little in terms of accuracy, helps to preserve tractability. Note also that we write the distribution in terms of the inter-spike-interval distribution p(yi|yi−1, x(t)) and not λ(t|x(t), H(t)), but the process could be considered equivalently in terms of conditional intensity. We now discretize x(t) : t ∈[0, T] by the time resolution of the experiment (∆, here 1 ms), to yield a series of n evenly spaced samples x = [x1, . . ., xn]′ (with n = T ∆). The events y become N + 1 time indices into x, with N much smaller than n. The discretized IGIP output process is now (ignoring terms that scale with ∆): 1The IGIP is one of a class of renewal models that works well for spike data (much better than inhomogeneous Poisson; see [6], [10]). Other log-concave renewal models such as the inhomogeneous inverse-Gaussian interval can be chosen, and the implementation details remain unchanged. 2 p(y | x) = N Y i=1 γxyi Γ(γ) γ yi−1 X k=yi−1 xk∆ γ−1 exp −γ yi−1 X k=yi−1 xk∆ · xy0exp − y0−1 X k=0 xk∆ · exp − n−1 X k=yN xk∆ , (3) where the final two terms are p0(·) and pT (·), respectively [11]. Our goal is to estimate a smoothly varying firing rate function from spike times. Loosely, instead of being restricted to only one family of functions, GP allows all functions to be possible; the choice of kernel determines which functions are more likely, and by how much. Here we use the standard squared exponential (SE) kernel. Thus, x ∼N (µ1, Σ), where Σ is the positive definite covariance matrix defined by Σ = K(ti, tj) i,j∈{1,...,n} where K(ti, tj) = σ2 fexp −κ 2 (ti −tj)2 + σ2 vδij. (4) For notational convenience, we define the hyperparameter set θ = [µ; γ; κ; σ2 f; σ2 v]. Typically, the GP mean µ is set to 0. Since our intensity function is nonnegative, however, it is sensible to treat µ instead as a hyperparameter and let it be optimized to a positive value. We note that other standard kernels - including the rational quadratic, Matern ν = 3 2, and Matern ν = 5 2 - performed similarly to the SE; thus we only present the SE here. For an in depth discussion of kernels and of GP, see [12]. As written, the model assumes only one observed spike train; it may be that we have m trials believed to be generated from the same firing rate profile. Our method naturally incorporates this case: define p({y}m 1 | x) = Ym i=1 p(y(i) | x), where y(i) denotes the ith spike train observed.2 Otherwise, the model is unchanged. 3 Finding an Optimal Firing Rate Estimate 3.1 Algorithmic Approach Ideally, we would calculate the posterior on firing rate p(x | y) = R θ p(x | y, θ)p(θ)dθ (integrating over the hyperparameters θ), but this problem is intractable. We consider two approximations: replacing the integral by evaluation at the modal θ, and replacing the integral with a sum over a discrete grid of θ values. We first consider choosing a modal hyperparameter set (ML-II model selection, see [12]), i.e. p(x | y) ≈q(x | y, θ∗) where q(·) is some approximate posterior, and θ∗= argmax θ p(θ | y) = argmax θ p(θ)p(y | θ) = argmax θ p(θ) Z x p(y | x, θ)p(x | θ)dx. (5) (This and the following equations hold similarly for a single observation y or multiple observations {y}m 1 , so we consider only the single observation for notational brevity.) Specific choices for the hyperprior p(θ) are discussed in Results. The integral in Eq. 5 is intractable under the distributions we are modelling, and thus we must use an approximation technique. Laplace approximation and Expectation Propagation (EP) are the most widely used techniques (see [13] for a comparison). The Laplace approximation fits an unnormalized Gaussian distribution to the integrand in Eq. 5. Below we show this integrand is log concave in x. This fact makes reasonable the Laplace approximation, since we know that the distribution being approximated is unimodal in x and shares log concavity with the normal distribution. Further, since we are modelling a non-zero mean GP, most of the Laplace approximated probability mass lies in the nonnegative orthant (as is the case with the true posterior). Accordingly, we write: 2Another reasonable approach would consider each trial as having a different rate function x that is a draw from a GP with a nonstationary mean function µ(t). Instead of inferring a mean rate function x∗, we would learn a distribution of means. We are considering this choice for future work. 3 p(y | θ) = Z x p(y | x, θ)p(x | θ)dx ≈p(y | x∗, θ)p(x∗| θ) (2π) n 2 |Λ∗+ Σ−1| 1 2 , (6) where x∗is the mode of the integrand and Λ∗= −∇2 xlog p(y | x, θ) |x=x∗. Note that in general both Σ and Λ∗(and x∗, implicitly) are functions of the hyperparameters θ. Thus, Eq. 6 can be differentiated with respect to the hyperparameter set, and an iterative gradient optimization (we used conjugate gradients) can be used to find (locally) optimal hyperparameters. Algorithmic details and the gradient calculations are typical for GP; see [12]. The Laplace approximation also naturally provides confidence intervals from the approximated posterior covariance (Σ−1 + Λ∗)−1. We can also consider approximate integration over θ using the Laplace approximation above. The Laplace approximation produces a posterior approximation q(x | y, θ) = N x∗, (Λ∗+ Σ−1)−1 and a model evidence approximation q(θ | y) (Eq. 6). The approximate integrated posterior can be written as p(x | y) = Eθ|y[p(x | y, θ)] ≈P j q(x | y, θj)q(θj | y) for some choice of samples θj (which again gives confidence intervals on the estimates). Since the dimensionality of θ is small, and since we find in practice that the posterior on θ is well behaved (well peaked and unimodal), we find that a simple grid of θj works very well, thereby obviating MCMC or another sampling scheme. This approximate integration consistently yields better results than a modal hyperparameter set, so we will only consider approximate integration for the remainder of this report. For the Laplace approximation at any value of θ, we require the modal estimate of firing rate x∗, which is simply the MAP estimator: x∗= argmax x⪰0 p(x | y) = argmax x⪰0 p(y | x)p(x). (7) Solving this problem is equivalent to solving an unconstrained problem where p(x) is a truncated multivariate normal (but this is not the same as individually truncating each marginal p(xi); see [14]). Typically a link or squashing function would be included to enforce nonnegativity in x, but this can distort the intensity space in unintended ways. We instead impose the constraint x ⪰0, which reduces the problem to being solved over the (convex) nonnegative orthant. To pose the problem as a convex program, we define f(x) = −log p(y | x)p(x): f(x) = N X i=1 −log xyi −(γ −1)log yi−1 X k=yi−1 xk∆ + yN −1 X k=y0 γxk∆ (8) −log xy0 + y0−1 X k=1 xk∆+ n−1 X k=yN xk∆+ 1 2(x −µ1)T Σ−1(x −µ1) + C, (9) where C represents constants with respect to x. From this form follows the Hessian ∇2 xf(x) = Σ−1 + Λ where Λ = −∇2 xlog p(y | x, θ) = B + D, (10) where D = diag(x−2 y0 , . . ., 0, . . ., x−2 yi . . ., 0, . . ., x−2 yN ) is positive semidefinite and diagonal. B is block diagonal with N blocks. Each block is rank 1 and associates its positive, nonzero eigenvalue with eigenvector [0, . . ., 0, bT i , 0, . . ., 0]T . The remaining n −N eigenvalues are zero. Thus, B has total rank N and is positive semidefinite. Since Σ is positive definite, it follows then that the Hessian is also positive definite, proving convexity. Accordingly, we can use a log barrier Newton method to efficiently solve for the global MAP estimator of firing rate x∗[15]. In the case of multiple spike train observations, we need only add extra terms of negative log likelihood from the observation model. This flows through to the Hessian, where ∇2 xf(x) = Σ−1 + Λ and Λ = Λ1 + . . . + Λm, with Λi ∀i ∈{1, . . ., m} defined for each observation as in Eq. 10. 4 3.2 Computational Practicality This method involves multiple iterative layers which require many Hessian inversions and other matrix operations (matrix-matrix products and determinants) that cost O(n3) in run-time complexity and O(n2) in memory, where (x ∈IRn). For any significant data size, a straightforward implementation is hopelessly slow. With 1 ms time resolution (or similar), this method would be restricted to spike trains lasting less than a second, and even this problem would be burdensome. Achieving computational improvements is critical, as a naive implementation is, for all practical purposes, intractable. Techniques to improve computational performance are a subject of study in themselves and are beyond the scope of this paper. We give a brief outline in the following paragraph. In the MAP estimation of x∗, since we have analytical forms of all matrices, we avoid explicit representation of any matrix, resulting in linear storage. Hessian inversions are avoided using the matrix inversion lemma and conjugate gradients, leaving matrix vector multiplications as the single costly operation. Multiplication of any vector by Λ can be done in linear time, since Λ is a (blockwise) vector outer product matrix. Since we have evenly spaced resolution of our data x in time indices ti, Σ is Toeplitz; thus multiplication by Σ can be done using Fast Fourier Transform (FFT) methods [16]. These techniques allow exact MAP estimation with linear storage and nearly linear run time performance. In practice, for example, this translates to solving MAP estimation problems of 103 variables in fractions of a second, with minimal memory load. For the modal hyperparameter scheme (as opposed to approximately integrating over the hyperparameters), gradients of Eq. 6 must also be calculated at each step of the model evidence optimization. In addition to using similar techniques as in the MAP estimation, log determinants and their derivatives (associated with the Laplace approximation) can be accurately approximated by exploiting the eigenstructure of Λ. In total, these techniques allow optimal firing rates functions of 103 to 104 variables to be estimated in seconds or minutes (on a modern workstation). These data sizes translate to seconds of spike data at 1 ms resolution, long enough for most electrophysiological trials. This algorithm achieves a reduction from a naive implementation which would require large amounts of memory and would require many hours or days to complete. 4 Results We tested the methods developed here using both simulated neural data, where the true firing rate was known by construction, and in real neural spike trains, where the true firing rate was estimated by a PSTH that averaged many similar trials. The real data used were recorded from macaque premotor cortex during a reaching task (see [17] for experimental method). Roughly 200 repeated trials per neuron were available for the data shown here. We compared the IGIP-likelihood GP method (hereafter, GP IGIP) to other rate estimators (kernel smoothers, Bayesian Adaptive Regressions Splines or BARS [3], and variants of the GP method) using root mean squared difference (RMS) to the true firing rate. PSTH and kernel methods approximate the mean conditional intensity λ(t) = EH(t)[λ(t|x(t), H(t))]. For a renewal process, we know (by the time rescaling theorem [7], [11]) that λ(t) = x(t), and thus we can compare the GP IGIP (which finds x(t)) directly to the kernel methods. To confirm that hyperparameter optimization improves performance, we also compared GP IGIP results to maximum likelihood (ML) estimates of x(t) using fixed hyperparameters θ. This result is similar in spirit to previously published likelihood methods with fixed bases or smoothness parameters. To evaluate the importance of an observation model with spike history dependence (the IGIP of Eq. 3), we also compared GP IGIP to an inhomogeneous Poisson (GP IP) observation model (again with a GP prior on x(t); simply γ = 1 in Eq. 3). The hyperparameters θ have prior distributions (p(θ) in Eq. 5). For σf, κ, and γ, we set lognormal priors to enforce meaningful values (i.e. finite, positive, and greater than 1 in the case of γ). Specifically, we set log(σ2 f) ∼N (5, 2) , log(κ) ∼N (2, 2), and log(γ −1) ∼N (0, 100). The variance σv can be set arbitrarily small, since the GP IGIP method avoids explicit inversions of Σ with the matrix inversion lemma (see 3.2). For the approximate integration, we chose a grid consisting of the empirical mean rate for µ (that is, total spike count N divided by total time T) and (γ, log(σ2 f), log(κ)) ∈[1, 2, 4] × [4, . . ., 8] × [0, . . ., 7]. We found this coarse grid (or similar) produced similar results to many other very finely sampled grids. 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 10 20 30 40 50 60 Time (sec) Firing Rate (spikes/sec) (a) Data Set L20061107.214.1; 1 spike train 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 10 20 30 40 50 60 70 Time (sec) Firing Rate (spikes/sec) (b) Data Set L20061107.14.1; 4 spike trains 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 5 10 15 20 25 30 35 40 45 50 Time (sec) Firing Rate (spikes/sec) (c) Data Set L20061107.151.5; 8 spike trains 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 2 4 6 8 10 12 14 16 Time (sec) Firing Rate (spikes/sec) (d) Data Set L20061107.46.3; 1 spike train Figure 1: Sample GP firing rate estimate. See full description in text. The four examples in Fig. 1 represent experimentally gathered firing rate profiles (according to the methods in [17]). In each of the plots, the empirical average firing rate of the spike trains is shown in bold red. For simulated spike trains, the spike trains were generated from each of these empirical average firing rates using an IGIP (γ = 4, comparable to fits to real neural data). For real neural data, the spike train(s) were selected as a subset of the roughly 200 experimentally recorded spike trains that were used to construct the firing rate profile. These spike trains are shown as a train of black dots, each dot indicating a spike event time (the y-axis position is not meaningful). This spike train or group of spike trains is the only input given to each of the fitting models. In thin green and magenta, we have two kernel smoothed estimates of firing rates; each represents the spike trains convolved with a normal distribution of a specified standard deviation (50 and 100 ms). We also smoothed these spike trains with adaptive kernel [18], fixed ML (as described above), BARS [3], and 150 ms kernel smoothers. We do not show these latter results in Fig. 1 for clarity of figures. These standard methods serve as a baseline from which we compare our method. In bold blue, we see x∗, the results of the GP IGIP method. The light blue envelopes around the bold blue GP firing rate estimate represent the 95% confidence intervals. Bold cyan shows the GP IP method. This color scheme holds for all of Fig. 1. We then ran all methods 100 times on each firing rate profile, using (separately) simulated and real neural spike trains. We are interested in the average performance of GP IGIP vs. other GP methods (a fixed ML or a GP IP) and vs. kernel smoothing and spline (BARS) methods. We show these results in Fig. 2. The four panels correspond to the same rate profiles shown in Fig. 1. In each panel, the top, middle, and bottom bar graphs correspond to the method on 1, 4, and 8 spike trains, respectively. GP IGIP produces an average RMS error, which is an improvement (or, less often, a deterioration) over a competing method. Fig. 2 shows the percent improvement of the GP IGIP method vs. the competing method listed. Only significant results are shown (paired t-test, p < 0.05). 6 0 50 0 50 GP IP Fixed ML short medium long adaptive BARS GP Methods Kernel Smoothers 0 50 % % % (a) L20061107.214.1; 1,4,8 spike trains GP IP Fixed ML short medium long adaptive BARS GP Methods Kernel Smoothers 0 50 0 50 0 50 % % % (b) L20061107.14.1; 1,4,8 spike trains GP IP Fixed ML short medium long adaptive BARS GP Methods Kernel Smoothers 0 50 0 50 0 50 % % % (c) L20061107.151.5; 1,4,8 spike trains GP IP Fixed ML short medium long adaptive BARS GP Methods Kernel Smoothers 0 50 0 50 0 50 % % % (d) L20061107.46.3; 1,4,8 spike trains Figure 2: Average percent RMS improvement of GP IGIP method (with model selection) vs. method indicated in the column title. See full description in text. Blue improvement bars are for simulated spike trains; red improvement bars are for real neural spike trains. The general positive trend indicates improvements, suggesting the utility of this approach. Note that, in the few cases where a kernel smoother performs better (e.g. the long bandwidth kernel in panel (b), real spike trains, 4 and 8 spike trains), outperforming the GP IGIP method requires an optimal kernel choice, which can not be judged from the data alone. In particular, the adaptive kernel method generally performed more poorly than GP IGIP. The relatively poor performance of GP IGIP vs. different techniques in panel (d) is considered in the Discussion section. The data sets here are by no means exhaustive, but they indicate how this method performs under different conditions. 5 Discussion We have demonstrated a new method that accurately estimates underlying neural firing rate functions and provides confidence intervals, given one or a few spike trains as input. This approach is not without complication, as the technical complexity and computational effort require special care. Estimating underlying firing rates is especially challenging due to the inherent noise in spike trains. Having only a few spike trains deprives the method of many trials to reduce spiking noise. It is important here to remember why we care about single trial or small number of trial estimates, since we believe that in general the neural processing on repeated trials is not identical. Thus, we expect this signal to be difficult to find with or without trial averaging. In this study we show both simulated and real neural spike trains. Simulated data provides a good test environment for this method, since the underlying firing rate is known, but it lacks the experimental proof of real neural spike trains (where spiking does not exactly follow a gamma-interval process). For the real neural spike trains, however, we do not know the true underlying firing rate, and thus we can only make comparisons to a noisy, trial-averaged mean rate, which may or may not accurately reflect the true underlying rate of an individual spike train (due to different cognitive processing on different trials). Taken together, however, we believe the real and simulated data give good evidence of the general improvements offered by this method. Panels (a), (b), and (c) in Fig. 2 show that GP IGIP offers meaningful improvements in many cases and a small loss in performance in a few cases. Panel (d) tells a different story. In simulation, GP IGIP generally outperforms the other smoothers (though, by considerably less than in other panels). In real neural data, however, GP IGIP performs the same or relatively worse than other methods. This may indicate that, in the low firing rate regime, the IGIP is a poor model for real neural spiking. 7 It may also be due to our algorithmic approximations (namely, the Laplace approximation, which allows density outside the nonnegative orthant). We will report on this question in future work. Furthermore, some neural spike trains may be inherently ill-suited to analysis. A problem with this and any other method is that of very low firing rates, as only occasional insight is given into the underlying generative process. With spike trains of only a few spikes/sec, it will be impossible for any method to find interesting structure in the firing rate. In these cases, only with many trial averaging can this structure be seen. Several studies have investigated the inhomogeneous gamma and other more general models (e.g. [6], [19]), including the inhomogeneous inverse gaussian (IIG) interval and inhomogeneous Markov interval (IMI) processes. The methods of this paper apply immediately to any log-concave inhomogeneous renewal process in which inhomogeneity is generated by time-rescaling (this includes the IIG and several others). The IMI (and other more sophisticated models) will require some changes in implementation details; one possibility is a variational Bayes approach. Another direction for this work is to consider significant nonstationarity in the spike data. The SE kernel is standard, but it is also stationary; the method will have to compromise between areas of categorically different covariance. Nonstationary covariance is an important question in modelling and remains an area of research [20]. Advances in that field should inform this method as well. Acknowledgments This work was supported by NIH-NINDS-CRCNS-R01, the Michael Flynn SGF, NSF, NDSEGF, Gatsby, CDRF, BWF, ONR, Sloan, and Whitaker. This work was conceived at the UK Spike Train Workshop, Newcastle, UK, 2006; we thank Stuart Baker for helpful discussions during that time. We thank Vikash Gilja, Stephen Ryu, and Mackenzie Risch for experimental, surgical, and animal care assistance. We thank also Araceli Navarro. References [1] B. Yu, A. Afshar, G. Santhanam, S. Ryu, K. Shenoy, and M. Sahani. Advances in NIPS, 17, 2005. [2] R. Kass, V. Ventura, and E. Brown. J. Neurophysiol, 94:8–25, 2005. [3] I. DiMatteo, C. Genovese, and R. Kass. Biometrika, 88:1055–1071, 2001. [4] H. Shimazaki and S. Shinomoto. Neural Computation, 19(6):1503–1527, 2007. [5] D. Endres, M. Oram, J. Schindelin, and P. Foldiak. Advances in NIPS, 20, 2008. [6] R. Barbieri, M. Quirk, L. Frank, M. Wilson, and E. Brown. J Neurosci Methods, 105:25–37, 2001. [7] E. Brown, R. Barbieri, V. Ventura, R. Kass, and L. Frank. Neural Comp, 2002. [8] W. Truccolo, U. Eden, M. Fellows, J. Donoghue, and E. Brown. J Neurophysiol., 93:1074– 1089, 2004. [9] J. Moller, A. Syversveen, and R. Waagepetersen. Scandanavian J. of Stats., 1998. [10] K. Miura, Y. Tsubo, M. Okada, and T. Fukai. J Neurosci., 27:13802–13812, 2007. [11] D. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes. Springer, 2002. [12] C. Rasmussen and C. Williams. Gaussian Processes for Machine Learning. MIT Press, 2006. [13] M. Kuss and C. Rasmussen. Journal of Machine Learning Res., 6:1679–1704, 2005. [14] W. Horrace. J Multivariate Analysis, 94(1):209–221, 2005. [15] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004. [16] B. Silverman. Journal of Royal Stat. Soc. Series C: Applied Stat., 33, 1982. [17] C. Chestek, A. Batista, G. Santhanam, B. Yu, A. Afshar, J. Cunningham, V. Gilja, S. Ryu, M. Churchland, and K. Shenoy. J Neurosci., 27:10742–10750, 2007. [18] B. Richmond, L. Optican, and H. Spitzer. J. Neurophys., 64(2), 1990. [19] R. Kass and V. Ventura. Neural Comp, 14:5–15, 2003. [20] C. Paciorek and M. Schervish. 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Stable Dual Dynamic Programming Tao Wang∗ Daniel Lizotte Michael Bowling Dale Schuurmans Department of Computing Science University of Alberta {trysi,dlizotte,bowling,dale}@cs.ualberta.ca Abstract Recently, we have introduced a novel approach to dynamic programming and reinforcement learning that is based on maintaining explicit representations of stationary distributions instead of value functions. In this paper, we investigate the convergence properties of these dual algorithms both theoretically and empirically, and show how they can be scaled up by incorporating function approximation. 1 Introduction Value function representations are dominant in algorithms for dynamic programming (DP) and reinforcement learning (RL). However, linear programming (LP) methods clearly demonstrate that the value function is not a necessary concept for solving sequential decision making problems. In LP methods, value functions only correspond to the primal formulation of the problem, while in the dual they are replaced by the notion of state (or state-action) visit distributions [1, 2, 3]. Despite the well known LP duality, dual representations have not been widely explored in DP and RL. Recently, we have showed that it is entirely possible to solve DP and RL problems in the dual representation [4]. Unfortunately, [4] did not analyze the convergence properties nor implement the proposed ideas. In this paper, we investigate the convergence properties of these newly proposed dual solution techniques, and show how they can be scaled up by incorporating function approximation. The proof techniques we use to analyze convergence are simple, but lead to useful conclusions. In particular, we find that the standard convergence results for value based approaches also apply to the dual case, even in the presence of function approximation and off-policy updating. The dual approach appears to hold an advantage over the standard primal view of DP/RL in one major sense: since the fundamental objects being represented are normalized probability distributions (i.e., belong to a bounded simplex), dual updates cannot diverge. In particular, we find that dual updates converge (i.e. avoid oscillation) in the very circumstance where primal updates can and often do diverge: gradient-based off-policy updates with linear function approximation [5, 6]. 2 Preliminaries We consider the problem of computing an optimal behavior strategy in a Markov decision process (MDP), defined by a set of actions A, a set of states S, a |S||A| by |S| transition matrix P, a reward vector r and a discount factor γ, where we assume the goal is to maximize the infinite horizon discounted reward r0 + γr1 + γ2r2 + · · · = P∞ t=0 γtrt. It is known that an optimal behavior strategy can always be expressed by a stationary policy, whose entries π(sa) specify the probability of taking action a in state s. Below, we represent a policy π by an equivalent representation as an |S|×|S||A| matrix Π where Π(s,s′a) = π(sa) if s′ = s, otherwise 0. One can quickly verify that the matrix product ΠP gives the state-to-state transition probabilities induced by the policy π in the environment P, and that PΠ gives the state-action to state-action transition probabilities induced by policy π in P. The problem is to compute an optimal policy given either (a) a complete ∗Current affiliation: Computer Sciences Laboratory, Australian National University, tao.wang@anu.edu.au. specification of the environmental variables P and r (the “planning problem”), or (b) limited access to the environment through observed states and rewards and the ability to select actions to cause further state transitions (the “learning problem”). The first problem is normally tackled by LP or DP methods, and the second by RL methods. In this paper, we will restrict our attention to scenario (a). 3 Dual Representations Traditionally, DP methods for solving the MDP planning problem are typically expressed in terms of the primal value function. However, [4] demonstrated that all the classical algorithms have natural duals expressed in terms of state and state-action probability distributions. In the primal representation, the policy state-action value function can be specified by an |S||A|×1 vector q = P∞ i=0 γi(PΠ)ir which satisfies q = r + γPΠq. To develop a dual form of stateaction policy evaluation, one considers the linear system d⊤= (1 −γ)ν⊤+ γd⊤PΠ, where ν is the initial distribution over state-action pairs. Not only is d a proper probability distribution over state-action pairs, it also allows one to easily compute the expected discounted return of the policy π. However, recovering the state-action distribution d is inadequate for policy improvement. Therefore, one considers the following |S||A|×|S||A| matrix H = (1 −γ)I + γPΠH. The matrix H that satisfies this linear relation is similar to d⊤, in that each row is a probability distribution and the entries H(sa,s′a′) correspond to the probability of discounted state-action visits to (s′a′) for a policy π starting in state-action pair (sa). Unlike d⊤, however, H drops the dependence on ν, giving (1 −γ)q = Hr. That is, given H we can easily recover the state-action values of π. For policy improvement, in the primal representation one can derive an improved policy π′ via the update a∗(s) = arg maxa q(sa) and π′ (sa) = 1 if a = a∗(s), otherwise 0. The dual form of the policy update can be expressed in terms of the state-action matrix H for π is a∗(s) = arg maxa H(sa,:)r. In fact, since (1 −γ)q = Hr, the two policy updates given in the primal and dual respectively, must lead to the same resulting policy π′. Further details are given in [4]. 4 DP algorithms and convergence We first investigate whether dynamic programming operators with the dual representations exhibit the same (or better) convergence properties to their primal counterparts. These questions will be answered in the affirmative. In the tabular case, dynamic programming algorithms can be expressed by operators that are successively applied to current approximations (vectors in the primal case, matrices in the dual), to bring them closer to a target solution; namely, the fixed point of a desired Bellman equation. Consider two standard operators, the on-policy update and the max-policy update. For a given policy Π, the on-policy operator O is defined as Oq = r + γPΠq and OH = (1 −γ)I + γPΠH, for the primal and dual cases respectively. The goal of the on-policy update is to bring current representations closer to satisfying the policy-specific Bellman equations, q = r + γPΠq and H = (1 −γ)I + γPΠH The max-policy operator M is different in that it is neither linear nor defined by any reference policy, but instead applies a greedy max update to the current approximations Mq = r + γPΠ∗[q] and MH = (1 −γ)I + γPΠ∗ r[H], where Π∗[q](s) = maxa q(sa) and Π∗ r[H](s,:) = H(sa′(s),:) such that a′(s) = arg maxa[Hr](sa). The goal of this greedy update is to bring the representations closer to satisfying the optimal-policy Bellman equations q = r + γPΠ∗[q] and H = (1 −γ)I + γPΠ∗ r[H]. 4.1 On-policy convergence For the on-policy operator O, convergence to the Bellman fixed point is easily proved in the primal case, by establishing a contraction property of O with respect to a specific norm on q vectors. In particular, one defines a weighted 2-norm with weights given by the stationary distribution determined by the policy Π and transition model P: Let z ≥0 be a vector such that z⊤PΠ = z⊤; that is, z is the stationary state-action visit distribution for PΠ. Then the norm is defined as ∥q∥z 2 = q⊤Zq = P (sa) z(sa)q2 (sa), where Z = diag(z). It can be shown that ∥PΠq∥z ≤∥q∥z and ∥Oq1 −Oq2∥z ≤γ∥q1 −q2∥z (see [7]). Crucially, for this norm, a state-action transition is not an expansion [7]. By the contraction map fixed point theorem [2] there exists a unique fixed point of O in the space of vectors q. Therefore, repeated applications of the on-policy operator converge to a vector qΠ such that qΠ = OqΠ; that is, qΠ satisfies the policy based Bellman equation. Analogously, for the dual representation H, one can establish convergence of the on-policy operator by first defining an approximate weighted norm over matrices and then verifying that O is a contraction with respect to this norm. Define ∥H∥z,r 2 = ∥Hr∥z 2 = X (sa) z(sa)( X (s′a′) H(sa,s′a′)r(s′a′))2 (1) It is easily verified that this definition satisfies the property of a pseudo-norm, and in particular, satisfies the triangle inequality. This weighted 2-norm is defined with respect to the stationary distribution z, but also the reward vector r. Thus, the magnitude of a row normalized matrix is determined by the magnitude of the weighted reward expectations it induces. Interestingly, this definition allows us to establish the same non-expansion and contraction results as the primal case. We can have ∥PΠH∥z,r ≤∥H∥z,r by arguments similar to the primal case. Moreover, the on-policy operator is a contraction with respect to ∥·∥z,r. Lemma 1 ∥OH1 −OH2∥z,r ≤γ∥H1 −H2∥z,r Proof: ∥OH1 −OH2∥z,r = γ∥PΠ(H1 −H2)∥z,r ≤γ∥H1 −H2∥z,r since ∥PΠH∥z,r ≤ ∥H∥z,r. Thus, once again by the contraction map fixed point theorem there exists a fixed point of O among row normalized matrices H, and repeated applications of O will converge to a matrix HΠ such that OHΠ = HΠ; that is, HΠ satisfies the policy based Bellman equation for dual representations. This argument shows that on-policy dynamic programming converges in the dual representation, without making direct reference to the primal case. We will use these results below. 4.2 Max-policy convergence The strategy for establishing convergence for the nonlinear max operator is similar to the on-policy case, but involves working with a different norm. Instead of considering a 2-norm weighted by the visit probabilities induced by a fixed policy, one simply uses the max-norm in this case: ∥q∥∞= max(sa) |q(sa)|. The contraction property of the M operator with respect to this norm can then be easily established in the primal case: ∥Mq1 −Mq2∥∞≤γ∥q1 −q2∥∞(see [2]). As in the on-policy case, contraction suffices to establish the existence of a unique fixed point of M among vectors q, and that repeated application of M converges to this fixed point q∗such that Mq∗= q∗. To establish convergence of the off-policy update in the dual representation, first define the maxnorm for state-action visit distribution as ∥H∥∞ = max (sa) | X (s′a′) H(sa,s′a′)r(s′a′)| (2) Then one can simply reduce the dual to the primal case by appealing to the relationship (1−γ)Mq = MHr to prove convergence of MH. Lemma 2 If (1−γ)q = Hr, then (1−γ)Mq = MHr. Proof: (1−γ)Mq = (1−γ)r+γPΠ∗[(1−γ)q]) = (1−γ)r+γPΠ∗[Hr] = (1−γ)r+γPΠ∗ r[H]r = MHr where the second equality holds since we assumed (1 −γ)q(sa) = [Hr](sa) for all (sa). Thus, given convergence of Mq to a fixed point Mq∗= q∗, the same must also hold for MH. However, one subtlety here is that the dual fixed point is not unique. This is not a contradiction because the norm on dual representations ∥·∥z,r is in fact just a pseudo-norm, not a proper norm. That is, the relationship between H and q is many to one, and several matrices can correspond to the same q. These matrices form a convex subspace (in fact, a simplex), since if H1r = (1 −γ)q and H2r = (1−γ)q then (αH1 +(1−α)H2)r = (1−γ)q for any α, where furthermore α must be restricted to 0 ≤α ≤1 to maintain nonnegativity. The simplex of fixed points {H∗: MH∗= H∗} is given by matrices H∗that satisfy H∗r = (1 −γ)q∗. 5 DP with function approximation Primal and dual updates exhibit strong equivalence in the tabular case, as they should. However, when we begin to consider approximation, differences emerge. We next consider the convergence properties of the dynamic programming operators in the context of linear basis approximation. We focus on the on-policy case here, because, famously, the max operator does not always have a fixed point when combined with approximation in the primal case [8], and consequently suffers the risk of divergence [5, 6]. Note that the max operator cannot diverge in the dual case, even with basis approximation, by boundedness alone; although the question of whether max updates always converge in this case remains open. Here we establish that a similar bound on approximation error in the primal case can be proved for the dual approach with respect to the on-policy operator. In the primal case, linear approximation proceeds by fixing a small set of basis functions, forming a |S||A|×k matrix Φ, where k is the number of bases. The approximation of q can be expressed by a linear combination of bases ˆq = Φw where w is a k×1 vector of adjustable weights. This is equivalent to maintaining the constraint that ˆq ∈col span(Φ). In the dual, a linear approximation to H can be expressed as vec( ˆH) = Ψw, where the vec operator creates a column vector from a matrix by stacking the column vectors of the matrix below one another, w is a k ×1 vector of adjustable weights as it is in the primal case, and Ψ is a (|S||A|)2 × k matrix of basis functions. To ensure that ˆH remains a nonnegative, row normalized approximation to H, we simply add the constraints that ˆH ∈simplex(Ψ) ≡{ ˆH : vec( ˆH)=Ψw, Ψ≥0,(1⊤⊗I)Ψ = 11⊤,w≥0, w⊤1 = 1} where the operator ⊗is the Kronecker product. In this section, we first introduce operators (projection and gradient step operators) that ensure the approximations stay representable in the given basis. Then we consider their composition with the on-policy and off-policy updates, and analyze their convergence properties. For the composition of the on-policy update and projection operators, we establish a similar bound on approximation error in the dual case as in the primal case. 5.1 Projection Operator Recall that in the primal, the action value function q is approximated by a linear combination of bases in Φ. Unfortunately, there is no reason to expect Oq or Mq to stay in the column span of Φ, so a best approximation is required. The subtlety resolved by Tsitsiklis and Van Roy [7] is to identify a particular form of best approximation—weighted least squares—that ensures convergence is still achieved when combined with the on-policy operator O. Unfortunately, the fixed point of this combined update operator is not guaranteed to be the best representable approximation of O’s fixed point, qΠ. Nevertheless, a bound can be proved on how close this altered fixed point is to the best representable approximation. We summarize a few details that will be useful below: First, the best least squares approximation is computed with respect to the distribution z. The map from a general q vector onto its best approximation in col span(Φ) is defined by another operator, P, which projects q into the column span of Φ, Pq = argminˆq∈col span(Φ) ∥q −ˆq∥z 2 = Φ(Φ⊤ZΦ)−1Φ⊤Zq, where ˆq is an approximation for value function q. The important property of this weighted projection is that it is a non-expansion operator in ∥·∥z, i.e., ∥Pq∥z ≤∥q∥z, which can be easily obtained from the generalized Pythagorean theorem. Approximate dynamic programming then proceeds by composing the two operators—the on-policy update O with the subspace projection P—to compute the best representable approximation of the one step update. This combined operator is guaranteed to converge, since composing a non-expansion with a contraction is still a contraction, i.e., ∥q+ −qΠ∥z ≤ 1 1−γ ∥qΠ −PqΠ∥z [7]. Linear function approximation in the dual case is a bit more complicated because matrices are being represented, not vectors, and moreover the matrices need to satisfy row normalization and nonnegativity constraints. Nevertheless, a very similar approach to the primal case can be successfully applied. Recall that in the dual, the state-action visit distribution H is approximated by a linear combination of bases in Ψ. As in the primal case, there is no reason to expect that an update like OH should keep the matrix in the simplex. Therefore, a projection operator must be constructed that determines the best representable approximation to OH. One needs to be careful to define this projection with respect to the right norm to ensure convergence. Here, the pseudo-norm ∥·∥z,r defined in Equation 1 suits this purpose. Define the weighted projection operator P over matrices PH = argmin ˆ H∈simplex(Ψ) ∥H −ˆH∥z,r 2 (3) The projection could be obtained by solving the above quadratic program. A key result is that this projection operator is a non-expansion with respect to the pseudo-norm ∥·∥z,r. Theorem 1 ∥PH∥z,r ≤∥H∥z,r Proof: The easiest way to prove the theorem is to observe that the projection operator P is really a composition of three orthogonal projections: first, onto the linear subspace span(Ψ), then onto the subspace of row normalized matrices span(Ψ) ∩{H : H1 = 1}, and finally onto the space of nonnegative matrices span(Ψ) ∩{H : H1 = 1} ∩{H : H ≥0}. Note that the last projection into the nonnegative halfspace is equivalent to a projection into a linear subspace for some hyperplane tangent to the simplex. Each one of these projections is a non-expansion in ∥·∥z,r in the same way: a generalized Pythagorean theorem holds. Consider just one of these linear projections P1 ∥H∥z,r 2 = ∥P1H + H −P1H∥z,r 2 = ∥P1Hr + Hr −P1Hr∥z 2 = ∥P1Hr∥z 2 + ∥Hr −P1Hr∥z 2 = ∥P1H∥z,r 2 + ∥H −P1H∥z,r 2 Since the overall projection is just a composition of non-expansions, it must be a non-expansion. As in the primal, approximate dynamic programming can be implemented by composing the onpolicy update O with the projection operator P. Since O is a contraction and P a non-expansion, PO must also be a contraction, and it then follows that it has a fixed point. Note that, as in the tabular case, this fixed point is only unique up to Hr-equivalence, since the pseudo-norm ∥·∥z,r does not distinguish H1 and H2 such that H1r = H2r. Here too, the fixed point is actually a simplex of equivalent solutions. For simplicity, we denote the simplex of fixed points for PO by some representative H+ = POH+. Finally, we can recover an approximation bound that is analogous to the primal bound, which bounds the approximation error between H+ and the best representable approximation to the on-policy fixed point HΠ = OHΠ. Theorem 2 ∥H+ −HΠ∥z,r ≤ 1 1−γ ∥PHΠ −HΠ∥z,r Proof: First note that ∥H+−HΠ∥z,r = ∥H+−PHΠ+PHΠ−HΠ∥z,r ≤∥H+−PHΠ∥z,r + ∥PHΠ−HΠ∥z,r by generalized Pythagorean theorem. Then since H+ = POH+ and P is a non-expansion operator, we have ∥H+−PHΠ∥z,r = ∥POH+−PHΠ∥z,r ≤∥OH+−HΠ∥z,r. Finally, using HΠ = OHΠ and Lemma 1, we obtain ∥OH+−HΠ∥z,r = ∥OH+−OHΠ∥z,r ≤ γ∥H+−HΠ∥z,r. Thus (1−γ)∥H+−HΠ∥z,r ≤∥PHΠ−HΠ∥z,r. To compare the primal and dual results, note that despite the similarity of the bounds, the projection operators do not preserve the tight relationship between primal and dual updates. That is, even if (1−γ)q = Hr and (1−γ)(Oq) = (OH)r, it is not true in general that (1−γ)(POq) = (POH)r. The most obvious difference comes from the fact that in the dual, the space of H matrices has bounded diameter, whereas in the primal, the space of q vectors has unbounded diameter in the natural norms. Automatically, the dual updates cannot diverge with compositions like PO and PM; yet, in the primal case, the update PM is known to not have fixed points in some circumstances [8]. 5.2 Gradient Operator In large scale problems one does not normally have the luxury of computing full dynamic programming updates that evaluate complete expectations over the entire domain, since this requires knowing the stationary visit distribution z for PΠ (essentially requiring one to know the model of the MDP). Moreover, full least squares projections are usually not practical to compute. A key intermediate step toward practical DP and RL algorithms is to formulate gradient step operators that only approximate full projections. Conveniently, the gradient update and projection operators are independent of the on-policy and off-policy updates and can be applied in either case. However, as we will see below, the gradient update operator causes significant instability in the off-policy update, to the degree that divergence is a common phenomenon (much more so than with full projections). Composing approximation with an off-policy update (max operator) in the primal case can be very dangerous. All other operator combinations are better behaved in practice, and even those that are not known to converge usually behave reasonably. Unfortunately, composing the gradient step with an off-policy update is a common algorithm attempted in reinforcement learning (Q-learning with function approximation), despite being the most unstable. In the dual representation, one can derive a gradient update operator in a similar way to the primal, except that it is important to maintain the constraints on the parameters w, since the basis functions are probability distributions. We start by considering the projection objective JH = 1 2∥H −ˆH∥z,r 2 subject to vec( ˆH) = Ψw, w ≥0, w⊤1 = 1 The unconstrained gradient of the above objective with respect to w is ∇wJH = Ψ⊤(r⊤⊗I)⊤Z(r⊤⊗I)(Ψw −h) = Γ⊤Z(r⊤⊗I)(ˆh −h) where Γ = (r⊤⊗I)Ψ, h = vec(H), and ˆh = vec( ˆH). However, this gradient step cannot be followed directly because we need to maintain the constraints. The constraint w⊤1 = 1 can be maintained by first projecting the gradient onto it, obtaining δw = (I −1 k11⊤)∇wJH. Thus, the weight vector can be updated by wt+1 = wt −αδw = wt −α(I −1 k 11⊤)Γ⊤Z(r⊤⊗I)(ˆh −h) where α is a step-size parameter. Then the gradient operator can then be defined by Gˆhh = ˆh −αΨδw = ˆh −αΨ(I −1 k 11⊤)Γ⊤Z(r⊤⊗I)(ˆh −h) (Note that to further respect the box constraints, 0 ≤h ≤1, the stepsize might need to be reduced and additional equality constraints might have to be imposed on some of the components of h that are at the boundary values.) Similarly as in the primal, since the target vector H (i.e., h) is determined by the underlying dynamic programming update, this gives the composed updates GOˆh = ˆh −αΨ(I−1 k 11⊤)Γ⊤Z(r⊤⊗I)(ˆh−Oˆh) and GMˆh = ˆh −αΨ(I−1 k 11⊤)Γ⊤(r⊤⊗I)(ˆh−Mˆh) respectively for the on-policy and off-policy cases (ignoring the additional equality constraints). Thus far, the dual approach appears to hold an advantage over the standard primal approach, since convergence holds in every circumstance where the primal updates converge, and yet the dual updates are guaranteed never to diverge because the fundamental objects being represented are normalized probability distributions (i.e., belong to a bounded simplex). We now investigate the convergence properties of the various updates empirically. 6 Experimental Results To investigate the effectiveness of the dual representations, we conducted experiments on various domains, including randomly synthesized MDPs, Baird’s star problem [5], and on the mountain car problem. The randomly synthesized MDP domains allow us to test the general properties of the algorithms. The star problem is perhaps the most-cited example of a problem where Q-learning with linear function approximation diverges [5], and the mountain car domain has been prone to divergence with some primal representations [9] although successful results were reported when bases are selected by sparse tile coding [10]. For each problem domain, twelve algorithms were run over 100 repeats with a horizon of 1000 steps. The algorithms were: tabular on-policy (O), projection on-policy (PO), gradient on-policy (GO), tabular off-policy (M), projection off-policy (PM), and gradient off-policy (GM), for both the primal and the dual. The discount factor was set to γ = 0.9. For on-policy algorithms, we measure the difference between the values generated by the algorithms and those generated by the analytically determined fixed-point. For off-policy algorithms, we measure the difference between the values generated by the resulting policy and the values of the optimal policy. The step size for the gradient updates was 0.1 for primal representations and 100 for dual representations. The initial values of state-action value functions q are set according to the standard normal distribution, and state-action visit distributions H are chosen uniformly randomly with row normalization. Since the goal is to investigate the convergence of the algorithms without carefully crafting features, we also choose random basis functions according to a standard normal distribution for the primal representations, and random basis distributions according to a uniform distribution for the dual representations. Randomly Synthesized MDPs. For the synthesized MDPs, we generated the transition and reward functions of the MDPs randomly—the transition function is uniformly distributed between 0 and 1 and the reward function is drawn from a standard normal. Here we only reported the results of random MDPs with 100 states, 5 actions, and 10 bases, observed consistent convergence of the dual representations on a variety of MDPs, with different numbers of states, actions, and bases. In Figure 1(right), the curve for the gradient off-policy update (GM) in the primal case (dotted line with the circle marker) blows up (diverges), while all the other algorithms in Figure 1 converge. Interestingly, the approximate error of the dual algorithm POH (4.60×10−3) is much smaller than the approximate error of the corresponding primal algorithm POq (4.23×10−2), even though their theoretical bounds are the same (see Figure 1(left)). 100 200 300 400 500 600 700 800 900 1000 10 −10 10 −5 10 0 10 5 10 10 Number of Steps Difference from Reference Point On−Policy Update on Random MDPs Oq POq G Oq OH POH G OH 100 200 300 400 500 600 700 800 900 1000 10 −10 10 −5 10 0 10 5 10 10 Number of Steps Difference from Reference Point Off−Policy Update on Random MDPs Mq PMq G Mq MH PMH G MH Figure 1: Updates of state-action value q and visit distribution H on randomly synthesized MDPs The Star Problem. The star problem has 7 states and 2 actions. The reward function is zero for each transition. In these experiments, we used the same fixed policy and linear value function approximation as in [5]. In the dual, the number of bases is also set to 14 and the initial values of the state-action visit distribution matrix H are uniformly distributed random numbers between 0 and 1 with row normalization. The gradient off-policy update in the primal case diverges (see the dotted line with the circle marker in Figure 2(right)). However, all the updates with the dual representation algorithms converge. 100 200 300 400 500 600 700 800 900 1000 10 −10 10 −5 10 0 10 5 10 10 Number of Steps Difference from Reference Point On−Policy Update on Star Problem Oq POq G Oq OH POH G OH 100 200 300 400 500 600 700 800 900 1000 10 −10 10 −5 10 0 10 5 10 10 Number of Steps Difference from Reference Point Off−Policy Update on Star Problem Mq PMq G Mq MH PMH G MH Figure 2: Updates of state-action value q and visit distribution H on the star problem The Mountain Car Problem The mountain car domain has continuous state and action spaces, which we discretized with a simple grid, resulting in an MDP with 222 states and 3 actions. The number of bases was chosen to be 5 for both the primal and dual algorithms. For the same reason as before, we chose the bases for the algorithms randomly. In the primal representations with linear function approximation, we randomly generated basis functions according to the standard normal distribution. In the dual representations, we randomly picked the basis distributions according to the uniform distribution. In Figure 3(right), we again observed divergence of the gradient off-policy update on state-action values in the primal, and the convergence of all the dual algorithms (see Figure 3). Again, the approximation error of the projected on-policy update POH in the dual (1.90×101) is also considerably smaller than POq (3.26×102) in the primal. 100 200 300 400 500 600 700 800 900 1000 10 −10 10 −5 10 0 10 5 10 10 Number of Steps Difference from Reference Point On−Policy Update on Mountain Car Oq POq G Oq OH POH G OH 100 200 300 400 500 600 700 800 900 1000 10 −10 10 −5 10 0 10 5 10 10 Number of Steps Difference from Reference Point Off−Policy Update on Mountain Car Mq PMq G Mq MH PMH G MH Figure 3: Updates of state-action value q and visit distribution H on the mountain car problem 7 Conclusion Dual representations maintain an explicit representation of visit distributions as opposed to value functions [4]. We extended the dual dynamic programming algorithms with linear function approximation, and studied the convergence properties of the dual algorithms for planning in MDPs. We demonstrated that dual algorithms, since they are based on estimating normalized probability distributions rather than unbounded value functions, avoid divergence even in the presence of approximation and off-policy updates. Moreover, dual algorithms remain stable in situations where standard value function estimation diverges. References [1] M. Puterman. Markov Decision Processes: Discrete Dynamic Programming. Wiley, 1994. [2] D. Bertsekas. Dynamic Programming and Optimal Control, volume 2. Athena Scientific, 1995. [3] D. Bertsekas and J. Tsitsiklis. Neuro-Dynamic Programming. Athena Scientific, 1996. [4] T. Wang, M. Bowling, and D. Schuurmans. Dual representations for dynamic programming and reinforcement learning. In Proceeding of the IEEE International Symposium on ADPRL, pages 44–51, 2007. [5] L. C. Baird. Residual algorithms: Reinforcement learning with function approximation. In International Conference on Machine Learning, pages 30–37, 1995. [6] R. Sutton and A. Barto. Reinforcement Learning: An Introduction. MIT Press, 1998. [7] J. Tsitsiklis and B. Van Roy. An analysis of temporal-difference learning with function approximation. IEEE Trans. Automat. Control, 42(5):674–690, 1997. [8] D. de Farias and B. Van Roy. On the existence of fixed points for approximate value iteration and temporal-difference learning. J. Optimization Theory and Applic., 105(3):589–608, 2000. [9] J. A. Boyan and A. W. Moore. Generalization in reinforcement learning: Safely approximating the value function. In NIPS 7, pages 369–376, 1995. [10] R. S. Sutton. Generalization in reinforcement learning: Successful examples using sparse coarse coding. In Advances in Neural Information Processing Systems, pages 1038–1044, 1996.
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People Tracking with the Laplacian Eigenmaps Latent Variable Model Zhengdong Lu CSEE, OGI, OHSU zhengdon@csee.ogi.edu Miguel ´A. Carreira-Perpi˜n´an EECS, UC Merced http://eecs.ucmerced.edu Cristian Sminchisescu University of Bonn sminchisescu.ins.uni-bonn.de Abstract Reliably recovering 3D human pose from monocular video requires models that bias the estimates towards typical human poses and motions. We construct priors for people tracking using the Laplacian Eigenmaps Latent Variable Model (LELVM). LELVM is a recently introduced probabilistic dimensionality reduction model that combines the advantages of latent variable models—a multimodal probability density for latent and observed variables, and globally differentiable nonlinear mappings for reconstruction and dimensionality reduction—with those of spectral manifold learning methods—no local optima, ability to unfold highly nonlinear manifolds, and good practical scaling to latent spaces of high dimension. LELVM is computationally efficient, simple to learn from sparse training data, and compatible with standard probabilistic trackers such as particle filters. We analyze the performance of a LELVM-based probabilistic sigma point mixture tracker in several real and synthetic human motion sequences and demonstrate that LELVM not only provides sufficient constraints for robust operation in the presence of missing, noisy and ambiguous image measurements, but also compares favorably with alternative trackers based on PCA or GPLVM priors. Recent research in reconstructing articulated human motion has focused on methods that can exploit available prior knowledge on typical human poses or motions in an attempt to build more reliable algorithms. The high-dimensionality of human ambient pose space—between 30-60 joint angles or joint positions depending on the desired accuracy level, makes exhaustive search prohibitively expensive. This has negative impact on existing trackers, which are often not sufficiently reliable at reconstructing human-like poses, self-initializing or recovering from failure. Such difficulties have stimulated research in algorithms and models that reduce the effective working space, either using generic search focusing methods (annealing, state space decomposition, covariance scaling) or by exploiting specific problem structure (e.g. kinematic jumps). Experience with these procedures has nevertheless shown that any search strategy, no matter how effective, can be made significantly more reliable if restricted to low-dimensional state spaces. This permits a more thorough exploration of the typical solution space, for a given, comparatively similar computational effort as a high-dimensional method. The argument correlates well with the belief that the human pose space, although high-dimensional in its natural ambient parameterization, has a significantly lower perceptual (latent or intrinsic) dimensionality, at least in a practical sense—many poses that are possible are so improbable in many real-world situations that it pays off to encode them with low accuracy. A perceptual representation has to be powerful enough to capture the diversity of human poses in a sufficiently broad domain of applicability (the task domain), yet compact and analytically tractable for search and optimization. This justifies the use of models that are nonlinear and low-dimensional (able to unfold highly nonlinear manifolds with low distortion), yet probabilistically motivated and globally continuous for efficient optimization. Reducing dimensionality is not the only goal: perceptual representations have to preserve critical properties of the ambient space. Reliable tracking needs locality: nearby regions in ambient space have to be mapped to nearby regions in latent space. If this does not hold, the tracker is forced to make unrealistically large, and difficult to predict jumps in latent space in order to follow smooth trajectories in the joint angle ambient space. 1 In this paper we propose to model priors for articulated motion using a recently introduced probabilistic dimensionality reduction method, the Laplacian Eigenmaps Latent Variable Model (LELVM) [1]. Section 1 discusses the requirements of priors for articulated motion in the context of probabilistic and spectral methods for manifold learning, and section 2 describes LELVM and shows how it combines both types of methods in a principled way. Section 3 describes our tracking framework (using a particle filter) and section 4 shows experiments with synthetic and real human motion sequences using LELVM priors learned from motion-capture data. Related work: There is significant work in human tracking, using both generative and discriminative methods. Due to space limitations, we will focus on the more restricted class of 3D generative algorithms based on learned state priors, and not aim at a full literature review. Deriving compact prior representations for tracking people or other articulated objects is an active research field, steadily growing with the increased availability of human motion capture data. Howe et al. and Sidenbladh et al. [2] propose Gaussian mixture representations of short human motion fragments (snippets) and integrate them in a Bayesian MAP estimation framework that uses 2D human joint measurements, independently tracked by scaled prismatic models [3]. Brand [4] models the human pose manifold using a Gaussian mixture and uses an HMM to infer the mixture component index based on a temporal sequence of human silhouettes. Sidenbladh et al. [5] use similar dynamic priors and exploit ideas in texture synthesis—efficient nearest-neighbor search for similar motion fragments at runtime—in order to build a particle-filter tracker with observation model based on contour and image intensity measurements. Sminchisescu and Jepson [6] propose a low-dimensional probabilistic model based on fitting a parametric reconstruction mapping (sparse radial basis function) and a parametric latent density (Gaussian mixture) to the embedding produced with a spectral method. They track humans walking and involved in conversations using a Bayesian multiple hypotheses framework that fuses contour and intensity measurements. Urtasun et al. [7] use a dynamic MAP estimation framework based on a GPLVM and 2D human joint correspondences obtained from an independent image-based tracker. Li et al. [8] use a coordinated mixture of factor analyzers within a particle filtering framework, in order to reconstruct human motion in multiple views using chamfer matching to score different configuration. Wang et al. [9] learn a latent space with associated dynamics where both the dynamics and observation mapping are Gaussian processes, and Urtasun et al. [10] use it for tracking. Taylor et al. [11] also learn a binary latent space with dynamics (using an energy-based model) but apply it to synthesis, not tracking. Our work learns a static, generative low-dimensional model of poses and integrates it into a particle filter for tracking. We show its ability to work with real or partially missing data and to track multiple activities. 1 Priors for articulated human pose We consider the problem of learning a probabilistic low-dimensional model of human articulated motion. Call y ∈RD the representation in ambient space of the articulated pose of a person. In this paper, y contains the 3D locations of anywhere between 10 and 60 markers located on the person’s joints (other representations such as joint angles are also possible). The values of y have been normalised for translation and rotation in order to remove rigid motion and leave only the articulated motion (see section 3 for how we track the rigid motion). While y is high-dimensional, the motion pattern lives in a low-dimensional manifold because most values of y yield poses that violate body constraints or are simply atypical for the motion type considered. Thus we want to model y in terms of a small number of latent variables x given a collection of poses {yn}N n=1 (recorded from a human with motion-capture technology). The model should satisfy the following: (1) It should define a probability density for x and y, to be able to deal with noise (in the image or marker measurements) and uncertainty (from missing data due to occlusion or markers that drop), and to allow integration in a sequential Bayesian estimation framework. The density model should also be flexible enough to represent multimodal densities. (2) It should define mappings for dimensionality reduction F : y →x and reconstruction f : x →y that apply to any value of x and y (not just those in the training set); and such mappings should be defined on a global coordinate system, be continuous (to avoid physically impossible discontinuities) and differentiable (to allow efficient optimisation when tracking), yet flexible enough to represent the highly nonlinear manifold of articulated poses. From a statistical machine learning point of view, this is precisely what latent variable models (LVMs) do; for example, factor analysis defines linear mappings and Gaussian densities, while the generative topographic mapping (GTM; [12]) defines nonlinear mappings and a Gaussian-mixture density in ambient space. However, factor analysis is too limited to be of practical use, and GTM— 2 while flexible—has two important practical problems: (1) the latent space must be discretised to allow tractable learning and inference, which limits it to very low (2–3) latent dimensions; (2) the parameter estimation is prone to bad local optima that result in highly distorted mappings. Another dimensionality reduction method recently introduced, GPLVM [13], which uses a Gaussian process mapping f(x), partly improves this situation by defining a tunable parameter xn for each data point yn. While still prone to local optima, this allows the use of a better initialisation for {xn}N n=1 (obtained from a spectral method, see later). This has prompted the application of GPLVM for tracking human motion [7]. However, GPLVM has some disadvantages: its training is very costly (each step of the gradient iteration is cubic on the number of training points N, though approximations based on using few points exist); unlike true LVMs, it defines neither a posterior distribution p(x|y) in latent space nor a dimensionality reduction mapping E {x|y}; and the latent representation it obtains is not ideal. For example, for periodic motions such as running or walking, repeated periods (identical up to small noise) can be mapped apart from each other in latent space because nothing constrains xn and xm to be close even when yn = ym (see fig. 3 and [10]). There exists a different type of dimensionality reduction methods, spectral methods (such as Isomap, LLE or Laplacian eigenmaps [14]), that have advantages and disadvantages complementary to those of LVMs. They define neither mappings nor densities but just a correspondence (xn, yn) between points in latent space xn and ambient space yn. However, the training is efficient (a sparse eigenvalue problem) and has no local optima, and often yields a correspondence that successfully models highly nonlinear, convoluted manifolds such as the Swiss roll. While these attractive properties have spurred recent research in spectral methods, their lack of mappings and densities has limited their applicability in people tracking. However, a new model that combines the advantages of LVMs and spectral methods in a principled way has been recently proposed [1], which we briefly describe next. 2 The Laplacian Eigenmaps Latent Variable Model (LELVM) LELVM is based on a natural way of defining an out-of-sample mapping for Laplacian eigenmaps (LE) which, in addition, results in a density model. In LE, typically we first define a k-nearestneighbour graph on the sample data {yn}N n=1 and weigh each edge yn ∼ym by a Gaussian affinity function K(yn, ym) = wnm = exp (−1 2 ∥(yn −ym)/σ∥2). Then the latent points X result from: min tr XLX⊤ s.t. X ∈RL×N, XDX⊤= I, XD1 = 0 (1) where we define the matrix XL×N = (x1, . . . , xN), the symmetric affinity matrix WN×N, the degree matrix D = diag (PN n=1 wnm), the graph Laplacian matrix L = D−W, and 1 = (1, . . . , 1)⊤. The constraints eliminate the two trivial solutions X = 0 (by fixing an arbitrary scale) and x1 = · · · = xN (by removing 1, which is an eigenvector of L associated with a zero eigenvalue). The solution is given by the leading u2, . . . , uL+1 eigenvectors of the normalised affinity matrix N = D−1 2 WD−1 2 , namely X = V⊤D−1 2 with VN×L = (v2, . . . , vL+1) (an a posteriori translated, rotated or uniformly scaled X is equally valid). Following [1], we now define an out-of-sample mapping F(y) = x for a new point y as a semisupervised learning problem, by recomputing the embedding as in (1) (i.e., augmenting the graph Laplacian with the new point), but keeping the old embedding fixed: min x∈RL tr ( X x ) L K(y) K(y)⊤1⊤K(y) X⊤ x⊤ (2) where Kn(y) = K(y, yn) = exp (−1 2 ∥(y −yn)/σ∥2) for n = 1, . . . , N is the kernel induced by the Gaussian affinity (applied only to the k nearest neighbours of y, i.e., Kn(y) = 0 if y ≁yn). This is one natural way of adding a new point to the embedding by keeping existing embedded points fixed. We need not use the constraints from (1) because they would trivially determine x, and the uninteresting solutions X = 0 and X = constant were already removed in the old embedding anyway. The solution yields an out-of-sample dimensionality reduction mapping x = F(y): x = F(y) = X K(y) 1⊤K(y) = PN n=1 K(y,yn) PN n′=1 K(y,yn′)xn (3) applicable to any point y (new or old). This mapping is formally identical to a Nadaraya-Watson estimator (kernel regression; [15]) using as data {(xn, yn)}N n=1 and the kernel K. We can take this a step further by defining a LVM that has as joint distribution a kernel density estimate (KDE): p(x, y) = 1 N PN n=1 Ky(y, yn)Kx(x, xn) p(y) = 1 N PN n=1 Ky(y, yn) p(x) = 1 N PN n=1 Kx(x, xn) 3 where Ky is proportional to K so it integrates to 1, and Kx is a pdf kernel in x–space. Consequently, the marginals in observed and latent space are also KDEs, and the dimensionality reduction and reconstruction mappings are given by kernel regression (the conditional means E {y|x}, E {x|y}): F(y) = PN n=1 p(n|y)xn f(x) = PN n=1 Kx(x,xn) PN n′=1 Kx(x,xn′)yn = PN n=1 p(n|x)yn. (4) We allow the bandwidths to be different in the latent and ambient spaces: Kx(x, xn) ∝exp (−1 2 ∥(x −xn)/σx∥2) and Ky(y, yn) ∝exp (−1 2 ∥(y −yn)/σy∥2). They may be tuned to control the smoothness of the mappings and densities [1]. Thus, LELVM naturally extends a LE embedding (efficiently obtained as a sparse eigenvalue problem with a cost O(N 2)) to global, continuous, differentiable mappings (NW estimators) and potentially multimodal densities having the form of a Gaussian KDE. This allows easy computation of posterior probabilities such as p(x|y) (unlike GPLVM). It can use a continuous latent space of arbitrary dimension L (unlike GTM) by simply choosing L eigenvectors in the LE embedding. It has no local optima since it is based on the LE embedding. LELVM can learn convoluted mappings (e.g. the Swiss roll) and define maps and densities for them [1]. The only parameters to set are the graph parameters (number of neighbours k, affinity width σ) and the smoothing bandwidths σx, σy. 3 Tracking framework We follow the sequential Bayesian estimation framework, where for state variables s and observation variables z we have the recursive prediction and correction equations: p(st|z0:t−1) = R p(st|st−1) p(st−1|z0:t−1) dst−1 p(st|z0:t) ∝p(zt|st) p(st|z0:t−1). (5) We define the state variables as s = (x, d) where x ∈RL is the low-dim. latent space (for pose) and d ∈R3 is the centre-of-mass location of the body (in the experiments our state also includes the orientation of the body, but for simplicity here we describe only the translation). The observed variables z consist of image features or the perspective projection of the markers on the camera plane. The mapping from state to observations is (for the markers’ case, assuming M markers): x ∈RL f −−−−→y ∈R3M −−→⊕ P −−−−−→z ∈R2M d ∈R3 (6) where f is the LELVM reconstruction mapping (learnt from mocap data); ⊕shifts each 3D marker by d; and P is the perspective projection (pinhole camera), applied to each 3D point separately. Here we use a simple observation model p(zt|st): Gaussian with mean given by the transformation (6) and isotropic covariance (set by the user to control the influence of measurements in the tracking). We assume known correspondences and observations that are obtained either from the 3D markers (for tracking synthetic data) or 2D tracks obtained from a 2D tracker. Our dynamics model is p(st|st−1) ∝pd(dt|dt−1) px(xt|xt−1) p(xt) (7) where both dynamics models for d and x are random walks: Gaussians centred at the previous step value dt−1 and xt−1, respectively, with isotropic covariance (set by the user to control the influence of dynamics in the tracking); and p(xt) is the LELVM prior. Thus the overall dynamics predicts states that are both near the previous state and yield feasible poses. Of course, more complex dynamics models could be used if e.g. the speed and direction of movement are known. As tracker we use the Gaussian mixture Sigma-point particle filter (GMSPPF) [16]. This is a particle filter that uses a Gaussian mixture representation for the posterior distribution in state space and updates it with a Sigma-point Kalman filter. This Gaussian mixture will be used as proposal distribution to draw the particles. As in other particle filter implementations, the prediction step is carried out by approximating the integral (5) with particles and updating the particles’ weights. Then, a new Gaussian mixture is fitted with a weighted EM algorithm to these particles. This replaces the resampling stage needed by many particle filters and mitigates the problem of sample depletion while also preventing the number of components in the Gaussian mixture from growing over time. The choice of this particular tracker is not critical; we use it to illustrate the fact that LELVM can be introduced in any probabilistic tracker for nonlinear, nongaussian models. Given the corrected distribution p(st|z0:t), we choose its mean as recovered state (pose and location). It is also possible to choose instead the mode closest to the state at t −1, which could be found by mean-shift or Newton algorithms [17] since we are using a Gaussian-mixture representation in state space. 4 4 Experiments We demonstrate our low-dimensional tracker on image sequences of people walking and running, both synthetic (fig. 1) and real (fig. 2–3). Fig. 1 shows the model copes well with persistent partial occlusion and severely subsampled training data (A,B), and quantitatively evaluates temporal reconstruction (C). For all our experiments, the LELVM parameters (number of neighbors k, Gaussian affinity σ, and bandwidths σx and σy) were set manually. We mainly considered 2D latent spaces (for pose, plus 6D for rigid motion), which were expressive enough for our experiments. More complex, higher-dimensional models are straightforward to construct. The initial state distribution p(s0) was chosen a broad Gaussian, the dynamics and observation covariance were set manually to control the tracking smoothness, and the GMSPPF tracker used a 5-component Gaussian mixture in latent space (and in the state space of rigid motion) and a small set of 500 particles. The 3D representation we use is a 102-D vector obtained by concatenating the 3D markers coordinates of all the body joints. These would be highly unconstrained if estimated independently, but we only use them as intermediate representation; tracking actually occurs in the latent space, tightly controlled using the LELVM prior. For the synthetic experiments and some of the real experiments (figs. 2–3) the camera parameters and the body proportions were known (for the latter, we used the 2D outputs of [6]). For the CMU mocap video (fig. 2B) we roughly guessed. We used mocap data from several sources (CMU, OSU). As observations we always use 2D marker positions, which, depending on the analyzed sequence were either known (the synthetic case), or provided by an existing tracker [6] or specified manually (fig. 2B). Alternatively 2D point trackers similar to the ones of [7] can be used. The forward generative model is obtained by combining the latent to ambient space mapping (this provides the position of the 3D markers) with a perspective projection transformation. The observation model is a product of Gaussians, each measuring the probability of a particular marker position given its corresponding image point track. Experiments with synthetic data: we analyze the performance of our tracker in controlled conditions (noise perturbed synthetically generated image tracks) both under regular circumstances (reasonable sampling of training data) and more severe conditions with subsampled training points and persistent partial occlusion (the man running behind a fence, with many of the 2D marker tracks obstructed). Fig. 1B,C shows both the posterior (filtered) latent space distribution obtained from our tracker, and its mean (we do not show the distribution of the global rigid body motion; in all experiments this is tracked with good accuracy). In the latent space plot shown in fig. 1B, the onset of running (two cycles were used) appears as a separate region external to the main loop. It does not appear in the subsampled training set in fig. 1B, where only one running cycle was used for training and the onset of running was removed. In each case, one can see that the model is able to track quite competently, with a modest decrease in its temporal accuracy, shown in fig. 1C, where the averages are computed per 3D joint (normalised wrt body height). Subsampling causes some ambiguity in the estimate, e.g. see the bimodality in the right plot in fig. 1C. In another set of experiments (not shown) we also tracked using different subsets of 3D markers. The estimates were accurate even when about 30% of the markers were dropped. Experiments with real images: this shows our tracker’s ability to work with real motions of different people, with different body proportions, not in its latent variable model training set (figs. 2–3). We study walking, running and turns. In all cases, tracking and 3D reconstruction are reasonably accurate. We have also run comparisons against low-dimensional models based on PCA and GPLVM (fig. 3). It is important to note that, for LELVM, errors in the pose estimates are primarily caused by mismatches between the mocap data used to learn the LELVM prior and the body proportions of the person in the video. For example, the body proportions of the OSU motion captured walker are quite different from those of the image in fig. 2–3 (e.g. note how the legs of the stick man are shorter relative to the trunk). Likewise, the style of the runner from the OSU data (e.g. the swinging of the arms) is quite different from that of the video. Finally, the interest points tracked by the 2D tracker do not entirely correspond either in number or location to the motion capture markers, and are noisy and sometimes missing. In future work, we plan to include an optimization step to also estimate the body proportions. This would be complicated for a general, unconstrained model because the dimensions of the body couple with the pose, so either one or the other can be changed to improve the tracking error (the observation likelihood can also become singular). But for dedicated prior pose models like ours these difficulties should be significantly reduced. The model simply cannot assume highly unlikely stances—these are either not representable at all, or have reduced probability—and thus avoids compensatory, unrealistic body proportion estimates. 5 n = 15 n = 40 n = 65 n = 90 n = 115 n = 140 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 n = 1 n = 13 n = 25 n = 37 n = 49 n = 60 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 0 50 100 150 0 0.02 0.04 0.06 0.08 0.1 time step n RMSE 0 10 20 30 40 50 60 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 time step n RMSE Figure 1: A B C OSU running man motion capture data. A: we use 217 datapoints for training LELVM (with added noise) and for tracking. Row 1: tracking in the 2D latent space. The contours (very tight in this sequence) are the posterior probability. Row 2: perspective-projection-based observations with occlusions. Row 3: each quadruplet (a, a′, b, b′) show the true pose of the running man from a front and side views (a, b), and the reconstructed pose by tracking with our model (a′, b′). B: we use the first running cycle for training LELVM and the second cycle for tracking. C: RMSE errors for each frame, for the tracking of A (left plot) and B (middle plot), normalised so that 1 equals the height of the stick man. RMSE(n) = 1 M PM j=1 ∥ynj −ˆynj∥2−1/2 for all 3D locations of the M markers, i.e., comparison of reconstructed stick man ˆyn with ground-truth stick man yn. Right plot: multimodal posterior distribution in pose space for the model of A (frame 42). Comparison with PCA and GPLVM (fig. 3): for these models, the tracker uses the same GMSPPF setting as for LELVM (number of particles, initialisation, random-walk dynamics, etc.) but with the mapping y = f(x) provided by GPLVM or PCA, and with a uniform prior p(x) in latent space (since neither GPLVM nor the non-probabilistic PCA provide one). The LELVM-tracker uses both its f(x) and latent space prior p(x), as discussed. All methods use a 2D latent space. We ensured the best possible training of GPLVM by model selection based on multiple runs. For PCA, the latent space looks deceptively good, showing non-intersecting loops. However, (1) individual loops do not collect together as they should (for LELVM they do); (2) worse still, the mapping from 2D to pose space yields a poor observation model. The reason is that the loop in 102-D pose space is nonlinearly bent and a plane can at best intersect it at a few points, so the tracker often stays put at one of those (typically an “average” standing position), since leaving it would increase the error a lot. Using more latent dimensions would improve this, but as LELVM shows, this is not necessary. For GPLVM, we found high sensitivity to filter initialisation: the estimates have high variance across runs and are inaccurate ≈80% of the time. When it fails, the GPLVM tracker often freezes in latent space, like PCA. When it does succeed, it produces results that are comparable with LELVM, although somewhat less accurate visually. However, even then GPLVM’s latent space consists of continuous chunks spread apart and offset from each other; GPLVM has no incentive to place nearby two xs mapping to the same y. This effect, combined with the lack of a data-sensitive, realistic latent space density p(x), makes GPLVM jump erratically from chunk to chunk, in contrast with LELVM, which smoothly follows the 1D loop. Some GPLVM problems might be alleviated using higher-order dynamics, but our experiments suggest that such modeling sophistication is less 6 n = 1 n = 15 n = 29 n = 43 n = 55 n = 69 −50 −40 −30 −20 −10 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 10 20 30 40 50 −100 −50 0 50 100 −50 −40 −30 −20 −10 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 10 20 30 40 50 −100 −50 0 50 100 −50 −40 −30 −20 −10 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 10 20 30 40 50 −100 −50 0 50 100 −50 −40 −30 −20 −10 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 10 20 30 40 50 −100 −50 0 50 100 −50 −40 −30 −20 −10 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 10 20 30 40 50 −100 −50 0 50 100 −50 −40 −30 −20 −10 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 10 20 30 40 50 −100 −50 0 50 100 n = 4 n = 9 n = 14 n = 19 n = 24 n = 29 50 100 150 200 250 300 350 20 40 60 80 100 120 140 160 180 200 220 50 100 150 200 250 300 350 20 40 60 80 100 120 140 160 180 200 220 50 100 150 200 250 300 350 20 40 60 80 100 120 140 160 180 200 220 50 100 150 200 250 300 350 20 40 60 80 100 120 140 160 180 200 220 50 100 150 200 250 300 350 20 40 60 80 100 120 140 160 180 200 220 50 100 150 200 250 300 350 20 40 60 80 100 120 140 160 180 200 220 −50 −40 −30 −20 −10 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 10 20 30 40 50 −100 −50 0 50 100 −50 −40 −30 −20 −10 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 10 20 30 40 50 −100 −50 0 50 100 −50 −40 −30 −20 −10 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 10 20 30 40 50 −100 −50 0 50 100 −50 −40 −30 −20 −10 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 10 20 30 40 50 −100 −50 0 50 100 −50 −40 −30 −20 −10 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 10 20 30 40 50 −100 −50 0 50 100 −50 −40 −30 −20 −10 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 10 20 30 40 50 −100 −50 0 50 100 Figure 2: A B A: tracking of a video from [6] (turning & walking). We use 220 datapoints (3 full walking cycles) for training LELVM. Row 1: tracking in the 2D latent space. The contours are the estimated posterior probability. Row 2: tracking based on markers. The red dots are the 2D tracks and the green stick man is the 3D reconstruction obtained using our model. Row 3: our 3D reconstruction from a different viewpoint. B: tracking of a person running straight towards the camera. Notice the scale changes and possible forward-backward ambiguities in the 3D estimates. We train the LELVM using 180 datapoints (2.5 running cycles); 2D tracks were obtained by manually marking the video. In both A–B the mocap training data was for a person different from the video’s (with different body proportions and motions), and no ground-truth estimate was available for favourable initialisation. LELVM GPLVM PCA −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.025 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 38 0.99 tracking in latent space −2 −1 0 1 2 3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 38 tracking in latent space −80 −60 −40 −20 0 20 40 60 80 −30 −20 −10 0 10 20 30 38 tracking in latent space Figure 3: Method comparison, frame 38. PCA and GPLVM map consecutive walking cycles to spatially distinct latent space regions. Compounded by a data independent latent prior, the resulting tracker gets easily confused: it jumps across loops and/or remains put, trapped in local optima. In contrast, LELVM is stable and follows tightly a 1D manifold (see videos). crucial if locality constraints are correctly modeled (as in LELVM). We conclude that, compared to LELVM, GPLVM is significantly less robust for tracking, has much higher training overhead and lacks some operations (e.g. computing latent conditionals based on partly missing ambient data). 7 5 Conclusion and future work We have proposed the use of priors based on the Laplacian Eigenmaps Latent Variable Model (LELVM) for people tracking. LELVM is a probabilistic dim. red. method that combines the advantages of latent variable models and spectral manifold learning algorithms: a multimodal probability density over latent and ambient variables, globally differentiable nonlinear mappings for reconstruction and dimensionality reduction, no local optima, ability to unfold highly nonlinear manifolds, and good practical scaling to latent spaces of high dimension. LELVM is computationally efficient, simple to learn from sparse training data, and compatible with standard probabilistic trackers such as particle filters. Our results using a LELVM-based probabilistic sigma point mixture tracker with several real and synthetic human motion sequences show that LELVM provides sufficient constraints for robust operation in the presence of missing, noisy and ambiguous image measurements. Comparisons with PCA and GPLVM show LELVM is superior in terms of accuracy, robustness and computation time. The objective of this paper was to demonstrate the ability of the LELVM prior in a simple setting using 2D tracks obtained automatically or manually, and single-type motions (running, walking). Future work will explore more complex observation models such as silhouettes; the combination of different motion types in the same latent space (whose dimension will exceed 2); and the exploration of multimodal posterior distributions in latent space caused by ambiguities. Acknowledgments This work was partially supported by NSF CAREER award IIS–0546857 (MACP), NSF IIS–0535140 and EC MCEXT–025481 (CS). CMU data: http://mocap.cs.cmu.edu (created with funding from NSF EIA–0196217). OSU data: http://accad.osu.edu/research/mocap/mocap data.htm. References [1] M. ´A. Carreira-Perpi˜n´an and Z. Lu. The Laplacian Eigenmaps Latent Variable Model. In AISTATS, 2007. [2] N. R. Howe, M. E. Leventon, and W. T. Freeman. Bayesian reconstruction of 3D human motion from single-camera video. In NIPS, volume 12, pages 820–826, 2000. [3] T.-J. Cham and J. M. Rehg. A multiple hypothesis approach to figure tracking. In CVPR, 1999. [4] M. Brand. Shadow puppetry. In ICCV, pages 1237–1244, 1999. [5] H. Sidenbladh, M. J. Black, and L. Sigal. Implicit probabilistic models of human motion for synthesis and tracking. In ECCV, volume 1, pages 784–800, 2002. [6] C. Sminchisescu and A. Jepson. Generative modeling for continuous non-linearly embedded visual inference. In ICML, pages 759–766, 2004. [7] R. Urtasun, D. J. Fleet, A. Hertzmann, and P. Fua. Priors for people tracking from small training sets. In ICCV, pages 403–410, 2005. [8] R. Li, M.-H. Yang, S. Sclaroff, and T.-P. Tian. Monocular tracking of 3D human motion with a coordinated mixture of factor analyzers. In ECCV, volume 2, pages 137–150, 2006. [9] J. M. Wang, D. Fleet, and A. Hertzmann. Gaussian process dynamical models. In NIPS, volume 18, 2006. [10] R. Urtasun, D. J. Fleet, and P. Fua. Gaussian process dynamical models for 3D people tracking. In CVPR, pages 238–245, 2006. [11] G. W. Taylor, G. E. Hinton, and S. Roweis. Modeling human motion using binary latent variables. In NIPS, volume 19, 2007. [12] C. M. Bishop, M. Svens´en, and C. K. I. Williams. GTM: The generative topographic mapping. Neural Computation, 10(1):215–234, January 1998. [13] N. Lawrence. Probabilistic non-linear principal component analysis with Gaussian process latent variable models. Journal of Machine Learning Research, 6:1783–1816, November 2005. [14] M. Belkin and P. Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15(6):1373–1396, June 2003. [15] B. W. Silverman. Density Estimation for Statistics and Data Analysis. Chapman & Hall, 1986. [16] R. van der Merwe and E. A. Wan. Gaussian mixture sigma-point particle filters for sequential probabilistic inference in dynamic state-space models. In ICASSP, volume 6, pages 701–704, 2003. [17] M. ´A. Carreira-Perpi˜n´an. Acceleration strategies for Gaussian mean-shift image segmentation. In CVPR, pages 1160–1167, 2006. 8
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The Distribution Family of Similarity Distances Gertjan J. Burghouts∗ Arnold W. M. Smeulders Intelligent Systems Lab Amsterdam Informatics Institute University of Amsterdam Jan-Mark Geusebroek † Abstract Assessing similarity between features is a key step in object recognition and scene categorization tasks. We argue that knowledge on the distribution of distances generated by similarity functions is crucial in deciding whether features are similar or not. Intuitively one would expect that similarities between features could arise from any distribution. In this paper, we will derive the contrary, and report the theoretical result that Lp-norms –a class of commonly applied distance metrics– from one feature vector to other vectors are Weibull-distributed if the feature values are correlated and non-identically distributed. Besides these assumptions being realistic for images, we experimentally show them to hold for various popular feature extraction algorithms, for a diverse range of images. This fundamental insight opens new directions in the assessment of feature similarity, with projected improvements in object and scene recognition algorithms. 1 Introduction Measurement of similarity is a critical asset of state of the art in computer vision. In all three major streams of current research - the recognition of known objects [13], assigning an object to a class [8, 24], or assigning a scene to a type [6, 25] - the problem is transposed into the equality of features derived from similarity functions. Hence, besides the issue of feature distinctiveness, comparing two images heavily relies on such similarity functions. We argue that knowledge on the distribution of distances generated by such similarity functions is even more important, as it is that knowledge which is crucial in deciding whether features are similar or not. For example, Nowak and Jurie [21] establish whether one can draw conclusions on two never seen objects based on the similarity distances from known objects. Where they build and traverse a randomized tree to establish region correspondence, one could alternatively use the distribution of similarity distances to establish whether features come from the mode or the tail of the distribution. Although this indeed only hints at an algorithm, it is likely that knowledge of the distance distribution will considerably improve or speed-up such tasks. As a second example, consider the clustering of features based on their distances. Better clustering algorithms significantly boost performance for object and scene categorization [12]. Knowledge on the distribution of distances aids in the construction of good clustering algorithms. Using this knowledge allows for the exact distribution shape to be used to determine cluster size and centroid, where now the Gaussian is often groundlessly assumed. We will show that in general distance distributions will strongly deviate from the Gaussian probability distribution. A third example is from object and scene recognition. Usually this is done by measuring invariant feature sets [9, 13, 24] at a predefined raster of regions in the image or at selected key points in the image [11, 13] as extensively evaluated [17]. Typically, an image contains a hundred regions or a ∗Dr. Burghouts is now with TNO Defense, The Netherlands, gertjan.burghouts@tno.nl. †Corresponding author. Email: mark@science.uva.nl. 1 thousand key points. Then, the most expensive computational step is to compare these feature sets to the feature sets of the reference objects, object classes or scene types. Usually this is done by going over all entries in the image to all entries in the reference set and select the best matching pair. Knowledge of the distribution of similarity distances and having established its parameters enables a remarkable speed-up in the search for matching reference points and hence for matching images. When verifying that a given reference key-point or region is statistically unlikely to occur in this image, one can move on to search in the next image. Furthermore, this knowledge can well be applied in the construction of fast search trees, see e.g. [16]. Hence, apart from obtaining theoretical insights in the general distribution of similarities, the results derived in this paper are directly applicable in object and scene recognition. Intuitively one would expect that the set of all similarity values to a key-point or region in an image would assume any distribution. One would expect that there is no preferred probability density distribution at stake in measuring the similarities to points or regions in one image. In this paper, we will derive the contrary. We will prove that under broad conditions the similarity values to a given reference point or region adhere to a class of distributions known as the Weibull distribution family. The density function has only three parameters: mean, standard deviation and skewness. We will verify experimentally that the conditions under which this result from mathematical statistics holds are present in common sets of images. It appears the theory predicts the resulting density functions accurately. Our work on density distributions of similarity values compares to the work by Pekalska and Duin [23] assuming a Gaussian distribution for similarities. It is based on an original combination of two facts from statistical physics. An old fact regards the statistics of extreme values [10], as generated when considering the minima and maxima of many measurements. The major result of the field of extreme value statistics is that the probability density in this case can only be one out of three different types, independent of the underlying data or process. The second fact is a new result, which links these extreme value statistics to sums of correlated variables [2, 3]. We exploit these two facts in order to derive the distribution family of similarity measures. This paper is structured as follows. In Section 2, we overview literature on similarity distances and distance distributions. In Section 3, we discuss the theory of distributions of similarity distances from one to other feature vectors. In Section 4, we validate the resulting distribution experimentally for image feature vectors. Finally, conclusions are given in Section 5. 2 Related work 2.1 Similarity distance measures To measure the similarity between two feature vectors, many distance measures have been proposed [15]. A common metric class of measures is the Lp-norm [1]. The distance from one reference feature vector s to one other feature vector t can be formalized as: d(s, t) = ( n X i=1 |si −ti|p)1/p, (1) where n and i are the dimensionality and indices of the vectors. Let the random variable Dp represent distances d(s, t) where t is drawn from the random variable T representing feature vectors. Independent of the reference feature vector s, the probability density function of Lp-distances will be denoted by f(Dp = d). 2.2 Distance distributions Ferencz et al. [7] have considered the Gamma distribution to model the L2-distances from image regions to one reference region: f(D2 = d) = 1 βγ Γ(γ) dγ−1 e−d/β, where γ is the shape parameter, and β the scale parameter; Γ(·) denotes the Gamma function. In [7], the distance function was fitted efficiently from few examples of image regions. Although the distribution fits were shown to represent the region distances to some extent, the method lacks a theoretical motivation. 2 Based on the central limit theorem, Pekalska and Duin [23] assumed that Lp-distances between feature vectors are normally distributed: f(Dp = d) = 1 √ 2π β e−(d2/β2)/2. As the authors argue, the use of the central limit theorem is theoretically justified if the feature values are independent, identically distributed, and have limited variance. Although feature values generally have limited variance, unfortunately, they cannot be assumed to be independent and/or identically distributed as we will show below. Hence, an alternative derivation of the distance distribution function has to be followed. 2.3 Contribution of this paper Our contribution is the theoretical derivation of a parameterized distribution for Lp-norm distances between feature vectors. In the experiments, we establish whether distances to image features adhere to this distribution indeed. We consider SIFT-based features [17], computed from various interest region types [18]. 3 Statistics of distances between features In this section, we derive the distribution function family of Lp-distances from a reference feature vector to other feature vectors. We consider the notation as used in the previous section, with t a feature vector drawn from the random variable T. For each vector t, we consider the value at index i, ti, resulting in a random variable Ti. The value of the reference vector at index i, si, can be interpreted as a sample of the random variable Ti. The computation of distances from one to other vectors involves manipulations of the random variable Ti resulting in a new random variable: Xi = |si−Ti|p. Furthermore, the computation of the distances D requires the summation of random variables, and a reparameterization: D = (PI i=1 Xi)1/p. In order to derive the distribution of D, we start with the statistics of the summation of random variables, before turning to the properties of Xi. 3.1 Statistics of sums As a starting point to derive the Lp-distance distribution function, we consider a lemma from statistics about the sum of random variables. Lemma 1 For non-identical and correlated random variables Xi, the sum PN i=1 Xi, with finite N, is distributed according to the generalized extreme value distribution, i.e. the Gumbel, Frechet or Weibull distribution. For a proof, see [2, 3]. Note that the lemma is an extension of the central limit theorem to nonidentically distributed random variables. And, indeed, the proof follows the path of the central limit theorem. Hence, the resulting distribution of sums is different from a normal distribution, and rather one of the Gumbel, Frechet or Weibull distributions instead. This lemma is important for our purposes, as later the feature values will turn out to be non-identical and correlated indeed. To confine the distribution function further, we also need the following lemma. Lemma 2 If in the above lemma the random variable Xi are upper-bounded, i.e. Xi < max, the sum of the variables is Weibull distributed (and not Gumbel nor Frechet): f(Y = y) = γ β ( y β )γ−1 e−( y β )γ , (2) with γ the shape parameter and β the scale parameter. For a proof, see [10]. Figure 1 illustrates the Weibull distribution for various shape parameters γ. This lemma is equally important to our purpose, as later the feature values will turn out to be upper-bounded indeed. The combination of Lemmas 1 and 2 yields the distribution of sums of non-identical, correlated and upper-bounded random variables, summarized in the following theorem. 3 1 2 3 4 5 distance 0.2 0.4 0.6 0.8 p Γ=8 Γ=6 Γ=4 Γ=2 shape parameter Figure 1: Examples of the Weibull distribution for various shape parameters γ. Theorem 1 For non-identical, correlated and upper-bounded random variables Xi, the random variable Y = PN i=1 Xi, with finite N, adheres to the Weibull distribution. The proof follows trivially from combining the different findings of statistics as laid down in Lemmas 1 and 2. Theorem 1 is the starting point to derive the distribution of Lp-norms from one reference vector to other feature vectors. 3.2 Lp-distances from one to other feature vectors Theorem 1 states that Y is Weibull-distributed, given that {Xi = |si −Ti|p}i∈[1,...,I] are nonidentical, correlated and upper-bounded random variables. We transform Y such that it represents Lp-distances, achieved by the transformation (·)1/p: Y 1/p = ( N X i=1 |si −Ti|p)1/p. (3) The consequence of the substitution Z = Y 1/p for the distribution of Y is a change of variables z = y1/p in Equation 2 [22]: g(Z = z) = f(zp) (1/p−1)z(1−p) . This transformation yields a different distribution still of the Weibull type: g(Z = z) = 1 (1/p −1) γ β1/p ( z β1/p )pγ−1 e −( z β1/p )pγ , (4) where γ′ = pγ is the new shape parameter and β′ = β1/p is the new scale parameter, respectively. Thus, also Y 1/p and hence Lp-distances are Weibull-distributed under the assumed case. We argue that the random variables Xi = |si −Ti|p and Xj (i ̸= j) are indeed non-identical, correlated and upper-bounded random variables when considering a set of values extracted from feature vectors at indices i and j: • Xi and Xj are upper-bounded. Features are usually an abstraction of a particular type of finite measurements, resulting in a finite feature. Hence, for general feature vectors, the values at index i, Ti, are finite. And, with finite p, it follows trivially that Xi is finite. • Xi and Xj are correlated. The experimental verification of this assumption is postponed to Section 4.1. • Xi and Xj are non-identically distributed. The experimental verification of this assumption is postponed to Section 4.1. We have obtained the following result. Corollary 1 For finite-length feature vectors with non-identical, correlated and upper-bounded values, Lp distances, for limited p, from one reference feature vector to other feature vectors adhere to the Weibull distribution. 4 3.3 Extending the class of features We extend the class of features for which the distances are Weibull-distributed. From now on, we allow the possibility that the vectors are preprocessed by a PCA transformation. We denote the PCA transform g(·) applied to a single feature vector as s′ = g(s). For the random variable Ti, we obtain T ′ i. We are still dealing with upper-bounded variables X′ i = |s′ i −T ′ i|p as PCA is a finite transform. The experimental verification of the assumption that PCA-transformed feature values T ′ i and T ′ j, i ̸= j are non-identically distributed is postponed to Section 4.1. Our point here, is that we have assumed originally correlating feature values, but after the decorrelating PCA transform we are no longer dealing with correlated feature values T ′ i and T ′ j. In Section 4.1, we will verify experimentally whether X′ i and X′ j correlate. The following observation is hypothesized. PCA translates the data to the origin, before applying an affine transformation that yields data distributed along orthogonal axes. The tuples (X′ i, X′ j) will be in the first quadrant due to the absolute value transformation. Obviously, variances σ(X′ i) and σ(X′ j) are limited and means µ(X′ i) > 0 and µ(X′ j) > 0. For data constrained to the first quadrant and distributed along orthogonal axes, a negative covariance is expected to be observed. Under the assumed case, we have obtained the following result. Corollary 2 For finite-length feature vectors with non-identical, correlated and upper-bounded values, and for PCA-transformations thereof, Lp distances, for limited p, from one to other features adhere to the Weibull distribution. 3.4 Heterogeneous feature vector data We extend the corollary to hold also for composite datasets of feature vectors. Consider the composite dataset modelled by random variables {Tt}, where each random variable Tt represents nonidentical and correlated feature values. Hence, from Corollary 2 it follows that feature vectors from each of the Tt can be fitted by a Weibull function f β,γ(d). However, the distances to each of the Tt may have a different range and modus, as we will verify by experimentation in Section 4.1. For heterogeneous distance data {Tt}, we obtain a mixture of Weibull functions [14]. Corollary 3 (Distance distribution) For feature vectors that are drawn from a mixture of datasets, of which each results in non-identical and correlated feature values, finite-length feature vectors with non-identical, correlated and upper-bounded values, and for PCA-transformations thereof, Lp distances, for limited p, from one reference feature vector to other feature vectors adhere to the Weibull mixture distribution: f(D = d) = Pc i=1 ρi · f βi,γi i (d), where fi are the Weibull functions and ρi are their respective weights such that Pc i=1 ρi = 1. 4 Experiments In our experiments, we validate assumptions and Weibull goodness-of-fit for the region-based SIFT, GLOH, SPIN, and PCA-SIFT features on COREL data [5]. We include these features for two reasons as: a) they are performing well for realistic computer vision tasks and b) they provide different mechanisms to describe an image region [17]. The region features are computed from regions detected by the Harris- and Hessian-affine regions, maximally stable regions (MSER), and intensity extrema-based regions (IBR) [18]. Also, we consider PCA-transformed versions for each of the detector/feature combinations. For reason of its extensive use, the experimentation is based on the L2-distance. We consider distances from 1 randomly drawn reference vector to 100 other randomly drawn feature vectors, which we repeat 1,000 times for generalization. In all experiments, the features are taken from multiple images, except for the illustration in Section 4.1.2 to show typical distributions of distances between features taken from single images. 4.1 Validation of the corollary assumptions for image features 4.1.1 Intrinsic feature assumptions Corollary 2 rests on a few explicit assumptions. Here we will verify whether the assumptions occur in practice. 5 Differences between feature values are correlated. We consider a set of feature vectors Tj and the differences at index i to a reference vector s: Xi = |si −Tji|p. We determine the significance of Pearson’s correlation [4] between the difference values Xi and Xj, i ̸= j. We establish the percentage of significantly correlating differences at a confidence level of 0.05. We report for each feature the average percentage of difference values that correlate significantly with difference values at an other feature vector index. As expected, the feature value differences correlate. For SIFT, 99% of the difference values are significantly correlated. For SPIN and GLOH, we obtain 98% and 96%, respectively. Also PCASIFT contains significantly correlating difference values: 95%. Although the feature’s name hints at uncorrelated values, it does not achieve a decorrelation of the values in practice. For each of the features, a low standard deviation < 5% is found. This expresses the low variation of correlations across the random samplings and across the various region types. We repeat the experiment for PCA-transformed feature values. Although the resulting values are uncorrelated by construction, their differences are significantly correlated. For SIFT, SPIN, GLOH, and PCA-SIFT, the percentages of significantly correlating difference values are: 94%, 86%, 95%, and 75%, respectively. Differences between feature values are non-identically distributed. We repeat the same procedure as above, but instead of measuring the significance of correlation, we establish the percentage of significantly differently distributed difference values Xi by the Wilcoxon rank sum test [4] at a confidence level of 0.05. For SIFT, SPIN, GLOH, and PCA-SIFT, the percentages of significantly differently distributed difference values are: 99%, 98%, 92%, and 87%. For the PCA-transformed versions of SIFT, SPIN, GLOH, and PCA-SIFT, we find: 62%, 40%, 64%, and 51%, respectively. Note that in all cases, correlation is sufficient to fulfill the assumptions of Corollary 2. We have illustrated that feature value differences are significantly correlated and significantly non-identically distributed. We conclude that the assumptions of Corollary 2 about properties of feature vectors are realistic in practice, and that Weibull functions are expected to fit distance distributions well. 4.1.2 Inter-feature assumptions In Corollary 3, we have assumed that distances from one to other feature vectors are described well by a mixture of Weibulls, if the features are taken from different clusters in the data. Here, we illustrate that clusters of feature vectors, and clusters of distances, occur in practice. Figure 2a shows Harris-affine regions from a natural scene which are described by the SIFT feature. The distances are described well by a single Weibull distribution. The same hold for distances from one to other regions computed from a man-made object, see Figure 2b. In Figure 2c, we illustrate the distances of one to other regions computed from a composite image containing two types of regions. This results in two modalitites of feature vectors hence of similarity distances. The distance distribution is therefore bimodal, illustrating the general case of multimodality to be expected in realistic, heterogeneous image data. We conclude that the assumptions of Corollary 3 are realistic in practice, and that the Weibull function, or a mixture, fits distance distributions well. 4.2 Validation of Weibull-shaped distance distributions In this experiment, we validate the fitting of Weibull distributions of distances from one reference feature vector to other vectors. We consider the same data as before. Over 1,000 repetitions we consider the goodness-of-fit of L2-distances by the Weibull distribution. The parameters of the Weibull distribution function are obtained by maximum likelihood estimation. The established fit is assessed by the Anderson-Darling test at a confidence level of α = 0.05 [20]. The Anderson-Darling test has also proven to be suited to measure the goodness-of-fit of mixture distributions [19]. Table 1 indicates that for most of the feature types computed from various regions, more than 90% of the distance distributions is fit by a single Weibull function. As expected, distances between each of the SPIN, SIFT, PCA-SIFT and GLOH features, are fitted well by Weibull distributions. The exception here is the low number of fits for the SIFT and SPIN features computed from Hessianaffine regions. The distributions of distances between these two region/feature combinations tend to have multiple modes. Likewise, there is a low percentage of fits of L2-distance distributions of the 6 250 300 350 400 450 500 550 600 650 700 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 distances probability (a) 250 300 350 400 450 500 550 600 650 700 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 distances probability (b) 250 300 350 400 450 500 550 600 650 700 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 distances probability (c) Figure 2: Distance distributions from one randomly selected image region to other regions, each described by the SIFT feature. The distance distribution is described by a single Weibull function for a natural scene (a) and a man-made object (b). For a composite image, the distance distribution is bimodal (c). Samples from each of the distributions are shown in the upper images. Table 1: Accepted Weibull fits for COREL data [5]. Harris-affine Hessian-affine MSER IBR c = 1 c ≤2 c = 1 c ≤2 c = 1 c ≤2 c = 1 c ≤2 SIFT 95% 100% 60% 99% 98% 100% 92% 100% SIFT (g =PCA) 95% 99% 60% 98% 98% 100% 92% 99% PCA-SIFT 89% 100% 96% 100% 94% 100% 95% 100% PCA-SIFT (g =PCA) 89% 100% 96% 100% 94% 100% 95% 100% SPIN 71% 99% 12% 99% 77% 99% 45% 98% SPIN (g =PCA) 71% 100% 12% 97% 77% 99% 45% 98% GLOH 87% 100% 91% 100% 82% 99% 86% 100% GLOH (g =PCA) 87% 100% 91% 99% 82% 99% 86% 100% Percentages of L2-distance distributions fitted by a Weibull function (c = 1) and a mixture of two Weibull functions (c ≤2) are given. SPIN feature computed from IBR regions. Again, multiple modes in the distributions are observed. For these distributions, a mixture of two Weibull functions provides a good fit (≥97%). 5 Conclusion In this paper, we have derived that similarity distances between one and other image features in databases are Weibull distributed. Indeed, for various types of features, i.e. the SPIN, SIFT, GLOH and PCA-SIFT features, and for a large variety of images from the COREL image collection, we have demonstrated that the similarity distances from one to other features, computed from Lp norms, are Weibull-distributed. These results are established by the experiments presented in Table 1. Also, between PCA-transformed feature vectors, the distances are Weibull-distributed. The Malahanobis distance is very similar to the L2-norm computed in the PCA-transformed feature space. Hence, we expect Mahalanobis distances to be Weibull distributed as well. Furthermore, when the dataset is a composition, a mixture of few (typically two) Weibull functions suffices, as established by the experiments presented in Table 1. The resulting Weibull distributions are distinctively different from the distributions suggested in literature, as they are positively or negatively skewed while the Gamma [7] and normal [23] distributions are positively and non-skewed, respectively. We have demonstrated that the Weibull distribution is the preferred choice for estimating properties of similarity distances. The assumptions under which the theory is valid are realistic for images. We experimentally have shown them to hold for various popular feature extraction algorithms, and for a diverse range of images. This fundamental insight opens new directions in the assessment of feature similarity, with projected improvements and speed-ups in object/scene recognition algorithms. 7 Acknowledgments This work is partly sponsored by the EU funded NEST project PERCEPT, by the Dutch BSIK project Multimedian, and by the EU Network of Excellence MUSCLE. References [1] B. G. Batchelor. Pattern Recognition: Ideas in Practice. Plenum Press, New York, 1995. [2] E. Bertin. Global fluctuations and Gumbel statistics. Physical Review Letters, 95(170601):1–4, 2005. [3] E. Bertin and M. Clusel. Generalised extreme value statistics and sum of correlated variables. Journal of Physics A, 39:7607, 2006. [4] W. J. Conover. Practical nonparametric statistics. Wiley, New York, 1971. [5] Corel Gallery. www.corel.com. [6] L. Fei-Fei and P. Perona. A bayesian hierarchical model for learning natural scene categories. In CVPR, 2005. [7] A. Ferencz, E.G. Learned-Miller, and J. Malik. Building a classification cascade for visual identification from one example. In Proceedings of the International Conference Computer Vision, pages 286–293. IEEE Computer Society, 2003. [8] R. Fergus, P. Perona, and A. Zisserman. A sparse object category model for efficient learning and exhaustive recognition. In Proceedings of the Computer Vision and Pattern Recognition. IEEE, 2005. [9] J. M. Geusebroek, R. van den Boomgaard, A. W. M. Smeulders, and H. Geerts. Color invariance. IEEE Transactions on Pattern Analysis and Machine Intelligence, 23(12):1338–1350, 2001. [10] E. J. Gumbel. Statistics of Extremes. Columbia University Press, New York, 1958. [11] C. Harris and M. Stephans. A combined corner and edge detector. In Proceedings of the 4th Alvey Vision Conference, pages 189–192, Manchester, 1988. [12] F. Jurie and B. Triggs. Creating efficient codebooks for visual recognition. In ICCV, pages 604–610, 2005. [13] D. G. Lowe. Distinctive image features from scale-invariant keypoints. International Journal of Computer Vision, 60(2):91–110, 2004. [14] J. M. Marin, M. T. Rodriquez-Bernal, and M. P. Wiper. Using weibull mixture distributions to model heterogeneous survival data. Communications in statistics, 34(3):673–684, 2005. [15] R. S. Michalski, R. E. Stepp, and E. Diday. A recent advance in data analysis: Clustering objects into classes characterized by conjunctive concepts. In L. N. Kanal and A. Rosenfeld, editors, Progress in Pattern Recognition, pages 33–56. North-Holland Publishing Co., Amsterdam, 1981. [16] K. Mikolajczyk, B. Leibe, and B. Schiele. Multiple object class detection with a generative model. In CVPR, 2006. [17] K. Mikolajczyk and C. Schmid. A performance evaluation of local descriptors. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27(10):1615–1630, 2005. [18] K. Mikolajczyk, T. Tuytelaars, C. Schmid, A. Zisserman, J. Matas, F. Schaffalitzky, T. Kadir, and L. Van Gool. A comparison of affine region detectors. International Journal of Computer Vision, 65(1/2):43–72, 2005. [19] K. Mosler. Mixture models in econometric duration analysis. Applied Stochastic Models in Business and Industry, 19(2):91–104, 2003. [20] NIST/SEMATECH. e-Handbook of Statistical Methods. NIST, http://www.itl.nist.gov/div898/handbook/, 2006. [21] E. Nowak and F. Jurie. Learning visual similarity measures for comparing never seen objects. In CVPR, 2007. [22] A. Papoulis and S. U. Pillai. Probability, Random Variables and Stochastic Processes. McGraw-Hill, New York, 4 edition, 2002. [23] E. Pekalska and R. P. W. Duin. Classifiers for dissimilarity-based pattern recognition. In Proceedings of the International Conference on Pattern Recognition, volume 2, page 2012, 2000. [24] C. Schmid and R. Mohr. Local grayvalue invariants for image retrieval. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19(5):530–535, 1997. [25] J.C. van Gemert, J.M. Geusebroek, C.J. Veenman, C.G.M. Snoek, and Arnold W.M. Smeulders. Robust scene categorization by learning image statistics in context. In CVPR Workshop on Semantic Learning Applications in Multimedia (SLAM), 2006. 8
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Congruence between model and human attention reveals unique signatures of critical visual events Robert J. Peters∗ Department of Computer Science University of Southern California Los Angeles, CA 90089 rjpeters@usc.edu Laurent Itti Departments of Neuroscience and Computer Science University of Southern California Los Angeles, CA 90089 itti@usc.edu Abstract Current computational models of bottom-up and top-down components of attention are predictive of eye movements across a range of stimuli and of simple, fixed visual tasks (such as visual search for a target among distractors). However, to date there exists no computational framework which can reliably mimic human gaze behavior in more complex environments and tasks, such as driving a vehicle through traffic. Here, we develop a hybrid computational/behavioral framework, combining simple models for bottom-up salience and top-down relevance, and looking for changes in the predictive power of these components at different critical event times during 4.7 hours (500,000 video frames) of observers playing car racing and flight combat video games. This approach is motivated by our observation that the predictive strengths of the salience and relevance models exhibit reliable temporal signatures during critical event windows in the task sequence—for example, when the game player directly engages an enemy plane in a flight combat game, the predictive strength of the salience model increases significantly, while that of the relevance model decreases significantly. Our new framework combines these temporal signatures to implement several event detectors. Critically, we find that an event detector based on fused behavioral and stimulus information (in the form of the model’s predictive strength) is much stronger than detectors based on behavioral information alone (eye position) or image information alone (model prediction maps). This approach to event detection, based on eye tracking combined with computational models applied to the visual input, may have useful applications as a less-invasive alternative to other event detection approaches based on neural signatures derived from EEG or fMRI recordings. 1 Introduction The human visual system provides an arena in which objects compete for our visual attention, and a given object may win the competition with support from a number of influences. For an example, an moving object in our visual periphery may capture our attention because of its salience, or the degree to which it is unusual or surprising given the overall visual scene [1]. On the other hand, a piece of fruit in a tree may capture our attention because of its relevance to our current foraging task, in which we expect rewarding items to be found in certain locations relative to tree trunks, and to have particular visual features such as a reddish color [2, 3]. Computational models of each of these influences have been developed and have individually been extensively characterized in terms of their ability to predict an overt measure of attention, namely gaze position [4, 5, 6, 3, 7, 8, 9]. Yet how do the real biological factors modeled by such systems interact in real-world settings [10]? Often salience and relevance are competing factors, and sometimes one factor is so strong that it ∗webpage: http://ilab.usc.edu/rjpeters/ game visual stimulus human game player event annotation BU model TD model eye position event-locked TD map template BU NSS TD NSS event-locked eye position template event-locked BU map template event-locked BU NSS template event-locked TD NSS template Figure 1: Our computational framework for generating detector templates which can be used to detect key events in video sequences. A human game player interacts with a video game, generating a sequence of video frames from the game, and a sequence of eye position samples from the game player. The video frames feed into computational models for predicting bottom-up (BU) salience and top-down (TD) relevance influences on attention. These predictions are then compared with the observed eye position using a “normalized scanpath saliency” (NSS) metric. Finally, the video game sequence is annotated with key event times, and these are used to generate event-locked templates from each of the game-related signals. These templates are used to try to detect the events in the original game sequences, and the results are quantified with metrics from signal detection theory. overrides our best efforts to ignore it, as in the case of oculomotor capture [11]. How does the visual system decide which factor dominates, and how does this vary as a function of the current task? We propose that one element of learning sophisticated visual or visuomotor tasks may be learning which attentional influences are important for each phase of the task. A key question is how to build models that can capture the effects of rapidly changing task demands on behavior. Here we address that question in the context of playing challenging video games, by comparing eye movements recorded during game play with the predictions of a combined salience/relevance computational model. Figure 1 illustrates the overall framework. The important factor in our approach is that we identify key game events (such as destroying an enemy plane, or crashing the car during driving race) which can be used as proxy indicators of likely transitions in the observer’s task set. Then we align subsequent analysis on these event times, such that we can detect repeatable changes in model predictive strength within temporal windows around the key events. Indeed, we find significant changes in the predictive strength of both salience and relevance models within these windows, including more than 8-fold increases in predictive strength as well as complete shifts from predictive to anti-predictive behavior. Finally we show that the predictive strength signatures formed in these windows can be used to detect the occurrence of the events themselves. 2 Psychophysics and eye tracking Five subjects (four male, one female) participated under a protocol approved by the Institutional Review Board of the University of Southern California. Subjects played two challenging games on a Nintendo GameCube: Need For Speed Underground (a car racing game) and Top Gun (a flight combat game). All of the subjects had at least some prior experience with playing video games in general, but none of the subjects had prior experience with the particular games involved in our experiment. For each game, subjects first practiced the game for several one-hour sessions on different days until reaching a success criterion (definition follows), and then returned for a one-hour eye tracking session with that game. Within each game, subjects learned to play three game levels, and during eye tracking, each subject played each game level twice. Thus, in total, our recorded data set consists of video frames and eye tracking data from 60 clips (5 subjects × 2 games per subject × 3 levels per game × 2 clips per level) covering 4.7 hours. Need For Speed: Underground (NFSU). In this game, players control a car in a race against three other computer-controlled racers in a three-lap race, with a different race course for each game level. The game display consists of a first-person view, as if the player were looking out the windshield from the driver’s seat of the vehicle, with several “heads-up display” elements showing current elapsed time, race position, and vehicle speed, as well as a race course map (see Figure 2 for sample game frames). The game controller joystick is used simply to steer the vehicle, and a pair of controller buttons are used to apply acceleration or brakes. Our “success” criterion for NFSU was finishing the race in third place or better out of the four racers. The main challenge for players was learning to be able to control the vehicle at a high rate of simulated speed (100+ miles per hour) while avoiding crashes with slow-moving non-race traffic and also avoiding the attempts of competing racers to knock the player’s vehicle offcourse. During eye tracking, the average length of an NFSU level was 4.11 minutes, with a range of 3.14–4.89 minutes across the 30 NFSU recordings. Top Gun (TG). In this game, players control a simulated fighter plane with a success criterion of destroying 12 specific enemy targets in 10 minutes or less. The game controller provides a simple set of flight controls: the joystick controls pitch (forward–backward axis) and combined yaw/roll (left– right axis), a pair of buttons controls thrust level up and down, and another button triggers missile firings toward enemy targets. Two onscreen displays aid the players in finding enemy targets: one is a radar map with enemy locations indicated by red triangles, and another is a direction finder running along the bottom screen showing the player’s current compass heading along with the headings to each enemy target. Players’ challenges during training involved first becoming familiar with the flight controls, and then learning a workable strategy for using the radar and direction finder to efficiently navigate the combat arena. During eye tracking, the average length of a TG level was 5.29 minutes, with a range of 2.96–8.78 minutes across the 30 TG recordings. Eye tracking. Stimuli were presented on a 22” computer monitor at a resolution of 640×480 pixels and refresh rate of 75 Hz. Subjects were seated at a viewing distance of 80 cm and used a chin-rest to stabilize their head position during eye tracking. Video game frames were captured at 29.97Hz from the GameCube using a Linux computer under SCHED_FIFO scheduling, which then displayed the captured frames onscreen for the player’s viewing and while simultaneously streaming the frames to disk for subsequent processing. Finally, subjects’ eye position was recorded at 240Hz with a hardware-based eye-tracking system (ISCAN, Inc.). In total, we obtained roughly 500,000 video game frames and 4,000,000 eye position samples during 4.7 hours of recording. 3 Computational attention prediction models We developed a computational model which uses existing building blocks for bottom-up and topdown components of attention to generate new eye position prediction maps for each of the recorded video game frames. Then, for each frame, we quantified the degree of correspondence between those maps and the actual eye position recorded from the game player. Although the individual models form the underlying computational foundation of our current study, our focus is not on testing their individual validity for predicting eye movements (which has already been established by prior studies), but rather on using them as components of a new model for investigating relationships between task structure and the relative strength of competing influences on visual attention; therefore we provide only a coarse summary of the workings of the models here and refer the reader to original sources for full details. Salience. Bottom-up salience maps were generated using a model based on detecting outliers in space and spatial frequency according to low-level features intensity, color, orientation, flicker and motion [4]. This model has been previously reported to be significantly predictive of eye positions across a range of stimuli and tasks [5, 6, 7, 8]. Relevance. Top-down task-relevance maps were generated using a model [9] which is trained to associate low-level “gist” signatures with relevant eye positions (see also [3]). We trained the taskrelevance model with a leave-one-out approach: for each of the 60 game clips, the task-relevance model used for testing against that clip was trained on the video frames and eye position samples from the remaining 59 clips. Model/human agreement. For each video game frame, we used the normalized scanpath saliency (NSS) metric [6] to quantify the agreement between the corresponding human eye position and the TD @ 113.41s TD @ 114.41s TD @ 115.42s BU @ 113.41s BU @ 114.41s BU @ 115.42s NSS BU model prediction strength 0 2 4 NSS TD model prediction strength time (s) in video game session 0 50 100 150 200 250 0 2 4 time (s) relative to event NSS start speed−up (# events = 522) BU TD −10 −5 0 5 10 0.5 1 1.5 horizontal eye position H eye pos 160 320 480 vertical eye position V eye pos 120 240 360 speedometer start speed−up (discrete events) speedometer (continuous signal) 0 20 40 60 80 100 120 140 eye position eye position eye position eye position eye position eye position eye position eye position eye position eye position 113.41s 114.41s 115.42s TD @ 216.08s TD @ 216.68s TD @ 217.28s BU @ 216.08s BU @ 216.68s BU @ 217.28s NSS BU model prediction strength 0 2 4 NSS TD model prediction strength time (s) in video game session 0 50 100 150 200 250 300 0 2 4 time (s) relative to event NSS target destroyed (# events = 328) BU TD −10 −5 0 5 10 0 1 2 horizontal eye position H eye pos 160 320 480 vertical eye position V eye pos 120 240 360 # targets destroyed target destroyed (discrete events) # targets destroyed (continuous signal) 2 4 6 8 10 12 eye position eye position eye position eye position eye position eye position eye position eye position eye position eye position 216.08s 216.68s 217.28s TD @ 130.26s TD @ 131.77s TD @ 133.27s BU @ 130.26s BU @ 131.77s BU @ 133.27s NSS BU model prediction strength 0 2 4 NSS TD model prediction strength time (s) in video game session 0 50 100 150 200 0 2 4 time (s) relative to event NSS missile fired (# events = 658) BU TD −10 −5 0 5 10 0 1 horizontal eye position H eye pos 160 320 480 vertical eye position V eye pos 120 240 360 # missiles fired missile fired (discrete events) # missiles fired (continuous signal) 5 10 15 eye position eye position eye position eye position eye position eye position eye position eye position eye position eye position 130.26s 131.77s 133.27s Game frames surrounding event at times t-δ, t, t+δ Game events extracted from a continuous signal across the full game session Eye position (in screen coordinates) recorded from observer playing the game Prediction strength of BU and TD models in predicting observers’ eye position Event-locked prediction strength of BU and TD models, averaged across all events of a given type shown here for a single example event shown here for a single example session shown here for a single example session shown here for a single example session shown here for all events of a given type shaded area represents 98% confidence interval BU prediction map of gaze position at times t-δ, t, t+δ, relative to an event time shown here for a single example event TD prediction map of gaze position at times t-δ, t, t+δ, relative to an event time shown here for a single example event (a) (b) (c) (d) (e) (f) (g) Figure 2: Event-locked analysis of agreement between model-predicted attention maps and observed human eye position. See Section 4 for details. 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 BU&TD NSS template match strength frequency (a) detector histograms for "missile fired" events non−events events 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 eye position template match strength frequency non−events events 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 BU&TD maps template match strength frequency non−events events 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 false positives true positives (b) ROC for "missile fired" events BU&TD NSS: AUC=0.837; d’=1.476 eye position: AUC=0.772; d’=1.093 BU&TD maps: AUC=0.585; d’=0.287 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 recall precision (c) precision/recall for "missile fired" events BU&TD NSS: max(F1)=0.373 eye position: max(F1)=0.278 BU&TD maps: max(F1)=0.153 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 BU&TD NSS template match strength frequency (d) detector histograms for "target destroyed " events non−events events 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 eye position template match strength frequency non−events events 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 BU&TD maps template match strength frequency non−events events 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 false positives true positives (e) ROC for "target destroyed " events BU&TD NSS: AUC=0.891; d’=1.891 eye position: AUC=0.807; d’=1.391 BU&TD maps: AUC=0.685; d’=0.576 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 recall precision (f) precision/recall for "target destroyed " events BU&TD NSS: max(F1)=0.377 eye position: max(F1)=0.250 BU&TD maps: max(F1)=0.142 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 BU&TD NSS template match strength frequency (g) detector histograms for "start speed−up" events non−events events 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 eye position template match strength frequency non−events events 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 BU&TD maps template match strength frequency non−events events 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 false positives true positives (h) ROC for "start speed−up" events BU&TD NSS: AUC=0.622; d’=0.445 eye position: AUC=0.600; d’=0.387 BU&TD maps: AUC=0.502; d’=−0.152 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 recall precision (i) precision/recall for "start speed−up" events BU&TD NSS: max(F1)=0.173 eye position: max(F1)=0.159 BU&TD maps: max(F1)=0.126 Figure 3: Signal detection results of using event-locked signatures to detect visual events in video game frame sequences. See Section 5 for details. model maps derived from that frame. Computing the NSS simply involves normalizing the model prediction map to have a mean of zero and a variance of one, and then finding the value in that normalized map at the location of the human eye position. An NSS value of 0 would represent a model at chance in predicting human eye position, while an NSS value of 1 would represent a model for which human eye positions fell at locations with salience (or relevance) one standard deviation above average. Previous studies have typically used the NSS as a summary statistic to describe the predictive strength of a model across an entire sequence of fixations [6, 12]; here, we use it instead as a continuous measure of the instantaneous predictive strength of the models. 4 Event-locked analyses We annotated the video game clips with several pieces of additional information that we could use to identify interesting events (see Figure 2) which would serve as the basis for event-locked analyses. These events were selected on the basis of representing transitions between different task phases. We hypothesized that such events should correlate with changes in the relative strengths of different influences on visual attention, and that we should be able to detect such changes using the previously described models as diagnostic tools. Therefore, after annotating the video clips with the times of each event of interest, we subsequently aligned further analyses on a temporal window of -5s/+5s around each event (shaded background regions in Figure 2, rows b–e). From those windows we extract the time courses of NSS scores from the salience and relevance models and then compute the average time course across all of the windows, giving an event-locked template showing the NSS signature of that event type (Figure 2e). TG “missile fired” events. In the TG game we looked for times when the player fired missiles (Figure 2, column 1). We selected these events because they represent transitions into a unique task phase, namely the phase of direct engagement with an enemy plane. During most of the TG game playing time, the player’s primary task involves actively using the radar and direction finder to locate enemy targets; however, during the time when a missile is in flight the player’s only task is to await visual confirmation of the missile destroying its target. Figure 2, column 1, row a illustrates one of the “missile fired” events with captured video frames at -1500ms, 0ms, and +1500ms relative to the event time. Row b uses one of the 30 TG clips to show how the event times represent transitions in a continuous signal (number of missiles fired); a -5s/+5s window around each event is highlighted by the shaded background regions. These windows then propagate through our model-based analysis, where we compare the eye position traces (row c) with the maps predicted by the BU salience (row f) and TD relevance (row g) to generate a continuous sequence of NSS values for each model (row d). Finally, all of the 658 event windows are pooled and we compute the average NSS value along with a 98% confidence interval at each time point in the window, giving event-locked template NSS signatures for the “missile fired” event type (row e). Those signatures show a strong divergence in the predictiveness of the BU and TD models: outside the event window, both models are significantly but weakly predictive of observers’ eye positions, with NSS values around 0.3, while inside the event window the BU NSS score increases to an NSS value around 1.0, while the TD NSS score drops below zero for several seconds. We believe this reflects the task phase transition. In general, the TD model has learned that the radar screen and direction finder (toward the bottom left of the game screens) are usually the relevant locations, as illustrated by the elevated activity at those locations in the sample TD maps in row g. Most of the time, that is indeed a good prediction of eye position, reflected by the fact that the TD NSS scores are typically higher than the BU NSS scores outside the event window. However, within the event window, players shift their attention away from the target search task to instead follow the salient objects on the display (enemy target, the missile in flight), which is reflected in the transient upswing in BU NSS scores. TG “target destroyed” events. In the TG game we also considered times when enemy targets were destroyed (Figure 2, column 2). Like the “missile fired” events, these represent transitions between task phases, but whereas the “missile fired” represented transitions from the enemy target search phase into a direct engagement phase, the “target destroyed” events represent the reverse transition; once the player sees that the enemy target has been destroyed, he or she can quickly begin searching the radar and direction finder for the next enemy target to engage. This is reflected in the sample frames shown in Figure 2, column 2, row a, where leading up to the event (at -600ms and 0ms) the player is watching the enemy target, but by +600ms after the event the player has switched back to looking at the direction finder to find a new target. The analysis proceeds as before, using -5s/+5s windows around each of the 328 events to generate average event-locked NSS signatures for the two models (row e). These signatures represent the end of the direct engagement phase whose beginning was represented by the “missile fired” events; here, the BU NSS score reaches an even higher peak of around 1.75 within 50ms after the target being destroyed, and then quickly drops to almost zero by 600ms after the event. Conversely, the TD NSS score is below zero leading up to the event, but then quickly rebounds after the event and transiently goes above its baseline level. Again, we believe these characteristic NSS traces reflect the observer’s task transitions. NFSU “start speed-up” events. In the NFSU game, we considered times at which the player just begins recovering from a crash (Figure 2, column 3); players’ task is generally to drive as fast as possible while avoiding obstacles, but when players inevitably crash they must transiently shift to a task of trying to recover from the crash. The general driving task typically involves inspecting the horizon line and and focus of expansion for oncoming obstacles, while the crash-recovery task typically involves examining the foreground scene to determine how to get back on course. To automatically identify crash recovery phases, we extracted the speedometer value from each video game frame to form a continuous speedometer history (Figure 2, column 3, row b); we identified “start speed-up” events as upward-turning zero crossings in the acceleration, represented again by shaded background bars in the figure. Again we computed average event-locked NSS signatures for the BU and TD models from -5s/+5s windows around each of the 522 events, giving the traces in row e. These traces reveal a significant drop in TD NSS scores during the event window, but no significant change in BU NSS scores. The drop in TD NSS scores likely reflects the players’ shift of attention away from the usual relevant locations (horizon line, focus of expansion) and toward other regions relevant to the crash-recovery task. However, the lack of change in BU NSS scores indicates that the locations attended during crash recovery where neither more nor less salient than locations attended in general; together, these results suggest that during crash recovery players’ attention is more strongly driven by some influence that is not captured well by either of the current BU and TD models. 5 Event detectors Having seen that critical game events are linked with highly significant signatures in the time course of BU and TD model predictiveness, we next asked whether these signatures could be used in turn to predict the events themselves. To test this question, we built event-locked “detector” templates from three sources (see Figure 1): (1) the raw BU and TD prediction maps (which carry explicit information only from the visual input image); (2) the raw eye position traces (which carry explicit information only from the player’s behavior); and (3) the BU and TD NSS scores, which represent a fusion of information from the image (BU and TD maps) and from the observer (eye position). For each of these detector types and for each event type, we compute event-locked signatures just as described in the previous section. For the BU and TD NSS scores, this is exactly what is represented in Figure 2, row e, and for the other two detector types the analysis is analogous. For the BU and TD prediction maps, we compute the event-locked average BU and TD prediction map at each time point within the event window, and for the eye position traces we compute the event-locked average x and y eye position coordinate at each time point. Thus we have signatures for how each of these detector signals is expected to look during the critical event intervals. Next, we go back to the original detector traces (that is, the raw eye position traces as in Figure 2 row c, or the raw BU and TD maps as in rows f and g, or the raw BU and TD NSS scores as in row d). At each point in those original traces, we compute the correlation coefficient between a temporal window in the trace and the corresponding event-locked detector signature. To combine each pair of correlation coefficients (from BU and TD maps, or from BU and TD NSS, or from x and y eye position) into a single match strength, we scale the individual correlation coefficients to a range of [0...1] and then multiply, to produce a soft logical “and” operation, where both components must have high values in order to produce a high output: BU,TD maps match strength = r ⟨BU⟩event,BU ·r ⟨TD⟩event,TD (1) eye position match strength = r ⟨x⟩event, x ·r ⟨y⟩event,y (2) BU,TD NSS match strength = r ⟨NSSBU⟩event,NSSBU ·r ⟨NSSTD⟩event,NSSTD , (3) where ⟨·⟩event represents the event-locked template for that signal, and r(·,·) represents the correlation coefficient between the two sequences of values, rescaled from the natural [-1...1] range to a [0...1] range. This yields continuous traces of match strength between the event detector templates and the current signal values, for each video game frame in the data set. Finally, we adopt a signal detection approach. For each event type, we label every video frame as “during event” if it falls within a -500ms/+500ms window around the event instant, and label it as “during non-event” otherwise. Then we ask how well the match strengths can predict the label, for each of the three detector types (BU and TD maps alone, eye position alone, or BU and TD NSS). Figure 3 shows the results using several signal detection metrics. Each row represents one of the three event types (“missile fired,” “target destroyed,” and “start speed-up”). The first column (panels a, d, and g) shows the histograms of the match strength values during events and during nonevents, for each of the three detector types; this gives a qualitative sense for how well each detector can distinguish events from non-events. The strongest separation between events and non-events is clearly obtained by the BU&TD NSS and eye position detectors for the “missile fired” and “target destroyed” events. Panels b, e, and h show ROC curves for each detector type and event type, along with values for area-under-the-curve (AUC) and d-prime (d’); panels c, f, and i show precision/recall curves with values with for the maximum F1 measure along the curve (F1 = (2· p·r)/(p+r), where p and r represent precision and recall). Each metric reflects the same qualitative trends. The highest scores overall occur for “target destroyed” events, followed by “missile fired” and “start speedup” events. Within each event type, the highest scores are obtained by the BU&TD NSS detector (representing fused image/behavioral information), followed by the eye position detector (behavioral information only) and then the BU&TD maps detector (image information only). 6 Discussion and Conclusion Our contributions here are twofold: First, we reported several instances in which the degree of correspondence between computational models of attention and human eye position varies systematically as a function of the current task phase. This finding suggests a direct means for integrating low-level computational models of visual attention with higher-level models of general cognition and task performance: the current task state could be linked through a weight matrix to determine the degree to which competing low-level signals may influence overall system behavior. Second, we reported that variations in the predictive strength of the salience and relevance models are systematic enough that the signals can be used to form template-based detectors of the key game events. Here, the detection is based on signals that represent a fusion of image-derived information (salience/relevance maps) with observer-derived behavior (eye position), and we found that such a combined signal is more powerful than a signal based on image-derived or observer-derived information alone. For event-detection or object-detection applications, this approach may have the advantage of being more generally applicable than a pure computer vision approach (which might require development of algorithms specifically tailored to the object or event of interest), by virtue of its reliance on human/model information fusion. Conversely, the approach of deriving human behavioral information only from eye movements has the advantage of being less invasive and cumbersome than other neurally-based event-detection approaches using EEG or fMRI [13]. Further, although an eye tracker’s x/y traces amounts to less raw information than EEG’s dozens of leads or fMRI’s 10,000s of voxels, the eye-tracking signals also contain a denser and less redundant representation of cognitive information, as they are a manifestation of whole-brain output. Together, these advantages could make our proposed method a useful approach in a number of applications. References [1] L. Itti and P. Baldi. A principled approach to detecting surprising events in video. In Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 631–637, San Siego, CA, Jun 2005. [2] V. Navalpakkam and L. Itti. Modeling the influence of task on attention. Vision Research, 45(2):205–231, January 2005. [3] A. Torralba, A. Oliva, M.S. Castelhano, and J.M. Henderson. Contextual guidance of eye movements and attention in real-world scenes: the role of global features in object search. Psychological Review, 113(4):766–786, October 2006. [4] L. Itti, C. Koch, and E. Niebur. A model of saliency-based visual attention for rapid scene analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, 20(11):1254–1259, November 1998. [5] D. Parkhurst, K. Law, and E. Niebur. Modeling the role of salience in the allocation of overt visual attention. Vision Research, 42(1):107–123, 2002. [6] R.J. Peters, A. Iyer, L. Itti, and C. Koch. Components of bottom-up gaze allocation in natural images. Vision Research, 45(18):2397–2416, 2005. [7] R. Carmi and L. Itti. Visual causes versus correlates of attentional selection in dynamic scenes. Vision Research, 46(26):4333–4345, Dec 2006. [8] R.J. Peters and L. Itti. Computational mechanisms for gaze direction in interactive visual environments. In Proc. ACM Eye Tracking Research and Applications, pages 27–32, Mar 2006. [9] R.J. Peters and L. Itti. Beyond bottom-up: Incorporating task-dependent influences into a computational model of spatial attention. In Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2007), Minneapolis, MN, Jun 2007. [10] M. Hayhoe and D. Ballard. Eye movements in natural behavior. Trends in Cognitive Sciences, 9(4):188– 194, April 2005. [11] J. Theeuwes, A.F. Kramer, S. Hahn, D.E. Irwin, and G.J. Zelinsky. Influence of attentional capture on oculomotor control. Journal of Experimental Psychology—Human Perception and Performance, 25(6):1595– 1608, December 1999. [12] W. Einhauser, W. Kruse, K.P. Hoffmann, and P. Konig. Differences of monkey and human overt attention under natural conditions. Vision Research, 46(8-9):1194–1209, April 2006. [13] A.D. Gerson, L.C. Parra, and P. Sajda. Cortically coupled computer vision for rapid image search. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 14(2):174–179, June 2006.
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Multi-task Gaussian Process Prediction Edwin V. Bonilla, Kian Ming A. Chai, Christopher K. I. Williams School of Informatics, University of Edinburgh, 5 Forrest Hill, Edinburgh EH1 2QL, UK edwin.bonilla@ed.ac.uk, K.M.A.Chai@sms.ed.ac.uk, c.k.i.williams@ed.ac.uk Abstract In this paper we investigate multi-task learning in the context of Gaussian Processes (GP). We propose a model that learns a shared covariance function on input-dependent features and a “free-form” covariance matrix over tasks. This allows for good flexibility when modelling inter-task dependencies while avoiding the need for large amounts of data for training. We show that under the assumption of noise-free observations and a block design, predictions for a given task only depend on its target values and therefore a cancellation of inter-task transfer occurs. We evaluate the benefits of our model on two practical applications: a compiler performance prediction problem and an exam score prediction task. Additionally, we make use of GP approximations and properties of our model in order to provide scalability to large data sets. 1 Introduction Multi-task learning is an area of active research in machine learning and has received a lot of attention over the past few years. A common set up is that there are multiple related tasks for which we want to avoid tabula rasa learning by sharing information across the different tasks. The hope is that by learning these tasks simultaneously one can improve performance over the “no transfer” case (i.e. when each task is learnt in isolation). However, as pointed out in [1] and supported empirically by [2], assuming relatedness in a set of tasks and simply learning them together can be detrimental. It is therefore important to have models that will generally benefit related tasks and will not hurt performance when these tasks are unrelated. We investigate this in the context of Gaussian Process (GP) prediction. We propose a model that attempts to learn inter-task dependencies based solely on the task identities and the observed data for each task. This contrasts with approaches in [3, 4] where task-descriptor features t were used in a parametric covariance function over different tasks—such a function may be too constrained by both its parametric form and the task descriptors to model task similarities effectively. In addition, for many real-life scenarios task-descriptor features are either unavailable or difficult to define correctly. Hence we propose a model that learns a “free-form” task-similarity matrix, which is used in conjunction with a parameterized covariance function over the input features x. For scenarios where the number of input observations is small, multi-task learning augments the data set with a number of different tasks, so that model parameters can be estimated more confidently; this helps to minimize over-fitting. In our model, this is achieved by having a common covariance function over the features x of the input observations. This contrasts with the semiparametric latent factor model [5] where, with the same set of input observations, one has to estimate the parameters of several covariance functions belonging to different latent processes. For our model we can show the interesting theoretical property that there is a cancellation of intertask transfer in the specific case of noise-free observations and a block design. We have investigated both gradient-based and EM-based optimization of the marginal likelihood for learning the hyperparameters of the GP. Finally, we make use of GP approximations and properties of our model in order to scale our approach to large multi-task data sets, and evaluate the benefits of our model on two practical multi-task applications: a compiler performance prediction problem and a exam score prediction task. The structure of the paper is as follows: in section 2 we outline our model for multi-task learning, and discuss some approximations to speed up computations in section 3. Related work is described in section 4. We describe our experimental setup in section 5 and give results in section 6. 2 The Model Given a set X of N distinct inputs x1, . . . , xN we define the complete set of responses for M tasks as y = (y11, . . . , yN1, . . . , y12, . . . , yN2, . . . , y1M, . . . , yNM)T, where yil is the response for the lth task on the ith input xi. Let us also denote the N × M matrix Y such that y = vec Y . Given a set of observations yo, which is a subset of y, we want to predict some of the unobserved response-values yu at some input locations for certain tasks. We approach this problem by placing a GP prior over the latent functions {fl} so that we directly induce correlations between tasks. Assuming that the GPs have zero mean we set ⟨fl(x)fk(x′)⟩= Kf lkkx(x, x′) yil ∼N(fl(xi), σ2 l ), (1) where Kf is a positive semi-definite (PSD) matrix that specifies the inter-task similarities, kx is a covariance function over inputs, and σ2 l is the noise variance for the lth task. Below we focus on stationary covariance functions kx; hence, to avoid redundancy in the parametrization, we further let kx be only a correlation function (i.e. it is constrained to have unit variance), since the variance can be explained fully by Kf. The important property of this model is that the joint Gaussian distribution over y is not blockdiagonal wrt tasks, so that observations of one task can affect the predictions on another task. In [4, 3] this property also holds, but instead of specifying a general PSD matrix Kf, these authors set Kf lk = kf(tl, tk), where kf(·, ·) is a covariance function over the task-descriptor features t. One popular setup for multi-task learning is to assume that tasks can be clustered, and that there are inter-task correlations between tasks in the same cluster. This can be easily modelled with a general task-similarity Kf matrix: if we assume that the tasks are ordered with respect to the clusters, then Kf will have a block diagonal structure. Of course, as we are learning a “free form” Kf the ordering of the tasks is irrelevant in practice (and is only useful for explanatory purposes). 2.1 Inference Inference in our model can be done by using the standard GP formulae for the mean and variance of the predictive distribution with the covariance function given in equation (1). For example, the mean prediction on a new data-point x∗for task l is given by ¯fl(x∗) = (kf l ⊗kx ∗)T Σ−1y Σ = Kf ⊗Kx + D ⊗I (2) where ⊗denotes the Kronecker product, kf l selects the lth column of Kf, kx ∗is the vector of covariances between the test point x∗and the training points, Kx is the matrix of covariances between all pairs of training points, D is an M × M diagonal matrix in which the (l, l)th element is σ2 l , and Σ is an MN × MN matrix. In section 2.3 we show that when there is no noise in the data (i.e. D = 0), there will be no transfer between tasks. 2.2 Learning Hyperparameters Given the set of observations yo, we wish to learn the parameters θx of kx and the matrix Kf to maximize the marginal likelihood p(yo|X, θx, Kf). One way to achieve this is to use the fact that y|X ∼N(0, Σ). Therefore, gradient-based methods can be readily applied to maximize the marginal likelihood. In order to guarantee positive-semidefiniteness of Kf, one possible parametrization is to use the Cholesky decomposition Kf = LLT where L is lower triangular. Computing the derivatives of the marginal likelihood with respect to L and θx is straightforward. A drawback of this approach is its computational cost as it requires the inversion of a matrix of potential size MN × MN (or solving an MN × MN linear system) at each optimization step. Note, however, that one only needs to actually compute the Gram matrix and its inverse at the visible locations corresponding to yo. Alternatively, it is possible to exploit the Kronecker product structure of the full covariance matrix as in [6], where an EM algorithm is proposed such that learning of θx and Kf in the M-step is decoupled. This has the advantage that closed-form updates for Kf and D can be obtained (see equation (5)), and that Kf is guaranteed to be positive-semidefinite. The details of the EM algorithm are as follows: Let f be the vector of function values corresponding to y, and similarly for F wrt Y . Further, let y·l denote the vector (y1l, . . . , yNl)T and similarly for f ·l. Given the missing data, which in this case is f, the complete-data log-likelihood is Lcomp = −N 2 log |Kf| −M 2 log |Kx| −1 2 tr h Kf−1 F T (Kx)−1 F i −N 2 M X l=1 log σ2 l −1 2 tr (Y −F)D−1(Y −F)T −MN 2 log 2π (3) from which we have following updates: bθx = arg min θx N log D F T (Kx(θx))−1 F E + M log |Kx(θx)| (4) bKf = N −1 F T Kx(c θx) −1 F bσ2 l = N −1 D (y·l −f ·l)T (y·l −f ·l) E (5) where the expectations ⟨·⟩are taken with respect to p f|yo, θx, Kf , and b· denotes the updated parameters. For clarity, let us consider the case where yo = y, i.e. a block design. Then p f|y, θx, Kf = N (Kf ⊗Kx)Σ−1y, (Kf ⊗Kx) −(Kf ⊗Kx)Σ−1(Kf ⊗Kx) . We have seen that Σ needs to be inverted (in time O(M 3N 3)) for both making predictions and learning the hyperparameters (when considering noisy observations). This can lead to computational problems if MN is large. In section 3 we give some approximations that can help speed up these computations. 2.3 Noiseless observations and the cancellation of inter-task transfer One particularly interesting case to consider is noise-free observations at the same locations for all tasks (i.e. a block-design) so that y|X ∼Normal(0, Kf ⊗Kx). In this case maximizing the marginal likelihood p(y|X) wrt the parameters θx of kx reduces to maximizing −M log |Kx| − N log |Y T (Kx)−1Y |, an expression that does not depend on Kf. After convergence we can obtain Kf as ˆKf = 1 N Y T (Kx)−1Y . The intuition behind is this: The responses Y are correlated via Kf and Kx. We can learn Kf by decorrelating Y with (Kx)−1 first so that only correlation with respect to Kf is left. Then Kf is simply the sample covariance of the de-correlated Y . Unfortunately, in this case there is effectively no transfer between the tasks (given the kernels). To see this, consider making predictions at a new location x∗for all tasks. We have (using the mixedproduct property of Kronecker products) that f(x∗) = Kf ⊗kx ∗ T Kf ⊗Kx−1 y (6) = (Kf)T ⊗(kx ∗)T (Kf)−1 ⊗(Kx)−1 y (7) = Kf(Kf)−1 ⊗ (kx ∗)T(Kx)−1 y (8) = (kx ∗)T(Kx)−1y·1 ... (kx ∗)T(Kx)−1y·M , (9) and similarly for the covariances. Thus, in the noiseless case with a block design, the predictions for task l depend only on the targets y·l. In other words, there is a cancellation of transfer. One can in fact generalize this result to show that the cancellation of transfer for task l does still hold even if the observations are only sparsely observed at locations X = (x1, . . . , xN) on the other tasks. After having derived this result we learned that it is known as autokrigeability in the geostatistics literature [7], and is also related to the symmetric Markov property of covariance functions that is discussed in [8]. We emphasize that if the observations are noisy, or if there is not a block design, then this result on cancellation of transfer will not hold. This result can also be generalized to multidimensional tensor product covariance functions and grids [9]. 3 Approximations to speed up computations The issue of dealing with large N has been much studied in the GP literature, see [10, ch. 8] and [11] for overviews. In particular, one can use sparse approximations where only Q out of N data points are selected as inducing inputs[11]. Here, we use the Nystr¨om approximation of Kx in the marginal likelihood, so that Kx ≈eKx def= Kx ·I(Kx II)−1Kx I·, where I indexes Q rows/columns of Kx. In fact for the posterior at the training points this result is obtained from both the subset of regressors (SoR) and projected process (PP) approximations described in [10, ch. 8]. Specifying a full rank Kf requires M(M + 1)/2 parameters, and for large M this would be a lot of parameters to estimate. One parametrization of Kf that reduces this problem is to use a PPCA model [12] Kf ≈eKf def= UΛU T + s2IM, where U is an M × P matrix of the P principal eigenvectors of Kf, Λ is a P × P diagonal matrix of the corresponding eigenvalues, and s2 can be determined analytically from the eigenvalues of Kf (see [12] and references therein). For numerical stability, we may further use the incomplete-Cholesky decomposition setting UΛU T = ˜L˜LT, where ˜L is a M × P matrix. Below we consider the case s = 0, i.e. a rank-P approximation to Kf. Applying both approximations to get Σ ≈eΣ def= ˜Kf ⊗˜Kx + D ⊗IN, we have, after using the Woodbury identity, eΣ−1 = ∆−1 −∆−1B I ⊗Kx II + BT∆−1B −1 BT∆−1 where B def= (˜L ⊗ Kx ·I), and ∆ def= D ⊗IN is a diagonal matrix. As ˜Kf ⊗˜Kx has rank PQ, we have that computation of ˜Σ−1y takes O(MNP 2Q2). For the EM algorithm, the approximation of eKx poses a problem in (4) because for the rank-deficient matrix eKx, its log-determinant is negative infinity, and its matrix inverse is undefined. We overcome this by considering eKx = limξ→0(Kx ·I(Kx II)−1Kx I·+ξ2I), so that we solve an equivalent optimization problem where the log-determinant is replaced by the well-defined log |Kx I·Kx ·I| −log |Kx II|, and the matrix inverse is replaced by the pseudo-inverse. With these approximations the computational complexity of hyperparameter learning can be reduced to O(MNP 2Q2) per iteration for both the Cholesky and EM methods. 4 Related work There has been a lot of work in recent years on multi-task learning (or inductive transfer) using methods such as Neural Networks, Gaussian Processes, Dirichlet Processes and Support Vector Machines, see e.g. [2, 13] for early references. The key issue concerns what properties or aspects should be shared across tasks. Within the GP literature, [14, 15, 16, 17, 18] give models where the covariance matrix of the full (noiseless) system is block diagonal, and each of the M blocks is induced from the same kernel function. Under these models each y·i is conditionally independent, but inter-task tying takes place by sharing the kernel function across tasks. In contrast, in our model and in [5, 3, 4] the covariance is not block diagonal. The semiparametric latent factor model (SLFM) of Teh et al [5] involves having P latent processes (where P ≤M) and each of these latent processes has its own covariance function. The noiseless outputs are obtained by linear mixing of these processes with a M × P matrix Φ. The covariance matrix of the system under this model has rank at most PN, so that when P < M the system corresponds to a degenerate GP. Our model is similar to [5] but simpler, in that all of the P latent processes share the same covariance function; this reduces the number of free parameters to be fitted and should help to minimize overfitting. With a common covariance function kx, it turns out that Kf is equal to ΦΦT, so a Kf that is strictly positive definite corresponds to using P = M latent processes. Note that if P > M one can always find an M × M matrix Φ′ such that Φ′Φ′T = ΦΦT. We note also that the approximation methods used in [5] are different to ours, and were based on the subset of data (SoD) method using the informative vector machine (IVM) selection heuristic. In the geostatistics literature, the prior model for f· given in eq. (1) is known as the intrinsic correlation model [7], a specific case of co-kriging. A sum of such processes is known as the linear coregionalization model (LCM) [7] for which [6] gives an EM-based algorithm for parameter estimation. Our model for the observations corresponds to an LCM model with two processes: the process for f· and the noise process. Note that SLFM can also be seen as an instance of the LCM model. To see this, let Epp be a P ×P diagonal matrix with 1 at (p, p) and zero elsewhere. Then we can write the covariance in SLFM as (Φ⊗I)(PP p=1 Epp⊗Kx p )(Φ⊗I)T = PP p=1(ΦEppΦT)⊗Kx p , where ΦEppΦT is of rank 1. Evgeniou et al. [19] consider methods for inducing correlations between tasks based on a correlated prior over linear regression parameters. In fact this corresponds to a GP prior using the kernel k(x, x′) = xT Ax′ for some positive definite matrix A. In their experiments they use a restricted form of Kf with Kf lk = (1 −λ) + λMδlk (their eq. 25), i.e. a convex combination of a rank-1 matrix of ones and a multiple of the identity. Notice the similarity to the PPCA form of Kf given in section 3. 5 Experiments We evaluate our model on two different applications. The first application is a compiler performance prediction problem where the goal is to predict the speed-up obtained in a given program (task) when applying a sequence of code transformations x. The second application is an exam score prediction problem where the goal is to predict the exam score obtained by a student x belonging to a specific school (task). In the sequel, we will refer to the data related to the first problem as the compiler data and the data related to the second problem as the school data. We are interested in assessing the benefits of our approach not only with respect to the no-transfer case but also with respect to the case when a parametric GP is used on the joint input-dependent and task-dependent space as in [3]. To train the parametric model note that the parameters of the covariance function over task descriptors kf(t, t′) can be tuned by maximizing the marginal likelihood, as in [3]. For the free-form Kf we initialize this (given kx(·, ·)) by using the noise-free expression ˆKf = 1 N Y T (Kx)−1Y given in section 2.3 (or the appropriate generalization when the design is not complete). For both applications we have used a squared-exponential (or Gaussian) covariance function kx and a non-parametric form for Kf. Where relevant the parametric covariance function kf was also taken to be of squared-exponential form. Both kx and kf used an automatic relevance determination (ARD) parameterization, i.e. having a length scale for each feature dimension. All the length scales in kx and kf were initialized to 1, and all σ2 l were constrained to be equal for all tasks and initialized to 0.01. 5.1 Description of the Data Compiler Data. This data set consists of 11 C programs for which an exhaustive set of 88214 sequences of code transformations have been applied and their corresponding speed-ups have been recorded. Each task is to predict the speed-up on a given program when applying a specific transformation sequence. The speed-up after applying a transformation sequence on a given program is defined as the ratio of the execution time of the original program (baseline) over the execution time of the transformed program. Each transformation sequence is described as a 13-dimensional vector x that records the absence/presence of one-out-of 13 single transformations. In [3] the taskdescriptor features (for each program) are based on the speed-ups obtained on a pre-selected set of 8 transformations sequences, so-called “canonical responses”. The reader is referred to [3, section 3] for a more detailed description of the data. School Data. This data set comes from the Inner London Education Authority (ILEA) and has been used to study the effectiveness of schools. It is publicly available under the name of “school effectiveness” at http://www.cmm.bristol.ac.uk/learning-training/ multilevel-m-support/datasets.shtml. It consists of examination records from 139 secondary schools in years 1985, 1986 and 1987. It is a random 50% sample with 15362 students. This data has also been used in the context of multi-task learning by Bakker and Heskes [20] and Evgeniou et al. [19]. In [20] each task is defined as the prediction of the exam score of a student belonging to a specific school based on four student-dependent features (year of the exam, gender, VR band and ethnic group) and four school-dependent features (percentage of students eligible for free school meals, percentage of students in VR band 1, school gender and school denomination). For comparison with [20, 19] we evaluate our model following the set up described above and similarly, we have created dummy variables for those features that are categorical forming a total of 19 student-dependent features and 8 school-dependent features. However, we note that school-descriptor features such as the percentage of students eligible for free school meals and the percentage of students in VR band 1 actually depend on the year the particular sample was taken. It is important to emphasize that for both data sets there are task-descriptor features available. However, as we have described throughout this paper, our approach learns task similarity directly without the need for task-dependent features. Hence, we have neglected these features in the application of our free-form Kf method. 6 Results For the compiler data we have M = 11 tasks and we have used a Cholesky decomposition Kf = LLT . For the school data we have M = 139 tasks and we have preferred a reduced rank parameterization of Kf ≈eKf = ˜L˜LT , with ranks 1, 2, 3 and 5. We have learnt the parameters of the models so as to maximize the marginal likelihood p(yo|X, Kf, θx) using gradient-based search in MATLAB with Carl Rasmussen’s minimize.m. In our experiments this method usually outperformed EM in the quality of solutions found and in the speed of convergence. Compiler Data: For this particular application, in a real-life scenario it is critical to achieve good performance with a low number of training data-points per task given that a training data-point requires the compilation and execution of a (potentially) different version of a program. Therefore, although there are a total of 88214 training points per program we have followed a similar set up to [3] by considering N = 16, 32, 64 and 128 transformation sequences per program for training. All the M = 11 programs (tasks) have been used for training, and predictions have been done at the (unobserved) remaining 88214 −N inputs. For comparison with [3] the mean absolute error (between the actual speed-ups of a program and the predictions) has been used as the measure of performance. Due to the variability of the results depending on training set selection we have considered 10 different replications. Figure 1 shows the mean absolute errors obtained on the compiler data for some of the tasks (top row and bottom left) and on average for all the tasks (bottom right). Sample task 1 (histogram) is an example where learning the tasks simultaneously brings major benefits over the no transfer case. Here, multi-task GP (transfer free-form) provides a reduction on the mean absolute error of up to 6 times. Additionally, it is consistently (although only marginally) superior to the parametric approach. For sample task 2 (fir), our approach not only significantly outperforms the no transfer case but also provides greater benefits over the parametric method (which for N = 64 and 128 is worse than no transfer). Sample task 3 (adpcm) is the only case out of all 11 tasks where our approach degrades performance, although it should be noted that all the methods perform similarly. Further analysis of the data indicates that learning on this task is hard as there is a lot of variability that cannot be explained by the 1-out-of-13 encoding used for the input features. Finally, for all tasks on average (bottom right) our approach brings significant improvements over single task learning and consistently outperforms the parametric method. For all tasks except one our model provides better or roughly equal performance than the non-transfer case and the parametric model. School Data: For comparison with [20, 19] we have made 10 random splits of the data into training (75%) data and test (25%) data. Due to the categorical nature of the data there are a maximum of N = 202 different student-dependent feature vectors x. Given that there can be multiple observations of a target value for a given task at a specific input x, we have taken the mean of these observations and corrected the noise variances by dividing them over the corresponding number of observations. As in [19], the percentage explained variance is used as the measure of performance. This measure can be seen as the percentage version of the well known coefficient of determination r2 between the actual target values and the predictions. 16 32 64 128 0 0.04 0.08 0.12 0.16 0.2 SAMPLE TASK 1 N MAE NO TRANSFER TRANSFER PARAMETRIC TRANSFER FREE−FORM 16 32 64 128 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 SAMPLE TASK 2 N MAE NO TRANSFER TRANSFER PARAMETRIC TRANSFER FREE−FORM (a) (b) 16 32 64 128 0 0.02 0.04 0.06 0.08 0.1 0.12 SAMPLE TASK 3 N MAE NO TRANSFER TRANSFER PARAMETRIC TRANSFER FREE−FORM 16 32 64 128 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 ALL TASKS N MAE NO TRANSFER TRANSFER PARAMETRIC TRANSFER FREE−FORM (c) (d) Figure 1: Panels (a), (b) and (c) show the average mean absolute error on the compiler data as a function of the number of training points for specific tasks. no transfer stands for the use of a single GP for each task separately; transfer parametric is the use of a GP with a joint parametric (SE) covariance function as in [3]; and transfer free-form is multi-task GP with a “free form” covariance matrix over tasks. The error bars show ± one standard deviation taken over the 10 replications. Panel (d) shows the average MAE over all 11 tasks, and the error bars show the average of the standard deviations over all 11 tasks. The results are shown in Table 1; note that larger figures are better. The parametric result given in the table was obtained from the school-descriptor features; in the cases where these features varied for a given school over the years, an average was taken. The results show that better results can be obtained by using multi-task learning than without. For the non-parametric Kf, we see that the rank-2 model gives best performance. This performance is also comparable with the best (29.5%) found in [20]. We also note that our no transfer result of 21.1% is much better than the baseline of 9.7% found in [20] using neural networks. no transfer parametric rank 1 rank 2 rank 3 rank 5 21.05 (1.15) 31.57 (1.61) 27.02 (2.03) 29.20 (1.60) 24.88 (1.62) 21.00 (2.42) Table 1: Percentage variance explained on the school dataset for various situations. The figures in brackets are standard deviations obtained from the ten replications. On the school data the parametric approach for Kf slightly outperforms the non-parametric method, probably due to the large size of this matrix relative to the amount of data. One can also run the parametric approach creating a task for every unique school-features descriptor1; this gives rise to 288 tasks rather than 139 schools, and a performance of 33.08% (±1.57). Evgeniou et al [19] use a linear predictor on all 8 features (i.e. they combine both student and school features into x) and then introduce inter-task correlations as described in section 4. This approach uses the same information as our 288 task case, and gives similar performance of around 34% (as shown in Figure 3 of [19]). 1Recall from section 5.1 that the school features can vary over different years. 7 Conclusion In this paper we have described a method for multi-task learning based on a GP prior which has inter-task correlations specified by the task similarity matrix Kf. We have shown that in a noisefree block design, there is actually a cancellation of transfer in this model, but not in general. We have successfully applied the method to the compiler and school problems. An advantage of our method is that task-descriptor features are not required (c.f. [3, 4]). However, such features might be beneficial if we consider a setup where there are only few datapoints for a new task, and where the task-descriptor features convey useful information about the tasks. Acknowledgments CW thanks Dan Cornford for pointing out the prior work on autokrigeability. KMC thanks DSO NL for support. This work is supported under EPSRC grant GR/S71118/01 , EU FP6 STREP MILEPOST IST-035307, and in part by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST2002-506778. This publication only reflects the authors’ views. References [1] Jonathan Baxter. A Model of Inductive Bias Learning. JAIR, 12:149–198, March 2000. [2] Rich Caruana. Multitask Learning. Machine Learning, 28(1):41–75, July 1997. [3] Edwin V. Bonilla, Felix V. Agakov, and Christopher K. I. Williams. Kernel Multi-task Learning using Task-specific Features. In Proceedings of the 11th AISTATS, March 2007. [4] Kai Yu, Wei Chu, Shipeng Yu, Volker Tresp, and Zhao Xu. Stochastic Relational Models for Discriminative Link Prediction. In NIPS 19, Cambridge, MA, 2007. MIT Press. [5] Yee Whye Teh, Matthias Seeger, and Michael I. Jordan. Semiparametric latent factor models. In Proceedings of the 10th AISTATS, pages 333–340, January 2005. [6] Hao Zhang. Maximum-likelihood estimation for multivariate spatial linear coregionalization models. Environmetrics, 18(2):125–139, 2007. [7] Hans Wackernagel. Multivariate Geostatistics: An Introduction with Applications. Springer-Verlag, Berlin, 2nd edition, 1998. [8] A. O’Hagan. A Markov property for covariance structures. Statistics Research Report 98-13, Nottingham University, 1998. [9] C. K. I. Williams, K. M. A. Chai, and E. V. Bonilla. A note on noise-free Gaussian process prediction with separable covariance functions and grid designs. Technical report, University of Edinburgh, 2007. [10] C. E. Rasmussen and C. K. I. Williams. Gaussian Processes for Machine Learning. MIT Press, Cambridge, Massachusetts, 2006. [11] Joaquin Qui˜nonero-Candela, Carl Edward Rasmussen, and Christopher K. I. Williams. Approximation Methods for Gaussian Process Regression. In Large Scale Kernel Machines. MIT Press, 2007. To appear. [12] Michael E. Tipping and Christopher M. Bishop. Probabilistic principal component analysis. Journal of the Royal Statistical Society, Series B, 61(3):611–622, 1999. [13] S. Thrun. Is Learning the n-th Thing Any Easier Than Learning the First? In NIPS 8, 1996. [14] Thomas P. Minka and Rosalind W. Picard. Learning How to Learn is Learning with Point Sets. 1999. [15] Neil D. Lawrence and John C. Platt. Learning to learn with the Informative Vector Machine. In Proceedings of the 21st International Conference on Machine Learning, July 2004. [16] Kai Yu, Volker Tresp, and Anton Schwaighofer. Learning Gaussian Processes from Multiple Tasks. In Proceedings of the 22nd International Conference on Machine Learning, 2005. [17] Anton Schwaighofer, Volker Tresp, and Kai Yu. Learning Gaussian Process Kernels via Hierarchical Bayes. In NIPS 17, Cambridge, MA, 2005. MIT Press. [18] Shipeng Yu, Kai Yu, Volker Tresp, and Hans-Peter Kriegel. Collaborative Ordinal Regression. In Proceedings of the 23rd International Conference on Machine Learning, June 2006. [19] Theodoros Evgeniou, Charles A. Micchelli, and Massimiliano Pontil. Learning Multiple Tasks with Kernel Methods. Journal of Machine Learning Research, 6:615–537, April 2005. [20] Bart Bakker and Tom Heskes. Task Clustering and Gating for Bayesian Multitask Learning. Journal of Machine Learning Research, 4:83–99, May 2003.
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Multi-Task Learning via Conic Programming Tsuyoshi Kato⋆,◦, Hisashi Kashima†, Masashi Sugiyama‡, Kiyoshi Asai⋆,⋄ ⋆Graduate School of Frontier Sciences, The University of Tokyo, ◦Institute for Bioinformatics Research and Development (BIRD), Japan Science and Technology Agency (JST) † Tokyo Research Laboratory, IBM Research, ‡ Department of Computer Science, Tokyo Institute of Technology, ⋄AIST Computational Biology Research Center, kato-tsuyoshi@cb.k.u-tokyo.ac.jp, kashi pong@yahoo.co.jp, sugi@cs.titech.ac.jp, asai@cbrc.jp Abstract When we have several related tasks, solving them simultaneously is shown to be more effective than solving them individually. This approach is called multi-task learning (MTL) and has been studied extensively. Existing approaches to MTL often treat all the tasks as uniformly related to each other and the relatedness of the tasks is controlled globally. For this reason, the existing methods can lead to undesired solutions when some tasks are not highly related to each other, and some pairs of related tasks can have significantly different solutions. In this paper, we propose a novel MTL algorithm that can overcome these problems. Our method makes use of a task network, which describes the relation structure among tasks. This allows us to deal with intricate relation structures in a systematic way. Furthermore, we control the relatedness of the tasks locally, so all pairs of related tasks are guaranteed to have similar solutions. We apply the above idea to support vector machines (SVMs) and show that the optimization problem can be cast as a second order cone program, which is convex and can be solved efficiently. The usefulness of our approach is demonstrated through simulations with protein super-family classification and ordinal regression problems. 1 Introduction In many practical situations, a classification task can often be divided into related sub-tasks. Since the related sub-tasks tend to share common factors, solving them together is expected to be more advantageous than solving them independently. This approach is called multi-task learning (MTL, a.k.a. inductive transfer or learning to learn) and has theoretically and experimentally proven to be useful [4, 5, 8]. Typically, the ‘relatedness’ among tasks is implemented as imposing the solutions of related tasks to be similar (e.g. [5]). However, the MTL methods developed so far have several limitations. First, it is often assumed that all sub-tasks are related to each other [5]. However, this may not be always true in practice—some are related but others may not be. The second problem is that the related tasks are often imposed to be close in the sense that the sum of the distances between solutions over all pairs of related tasks is upper-bounded [8] (which is often referred to as the global constraint [10]). This implies that all the solutions of related tasks are not necessarily close, but some can be quite different. In this paper, we propose a new MTL method which overcomes the above limitations. We settle the first issue by making use of a task network that describes the relation structure among tasks. This enables us to deal with intricate relation structures in a systematic way. We solve the second problem 1 by directly upper-bounding each distance between the solutions of related task pairs (which we call local constraints). We apply this ideas in the framework of support vector machines (SVMs) and show that linear SVMs can be trained via a second order cone program (SOCP) [3] in the primal. An SOCP is a convex problem and the global solution can be computed efficiently. We further show that the kernelized version of the proposed method can be formulated as a matrix-fractional program (MFP) [3] in the dual, which can be again cast as an SOCP; thus the optimization problem of the kernelized variant is still convex and the global solution can be computed efficiently. Through experiments with artificial and real-world protein super-family classification data sets, we show that the proposed MTL method compares favorably with existing MTL methods. We further test the performance of the proposed approach in ordinal regression scenarios [9], where the goal is to predict ordinal class labels such as users’ preferences (‘like’/‘neutral’/‘dislike’) or students’ grades (from ‘A’ to ‘F’). The ordinal regression problems can be formulated as a set of one-versus-one classification problems, e.g., ‘like’ vs. ‘neutral’ and ‘neutral’ vs. ‘dislike’. In ordinal regression, the relatedness among tasks is highly structured. That is, the solutions (decision boundaries) of adjacent problems are expected to be similar, but others may not be related, e.g., ‘A’ vs. ‘B’ and ‘B’ vs. ‘C’ would be related, but ‘A’ vs. ‘B’ and ‘E’ vs. ‘F’ may not be. Our experiments demonstrate that the proposed method is also useful in the ordinal regression scenarios and tends to outperform existing approaches [9, 8] 2 Problem Setting In this section, we formulate the MTL problem. Let us consider M binary classification tasks, which all share the common input-output space X × {±1}. For the time being, we assume X ⊂Rd for simplicity; later in Section 4, we extend it to reproducing kernel Hilbert spaces. Let {xt, yt}ℓ t=1 be the training set, where xt ∈X and yt ∈{±1} for t = 1, . . . , ℓ. Each data sample (xt, yt) has its target task; we denote the set of sample indices of the i-th task by Ii. We assume that each sample belongs only to a single task, i.e., the index sets are exclusive: M i=1 |Ii| = ℓand Ii ∩Ij = null, ∀i ̸= j. The goal is to learn the score function of each classification task: f i(x; wi, bi) = w⊤ i x + bi, for i = 1, . . . , M, where wi ∈Rd and bi ∈R are the model parameters of the i-th task. We assume that a task network is available. The task network describes the relationships among tasks, where each node represents a task and two nodes are connected by an edge if they are related to each other 1. We denote the edge set by E ≡{(ik, jk)}K k=1. 3 Local MTL with Task Network: Linear Version In this section, we propose a new MTL method. 3.1 Basic Idea When the relation among tasks is not available, we may just solve M penalized fitting problems individually: 1 2∥wi∥2 + Cα t∈Ii Hinge(fi(xt; wi, bi), yt), for i = 1, . . . , M, (1) where Cα ∈ R+ is a regularization constant and Hinge(·, ·) is the hinge loss function: Hinge(f, y) ≡max(1 −fy, 0). This individual approach tends to perform poorly if the number of training samples in each task is limited—the performance is expected to be improved if more training samples are available. Here, we can exploit the information of the task network. A naive 1More generally, the tasks can be related in an inhomogeneous way, i.e., the strength of the relationship among tasks can be dependent on tasks. This general setting can be similarly formulated by a weighted network, where edges are weighted according to the strength of the connections. All the discussions in this paper can be easily extended to weighted networks, but for simplicity we focus on unweighted networks. 2 idea would be to use the training samples of neighboring tasks in the task network for solving the target fitting problem. However, this does not fully make use of the network structure since there are many other indirectly connected tasks via some paths on the network. To cope with this problem, we take another approach here, which is based on the expectation that the solutions of related tasks are close to each other. More specifically, we impose the following constraint on the optimization problem (1): 1 2 ∥wik −wjk∥2 ≤ρ, for ∀k = 1, . . . , K. (2) Namely, we upper-bound each difference between the solutions of related task pairs by a positive scalar ρ ∈R+. We refer to this constraint as local constraint following [10]. Note that we do not impose a constraint on the bias parameter bi since the bias could be significantly different even among related tasks. The constraint (2) allows us to implicitly increase the number of training samples over the task network in a systematic way through the solutions of related tasks. Following the convention [8], we blend Eqs.(1) and (2) as 1 2M M i=1 ∥wi∥2 + Cα M i=1 t∈Ii Hinge(fi(xt; θ), yt) + Cρρ, (3) where Cρ is a positive trade-off parameter. Then our optimization problem is summarized as follows: Problem 1. min 1 2M M i=1 ∥wi∥2 + Cα∥ξ∥1 + Cρρ, wrt. w ∈RMd, b ∈RM, ξα ∈Rℓ +, and ρ ∈R+, subj. to 1 2 ∥wik −wjk∥2 ≤ρ, ∀k, and yt w⊤ i xt + bi ≥1 −ξα t , ∀t ∈Ii, ∀i where w ≡ w⊤ 1 , . . . , w⊤ M ⊤, and ξα ≡[ξα 1 , . . . , ξα ℓ]⊤. (4) 3.2 Primal MTL Learning by SOCP The second order cone program (SOCP) is a class of convex programs of minimizing a linear function over an intersection of second-order cones [3]: 2 Problem 2. min f ⊤z wrt z ∈Rn subj. to ∥Aiz + bi∥≤c⊤ i z + di, for i = 1, . . . , N, (5) where f ∈Rn, Ai ∈R(ni−1)×n, bi ∈Rni−1, ci ∈Rn, di ∈R. Linear programs, quadratic programs, and quadratically-constrainedquadratic programs are actually special cases of SOCPs. SOCPs are a sub-class of semidefinite programs (SDPs) [3], but SOCPs can be solved more efficiently than SDPs. Successful optimization algorithms for both SDP and SOCP are interior-point algorithms. The SDP solvers (e.g. [2]) consume O(n 2 i n2 i ) time complexity for solving Problem 2, but the SOCP-specialized solvers that directly solve Problem 2 take only O(n2 i ni) computation [7]. Thus, SOCPs can be solved more efficiently than SDPs. We can show that Problem 1 is cast as an SOCP using hyperbolic constraints [3]. Theorem 1. Problem 1 can be reduced to an SOCP and it can be solved with O((Md+ℓ) 2(Kd+ℓ)) computation. 4 Local MTL with Task Network: Kernelization The previous section showed that a linear version of the proposed MTL method can be cast as an SOCP. In this section, we show how the kernel trick could be employed for obtaining a non-linear variant. 2More generally, an SOCP can include linear equality constraints, but they can be eliminated, for example, by some projection method. 3 4.1 Dual Formulation Let Kfea be a positive semidefinite matrix with the (s, t)-th element being the inner-product of feature vectors xs and xt: Kfea s,t ≡⟨xs, xt⟩. This is a kernel matrix of feature vectors. We also introduce a kernel among tasks. Using a new K-dimensional non-negative parameter vector λ ∈ RK + , we define the kernel matrix of tasks by Knet(λ) ≡ 1 M IM + Uλ −1 , where Uλ ≡K k=1 λkUk, Uk ≡Eikik + Ejkjk −Eikjk −Ejkik, and E(i,j) ∈RM×M is the sparse matrix whose (i, j)-th element is one and all the others are zero. Note that this is the graph Laplacian kernel [11], where the k-th edge is weighted according to λ k. Let Z ∈NM×ℓbe the indicator of a task and a sample such that Zi,t = 1 if t ∈Ii and Zi,t = 0 otherwise. Then the information about the tasks are expressed by the ℓ× ℓkernel matrix Z ⊤Knet(λ) Z. We integrate the two kernel matrices Kfea and Z⊤Knet(λ) Z by Kint(λ) ≡Kfea ◦ Z⊤Knet(λ) Z , (6) where ◦denotes the Hadamard product (a.k.a element-wise product). This parameterized matrix Kint(λ) is guaranteed to be positive semidefinite [6]. Based on the above notations, the dual formulation of Problem 1 can be expressed using the parameterized integrated kernel matrix Kint(λ) as follows: Problem 3. min 1 2α⊤diag(y)Kint(λ) diag(y)α −∥α∥1, wrt. α ∈Rℓ +, and λ ∈RM + , subj. to α ≤Cα1ℓ, Z diag(y) α = 0M, ∥λ∥1 ≤Cρ. (7) We note that the solutions α and λ tend to be sparse due to the ℓ1 norm. Changing the definition of Kfea from the linear kernel to an arbitrary kernel, we can extend the proposed linear MTL method to non-linear domains. Furthermore, we can also deal with nonvectorial structured data by employing a suitable kernel such as the string kernel and the Fisher kernel. In the test stage, a new sample x in the j-th task is classified by fj(x) = ℓ t=1 M i=1 αtytkfea(xt, x)knet(i, j)Zi,t + bj, (8) where kfea(·, ·) and knet(·, ·) are the kernel functions over features and tasks, respectively. 4.2 Dual MTL Learning by SOCP Here, we show that the above dual problem can also be reduced to an SOCP. To this end, we first introduce a matrix-fractional program (MFP) [7]: Problem 4. min (F z + g)⊤P (z)−1 (F z + g) wrt. z ∈Rp + subj. to P (z) ≡P0 + p i=1 ziPi ∈Sn ++, where Pi ∈Sn +, F ∈Rn×p, and g ∈Rn. Here Sn + and Sn ++ denote the positive semidefinite cone and strictly positive definite cone of n × n matrices, respectively. Let us re-define d as the rank of the feature kernel matrix K fea. We introduce a matrix Vfea ∈Rℓ×d which decomposes the feature kernel matrix as Kfea = VfeaVfea ⊤. Define the ℓ-dimensional vectors fh ∈Rℓof the h-th feature as Vfea ≡[f1, . . . , fd] ∈Rℓ×d and the matrices Fh ≡Z diag(fh ◦ y), for h = 1, . . . , d. Using those variables, the objective function in Problem 3 can be rewritten as JD = 1 2 d h=1 α⊤F ⊤ h 1 M IM + Uλ −1 Fhα −α⊤1ℓ. (9) 4 This implies that Problem 3 can be transformed into the combination of a linear program and d MFPs. Let us further introduce the vector vk ∈RM for each edge: vk = eik−ejk, where eik is a unit vector with the ik-th element being one. Let Vlap be a matrix defined by Vlap = [v1, . . . , vK] ∈RM×K. Then we can re-express the graph Lagrangian matrix of tasks as Uλ = V lap diag(λ)Vlap ⊤. Given the fact that an MFP can be reduced to an SOCP [7], we can reduce Problem 3 to the following SOCP: Problem 5. min −1⊤ ℓα + 1 2 d h=1 s0,h + s1,h + · · · + sK,h, (10) wrt s0,h ∈R, sk,h ∈R, u0,h ∈RM, uh = [u1,h, . . . , uK,h]⊤∈RK ∀k, ∀h (11) λ ∈RK + , α ∈Rℓ +, (12) subj. to α ≤Cα1ℓ, Z diag(y) α = 0M, 1⊤ Kλ ≤Cρ, (13) M −1/2u0,h + Vlapuh = Fhα, 2u0,h s0,h −1 ≤s0,h + 1, ∀h (14) 2uk,h sk,h −λk ≤sk,h + λk ∀k, ∀h (15) Consequently, we obtain the following result: Theorem 2. The dual problem of CoNs learning (Problem 3) can be reduced to the SOCP in Problem 5 and it can be solved with O((Kd + ℓ)2((M + K)d + ℓ)) computation. 5 Discussion In this section, we discuss the properties of the proposed MTL method and the relation to existing methods. MTL with Common Bias A possible variant of the proposed MTL method would be to share the common bias parameter with all tasks (i.e. b1 = b2 = · · · = bM). The idea is expected to be useful particularly when the number of samples in each task is very small. We can also apply the common bias idea in the kernelized version just by replacing the constraint Z diag(y)α = 0 M in Problem 3 by y⊤α = 0. Global vs. Local Constraints Micchelli and Pontil [8] have proposed a related MTL method which upper-bounds the sum of the differences of K related task pairs, i.e., 1 2 K k=1 ∥wik −wjk∥2 ≤ρ. We call it the global constraint. This global constraint can also have a similar effect to our local constraint (2), i.e., the related task pairs tend to have close solutions. However, the global constraint can allow some of the distances to be large since only the sum is upper-bounded. This actually causes a significant performance degradation in practice, which will be experimentally demonstrated in Section 6. We note that the idea of local constraints is also used in the kernel learning problem [10]. Relation to Standard SVMs By construction, the proposed MTL method includes the standard SVM learning algorithm a special case. Indeed, when the number of tasks is one, Problem 3 is reduced to the standard SVM optimization problem. Thus, the proposed method may be regarded as a natural extension of SVMs. Ordinal Regression As we mentioned in Section 1, MTL approaches are useful in ordinal regression problems. Ordinal regression is a task of learning multiple quantiles, which can be formulated as a set of one-versus-one classification problems. A naive approach to ordinal regression is to individually train M SVMs with score functions fi(x) = ⟨wi, x⟩+ bi, i = 1, . . . , M. Shashua 5 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 (a) True classification boundaries (b) IL-SVMs -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 (c) MTL-SVM(global/full) (d) MTL-SVM(local/network) Figure 1: Toy multi classification tasks. Each subfigure contains the 10-th, 30-th, 50-th, 70-th, and 90-th tasks in the top row and the 110-th, 130-th, 150-th, 170-th, and 190-th tasks in the bottom row. and Levin [9] proposed an ordinal regression method called the support vector ordinal regression (SVOR), where the weight vectors are shared by all SVMs (i.e. w1 = w2 = · · · = wM) and only the bias parameter is learned individually. The proposed MTL method can be naturally employed in ordinal regression by constraining the weight vectors as ∥wi −wi+1∥2 ≤ρ, i = 1, . . . , M −1, i.e., the task network only has a weight between consecutive tasks. This method actually includes the above two ordinal regression approaches as special cases—Cρ = 0 (i.e., ignoring the task network) yields the independent training of SVMs and Cρ = ∞(i.e., the weight vectors of all SVMs agree) is reduced to SVOR. Thus, in the context of ordinal regression, the proposed method smoothly bridges two extremes and allows us to control the belief of task constraints. 6 Experiments In this section, we show the usefulness of the proposed method through experiments. 6.1 Toy Multiple Classification Tasks First, we illustrate how the proposed method behaves using a 2-dimensional toy data set, which includes 200 tasks (see Figure 1(a)). Each task possesses a circular-shaped classification boundary with different centers and a fixed radius 0.5. The location of the center in the i-th task is (−1 + 0.02(i −1), 0) for 1 ≤i ≤100 and (0, −1 + 0.02(i −101)) for 101 ≤i ≤200. For each task, only two positive and two negative samples are generated following the uniform distribution. We construct a task network where consecutive tasks are connected in a circular manner, i.e., (1, 2), (2, 3), . . ., (99, 100), and (100, 1) for the first 100 tasks and (101, 102), (102, 103), . . ., (199, 200), and (200, 1) for the last 100 tasks; we further add (50, 150), which connects the clusters of the first 100 and the last 100 nodes. We compare the following methods: a naive method where 200 SVMs are trained indivisually (individually learned SVM, ‘IL-SVM’), the MTL-SVM algorithm where the global constraint and the fully connected task network are used [5] (‘MTL-SVM(global/full)’),and the proposed method which uses local constraints and the properly defined task network (‘MTL-SVM(local/network)’). The results are exhibited in Figure 1, showing that IL-SVM can not capture the circular shape due to the small sample size in each task. MTL-SVM(global/full) can successfully capture closed-loop boundaries by making use of the information from other tasks. However, the result is still not so reliable since non-consecutive unrelated tasks heavily damage the solutions. On the other hand, MTL-SVM(local/network) nicely captures the circular boundaries and the results are highly reliable. Thus, given an appropriate task network, the proposed MTL-SVM(local/network) can effectively exploit information of the related tasks. 6 Table 1: The accuracy of each method in the protein super-family classification task. Dataset IL-SVM One-SVM MTL-SVM (global/full) MTL-SVM (global/network) MTL-SVM (local/network) d-f 0.908 (0.023) 0.941 (0.015) 0.945 (0.013) 0.933 (0.017) 0.952 (0.015) d-s 0.638 (0.067) 0.722 (0.030) 0.698 (0.036) 0.695 (0.032) 0.747 (0.020) d-o 0.725 (0.032) 0.747 (0.017) 0.748 (0.021) 0.749 (0.023) 0.764 (0.028) f-s 0.891 (0.036) 0.886 (0.021) 0.918 (0.020) 0.911 (0.022) 0.918 (0.025) f-o 0.792 (0.046) 0.819 (0.029) 0.834 (0.021) 0.828 (0.015) 0.838 (0.018) s-o 0.663 (0.034) 0.695 (0.034) 0.692 (0.050) 0.663 (0.068) 0.703 (0.036) 6.2 Protein Super-Family Classification Next, we test the performance of the proposed method with real-world protein super-family classification problems. The input data are amino acid sequences from the SCOP database [1] (not SOCP). We counted 2-mers for extraction of feature vectors. There are 20 kinds of amino acids. Hence, the number of features is 202 = 400. We use RBF kernels, where the kernel width σ2 rbf is set to the average of the squared distances to the fifth nearest neighbors. Each data set consists of two folds. Each fold is divided into several super-families. We here consider the classification problem into the super-families. A positive class is chosen from one fold, and a negative class is chosen from the other fold. We perform multi-task learning from all the possible combinations. For example, three super-families are in DNA/RNA binding, and two in SH3. The number of combinations is 3 · 2 = 6. So the data set d-s has the six binary classification tasks. We used four folds: DNA/RNA binding, Flavodoxin, OB-fold and SH3. From these folds, we generate six data sets: d-f, d-f, d-o, f-o, f-s, and o-s, where the fold names are abbreviated to d, f, o, and s, respectively. The task networks are constructed as follows: if the positive super-family or the negative superfamily is common to two tasks, the two tasks are regarded as a related task pair and connected by an edge. We compare the proposed MTL-SVM(local/network) with IL-SVM, ‘One-SVM’, MTLSVM(global/full), and MTL-SVM(global/network). One-SVM regards the multiple tasks as one big task and learns the big task once by a standard SVM. We set Cα = 1 for all the approaches. The value of the parameter Cρ for three MTL-SVM approaches is determined by cross-validation over the training set. We randomly pick ten training sequences from each super-family, and use them for training. We compute the classification accuracies of the remaining test sequences. We repeat this procedure 10 times and take the average of the accuracies. The results are described in Table 1, showing that the proposed MTL-SVM(local/network) compares favorably with the other methods. In this simulation, the task network is constructed rather heuristically. Even so, the proposed MTL-SVM(local/network) is shown to significantly outperform MTL-SVM(global/full), which does not use the network structure. This implies that the proposed method still works well even when the task network contains small errors. It is interesting to note that MTL-SVM(global/network) actually does not work well in this simulation, implying that the task relatedness are not properly controlled by the global constraint. Thus the use of the local constraints would be effective in MTL scenarios. 6.3 Ordinal Regression As discussed in Section 5, MTL methods are useful in ordinal regression. Here we create five ordinal regression data sets described in Table 2, where all the data sets are originally regression and the output values are divided into five quantiles. Therefore, the overall task can be divided into four isolated classification tasks, each of which estimates a quantile. We compare MTL-SVM(local/network)with IL-SVM, SVOR [9] (see Section 5), MTL-SVM(full/network) and MTL-SVM(global/network). The value of the parameter Cρ for three MTL-SVM approaches is determined by cross-validation over the training set. We set Cα = 1 for all the approaches. We use RBF kernels, where the parameter σ2 rbf is set to the average of the squared distances to the fifth nearest neighbors. We randomly picked 200 samples for training. The remaining samples are used for evaluating the classification accuracies. 7 Table 2: The accuracy of each method in ordinal regression tasks. Data set IL-SVM SVOR MTL-SVM (global/full) MTL-SVM (global/network) MTL-SVM (local/network) pumadyn 0.643 (0.007) 0.661 (0.006) 0.629 (0.025) 0.645 (0.018) 0.661 (0.007) stock 0.894 (0.012) 0.878 (0.011) 0.872 (0.010) 0.888 (0.010) 0.902 (0.007) bank-8fh 0.781 (0.003) 0.777 (0.006) 0.772 (0.006) 0.773 (0.006) 0.779 (0.002) bank-8fm 0.854 (0.004) 0.845 (0.010) 0.832 (0.013) 0.847 (0.009) 0.854 (0.009) calihouse 0.648 (0.003) 0.642 (0.008) 0.640 (0.005) 0.646 (0.007) 0.650 (0.004) The averaged performance over five runs is described in Table 2, showing that the proposed MTLSVM(local/network) is also promising in ordinal regression scenarios. 7 Conclusions In this paper, we proposed a new multi-task learning method, which overcomes the limitation of existing approaches by making use of a task network and local constraints. We demonstrated through simulations that the proposed method is useful in multi-task learning scenario; moreover, it also works excellently in ordinal regression scenarios. The standard SVMs have a variety of extensions and have been combined with various techniques, e.g., one-class SVMs, SV regression, and the ν-trick. We expect that such extensions and techniques can also be applied similarly to the proposed method. Other possible future works include the elucidation of the entire regularization path and the application to learning from multiple networks; developing algorithms for learning probabilistic models with a task network is also a promising direction to be explored. Acknowledgments This work was partially supported by a Grant-in-Aid for Young Scientists (B), number 18700287, from the Ministry of Education, Culture, Sports, Science and Technology, Japan. References [1] A. Andreeva, D. Howorth, S. E. Brenner, T. J. P. Hubbard, C. Chothia, and A. G. Murzin. SCOP database in 2004: refinements integrate structure and sequence family data. Nucl. Acid Res., 32:D226–D229, 2004. [2] B. Borchers. CSDP, a C library for semidefinite programming. Optimization Methods and Software, 11(1):613–623, 1999. [3] Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004. [4] R. Caruana. Multitask learning. Machine Learning, 28(1):41–75, 1997. [5] T. Evgeniou and M. Pontil. Regularized multitask learning. In Proc. of 17-th SIGKDD Conf. on Knowledge Discovery and Data Mining, 2004. [6] D. Haussler. Convolution kernels on discrete structures. Technical Report UCSC-CRL-99-10, UC Santa Cruz, July 1999. [7] M. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret. Applications of second-order cone programming. Linear Algebra and its Applications, 284:193–228, 1998. [8] C. A. Micchelli and M. Pontil. Kernels for multi-task learning. In Lawrence K. Saul, Yair Weiss, and L´eon Bottou, editors, Advances in Neural Information Processing Systems 17, pages 921–928, Cambridge, MA, 2005. MIT Press. [9] A. Shashua and A. Levin. Ranking with large margin principle: two approaches. In Advances in Neural Information Processing Systems 15, pages 937–944, Cambridge, MA, 2003. MIT Press. [10] K. Tsuda and W.S. Noble. Learning kernels from biological networks by maximizing entropy. Bioinformatics, 20(Suppl. 1):i326–i333, 2004. [11] X. Zhu, J. Kandola, Z. Ghahramani, and J. Lafferty. Nonparametric transforms of graph kernels for semi-supervised learning. In Lawrence K. Saul, Yair Weiss, and Lon Bottou, editors, Advances in Neural Information Processing Systems 17, Cambridge, MA, 2004. MIT Press. 8
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Incremental Natural Actor-Critic Algorithms Shalabh Bhatnagar Department of Computer Science & Automation, Indian Institute of Science, Bangalore, India Richard S. Sutton, Mohammad Ghavamzadeh, Mark Lee Department of Computing Science, University of Alberta, Edmonton, Alberta, Canada Abstract We present four new reinforcement learning algorithms based on actor-critic and natural-gradient ideas, and provide their convergence proofs. Actor-critic reinforcement learning methods are online approximations to policy iteration in which the value-function parameters are estimated using temporal difference learning and the policy parameters are updated by stochastic gradient descent. Methods based on policy gradients in this way are of special interest because of their compatibility with function approximation methods, which are needed to handle large or infinite state spaces. The use of temporal difference learning in this way is of interest because in many applications it dramatically reduces the variance of the gradient estimates. The use of the natural gradient is of interest because it can produce better conditioned parameterizations and has been shown to further reduce variance in some cases. Our results extend prior two-timescale convergence results for actor-critic methods by Konda and Tsitsiklis by using temporal difference learning in the actor and by incorporating natural gradients, and they extend prior empirical studies of natural actor-critic methods by Peters, Vijayakumar and Schaal by providing the first convergence proofs and the first fully incremental algorithms. 1 Introduction Actor-critic (AC) algorithms are based on the simultaneous online estimation of the parameters of two structures, called the actor and the critic. The actor corresponds to a conventional actionselection policy, mapping states to actions in a probabilistic manner. The critic corresponds to a conventional value function, mapping states to expected cumulative future reward. Thus, the critic addresses a problem of prediction, whereas the actor is concerned with control. These problems are separable, but are solved simultaneously to find an optimal policy, as in policy iteration. A variety of methods can be used to solve the prediction problem, but the ones that have proved most effective in large applications are those based on some form of temporal difference (TD) learning (Sutton, 1988) in which estimates are updated on the basis of other estimates. Such bootstrapping methods can be viewed as a way of accelerating learning by trading bias for variance. Actor-critic methods were among the earliest to be investigated in reinforcement learning (Barto et al., 1983; Sutton, 1984). They were largely supplanted in the 1990’s by methods that estimate action-value functions and use them directly to select actions without an explicit policy structure. This approach was appealing because of its simplicity, but when combined with function approximation was found to have theoretical difficulties including in some cases a failure to converge. These problems led to renewed interest in methods with an explicit representation of the policy, which came to be known as policy gradient methods (Marbach, 1998; Sutton et al., 2000; Konda & Tsitsiklis, 2000; Baxter & Bartlett, 2001). Policy gradient methods without bootstrapping can be easily proved convergent, but converge slowly because of the high variance of their gradient estimates. Combining them with bootstrapping is a promising avenue toward a more effective method. Another approach to speeding up policy gradient algorithms was proposed by Kakade (2002) and then refined and extended by Bagnell and Schneider (2003) and by Peters et al. (2003). The idea 1 was to replace the policy gradient with the so-called natural policy gradient. This was motivated by the intuition that a change in the policy parameterization should not influence the result of the policy update. In terms of the policy update rule, the move to the natural gradient amounts to linearly transforming the gradient using the inverse Fisher information matrix of the policy. In this paper, we introduce four new AC algorithms, three of which incorporate natural gradients. All the algorithms are for the average reward setting and use function approximation in the state-value function. For all four methods we prove convergence of the parameters of the policy and state-value function to a local maximum of a performance function that corresponds to the average reward plus a measure of the TD error inherent in the function approximation. Due to space limitations, we do not present the convergence analysis of our algorithms here; it can be found, along with some empirical results using our algorithms, in the extended version of this paper (Bhatnagar et al., 2007). Our results extend prior AC methods, especially those of Konda and Tsitsiklis (2000) and of Peters et al. (2005). We discuss these relationships in detail in Section 6. Our analysis does not cover the use of eligibility traces but we believe the extension to that case would be straightforward. 2 The Policy Gradient Framework We consider the standard reinforcement learning framework (e.g., see Sutton & Barto, 1998), in which a learning agent interacts with a stochastic environment and this interaction is modeled as a discrete-time Markov decision process. The state, action, and reward at each time t ∈{0, 1, 2, . . .} are denoted st ∈S, at ∈A, and rt ∈R respectively. We assume the reward is random, realvalued, and uniformly bounded. The environment’s dynamics are characterized by state-transition probabilities p(s′|s, a) = Pr(st+1 = s′|st = s, at = a), and single-stage expected rewards r(s, a) = E[rt+1|st = s, at = a], ∀s, s′ ∈S, ∀a ∈A. The agent selects an action at each time t using a randomized stationary policy π(a|s) = Pr(at = a|st = s). We assume (B1) The Markov chain induced by any policy is irreducible and aperiodic. The long-term average reward per step under policy π is defined as J(π) = lim T →∞ 1 T E "T −1 X t=0 rt+1|π # = X s∈S dπ(s) X a∈A π(a|s)r(s, a), where dπ(s) is the stationary distribution of state s under policy π. The limit here is welldefined under (B1). Our aim is to find a policy π∗that maximizes the average reward, i.e., π∗= arg maxπ J(π). In the average reward formulation, a policy π is assessed according to the expected differential reward associated with states s or state–action pairs (s, a). For all states s ∈S and actions a ∈A, the differential action-value function and the differential state-value function under policy π are defined as1 Qπ(s, a) = ∞ X t=0 E[rt+1 −J(π)|s0 = s, a0 = a, π] , V π(s) = X a∈A π(a|s)Qπ(s, a). (1) In policy gradient methods, we define a class of parameterized stochastic policies {π(·|s; θ), s ∈S, θ ∈Θ}, estimate the gradient of the average reward with respect to the policy parameters θ from the observed states, actions, and rewards, and then improve the policy by adjusting its parameters in the direction of the gradient. Since in this setting a policy π is represented by its parameters θ, policy dependent functions such as J(π), dπ(·), V π(·), and Qπ(·, ·) can be written as J(θ), d(·; θ), V (·; θ), and Q(·, ·; θ), respectively. We assume (B2) For any state–action pair (s, a), policy π(a|s; θ) is continuously differentiable in the parameters θ. Previous works (Marbach, 1998; Sutton et al., 2000; Baxter & Bartlett, 2001) have shown that the gradient of the average reward for parameterized policies that satisfy (B1) and (B2) is given by2 ∇J(π) = X s∈S dπ(s) X a∈A ∇π(a|s)Qπ(s, a). (2) 1From now on in the paper, we use the terms state-value function and action-value function instead of differential state-value function and differential action-value function. 2Throughout the paper, we use notation ∇to denote ∇θ – the gradient w.r.t. the policy parameters. 2 Observe that if b(s) is any given function of s (also called a baseline), then X s∈S dπ(s) X a∈A ∇π(a|s)b(s) = X s∈S dπ(s)b(s)∇ X a∈A π(a|s) ! = X s∈S dπ(s)b(s)∇(1) = 0, and thus, for any baseline b(s), the gradient of the average reward can be written as ∇J(π) = X s∈S dπ(s) X a∈A ∇π(a|s)(Qπ(s, a) ± b(s)). (3) The baseline can be chosen such in a way that the variance of the gradient estimates is minimized (Greensmith et al., 2004). The natural gradient, denoted ˜∇J(π), can be calculated by linearly transforming the regular gradient, using the inverse Fisher information matrix of the policy: ˜∇J(π) = G−1(θ)∇J(π). The Fisher information matrix G(θ) is positive definite and symmetric, and is given by G(θ) = Es∼dπ,a∼π[∇log π(a|s)∇log π(a|s)⊤]. (4) 3 Policy Gradient with Function Approximation Now consider the case in which the action-value function for a fixed policy π, Qπ, is approximated by a learned function approximator. If the approximation is sufficiently good, we might hope to use it in place of Qπ in Eqs. 2 and 3, and still point roughly in the direction of the true gradient. Sutton et al. (2000) showed that if the approximation ˆQπ w with parameters w is compatible, i.e., ∇w ˆQπ w(s, a) = ∇log π(a|s), and minimizes the mean squared error Eπ(w) = X s∈S dπ(s) X a∈A π(a|s)[Qπ(s, a) −ˆQπ w(s, a)]2 (5) for parameter value w∗, then we can replace Qπ with ˆQπ w∗in Eqs. 2 and 3. Thus, we work with a linear approximation ˆQπ w(s, a) = w⊤ψ(s, a), in which the ψ(s, a)’s are compatible features defined according to ψ(s, a) = ∇log π(a|s). Note that compatible features are well defined under (B2). The Fisher information matrix of Eq. 4 can be written using the compatible features as G(θ) = Es∼dπ,a∼π[ψ(s, a)ψ(s, a)⊤]. (6) Suppose Eπ(w) denotes the mean squared error Eπ(w) = X s∈S dπ(s) X a∈A π(a|s)[Qπ(s, a) −w⊤ψ(s, a) −b(s)]2 (7) of our compatible linear parameterized approximation w⊤ψ(s, a) and an arbitrary baseline b(s). Let w∗= arg minw Eπ(w) denote the optimal parameter. Lemma 1 shows that the value of w∗ does not depend on the given baseline b(s); as a result the mean squared error problems of Eqs. 5 and 7 have the same solutions. Lemma 2 shows that if the parameter is set to be equal to w∗, then the resulting mean squared error Eπ(w∗) (now treated as a function of the baseline b(s)) is further minimized when b(s) = V π(s). In other words, the variance in the action-value-function estimator is minimized if the baseline is chosen to be the state-value function itself.3 Lemma 1 The optimum weight parameter w∗for any given θ (policy π) satisfies4 w∗= G−1(θ)Es∼dπ,a∼π[Qπ(s, a)ψ(s, a)]. Proof Note that ∇wEπ(w) = −2 X s∈S dπ(s) X a∈A π(a|s)[Qπ(s, a) −w⊤ψ(s, a) −b(s)]ψ(s, a). (8) Equating the above to zero, one obtains X s∈S dπ(s) X a∈A π(a|s)ψ(s, a)ψ(s, a)⊤w∗= X s∈S dπ(s) X a∈A π(a|s)Qπ(s, a)ψ(s, a)− X s∈S dπ(s) X a∈A π(a|s)b(s)ψ(s, a). 3It is important to note that Lemma 2 is not about the minimum variance baseline for gradient estimation. It is about the minimum variance baseline of the action-value-function estimator. 4This lemma is similar to Kakade’s (2002) Theorem 1. 3 The last term on the right-hand side equals zero because P a∈A π(a|s)ψ(s, a) = P a∈A ∇π(a|s) = 0 for any state s. Now, from Eq. 8, the Hessian ∇2 wEπ(w) evaluated at w∗can be seen to be 2G(θ). The claim follows because G(θ) is positive definite for any θ. □ Next, given the optimum weight parameter w∗, we obtain the minimum variance baseline in the action-value-function estimator corresponding to policy π. Thus we consider now E π(w∗) as a function of the baseline b, and obtain b∗= arg minb Eπ(w∗). Lemma 2 For any given policy π, the minimum variance baseline b∗(s) in the action-valuefunction estimator corresponds to the state-value function V π(s). Proof For any s ∈ S, let Eπ,s(w∗) = P a∈A π(a|s)[Qπ(s, a) −w∗⊤ψ(s, a) −b(s)]2. Then Eπ(w∗) = P s∈S dπ(s)Eπ,s(w∗). Note that by (B1), the Markov chain corresponding to any policy π is positive recurrent because the number of states is finite. Hence, dπ(s) > 0 for all s ∈S. Thus, one needs to find the baseline b(s) that minimizes E π,s(w∗) for each s ∈S. For any s ∈S, ∂Eπ,s(w∗) ∂b(s) = −2 X a∈A π(a|s)[Qπ(s, a) −w∗⊤ψ(s, a) −b(s)]. Equating the above to zero, we obtain b∗(s) = X a∈A π(a|s)Qπ(s, a) − X a∈A π(a|s)w∗⊤ψ(s, a). The rightmost term equals zero because P a∈A π(a|s)ψ(s, a) = 0. Hence b∗(s) = P a∈A π(a|s) Qπ(s, a) = V π(s). The second derivative of Eπ,s(w∗) w.r.t. b(s) equals 2. The claim follows. □ From Lemmas 1 and 2, w∗⊤ψ(s, a) is a least-squared optimal parametric representation for the advantage function Aπ(s, a) = Qπ(s, a) −V π(s) as well as for the action-value function Qπ(s, a). However, because Ea∼π[w⊤ψ(s, a)] = P a∈A π(a|s)w⊤ψ(s, a) = 0, ∀s ∈S, it is better to think of w⊤ψ(s, a) as an approximation of the advantage function rather than of the action-value function. The TD error δt is a random quantity that is defined according to δt = rt+1−ˆJt+1+ ˆV (st+1)−ˆV (st), where ˆV and ˆJ are consistent estimates of the state-value function and the average reward, respectively. Thus, these estimates satisfy E[ ˆV (st)|st, π] = V π(st) and E[ ˆJt+1|st, π] = J(π), for any t ≥0. The next lemma shows that δt is a consistent estimate of the advantage function Aπ. Lemma 3 Under given policy π, we have E[δt|st, at, π] = Aπ(st, at). Proof Note that E[δt|st, at, π] = E[rt+1−ˆJt+1+ ˆV (st+1)−ˆV (st)|st, at, π] = r(st, at)−J(π)+E[ ˆV (st+1)|st, at, π]−V π(st). Now E[ ˆV (st+1)|st, at, π] = E[E[ ˆV (st+1)|st+1, π]|st, at, π] = E[V π(st+1)|st, at] = X st+1∈S p(st+1|st, at)V π(st+1). Also r(st, at) −J(π) + P st+1∈S p(st+1|st, at)V π(st+1) = Qπ(st, at). The claim follows. □ By setting the baseline b(s) equal to the value function V π(s), Eq. 3 can be written as ∇J(π) = P s∈S dπ(s) P a∈A π(a|s)ψ(s, a)Aπ(s, a). From Lemma 3, δt is a consistent estimate of the advantage function Aπ(s, a). Thus, d ∇J(π) = δtψ(st, at) is a consistent estimate of ∇J(π). However, calculating δt requires having estimates, ˆJ, ˆV , of the average reward and the value function. While an average reward estimate is simple enough to obtain given the single-stage reward function, the same is not necessarily true for the value function. We use function approximation for the value function as well. Suppose f(s) is a feature vector for state s. One may then approximate V π(s) with v⊤f(s), where v is a parameter vector that can be tuned (for a fixed policy π) using a TD algorithm. In our algorithms, we use δt = rt+1 −ˆJt+1 + v⊤ t f(st+1) −v⊤ t f(st) as an estimate for the TD error, where vt corresponds to the value function parameter at time t. 4 Let ¯V π(s) = P a∈A π(a|s)[r(s, a) −J(π) + P s′∈S p(s′|s, a)vπ⊤f(s′)], where vπ⊤f(s′) is an estimate of the value function V π(s′) that is obtained upon convergence viz., limt→∞vt = vπ with probability one. Also, let δπ t = rt+1 −ˆJt+1 + vπ⊤f(st+1) −vπ⊤f(st), where δπ t corresponds to a stationary estimate of the TD error with function approximation under policy π. Lemma 4 E[δπ t ψ(st, at)|θ] = ∇J(π) + P s∈S dπ(s)[∇¯V π(s) −∇vπ⊤f(s)]. Proof of this lemma can be found in the extended version of this paper (Bhatnagar et al., 2007). Note that E[δtψ(st, at)|θ] = ∇J(π), provided δt is defined as δt = rt+1 −ˆJt+1 + ˆV (st+1)−ˆV (st) (as was considered in Lemma 3). For the case with function approximation that we study, from Lemma 4, the quantity P s∈S dπ(s)[∇¯V π(s) −∇vπ⊤f(s)] may be viewed as the error or bias in the estimate of the gradient of average reward that results from the use of function approximation. 4 Actor-Critic Algorithms We present four new AC algorithms in this section. These algorithms are in the general form shown in Table 1. They update the policy parameters along the direction of the average-reward gradient. While estimates of the regular gradient are used for this purpose in Algorithm 1, natural gradient estimates are used in Algorithms 2–4. While critic updates in our algorithms can be easily extended to the case of TD(λ), λ > 0, we restrict our attention to the case when λ = 0. In addition to assumptions (B1) and (B2), we make the following assumption: (B3) The step-size schedules for the critic {αt} and the actor {βt} satisfy X t αt = X t βt = ∞ , X t α2 t , X t β2 t < ∞ , lim t→∞ βt αt = 0. (9) As a consequence of Eq. 9, βt →0 faster than αt. Hence the critic has uniformly higher increments than the actor beyond some t0, and thus it converges faster than the actor. Table 1: A Template for Incremental AC Algorithms. 1: Input: • Randomized parameterized policy π(·|·; θ), • Value function feature vector f(s). 2: Initialization: • Policy parameters θ = θ0, • Value function weight vector v = v0, • Step sizes α = α0, β = β0, ξ = cα0, • Initial state s0. 3: for t = 0, 1, 2, . . . do 4: Execution: • Draw action at ∼π(at|st; θt), • Observe next state st+1 ∼p(st+1|st, at), • Observe reward rt+1. 5: Average Reward Update: ˆJt+1 = (1 −ξt) ˆJt + ξtrt+1 6: TD error: δt = rt+1 −ˆJt+1 + v⊤ t f(st+1) −v⊤ t f(st) 7: Critic Update: algorithm specific (see the text) 8: Actor Update: algorithm specific (see the text) 9 : endfor 10: return Policy and value-function parameters θ, v We now present the critic and the actor updates of our four AC algorithms. Algorithm 1 (Regular-Gradient AC): Critic Update: vt+1 = vt + αtδtf(st), Actor Update: θt+1 = θt + βtδtψ(st, at). 5 This is the only AC algorithm presented in the paper that is based on the regular gradient estimate. This algorithm stores two parameter vectors θ and v. Its per time-step computational cost is linear in the number of policy and value-function parameters. The next algorithm is based on the natural-gradient estimate ˜∇J(θt) = G−1(θt)δtψ(st, at) in place of the regular-gradient estimate in Algorithm 1. We derive a procedure for recursively estimating G−1(θ) and show in Lemma 5 that our estimate G−1 t converges to G−1(θ) as t →∞with probability one. This is required for proving convergence of this algorithm. The Fisher information matrix can be estimated in an online manner as Gt+1 = 1 t+1 Pt i=0 ψ(si, ai)ψ⊤(si, ai). One may obtain recursively Gt+1 = (1 − 1 t+1)Gt + 1 t+1ψ(st, at)ψ⊤(st, at), or more generally Gt+1 = (1 −ζt)Gt + ζtψ(st, at)ψ⊤(st, at). (10) Using the Sherman-Morrison matrix inversion lemma, one obtains G−1 t+1 = 1 1 −ζt G−1 t −ζt G−1 t ψ(st, at)(G−1 t ψ(st, at))⊤ 1 −ζt + ζtψ(st, at)⊤G−1 t ψ(st, at) (11) For our Alg. 2 and 4, we require the following additional assumption for the convergence analysis: (B4) The iterates Gt and G−1 t satisfy supt,θ,s,a ∥Gt ∥and supt,θ,s,a ∥G−1 t ∥< ∞. Lemma 5 For any given parameter θ, G−1 t in Eq. 11 satisfies G−1 t →G−1(θ) as t →∞with probability one. Proof It is easy to see from Eq. 10 that Gt →G(θ) as t →∞with probability one, for any given θ held fixed. For a fixed θ, ∥G−1 t −G−1(θ) ∥=∥G−1(θ)(G(θ)G−1 t −I) ∥=∥G−1(θ)(G(θ) −Gt)G−1 t ∥≤ sup θ ∥G−1(θ) ∥sup t,θ,s,a ∥G−1 t ∥· ∥G(θ) −Gt ∥→0 as t →∞ by assumption (B4). The claim follows. □ Our second algorithm stores a matrix G−1 and two parameter vectors θ and v. Its per timestep computational cost is linear in the number of value-function parameters and quadratic in the number of policy parameters. Algorithm 2 (Natural-Gradient AC with Fisher Information Matrix): Critic Update: vt+1 = vt + αtδtf(st), Actor Update: θt+1 = θt + βtG−1 t+1δtψ(st, at), with the estimate of the inverse Fisher information matrix updated according to Eq. 11. We let G−1 0 = kI, where k is a positive constant. Thus G−1 0 and G0 are positive definite and symmetric matrices. From Eq. 10, Gt, t > 0 can be seen to be positive definite and symmetric because these are convex combinations of positive definite and symmetric matrices. Hence, G−1 t , t > 0, are positive definite and symmetric as well. As mentioned in Section 3, it is better to think of the compatible approximation w⊤ψ(s, a) as an approximation of the advantage function rather than of the action-value function. In our next algorithm we tune the parameters w in such a way as to minimize an estimate of the least-squared error Eπ(w) = Es∼dπ,a∼π[(w⊤ψ(s, a) −Aπ(s, a))2]. The gradient of Eπ(w) is thus ∇wEπ(w) = 2Es∼dπ,a∼π[(w⊤ψ(s, a) −Aπ(s, a))ψ(s, a)], which can be estimated as \ ∇wEπ(w) = 2[ψ(st, at)ψ(st, at)⊤w −δtψ(st, at)]. Hence, we update advantage parameters w along with value-function parameters v in the critic update of this algorithm. As with Peters et al. (2005), we use the natural gradient estimate ˜∇J(θt) = wt+1 in the actor update of Alg. 3. This algorithm stores three parameter vectors, v, w, and θ. Its per time-step computational cost is linear in the number of value-function parameters and quadratic in the number of policy parameters. 6 Algorithm 3 (Natural-Gradient AC with Advantage Parameters): Critic Update: vt+1 = vt + αtδtf(st), wt+1 = [I −αtψ(st, at)ψ(st, at)⊤]wt + αtδtψ(st, at), Actor Update: θt+1 = θt + βtwt+1. Although an estimate of G−1(θ) is not explicitly computed and used in Algorithm 3, the convergence analysis of this algorithm shows that the overall scheme still moves in the direction of the natural gradient of average reward. In Algorithm 4, however, we explicitly estimate G−1(θ) (as in Algorithm 2), and use it in the critic update for w. The overall scheme is again seen to follow the direction of the natural gradient of average reward. Here, we let ˜∇wEπ(w) = 2G−1 t [ψ(st, at)ψ(st, at)⊤w −δtψ(st, at)] be the estimate of the natural gradient of the leastsquared error Eπ(w). This also simplifies the critic update for w. Algorithm 4 stores a matrix G−1 and three parameter vectors, v, w, and θ. Its per time-step computational cost is linear in the number of value-function parameters and quadratic in the number of policy parameters. Algorithm 4 (Natural-Gradient AC with Advantage Parameters and Fisher Information Matrix): Critic Update: vt+1 = vt + αtδtf(st), wt+1 = (1 −αt)wt + αtG−1 t+1δtψ(st, at), Actor Update: θt+1 = θt + βtwt+1, where the estimate of the inverse Fisher information matrix is updated according to Eq. 11. 5 Convergence of Our Actor-Critic Algorithms Since our algorithms are gradient-based, one cannot expect to prove convergence to a globally optimal policy. The best that one could hope for is convergence to a local maximum of J(θ). However, because the critic will generally converge to an approximation of the desired projection of the value function (defined by the value function features f) in these algorithms, the corresponding convergence results are necessarily weaker, as indicated by the following theorem. Theorem 1 For the parameter iterations in Algorithms 1-4,5 we have ( ˆJt, vt, θt) → {(J(θ∗), vθ∗, θ∗)|θ∗∈Z} as t →∞with probability one, where the set Z corresponds to the set of local maxima of a performance function whose gradient is E[δπ t ψ(st, at)|θ] (cf. Lemma 4). For the proof of this theorem, please refer to Section 6 (Convergence Analysis) of the extended version of this paper (Bhatnagar et al., 2007). This theorem indicates that the policy and state-value-function parameters converge to a local maximum of a performance function that corresponds to the average reward plus a measure of the TD error inherent in the function approximation. 6 Relation to Previous Algorithms Actor-Critic Algorithm of Konda and Tsitsiklis (2000): Unlike our Alg. 2–4, their algorithm does not use estimates of the natural gradient in its actor’s update. Their algorithm is similar to our Alg. 1, but with some key differences. 1) Konda’s algorithm uses the Markov process of state– action pairs, and thus its critic update is based on an action-value function. Alg. 1 uses the state process, and therefore its critic update is based on a state-value function. 2) Whereas Alg. 1 uses a TD error in both critic and actor recursions, Konda’s algorithm uses a TD error only in its critic update. The actor recursion in Konda’s algorithm uses an action-value estimate instead. Because the TD error is a consistent estimate of the advantage function (Lemma 3), the actor recursion in Alg. 1 uses estimates of advantages instead of action-values, which may result in lower variances. 3) The convergence analysis of Konda’s algorithm is based on the martingale approach and aims at bounding error terms and directly showing convergence; convergence to a local optimum is shown when a TD(1) critic is used. For the case where λ < 1, they show that given an ϵ > 0, there exists λ close enough to one such that when a TD(λ) critic is used, one gets lim inf t |∇J(θt)| < ϵ with 5The proof of this theorem requires another assumption viz., (A3) in the extended version of this paper (Bhatnagar et al., 2007), in addition to (B1)-(B3) (resp. (B1)-(B4)) for Algorithm 1 and 3 (resp. for Algorithm 2 and 4). This was not included in this paper due to space limitations. 7 probability one. Unlike Konda and Tsitsiklis, we primarily use the ordinary differential equation (ODE) based approach for our convergence analysis. Though we use martingale arguments in our analysis, these are restricted to showing that the noise terms asymptotically diminish; the resulting scheme can be viewed as an Euler-discretization of the associated ODE. Natural Actor-Critic Algorithm of Peters et al. (2005): Our Algorithms 2–4 extend their algorithm by being fully incremental and in that we provide convergence proofs. Peters’s algorithm uses a least-squares TD method in its critic’s update, whereas all our algorithms are fully incremental. It is not clear how to satisfactorily incorporate least-squares TD methods in a context in which the policy is changing, and our proof techniques do not immediately extend to this case. 7 Conclusions and Future Work We have introduced and analyzed four AC algorithms utilizing both linear function approximation and bootstrapping, a combination which seems essential to large-scale applications of reinforcement learning. All of the algorithms are based on existing ideas such as TD-learning, natural policy gradients, and two-timescale stochastic approximation, but combined in new ways. The main contribution of this paper is proving convergence of the algorithms to a local maximum in the space of policy and value-function parameters. Our Alg. 2–4 are explorations of the use of natural gradients within an AC architecture. The way we use natural gradients is distinctive in that it is totally incremental: the policy is changed on every time step, yet the gradient computation is never reset as it is in the algorithm of Peters et al. (2005). Alg. 3 is perhaps the most interesting of the three natural-gradient algorithms. It never explicitly stores an estimate of the inverse Fisher information matrix and, as a result, it requires less computation. In empirical experiments using our algorithms (not reported here) we observed that it is easier to find good parameter settings for Alg. 3 than it is for the other natural-gradient algorithms and, perhaps because of this, it converged more rapidly than the others and than Konda’s algorithm. All our algorithms performed better than Konda’s algorithm. There are a number of ways in which our results are limited and suggest future work. 1) It is important to characterize the quality of the converged solutions, either by bounding the performance loss due to bootstrapping and approximation error, or through a thorough empirical study. 2) The algorithms can be extended to incorporate eligibility traces and least-squares methods. As discussed earlier, the former seems straightforward whereas the latter requires more fundamental extensions. 3) Application of the algorithms to real-world problems is needed to assess their ultimate utility. References Bagnell, J., & Schneider, J. (2003). Covariant policy search. Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence. Barto, A. G., Sutton, R. S., & Anderson, C. (1983). Neuron-like elements that can solve difficult learning control problems. IEEE Transaction on Systems, Man and Cybernetics, 13, 835–846. Baxter, J., & Bartlett, P. (2001). Infinite-horizon policy-gradient estimation. JAIR, 15, 319–350. Bhatnagar, S., Sutton, R. S., Ghavamzadeh, M., & Lee, M. (2007). Natural actor-critic algorithms. Submitted to Automatica. Greensmith, E., Bartlett, P., & Baxter, J. (2004). Variance reduction techniques for gradient estimates in reinforcement learning. Journal of Machine Learning Research, 5, 1471–1530. Kakade, S. (2002). A natural policy gradient. Proceedings of NIPS 14. Konda, V., & Tsitsiklis, J. (2000). Actor-critic algorithms. Proceedings of NIPS 12 (pp. 1008–1014). Marbach, P. (1998). Simulated-based methods for Markov decision processes. Doctoral dissertation, MIT. Peters, J., Vijayakumar, S., & Schaal, S. (2003). Reinforcement learning for humanoid robotics. Proceedings of the Third IEEE-RAS International Conference on Humanoid Robots. Peters, J., Vijayakumar, S., & Schaal, S. (2005). Natural actor-critic. Proceedings of the Sixteenth European Conference on Machine Learning (pp. 280–291). Sutton, R. S. (1984). Temporal credit assignment in reinforcement learning. Doctoral dissertation, UMass Amherst. Sutton, R. S. (1988). Learning to predict by the methods of temporal differences. Machine Learning, 3, 9–44. Sutton, R. S., & Barto, A. G. (1998). Reinforcement learning: An introduction. MIT Press. Sutton, R. S., McAllester, D., Singh, S., & Mansour, Y. (2000). Policy gradient methods for reinforcement learning with function approximation. Proceedings of NIPS 12 (pp. 1057–1063). 8
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Collective Inference on Markov Models for Modeling Bird Migration Daniel Sheldon M. A. Saleh Elmohamed Dexter Kozen Cornell University Ithaca, NY 14853 {dsheldon,kozen}@cs.cornell.edu saleh@cam.cornell.edu Abstract We investigate a family of inference problems on Markov models, where many sample paths are drawn from a Markov chain and partial information is revealed to an observer who attempts to reconstruct the sample paths. We present algorithms and hardness results for several variants of this problem which arise by revealing different information to the observer and imposing different requirements for the reconstruction of sample paths. Our algorithms are analogous to the classical Viterbi algorithm for Hidden Markov Models, which finds the single most probable sample path given a sequence of observations. Our work is motivated by an important application in ecology: inferring bird migration paths from a large database of observations. 1 Introduction Hidden Markov Models (HMMs) assume a generative model for sequential data whereby a sequence of states (or sample path) is drawn from a Markov chain in a hidden experiment. Each state generates an output symbol from alphabet Σ, and these output symbols constitute the data or observations. A classical problem, solved by the Viterbi algorithm, is to find the most probable sample path given certain observations for a given Markov model. We call this the single path problem; it is well suited to labeling or tagging a single sequence of data. For example, HMMs have been successfully applied in speech recognition [1], natural language processing [2], and biological sequencing [3]. We introduce two generalizations of the single path problem for performing collective inference on Markov models, motivated by an effort to model bird migration patterns using a large database of static observations. The eBird database hosted by the Cornell Lab of Ornithology contains millions of bird observations from throughout North America, reported by the general public using the eBird web application.1 Observations report location, date, species and number of birds observed. The eBird data set is very rich; the human eye can easily discern migration patterns from animations showing the observations as they unfold over time on a map of North America.2 However, the eBird data are static, and they do not explicitly record movement, only the distributions at different points in time. Conclusions about migration patterns are made by the human observer. Our goal is to build a mathematical framework to infer dynamic migration models from the static eBird data. Quantitative migration models are of great scientific and practical import: for example, this problem arose out of an interdisciplinary project at Cornell University to model the possible spread of avian influenza in North America through wild bird migration. The migratory behavior for a species of birds can be modeled using a single generative process that independently governs how individual birds fly between locations, giving rise to the following 1http://ebird.org 2http://www.avianknowledge.net/visualization 1 inference problem: a hidden experiment simultaneously draws many independent sample paths from a Markov chain, and the observations reveal aggregate information about the collection of sample paths at each time step, from which the observer attempts to reconstruct the paths. For example, the eBird data estimate the geographical distribution of a species on successive days, but do not track individual birds. We discuss two problems within this framework. In the multiple path problem, we assume that exactly M independent sample paths are drawn from the Markov model, and the observations reveal the number of paths that output symbol α at time t, for each α and t. The observer seeks the most likely collection of paths given the observations. The fractional path problem is a further generalization in which paths are divisible entities. The observations reveal the fraction of paths that output symbol α at time t, and the observer’s job is to find the most likely (in a sense to be defined later) weighted collection of paths given the observations. Conceptually, the fractional path problem can be derived from the multiple path problem by letting M go to infinity; or it has a probabilistic interpretation in terms of distributions over paths. After discussing some preliminaries in section 2, sections 3 and 4 present algorithms for the multiple and fractional path problems, respectively, using network flow techniques on the trellis graph of the Markov model. The multiple path problem in its most general form is NP-hard, but can be solved as an integer program. The special case when output symbols uniquely identify their associated states can be solved efficiently as a flow problem; although the single path problem is trivial in this case, the multiple and fractional path problems remain interesting. The fractional path problem can be solved by linear programming. We also introduce a practical extension to the fractional path problem, including slack variables allowing the solution to deviate slightly from potentially noisy observations. In section 5, we demonstrate our techniques with visualizations for the migration of Archilochus colubris, the Ruby-throated Hummingbird, devoting some attention to a challenging problem we have neglected so far: estimating species distributions from eBird observations. We briefly mention some related work. Caruana et al. [4] and Phillips et al. [5] used machine learning techniques to model bird distributions from observations and environmental features. For problems on sequential data, many variants of HMMs have been proposed [3], and recently, conditional random fields (CRFs) have become a popular alternative [6]. Roth and Yih [7] present an integer programming inference framework for CRFs that is similar to our problem formulations. 2 Preliminaries 2.1 Data Model and Notation A Markov model (V, p, Σ, σ) is a Markov chain with state set V and transition probabilities p(u, v) for all u, v ∈V . Each state generates a unique output symbol from alphabet Σ, given by the mapping σ : V →Σ. Although some presentations allow each state to output multiple symbols with different emission probabilities, we lose no generality assuming that each state emits a unique symbol — to encode a model where state v output multiple symbols, we simply duplicate v for each symbol and encode the emission probabilities into the transitions. Of course, σ need not be one-to-one. It is useful to think of σ as a partition of the states, letting Vα = σ−1(α) be the set of all states that output α. We assume each model has a distinguished start state s and output symbol start. Let Y = V T be the set of all possible sample paths of length T. We represent a path y ∈Y as a row vector y = (y1, . . . , yT ), and a collection of M paths as the M × T matrix Y = (yit), with each row yi· representing an independent sample path. The transition probabilities induce a distribution λ on Y, where λ(y) = QT −1 t=1 p(yt, yt+1). We will also consider arbitrary distributions π over Y, letting Y = (Y1, . . . , YT ) denote a random path from π. Then, for example, we write Prπ [Yt = u] to be the probability under π that the tth state is u, and Eπ [f(Y )] to be the expected value of f(Y ) for any function f of a random path Y drawn from π. Note that Y (boldface) denotes a matrix of M paths, while Y denotes a random path. 2.2 The Trellis Graph and Viterbi as Shortest Path To develop our flow-based algorithms, it is instructive to build upon a shortest-path interpretation of the Viterbi algorithm [7]. In an instance of the single path problem we are given a model (V, p, Σ, σ) 2 p(u, u) p(u, w) p(s, u) u v w 0 1 V0 V1 V0 V0 V1 V1 0 start Observations s c(u, u) c(u, w) c(s, u) u v w 0 1 V0 V1 V0 V0 V1 V1 0 start Observations s (a) (b) Figure 1: Trellis graph for Markov model with states {s, u, v, w} and alphabet {start, 0, 1}. States u and v output the symbol 0, and state w outputs the symbol 1. (a) The bold path is feasible for the specified observations, with probability p(s, u)p(u, u)p(u, w). (b) Infeasible edges have been removed (indicated by light dashed lines), and probabilities changed to costs. The bold path has cost c(s, u) + c(u, u) + c(u, w). and observations α1, . . . , αT , and we seek the most probable path y given these observations. We call path y feasible if σ(yt) = αt for all t; then we wish to maximize λ(y) over feasible y. The problem is conveniently illustrated using the trellis graph of the Markov model (Figure 1). Here, the states are replicated for each time step, and edges connect a state at time t to its possible successors at time t + 1, labeled with the transition probability. A feasible path must pass through partition Vαt at step t, so we can prune all edges incident on other partitions, leaving only feasible paths. By defining the cost of an edge as c(u, v) = −log p(u, v), and letting the path cost c(y) be the sum of its edge costs, straightforward algebra shows that arg maxy λ(y) = arg miny c(y), i.e., the path of maximum probability becomes the path of minimum cost under this transformation. Thus the Viterbi algorithm finds the shortest feasible path in the trellis using edge lengths c(u, v). 3 Multiple Path Problem In the multiple path problem, M sample paths are drawn from the model and the observations reveal the number of paths Nt(α) that output α at time t, for all α and t; or, equivalently, the multiset At of output symbols at time t. The objective is to find the most probable collection Y that is feasible, meaning it produces multisets A1, . . . , AT . The probability λ(Y) is just the product of the path-wise probabilities: λ(Y) = M Y i=1 λ(yi) = M Y i=1 T −1 Y t=1 p(yi,t, yi,t+1). (1) Then the formal specification of this problem is max Y λ(Y) subject to |{i : yi,t ∈Vα}| = Nt(α) for all α, t. (2) 3.1 Reduction to the Single Path Problem A naive approach to the multiple path problem reduces it to the single path problem by creating a new Markov model on state set V M where state ⟨v1, . . . , vM⟩encodes an entire tuple of original states, and the transition probabilities are given by the product of the element-wise transition probabilities: p(⟨u1, . . . , uM⟩, ⟨v1, . . . , vM⟩) = M Y i=1 p(ui, vi). A state from the product space V M corresponds to an entire column of the matrix Y, and by changing the order of multiplication in (1), we see that the probability of a path in the new model is equal to the probability of the entire collection of paths in the old model. To complete the reduction, we form a new alphabet ˆΣ whose symbols represent multisets of size M on Σ. Then the solution to (2) can be found by running the Viterbi algorithm to find the most likely sequence of states from V M that produce output symbols (multisets) A1, . . . , AT . The running time is polynomial in |V M| and |ˆΣ|, but exponential in M. 3 3.2 Graph Flow Formulation Can we do better than the naive approach? Viewing the cost of a path as the cost of routing one unit of flow along that path in the trellis, a minimum cost collection of M paths is equivalent to a minimum cost flow of M units through the trellis — given M paths, we can route one unit along each to get a flow, and we can decompose any flow of M units into paths each carrying a single unit of flow. Thus we can write the optimization problem in (2) as the following flow integer program, with additional constraints that the flow paths generate the correct observations. The decision variable xt uv indicates the flow traveling from u to v at time t; or, the number of sample paths that transition from u to v at time t. (IP) min X u,v,t c(u, v)xt uv s.t. X u xt uv = X w xt+1 vw for all v, t, (3) X u∈Vα,v∈V xt uv = Nt(α) for all α, t, (4) xt uv ∈N for all u, v, t. The flow conservation constraints (3) are standard: the flow into v at time t is equal to the flow leaving v at time t + 1. The observation constraints (4) specify that Nt(α) units of flow leave partition Vα at time t. These also imply that exactly M units of flow pass through each level of the trellis, by summing over all α, X u,v xt uv = X α X u∈Vα,v∈V xt uv = X α Nt(α) = M. Without the observation constraints, IP would be an instance of the minimum-cost flow problem [8], which is solvable in polynomial time by a variety of algorithms [9]. However, we cannot hope to encode the observation constraints into the flow framework, due to the following result. Lemma 1. The multiple path problem is NP-hard. The proof of Lemma 1 is by reduction from SET COVER, and is omitted. One may use a general purpose integer program solver to solve IP directly; this may be efficient in some cases despite the lack of polynomial time performance guarantees. In the following sections we discuss alternatives that are efficiently solvable. 3.3 An Efficient Special Case In the special case when σ is one-to-one, the output symbols uniquely identify their generating states, so we may assume that Σ = V , and the output symbol is always the name of the current state. To see how the problem IP simplifies, we now have Vu = {u} for all u, so each partition consists of a single state, and the observations completely specify the flow through each node in the trellis: X v xt uv = Nt(u) for all u, t. (4′) Substituting the new observation constraints (4′) for time t+1 into the RHS of the flow conservation constraints (3) for time t yield the following replacements: X u xt uv = Nt+1(v) for all v, t. (3′) This gives an equivalent set of constraints, each of which refers only to variables xt uv for a single t. Hence the problem can be decomposed into T −1 disjoint subproblems for t = 1, . . . , T −1. The tth subproblem IPt is given in Figure 2(a), and illustrated on the trellis in Figure 2(b). State u at time t has a supply of Nt(u) units of flow coming from the previous step, and we must route Nt+1(v) units of flow to state v at time t+1, so we place a demand of Nt+1(v) at the corresponding node. Then the problem reduces to finding a minimum cost routing of the supply from time t to meet the demand at time t + 1, solved separately for all t = 1, . . . , T −1. The problem IPt an instance of the transportation problem [10], a special case of the minimum-cost flow problem. There are a variety of efficient algorithms to solve both problems [8,9], or one may use a general purpose linear program (LP) solver; any basic solution to the LP relaxation of IPt is guaranteed to be integral [8]. 4 (IPt) min X u,v c(u, v)xt uv s.t. X u xt uv = Nt+1(v) ∀v, (3′) X v xt uv = Nt(u) ∀u, (4′) xt uv ∈N ∀u, v. 0 4 1 3 2 0 Supply Demand t t + 1 v1 v2 v3 v1 v2 v3 c(v1, v1) Nt(·) Nt+1(·) (a) (b) Figure 2: (a) The definition of subproblem IPt. (b) Illustration on the trellis. 4 Fractional Path Problem In the fractional path problem, a path is a divisible entity. The observations specify qt(α), the fraction of paths that output α at time t, and the observer chooses π(y) fractional units of each path y, totaling one unit, such that qt(α) units output α at time t. The objective is to maximize Q y∈Y λ(y)π(y). Put another way, π is a distribution over paths such that Prπ [Yt ∈Vα] = qt(α), i.e., qt specifies the marginal distribution over symbols at time t. By taking the logarithm, an equivalent objective is to maximize Eπ [log λ(Y )], so we seek the distribution π that maximizes the expected log-probability of a path Y drawn from π. Conceptually, the fractional path problem arises by letting M →∞in the multiple path problem and normalizing to let qt(α) = Nt(α)/M specify the fraction of paths that output α at time t. Operationally, the fractional path problem is modeled by the LP relaxation of IP, which routes one splittable unit of flow through the trellis. (RELAX) min X u,v,t c(u, v)xt uv s.t. X u xt uv = X w xt+1 vw for all v, t, X u∈Vα X v∈V xt uv = qt(α) for all α, t, (5) xt uv ≥0 for all u, v, t. It is easy to see that a unit flow x corresponds to a probability distribution π. Given any distribution π, let xt uv = Prπ [Yt = u, Yt+1 = v]; then x is a flow because the probability a path enters v at time t is equal to the probability it leaves v at time t + 1. Conversely, given a unit flow x, any path decomposition assigning flow π(y) to each y ∈Y is a probability distribution because the total flow is one. In general, the decomposition is not unique, but any choice yields a distribution π with the same objective value. Furthermore, under this correspondence, x satisfies the marginal constraints (5) if and only if π has the correct marginals: X u∈Vα X v∈V xt uv = X u∈Vα X v∈V Pr [Yt = u, Yt+1 = v] = X u∈Vα Pr [Yt = u] = Pr [Yt ∈Vα] . Finally, we can rewrite the objective function in terms of paths: X u,v,t c(u, v)xt uv = X y∈Y π(y)c(y) = Eπ [c(Y )] = Eπ [−log λ(Y )] . By switching signs and changing from minimization to maximization, we see that RELAX solves the fractional path problem. This problem is very similar to maximum entropy or minimum cross entropy modeling, but the details are slightly different: such a model would typically find the distribution π with the correct marginals that minimizes the cross entropy or Kullback-Leibler divergence [11] between λ and π, which, after removing a constant term, reduces to minimizing Eλ [−log π(Y )]. Like IP, the RELAX problem also decomposes into subproblems in the case when σ is one-to-one, but this simplification is incompatible with the slack variables introduced in the following section. 5 4.1 Incorporating Slack In our application, the marginal distributions qt(·) are themselves estimates, and it is useful to allow the LP to deviate slightly from these marginals to find a better overall solution. To accomplish this, we add slack variables δt u into the marginal constraints (5), and charge for the slack in the objective function. The new marginal constraints are X u∈Vα X v∈V xt uv = qt(α) + δt α for all α, t, (5′) and we add the term P α,t γt α|δt α| into the objective function to charge for the slack, using a standard LP trick [8] to model the absolute value term. The slack costs γt α can be tailored to individual input values; for example, one may want to charge more to deviate from a confident estimate. This will depend on the specific application. We also add the necessary constraints to ensure that the new marginals q′ t(α) = qt(α) + δt α form a valid probability distribution for all t. 5 Demonstration In this section, we demonstrate our techniques by using the fractional path problem to create visualizations showing likely migration routes of Archilochus colubris, the Ruby-throated Hummingbird, a common bird whose range is relatively well covered by eBird observations. We work in discretized space and time, dividing the map into grid cells and the year into weeks. We must specify the Markov model governing transitions between locations (grid cells) in successive weeks; also, we require estimates qt(·) for the weekly distributions of hummingbirds across locations. Since the actual eBird observations are highly non-uniform in space and time, estimating weekly distributions requires significant inference for locations with few or no observations. In the appendix, we outline one approach based on harmonic energy minimization [12], but we may use any technique that produces weekly distributions qt(u) and slack costs γt u. Improving these estimates, say, by incorporating important side information such as climate and habitat features, could significantly improve the overall model. Finally, although our final observations qt(·) are distributions over states (locations) and not output symbols — i.e., σ is one-to-one — we cannot use the simplification from section 3.3 because we incorporate slack into the model. 5.1 eBird Data Launched in 2002, eBird is a citizen science project run by the Cornell Lab of Ornithology, leveraging the data gathering power of the public. On the eBird website, birdwatchers submit checklists of birds they observe, indicating a count for each species, along with the location, date, time and additional information. Our data set consists of the 428,648 complete checklists from 19953 through 2007, meaning the reporter listed all species observed. This means we can infer a count of zero, or a negative observation, for any species not listed. Using a land cover map from the United States Geological Survey (USGS), we divide North America into grid cells that are roughly 225 km on a side. All years of data are aggregated into one, and the year is divided into weeks so t = 1, . . . , 52 represents the week of the year. 5.2 Migration Inference Given weekly distributions qt(u) and slack costs γt u (see the appendix), it remains to specify the Markov model. We use a simple Gaussian model favoring short flights, letting p(u, v) ∝ exp(−d(u, v)2/σ2), where d(u, v) measures the distance between grid cell centers. This corresponds to a squared distance cost function. To reduce problem size, we omitted variables xt uv from the LP when d(u, v) > 1350 km, effectively setting p(u, v) = 0. We also found it useful to impose upper bounds δt u ≤qt(u) on the slack variables so no single value could increase by more than a factor of two. Our final LP, which was solved using the MOSEK optimization toolbox, had 78,521 constraints and 3,031,116 variables. Figure 3 displays the migration paths our model inferred for the four weeks starting on the dates indicated. The top row shows the distribution and paths inferred by the model; grid cells colored 3Users may enter historical observations. 6 Week 10 Week 20 Week 30 Week 40 March 5 May 14 July 28 October 1 Figure 3: Ruby-throated Hummingbird migration. See text for description. in lighter shades have more birds (higher values for q′ t(u)). Arrows indicate flight paths (xt uv) between the week shown and the following week, with line width proportional to flow xt uv. In the bottom row, the raw data is given for comparison. White dots indicate negative observations; black squares indicate positive observations, with size proportional to count. Locations with both positive and negative observations appear a charcoal color. The inferred distributions and paths are consistent with both seasonal ranges and written accounts of migration routes. For example, in the summary paragraph on migration from the Archilochus colubris species account in Birds of North America [13], Robinson et al. write “Many fly across Gulf of Mexico, but many also follow coastal route. Routes may differ for north- and southbound birds.” Acknowledgments We are grateful to Daniel Fink, Wesley Hochachka and Steve Kelling from the Cornell Lab of Ornithology for useful discussions. This work was supported in part by ONR Grant N00014-01-10968 and by NSF grant CCF-0635028. The views and conclusions herein are those of the authors and do not necessarily represent the official policies or endorsements of these organizations or the US Government. References [1] L. R. Rabiner. A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE, 77(2):257–286, 1989. [2] E. Charniak. Statistical techniques for natural language parsing. AI Magazine, 18(4):33–44, 1997. [3] R. Durbin, S. Eddy, A. Krogh, and G. Mitchison. Biological sequence analysis: Probabilistic models of proteins and nucleic acids. Cambridge University Press, 1998. [4] R. Caruana, M. Elhawary, A. Munson, M. Riedewald, D. Sorokina, D. Fink, W. M. Hochachka, and S. Kelling. Mining citizen science data to predict prevalence of wild bird species. In SIGKDD, 2006. [5] S. J. Phillips, M. Dud´ık, and R. E. Schapire. A maximum entropy approach to species distribution modeling. In ICML, 2004. [6] J. Lafferty, A. McCallum, and F. Pereira. Conditional random fields: Probabilistic models for segmenting and labeling sequence data. ICML, 2001. [7] D. Roth and W. Yih. Integer linear programming inference for conditional random fields. ICML, 2005. 7 [8] V. Chv´atal. Linear Programming. W.H. Freeman, New York, NY, 1983. [9] A. V. Goldberg, S. A. Plotkin, and E. Tardos. Combinatorial algorithms for the generalized circulation problem. Math. Oper. Res., 16(2):351–381, 1991. [10] G. B. Dantzig. Application of the simplex method to a transportation problem. In T. C. Koopmans, editor, Activity Analysis of Production and Allocation, volume 13 of Cowles Commission for Research in Economics, pages 359–373. Wiley, 1951. [11] J. Shore and R. Johnson. Properties of cross-entropy minimization. IEEE Trans. on Information Theory, 27:472–482, 1981. [12] X. Zhu, Z. Ghahramani, and J. Lafferty. Semi-supervised learning using Gaussian fields and harmonic functions. In ICML, 2003. [13] T. R. Robinson, R. R. Sargent, and M. B. Sargent. Ruby-throated Hummingbird (Archilochus colubris). In A. Poole and F. Gill, editors, The Birds of North America, number 204. The Academy of Natural Sciences, Philadelphia, and The American Ornithologists’ Union, Washington, D.C., 1996. [14] D. Aldous and J. Fill. Reversible Markov Chains and Random Walks on Graphs. Monograph in Preparation, http://www.stat.berkeley.edu/users/aldous/RWG/book.html. A Estimating Weekly Distributions from eBird Our goal is to estimate qt(u), the fraction of birds in grid cell u during week t. Given enough observations, we can estimate qt(u) using the average number of birds counted per checklist, a quantity we call the rate rt(u). However, even for a bird with good eBird coverage, there are cells with few or no observations during some weeks. To fill these gaps, we use the harmonic energy minimization technique [12] to determine values for empty cells based on neighbors in space and time. This technique uses a graph-based similarity structure, in our case the 3-dimensional lattice built on points ut, where ut represents cell u during week t. Edges are weighted, with weights representing similarity between points. Point ut is connected to its four grid neighbors in time slice t by edges of unit weight, excluding edges between cells separated by water (specifically, when the line connecting the centers is more than half water). Point ut is also connected to points ut−1 and ut+1 with weight 1/4, to achieve some temporal smoothing. Harmonic energy minimization learns a function f on the graph; the idea is to match rt(u) on points with sufficient data and find values for other points according to the similarity structure. To this end, we designate some boundary points for which the value of f is fixed by the data, while other points are interior points. The value of f at interior point ut is determined by the expected value of the following random experiment: perform a random walk starting from ut, following outgoing edges with probability proportional to their weight. When the walk first hits a boundary point vt′, terminate and accept the boundary value f(vt′). In this way, the values at interior points are a weighted average of nearby boundary values, where “nearness” is interpreted as the absorption probability in an absorbing random walk. We derive a measure of confidence in the value f(ut) from the same experiment: let h(ut) be the expected number of steps for the random walk from ut to hit the boundary (the hitting time of the boundary set [14]). When h(ut) is small, ut is close to the boundary and we are more confident in f(ut). Rather than choosing a threshold on the number of observations required to be a boundary point, we create a soft boundary by designating all points ut as interior points, and adding one boundary node to the graph structure for each observation, connected by an edge of unit weight to the cell in which it occurred, with value equal to the number of birds observed. As point ut gains more observations, its behavior approaches that of a hard boundary: with probability approaching one, the walk from ut will reach an observation in the first step, so f(ut) will approach rt(u), the average of the observations. As a conservative measure, each node is also connected to a sink with boundary value 0, to prevent values from propagating over very long distances. We compute h and f iteratively using standard techniques. Since f(ut) approximates the rate rt(u), we multiply by the land mass of cell u to get an estimate ˆqt(u) for the (relative) number of birds in cell u at time t. Finally, we normalize ˆq for each time slice t, taking qt(u) = ˆqt(u)/ P u ˆqt(u). For slack costs, we set γt u = γ0/h(ut) to be inversely proportional to boundary hitting time, with γ0 ≈261 chosen in conjunction with the transition costs in section 5.2 so the average cost for a unit of slack is the same as moving 600 km. 8
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EEG-Based Brain-Computer Interaction: Improved Accuracy by Automatic Single-Trial Error Detection Pierre W. Ferrez IDIAP Research Institute Centre du Parc Av. des Pr´es-Beudin 20 1920 Martigny, Switzerland pierre.ferrez@idiap.ch Jos´e del R. Mill´an IDIAP Research Institute Centre du Parc Av. des Pr´es-Beudin 20 1920 Martigny, Switzerland jose.millan@idiap.ch ∗ Abstract Brain-computer interfaces (BCIs), as any other interaction modality based on physiological signals and body channels (e.g., muscular activity, speech and gestures), are prone to errors in the recognition of subject’s intent. An elegant approach to improve the accuracy of BCIs consists in a verification procedure directly based on the presence of error-related potentials (ErrP) in the EEG recorded right after the occurrence of an error. Six healthy volunteer subjects with no prior BCI experience participated in a new human-robot interaction experiment where they were asked to mentally move a cursor towards a target that can be reached within a few steps using motor imagination. This experiment confirms the previously reported presence of a new kind of ErrP. These “Interaction ErrP” exhibit a first sharp negative peak followed by a positive peak and a second broader negative peak (∼290, ∼350 and ∼470 ms after the feedback, respectively). But in order to exploit these ErrP we need to detect them in each single trial using a short window following the feedback associated to the response of the classifier embedded in the BCI. We have achieved an average recognition rate of correct and erroneous single trials of 81.8% and 76.2%, respectively. Furthermore, we have achieved an average recognition rate of the subject’s intent while trying to mentally drive the cursor of 73.1%. These results show that it’s possible to simultaneously extract useful information for mental control to operate a brain-actuated device as well as cognitive states such as error potentials to improve the quality of the braincomputer interaction. Finally, using a well-known inverse model (sLORETA), we show that the main focus of activity at the occurrence of the ErrP are, as expected, in the pre-supplementary motor area and in the anterior cingulate cortex. 1 Introduction People with severe motor disabilities (spinal cord injury (SCI), amyotrophic lateral sclerosis (ALS), etc.) need alternative ways of communication and control for their everyday life. Over the past two decades, numerous studies proposed electroencephalogram (EEG) activity for direct brain-computer interaction [1]-[2]. EEG-based brain-computer interfaces (BCIs) provide disabled people with new tools for control and communication and are promising alternatives to invasive methods. However, as any other interaction modality based on physiological signals and body channels (e.g., muscular activity, speech and gestures), BCIs are prone to errors in the recognition of subject’s intent, and those errors can be frequent. Indeed, even well-trained subjects rarely reach 100% of success. In ∗This work is supported by the European IST Programme FET Project FP6-003758 and by the Swiss National Science Foundation NCCR “IM2”. This paper only reflects the authors’ views and funding agencies are not liable for any use that may be made of the information contained herein. 1 contrast to other interaction modalities, a unique feature of the “brain channel” is that it conveys both information from which we can derive mental control commands to operate a brain-actuated device as well as information about cognitive states that are crucial for a purposeful interaction, all this on the millisecond range. One of these states is the awareness of erroneous responses, which a number of groups have recently started to explore as a way to improve the performance of BCIs [3]-[6]. In particular, [6] recently reported the presence of a new kind of error potentials (ErrP) elicited by erroneous feedback provided by a BCI during the recognition of the subject’s intent. In this study subjects were asked to reach a target by sending repetitive manual commands to pass over several steps. The system was executing commands with an 80% accuracy, so that at each step there was a 20% probability that the system delivered an erroneous feedback. The main components of these “Interaction ErrP” are a negative peak 250 ms after the feedback, a positive peak 320 ms after the feedback and a second broader negative peak 450 ms after the feedback. To exploit these ErrP for BCIs, it is mandatory to detect them no more in grand averages but in each single trial using a short window following the feedback associated to the response of the BCI. The reported average recognition rates of correct and erroneous single trials are 83.5% and 79.2%, respectively. These results tend to show that ErrP could be a potential tool to improve the quality of the brain-computer interaction. However, it is to note that in order to isolate the issue of the recognition of ErrP out of the more difficult and general problem of a whole BCI where erroneous feedback can be due to nonoptimal performance of both the interface (i.e., the classifier embedded into the interface) and the user himself, the subject delivered commands manually. The key issue now is to investigate whether subjects also show ErrP while already engaged in tasks that require a high level of concentration such as motor imagination, and no more in easy tasks such as pressing a key. The objective of the present study is to investigate the presence of these ErrP in a real BCI task. Subjects don’t deliver manual commands anymore, but are focussing on motor imagination tasks to reach targets randomly selected by the system. In this paper we report new experimental results recorded with six healthy volunteer subjects with no prior BCI experience during a simple humanrobot interaction that confirm the previously reported existence of a new kind of ErrP [6], which is satisfactorily recognized in single trials using a short window just after the feedback. Furthermore, using a window just before the feedback, we report a 73.1% accuracy in the recognition of the subject’s intent during mental control of the BCI. This confirms the fact that EEG conveys simultaneously information from which we can derive mental commands as well as information about cognitive states and shows that both can be sufficiently well recognized in each single trials to provide the subject with an improved brain-computer interaction. Finally, using a well-known inverse model called sLORETA [7] that non-invasively estimates the intracranial activity from scalp EEG, we show that the main focus of activity at the occurrence of ErrP seems to be located in the presupplementary motor area (pre-SMA) and in the anterior cingulate cortex (ACC), as expected [8][9]. Figure 1: Illustration of the protocol. (1) The target (blue) appears 2 steps on the left side of the cursor (green). (2) The subject is imagining a movement of his/her left hand and the cursor moves 1 step to the left. (3) The subject still focuses on his/her left hand, but the system moves the cursor in the wrong direction. (4) Correct move to the left, compensating the error. (5) The cursor reaches the target. (6) A new target (red) appears 3 steps on the right side of the cursor, the subject will now imagine a movement of his/her right foot. The system moved the cursor with an error rate of 20%; i.e., at each step, there was a 20% probability that the robot made a movement in the wrong direction. 2 Experimental setup The first step to integrate ErrP detection in a BCI is to design a protocol where the subject is focussing on a mental task for device control and on the feedback delivered by the BCI for ErrP 2 detection. To test the ability of BCI users to concentrate simultaneously on a mental task and to be aware of the BCI feedback at each single trial, we have simulated a human-robot interaction task where the subject has to bring the robot to targets 2 or 3 steps either to the left or to the right. This virtual interaction is implemented by means of a green square cursor that can appear on any of 20 positions along an horizontal line. The goal with this protocol is to bring the cursor to a target that randomly appears either on the left (blue square) or on the right(red square) of the cursor. The target is no further away than 3 positions from the cursor (symbolizing the current position of the robot). This prevents the subject from habituation to one of the stimuli since the cursor reaches the target within a small number of steps. Figure 1 illustrates the protocol with the target (blue) initially positioned 2 steps away on the left side of the cursor (green). An error occurred at step 3) so that the cursor reaches the target in 5 steps. Each target corresponds to a specific mental task. The subjects were asked to imagine a movement of their left hand for the left target and to imagine a movement of their right foot for the right target (note that subject n◦1 selected left foot for the left target and right hand for the right target). However, since the subjects had no prior BCI experience, the system was not moving the cursor following the mental commands of the subject, but with an error rate of 20%, to avoid random or totally biased behavior of the cursor. Six healthy volunteer subjects with no prior BCI experience participated in these experiments. After the presentation of the target, the subject focuses on the corresponding mental task until the cursor reached the target. The system moved the cursor with an error rate of 20%; i.e., at each step, there was a 20% probability that the cursor moved in the opposite direction. When the cursor reached a target, it briefly turned from green to light green and then a new target was randomly selected by the system. If the cursor didn’t reach the target after 10 steps, a new target was selected. As shown in figure 2, while the subject focuses on a specific mental task, the system delivers a feedback about every 2 seconds. This provides a window just before the feedback for BCI classification and a window just after the feedback for ErrP detection for every single trial. Subjects performed 10 sessions of 3 minutes on 2 different days (the delay between the two days of measurements varied from 1 week to 1 month), corresponding to ∼75 single trials per session. The 20 sessions were split into 4 groups of 5, so that classifiers were built using a group and tested on the following group. The classification rates presented in Section 3 are therefore the average of 3 prediction performances: classification of group n + 1 using group n to build a classifier. This rule applies for both mental tasks classification and ErrP detection. Figure 2: Timing of the protocol. The system delivers a feedback about every 2 seconds, this provides a window just before the feedback for BCI classification and a window just after the feedback for ErrP detection for every single trial. As a new target is presented, the subject focuses on the corresponding mental task until the target is reached. EEG potentials were acquired with a portable system (Biosemi ActiveTwo) by means of a cap with 64 integrated electrodes covering the whole scalp uniformly. The sampling rate was 512 Hz and signals were measured at full DC. Raw EEG potentials were first spatially filtered by subtracting from each electrode the average potential (over the 64 channels) at each time step. The aim of this re-referencing procedure is to suppress the average brain activity, which can be seen as underlying background activity, so as to keep the information coming from local sources below each electrode. Then for off-line mental tasks classification, the power spectrum density (PSD) of EEG channels was estimated over a window of one second just before the feedback. PSD was estimated using the Welch method resulting in spectra with a 2 Hz resolution from 6 to 44 Hz. The most relevant EEG channels and frequencies were selected by a simple feature selection algorithm based on the overlap of the distributions of the different classes. For off-line ErrP detection, we applied a 1-10 3 Hz bandpass filter as ErrP are known to be a relatively slow cortical potential. EEG signals were then subsampled from 512 Hz to 64 Hz (i.e., we took one point out of 8) before classification, which was entirely based on temporal features. Indeed the actual input vector for the statistical classifier described below is a 150 ms window starting 250 ms after the feedback for channels FCz and Cz. The choice of these channels follows the fact that ErrP are characterized by a fronto-central distribution along the midline. For both mental tasks and ErrP classification, the two different classes (left or right for mental tasks and error or correct for ErrP) are recognized by a Gaussian classifier. The output of the statistical classifier is an estimation of the posterior class probability distribution for a single trial; i.e., the probability that a given single trial belongs to one of the two classes. In this statistical classifier, every Gaussian unit represents a prototype of one of the classes to be recognized, and we use several prototypes per class. During learning, the centers of the classes of the Gaussian units are pulled towards the trials of the class they represent and pushed away from the trials of the other class. No artifact rejection algorithm (for removing or filtering out eye or muscular movements) was applied and all trials were kept for analysis. It is worth noting, however, that after a visual a-posteriori check of the trials we found no evidence of muscular artifacts that could have contaminated one condition differently from the other. More details on the Gaussian classifier and the analysis procedure to rule out ocular/muscular artifacts as the relevant signals for both classifiers (BCI itself and ErrP) can be found in [10]. Figure 3: (Top) Discriminant power (DP) of frequencies. Sensory motor rhythm (12-16 Hz) and some beta components are discriminant for all subjects. (Bottom) Discriminant power (DP) of electrodes. The most relevant electrodes are in the central area (C3, C4 and Cz) according to the ERD/ERD location for hand and foot movement or imagination. 3 Experimental results 3.1 Mental tasks classification Subject were asked to imagine a movement of their left hand when the left target was proposed and to imagine a movement of their right foot when the right target was proposed (note that subject n◦1 4 was imagining left foot for the left target and right hand for the right target). The most relevant EEG channels and frequencies were selected by a simple feature selection algorithm based on the overlap of the distributions of the different classes. Figure 3 shows the discriminant power (DP) of frequencies (top) and electrodes (bottom) for the 6 subject. For frequencies, the DP is based on the best electrode, and for electrodes it is based on the best frequency. Table 1 shows the classification rates for the two mental tasks and the general BCI accuracy for the 6 subjects and the average of them, it also shows the features (electrodes and frequencies) used for classification. For all 6 subjects, the 12-16 Hz band (sensory motor rhythm (SMR)) appears to be relevant for classification. Subject 1, 3 and 5 show a peak in DP for frequencies around 25 Hz (beta band). For subject 2 this peak in the beta band is centered at 20 Hz and for subject 6 it is centered at 30 Hz. Finally subject 4 shows no particular discriminant power in the beta band. Previous studies confirm these results. Indeed, SMR and beta rhythm over left and/or right sensorimotor cortex have been successfully used for BCI control [11]. Event-related de-synchronization (ERD) and synchronization (ERS) refer to large-scale changes in neural processing. During periods of inactivity, brain areas are in a kind of idling state with large populations of neurons firing in synchrony resulting in an increase of amplitude of specific alpha (8-12 Hz) and beta (12-26 Hz) bands. During activity, populations of neurons work at their own pace and the power of this idling state is reduced, the cortex has become de-synchronized. [12]. In our case, the most relevant electrodes for all subjects are in the C3, C4 or Cz area. These locations confirm previous studies since C3 and C4 areas usually show ERD/ERS during hands movement or imagination whereas foot movement or imagination are focused in the Cz area [12]. Table 1: Percentages (mean and standard deviations) of correctly recognized single trials for the 2 motor imagination tasks for the 6 subjects and the average of them. All subjects show classification rates of about 70-75% for motor imagination and the general BCI accuracy is 73%. Features used for classification are also shown. Electrodes Frequencies Left hand Right foot Accuracy [Hz] [%] [%] [%] # 1* C3 CP3 CP1 CPz CP2 10 12 14 26 77.2 ± 3.7 70.4 ± 3.2 73.8 ± 4.8 # 2 C4 CP4 P4 10 12 14 18 20 22 71.8 ± 9.0 80.9 ± 7.1 76.4 ± 6.4 # 3 C3 C4 C6 CP6 CP4 14 16 26 76.4 ± 5.8 62.6 ± 6.7 69.5 ± 9.8 # 4 Cz C2 C4 12 14 79.6 ± 1.6 66.3 ± 10.1 73.0 ± 9.4 # 5 Cz C4 CP4 12 24 26 73.5 ± 16.1 71.9 ± 13.3 72.7 ± 1.1 # 6 CPz Cz CP6 CP4 12 14 28 30 32 77.9 ± 7.4 69.0 ± 13.7 73.5 ± 6.3 Avg 76.1 ± 2.9 70.2 ± 6.2 73.1 ± 4.2 * Left foot and Right hand All 6 subjects show classification rates of about 70-75% for motor imagination. These figures were achieved with a relatively low number of features (up to 5 electrodes and up to 6 frequencies) and the general BCI accuracy is 73%. This level of performance can appear relatively low for a 2-class BCI. However, keeping in mind that first all subjects had no prior BCI experience and second that these figures were obtained exclusively in prediction (i.e. classifiers were always tested on new data), the performance is satisfactory. 3.2 Error-related potentials Figure 4 shows the averages of error trials (red curve), of correct trials (green curve) and the difference error-minus-correct (blue curve) for channel FCz for the six subjects (top). A first small positive peak shows up about ∼230 ms after the feedback (t=0). A negative peak clearly appears ∼290 ms after the feedback for 5 subjects. This negative peak is followed by a positive peak ∼350 ms after the feedback. Finally a second broader negative peak occurs about ∼470 ms after the feedback. Figure 4 also shows the scalp potentials topographies (right) for the average of the six subjects, at the occurrence of the four previously described peaks: a first fronto-central positivity appears after ∼230 ms, followed by a fronto-central negativity at ∼290 ms, a fronto-central positivity at ∼350 ms and a fronto-central negativity at ∼470 ms. All six subjects show similar ErrP time courses whose amplitudes slightly differ from one subject to the other. These experiments seem to confirm the existence of a new kind of error-related potentials [6]. Furthermore, the fronto-central 5 focus at the occurrence of the different peaks tends to confirm the hypothesis that ErrP are generated in a deep brain region called anterior cingulate cortex [8][9] (see also Section 3.3). Table 2 reports the recognition rates (mean and standard deviations) for the six subjects plus the average of them. These results show that single-trial recognition of erroneous and correct responses are above 75% and 80%, respectively. Beside the crucial importance to integrate ErrP in the BCI in a way that the subject still feels comfortable, for example by reducing as much as possible the rejection of actually correct commands, a key point for the exploitation of the automatic recognition of interaction errors is that they translate into an actual improvement of the performance of the BCI. Table 2 also show the performance of the BCI in terms of bit rate (bits per trial) when detection of ErrP is used or not and the induced increase of performance (for details see [6]). The benefit of integrating ErrP detection is obvious since it at least doubles the bit rate for five of the six subjects and the average increase is 124%. Figure 4: (Top) Averages of error trials (red curve), of correct trials (green curve) and the difference errorminus-correct (blue curve) for channel FCz for the six subjects. All six subjects show similar ErrP time courses whose amplitudes slightly differ from one subject to the other. (Bottom) Scalp potentials topographies for the average of the six subjects, at the occurrence of the four described peaks. All focuses are located in frontocentral areas, over the anterior cingulate cortex (ACC). Table 2: Percentages (mean and standard deviations) of correctly recognized error trials and correct trials for the six subjects and the average of them. Table also show the BCI performance in terms of bit rate and its increase using ErrP detection. Classification rates are above 75% and 80% for error trials and correct trials, respectively. The benefit of integrating ErrP detection is obvious since it at least doubles the bit rate for five of the six subjects. Error Correct BCI accuracy [%] Bit rate [bits/trial] Increase [%] [%] (from Table 1) (no ErrP) (ErrP) [%] # 1 77.7 ± 13.9 76.8 ± 5.4 73.8 ± 4.8 0.170 0.345 103 # 2 75.4 ± 5.5 80.1 ± 7.9 76.4 ± 6.4 0.212 0.385 82 # 3 74.0 ± 12.9 85.9 ± 1.6 69.5 ± 9.8 0.113 0.324 187 # 4 84.3 ± 7.7 80.1 ± 5.5 73.0 ± 9.4 0.159 0.403 154 # 5 75.3 ± 6.0 85.6 ± 5.2 72.7 ± 1.1 0.154 0.371 141 # 6 70.7 ± 11.4 82.2 ± 5.1 73.5 ± 6.3 0.166 0.333 101 Avg 76.2 ± 4.6 81.8 ± 3.5 73.1 ± 4.2 0.160 0.359 124 3.3 Estimation of intracranial activity Estimating the neuronal sources that generate a given potential map at the scalp surface (EEG) requires the solution of the so-called inverse problem. This inverse problem is always initially undetermined, i.e. there is no unique solution since a given potential map at the surface can be 6 generated by many different intracranial activity map. The inverse problem requires supplementary a priori constraints in order to be univocally solved. The ultimate goal is to unmix the signals measured at the scalp and to attribute to each brain area its own estimated temporal activity. The sLORETA inverse model [7] is a standardized low resolution brain electromagnetic tomography. This software, known for its zero localization error, was used as a localization tool to estimate the focus of intracranial activity at the occurrence of the four ErrP peaks described in Section 3.2. Figure 5 shows Talairach slices of localized activity for the grand average of the six subjects at the occurrence of the four described peaks and at the occurrence of a late positive component showing up 650 ms after the feedback. As expected, the areas involved in error processing, namely the pre-supplementary motor area (pre-SMA, Brodmann area 6) and the rostral cingulate zone (RCZ, Brodmann areas 24 & 32) are systematically activated [8][9]. For the second positive peak (350 ms) and mainly for the late positive component (650 ms), parietal areas are also activated. These associative areas (somatosensory association cortex, Brodmann areas 5 & 7) could be related to the fact that the subject becomes aware of the error. It has been proposed that the positive peak was associated with conscious error recognition in case of error potentials elicited in reaction task paradigm [13]. In our case, activation of parietal areas after 350 ms after the feedback agrees with this hypothesis. Figure 5: Talairach slices of localized activity for the grand average of the six subjects at the occurrence of the four peaks described in Section 3.2 and at the occurrence of a late positive component showing up 650 ms after the feedback. Supplementary motor cortex and anterior cingulate cortex are systematically activated. Furthermore, for the second positive peak (350 ms) and mainly for the late positive component (650 ms), parietal areas are also activated. This parietal activation could reflect the fact that the subject is aware of the error. 4 Discussion In this study we have reported results on the detection of the neural correlate of error awareness for improving the performance and reliability of BCI. In particular, we have confirmed the existence of a new kind of error-related potential elicited in reaction to an erroneous recognition of the subject’s intention. More importantly, we have shown the feasibility of simultaneously and satisfactorily detecting erroneous responses of the interface and classifying motor imagination for device control at the level of single trials. However, the introduction of an automatic response rejection strongly interferes with the BCI. The user needs to process additional information which induces higher workload and may considerably slow down the interaction. These issues have to be investigated when running online BCI experiments integrating automatic error detection. Given the promising results obtained in this simulated human-robot interaction, we are currently working in the actual integration of online ErrP detection into our BCI system. The preliminary results are very promising and confirm that the online detection of errors is a tool of great benefit, especially for subjects with no prior BCI experience or showing low BCI performance. In parallel, we are exploring how to increase the recognition rate of single-trial erroneous and correct responses. 7 In this study we have also shown that, as expected, typical cortical areas involved in error processing such as pre-supplementary motor area and anterior cingulate cortex are systematically activated at the occurrence of the different peaks. The software used for the estimation of the intracranial activity (sLORETA) is only a localization tool. However, Babiloni et al. [14] have recently developed the so-called CCD (“cortical current density”) inverse model that estimates the activity of the cortical mantle. Since ErrP seems to be generated by cortical areas, we plan to use this method to best discriminate erroneous and correct responses of the interface. As a matter of fact, a key issue to improve classification is the selection of the most relevant current dipoles out of a few thousands. In fact, the very preliminary results using the CCD inverse model confirm the reported localization in the pre-supplementary motor area and in the anterior cingulate cortex and thus we may well expect a significant improvement in recognition rates by focusing on the dipoles estimated in those specific brain areas. More generally, the work described here suggests that it could be possible to recognize in real time high-level cognitive and emotional states from EEG (as opposed, and in addition, to motor commands) such as alarm, fatigue, frustration, confusion, or attention that are crucial for an effective and purposeful interaction. Indeed, the rapid recognition of these states will lead to truly adaptive interfaces that customize dynamically in response to changes of the cognitive and emotional/affective states of the user. References [1] J.R. Wolpaw, N. Birbaumer, D.J. McFarland, G. Pfurtscheller, and T.M. Vaughan. Brain-computer interfaces for communication and control. Clinical Neurophysiology, 113:767–791, 2002. [2] J. del R. Mill´an, F. Renkens, J. Mouri˜no, and W. Gerstner. Non-invasive brain-actuated control of a mobile robot by human EEG. IEEE Transactions on Biomedical Engineering, 51:1026–1033, 2004. [3] G. Schalk, J.R. Wolpaw, D.J. McFarland, and G. Pfurtscheller. EEG-based communication: presence of and error potential. Clinical Neurophysiology, 111:2138–2144, 2000. [4] B. Blankertz, G. Dornhege, C. Sch¨afer, R. Krepki, J. Kohlmorgen, K.-R. M¨uller, V. Kunzmann, F. Losch, and G. Curio. Boosting bit rates and error detection for the classification of fast-paced motor commands based on single-trial EEG analysis. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 11(2):127–131, 2003. [5] L.C. Parra, C.D. Spence, A.D. Gerson, and P. Sajda. Response error correction—a demonstration of improved human-machine performance using real-time EEG monitoring. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 11(2):173–177, 2003. [6] P.W. Ferrez and J. del R. Mill´an. You are wrong!—Automatic detection of interaction errors from brain waves. In Proc. 19th Int. Joint Conf. Artificial Intelligence, 2005. [7] R.D. Pascual-Marqui. Standardized low resolution brain electromagnetic tomography (sLORETA): Technical details. Methods & Findings in Experimental & Clinical Pharmacology, 24D:5–12, 2002. [8] C.B. Holroyd and M.G.H. Coles. The neural basis of human error processing: Reinforcement learning, dopamine and the error-related negativity. Psychological Review, 109:679–709, 2002. [9] K. Fiehler, M. Ullsperger, and Y. von Cramon. Neural correlates of error detection and error correction: Is there a common neuroanatomical substrate? European Journal of Neuroscience, 19:3081–3087, 2004. [10] P.W. Ferrez and J. del R. Mill´an. Error-related EEG potentials in brain-computer interfaces. In G. Dornhege, J. del R. Mill´an, T. Hinterberger, D. McFarland, and K.-R. M¨uller, editors, Toward Brain-Computing Interfacing, pages 291–301. The MIT Press, 2007. [11] D. McFarland and J.R. Wolpow. Sensorimotor rhythm-based brain-computer interface (BCI): Feature selection by regression improves performance. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 13(3):372–379, 2005. [12] G. Pfurtscheller and F.H. Lopes da Silva. Event-related EEG/MEG synchronization and desynchronization: Basic principles. Clinical Neurophysiology, 110:1842–1857, 1999. [13] S. Nieuwenhuis, K.R. Ridderinkhof, J. Blom, G.P.H. Band, and A. Kok. Error-related brain potentials are differently related to awareness of response errors: Evidence from an antisaccade task. Psychophysiology, 38:752–760, 2001. [14] F. Babiloni, C. Babiloni, L. Locche, F. Cincotti, P.M. Rossini, and F. Carducci. High-resolution electroencephalogram: Source estimates of laplacian-transformed somatosensory-evoked potentials using realistic subject head model constructed from magnetic resonance imaging. Medical & Biological Engineering and Computing, 38:512–519, 2000. 8
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Invariant Common Spatial Patterns: Alleviating Nonstationarities in Brain-Computer Interfacing Benjamin Blankertz1,2 Motoaki Kawanabe2 Ryota Tomioka3 Friederike U. Hohlefeld4 Vadim Nikulin5 Klaus-Robert Müller1,2 1TU Berlin, Dept. of Computer Science, Machine Learning Laboratory, Berlin, Germany 2Fraunhofer FIRST (IDA), Berlin, Germany 3Dept. Mathematical Informatics, IST, The University of Tokyo, Japan 4Berlin School of Mind and Brain, Berlin, Germany 5Dept. of Neurology, Campus Benjamin Franklin, Charité University Medicine Berlin, Germany {blanker,krm}@cs.tu-berlin.de Abstract Brain-Computer Interfaces can suffer from a large variance of the subject conditions within and across sessions. For example vigilance fluctuations in the individual, variable task involvement, workload etc. alter the characteristics of EEG signals and thus challenge a stable BCI operation. In the present work we aim to define features based on a variant of the common spatial patterns (CSP) algorithm that are constructed invariant with respect to such nonstationarities. We enforce invariance properties by adding terms to the denominator of a Rayleigh coefficient representation of CSP such as disturbance covariance matrices from fluctuations in visual processing. In this manner physiological prior knowledge can be used to shape the classification engine for BCI. As a proof of concept we present a BCI classifier that is robust to changes in the level of parietal α-activity. In other words, the EEG decoding still works when there are lapses in vigilance. 1 Introduction Brain-Computer Interfaces (BCIs) translate the intent of a subject measured from brain signals directly into control commands, e.g. for a computer application or a neuroprosthesis ([1, 2, 3, 4, 5, 6]). The classical approach to brain-computer interfacing is operant conditioning ([2, 7]) where a fixed translation algorithm is used to generate a feedback signal from the electroencephalogram (EEG). Users are not equipped with a mental strategy they should use, rather they are instructed to watch a feedback signal and using the feedback to find out ways to voluntarily control it. Successful BCI operation is reinforced by a reward stimulus. In such BCI systems the user adaption is crucial and typically requires extensive training. Recently machine learning techniques were applied to the BCI field and allowed to decode the subject’s brain signals, placing the learning task on the machine side, i.e. a general translation algorithm is trained to infer the specific characteristics of the user’s brain signals [8, 9, 10, 11, 12, 13, 14]. This is done by a statistical analysis of a calibration measurement in which the subject performs well-defined mental acts like imagined movements. Here, in principle no adaption of the user is required, but it is to be expected that users will adapt their behaviour during feedback operation. The idea of the machine learning approach is that a flexible adaption of the system relieves a good amount of the learning load from the subject. Most BCI systems are somewhere between those extremes. 1 Although the proof-of-concept of machine learning based BCI systems1 was given some years ago, several major challenges are still to be faced. One of them is to make the system invariant to non task-related fluctuations of the measured signals during feedback. These fluctuations may be caused by changes in the subject’s brain processes, e.g. change of task involvement, fatigue etc., or by artifacts such as swallowing, blinking or yawning. The calibration measurement that is used for training in machine learning techniques is recorded during 10-30 min, i.e. a relatively short period of time and typically in a monotone atmosphere, so this data does not contain all possible kinds of variations to be expected during on-line operation. The present contribution focusses on invariant feature extraction for BCI. In particular we aim to enhance the invariance properties of the common spatial patterns (CSP, [15]) algorithm. CSP is the solution of a generalized eigenvalue problem and has as such a strong link to the maximization of a Rayleigh coefficient, similar to Fisher’s discriminant analysis. Prior work by Mika et al. [16] in the context of kernel Fisher’s discriminant analysis contains the key idea that we will follow: noise and distracting signal aspects with respect to which we want to make our feature extractor invariant is added to the denominator of a Rayleigh coefficient. In other words, our prior knowledge about the noise type helps to re-design the optimization of CSP feature extraction. We demonstrate how our invariant CSP (iCSP) technique can be used to make a BCI system invariant to changes in the power of the parietal α-rhythm (see Section 2) reflecting, e.g. changes in vigilance. Vigilance changes are among the most pressing challenges when robustifying a BCI system for long-term real-world applications. In principle we could also use an adaptive BCI, however, adaptation typically has a limited time scale which might not allow to follow fluctuations quickly enough. Furthermore online adaptive BCI systems have so far only been operated with 4-9 channels. We would like to stress that adaptation and invariant classification are no mutually exclusive alternatives but rather complementary approaches when striving for the same goal: a BCI system that is invariant to undesired distortions and nonstationarities. 2 Neurophysiology and Experimental Paradigms Neurophysiological background. Macroscopic brain activity during resting wakefulness contains distinct ‘idle’ rhythms located over various brain areas, e.g. the parietal α-rhythm (7-13Hz) can be measured over the visual cortex [17] and the µ-rhythm can be measured over the pericentral sensorimotor cortices in the scalp EEG, usually with a frequency of about 8–14Hz ([18]). The strength of the parietal α-rhythm reflects visual processing load as well as attention and fatigue resp. vigilance. The moment-to-moment amplitude fluctuations of these local rhythms reflect variable functional states of the underlying neuronal cortical networks and can be used for brain-computer interfacing. Specifically, the pericentral µ- and β rythms are diminished, or even almost completely blocked, by movements of the somatotopically corresponding body part, independent of their active, passive or reflexive origin. Blocking effects are visible bilateral but with a clear predominance contralateral to the moved limb. This attenuation of brain rhythms is termed event-related desynchronization (ERD) and the dual effect of enhanced brain rhythms is called event-related synchronization (ERS) (see [19]). Since a focal ERD can be observed over the motor and/or sensory cortex even when a subject is only imagining a movement or sensation in the specific limb, this feature can be used for BCI control: The discrimination of the imagination of movements of left hand vs. right hand vs. foot can be based on the somatotopic arrangement of the attenuation of the µ and/or β rhythms. However the challenge is that due to the volume conduction EEG signal recorded at the scalp is a mixture of many cortical activities that have different spatial localizations; for example, at the electrodes over the mortor cortex, the signal not only contains the µ-rhythm that we are interested in but also the projection of parietal α-rhythm that has little to do with the motor-imagination. To this end, spatial filtering is an indispensable technique; that is to take a linear combination of signals recorded over EEG channels and extract only the component that we are interested in. In particular the CSP algorithm that optimizes spatial filters with respect to discriminability is a good candidate for feature extraction. Experimental Setup. In this paper we evaluate the proposed algorithm on off-line data in which the nonstationarity is induced by having two different background conditions for the same primary 1Note: In our exposition we focus on EEG-based BCI systems that does not rely on evoked potentials (for an extensive overview of BCI systems including invasive and systems based on evoked potentials see [1]). 2 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0 0.1 0.2 0.3 0.4 0.5 Figure 1: Topographies of r2–values (multiplied by the sign of the difference) quantifying the difference in log band-power in the alpha band (8–12 Hz) between different recording sessions: Left: Difference between imag_move and imag_lett. Due to lower visual processing demands, alpha power in occipital areas is stronger in imag_lett. Right: Difference between imag_move and sham_feedback. The latter has decreased alpha power in centro-parietal areas. Note the different sign in the colormaps. task. The ultimate challenge will be on-line feedback with strong fluctuations of task demands etc, a project envisioned for the near future. We investigate EEG recordings from 4 subjects (all from whom we have an ‘invariance measurement’, see below). Brain activity was recorded from the scalp with multi-channel amplifiers using 55 EEG channels. In the ‘calibration measurement’ all 4.5–6 seconds one of 3 different visual stimuli indicated for 3 seconds which mental task the subject should accomplish during that period. The investigated mental tasks were imagined movements of the left hand, the right hand, and the right foot. There were two types of visual stimulation: (1: imag_lett) targets were indicated by letters (L, R, F) appearing at a central fixation cross and (2: imag_move) a randomly moving small rhomboid with either its left, right or bottom corner filled to indicate left or right hand or foot movement, respectively. Since the movement of the object was independent from the indicated targets, target-uncorrelated eye movements are induced. Due to the different demands in visual processing, the background brain activity can be expected to differ substancially in those two types of recordings. The topography of the r2–values (bi-serial correlation coefficient of feature values with labels) of the log band-power difference between imag_move and imag_lett is shown in the left plot of Fig. 2. It shows a pronounced differene in parietal areas. A sham_feedback paradigm was designed in order to charaterize invariance properties needed for stable real-world BCI applications. In this measurement the subjects received a fake feedback sequence which was preprogrammed. The aim of this recording was to collect data during a large variety of mental states and actions that are not correlated with the BCI control states (motor imagery of hands and feet). Subjects were told that they could control the feedback in some way that they should find out, e.g. with eye movements or muscle activity. They were instructed not to perform movements of hands, arms, legs and feet. The type of feedback was a standard 1D cursor control. In each trial the cursor starts in the middle and should be moved to either the left or right side as indicated by a target cue. When the cursor touched the left or right border, a response (correct or false) was shown. Furthermore the number of hits and misses was shown. The preprogrammed ‘feedback’ signal was constructed such that it was random in the beginning and then alternating periods of increasingly more hits and periods with chance level performance. This was done to motivate the subjects to try a variety of different actions and to induce different states of mood (satisfaction during ‘successful’ periods and anger resp. disfavor during ‘failure’). The right plot of Fig. 2 visualizes the difference in log band-power between imag_move and sham_feedback. A decreased alpha power in centro-parietal areas during sham_feedback can be observed. Note that this recording includes much more variations of background mental activity than the difference between imag_move and imag_lett. 3 Methods Common Spatial Patterns (CSP) Analysis. The CSP technique ([15]) allows to determine spatial filters that maximize the variance of signals of one condition and at the same time minimize the variance of signals of another condition. Since variance of band-pass filtered signals is equal to bandpower, CSP filters are well suited to discriminate mental states that are characterized by ERD/ERS effects ([20]). As such it has been well used in BCI systems ([8, 14]) where CSP filters are calculated individually for each subject on the data of a calibration measurement. Technically the Common Spatial Pattern (CSP) [21] algorithm gives spatial filters based on a discriminative criterion. Let X1 and X2 be the (time × channel) data matrices of the band-pass filtered 3 EEG signals (concatenated trials) under the two conditions (e.g., right-hand or left-hand imagination, respectively2) and Σ1 and Σ2 be the corresponding estimates of the covariance matrices Σi = X⊤ i Xi. We define the two matrices Sd and Sc as follows: Sd = Σ(1) −Σ(2) : discriminative activity matrix, Sc = Σ(1) +Σ(2) : common activity matrix. The CSP spatial filter v ∈RC (C is the number of channels) can be obtained by extremizing the Rayleigh coefficient: {max, min}v∈RC v⊤Sdv v⊤Scv . (1) This can be done by solving a generalized eigenvalue problem. Sdv = λScv. (2) The eigenvalue λ is bounded between −1 and 1; a large positive eigenvalue corresponds to a projection of the signal given by v that has large power in the first condition but small in the second condition; the converse is true for a large negative eigenvalue. The largest and the smallest eigenvalues correspond to the maximum and the minimum of the Rayleigh coefficient problem (Eq. (1)). Note that v⊤Sdv = v⊤Σ1v −v⊤Σ2v is the average power difference in two conditions that we want to maximize. On the other hand, the projection of the activity that is common to two classes v⊤Scv should be minimized because it doesn’t contribute to the discriminability. Using the same idea from [16] we can rewrite the Rayleigh problem (Eq. (1)) as follows: min v∈RC v⊤Scv, s.t. v⊤Σ1v−v⊤Σ2v = λ, which can be interpreted as finding the minimum norm v with the condition that the average power difference between two conditions to be λ. The norm is defined by the common activity matrix Sc. In the next section, we extend the notion of Sc to incorporate any disturbances that is common to two classes that we can measure a priori. In this paper we call filter the generalized eigenvectorsvj (j = 1,...,C) of the generalized eigenvalue problem (Eq. (2)) or a similar problem discussed in the next section. Moreover we denote by V the matrix we obtain by putting the C generalized eigenvectors into columns, namely V = {vj}C j=1 ∈ RC×C and call patterns the row vectors of the inverse A = V −1. Note that a filter vj ∈RC has its corresponding pattern aj ∈RC; a filter vj extracts only the activity spanned by a j and cancels out all other activities spanned by ai (i ̸= j); therefore a pattern a j tells what the filter vj is extracting out (see Fig. 2). For classification the features of single-trials are calculated as the log-variance in CSP projected signals. Here only a few (2 to 6) patterns are used. The selection of patterns is typically based on eigenvalues. But when a large amount of calibration data is not available it is advisable to use a more refined technique to select the patterns or to manually choose them by visual inspection. The variance features are approximately chi-square distributed. Taking the logarithm makes them similar to gaussian distributions, so a linear classifier (e.g., linear discriminant analysis) is fine. For the evaluation in this paper we used the CSPs corresponding the the two largest and the two smallest eigenvalues and used linear disciminant analysis for classification. The CSP algorithm, several extentions as well as practical issues are reviewed in detail in [15]. Invariant CSP. The CSP spatial filters extracted as above are optimized for the calibration measurement. However, in online operation of the BCI system different non task-related modulations of brain signals may occur which are not suppressed by the CSP filters. The reason may be that these modulations have not been recorded in the calibration measurement or that they have been so infrequent that they are not consistently reflected in the statistics (e.g. when they are not equally distributed over the two conditions). The proposed iCSP method minimizes the influence of modulations that can be characterized in advance by a covariance matrix. In this manner we can code neurophysiological prior knowledge 2We use the term covariance for zero-delay second order statistics between channels and not for the statistical variability. Since we assume the signal to be band-pass filtered, the second order statistics reflects band power. 4 or further information such as the tangent covariance matrix ([22]) into such a covariante matrix Ξ. In the following motivation we assume that Ξ is the covariance matrix of a signal matrix Y. Using the notions from above, the objective is then to calculate spatial filters v(1) j such that var(X1v(1) j ) is maximized and var(X2v(1) j ) and var(Yv(1) j ) are minimized. Dually spatial filters v(2) j are determined that maximize var(X2v(2) j ) and minimize var(X1v(2) j ) and var(Yv(2) j ). Pratically this can be accomplished by solving the following two generalized eigenvalue problems: V (1)⊤Σ1V (1) = D(1) and V (1)⊤((1−ξ)(Σ1 +Σ2)+ξΞ)V(1) = I (3) V (2)⊤Σ2V (2) = D(2) and V (2)⊤((1−ξ)(Σ1 +Σ2)+ξΞ)V(2) = I (4) where ξ ∈[0,1] is a hyperparameter to trade-off the discrimination of the training classes (X1, X2 ) against invariance (as characterized by Ξ). Section 4 discusses the selection of parameter ξ. Filters v(1) j with high eigenvalues d(1) j provide not only high var(X1v(1) j ) but also small v(1) j ⊤ ((1 −ξ)Σ2 +ξΞ)v(1) j = 1 −(1 −ξ)d(1) j , i.e. small var(X2v(1) j ) and small var(Yv(1) j ). The dual is true for the selection of filters from v(2) j . Note that for ξ = 0.5 there is a strong connection to the one-vs-rest strategy for 3-class CSP ([23]). Features for classification are calculated as log-variance using the two filters from each of v(1) j and v(2) j corresponding to the largest eigenvalues. Note that the idea of iCSP is in the spirit of the invariance constraints in (kernel) Fisher’s Discriminant proposed in [16]. A Theoretical Investigation of iCSP by Influence Analysis. As mentioned, iCSP is aiming at robust spatial filtering against disturbances whose covariance Ξ can be anticipated from prior knowledge. Influence analysis is a statistical tool with which we can assess robustness of inference procedures [24]. Basically, it evaluates the effect in inference procedures, if we add a small perturbation of O(ε), where ε ≪1. For example, influence functions for the component analyses such as PCA and CCA have been discussed so far [25, 26]. We applied the machinery to iCSP, in order to check whether iCSP really reduces influence caused by the disturbance at least in local sense. For this purpose, we have the following lemma (its proof is included in the Appendix). Lemma 1 (Influence of generalized eigenvalue problems) Let λk and wk be k-th eigenvalue and eigenvector of the generalized eigvenvalue problem Aw = λBw, (5) respectively. Suppose that the matrices A and B are perturbed with small matrices ε∆and εP where ε ≪1. Then the eigenvalues ewk and eigenvectors eλk of the purterbed problem (A+ε∆)ew = eλ(B+εP)ew (6) can be expanded as λk +εχk +o(ε) and wk +εψk +o(ε), where χk = w⊤ k (∆−λkP)wk, ψk = −Mk(∆−λkP)wk −1 2(w⊤ k Pwk)wk, (7) Mk := B−1/2(B−1/2AB−1/2 −λkI)+B−1/2 and the suffix ’+’ denotes Moore-Penrose matrix inverse. The generalized eigenvalue problem eqns (3) and (4) can be rephrased as Σ1v = d{(1 −ξ)(Σ1 +Σ2)+ξΞ}v, Σ2u = c{(1 −ξ)(Σ1 +Σ2)+ξΞ}u. For simplicity, we consider here the simplest perturbation of the covariances as Σ1 →Σ1 + εΞ and Σ2 →Σ1 + εΞ. In this case, the perturbation matrices in the lemma can be expressed as ∆1 = Ξ, ∆2 = Ξ, P = 2(1 −ξ)Ξ. Therefore, we get the expansions of the eigenvalues and eigenvectors as dk +εχ1k, ck +εχ2k, vk +εψ1k and uk +εψ2k, where χ1k = {1 −2(1 −ξ)dk}v⊤ k Ξvk, χ2k = {1 −2(1 −ξ)ck}u⊤ k Ξuk, (8) ψ1k = −{1 −2(1 −ξ)dk}M1kΞvk −(1 −ξ)(v⊤ k Ξvk)vk, (9) ψ2k = −{1 −2(1 −ξ)ck}M2kΞuk −(1 −ξ)(u⊤ k Ξuk)uk, (10) 5 alpha=0.0 alpha=0.5 alpha=1.0 alpha=2.0 original CSP − error: 10.7% / 11.4% / 12.9% / 37.9% alpha=0.0 alpha=0.5 alpha=1.0 alpha=2.0 invariant CSP − error: 9.3% / 10.0% / 9.3% / 11.4% α=0 α=0 α=0.5 α=1 α=2 α=0.5 α=1 α=2 10.7% 11.4% 12.9% 37.9% 9.3% 10.0% 9.3% 11.4% filter filter pattern pattern errors original CSP invariant CSP Figure 2: Comparison of CSP and iCSP on test data with artificially increased occipital alpha. The upper plots show the classifier output on the test data with different degrees of alpha added (factors α= 0, 0.5, 1, 2). The lower panel shows the filter/pattern coefficients topographically mapped on the scalp from original CSP (left) and iCSP (right). Here the invariance property was defined with respect to the increase in the alpha activity in the visual cortex (occipital location) using an eyes open/eyes closed recording. See Section 3 for the definition of filter and pattern. M1k := Σ−1/2(Σ−1/2Σ1Σ−1/2 −dkI)+Σ−1/2, M2k := Σ−1/2(Σ−1/2Σ2Σ−1/2 −dkI)+Σ−1/2, and Σ := (1 −ξ)(Σ1 + Σ2) + ξΞ. The implication of the result is the following. If ξ = 1 − 1 2dk (resp. ξ = 1 − 1 2ck ) is satisfied, the O(ε) term χ1k (resp. χ2k) of the k-th eigenvalue vanishes and also the k-th eigenvector does coincide with the one for the original problem up to ε order, because the first term of ψ1k (resp. ψ2k) becomes zero (we note that dk and ck also depend on ξ). 4 Evaluation Test Case with Constructed Test Data. To validate the proposed iCSP, we first applied it to specifically constructed test data. iCSP was trained (ξ = 0.5) on motor imagery data with the invariance characterized by data from a measurement during ‘eyes open’ (approx. 40s) and ‘eyes closed’ (approx. 20 s). The motor imagery test data was used in its original form and variants that were modified in a controlled manner: From another data set during ‘eyes closed’ we extracted activity related to increased occipital alpha activity (backprojection of 5 ICA components) and added this with 3 different factors (α = 0.5, 1, 2) to the test data. The upper plots of Fig. 2 display the classifier output on the constructed test data. While the performance of the original CSP is more and more deteriorated with increased alpha mixed in, the proposed iCSP method maintains a stable performance independent of the amount of increased alpha activity. The spatial filters that were extracted by CSP analysis vs. the proposed iCSP often look quite similar. However, tiny but apparently important differences exist. In the lower panel of Fig. 2 the filter (vj) pattern (aj) pairs from original CSP (left) and iCSP (right) are shown. The filters from two approaches resemble each other strongly. However, the corresponding patterns reveal an important difference. While the pattern of the original CSP has positive weights at the right occipital side which might be susceptible to α modulations, the corresponding iCSP has not. A more detailed inspection shows that both filters have a focus over the right (sensori-) motor cortex, but only the invariant filter has a spot of opposite sign right posterior to it. This spot will filter out contributions coming from occipital/parietal sites. Model selection for iCSP. For each subject, a cross-validation was performed for different values of ξ on the training data (session imag_move) and the ξ resulting in minimum error was chosen. For the same values of ξ the iCSP filters + LDA classifier trained on imag_move were applied to calcu6 0 0.2 0.4 0.6 0.8 0 5 10 15 20 25 30 35 xi error [%] Subject cv test train 0 0.2 0.4 0.6 0.8 0 5 10 15 20 25 30 35 xi error [%] Subject zv test train 0 0.2 0.4 0.6 0.8 0 5 10 15 20 25 30 35 xi error [%] Subject zk test train 0 0.2 0.4 0.6 0.8 0 5 10 15 20 25 30 35 xi error [%] Subject zq test train CSP iCSP 0 5 10 15 20 25 error [%] cv zv zk zq Figure 3: Modelselection and evaluation. Left subplots: Selection of hyperparameter ξ of the iCSP method. For each subject, a cross-validation was performed for different values of ξ on the training data (session imag_move), see thin black line, and the ξ resulting in minimum error was chosen (circle). For the same values of ξ the iCSP filters + LDA classifier trained on imag_move were applied to calculate the test error on data from imag_lett (thick colorful line). Right plot: Test error in all four recordings for classical CSP and the proposed iCSP (with model parameter ξ chosen by cross-validation on the training set as described in Section 4). late the test error on data from imag_lett. Fig. 3 (left plots) shows the result of this procedure. The shape of the cross-validation error on the training set and the test error is very similar. Accordingly, the selection of values for parameter ξ is successful. For subject zq ξ = 0 was chosen, i.e. classical CSP. The case for subject zk shows that the selection of ξ may be a delicate issue. For larges values of ξ cross-validation error and test error differ dramatically. A choice of ξ > 0.5 would result in bad performance of iCSP, while this effect could have not been predicted so severely from the cross-validation of the training set. Evaluation of Performance with Real BCI Data. For evaluation we used the imag_move session (see Section 2) as training set and the imag_lett session as test set. Fig 3 (right plot) compares the classification error obtained by classical CSP and by the proposed method iCSP with model parameter ξ chosen by cross-validation on the training set as described above. Again an excellent improvement is visible. 5 Concluding discussion EEG data from Brain-Computer Interface experiments are highly challenging to evaluate due to noise, nonstationarity and diverse artifacts. Thus, BCI provides an excellent testbed for testing the quality and applicability of robust machine learning methods (cf. the BCI Competitions [27, 28]). Obviously BCI users are subject to variations in attention and motivation. These types of nonstationarities can considerably deteriorate the BCI classifier performance. In present paper we proposed a novel method to alleviate this problem. A limitation of our method is that variations need to be characterized in advance (by estimating an appropriate covariance matrix). At the same time this is also a strength of our method as neurophysiological prior knowledge about possible sources of non-stationarity is available and can thus be taken into account in a controlled manner. Also the selection of hyperparameter ξ needs more investigation, cf. the case of subject zk in Fig. 3. One strategy to pursue is to update the covariance matrix Ξ online with incoming test data. (Note that no label information is needed.) Online learning (learning algorithms for adaptation within a BCI session) could also be used to further stabilize the system against unforeseen changes. It remains to future research to explore this interesting direction. Appendix: Proof of Lemma 1. By substituting the expansions of eλk and ewk to Eq.(6) and taking the O(ε) term, we get Aψk +∆wk = λkBψk +λkPwk + χkBwk. (11) Eq.(7) can be obtained by multiplying w⊤ k to Eq.(11) and applying Eq.(5). Then, from Eq.(11), (A−λkB)ψk = −(∆−λkP)wk + χkBwk = −(A−λkB)Mk(∆−λkP)wk, 7 holds, where we used the constraints w⊤ j Bwk = δ jk and (A−λkB)Mk = ∑ j̸=k Bwjw⊤ j = I −Bwkw⊤ k . (12) Eq.(12) can be proven by B−1/2AB−1/2 −λkI = ∑j̸=k λ jB1/2w jw⊤ j B1/2 and (B−1/2AB−1/2 −λkI)+ = ∑j̸=k 1/λ jB1/2w jw⊤ j B1/2. Since span{wk} is the kernel of the operator A −λkB, ψk can be explained as ψk = −Mk(∆−λkP)wk + cwk. 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The Infinite Gamma-Poisson Feature Model Michalis K. Titsias School of Computer Science, University of Manchester, UK mtitsias@cs.man.ac.uk Abstract We present a probability distribution over non-negative integer valued matrices with possibly an infinite number of columns. We also derive a stochastic process that reproduces this distribution over equivalence classes. This model can play the role of the prior in nonparametric Bayesian learning scenarios where multiple latent features are associated with the observed data and each feature can have multiple appearances or occurrences within each data point. Such data arise naturally when learning visual object recognition systems from unlabelled images. Together with the nonparametric prior we consider a likelihood model that explains the visual appearance and location of local image patches. Inference with this model is carried out using a Markov chain Monte Carlo algorithm. 1 Introduction Unsupervised learning using mixture models assumes that one latent cause is associated with each data point. This assumption can be quite restrictive and a useful generalization is to consider factorial representations which assume that multiple causes have generated the data [11]. Factorial models are widely used in modern unsupervised learning algorithms; see e.g. algorithms that model text data [2, 3, 4]. Algorithms for learning factorial models should deal with the problem of specifying the size of the representation. Bayesian learning and especially nonparametric methods such as the Indian buffet process [7] can be very useful for solving this problem. Factorial models usually assume that each feature occurs once in a given data point. This is inefficient to model the precise generation mechanism of several data such as images. An image can contain views of multiple object classes such as cars and humans and each class may have multiple occurrences in the image. To deal with features having multiple occurrences, we introduce a probability distribution over sparse non-negative integer valued matrices with possibly an unbounded number of columns. Each matrix row corresponds to a data point and each column to a feature similarly to the binary matrix used in the Indian buffet process [7]. Each element of the matrix can be zero or a positive integer and expresses the number of times a feature occurs in a specific data point. This model is derived by considering a finite gamma-Poisson distribution and taking the infinite limit for equivalence classes of non-negative integer valued matrices. We also present a stochastic process that reproduces this infinite model. This process uses the Ewens’s distribution [5] over integer partitions which was introduced in population genetics literature and it is equivalent to the distribution over partitions of objects induced by the Dirichlet process [1]. The infinite gamma-Poisson model can play the role of the prior in a nonparametric Bayesian learning scenario where both the latent features and the number of their occurrences are unknown. Given this prior, we consider a likelihood model which is suitable for explaining the visual appearance and location of local image patches. Introducing a prior for the parameters of this likelihood model, we apply Bayesian learning using a Markov chain Monte Carlo inference algorithm and show results in some image data. 2 The finite gamma-Poisson model Let X = {X1, . . . , XN} be some data where each data point Xn is a set of attributes. In section 4 we specify Xn to be a collection of local image patches. We assume that each data point is associated with a set of latent features and each feature can have multiple occurrences. Let znk denote the number of times feature k occurs in the data point Xn. Given K features, Z = {znk} is a N × K non-negative integer valued matrix that collects together all the znk values so as each row corresponds to a data point and each column to a feature. Given that znk is drawn from a Poisson with a feature-specific parameter λk, Z follows the distribution P(Z|{λk}) = N Y n=1 K Y k=1 λznk k exp{−λk} znk! = K Y k=1 λmk k exp{−Nλk} QN n=1 znk! , (1) where mk = PN n=1 znk. We further assume that each λk parameter follows a gamma distribution that favors sparsity (in a sense that will be explained shortly): G(λk; α K , 1) = λ α K −1 k exp{−λk} Γ( α K ) . (2) The hyperparameter α itself is given a vague gamma prior G(α; α0, β0). Using the above equations we can easily integrate out the parameters {λk} as follows P(Z|α) = K Y k=1 Γ(mk + α K ) Γ( α K )(N + 1)mk+ α K QN n=1 znk! , (3) which shows that given the hyperparameter α the columns of Z are independent. Note that the above distribution is exchangeable since reordering the rows of Z does not alter the probability. Also as K increases the distribution favors sparsity. This can be shown by taking the expectation of the sum of all elements of Z. Since the columns are independent this expectation is K PN n=1 E(znk) and E(znk) is given by E(znk) = ∞ X znk=0 znkNB(znk; α K , 1 2) = α K , (4) where NB(znk; r, p), with r > 0 and 0 < p < 1, denotes the negative binomial distribution over positive integers NB(znk; r, p) = Γ(r + znk) znk!Γ(r) pr(1 −p)znk, (5) that has a mean equal to r(1−p) p . Using Equation (4) the expectation of the sum of znks is αN and is independent of the number of features. As K increases, Z becomes sparser and α controls the sparsity of this matrix. There is an alternative way of deriving the joint distribution P(Z|α) according to the following generative process: (θ1, . . . , θK) ∼D α K , λ ∼G(λ; α, 1), Ln ∼Poisson(λ), (zn1, . . . , znK) ∼ Ln zn1 . . . znK K Y k=1 θznk k , n = 1, . . . , N, where D( α K ) denotes the symmetric Dirichlet. Marginalizing out θ and λ gives rise to the same distribution P(Z|α). The above process generates a gamma random variable and multinomial parameters and then samples the rows of Z independently by using the Poisson-multinomial pair. The connection with the Dirichlet-multinomial pair implies that the infinite limit of the gamma-Poisson model must be related to the Dirichlet process. In the next section we see how this connection is revealed through the Ewens’s distribution [5]. Models that combine gamma and Poisson distributions are widely applied in statistics. We point out that the above finite model shares similarities with the techniques presented in [3, 4] that model text data. 3 The infinite limit and the stochastic process To express the probability distribution in (3) for infinite many features K we need to consider equivalence classes of Z matrices similarly to [7]. The association of columns in Z with features defines an arbitrary labelling of the features. Given that the likelihood p(X|Z) is not affected by relabelling the features, there is an equivalence class of matrices that all can be reduced to the same standard form after column reordering. We define the left-ordered form of non-negative integer valued matrices as follows. We assume that for any possible znk holds znk ≤c −1, where c is a sufficiently large integer. We define h = (z1k . . . zNk) as the integer number associated with column k that is expressed in a numeral system with basis c. The left-ordered form is defined so as the columns of Z appear from left to right in a decreasing order according to the magnitude of their numbers. Starting from Equation (3) we wish to define the probability distribution over matrices constrained in a left-ordered standard form. Let Kh be the multiplicity of the column with number h; for example K0 is the number of zero columns. An equivalence class [Z] consists of K! PcN −1 h=0 Kh! different matrices that they are generated from the distribution in (3) with equal probabilities and can be reduced to the same left-ordered form. Thus, the probability of [Z] is P([Z]) = K! PcN−1 h=0 Kh! K Y k=1 Γ(mk + α K ) Γ( α K )(N + 1)mk+ α K QN n=1 znk! . (6) We assume that the first K+ features are represented i.e. mk > 0 for k ≤K+, while the rest K−K+ features are unrepresented i.e. mk = 0 for k > K+. The infinite limit of (6) is derived by following a similar strategy with the one used for expressing the distribution over partitions of objects as a limit of the Dirichlet-multinomial pair [6, 9]. The limit takes the following form: P(Z|α) = 1 PcN−1 h=1 Kh! αK+ (N + 1)m+α QK+ k=1(mk −1)! QK+ k=1 QN n=1 znk! , (7) where m = PK+ k=1 mk. This expression defines an exchangeable joint distribution over non-negative integer valued matrices with infinite many columns in a left-ordered form. Next we present a sequential stochastic process that reproduces this distribution. 3.1 The stochastic process The distribution in Equation (7) can be derived from a simple stochastic process that constructs the matrix Z sequentially so as the data arrive one at each time in a fixed order. The steps of this stochastic process are discussed below. When the first data point arrives all the features are currently unrepresented. We sample feature occurrences from the set of unrepresented features as follows. Firstly, we draw an integer number g1 from the negative binomial NB(g1; α, 1 2) which has a mean value equal to α. g1 is the total number of feature occurrences for the first data point. Given g1, we randomly select a partition (z11, . . . , z1K1) of the integer g1 into parts1, i.e. z11 + . . . + z1K1 = g1 and 1 ≤K1 ≤g1, by drawing from Ewens’s distribution [5] over integer partitions which is given by P(z11, . . . , z1K1) = αK1 Γ(α) Γ(g1 + α) g1! z11 × . . . × z1K1 g1 Y i=1 1 v(1) i ! , (8) where v(1) i is the multiplicity of integer i in the partition (z11, . . . , z1K1). The Ewens’s distribution is equivalent to the distribution over partitions of objects induced by the Dirichlet process and the Chinese restaurant process since we can derive the one from the other using simple combinatorics arguments. The difference between them is that the former is a distribution over integer partitions while the latter is a distribution over partitions of objects. Let Kn−1 be the number of represented features when the nth data point arrives. For each feature k, with k ≤Kn−1, we choose znk based on the popularity of this feature in the previous n −1 data 1The partition of a positive integer is a way of writing this integer as a sum of positive integers where order does not matter, e.g. the partitions of 3 are: (3),(2,1) and (1,1,1). points. This popularity is expressed by the total number of occurrences for the feature k which is given by mk = Pn−1 i=1 zik. Particularly, we draw znk from NB(znk; mk, n n+1) which has a mean value equal to mk n . Once we have sampled from all represented features we need to consider a sample from the set of unrepresented features. Similarly to the first data point, we first draw an integer gn from NB(gn; α, n n+1), and subsequently we select a partition of that integer by drawing from the Ewens’s formula. This process produces the following distribution: P(Z|α) = 1 Qg1 i=1 v(1) i ! × . . . × QgN i=1 v(N) i ! αK+ (N + 1)m+α QK+ k=1(mk −1)! QK+ k=1 QN n=1 znk! , (9) where {v(n) i } are the integer-multiplicities for the nth data point which arise when we draw from the Ewens’s distribution. Note that the above expression does not have exactly the same form as the distribution in Equation (7) and is not exchangeable since it depends on the order the data arrive. However, if we consider only the left-ordered class of matrices generated by the stochastic process then we obtain the exchangeable distribution in Equation (7). Note that a similar situation arises with the Indian buffet process. 3.2 Conditional distributions When we combine the prior P(Z|α) with a likelihood model p(X|Z) and we wish to do inference over Z using Gibbs-type sampling, we need to express the conditionals of the form P(znk|Z−(nk), α) where Z−(nk) = Z \ znk. We can derive such conditionals by taking limits of the conditionals for the finite model or by using the stochastic process. Suppose that for the current value of Z, there exist K+ represented features i.e. mk > 0 for k ≤K+. Let m−n,k = P en̸=n zenk. When m−n,k > 0, the conditional of znk is given by NB(znk; m−n,k, N N+1). In all different cases, we need a special conditional that samples from new features2 and accounts for all k such that m−n,k = 0. This conditional draws an integer number from NB(gn; a, N N+1) and then determines the occurrences for the new features by choosing a partition of the integer gn using the Ewens’s distribution. Finally the conditional p(α|Z), which can be directly expressed from Equation (7) and the prior of α, is given by p(α|Z) ∝G(α; α0, β0) αK+ (N + 1)α . (10) Typically the likelihood model does not depend on α and thus the above quantity is also the posterior conditional of α given data and Z. 4 A likelihood model for images An image can contain multiple objects of different classes. Each object class can have more than one occurrences, i.e. multiple instances of the class may appear simultaneously in the image. Unsupervised learning should deal with the unknown number of object classes in the images and also the unknown number of occurrences of each class in each image separately. If object classes are the latent features, what we wish to infer is the underlying feature occurrence matrix Z. We consider an observation model that is a combination of latent Dirichlet allocation [2] and Gaussian mixture models. Such a combination has been used before [12]. Each image n is represented by dn local patches that are detected in the image so as Xn = (Yn, Wn) = {(yni, wni), i = 1, . . . , dn}. yni is the two-dimensional location of patch i and wni is an indicator vector (i.e. is binary and satisfies PL ℓ=1 wℓ ni = 1) that points into a set of L possible visual appearances. X, Y , and W denote all the data the locations and the appearances, respectively. We will describe the probabilistic model starting from the joint distribution of all variables which is given by joint = p(α)P(Z|α)p({θk}|Z)× N Y n=1 " p(πn|Zn)p(mn, Σn|Zn) dn Y i=1 P(sni|πn)P(wni|sni, {θk})p(yni|sni, mn, Σn) # . (11) 2Features of this kind are the unrepresented features (k > K+) as well as all the unique features that occur only in the data point n (i.e. m−n,k = 0, but znk > 0). N Z dn sni (mn, Σn) πn yni {θk} α wni Figure 1: Graphical model for the joint distribution in Equation (11). The graphical representation of this distribution is depicted in Figure 1. We now explain all the pieces of this joint distribution following the causal structure of the graphical model. Firstly, we generate α from its prior and then we draw the feature occurrence matrix Z using the infinite gamma-Poisson prior P(Z|α). The matrix Z defines the structure for the remaining part of the model. The parameter vector θk = {θk1, . . . , θkL} describes the appearance of the local patches W for the feature (object class) k. Each θk is generated from a symmetric Dirichlet so as the whole set of {θk} vectors is drawn from p({θk}|Z) = QK+ k=1 D(θk|γ), where γ is the hyperparameter of the symmetric Dirichlet and it is common for all features. Note that the feature appearance parameters {θk} depend on Z only through the number of represented features K+ which is obtained by counting the non-zero columns of Z. The parameter vector πn = {πnkj} defines the image-specific mixing proportions for the mixture model associated with image n. To see how this mixture model arises, notice that a local patch in image n belongs to a certain occurrence of a feature. We use the double index kj to denote the j occurrence of feature k where j = 1, . . . , znk and k ∈{ek : znek > 0}. This mixture model has Mn = PK+ k=1 znk components, i.e. as many as the total number of feature occurrences in image n. The assignment variable sni = {skj ni}, which takes Mn values, indicates the feature occurrence of patch i. πn is drawn from a symmetric Dirichlet given by p(πn|Zn) = D(πn|β/Mn), where Zn denotes the nth row of Z and β is a hyperparameter shared by all images. Notice that πn depends only on the nth row of Z. The parameters (mn, Σn) determine the image-specific distribution for the locations {yni} of the local patches in image n. We assume that each occurrence of a feature forms a Gaussian cluster of patch locations. Thus yni follows a image-specific Gaussian mixture with Mn components. We assume that the component kj has mean mnkj and covariance Σnkj. mnkj describes object location and Σnkj object shape. mn and Σn collect all the means and covariances of the clusters in the image n. Given that any object can be anywhere in the image and have arbitrary scale and orientation, (mnkj, Σnkj) should be drawn from a quite vague prior. We use a conjugate normal-Wishart prior for the pair (mnkj, Σnkj) so as p(mn, Σn|Zn) = Y k:znk>0 znk Y j=1 N(mnkj|µ, τΣnkj)W(Σ−1 nkj|v, V ), (12) where (µ, τ, v, V ) are the hyperparameters shared by all features and images. The assignment sni which determines the allocation of a local patch in a certain feature occurrence follows a multinomial: P(sni|πn) = Q k:znk>0 Qznk j=1(πnkj)skj ni. Similarly the observed data pair (wni, yni) of a local image patch is generated according to P(wni|sni, {θk}) = K+ Y k=1 L Y ℓ=1 θ wℓ ni Pznk j=1 skj ni kℓ and p(yni|sni, mn, Σn) = Y k:znk>0 znk Y j=1 [N(yni|mnkj, Σnkj)]skj ni . The hyperparameters (γ, β, µ, τ, v, V ) take fixed values that give vague priors and they are not depicted in the graphical model shown in Figure 1. Since we have chosen conjugate priors, we can analytically marginalize out from the joint distribution all the parameters {πn}, {θk}, {mn} and {Σn} and obtain p(X, S, Z, α). Marginalizing out the assignments S is generally intractable and the MCMC algorithm discussed next produces samples from the posterior P(S, Z, α|X). 4.1 MCMC inference Inference with our model involves expressing the posterior P(S, Z, α|X) over the feature occurrences Z, the assignments S and the parameter α. Note that the joint P(S, Z, α, X) factorizes according to p(α)P(Z|α)P(W|S, Z) QN n=1 P(Sn|Zn)p(Yn|Sn, Zn) where Sn denotes the assignments associated with image n. Our algorithm uses mainly Gibbs-type sampling from conditional posterior distributions. Due to space limitations we briefly discuss the main points of this algorithm. The MCMC algorithm processes the rows of Z iteratively and updates its values. A single step can change an element of Z by one so as |znew nk −zold nk | ≤1. Initially Z is such that Mn = PK+ k=1 znk ≥ 1, for any n which means that at least one mixture component explains the data of each image. The proposal distribution for changing znks ensures that this constraint is satisfied. Suppose we wish to sample a new value for znk using the joint model p(S, Z, α, X). Simply witting P(znk|S, Z−(nk), α, X) is not useful since when znk changes the number of states the assignments Sn can take also changes. This is clear since znk is a structural variable that affects the number of components Mn = PK+ k=1 znk of the mixture model associated with image n and assignments Sn. On the other hand the dimensionality of the assignments S−n = S \ Sn of all other images is not affected when znk changes. To deal with the above we marginalize out Sn and we sample znk from the marginalized posterior conditional P(znk|S−n, Z−(nk), α, X) which is computed according to P(znk|S−n, Z−(nk), α, X) ∝P(znk|Z−(nk), α) X Sn P(W|S, Z)p(Yn|Sn, Zn)P(Sn|Zn), (13) where P(znk|Z−n,k, α) for the infinite case is computed as described in section 3.2 while computing the sum requires an approximation. This sum is a marginal likelihood and we apply importance sampling using as an importance distribution the posterior conditional P(Sn|S−n, Z, W, Yn) [10]. Sampling from P(Sn|S−n, Z, W, Yn) is carried out by applying local Gibbs sampling moves and global Metropolis moves that allow two occurrences of different features to exchange their data clusters. In our implementation we consider a single sample drawn from this posterior distribution so that the sum is approximated by P(W|S∗ n, S−n, Z)p(Yn|S∗ n, Zn) and S∗ n is a sample accepted after a burn in period. Additionally to scans that update Z and S we add few Metropolis-Hastings steps that update the hyperparameter α using the posterior conditional given by Equation (10). 5 Experiments In the first experiment we use a set of 10 artificial images. We consider four features that have the regular shapes shown in Figure 2. The discrete patch appearances correspond to pixels and can take 20 possible grayscale values. Each feature has its own multinomial distribution over the appearances. To generate an image we first decide to include each feature with probability 0.5. Then for each included feature we randomly select the number of occurrences from the range [1, 3]. For each feature occurrence we select the pixels using the appearance multinomial and place the respective feature shape in a random location so that feature occurrences do not occlude each other. The first row of Figure 2 shows a training image (left), the locations of pixels (middle) and the discrete appearances (right). The MCMC algorithm was initialized with K+ = 1, α = 1 and zn1 = 1, n = 1, . . . , 10. The third row of Figure 2 shows how K+ (left) and the sum of all znks (right) evolve through the first 500 MCMC iterations. The algorithm in the first 20 iterations has training image n locations Yn appearances Wn 1 3 3 1 3 2 3 0 0 2 1 2 Figure 2: The first row shows a training image (left), the locations of pixels (middle) and the discrete appearances (right). The second row shows the localizations of all feature occurrences in three images. Below of each image the corresponding row of Z is also shown. The third row shows how K+ (left) and the sum of all znks (right) evolve through the first 500 MCMC iterations. Figure 3: The left most plot on the first row shows the locations of detected patches and the bounding boxes in one of the annotated images. The remaining five plots show examples of detections and localizations of the three most dominant features (including the car-category) in five non-annotated images. visited the matrix Z that was used to generate the data and then stabilizes. For 86% of the samples K+ is equal to four. For the state (Z, S) that is most frequently visited, the second row of Figure 2 shows the localizations of all different feature occurrences in three images. Each ellipse is drawn using the posterior mean values for a pair (mnkj, Σnkj) and illustrates the predicted location and shape of a feature occurrence. Note that ellipses with the same color correspond to the different occurrences of the same feature. In the second experiment we consider 25 real images from the UIUC3 cars database. We used the patch detection method presented in [8] and we constructed a dictionary of 200 visual appearances by clustering the SIFT [8] descriptors of the patches using K-means. Locations of detected patches are shown in the first row (left) of Figure 3. We partially labelled some of the images. Particularly, for 7 out of 25 images we annotated the car views using bounding boxes (Figure 3). This allows us to specify seven elements of the first column of the matrix Z (the first feature will correspond to the car-category). These znks values plus the assignments of all patches inside the boxes do not change during sampling. Also the patches that lie outside the boxes in all annotated images are not allowed to be part of car occurrences. This is achieved by applying partial Gibbs sampling updates and Metropolis moves when sampling the assignments S. The algorithm is initialized with K+ = 1, after 30 iterations stabilizes and then fluctuates between nine to twelve features. To keep the plots uncluttered, Figure 3 shows the detections and localizations of only the three most dominant features (including the car-category) in five non-annotated images. The red ellipses correspond to different occurrences of the car-feature, the green ones to a tree-feature and the blue ones to a street-feature. 6 Discussion We presented the infinite gamma-Poisson model which is a nonparametric prior for non-negative integer valued matrices with infinite number of columns. We discussed the use of this prior for unsupervised learning where multiple features are associated with our data and each feature can have multiple occurrences within each data point. The infinite gamma-Poisson prior can be used for other purposes as well. For example, an interesting application can be Bayesian matrix factorization where a matrix of observations is decomposed into a product of two or more matrices with one of them being a non-negative integer valued matrix. References [1] C. Antoniak. Mixture of Dirichlet processes with application to Bayesian nonparametric problems. The Annals of Statistics, 2:1152–1174, 1974. [2] D. M. Blei, A. Y. Ng, and M. I. Jordan. Latent Dirichlet allocation. JMLR, 3, 2003. [3] W. Buntime and A. Jakulin. Applying discrete PCA in data analysis. In UAI, 2004. [4] J. Canny. GaP: A factor model for discrete data. In SIGIR, pages 122–129. ACM Press, 2004. [5] W. Ewens. The sampling theory of selectively neutral alleles. Theoretical Population Biology, 3:87–112, 1972. [6] P. Green and S. Richardson. Modelling heterogeneity with and without the Dirichlet process. Scandinavian Journal of Statistics, 28:355–377, 2001. [7] T. Griffiths and Z. Ghahramani. Infinite latent feature models and the Indian buffet process. In NIPS 18, 2006. [8] D. G. Lowe. Distinctive image features from scale-invariant keypoints. International Journal of Computer Vision, 60(2):91–110, 2004. [9] R. M. Neal. Bayesian mixture modeling. In 11th International Workshop on Maximum Entropy and Bayesian Methods of Statistical Analysis, pages 197–211, 1992. [10] M. A. Newton and A. E Raftery. Approximate Bayesian inference by the weighted likelihood bootstrap. Journal of the Royal Statistical Society, Series B, 3:3–48, 1994. [11] E. Saund. A multiple cause mixture model for unsupervised learning. Neural Computation, 7:51–71, 1995. [12] E. Sudderth, A. Torralba, W. T. Freeman, and A. Willsky. Describing Visual Scenes using Transformed Dirichlet Processes. In NIPS 18, 2006. 3available from http://l2r.cs.uiuc.edu/∼cogcomp/Data/Car/.
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FilterBoost: Regression and Classification on Large Datasets Joseph K. Bradley Machine Learning Department Carnegie Mellon University Pittsburgh, PA 15213 jkbradle@cs.cmu.edu Robert E. Schapire Department of Computer Science Princeton University Princeton, NJ 08540 schapire@cs.princeton.edu Abstract We study boosting in the filtering setting, where the booster draws examples from an oracle instead of using a fixed training set and so may train efficiently on very large datasets. Our algorithm, which is based on a logistic regression technique proposed by Collins, Schapire, & Singer, requires fewer assumptions to achieve bounds equivalent to or better than previous work. Moreover, we give the first proof that the algorithm of Collins et al. is a strong PAC learner, albeit within the filtering setting. Our proofs demonstrate the algorithm’s strong theoretical properties for both classification and conditional probability estimation, and we validate these results through extensive experiments. Empirically, our algorithm proves more robust to noise and overfitting than batch boosters in conditional probability estimation and proves competitive in classification. 1 Introduction Boosting provides a ready method for improving existing learning algorithms for classification. Taking a weaker learner as input, boosters use the weak learner to generate weak hypotheses which are combined into a classification rule more accurate than the weak hypotheses themselves. Boosters such as AdaBoost [1] have shown considerable success in practice. Most boosters are designed for the batch setting where the learner trains on a fixed example set. This setting is reasonable for many applications, yet it requires collecting all examples before training. Moreover, most batch boosters maintain distributions over the entire training set, making them computationally costly for very large datasets. To make boosting feasible on larger datasets, learners can be designed for the filtering setting. The batch setting provides the learner with a fixed training set, but the filtering setting provides an oracle which can produce an unlimited number of labeled examples, one at a time. This idealized model may describe learning problems with on-line example sources, including very large datasets which must be loaded piecemeal into memory. By using new training examples each round, filtering boosters avoid maintaining a distribution over a training set and so may use large datasets much more efficiently than batch boosters. The first polynomial-time booster, by Schapire, was designed for filtering [2]. Later filtering boosters included two more efficient ones proposed by Freund, but both are non-adaptive, requiring a priori bounds on weak hypothesis error rates and combining weak hypotheses via unweighted majority votes [3,4]. Domingo & Watanabe’s MadaBoost is competitive with AdaBoost empirically but theoretically requires weak hypotheses’ error rates to be monotonically increasing, an assumption we found to be violated often in practice [5]. Bshouty & Gavinsky proposed another, but, like Freund’s, their algorithm requires an a priori bound on weak hypothesis error rates [6]. Gavinsky’s AdaFlatfilt algorithm and Hatano’s GiniBoost do not have these limitations, but the former has worse bounds than other adaptive algorithms while the latter explicitly requires finite weak hypothesis spaces [7,8]. 1 This paper presents FilterBoost, an adaptive boosting-by-filtering algorithm. We show it is applicable to both conditional probability estimation, where the learner predicts the probability of each label given an example, and classification. In Section 2, we describe the algorithm, after which we interpret it as a stepwise method for fitting an additive logistic regression model for conditional probabilities. We then bound the number of rounds and examples required to achieve any target error in (0, 1). These bounds match or improve upon those for previous filtering boosters but require fewer assumptions. We also show that FilterBoost can use the confidence-rated predictions from weak hypotheses described by Schapire & Singer [9]. In Section 3, we give results from extensive experiments. For conditional probability estimation, we show that FilterBoost often outperforms batch boosters, which prove less robust to overfitting. For classification, we show that filtering boosters’ efficiency on large datasets allows them to achieve higher accuracies faster than batch boosters in many cases. FilterBoost is based on a modification of AdaBoost by Collins, Schapire & Singer designed to minimize logistic loss [10]. Their batch algorithm has yet to be shown to achieve arbitrarily low test error, but we use techniques similar to those of MadaBoost to adapt the algorithm to the filtering setting and prove generalization bounds. The result is an adaptive algorithm with realistic assumptions and strong theoretical properties. Its robustness and efficiency on large datasets make it competitive with existing methods for both conditional probability estimation and classification. 2 The FilterBoost Algorithm Let X be the set of examples and Y a discrete set of labels. For simplicity, assume X is countable, and consider only binary labels Y = {−1, +1}. We assume there exists an unknown target distribution D over labeled examples (x, y) ∈X × Y from which training and test examples are generated. The goal in classification is to choose a hypothesis h : X →Y which minimizes the classification error PrD[h(x) ̸= y], where the subscript indicates that the probability is with respect to (x, y) sampled randomly from D. In the batch setting, a booster is given a fixed training set S and a weak learner which, given any distribution Dt over training examples S, is guaranteed to return a weak hypothesis ht : X →R such that the error ϵt ≡PrDt[sign(ht(x)) ̸= y] < 1/2. For T rounds t, the booster builds a distribution Dt over S, runs the weak learner on S and Dt, and receives ht. The booster usually then estimates ϵt using S and weights ht with αt = αt(ϵt). After T rounds, the booster outputs a final hypothesis H which is a linear combination of the weak hypotheses (e.g. H(x) = P t αtht(x)). The sign of H(x) indicates the predicted label ˆy for x. Two key elements of boosting are constructing Dt over S and weighting weak hypotheses. Dt is built such that misclassified examples receive higher weights than in Dt−1, eventually forcing the weak learner to classify previously poorly classified examples correctly. Weak hypotheses ht are generally weighted such that hypotheses with lower errors receive higher weights. 2.1 Boosting-by-Filtering We describe a general framework for boosting-by-filtering which includes most existing algorithms as well as our algorithm Filterboost. The filtering setting assumes the learner has access to an example oracle, allowing it to use entirely new examples sampled i.i.d. from D on each round. However, while maintaining the distribution Dt is straightforward in the batch setting, there is no fixed set S on which to define Dt in filtering. Instead, the booster simulates examples drawn from Dt by drawing examples from D via the oracle and reweighting them according to Dt. Filtering boosters generally accept each example (x, y) from the oracle for training on round t with probability proportional to the example’s weight Dt(x, y). The mechanism which accepts examples from the oracle with some probability is called the filter. Thus, on each round, a boosting-by-filtering algorithm draws a set of examples from Dt via the filter, trains the weak learner on this set, and receives a weak hypothesis ht. Though a batch booster would estimate ϵt using the fixed set S, filtering boosters may use new examples from the filter. Like batch boosters, filtering boosters may weight ht using αt = αt(ϵt), and they output a linear combination of h1, . . . , hT as a final hypothesis. 2 Define Ft(x) ≡Pt−1 t′=1 αt′ht′(x) Algorithm FilterBoost accepts Oracle(), ε, δ, τ: For t = 1, 2, 3, . . . δt ←− δ 3t(t+1) Call Filter(t, δt, ε) to get mt examples to train WL; get ht ˆγ′ t ←−getEdge(t, τ, δt, ε) αt ←−1 2 ln 1/2+ˆγ′ t 1/2−ˆγ′ t Define Ht(x) = sign Ft+1(x) (Algorithm exits from Filter() function.) Function Filter(t, δt, ε) returns (x, y) Define r = # calls to Filter so far on round t δ′ t ←− δt r(r+1) For (i = 0; i < 2 ε ln( 1 δ′ t ); i = i + 1): (x, y) ←−Oracle() qt(x, y) ←− 1 1+eyFt(x) Return (x, y) with probability qt(x, y) End algorithm; return Ht−1 Function getEdge(t, τ, δt, ε) returns ˆγ′ t Let m ←−0, n ←−0, u ←−0, α ←−∞ While (|u| < α(1 + 1/τ)): (x, y) ←−Filter(t, δt, ε) n ←−n + 1 m ←−m + I(ht(x) = y) u ←−m/n −1/2 α ←− p (1/2n) ln(n(n + 1)/δt) Return u/(1 + τ) Figure 1: The algorithm FilterBoost. The filtering setting allows the learner to estimate the error of Ht to arbitrary precision by sampling from D via the oracle, so FilterBoost does this to decide when to stop boosting. 2.2 FilterBoost FilterBoost, given in Figure 1, is modeled after the aforementioned algorithm by Collins et al. [10] and MadaBoost [5]. Given an example oracle, weak learner, target error ε ∈(0, 1), and confidence parameter δ ∈(0, 1) upper-bounding the probability of failure, it iterates until the current combined hypothesis Ht has error ≤ε. On round t, FilterBoost draws mt examples from the filter to train the weak learner and get ht. The number mt must be large enough to ensure ht has error ϵt < 1/2 with high probability. The edge of ht is γt = 1/2 −ϵt, and this edge is estimated by the function getEdge(), discussed below, and is used to set ht’s weight αt. The current combined hypothesis is defined as Ht = sign(Pt t′=1 αt′ht′). The Filter() function generates (x, y) from Dt by repeatedly drawing (x, y) from the oracle, calculating the weight qt(x, y) ∝Dt(x, y), and accepting (x, y) with probability qt(x, y). Function getEdge() uses a modification of the Nonmonotonic Adaptive Sampling method of Watanabe [11] and Domingo, Galvad`a & Watanabe [12]. Their algorithm draws an adaptively chosen number of examples from the filter and returns an estimate ˆγt of the edge of ht within relative error τ of the true edge γt with high probability. The getEdge() function revises this estimate as ˆγ′ t = ˆγt/(1 + τ). 2.3 Analysis: Conditional Probability Estimation We begin our analysis of FilterBoost by interpreting it as an additive model for logistic regression, for this interpretation will later aid in the analysis for classification. Such models take the form log Pr[y = 1|x] Pr[y = −1|x] = X t ft(x) = F(x), which implies Pr[y = 1|x] = 1 1 + e−F (x) where, for FilterBoost, ft(x) = αtht(x). Dropping subscripts, we can write the expected negative log likelihood of example (x, y) after round t as π(Ft + αtht) = π(F + αh) = E −ln 1 1 + e−y(F (x)+αh(x)) = E h ln 1 + e−y(F (x)+αh(x))i . Taking a similar approach to the analysis of AdaBoost in [13], we show in the following theorem that FilterBoost performs an approximate stepwise minimization of this negative log likelihood. The proof is in the Appendix. 3 Theorem 1 Define the expected negative log likelihood π(F + αh) as above. Given F, FilterBoost chooses h to minimize a second-order Taylor expansion of π around h = 0. Given this h, it then chooses α to minimize an upper bound of π. The batch booster given by Collins et al. [10] which FilterBoost is based upon is guaranteed to converge to the minimum of this objective when working over a finite sample. Note that FilterBoost uses weak learners which are simple classifiers to perform regression. AdaBoost too may be interpreted as an additive logistic regression model of the form Pr[y = 1|x] = 1 1+e−2F (x) with E[exp(−yF(x))] as the optimization objective [13]. 2.4 Analysis: Classification In this section, we interpret FilterBoost as a traditional boosting algorithm for classification and prove bounds on its generalization error. We first give a theorem relating errt, the error rate of Ht over the target distribution D, to pt, the probability with which the filter accepts a random example generated by the oracle on round t. Theorem 2 Let errt = PrD[Ht(x) ̸= y], and let pt = ED[qt(x, y)]. Then errt ≤2pt. Proof: errt = PrD[Ht(x) ̸= y] = PrD[yFt−1(x) ≤0] = PrD[qt(x, y) ≥1/2] ≤2 · ED[qt(x, y)] = 2pt (using Markov’s inequality above) ■ We next use the expected negative log likelihood π from Section 2.3 as an auxiliary function to aid in bounding the required number of boosting rounds. Viewing π as a function of the boosting round t, we can write πt = −P (x,y) D(x, y) ln(1 −qt(x, y)). Our goal is then to minimize πt, and the following lemma captures the learner’s progress in terms of the decrease in πt on each round. This lemma assumes edge estimates returned by getEdge() are exact, i.e. ˆγ′ t = γt, which leads to a simpler bound on T in Theorem 3. We then consider the error in edge estimates and give a revised bound in Lemma 2 and Theorem 5. The proofs of Lemmas 1 and 2 are in the Appendix. Lemma 1 Assume for all t that γt ̸= 0 and γt is estimated exactly. Let πt = −P (x,y) D(x, y) ln(1 −qt(x, y)). Then πt −πt+1 ≥pt 1 −2 q 1/4 −γ2 t . Combining Theorem 2, which bounds the error of the current combined hypothesis in terms of pt, with Lemma 1 gives the following upper bound on the required rounds T. Theorem 3 Let γ = mint |γt|, and let ε be the target error. Given Lemma 1’s assumptions, if FilterBoost runs T > 2 ln(2) ε 1 −2 p 1/4 −γ2 rounds, then errt < ε for some t, 1 ≤t ≤T. In particular, this is true for T > ln(2) 2εγ2 . Proof: For all (x, y), since F1(x, y) = 0, then q1(x, y) = 1/2 and π1 = ln(2). Now, suppose errt ≥ε, ∀t ∈{1, ..., T}. Then, from Theorem 2, pt ≥ε/2, so Lemma 1 gives πt −πt+1 ≥1 2ε 1 −2 p 1/4 −γ2 Unraveling this recursion as PT t=1 (πt −πt+1) = π1 −πT +1 ≤π1 gives T ≤ 2 ln(2) ε 1 −2 p 1/4 −γ2 . 4 So, errt ≥ε, ∀t ∈{1, ..., T} is contradicted if T exceeds the theorem’s lower bound. The simplified bound follows from the first bound via the inequality 1 −√1 −x ≤x for x ∈[0, 1]. ■ Theorem 3 shows FilterBoost can reduce generalization error to any ε ∈(0, 1), but we have thus far overlooked the probabilities of failure introduced by three steps: training the weak learner, deciding when to stop boosting, and estimating edges. We bound the probability of each of these steps failing on round t with a confidence parameter δt = δ 3t(t+1) so that a simple union bound ensures the probability of some step failing to be at most FilterBoost’s confidence parameter δ. Finally, we revise Lemma 1 and Theorem 3 to account for error in estimating edges. The number mt of examples the weak learner trains on must be large enough to ensure weak hypothesis ht has a non-zero edge and should be set according to the choice of weak learner. To decide when to stop boosting (i.e. when errt ≤ε), we can use Theorem 2, which upper-bounds the error of the current combined hypothesis Ht in terms of the probability pt that Filter() accepts a random example from the oracle. If the filter rejects enough examples in a single call, we know pt is small, so Ht is accurate enough. Theorem 4 formalizes this intuition; the proof is in the Appendix. Theorem 4 In a single call to Filter(t), if n examples have been rejected, where n ≥2 ε ln(1/δ′ t), then errt ≤ε with probability at least 1 −δ′ t. Theorem 4 provides a stopping condition which is checked on each call to Filter(). Each check may fail with probability at most δ′ t = δt r(r+1) on the rth call to Filter() so that a union bound ensures FilterBoost stops prematurely on round t with probability at most δt. Theorem 4 uses a similar argument to that used for MadaBoost, giving similar stopping criteria for both algorithms. We estimate weak hypotheses’ edges γt using the Nonmonotonic Adaptive Sampling (NAS) algorithm [11,12] used by MadaBoost. To compute an estimate ˆγt of the true edge γt within relative error τ ∈(0, 1) with probability ≥1 −δt, the NAS algorithm uses at most 2(1+2τ)2 (τγt)2 ln( 1 τγtδt ) filtered examples. With this guarantee on edge estimates, we can rewrite Lemma 1 as follows: Lemma 2 Assume for all t that γt ̸= 0 and γt is estimated to within τ ∈(0, 1) relative error. Let πt = −P (x,y) D(x, y) ln(1 −qt(x, y)). Then πt −πt+1 ≥pt 1 −2 s 1/4 −γ2 t 1 −τ 1 + τ 2! . Using Lemma 2, the following theorem modifies Theorem 3 to account for error in edge estimates. Theorem 5 Let γ = mint |γt|. Let ε be the target error. Given Lemma 2’s assumptions, if FilterBoost runs T > 2 ln(2) ε 1 −2 q 1/4 −γ2( 1−τ 1+τ )2 rounds, then errt < ε for some t, 1 ≤t ≤T. The bounds from Theorems 3 and 5 show FilterBoost requires at most O(ε−1γ−2) boosting rounds. MadaBoost [5], which we test in our experiments, resembles FilterBoost but uses truncated exponential weights qt(x, y) = min{1, exp(yFt−1(x))} instead of the logistic weights qt(x, y) = (1 + exp(yFt(x)))−1 used by FilterBoost. The algorithms’ analyses differ, with MadaBoost requiring the edges γt to be monotonically decreasing, but both lead to similar bounds on the number of rounds T proportional to ε−1. The non-adaptive filtering boosters of Freund [3,4] and of Bshouty & Gavinsky [6] and the batch booster AdaBoost [1] have smaller bounds on T, proportional to log(ε−1). However, we can use boosting tandems, a technique used by Freund [4] and Gavinsky [7], to create a filtering booster with T bounded by O(log(ε−1)γ−2). Following Gavinsky, we can use FilterBoost to boost the accuracy of the weak learner to some constant and, in turn, treat FilterBoost as a weak learner and use an algorithm from Freund to achieve any target error. As with AdaFlatfilt, boosting tandems turn FilterBoost into an adaptive booster with a bound on T proportional to log(ε−1). (Without boosting tandems, AdaFlatfilt requires T ∝ε−2 rounds.) Note, however, that boosting tandems result in more complicated final hypotheses. 5 An alternate bound for FilterBoost may be derived using techniques from Shalev-Shwartz & Singer [14]. They use the framework of convex repeated games to define a general method for bounding the performance of online and boosting algorithms. For FilterBoost, their techniques, combined with Theorem 2, give a bound similar to that in Theorem 3 but proportional to ε−2 instead of ε−1. Schapire & Singer [9] show AdaBoost benefits from confidence-rated predictions, where weak hypotheses return predictions whose absolute values indicate confidence. These values are chosen to greedily minimize AdaBoost’s exponential loss function over training data, and this aggressive weighting can result in faster learning. FilterBoost may use confidence-rated predictions in an identical manner. In the proof of Lemma 1, the decrease in the negative log likelihood πt of the data (relative to Ht and the target distribution D) is lower-bounded by pt−pt P (x,y) Dt(x, y)e−αtyht(x). Since pt is fixed, maximizing this bound is equivalent to minimizing the exponential loss over Dt. 3 Experiments Vanilla FilterBoost accepts examples (x, y) from the oracle with probability qt(x, y), but it may instead accept all examples and weight each with qt(x, y). Weighting instead of filtering examples increases accuracy but also increases the size of the training set passed to the weak learner. For efficiency, we choose to filter when training the weak learner but weight when estimating edges γt. We also modify FilterBoost’s getEdge() function for efficiency. The Nonmonotonic Adaptive Sampling (NAS) algorithm used to estimate edges γt uses many examples, but using several orders of magnitude fewer sacrifices little accuracy. The same is true for MadaBoost. In all tests, we use Cn log(t + 1) examples to estimate γt, where Cn = 300 and the log factor scales the number as the NAS algorithm would. For simplicity, we train weak learners with Cn log(t + 1) examples as well. These modifications mean τ (error in edge estimates) and δ (confidence) have no effect on our tests. To simulate an oracle, we randomly permute the data and use examples in the new order. In practice, filtering boosters can achieve higher accuracy by cycling through training sets again instead of stopping once examples are depleted, and we use this “recycling” in our tests. We test FilterBoost with and without confidence-rated predictions (labeled “(C-R)” in our results). We compare FilterBoost against MadaBoost [5], which does not require an a priori bound on weak hypotheses’ edges and has similar bounds without the complication of boosting tandems. We implement MadaBoost with the same modifications as FilterBoost. We test FilterBoost against two batch boosters: the well-studied and historically successful AdaBoost [1] and the algorithm from Collins et al. [10] which is essentially a batch version of FilterBoost (labeled “AdaBoost-LOG”). We test both with and without confidence-rated predictions as well as with and without resampling (labeled “(resamp)”). In resampling, the booster trains weak learners on small sets of examples sampled from the distribution Dt over the training set S rather than on the entire set S, and this technique often increases efficiency with little effect on accuracy. Our batch boosters use sets of size Cm log(t + 1) for training, like the filtering boosters, but use all of S to estimate edges γt since this can be done efficiently. We test the batch boosters using confidence-rated predictions and resampling in order to compare FilterBoost with batch algorithms optimized for the efficiency which boosting-by-filtering claims as its goal. We test each booster using decision stumps and decision trees as weak learners to discern the effects of simple and complicated weak hypotheses. The decision stumps minimize training error, and the decision trees greedily maximize information gain and are pruned using 1/3 of the data. Both weak learners minimize exponential loss when outputing confidence-rated predictions. We use four datasets, described in the Appendix. Briefly, we use two synthetic sets: Majority (majority vote) and Twonorm [15], and two real sets from the UCI Machine Learning Repository [16]: Adult (census data; from Ron Kohavi) and Covertype (forestry data with 7 classes merged to 2; Copyr. Jock A. Blackard & Colorado State U.). We average over 10 runs, using new examples for synthetic data (with 50,000 test examples except where stated) and cross validation for real data. Figure 2 compares the boosters’ runtimes. As expected, filtering boosters run slower per round than batch boosters on small datasets but much faster on large ones. Interestingly, filtering boosters take longer on very small datasets in some cases (not shown), for the probability the filter accepts an example quickly shrinks when the booster has seen that example many times. 6 Figure 2: Running times: Ada/Filter/MadaBoost. Majority; WL = stumps. 3.1 Results: Conditional Probability Estimation Figure 3: Log (base e) loss & root mean squared error (RMSE). Majority; 10,000 train exs. Left two: WL = stumps (FilterBoost vs. AdaBoost); Right two: WL = trees (FilterBoost vs. AdaBoost-LOG). In Section 2.3, we discussed the interpretation of FilterBoost and AdaBoost as stepwise algorithms for conditional probability estimation. We test both algorithms and the variants discussed above on all four datasets. We do not test MadaBoost, as it is not clear how to use it to estimate conditional probabilities. As Figure 3 shows, both FilterBoost variants are competitive with batch algorithms when boosting decision stumps. With decision trees, all algorithms except for FilterBoost overfit badly, including FilterBoost(C-R). In each plot, we compare FilterBoost with the best of AdaBoost and AdaBoost-LOG: AdaBoost was best with decision stumps and AdaBoost-LOG with decisions trees. For comparison, batch logistic regression via gradient descent achieves RMSE 0.3489 and log (base e) loss .4259; FilterBoost, interpretable as a stepwise method for logistic regression, seems to be approaching these asymptotically. On Adult and Twonorm, FilterBoost generally outperforms the batch boosters, which tend to overfit when boosting decision trees, though AdaBoost slightly outperforms FilterBoost on smaller datasets when boosting decision stumps. The Covertype dataset is an exception to our results and highlights a danger in filtering and in resampling for batch learning: the complicated structure of some datasets seems to require decision trees to train on the entire dataset. With decision stumps, the filtering boosters are competitive, yet only the non-resampling batch boosters achieve high accuracies with decision trees. The first decision tree trained on the entire training set achieves about 94% accuracy, which is unachievable by any of the filtering or resampling batch boosters when using Cm = 300 as the base number of examples for training the weak learner. To compete with non-resampling batch boosters, the other boosters must use Cm on the order of 105, by which point they become very inefficient. 3.2 Results: Classification Vanilla FilterBoost and MadaBoost perform similarly in classification (Figure 4). Confidence-rated predictions allow FilterBoost to outperform MadaBoost when using decision stumps but sometimes cause FilterBoost to perform poorly with decision trees. Figure 5 compares FilterBoost with the best batch booster for each weak learner. With decision stumps, all boosters achieve higher accuracies with the larger dataset, on which filtering algorithms are much more efficient. Majority is represented well as a linear combination of decision stumps, so the boosters all learn more slowly 7 Figure 4: FilterBoost vs. MadaBoost. Figure 5: FilterBoost vs. AdaBoost & AdaBoost-LOG. Majority. when using the overly complicated decision trees. However, this problem generally affects filtering boosters less than most batch variants, especially on larger datasets. Adult and Twonorm gave similar results. As in Section 3.1, filtering and resampling batch boosters perform poorly on Covertype. Thus, while FilterBoost is competitive in classification, its best performance is in regression. 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Journal of Machine Learning Research, 4, pp. 101-117. [8] Hatano, K. (2006) Smooth boosting using an information-based criterion. 17th International Conference on Algorithmic Learning Theory, pp. 304-319. [9] Schapire, R. E., & Singer, Y. (1999) Improved boosting algorithms using confidence-rated predictions. Machine Learning, 37, 297-336. [10] Collins, M., Schapire, R. E., & Singer, Y. (2002) Logistic regression, AdaBoost and Bregman distances. Machine Learning, 48, pp. 253-285. [11] Watanabe, O. (2000) Simple sampling techniques for discovery science. IEICE Trans. Information and Systems, E83-D(1), 19-26. [12] Domingo, C., Galvad`a, R., & Watanabe, O. (2002) Adaptive sampling methods for scaling up knowledge discovery algorithms. Data Mining and Knowledge Discovery, 6, pp. 131-152. [13] Friedman, J., Hastie, T., & Tibshirani, R. (2000) Additive logistic regression: a statistical view of boosting. The Annals of Statistics, 28, 337-407. [14] Shalev-Shwartz, S., & Singer, Y. (2006) Convex repeated games and Fenchel duality. Advances in Neural Information Processing Systems 20. [15] Breiman, L. (1998) Arcing classifiers. The Annals of Statistics, 26, pp. 801-849. [16] Newman, D. J., Hettich, S., Blake, C. L., & Merz, C. J. (1998) UCI Repository of machine learning databases [http://www.ics.uci.edu/∼mlearn/MLRepository.html]. Irvine, CA: U. of California, Dept. of Information & Computer Science. 8
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A Unified Near-Optimal Estimator For Dimension Reduction in lα (0 < α ≤2) Using Stable Random Projections Ping Li Department of Statistical Science Faculty of Computing and Information Science Cornell University pingli@cornell.edu Trevor J. Hastie Department of Statistics Department of Health, Research and Policy Stanford University hastie@stanford.edu Abstract Many tasks (e.g., clustering) in machine learning only require the lα distances instead of the original data. For dimension reductions in the lα norm (0 < α ≤2), the method of stable random projections can efficiently compute the lα distances in massive datasets (e.g., the Web or massive data streams) in one pass of the data. The estimation task for stable random projections has been an interesting topic. We propose a simple estimator based on the fractional power of the samples (projected data), which is surprisingly near-optimal in terms of the asymptotic variance. In fact, it achieves the Cram´er-Rao bound when α = 2 and α = 0+. This new result will be useful when applying stable random projections to distancebased clustering, classifications, kernels, massive data streams etc. 1 Introduction Dimension reductions in the lα norm (0 < α ≤2) have numerous applications in data mining, information retrieval, and machine learning. In modern applications, the data can be way too large for the physical memory or even the disk; and sometimes only one pass of the data can be afforded for building statistical learning models [1, 2, 5]. We abstract the data as a data matrix A ∈Rn×D. In many applications, it is often the case that we only need the lα properties (norms or distances) of A. The method of stable random projections [9,18,22] is a useful tool for efficiently computing the lα (0 < α ≤2) properties in massive data using a small (memory) space. Denote the leading two rows in the data matrix A by u1, u2 ∈RD. The lα distance d(α) is d(α) = D X i=1 |u1,i −u2,i|α. (1) The choice of α is beyond the scope of this study; but basically, we can treat α as a tuning parameter. In practice, the most popular choice, i.e., the α = 2 norm, often does not work directly on the original (unweighted) data, as it is well-known that truly large-scale datasets (especially Internet data) are ubiquitously “heavy-tailed.” In machine learning, it is often crucial to carefully term-weight the data (e.g., taking logarithm or tf-idf) before applying subsequent learning algorithms using the l2 norm. As commented in [12, 21], the term-weighting procedure is often far more important than fine-tuning the learning parameters. Instead of weighting the original data, an alternative scheme is to choose an appropriate norm. For example, the l1 norm has become popular recently, e.g., LASSO, LARS, 1-norm SVM [23], Laplacian radial basis kernel [4], etc. But other norms are also possible. For example, [4] proposed a family of non-Gaussian radial basis kernels for SVM in the form K(x, y) = exp (−ρ P i |xi −yi|α), where x and y are data points in high-dimensions; and [4] showed that α ≤1 (even α = 0) in some cases produced better results in histogram-based image classifications. The lα norm with α < 1, which may initially appear strange, is now well-understood to be a natural measure of sparsity [6]. In the extreme case, when α →0+, the lα norm approaches the Hamming norm (i.e., the number of non-zeros in the vector). Therefore, there is the natural demand in science and engineering for dimension reductions in the lα norm other than l2. While the method of normal random projections for the l2 norm [22] has become very popular recently, we have to resort to more general methodologies for 0 < α < 2. The idea of stable random projections is to multiply A with a random projection matrix R ∈RD×k (k ≪D). The matrix B = A × R ∈Rn×k will be much smaller than A. The entries of R are (typically) i.i.d. samples from a symmetric α-stable distribution [24], denoted by S(α, 1), where α is the index and 1 is the scale. We can then discard the original data matrix A because the projected matrix B now contains enough information to recover the original lα properties approximately. A symmetric α-stable random variable is denoted by S(α, d), where d is the scale parameter. If x ∼S(α, d), then its characteristic function (Fourier transform of the density function) would be E exp √ −1xt = exp (−d|t|α) , (2) whose inverse does not have a closed-form except for α = 2 (i.e., normal) or α = 1 (i.e., Cauchy). Applying stable random projections on u1 ∈RD, u2 ∈RD yields, respectively, v1 = RTu1 ∈Rk and v2 = RTu2 ∈Rk. By the properties of Fourier transforms, the projected differences, v1,j −v2,j, j = 1, 2, ..., k, are i.i.d. samples of the stable distribution S(α, d(α)), i.e., xj = v1,j −v2,j ∼S(α, d(α)), j = 1, 2, ..., k. (3) Thus, the task is to estimate the scale parameter from k i.i.d. samples xj ∼S(α, d(α)). Because no closed-form density functions are available except for α = 1, 2, the estimation task is challenging when we seek estimators that are both accurate and computationally efficient. For general 0 < α < 2, a widely used estimator is based on the sample inter-quantiles [7,20], which can be simplified to be the sample median estimator by choosing the 0.75 - 0.25 sample quantiles and using the symmetry of S(α, d(α)). That is ˆd(α),me = median{|xj|α, j = 1, 2, ..., k} median{S(α, 1)}α . (4) It has been well-known that the sample median estimator is not accurate, especially when the sample size k is not too large. Recently, [13] proposed various estimators based on the geometric mean and the harmonic mean of the samples. The harmonic mean estimator only works for small α. The geometric mean estimator has nice properties including closed-form variances, closed-form tail bounds in exponential forms, and very importantly, an analog of the Johnson-Lindenstrauss (JL) Lemma [10] for dimension reduction in lα. The geometric mean estimator, however, can still be improved for certain α, especially for large samples (e.g., as k →∞). 1.1 Our Contribution: the Fractional Power Estimator The fractional power estimator, with a simple unified format for all 0 < α ≤2, is (surprisingly) near-optimal in the Cram´er-Rao sense (i.e., when k →∞, its variance is close to the Cram´er-Rao lower bound). In particularly, it achieves the Cram´er-Rao bound when α = 2 and α →0+. The basic idea is straightforward. We first obtain an unbiased estimator of dλ (α), denoted by ˆR(α),λ. We then estimate d(α) by ˆR(α),λ 1/λ, which can be improved by removing the O 1 k bias (this consequently also reduces the variance) using Taylor expansions. We choose λ = λ∗(α) to minimize the theoretical asymptotic variance. We prove that λ∗(α) is the solution to a simple convex program, i.e., λ∗(α) can be pre-computed and treated as a constant for every α. The main computation involves only Pk j=1 |xj|λ∗α1/λ∗ ; and hence this estimator is also computationally efficient. 1.2 Applications The method of stable random projections is useful for efficiently computing the lα properties (norms or distances) in massive data, using a small (memory) space. • Data stream computations Massive data streams are fundamental in many modern data processing application [1, 2, 5, 9]. It is common practice to store only a very small sketch of the streams to efficiently compute the lα norms of the individual streams or the lα distances between a pair of streams. For example, in some cases, we only need to visually monitor the time history of the lα distances; and approximate answers often suffice. One interesting special case is to estimate the Hamming norms (or distances) using the fact that, when α →0+, d(α) = PD i=1 |u1,i −u2,i|α approaches the total number of non-zeros in {|u1,i −u2,i|}D i=1, i.e., the Hamming distance [5]. One may ask why not just (binary) quantize the data and then apply normal random projections to the binary data. [5] considered that the data are dynamic (i.e., frequent addition/subtraction) and hence prequantizing the data would not work. With stable random projections, we only need to update the corresponding sketches whenever the data are updated. • Computing all pairwise distances In many applications including distanced-based clustering, classifications and kernels (e.g.) for SVM, we only need the pairwise distances. Computing all pairwise distances of A ∈Rn×D would cost O(n2D), which can be significantly reduced to O(nDk + n2k) by stable random projections. The cost reduction will be more considerable when the original datasets are too large for the physical memory. • Estimating lα distances online While it is often infeasible to store the original matrix A in the memory, it is also often infeasible to materialize all pairwise distances in A. Thus, in applications such as online learning, databases, search engines, online recommendation systems, and online market-basket analysis, it is often more efficient if we store B ∈Rn×k in the memory and estimate any pairwise distance in A on the fly only when it is necessary. When we treat α as a tuning parameter, i.e., re-computing the lα distances for many different α, stable random projections will be even more desirable as a cost-saving device. 2 Previous Estimators We assume k i.i.d. samples xj ∼S(α, d(α)), j = 1, 2, ..., k. We list several previous estimators. • The geometric mean estimator is recommended in [13] for α < 2. ˆd(α),gm = Qk j=1 |xj|α/k 2 π Γ α k Γ 1 −1 k sin π 2 α k k . (5) Var ˆd(α),gm = d2 (α) ( 2 π Γ 2α k Γ 1 −2 k sin π α k k 2 π Γ α k Γ 1 −1 k sin π 2 α k 2k −1 ) (6) = d2 (α) 1 k π2 12 α2 + 2 + O 1 k2 . (7) • The harmonic mean estimator is recommended in [13] for 0 < α ≤0.344. ˆd(α),hm = −2 π Γ(−α) sin π 2 α Pk j=1 |xj|−α k − −πΓ(−2α) sin (πα) Γ(−α) sin π 2 α 2 −1 !! , (8) Var ˆd(α),hm = d2 (α) 1 k −πΓ(−2α) sin (πα) Γ(−α) sin π 2 α 2 −1 ! + O 1 k2 . (9) • For α = 2, the arithmetic mean estimator, 1 k Pk j=1 |xj|2, is commonly used, which has variance = 2 kd2 (2). It can be improved by taking advantage of the marginal l2 norms [17]. 3 The Fractional Power Estimator The fractional power estimator takes advantage of the following statistical result in Lemma 1. Lemma 1 Suppose x ∼S α, d(α) . Then for −1 < λ < α, E |x|λ = dλ/α (α) 2 π Γ 1 −λ α Γ(λ) sin π 2 λ . (10) If α = 2, i.e., x ∼S(2, d(2)) = N(0, 2d(2)), then for λ > −1, E |x|λ = dλ/2 (2) 2 π Γ 1 −λ 2 Γ(λ) sin π 2 λ = dλ/2 (2) 2Γ (λ) Γ λ 2 . (11) Proof: For 0 < α ≤2 and −1 < λ < α, (10) can be inferred directly from [24, Theorem 2.6.3]. For α = 2, the moment E |x|λ exists for any λ > −1. (11) can be shown by directly integrating the Gaussian density (using the integral formula [8, 3.381.4]). The Euler’s reflection formula Γ(1 −z)Γ(z) = π sin(πz) and the duplication formula Γ(z)Γ z + 1 2 = 21−2z√πΓ(2z) are handy. The fractional power estimator is defined in Lemma 2. See the proof in Appendix A. Lemma 2 Denoted by ˆd(α),fp, the fractional power estimator is defined as ˆd(α),fp = 1 k Pk j=1 |xj|λ∗α 2 π Γ(1 −λ∗)Γ(λ∗α) sin π 2 λ∗α !1/λ∗ × 1 −1 k 1 2λ∗ 1 λ∗−1 2 π Γ(1 −2λ∗)Γ(2λ∗α) sin (πλ∗α) 2 π Γ(1 −λ∗)Γ(λ∗α) sin π 2 λ∗α 2 −1 !! , (12) where λ∗= argmin −1 2α λ< 1 2 g (λ; α) , g (λ; α) = 1 λ2 2 π Γ(1 −2λ)Γ(2λα) sin (πλα) 2 πΓ(1 −λ)Γ(λα) sin π 2 λα 2 −1 ! . (13) Asymptotically (i.e., as k →∞), the bias and variance of ˆd(α),fp are E ˆd(α),fp −d(α) = O 1 k2 , (14) Var ˆd(α),fp = d2 (α) 1 k 1 λ∗2 2 π Γ(1 −2λ∗)Γ(2λ∗α) sin (πλ∗α) 2 π Γ(1 −λ∗)Γ(λ∗α) sin π 2 λ∗α 2 −1 ! + O 1 k2 . (15) Note that in calculating ˆd(α),fp, the real computation only involves Pk j=1 |xj|λ∗α1/λ∗ , because all other terms are basically constants and can be pre-computed. Figure 1(a) plots g (λ; α) as a function of λ for many different values of α. Figure 1(b) plots the optimal λ∗as a function of α. We can see that g (λ; α) is a convex function of λ and −1 < λ∗< 1 2 (except for α = 2), which will be proved in Lemma 3. −1 −.8−.6−.4−.2 0 .2 .4 .6 .8 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 2 1.5 1.2 1 0.8 0.5 λ Variance factor 2e−16 0.3 1.95 1.999 1.9 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 α λopt Figure 1: Left panel plots the variance factor g (λ; α) as functions of λ for different α, illustrating g (λ; α) is a convex function of λ and the optimal solution (lowest points on the curves) are between -1 and 0.5 (α < 2). Note that there is a discontinuity between α →2−and α = 2. Right panel plots the optimal λ∗as a function of α. Since α = 2 is not included, we only see λ∗< 0.5 in the figure. 3.1 Special cases The discontinuity, λ∗(2−) = 0.5 and λ∗(2) = 1, reflects the fact that, for x ∼S(α, d), E |x|λ exists for −1 < λ < α when α < 2 and exists for any λ > −1 when α = 2. When α = 2, since λ∗(2) = 1, the fractional power estimator becomes 1 k Pk j=1 |xj|2, i.e., the arithmetic mean estimator. We will from now on only consider 0 < α < 2. when α →0+, since λ∗(0+) = −1, the fractional power estimator approaches the harmonic mean estimator, which is asymptotically optimal when α = 0+ [13]. When α →1, since λ∗(1) = 0 in the limit, the fractional power estimator has the same asymptotic variance as the geometric mean estimator. 3.2 The Asymptotic (Cram´er-Rao) Efficiency For an estimator ˆd(α), its variance, under certain regularity condition, is lower-bounded by the Information inequality (also known as the Cram´er-Rao bound) [11, Chapter 2], i.e., Var ˆd(α) ≥ 1 kI(α). The Fisher Information I(α) can be approximated by computationally intensive procedures [19]. When α = 2, it is well-known that the arithmetic mean estimator attains the Cram´er-Rao bound. When α = 0+, [13] has shown that the harmonic mean estimator is also asymptotically optimal. Therefore, our fractional power estimator achieves the Cram´er-Rao bound, exactly when α = 2, and asymptotically when α = 0+. The asymptotic (Cram´er-Rao) efficiency is defined as the ratio of 1 kI(α) to the asymptotic variance of ˆd(α) (d(α) = 1 for simplicity). Figure 2 plots the efficiencies for all estimators we have mentioned, illustrating that the fractional power estimator is near-optimal in a wide range of α. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.4 0.5 0.6 0.7 0.8 0.9 1 α Efficiency Fractional Geometric Harmonic Median Figure 2: The asymptotic Cram´er-Rao efficiencies of various estimators for 0 < α < 2, which are the ratios of 1 kI(α) to the asymptotic variances of the estimators. Here k is the sample size and I(α) is the Fisher Information (we use the numeric values in [19]). The asymptotic variance of the sample median estimator ˆd(α),me is computed from known statistical theory for sample quantiles. We can see that the fractional power estimator ˆd(α),fp is close to be optimal in a wide range of α; and it always outperforms both the geometric mean and the harmonic mean estimators. Note that since we only consider α < 2, the efficiency of ˆd(α),fp does not achieve 100% when α →2−. 3.3 Theoretical Properties We can show that, when computing the fractional power estimator ˆd(α),fp, to find the optimal λ∗only involves searching for the minimum on a convex curve in the narrow range λ∗∈ max −1, −1 2α , 0.5 . These properties theoretically ensure that the new estimator is well-defined and is numerically easy to compute. The proof of Lemma 3 is briefly sketched in Appendix B. Lemma 3 Part 1: g (λ; α) = 1 λ2 2 πΓ(1 −2λ)Γ(2λα) sin (πλα) 2 πΓ(1 −λ)Γ(λα) sin π 2 λα 2 −1 ! , (16) is a convex function of λ. Part 2: For 0 < α < 2, the optimal λ∗= argmin −1 2α λ< 1 2 g (λ; α), satisfies −1 < λ∗< 0.5. 3.4 Comparing Variances at Finite Samples It is also important to understand the small sample performance of the estimators. Figure 3 plots the empirical mean square errors (MSE) from simulations for the fractional power estimator, the harmonic mean estimator, and the sample median estimator. The MSE for the geometric mean estimators can be computed exactly without simulations. Figure 3 indicates that the fractional power estimator ˆd(α),fp also has good small sample performance unless α is close to 2. After k ≥50, the advantage of ˆd(α),fp becomes noticeable even when α is very close to 2. It is also clear that the sample median estimator has poor small sample performance; but even at very large k, its performance is not that good except when α is about 1. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 k = 10 α Mean square error (MSE) Fractional Geometric Harmonic Median 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.02 0.04 0.06 0.08 0.1 0.12 k = 50 α Mean square error (MSE) Fractional Geometric Harmonic Median 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.01 0.02 0.03 0.04 0.05 0.06 k = 100 α Mean square error (MSE) Fractional Geometric Harmonic Median 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.0109 k = 500 α Mean square error (MSE) Fractional Geometric Harmonic Median Figure 3: We simulate the mean square errors (MSE) (106 simulations at every α and k) for the harmonic mean estimator (0 < α ≤0.344 only) and the fractional power estimator. We compute the MSE exactly for the geometric mean estimator (for 0.344α < 2). The fractional power has good accuracy (small MSE) at reasonable sample sizes (e.g., k ≥50). But even at small samples (e.g., k = 10), it is quite accurate except when α approaches 2. 4 Discussion The fractional power estimator ˆd(α),fp ∝ Pk j=1 |xj|λ∗α1/λ∗ can be treated as a linear estimator in because the power 1/λ∗is just a constant. However, Pk j=1 |xj|λ∗α is not a metric because λ∗α < 1, as shown in Lemma 3. Thus our result does not conflict the celebrated impossibility result [3], which proved that there is no hope to recover the original l1 distances using linear projections and linear estimators without incurring large errors. Although the fractional power estimator achieves near-optimal asymptotic variance, analyzing its tail bounds does not appear straightforward. In fact, when α approaches 2, this estimator does not have finite moments much higher than the second order, suggesting poor tail behavior. Our additional simulations (not included in this paper) indicate that ˆd(α),fp still has comparable tail probability behavior as the geometric mean estimator, when α ≤1. Finally, we should mention that the method of stable random projections does not take advantage of the data sparsity while high-dimensional data (e.g., text data) are often highly sparse. A new method call Conditional Random Sampling (CRS) [14–16] may be more preferable in highly sparse data. 5 Conclusion In massive datasets such as the Web and massive data streams, dimension reductions are often critical for many applications including clustering, classifications, recommendation systems, and Web search, because the data size may be too large for the physical memory or even for the hard disk and sometimes only one pass of the data can be afforded for building statistical learning models. While there are already many papers on dimension reductions in the l2 norm, this paper focuses on the lα norm for 0 < α ≤2 using stable random projections, as it has become increasingly popular in machine learning to consider the lα norm other than l2. It is also possible to treat α as an additional tuning parameter and re-run the learning algorithms many times for better performance. Our main contribution is the fractional power estimator for stable random projections. This estimator, with a unified format for all 0 < α ≤2, is computationally efficient and (surprisingly) is also near-optimal in terms of the asymptotic variance. We also prove some important theoretical properties (variance, convexity, etc.) to show that this estimator is well-behaved. We expect that this work will help advance the state-of-the-art of dimension reductions in the lα norms. A Proof of Lemma 2 By Lemma 1, we first seek an unbiased estimator of of dλ (α), denoted by ˆR(α),λ, ˆ R(α),λ = 1 k P k j=1 |xj|λα 2 π Γ(1 −λ)Γ(λα) sin π 2 λα , −1/α < λ < 1 whose variance is Var ˆ R(α),λ = d2λ (α) k 2 π Γ(1 −2λ)Γ(2λα) sin (πλα) 2 π Γ(1 −λ)Γ(λα) sin π 2 λα2 −1 ! , −1 2α < λ < 1 2 A biased estimator of d(α) would be simply ˆR(α),λ 1/λ, which has O 1 k bias. This bias can be removed to an extent by Taylor expansions [11, Theorem 6.1.1]. While it is well-known that bias-corrections are not always beneficial because of the bias-variance trade-off phenomenon, in our case, it is a good idea to conduct the bias-correction because the function f(x) = x1/λ is convex for x > 0. Note that f ′(x) = 1 λx1/λ−1 and f ′′(x) = 1 λ 1 λ −1 x1/λ−2 > 0, assuming −1 2α < λ < 1 2. Because f(x) is convex, removing the O 1 k bias will also lead to a smaller variance. We call this new estimator the “fractional power” estimator: ˆd(α),fp,λ = ˆ R(α),λ 1/λ − Var ˆ R(α),λ 2 1 λ 1 λ −1 dλ (α) 1/λ−2 = 1 k P k j=1 |xj|λα 2 π Γ(1 −λ)Γ(λα) sin π 2 λα ! 1/λ 1 −1 k 1 2λ 1 λ −1 2 π Γ(1 −2λ)Γ(2λα) sin (πλα) 2 π Γ(1 −λ)Γ(λα) sin π 2 λα 2 −1 !! , where we plug in the estimated dλ (α). The asymptotic variance would be Var ˆd(α),fp,λ = Var ˆ R(α),λ 1 λ dλ (α) 1/λ−12 + O 1 k2 = d2 (α) 1 λ2k 2 π Γ(1 −2λ)Γ(2λα) sin (πλα) 2 π Γ(1 −λ)Γ(λα) sin π 2 λα 2 −1 ! + O 1 k2 . The optimal λ, denoted by λ∗, is then λ∗= argmin −1 2α λ< 1 2 ( 1 λ2 2 π Γ(1 −2λ)Γ(2λα) sin (πλα) 2 π Γ(1 −λ)Γ(λα) sin π 2 λα 2 −1 !) . B Proof of Lemma 3 We sketch the basic steps; and we direct readers to the additional supporting material for more detail. We use the infinite-product representations of the Gamma and sine functions [8, 8.322,1.431.1], Γ(z) = exp (−γez) z ∞ Y s=1 1 + z s −1 exp z s , sin(z) = z ∞ Y s=1 1 − z2 s2π2 ! , to re-write g (λ; α) as g(λ; α) = 1 λ2 (M (λ; α) −1) = 1 λ2 ∞ Y s=1 fs(λ; α) −1 ! , fs(λ; α) = 1 −λ s 2 1 + 2λα s −1 1 −λα s 1 + λα s 3 1 −λ2α2 4s2 ! −2 1 −2λ s −1 . With respect to λ, the first two derivatives of g(λ; α) are ∂g ∂λ = 1 λ2 −2 λ (M −1) + ∞ X s=1 ∂log fs ∂λ M ! . ∂2g ∂λ2 = M λ2 6 λ2 + ∞ X s=1 ∂2 log fs ∂λ2 + ∞ X s=1 ∂log fs ∂λ ! 2 −4 λ ∞ X s=1 ∂log fs ∂λ ! −6 λ4 . Also, ∞ X s=1 ∂log fs ∂λ = 2λ ∞ X s=1 1 s2 −3sλ + 2λ2 + α2 2 4s2 −λ2α2 + 1 s2 + 3sλα + 2λ2α2 − 1 s2 −λ2α2 , ∞ X s=1 ∂2 log fs ∂λ2 = ∞ X s=1 −2 (s −λ)2 + 4 (s −2λ)2 + 2α2 (2s −λα)2 − α2 (s −λα)2 − 3α2 (s + λα)2 + 4α2 (s + 2λα)2 + 2α2 (2s + λα)2 ∞ X s=1 ∂3 log fs ∂λ3 = ∞ X s=1 4 (s −λ)3 + 16 (s −2λ)3 + 2α3 2 (2s −λα)3 − 1 (s −λα)3 + 3 (s + λα)3 − 8 (s + 2λα)3 − 2 (2s + λα)3 , ∞ X s=1 ∂4 log fs ∂λ4 = ∞ X s=1 −12 (s −λ)4 + 96 (s −2λ)4 + 6α4 2 (2s −λα)4 − 1 (s −λα)4 − 3 (s + λα)4 + 16 (s + 2λα)4 + 2 (2s + λα)4 . To show ∂2g ∂λ2 > 0, it suffices to show λ4 ∂2g ∂λ2 > 0, which can shown based on its own second derivative (and hence we need P∞ s=1 ∂4 log fs ∂λ4 ). Here we consider λ ̸= 0 to avoid triviality. To complete the proof, we use some properties of the Riemann’s Zeta function and the infinite countability. 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Continuous Time Particle Filtering for fMRI Lawrence Murray School of Informatics University of Edinburgh lawrence.murray@ed.ac.uk Amos Storkey School of Informatics University of Edinburgh a.storkey@ed.ac.uk Abstract We construct a biologically motivated stochastic differential model of the neural and hemodynamic activity underlying the observed Blood Oxygen Level Dependent (BOLD) signal in Functional Magnetic Resonance Imaging (fMRI). The model poses a difficult parameter estimation problem, both theoretically due to the nonlinearity and divergence of the differential system, and computationally due to its time and space complexity. We adapt a particle filter and smoother to the task, and discuss some of the practical approaches used to tackle the difficulties, including use of sparse matrices and parallelisation. Results demonstrate the tractability of the approach in its application to an effective connectivity study. 1 Introduction Functional Magnetic Resonance Imaging (fMRI) poses a large-scale, noisy and altogether difficult problem for machine learning algorithms. The Blood Oxygen Level Dependent (BOLD) signal, from which fMR images are produced, is a measure of hemodynamic activity in the brain – only an indirect indicator of the neural processes which are of primary interest in most cases. For studies of higher level patterns of activity, such as effective connectivity [1], it becomes necessary to strip away the hemodynamic activity to reveal the underlying neural interactions. In the first instance, this is because interactions between regions at the neural level are not necessarily evident at the hemodynamic level [2]. In the second, analyses increasingly benefit from the temporal qualities of the data, and the hemodynamic response itself is a form of temporal blurring. We are interested in the application of machine learning techniques to reveal meaningful patterns of neural activity from fMRI. In this paper we construct a model of the processes underlying the BOLD signal that is suitable for use in a filtering framework. The model proposed is close to that of Dynamic Causal Modelling (DCM) [3]. The main innovation over these deterministic models is the incorporation of stochasticity at all levels of the system. This is important; under fixed inputs, DCM reduces to a generative model with steady state equilibrium BOLD activity and independent noise at each time point. Incorporating stochasticity allows proper statistical characterisation of the dependence between brain regions, rather than relying on relating decay rates1. Our work has involved applying a number of filtering techniques to estimate the parameters of the model, most notably the Unscented Kalman Filter [4] and various particle filtering techniques. This paper presents the application of a simple particle filter. [5] take a similar filtering approach, applying a local linearisation filter [6] to a model of individual regions. In contrast, the approach here is applied to multiple regions and their interactions, not single regions in isolation. Other approaches to this type of problem are worth noting. Perhaps the most commonly used technique to date is Structural Equation Modelling (SEM) [7; 8] (e.g. [9; 10; 11]). SEM is a multivariate 1A good analogy is the fundamental difference between modelling time series data yt using an exponentially decaying curve with observational noise xt = axt−1 +c, yt = xt +ϵt, and using a much more flexible Kalman filter xt = axt−1 + c + ωt, yt = xt + ϵt (where xt is a latent variable, a a decay constant, c a constant and ϵ and ω Gaussian variables). 1 regression technique where each dependent variable may be a linear combination of both independent and other dependent variables. Its major limitation is that it is static, assuming that all observations are temporally independent and that interactions are immediate and wholly evident within each single observation. Furthermore, it does not distinguish between neural and hemodynamic activity, and in essence identifies interactions only at the hemodynamic level. The major contributions of this paper are establishing a stochastic model of latent neural and hemodynamic activity, formulating a filtering and smoothing approach for inference in this model, and overcoming the basic practical difficulties associated with this. The estimated neural activity relates to the domain problem and is temporally consistent with the stimulus. The approach is also able to establish connectivity relationships. The ability of this model to establish such connectivity relationships on the basis of stochastic temporal relationships is significant. One problem in using structural equation models for effective connectivity analysis is the statistical equivalence of different causal models. By presuming a temporal causal order, temporal models of this form have no such equivalence problems. Any small amount of temporal connectivity information available in fMRI data is of significant benefit, as it can disambiguate between statically equivalent models. Section 2 outlines the basis of the hemodynamic model that is used. This is combined with neural, input and measurement models in Section 3 to give the full framework. Inference and parameter estimation are discussed in Section 4, before experiments and analysis in Sections 5 and 6. 2 Hemodynamics Temporal analysis of fMRI is significantly confounded by the fact that it does not measure brain activity directly, but instead via hemodynamic activity, which (crudely) temporally smooths the activity signal. The quality of temporal analysis therefore depends significantly on the quality of model used to relate neural and hemodynamic activity. This relationship may be described using the now well established Balloon model [12]. This models a venous compartment as a balloon using Windkessel dynamics. The state of the compartment is represented by its blood volume normalised to the volume at rest, v = V/V0 (blood volume V , rest volume V0), and deoxyhemoglobin (dHb) content normalised to the content at rest, q = Q/Q0 (dHb content Q, rest content Q0). The compartment receives inflow of fully oxygenated arterial blood fin(t), extracts oxygen from the blood, and expels partially deoxygenated blood fout(t). The full dynamics may be represented by the differential system: dq dt = 1 τ0 · fin(t)E(t) E0 −fout(v)q v ¸ (1) dv dt = 1 τ0 [fin(t) −fout(v)] (2) E(t) ≈ 1 −(1 −E0) 1 fin(t) (3) fout(v) ≈ v 1 α (4) where τ0 and α are constants, and E0 is the oxygen extraction fraction at rest. This base model is driven by the independent input fin(t). It may be further extended to couple in neural activity z(t) via an abstract vasodilatory signal s [13]: df dt = s (5) ds dt = ϵz(t) −s τs −(f −1) τf . (6) The complete system defined by Equations 1-6, with fin(t) = f, is now driven by the independent input z(t). From the balloon model, the relative BOLD signal change over the baseline S at any time may be predicted using [12]: ∆S S = V0 h k1(1 −q) + k2 ³ 1 −q v ´ + k3(1 −v) i . (7) Figure 1 illustrates the system dynamics. Nominal values for constants are given in Table 1. 2 0.84 1.04 0 30 q 0.9 1.35 0 30 v 0.8 1.9 0 30 f -0.3 0.7 0 30 s -0.2 1 0 30 BOLD (%) Figure 1: Response of the balloon model to a 1s burst of neural activity at magnitude 1 (time on x axis, response on y axis). 3 Model We define a model of the neural and hemodynamic interactions between M regions of interest. A region consists of neural tissue and a venous compartment. The state xi(t) of region i at time t is given by: xi(t) = zi(t) neural activity fi(t) normalised blood flow into the venous compartment si(t) vasodilatory signal qi(t) normalised dHb content of the venous compartment vi(t) normalised blood volume of the venous compartment The complete state at time t is given by x(t) = (x1(t)T , . . . , xM(t)T )T . We construct a model of the interactions between regions in four parts – the input model, the neural model, the hemodynamic model and the measurement model. 3.1 Input model The input model represents the stimulus associated with the experimental task during an fMRI session. In general this is a function u(t) with U dimensions. For a simple block design paradigm a one-dimensional box-car function is sufficient. 3.2 Neural model Neural interactions between the regions are given by: dz = Az dt + Cu dt + c + Σz dW, (8) where dW is the M-dimensional standard (zero mean, unit variance) Wiener process, A an M ×M matrix of efficacies between regions, C an M × U matrix of efficacies between inputs and regions, c an M-dimensional vector of constant terms and Σz an M × M diagonal diffusion matrix with σz1, . . . , σzM along the diagonal. This is similar to the deterministic neural model of DCM expressed as a stochastic differential equation, but excludes the bilinear components allowing modulation of connections between seeds. In theory these can be added, we simply limit ourselves to a simpler model for this early work. In addition, and unlike DCM, nonlinear interactions between regions could also be included to account for modulatory activity. Again it seems sensible to keep the simplest linear case at this stage of the work, but the potential for nonlinear generalisation is one of the longer term benefits of this approach. 3.3 Hemodynamic model Within each region, the variables fi, si, qi, vi and zi interact according to a stochastic extension of the balloon model (c.f. Equations 1-6). It is assumed that regions are sufficiently separate that their Constant τ0 τf τs α ϵ V0 E0 k1 k2 k3 Value 0.98 1/0.65 1/0.41 0.32 0.8 0.018 0.4 7E0 2 2E0 −0.2 Table 1: Nominal values for constants of the balloon model [12; 13]. 3 hemodynamic activity is independent given neural activity[14]. Noise in the form of the Wiener process is introduced to si and the log space of fi, qi and vi, in the latter three cases to ensure positivity: d ln fi = 1 fi si dt + σfi dW (9) dsi = · ϵzi −s τs −(f −1) τf ¸ dt + σsi dW (10) d ln qi = 1 qiτ0 " fi 1 −(1 −E0) 1 fi E0 −v 1 α −1 i qi # dt + σqi dW (11) d ln vi = 1 viτ0 h fi −v 1 α i i dt + σvi dW. (12) 3.4 Measurement model The relative BOLD signal change at any time for a particular region is given by (c.f. Equation 7): ∆yi = V0 · k1(1 −qi) + k2 µ 1 −qi vi ¶ + k3(1 −vi) ¸ . (13) This may be converted to an absolute measurement y∗ i for comparison with actual observations by using the baseline signal bi for each seed and an independent noise source ξ ∼N(0, 1): y∗ i = bi(1 + ∆yi) + σyiξ. (14) 4 Estimation The model is completely defined by Equations 8 to 14. This fits nicely into a filtering framework, whereby the input, neural and hemodynamic models define state transitions, and the measurement model predicted observations. For i = 1, . . . , M, σzi, σfi, σsi, σqi and σvi define the system noise and σyi the measurement noise. Parameters to estimate are the elements of A, C, c and b. For a sequence of time points t1, . . . , tT , we are given observations y(t1), . . . , y(tT ), where y(t) = (y1(t), . . . , yM(t))T . We seek to exploit the data as much as possible by estimating P(x(tn) | y(t1), . . . , y(tT )) for n = 1, . . . , T – the distribution over the state at each time point given all the data. Because of non-Gaussianity and nonlinearity of the transitions and measurements, a two-pass particle filter is proposed to solve the problem. The forward pass is performed using a sequential importance resampling technique similar to CONDENSATION [15], obtaining P(x(tn) | y(t1), . . . , y(tn)) for n = 1, . . . , T. Resampling at each step is handled using a deterministic resampling method [16]. The transition of particles through the differential system uses a 4th/5th order Runge-Kutta-Fehlberg method, the adaptive step size maintaining fixed error bounds. The backwards pass is substantially more difficult. Naively, we can simply negate the derivatives of the differential system and step backwards to obtain P(x(tn) | y(tn+1), . . . , y(tT )), then fuse these with the results of the forwards pass to obtain the desired posterior. Unfortunately, such a backwards model is divergent in q and v, so that the accumulated numerical errors of the Runge-Kutta can easily cause an explosion to implausible values and a tip-toe adaptive step size to maintain error bounds. This can be mitigated by tightening the error bounds, but the task becomes computationally prohibitive well before the system is tamed. An alternative is a two-pass smoother that reuses particles from the forwards pass [17], reweighting them on the backwards pass so that no explicit backwards dynamics are required. This sidesteps the divergence issue completely, but is computationally and spatially expensive and requires computation of p(x(tn) = s(i) tn | x(tn−1) = s(j) tn−1) for particular particles s(i) tn and s(j) tn−1. This imposes some limitations, but is nevertheless the method used here. The forwards pass provides a weighted sample set {(s(i) t , π(i) t )} at each time point t = t1, . . . , tT for i = 1, . . . , P. Initialising with ψtT = πtT , the backwards step to calculate weights at time tn is 4 as follows [17]2: α(i,j) tn = p(x(tn+1) = s(i) tn+1 | x(tn) = s(j) tn ) for i, j = 1, . . . , P γtn = αtnπtn δtn = αT tn(ψtn+1 ⊘γtn) where ⊘is element-wise division, ψtn = πtn ⊗δtn where ⊗is element-wise multiplication. These are then normalised so that P ψ(i) tn = 1 and the smoothed result {(s(i) tn , ψ(i) tn )} for i = 1, . . . , P is stored. There are numerous means of propagating particles through the forwards pass that accommodate the resampling step and propagation of the Wiener noise through the nonlinearity. These include various stochastic Runge-Kutta methods, the Unscented Transformation [4] or a simple Euler scheme using fixed time steps and adding an appropriate portion of noise after each step. The requirement to efficiently make P 2 density calculations of p(x(tn+1) = s(i) tn+1 | x(tn) = s(j) tn ) during the backwards pass is challenging with such approaches, however. To keep things simple, we instead simply propagate particles noiselessly through the transition function, and add noise from the Wiener process only at times t1, . . . , tT as if the transition were linear. This reasonably approximates the noise of the system while keeping the density calculations very simple – transition s(j) tn noiselessly to obtain the mean value of a Gaussian with covariance equal to that of the system noise, then calculate the density of this Gaussian at s(i) tn+1. Observe that if system noise is sufficiently tight, αtn becomes sparse as negligibly small densities round to zero. Implementing αtn as a sparse matrix can provide significant time and space savings. Propagation of particles through the transition function and density calculations can be performed in parallel. This applies during both passes. For the backwards pass, each particle at tn need only be transitioned once to produce a Gaussian from which the density of all particles at tn+1 can be calculated, filling in one column of αtn. Finally, the parameters A, C, c and b may be estimated by adding them to the state with artificial dynamics (c.f. [18]), applying a broad prior and small system noise to suggest that they are generally constant. The same applies to parameters of the balloon model, which may be included to allow variation in the hemodynamic response across the brain. 5 Experiments We apply the model to data collected during a simple finger tapping exercise. Using a Siemens Vision at 2T with a TR of 4.1s, a healthy 23-year-old right-handed male was scanned on 33 separate days over a period of two months. In each session, 80 whole volumes were taken, with the first two discarded to account for T1 saturation effects. The experimental paradigm consists of alternating 6TR blocks of rest and tapping of the right index finger at 1.5Hz, where tapping frequency is provided by a constant audio cue, present during both rest and tapping phases. All scans across all sessions were realigned using SPM5 [19] and a two-level random effects analysis performed, from which 13 voxels were selected to represent regions of interest. No smoothing or normalisation was applied to the data. Of the 13 voxels, four are selected for use in this experiment – located in the left posterior parietal cortex, left M1, left S1 and left premotor cortex. The mean of all sessions is used as the measurement y(t), which consists of M = 4 elements, one for each region. We set t1 = 1TR = 4.1s, . . . , tT = 78TR = 319.8s as the sequence of times, corresponding to the times at which measurements are taken after realignment. The experimental input function u(t) is plotted in Figure 2, taking a value of 0 at rest and 1 during tapping. The error bounds on the Runge-Kutta are set to 10−4. Measurement noise is set to σyi = 2 for i = 1, . . . , M and the prior and system noise as in Table 2. With the elements of A, C, c and b included in the state, the state size is 48. P = 106 particles are used for the forwards pass, downsampling to 2.5 × 104 particles for the more expensive backwards pass. 2We have expressed this in matrix notation rather than the original notation in [17] 5 -1 0 1 2 0 6 12 18 Figure 2: Experimental input u(t), x axis is time t expressed in TRs. 0 1x108 2x108 3x108 4x108 0 77 Figure 3: Number of nonzero elements in αtn for n = 1, . . . , 77. Prior Noise µ σ σ Ai,i i = 1, . . . , N −1 1/2 10−2 Ai,j i, j = 1, . . . , N, i ̸= j 0 1/2 10−2 Ci,1 i = 1, . . . , N 0 1/2 10−2 zi i = 1, . . . , N 0 1/2 10−1 fi, si, qi, vi, ci i = 1, . . . , N 0 1/2 10−2 bi i = 1, . . . , N ¯yi 10 10−2 Table 2: Prior and system noise. The experiment is run on the Eddie cluster of the Edinburgh Compute and Data Facility (ECDF) 3 over 200 nodes, taking approximately 10 minutes real time. The particle filter and smoother are distributed across nodes and run in parallel using the dysii Dynamic Systems Library 4. After application of the filter, the predicted neural activity is given in Figure 4 and parameter estimates in Figures 6 and 7. The predicted output obtained from the model is in Figure 5, where it is compared to actual measurements acquired during the experiment to assess model fit. 6 Discussion The model captures the expected underlying form for neural activity, with all regions distinctly correlated with the experimental stimulus. Parameter estimates are generally constant throughout the length of the experiment and some efficacies are significant enough in magnitude to provide biological insight. The parameters found typically match those expected for this form of finger tapping task. However, as the focus of this paper is the development of the filtering approach we will reserve a real analysis of the results for a future paper, and focus on the issues surrounding the filter and its capabilities and deficiencies. A number of points are worth making in this regard. Particles stored during the forwards pass do not necessarily support the distributions obtained during the backwards pass. This is particularly obvious towards the extreme left of Figure 4, where the smoothed results appear to become erratic, essentially due to degeneracy in the backwards pass. Furthermore, while the smooth weighting of particles in the forwards pass is informative, that of the backwards pass is often not, potentially relying on heavy weighting of outlying particles and shedding little light on the actual nature of the distributions involved. Figure 3 provides empirical results as to the sparseness of αtn. At worst at least 25% of elements are zero, demonstrating the advantages of a sparse matrix implementation in this case. The particle filter is able to establish consistent neural activity and parameter estimates across runs. These estimates also come with distributions in the form of weighted sample sets which enable the uncertainty of the estimates to be understood. This certainly shows the stochastic model and particle filter to be a promising approach for systematic connectivity analysis. 3http://www.is.ed.ac.uk/ecdf/ 4http://www.indii.org/software/dysii/ 6 0 0.14 -1 0 1 0 0.14 -1 0 1 0 0.14 -1 0 1 0 0.14 0 319.8 -1 0 1 Figure 4: Neural activity predictions z (y axis) over time (x axis). Forwards pass results as shaded histogram, smoothed results as solid line with 2σ error. 0 0.06 180 190 200 210 0 0.06 180 190 200 210 0 0.06 180 190 200 210 0 0.06 0 319.8 180 190 200 210 Figure 5: Measurement predictions y∗(y axis) over time (x axis). Forwards pass results as shaded histogram, smoothed results as solid line with 2σ error, circles actual measurements. -2 -1 0 1 -2 -1 0 1 -2 -1 0 1 -2 -1 0 1 0 0.2 0 0.2 0 0.2 0 0.2 Figure 6: Parameter estimates A (y axis) over time (x axis). Forwards pass results as shaded histogram, smoothed results as solid line with 2σ error. The authors would like to thank David McGonigle for helpful discussions and detailed information regarding the data set. 7 -1 0 1 0 0.2 Figure 7: Parameter estimates of C (y axis) over time (x axis). Forwards pass results as shaded histogram, smoothed results as solid line with 2σ error. References [1] Friston, K. and Buchel, C. (2004) Human Brain Function, chap. 49, pp. 999–1018. Elsevier. [2] Gitelman, D. R., Penny, W. D., Ashburner, J., and Friston, K. J. (2003) Modeling regional and psychophysiologic interactions in fMRI: the importance of hemodynamic deconvolution. NeuroImage, 19, 200–207. [3] Friston, K., Harrison, L., and Penny, W. (2003) Dynamic causal modelling. NeuroImage, 19, 1273–1302. [4] Julier, S. J. and Uhlmann, J. K. (1997) A new extension of the Kalman filter to nonlinear systems. The Proceedings of AeroSense: The 11th International Symposium on Aerospace/Defense Sensing, Simulation and Controls, Multi Sensor Fusion, Tracking and Resource Management. [5] Riera, J. J., Watanabe, J., Kazuki, I., Naoki, M., Aubert, E., Ozaki, T., and Kawashim, R. (2004) A state-space model of the hemodynamic approach: nonlinear filtering of BOLD signals. NeuroImage, 21, 547–567. [6] Ozaki, T. (1993) A local linearization approach to nonlinear filtering. International Journal on Control, 57, 75–96. [7] Bentler, P. M. and Weeks, D. G. (1980) Linear structural equations with latent variables. Psychometrika, 45, 289–307. [8] McArdle, J. J. and McDonald, R. P. (1984) Some algebraic properties of the reticular action model for moment structures. British Journal of Mathematical and Statistical Psychology, 37, 234–251. [9] Schlosser, R., Gesierich, T., Kaufmann, B., Vucurevic, G., Hunsche, S., Gawehn, J., and Stoeterb, P. (2003) Altered effective connectivity during working memory performance in schizophrenia: a study with fMRI and structural equation modeling. NeuroImage, 19, 751–763. [10] Au Duong, M., et al. (2005) Modulation of effective connectivity inside the working memory network in patients at the earliest stage of multiple sclerosis. NeuroImage, 24, 533–538. [11] Storkey, A. J., Simonotto, E., Whalley, H., Lawrie, S., Murray, L., and McGonigle, D. (2007) Learning structural equation models for fMRI. Advances in Neural Information Processing Systems, 19. [12] Buxton, R. B., Wong, E. C., and Frank, L. R. (1998) Dynamics of blood flow and oxygenation changes during brain activation: The balloon model. Magnetic Resonance in Medicine, 39, 855–864. [13] Friston, K. J., Mechelli, A., Turner, R., and Price, C. J. (2000) Nonlinear responses in fMRI: The balloon model, Volterra kernels, and other hemodynamics. NeuroImage, 12, 466–477. [14] Zarahn, E. (2001) Spatial localization and resolution of BOLD fMRI. Current Opinion in Neurobiology, 11, 209–212. [15] Isard, M. and Blake, A. (1998) Condensation – conditional density propagation for visual tracking. International Journal of Computer Vision, 29, 5–28. [16] Kitagawa, G. (1996) Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. Journal of Computational and Graphical Statistics, 5, 1–25. [17] Isard, M. and Blake, A. (1998) A smoothing filter for condensation. Proceedings of the 5th European Conference on Computer Vision, 1, 767–781. [18] Kitagawa, G. (1998) A self-organising state-space model. Journal of the American Statistical Association, 93, 1203–1215. [19] Wellcome Department of Imaging Neuroscience (2006), Statistical parametric mapping. Online at www.fil.ion.ucl.ac.uk/spm/. 8
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Computing Robust Counter-Strategies Michael Johanson johanson@cs.ualberta.ca Martin Zinkevich maz@cs.ualberta.ca Michael Bowling Computing Science Department University of Alberta Edmonton, AB Canada T6G2E8 bowling@cs.ualberta.ca Abstract Adaptation to other initially unknown agents often requires computing an effective counter-strategy. In the Bayesian paradigm, one must find a good counterstrategy to the inferred posterior of the other agents’ behavior. In the experts paradigm, one may want to choose experts that are good counter-strategies to the other agents’ expected behavior. In this paper we introduce a technique for computing robust counter-strategies for adaptation in multiagent scenarios under a variety of paradigms. The strategies can take advantage of a suspected tendency in the decisions of the other agents, while bounding the worst-case performance when the tendency is not observed. The technique involves solving a modified game, and therefore can make use of recently developed algorithms for solving very large extensive games. We demonstrate the effectiveness of the technique in two-player Texas Hold’em. We show that the computed poker strategies are substantially more robust than best response counter-strategies, while still exploiting a suspected tendency. We also compose the generated strategies in an experts algorithm showing a dramatic improvement in performance over using simple best responses. 1 Introduction Many applications for autonomous decision making (e.g., assistive technologies, electronic commerce, interactive entertainment) involve other agents interacting in the same environment. The agents’ choices are often not independent, and good performance may necessitate adapting to the behavior of the other agents. A number of paradigms have been proposed for adaptive decision making in multiagent scenarios. The agent modeling paradigm proposes to learn a predictive model of other agents’ behavior from observations of their decisions. The model is then used to compute or select a counter-strategy that will perform well given the model. An alternative paradigm is the mixture of experts. In this approach, a set of expert strategies is identified a priori. These experts can be thought of as counter-strategies for the range of expected tendencies in the other agents’ behavior. The decision maker, then, chooses amongst the counter-strategies based on their online performance, commonly using techniques for regret minimization (e.g., UCB1 [ACBF02]). In either approach, finding counter-strategies is an important subcomponent. The most common approach to choosing a counter-strategy is best response: the performance maximizing strategy if the other agents’ behavior is known [Rob51, CM96]. In large domains where best response computations are not tractable, they are often approximated with “good responses” from a computationally tractable set, where performance maximization remains the only criterion [RV02]. The problem with this approach is that best response strategies can be very brittle. While max1 imizing performance against the model, they can (and often do) perform poorly when the model is wrong. The use of best response counter-strategies, therefore, puts an impossible burden on a priori choices, either the agent model bias or the set of expert counter-strategies. McCracken and Bowling [MB04] proposed ϵ-safe strategies to address this issue. Their technique chooses the best performance maximizing strategy from the set of strategies that don’t lose more than ϵ in the worstcase. The strategy balances exploiting the agent model with a safety guarantee in case the model is wrong. Although conceptually appealing, it is computationally infeasible even for moderately sized domains and has only been employed in the simple game of Ro-Sham-Bo. In this paper, we introduce a new technique for computing robust counter-strategies. The counterstrategies, called restricted Nash responses, balance performance maximization against the model with reasonable performance even when the model is wrong. The technique involves computing a Nash equilibrium of a modified game, and therefore can exploit recent advances in solving large extensive games [GHPS07, ZBB07, ZJBP08]. We demonstrate the practicality of the approach in the challenging domain of poker. We begin by reviewing the concepts of extensive form games, best responses, and Nash equilibria, as well as describing how these concepts apply in the poker domain. We then describe a technique for computing an approximate best response to an arbitrary poker strategy, and show that this, indeed, produces brittle counter-strategies. We then introduce restricted Nash responses, describe how they can be computed efficiently, and show that they are significantly more robust while still being effective counter-strategies. Finally, we demonstrate that these strategies can be used in an experts algorithm to make a more effective adaptive player than when using simple best response. 2 Background A perfect information extensive game consists of a tree of game states. At each game state, an action is made either by nature, or by one of the players, or the state is a terminal state where each player receives a fixed utility. A strategy for a player consists of a distribution over actions for every game state. In an imperfect information extensive game, the states where a player makes an action are divided into information sets. When a player chooses an action, it does not know the state of the game, only the information set, and therefore its strategy is a mapping from information sets to distributions over actions. A common restriction on imperfect information extensive games is perfect recall, where two states can only be in the same information set for a player if that player took the same actions from the same information sets to reach the two game states. In the remainder of the paper, we will be considering imperfect information extensive games with perfect recall. Let σi be a strategy for player i where σi(I, a) is the probability that strategy assigns to action a in information set I. Let Σi be the set of strategies for player i, and define ui(σ1, σ2) to be the expected utility of player i if player 1 uses σ1 ∈Σ1 and player 2 uses σ2 ∈Σ2. Define BR(σ2) ⊆Σ1 to be the set of best responses to σ2, i.e.: BR(σ2) = argmax σ1∈Σ1 u1(σ1, σ2) (1) and define BR(σ1) ⊆Σ2 similarly. If σ1 ∈BR(σ2) and σ2 ∈BR(σ1), then (σ1, σ2) is a Nash equilibrium. A zero-sum extensive game is an extensive game where u1 = −u2. In this type of game, for any two equilibria (σ1, σ2) and (σ′ 1, σ′ 2), u1(σ1, σ2) = u1(σ′ 1, σ′ 2) and (σ1, σ′ 2) (as well as (σ′ 1, σ2)) are also equilibria. Define the value of the game to player 1 (v1) to be the expected utility of player 1 in equilibrium. In a zero-sum extensive game, the exploitability of a strategy σ1 ∈Σ1 is: ex(σ1) = max σ2∈Σ2(v1 −u1(σ1, σ2)). (2) The value of the game to player 2 (v2) and the exploitability of a strategy σ2 ∈Σ2 are defined similarly. A strategy which can be exploited for no more than ϵ is ϵ-safe. An ϵ-Nash equilibrium in a zero-sum extensive game is a strategy pair where both strategies are ϵ-safe. In the remainder of the work, we will be dealing with mixing two strategies. Informally, one can think of mixing two strategies as performing the following operation: first, flip a (possibly biased) coin; if it comes up heads, use the first strategy, otherwise use the second strategy. Formally, define πσi(I) to be the probability that player i when following strategy σi chooses the actions necessary to 2 make information set I reachable from the root of the game tree. Given σ1, σ′ 1 ∈Σ1 and p ∈[0, 1], define mixp(σ1, σ′ 1) ∈Σ1 such that for any information set I of player 1, for all actions a: mixp(σ1, σ′ 1)(I, a) = p × πσ1(I)σ1(I, a) + (1 −p) × πσ′ 1(I)σ1(I, a) p × πσ1(I) + (1 −p) × πσ′ 1(I) . (3) Given an event E, define Prσ1,σ2[E] to be the probability of the event E given player 1 uses σ1, and player 2 uses σ2. Given the above definition of mix, it is the case that for all σ1, σ′ 1 ∈Σ1, all σ2 ∈Σ2, all p ∈[0, 1], and all events E: Pr mixp(σ1,σ′ 1),σ2[E] = p Pr σ1,σ2[E] + (1 −p) Pr σ′ 1,σ2[E] (4) So probabilities of outcomes can simply be combined linearly. As a result the utility of a mixture of strategies is just u(mixp(σ1, σ′ 1), σ2) = pu(σ1, σ2) + (1 −p)u(σ′ 1, σ2). 3 Texas Hold’Em While the techniques in this paper apply to general extensive games, our empirical results will focus on the domain of poker. In particular, we look at heads-up limit Texas Hold’em, the game used in the AAAI Computer Poker Competition [ZL06]. A single hand of this poker variant consists of two players each being dealt two private cards, followed by five community cards being revealed. Each player tries to form the best five-card poker hand from the community cards and her private cards: if the hand goes to a showdown, the player with the best five-card hand wins the pot. The key to good play is on average to have more chips in the pot when you win than are in the pot when you lose. The players’ actions control the pot size through betting. After the private cards are dealt, a round of betting occurs, followed by additional betting rounds after the third (flop), fourth (turn), and fifth (river) community cards are revealed. Betting rounds involve players alternately deciding to either fold (letting the other player win the chips in the pot), call (matching the opponent’s chips in the pot), or raise (matching, and then adding an additional fixed amount into the pot). No more than four raises are allowed in a single betting round. Notice that heads-up limit Texas Hold’em is an example of a finite imperfect information extensive game with perfect recall. When evaluating the results of a match (several hands of poker) between two players, we find it convenient to state the result in millibets won per hand. A millibet is one thousandth of a small-bet, the fixed magnitude of bets used in the first two rounds of betting. To provide some intuition for these numbers, a player that always folds will lose 750 mb/h while a typical player that is 10 mb/h stronger than another would require over one million hands to be 95% certain to have won overall. Abstraction. While being a relatively small variant of poker, the game tree for heads-up limit Texas Hold’em is still very large, having approximately 9.17×1017 states. Fundamental operations, such as computing a best response strategy or a Nash equilibrium as described in Section 2, are intractable on the full game. Common practice is to define a more reasonably sized abstraction by merging information sets (e.g., by treating certain hands as indistinguishable). If the abstraction involves the same betting structure, a strategy for an abstract game can be played directly in the full game. If the abstraction is small enough Nash equilibria and best response computations become feasible. Finding an approximate Nash equilibrium in an abstract game has proven to be an effective way to construct a strong program for the full game [BBD+03, GS06]. Recent solution techniques have been able to compute approximate Nash equilibria for abstractions with as many as 1010 game states [ZBB07, GHPS07]. Given a strategy defined in a small enough abstraction, it is also possible to compute a best response to the strategy in the abstract game. This can be done in time linear in the size of the extensive game. The abstraction used in this paper has approximately 6.45 × 109 game states, and is described in an accompanying technical report [JZB07]. The Competitors. Since this work focuses on adapting to other agents’ behavior, our experiments make use of a battery of different poker playing programs. We give a brief description of these programs here. PsOpti4 [BBD+03] is one of the earliest successful near equilibrium programs for poker and is available as “Sparbot” in the commercial title Poker Academy. PsOpti6 is a later and weaker variant, but whose weaknesses are thought to be less obvious to human players. Together, PsOpti4 and PsOpti6 formed Hyperborean, the winner of the AAAI 2006 Computer Poker Competition. S1239, S1399, and S2298 are similar near equilibrium strategies generated by a new 3 equilibrium computation method [ZBB07] using a much larger abstraction than is used in PsOpti4 and PsOpti6. A60 and A80 are two past failed attempts at generating interesting exploitive strategies, and are highly exploitable for over 1000 mb/h. CFR5 is a new near Nash equilibrium [ZJBP08], and uses the abstraction described in the accompanying technical report [JZB07]. We will also experiment with two programs Bluffbot and Monash, who placed second and third respectively in the AAAI 2006 Computer Poker Competition’s bankroll event [ZL06]. 4 Frequentist Best Response In the introduction, we described best response counter-strategies as brittle, performing poorly when playing against a different strategy from the one which they were computed to exploit. In this section, we examine this claim empirically in the domain of poker. Since a best response computation is intractable in the full game, we first describe a technique, called frequentist best response, for finding a “good response” using an abstract game. As described in the previous section, given a strategy in an abstract game we can compute a best response to that strategy within the abstraction. The challenge is that the abstraction used by an arbitrary opponent is not known. In addition, it may be beneficial to find a best response in an alternative, possible more powerful, abstraction. Suppose we want to find a “good response” to some strategy P. The basic idea of frequentist best response (FBR) is to observe P playing the full game of poker, construct a model of it in an abstract game (unrelated to that P’s own abstraction), and then compute a best-response in this abstraction. FBR first needs many examples of the strategy playing the full, unabstracted game. It then iterates through every one of P’s actions for every hand. It finds the action’s associated information set in the abstract game and increments a counter associated with that information set and action. After observing a sufficient number of hands, we can construct a strategy in the abstract game based on the frequency counts. At each information set, we set the strategy’s probability for performing each action to be the number of observations of that action being chosen from that information set, divided by the total number of observations in the information set. If an information set was never observed, the strategy defaults to the call action. Since this strategy is defined in a known abstraction, FBR can simply calculate a best response to this frequentist strategy. P’s opponent in the observed games greatly affects the quality of the model. We have found it most effective to have P play against a trivial strategy that calls and raises with equal probability. This provides with us the most observations of P’s decisions that are well distributed throughout the possible betting sequences. Observing P in self-play or against near equilibrium strategies has shown to require considerably more observed hands. We typically use 5 million hands of training data to compute the model strategy, although reasonable responses can still be computed with as few as 1 million hands. Evaluation. We computed frequentist best response strategies against seven different opponents. We played the resulting responses both against the opponent it was designed to exploit as well as the other six opponents and an approximate equilibrium strategy computed using the same abstraction. The results of this tournament are shown as a crosstable in Table 1. Positive numbers (appearing with a green background) are in favor of the row player (FBR strategies, in this case). The first thing to notice is that FBR is very successful at exploiting the opponent it was designed to exploit, i.e., the diagonal of the crosstable is positive and often large. In some cases, FBR identified strategies exploiting the opponent for more than previously known to be possible, e.g., PsOpti4 had only previously been exploited for 75 mb/h [Sch06], while FBR exploits it for 137 mb/h. The second thing to notice is that when FBR strategies play against other opponents their performance is poor, i.e., the off-diagonal of the crosstable is generally negative and occasionally by a large amount. For example, A60 is not a strong program. It is exploitable for over 2000 mb/h (note that always fold only loses 750 mb/h) and an approximate equilibrium strategy defeats it by 93 mb/h. Yet, every FBR strategy besides the one trained on it, loses to it, sometimes by a substantial amount. These results give evidence that best response is, in practice, a brittle computation, and can perform poorly when the model is wrong. One exception to this trend is play within the family of S-bots. In particular, consider S1399 and S1239, which are very similar programs, using the same technique for equilibrium computation with the same abstract game. They only differ in the number of iterations the algorithm was afforded. The 4 Opponents PsOpti4 PsOpti6 A60 A80 S1239 S1399 S2298 CFR5 Average FBR-PsOpti4 137 -163 -227 -231 -106 -85 -144 -210 -129 FBR-PsOpti6 -79 330 -68 -89 -36 -23 -48 -97 -14 FBR-A60 -442 -499 2170 -701 -359 -305 -377 -620 -142 FBR-A80 -312 -281 -557 1048 -251 -231 -266 -331 -148 FBR-S1239 -20 105 -89 -42 106 91 -32 -87 3 FBR-S1399 -43 38 -48 -77 75 118 -46 -109 -11 FBR-S2298 -39 51 -50 -26 42 50 33 -41 2 CFR5 36 123 93 41 70 68 17 0 56 Max 137 330 2170 1048 106 118 33 0 Table 1: Results of frequentist best responses (FBR) against a variety of opponent programs in full Texas Hold’em, with winnings in mb/h for the row player. Results involving PsOpti4 or PsOpti6 used 10 duplicate matches of 10,000 hands and are significant to 20 mb/h. Other results used 10 duplicate matches of 500,000 hands and are significant to 2 mb/h. results show they do share weaknesses as FBR-S1399 does beat S1239 by 75 mb/h. However, this is 30% less than 106 mb/h, the amount that FBR-S1239 beats the same opponent. Considering the similarity of these opponents, even this apparent exception is actually suggestive that best response is not robust to even slight changes in the model. Finally, consider the performance of the approximate equilibrium player, CFR5. As it was computed from a relatively large abstraction it performs comparably well, not losing to any of the seven opponents. However, it also does not win by the margins of the correct FBR strategy. As noted, against the highly exploitable A60, it wins by a mere 93 mb/h. What we really want is a compromise. We would like a strategy that can exploit an opponent successfully like FBR, but without the large penalty when playing against a different opponent. The remainder of the paper examines Restricted Nash Response, a technique for creating such strategies. 5 Restricted Nash Response Imagine that you had a model of your opponent, but did not believe that this model was perfect. The model may capture the general idea of the adversary you expect to face, but most likely is not identical. For example, maybe you have played a previous version of the same program, have a model of its play, but suspect that the designer is likely to have made some small improvements in the new version. One way to explicitly define our situation is that with the new version we might expect that 75 percent of the hands will be played identically to the old version. The other 25 percent is some new modification, for which we want to be robust. This, in itself, can be thought of as a game for which we can apply the usual game theoretic machinery of equilibria. Let our model of our opponent be some strategy σfix ∈Σ2. Define Σp,σfix 2 to be those strategies of the form mixp(σfix, σ′ 2), where σ′ 2 is an arbitrary strategy in Σ2. Define the set of restricted best responses to σ1 ∈Σ1 to be: BRp,σfix(σ1) = argmax σ2∈Σ p,σfix 2 u2(σ1, σ2) (5) A (p, σfix) restricted Nash equilibrium is a pair of strategies (σ∗ 1, σ∗ 2) where σ∗ 2 ∈BRp,σfix(σ∗ 1) and σ∗ 1 ∈BR(σ∗ 2). In this pair, the strategy σ∗ 1 is a p-restricted Nash response (RNR) to σfix. We propose these RNRs would be ideal counter-strategies for σfix, where p provides a balance between exploitation and exploitability. This concept is closely related to ϵ-safe best responses [MB04]. Define Σϵ-safe 1 ⊆Σ1 to be the set of all strategies which are ϵ-safe (with an exploitability less than ϵ). Then the set of ϵ-safe best responses are: BRϵ-safe(σ2) = argmax σ1∈Σϵ-safe u1(σ1, σ2) (6) Theorem 1 For all σ2 ∈Σ2, for all p ∈(0, 1], if σ1 is a p-RNR to σ2, then there exists an ϵ such that σ1 is an ϵ-safe best response to σ2. 5 80 100 120 140 160 180 200 220 240 260 0 1000 2000 3000 4000 5000 6000 7000 8000 Exploitation (mb/h) Exploitability (mb/h) (0.00) (0.50) (0.75) (0.82) (0.85) (0.90) (0.95) (0.99) (1.00) (a) Versus PsOpti4 0 100 200 300 400 500 600 700 800 900 1000 1100 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Exploitation (mb/h) Exploitability (mb/h) (0.00) (0.25) (0.40) (0.45) (0.50) (0.55) (0.60) (0.80) (0.90) (0.95) (1.00) (b) Versus A80 Figure 1: The tradeoff between ϵ and utility. For each opponent, we varied p ∈[0, 1] for the RNR. The labels at each datapoint indicate the value of p used. The proof of Theorem 1 is in an accompanying technical report [JZB07]. The significance of Theorem 1 is that, among all strategies that are at most ϵ suboptimal, the RNR strategies are among the best responses. Thus, if we want a strategy that is at most ϵ suboptimal, we can vary p to produce a strategy that is the best response among all such ϵ-safe strategies. Unlike safe best responses, a RNR can be computed by just solving a modification of the original abstract game. For example, if using a sequence form representation of linear programming then one just needs to add lower bound constraints for the restricted player’s realization plan probabilities. In our experiments we use a recently developed solution technique based on regret minimization [ZJBP08] with a modified game that starts with an unobserved chance node deciding whether the restricted player is forced to use strategy σfix on the current hand. The RNRs used in our experiments were computed with less than a day of computation on a 2.4Ghz AMD Opteron. Choosing p. In order to compute a RNR we have to choose a value of p. By varying the value p ∈[0, 1], we can produce poker strategies that are closer to a Nash equilibrium (when p is near 0) or are closer to the best response (when p is near 1). When producing an RNR to a particular opponent, it is useful to consider the tradeoff between the utility of the response against that opponent and the exploitability of the response itself. We explore this tradeoff in Figure 1. In 1a we plot the results of using RNR with various values of p against the model of PsOpti4. The x-axis shows the exploitability of the response, ϵ. The y-axis shows the exploitation of the model by the response in the abstract game. Note that the actual exploitation and exploitability in the full game may be different, as we explore later. Figure 1b shows this tradeoff against A80. Notice that by selecting values of p, we can control the tradeoff between ϵ and the response’s exploitation of the strategy. More importantly, the curves are highly concave meaning that dramatic reductions in exploitability can be achieved with only a small sacrifice in the ability to exploit the model. Evaluation. We used RNR to compute a counter-strategy to the same seven opponents used in the FBR experiments, with the p value used for each opponent selected such that the resulting ϵ is close to 100 mb/h. The RNR strategies were played against these seven opponents and the equilibrium CFR5 in the full game of Texas Hold’em. The results of this tournament are displayed as a crosstable in Table 2. The first thing to notice is that RNR is capable of exploiting the opponent for which it was designed as a counter-strategy, while still performing well against the other opponents. In other words, not only is the diagonal positive and large, most of the crosstable is positive. For the highly exploitable opponents, such as A60 and A80, the degree of exploitation is much reduced from FBR, which is a consequence of choosing p such that ϵ is at most 100 mb/h. Notice, though, that it does exploit these opponents significantly more than the approximate Nash strategy (CFR5). 6 Opponents PsOpti4 PsOpti6 A60 A80 S1239 S1399 S2298 CFR5 Average RNR-PsOpti4 85 112 39 9 63 61 -1 -23 43 RNR-PsOpti6 26 234 72 34 59 59 1 -28 57 RNR-A60 -17 63 582 -22 37 39 -9 -45 78 RNR-A80 -7 66 22 293 11 12 0 -29 46 RNR-S1239 38 130 68 31 111 106 9 -20 59 RNR-S1399 31 136 66 29 105 112 6 -24 58 RNR-S2298 21 137 72 30 77 76 31 -11 54 CFR5 36 123 93 41 70 68 17 0 56 Max 85 234 582 293 111 112 31 0 Table 2: Results of restriced Nash response (RNR) against a variety of opponent programs in full Texas Hold’em, with winnings in mb/h for the row player. See the caption of Table 1 for match details. -100 0 100 200 300 400 500 600 700 800 Holdout Average Monash BluffBot Training Average S2298 S1399 Attack80 Opti4 Performance (mb/h) FBR Experts RNR Experts 5555hs2 Figure 2: Performance of FBR-experts, RNR-experts, and a near Nash equilibrium strategy (CFR5) against “training” opponents and “hold out” opponents in 50 duplicate matches of 1000 hands. Revisiting the family of S-bots, we notice that the known similarity of S1239 and S1399 is more apparent with RNR. The performance of RNR with the correct model against these two players is close to that of FBR, while the performance with the similar model is only a 6mb/h drop. Essentially, RNR is forced to exploit only the weaknesses that are general and is robust to small changes. Overall, RNR offers a similar degree of exploitation to FBR, but with far more robustness. 6 Restricted Nash Experts We have shown that RNR can be used to find robust counter-strategies. In this section we investigate their use in an adaptive poker program. We generated four counter-strategies based on the opponents PsOpti4, A80, S1399, and S2298, and then used these as experts which UCB1 [ACBF02] (a regret minimizing algorithm) selected amongst. The FBR-experts algorithm used a FBR to each opponent, and the RNR-experts used RNR to each opponent. We then played these two expert mixtures in 1000 hand matches against both the four programs used to generate the counter strategies as well as two programs from the 2006 AAAI Computer Poker Competition, which have an unknown origin and were developed independently of the other programs. We call the first four programs “training opponents” and the other two programs “holdout opponents”, as they are similar to training error and holdout error in supervised learning. The results of these matches are shown in Figure 2. As expected, when the opponent matches one of the training models, FBR-experts and RNR-experts perform better, on average, than a near equilibrium strategy (see “Training Average” in Figure 2). However, if we look at the break down against individual opponents, we see that all of FBR’s performance comes from its ability to significantly exploit one single opponent. Against the other opponents, it actually performs worse than the nonadaptive near equilibrium strategy. RNR does not exploit A80 to the same degree as FBR, but also does not lose to any opponent. 7 The comparison with the holdout opponents, though, is more realistic and more telling. Since it is unlikely a player will have a model of the exact program its likely to face in a competition, it is important for its counter-strategies to exploit general weaknesses that might be encountered. Our holdout programs have no explicit relationship to the training programs, yet the RNR counterstrategies are still effective at exploiting these programs as demonstrated by the expert mixture being able to exploit these programs by more than the near equilibrium strategy. The FBR counterstrategies, on the other hand, performed poorly outside of the training programs, demonstrating once again that RNR counter-strategies are both more robust and more suitable as a basis for adapting behavior to other agents in the environment. 7 Conclusion We proposed a new technique for generating robust counter-strategies in multiagent scenarios. The restricted Nash responses balance exploiting suspected tendencies in other agents’ behavior, while bounding the worst-case performance when the tendency is not observed. The technique involves computing an approximate equilibrium to a modification of the original game, and therefore can make use of recently developed algorithms for solving very large extensive games. We demonstrated the technique in the domain of poker, showing it to generate more robust counter-strategies than traditional best response. We also showed that a simple mixture of experts algorithm based on restricted Nash response counter-strategies was far superior to using best response counter-strategies if the exact opponent was not used in training. Further, the restricted Nash experts algorithm outperformed a static non-adaptive near equilibrium at exploiting the previously unseen programs. References [ACBF02] P. Auer, N. Cesa-Bianchi, and P. Fischer. Finite time analysis of the multiarmed bandit problem. Machine Learning, 47:235–256, 2002. [BBD+03] D. Billings, N. Burch, A. Davidson, R. Holte, J. Schaeffer, T. Schauenberg, and D. Szafron. Approximating game-theoretic optimal strategies for full-scale poker. In International Joint Conference on Artificial Intelligence, pages 661–668, 2003. [CM96] David Carmel and Shaul Markovitch. Learning models of intelligent agents. In Proceedings of the Thirteenth National Conference on Artificial Intelligence, Menlo Park, CA, 1996. AAAI Press. [GHPS07] A. Gilpin, S. Hoda, J. Pena, and T. Sandholm. Gradient-based algorithms for finding nash equilibria in extensive form games. In Proceedings of the Eighteenth International Conference on Game Theory, 2007. [GS06] A. Gilpin and T. Sandholm. A competitive texas hold’em poker player via automated abstraction and real-time equilibrium computation. In National Conference on Artificial Intelligence, 2006. [JZB07] Michael Johanson, Martin Zinkevich, and Michael Bowling. Computing robust counter-strategies. Technical Report TR07-15, Department of Computing Science, University of Alberta, 2007. [MB04] Peter McCracken and Michael Bowling. Safe strategies for agent modelling in games. In AAAI Fall Symposium on Artificial Multi-agent Learning, October 2004. [Rob51] Julia Robinson. An iterative method of solving a game. Annals of Mathematics, 54:296–301, 1951. [RV02] Patrick Riley and Manuela Veloso. Planning for distributed execution through use of probabilistic opponent models. In Proceedings of the Sixth International Conference on AI Planning and Scheduling, pages 77–82, April 2002. [Sch06] T.C. Schauenberg. Opponent modelling and search in poker. Master’s thesis, University of Alberta, 2006. [ZBB07] M. Zinkevich, M. Bowling, and N. Burch. A new algorithm for generating strong strategies in massive zero-sum games. In Proceedings of the Twenty-Seventh Conference on Artificial Intelligence (AAAI), 2007. To Appear. [ZJBP08] M. Zinkevich, M. Johanson, M. Bowling, and C. Piccione. Regret minimization in games with incomplete information. In Neural Information Processing Systems 21, 2008. [ZL06] M. Zinkevich and M. Littman. The AAAI computer poker competition. Journal of the International Computer Games Association, 29, 2006. News item. 8
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Random Sampling of States in Dynamic Programming Christopher G. Atkeson and Benjamin Stephens Robotics Institute, Carnegie Mellon University cga@cmu.edu, bstephens@cmu.edu www.cs.cmu.edu/∼cga, www.cs.cmu.edu/∼bstephe1 Abstract We combine three threads of research on approximate dynamic programming: sparse random sampling of states, value function and policy approximation using local models, and using local trajectory optimizers to globally optimize a policy and associated value function. Our focus is on finding steady state policies for deterministic time invariant discrete time control problems with continuous states and actions often found in robotics. In this paper we show that we can now solve problems we couldn’t solve previously. 1 Introduction Optimal control provides a potentially useful methodology to design nonlinear control laws (policies) u = u(x) which give the appropriate action u for any state x. Dynamic programming provides a way to find globally optimal control laws, given a one step cost (a.k.a. “reward” or “loss”) function and the dynamics of the problem to be optimized. We focus on control problems with continuous states and actions, deterministic time invariant discrete time dynamics xk+1 = f(xk,uk), and a time invariant one step cost function L(x,u). Policies for such time invariant problems will also be time invariant. We assume we know the dynamics and one step cost function. Future work will address simultaneously learning a dynamic model, finding a robust policy, and performing state estimation with an erroneous partially learned model. One approach to dynamic programming is to approximate the value function V(x) (the optimal total future cost from each state V(x) = ∑∞ k=0 L(xk,uk)), and to repeatedly solve the Bellman equation V(x) = minu(L(x,u) +V(f(x,u))) at sampled states x until the value function estimates have converged to globally optimal values. We explore approximating the value function and policy using many local models. An example problem: We use one link pendulum swingup as an example problem in this introduction to provide the reader with a visualizable example of a value function and policy. In one link pendulum swingup a motor at the base of the pendulum swings a rigid arm from the downward stable equilibrium to the upright unstable equilibrium and balances the arm there (Figure 1). What makes this challenging is that the one step cost function penalizes the amount of torque used and the deviation of the current position from the goal. The controller must try to minimize the total cost of the trajectory. The one step cost function for this example is a weighted sum of the squared position errors (θ: difference between current angles and the goal angles) and the squared torques τ: L(x,u) = 0.1θ2T + τ2T where 0.1 weights the position error relative to the torque penalty, and T is the time step of the simulation (0.01s). There are no costs associated with the joint velocity. Figure 2 shows the value function and policy generated by dynamic programming. One important thread of research on approximate dynamic programming is developing representations that adapt to the problem being solved and extend the range of problems that can be solved with a reasonable amount of memory and time. Random sampling of states has been proposed by a number of researchers [1, 2, 3, 4, 5, 6, 7]. In our case we add new randomly selected states as we 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Figure 1: Configurations from the simulated one link pendulum optimal trajectory every half a second and at the end of the trajectory. solve the problem, allowing the “grid” that results to reflect the local complexity of the value function as we generate it. Figure 2:right shows such a randomly generated set of states superimposed on a contour plot of the value function for one link swingup. Another important thread in our work on applied dynamic programming is developing ways for grids or random samples to be as sparse as possible. One technique that we apply here is to represent full trajectories from each sampled state to the goal, and to refine each trajectory using local trajectory optimization [8]. Figure 2:right shows a set of optimized trajectories from the sampled states to the goal. One key aspect of the local trajectory optimizer we use is that it provides a local quadratic model of the value function and a local linear model of the policy at the sampled state. These local models help our function approximators handle sparsely sampled states. To obtain globally optimal solutions, we incorporate exchange of information between non-neighboring sampled states. On what problems will the proposed approach work? We believe our approach can discover underlying simplicity in many typical problems. An example of a problem that appears complex but is actually simple is a problem with linear dynamics and a quadratic one step cost function. Dynamic programming can be done for linear quadratic regulator (LQR) problems even with hundreds of dimensions and it is not necessary to build a grid of states [9]. The cost of representing the value function is quadratic in the dimensionality of the state. The cost of performing a “sweep” or update of the value function is at most cubic in the state dimensionality. Continuous states and actions are easy to handle. Perhaps many problems, such as the examples in this paper, have simplifying characteristics similar to LQR problems. For example, problems that are only “slightly” nonlinear and have a locally quadratic cost function may be solvable with quite sparse representations. One goal of our work is to develop methods that do not immediately build a hugely expensive representation if it is not necessary, and attempt to harness simple and inexpensive parallel local planning to solve complex planning problems. Another goal of our work is to develop methods that can take advantage of situations where only a small amount of global interaction is necessary to enable local planners capable of solving local problems to find globally optimal solutions. 2 Related Work Random state selection: Random grids and random sampling are well known in numerical integration, finite element methods, and partial differential equations. Rust applied random sampling of states to dynamic programming [1, 10]. He showed that random sampling of states can avoid the curse of dimensionality for stochastic dynamic programming problems with a finite set of discrete actions. This theoretical result focused on the cost of computing the expectation term in the stochastic version of the Bellman equation. [11] claim the assumptions used in [1] are unrealistically restrictive, and [12] point out that the complexity of Rust’s approach is proportional to the Lipschitz constant of the problem data, which often increases exponentially with increasing dimensions. The practicality and usefulness of random sampling of states in deterministic dynamic programming with continuous actions (the focus of our paper) remains an open question. We note that deterministic problems are usually more difficult to solve since the random element in the stochastic dynamics smooths the dynamics and makes them easier to sample. Alternatives to random sampling of states are irregular or adaptive grids [13], but in our experience they still require too many representational resources as the problem dimensionality increases. In reinforcement learning random sampling of states is sometimes used to provide training data for function approximation of the value function. Reinforcement learning also uses random exploration for several purposes. In model-free approaches exploration is used to find actions and states that lead to better outcomes. This process is somewhat analogous to the random state sampling described in this paper for model-based approaches. In model-based approaches, exploration is also used to improve the model of the task. In our paper it is assumed a model of the task is available, so this type of exploration is not necessary. 2 −6 −5 −4 −3 −2 −1 0 1 2 3 −20 −15 −10 −5 0 5 10 15 20 0 10 20 velocity (r/s) Value function for one link example position (r) value −6 −5 −4 −3 −2 −1 0 1 2 3 −20 −15 −10 −5 0 5 10 15 20 −10 0 10 velocity (r/s) Policy for one link example position (r) torque (Nm) position (r) velocity (r/s) random initial states and trajectories for one link example −6 −5 −4 −3 −2 −1 0 1 2 3 −10 −8 −6 −4 −2 0 2 4 6 8 10 Figure 2: Left and Middle: The value function and policy for a one link pendulum swingup. The optimal trajectory is shown as a yellow line in the value function plot, and as a black line with a yellow border in the policy plot. The value function is cut off above 20 so we can see the details of the part of the value function that determines the optimal trajectory. The goal is at the state (0,0). Right: Random states (dots) and trajectories (black lines) used to plan one link swingup, superimposed on a contour map of the value function. In the field of Partially Observable Markov Decision Processes (POMDPs) there has been some work on randomly sampling belief states, and also using local models of the value function and its first derivative at each randomly sampled belief state (for example [2, 3, 4, 5, 6, 7]). Thrun explored random sampling of belief states where the underlying states and actions were continuous [7]. He used a nearest neighbor scheme to perform value function interpolation, and a coverage test to decide whether to accept a new random state (is a new random state far enough from existing states?) rather than a surprise test (is the value of the new random state predicted incorrectly?). In robot planning for obstacle avoidance random sampling of states is now quite popular [14]. Probabilistic Road Map (PRM) methods build a graph of plans between randomly selected states. Rapidly Exploring Random Trees (RRTs) grow paths or trajectories towards randomly selected states. In general it is difficult to modify PRM and RRT approaches to find optimal paths, and the resulting algorithms based on RRTs are very similar to A* search. 3 Combining Random State Sampling With Local Optimization The process of using the Bellman equation to update a representation of the value function by minimizing over all actions at a state is referred to as value iteration. Standard value iteration represents the value function and associated policy using multidimensional tables, with each entry in the table corresponding to a particular state. In our approach we randomly select states, and associate with each state a local quadratic model of the value function and a local linear model of the policy. Our approach generalizes value iteration, and has the following components: 1. There is a “global” function approximator for both the value function and the policy. In our current implementation the value function and policy are represented through a combination of sampled and parametric representations, building global approximations by combining local models. 2. It is possible to estimate the value of a state in two ways. The first is to use the approximated value function. The second is our analog of using the Bellman equation: use the cost of a trajectory starting from the state under consideration and following the current global policy. The trajectory is optimized using local trajectory optimization. 3. As in a Bellman update, there is a way to globally optimize the value of a state by considering many possible “actions”. In our approach we consider many local policies associated with different stored states. Taking advantage of goal states: For problems with goal states there are several ways to speed up convergence. In cases where LQR techniques apply [9], we use the policy obtained by solving the corresponding LQR control problem at the goal as the default policy everywhere, to which the policy computed by dynamic programming is added. [15] plots an example of a default policy and the policy generated by dynamic programming for comparison. We limit the outputs of this default policy. In setting up the goal LQR controller, a radius is established and tested within which the goal LQR controller always works and achieves close to the predicted optimal cost. This has the effect of making of enlarging the goal. If the dynamic programming process can get within the LQR radius of the goal, it can use only the default policy to go the rest of the way. If it is not possible to create a goal LQR controller due to a hard nonlinearity, or if there is no goal state, it does not have to be done as the goal controller merely accelerates the solution process. The proposed technique can be generalized in a straightforward way to use any default goal policy. In this paper the swingup 3 problems use an LQR default policy, which was limited for each action dimension to ±5Nm. For the balance problem we did not use a default policy. We note that for the swingup problems shown here the default LQR policy is capable of balancing the inverted pendulum at the goal, but is not capable of swinging up the pendulum to the goal. We also initially only generate the value function and policy in the region near the goal. This solved region is gradually increased in size by increasing a value function threshold. Examples of regions bounded by a constant value are shown by the value function contours in Figure 2. [16] describes how to handle periodic tasks which have no goal states, and also discontinuities in the dynamics. Local models of the value function and policy: We need to represent value functions as sparsely as possible. We propose a hybrid tabular and parametric approach: parametric local models of the value function and policy are represented at sampled locations. This representation is similar to using many Taylor series approximations of a function at different points. At each sampled state xp the local quadratic model for the value function is: V p(x) ≈V p 0 +V p x ˆx+ 1 2 ˆxTV p xxˆx (1) where ˆx = x−xp is the vector from the stored state xp, V p 0 is the constant term of the local model, V p x is the first derivative of the local model (and the value function) at xp, and V p xx is the second derivative of the local model (and the value function) at xp. The local linear model for the policy is: up(x) = up 0 −Kpˆx (2) where up 0 is the constant term of the local policy, and Kp is the first derivative of the local policy and also the gain matrix for a local linear controller. Creating the local model: These local models of the value function can be created using Differential Dynamic Programming (DDP) [17, 18, 8, 16]. This local trajectory optimization process is similar to linear quadratic regulator design in that a local model of the value function is produced. In DDP, value function and policy models are produced at each point along a trajectory. Suppose at a point (xi,ui) we have 1) a local second order Taylor series approximation of the optimal value function: V i(x) ≈V i 0 +V i x ˆx + 1 2 ˆxTV i xxˆx where ˆx = x −xi. 2) a local second order Taylor series approximation of the robot dynamics, which can be learned using local models of the dynamics (fi x and fi u correspond to A and B of the linear plant model used in linear quadratic regulator (LQR) design): xk+1 = fi(x,u) ≈fi 0 +fi xˆx+fi u ˆu+ 1 2 ˆxTfi xxˆx+ ˆxTfi xu ˆu+ 1 2 ˆuTfi uu ˆu where ˆu = u−ui, and 3) a local second order Taylor series approximation of the one step cost, which is often known analytically for human specified criteria (Lxx and Luu correspond to Q and R of LQR design): Li(x,u) ≈Li 0 +Li xˆx+Li u ˆu+ 1 2 ˆxTLi xxˆx+ ˆxTLi xu ˆu+ 1 2 ˆuTLi uu ˆu Given a trajectory, one can integrate the value function and its first and second spatial derivatives backwards in time to compute an improved value function and policy. We utilize the “Q function” notation from reinforcement learning: Q(x,u) = L(x,u) +V(f(x,u)). The backward sweep takes the following form (in discrete time): Qi x = Li x +V i xfi x; Qi u = Li u +V i xfi u (3) Qi xx = Li xx +V i xfi xx +(fi x)TV i xxfi x; Qi ux = Li ux +V i xfi ux +(fi u)TV i xxfi x; Qi uu = Li uu +V i xfi uu +(fi u)TV i xxfi u (4) ∆ui = (Qi uu)−1Qi u; Ki = (Qi uu)−1Qi ux (5) V i−1 x = Qi x −Qi uKi; V i−1 xx = Qi xx −Qi xuKi (6) where subscripts indicate derivatives and superscripts indicate the trajectory index. After the backward sweep, forward integration can be used to update the trajectory itself: ui new = ui −∆ui − Ki(xi new −xi). We note that the cost of this approach grows at most cubically rather than exponentially with respect to the dimensionality of the state. In problems that have a goal state, we can generate a trajectory from each stored state all the way to the goal. The cost of this trajectory is an upper bound on the true value of the state, and is used to bound the estimated value for the old state. Utilizing the local models: For the purpose of explaining our algorithm, let’s assume we already have a set of sampled states, each of which has a local model of the value function and the policy. 4 How should we use these multiple local models? The simplest approach is to just use the predictions of the nearest sampled state, which is what we currently do. We use a kd-tree to efficiently find nearest neighbors, but there are many other approaches that will find nearby stored states efficiently. In the future we will investigate using other methods to combine local model predictions from nearby stored states: distance weighted averaging (kernel regression), linear locally weighted regression, and quadratic locally weighted regression for value functions. Creating new random states: For tasks with a goal state, we initialize the set of stored states by storing the goal state itself. We have explored a number of distributions to select additional states from: uniform within bounds on the states; Gaussian with the mean at the goal; sampling near existing states; and sampling from an underlying low resolution regular grid. The uniform approach is a useful default approach, which we use in the swingup examples, the Gaussian approach provides a simple way to tune the distribution, sampling near existing states provides a way to efficiently sample while growing the solved region in high dimensions, and sampling from an underlying low resolution grid seems to perform well when only a small number of stored states are used (similar to using low dispersion sequences [1, 14]). A key point of our approach is that we do not generate the random states in advance but instead select them as the algorithm progresses. This allows us to apply an acceptance criteria to candidate states, which we describe in the next paragraph. We have also explored changing the distribution we generate candidate states from as the algorithm progresses, for example using a mixture of Gaussians with the Gaussians centered on existing stored states. Another reasonable hybrid approach would be to initially sample from a grid, and then bias more general sampling to regions of higher value function approximation error. Acceptance criteria for candidate states: We have several criteria to accept or reject states to be permanently stored. In the future we will explore “forgetting” or removing stored states, but at this point we apply all memory control techniques at the storage event. To focus the search and limit the volume considered, a steadily increasing value limit is maintained (Vlimit), which is increased slightly after each use. The approximated value function is used to predict the value of the candidate state. If the prediction is above Vlimit, the candidate state is rejected. Otherwise, a trajectory is created from the candidate state using the current approximated policy, and then locally optimized. If the value of that trajectory is above Vlimit, the candidate state is rejected. If the value of the trajectory is within 10% of the predicted value, the candidate state is rejected. Only “surprises” are stored. For problems with a goal state, if the trajectory does not reach the goal the candidate state is rejected. Other criteria such as an A* like criteria (cost-to-go(x) + cost-from-start(x) > threshold) can be used to reject candidate states. All of the thresholds mentioned can be changed as the algorithm progresses. For example, Vlimit is gradually increased during the solution process, to increase the volume considered by the algorithm. We currently use a 10% “surprise” threshold. In future work we will explore starting with a larger threshold and decreasing this threshold with time, to further reduce the number of samples accepted and stored while improving convergence. It is possible to take the distance to the nearest sampled state into account in the acceptance criteria for new samples. The common approach of accepting states beyond a distance threshold enforces a minimum resolution, and leads to potentially severe curse of dimensionality effects. Rejecting states that are too close to existing states will increase the error in representing the value function, but may be a way for preventing too many samples near complex regions of the value functions that have little practical effect. For example, we often do not need much accuracy in representing the value function near policy discontinuities where the value function has discontinuities in its spatial derivative and “creases”. In these areas the trajectories typically move away from the discontinuities, and the details of the value function have little effect. In the current implementation, after a candidate state is accepted, the state in the database whose local model was used to make the prediction is re-optimized including information from the newly added point, since the prediction was wrong and the new point’s policy may lead to a better value for that state. Creating a trajectory from a state: We create a trajectory from a candidate state or refine a trajectory from a stored state in the same way. The first step is to use the current approximated policy until the goal or a time limit is reached. In the current implementation this involves finding the stored state nearest to the current state in the trajectory and using its locally linear policy to compute the action on each time step. The second step is to locally optimize the trajectory. We use Differential Dynamic Programming (DDP) in the current implementation [17, 18, 8, 16]. In the current implementation we do not save the trajectory but only the local models from its start. If the cost of the 5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Figure 3: Configurations from the simulated two link pendulum optimal swing up trajectory every fifth of a second and the end of the trajectory. trajectory is more than the currently stored value for the state, we reject the new value, as the values all come from actual trajectories and are upper bounds for the true value. We always keep the lowest upper bound. Combining parallel greedy local optimizers to perform global optimization: As currently described, the algorithm finds a locally optimal policy, but not necessarily a globally optimal policy. For example, the stored states could be divided into two sets of nearest neighbors. One set could have a suboptimal policy, but have no interaction with the other set of states that had a globally optimal policy since no nearest neighbor relations joined the two sets. We expect the locally optimal policies to be fairly good because we 1) gradually increase the solved volume and 2) use local optimizers. Given local optimization of actions, gradually increasing the solved volume will result in a globally optimal policy if the boundary of this volume never touches a non adjacent section of itself. Figures 2 and 2 show the creases in the value function (discontinuities in the spatial derivative) and corresponding discontinuities in the policy that typically result when the constant cost contour touches a non adjacent section of itself as Vlimit is increased. In theory, the approach we have described will produce a globally optimal policy if it has infinite resolution and all the stored states form a densely connected set in terms of nearest neighbor relations [8]. By enforcing consistency of the local value function models across all nearest neighbor pairs, we can create a globally consistent value function estimate. Consistency means that any state’s local model correctly predicts values of nearby states. If the value function estimate is consistent everywhere, the Bellman equation is solved and we have a globally optimal policy. We can enforce consistency of nearest neighbor value functions by 1) using the policy of one state of a pair to reoptimize the trajectory of the other state of the pair and vice versa, and 2) adding more stored states in between nearest neighbors that continue to disagree [8]. This approach is similar to using the method of characteristics to solve partial differential equations and finding value functions for games. In practice, we cannot achieve infinite resolution. To increase the likelihood of finding a globally optimal policy with a limited resolution of stored states, we need an analog to exploration and to global minimization with respect to actions found in the Bellman equation. We approximate this process by periodically reoptimizing each stored state using the policies of other stored states. As more and more states are stored, and many alternative stored states are considered in optimizing any given stored state, a wide range of actions are considered for each state. We run a reoptimization phase of the algorithm after every N (typically 100) states have been stored. There are several ways to design this reoptimization phase. Each state could use the policy of a nearest neighbor, or a randomly chosen neighbor with the distribution being distance dependent, or just choosing another state randomly with no consideration of distance (what we currently do). [8] describes how to follow a policy of another stored state if its trajectory is stored, or can be recomputed as needed. In this work we explored a different approach that does not require each stored state to save its trajectory or recompute it. To “follow” the policy of another state, we follow the locally linear policy for that state until the trajectory begins to go away from the state. At that point we switch to following the globally approximated policy. Since we apply this reoptimization process periodically with different randomly selected policies, over time we explore using a wide range of actions from each state. 6 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Figure 4: Configurations from the simulated three link pendulum optimal trajectory every tenth of a second and at the end of the trajectory. 4 Results In addition to the one link swingup example presented in the introduction, we present results on two link swingup (4 dimensional state) and three link swingup (6 dimensional state). A companion paper using these techniques to explore how multiple balance strategies can be generated from one optimization criterion is [19]. Further results, including some for a four link (8 dimensional state) standing robot are presented. One link pendulum swingup: For the one link swingup case, the random state approach found a globally optimal trajectory (the same trajectory found by our grid based approaches [15]) after adding only 63 random states. Figure 2:right shows the distribution of states and their trajectories superimposed on a contour map of the value function for one link swingup. Two link pendulum swingup: For the two link swingup case, the random state approach finds what we believe is a globally optimal trajectory (the same trajectory found by our grid based approaches [15]) after storing an average of 12000 random states (Figure 3). In this case the state has four dimensions (a position and velocity for each joint) and a two dimensional action (a torque at each joint). The one step cost function was a weighted sum of the squared position errors and the squared torques: L(x,u) = 0.1(θ2 1 + θ2 2)T + (τ2 1 + τ2 2)T. 0.1 weights the position errors relative to the torque penalty, T is the time step of the simulation (0.01s), and there were no costs associated with joint velocities. The approximately 12000 sampled states should be compared to the millions of states used in grid-based approaches. A 60x60x60x60 grid with almost 13 million states failed to find a trajectory as good as this one, while a 100x100x100x100 grid with 100 million states did find the same trajectory. In 13 runs with different random number generator seeds, the mean number of states stored at convergence is 11430. All but two of the runs converged after storing less than 13000 states, and all runs converged after storing 27000 states. Three link pendulum swingup: For the three link swingup case, the random state approach found a good trajectory after storing less than 22000 random states (Figure 4). We have not yet solved this problem a sufficient number of times to be convinced this is a global optimum, and we do not have a solution based on a regular grid available for comparison. We were not able to solve this problem using regular grid-based approaches due to limited state resolution: 22x22x22x22x38x44 = 391,676,032 states filled our largest memory. As in the previous examples, the one step cost function was a weighted sum of the squared position errors and the squared torques: L(x,u) = 0.1(θ2 1 +θ2 2 +θ2 3)T+(τ2 1 +τ2 2 +τ2 3)T. 5 Conclusion We have combined random sampling of states and local trajectory optimization to create a promising approach to practical dynamic programming for robot control problems. We are able to solve problems we couldn’t solve before due to memory limitations. Future work will optimize aspects and variants of this approach. 7 Acknowledgments This material is based upon work supported in part by the DARPA Learning Locomotion Program and the National Science Foundation under grants CNS-0224419, DGE-0333420, ECS-0325383, and EEC-0540865. References [1] J. Rust. Using randomization to break the curse of dimensionality. Econometrica, 65(3):487– 516, 1997. [2] M. Hauskrecht. Incremental methods for computing bounds in partially observable Markov decision processes. In Proceedings of the 14th National Conference on Artificial Intelligence (AAAI-97), pages 734–739, Providence, Rhode Island, 1997. AAAI Press / MIT Press. [3] N.L. Zhang and W. Zhang. Speeding up the convergence of value iteration in partially observable Markov decision processes. JAIR, 14:29–51, 2001. [4] J. Pineau, G. Gordon, and S. Thrun. Point-based value iteration: An anytime algorithm for POMDPs. In International Joint Conference on Artificial Intelligence (IJCAI), 2003. [5] T. Smith and R. Simmons. Heuristic search value iteration for POMDPs. In Uncertainty in Artificial Intelligence, 2004. [6] M.T.J. Spaan and Nikos V. A point-based POMDP algorithm for robot planning. In Proceedings of the IEEE International Conference on Robotics and Automation, pages 2399–2404, New Orleans, Louisiana, April 2004. [7] S. Thrun. Monte Carlo POMDPs. In S.A. Solla, T.K. Leen, and K.-R. M¨uller, editors, Advances in Neural Information Processing 12, pages 1064–1070. MIT Press, 2000. [8] C. G. Atkeson. Using local trajectory optimizers to speed up global optimization in dynamic programming. In Jack D. Cowan, Gerald Tesauro, and Joshua Alspector, editors, Advances in Neural Information Processing Systems, volume 6, pages 663–670. Morgan Kaufmann Publishers, Inc., 1994. [9] F. L. Lewis and V. L. Syrmos. Optimal Control, 2nd Edition. Wiley-Interscience, 1995. [10] C. Szepesv´ari. Efficient approximate planning in continuous space Markovian decision problems. AI Communications, 13(3):163–176, 2001. [11] J. N. Tsitsiklis and Van B. Roy. Regression methods for pricing complex American-style options. IEEE-NN, 12:694–703, July 2001. [12] V. D. Blondel and J. N. Tsitsiklis. A survey of computational complexity results in systems and control, 2000. [13] R. Munos and A. W. Moore. Variable resolution discretization in optimal control. Machine Learning Journal, 49:291–323, 2002. [14] S. M. LaValle. Planning Algorithms. Cambridge University Press, 2006. [15] C. G. Atkeson. Randomly sampling actions in dynamic programming. In 2007 IEEE International Symposium on Approximate Dynamic Programming and Reinforcement Learning (ADPRL), 2007. [16] C. G. Atkeson and J. Morimoto. Nonparametric representation of a policies and value functions: A trajectory based approach. In Advances in Neural Information Processing Systems 15. MIT Press, 2003. [17] P. Dyer and S. R. McReynolds. The Computation and Theory of Optimal Control. Academic Press, New York, NY, 1970. [18] D. H. Jacobson and D. Q. Mayne. Differential Dynamic Programming. Elsevier, New York, NY, 1970. [19] C. G. Atkeson and B. Stephens. Multiple balance strategies from one optimization criterion. In Humanoids, 2007. 8
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Predicting human gaze using low-level saliency combined with face detection Moran Cerf Computation and Neural Systems California Institute of Technology Pasadena, CA 91125 moran@klab.caltech.edu Jonathan Harel Electrical Engineering California Institute of Technology Pasadena, CA 91125 harel@klab.caltech.edu Wolfgang Einh¨auser Institute of Computational Science Swiss Federal Institute of Technology (ETH) Zurich, Switzerland wolfgang.einhaeuser@inf.ethz.ch Christof Koch Computation and Neural Systems California Institute of Technology Pasadena, CA 91125 koch@klab.caltech.edu Abstract Under natural viewing conditions, human observers shift their gaze to allocate processing resources to subsets of the visual input. Many computational models try to predict such voluntary eye and attentional shifts. Although the important role of high level stimulus properties (e.g., semantic information) in search stands undisputed, most models are based on low-level image properties. We here demonstrate that a combined model of face detection and low-level saliency significantly outperforms a low-level model in predicting locations humans fixate on, based on eye-movement recordings of humans observing photographs of natural scenes, most of which contained at least one person. Observers, even when not instructed to look for anything particular, fixate on a face with a probability of over 80% within their first two fixations; furthermore, they exhibit more similar scanpaths when faces are present. Remarkably, our model’s predictive performance in images that do not contain faces is not impaired, and is even improved in some cases by spurious face detector responses. 1 Introduction Although understanding attention is interesting purely from a scientific perspective, there are numerous applications in engineering, marketing and even art that can benefit from the understanding of both attention per se, and the allocation of resources for attention and eye movements. One accessible correlate of human attention is the fixation pattern in scanpaths [1], which has long been of interest to the vision community [2]. Commonalities between different individuals’ fixation patterns allow computational models to predict where people look, and in which order [3]. There are several models for predicting observers’ fixations [4], some of which are inspired by putative neural mechanisms. A frequently referenced model for fixation prediction is the Itti et al. saliency map model (SM) [5]. This “bottom-up” approach is based on contrasts of intrinsic images features such as color, orientation, intensity, flicker, motion and so on, without any explicit information about higher order scene structure, semantics, context or task-related (“top-down”) factors, which may be crucial for attentional allocation [6]. Such a bottom-up saliency model works well when higher order semantics are reflected in low-level features (as is often the case for isolated objects, and even for reasonably cluttered scenes), but tends to fail if other factors dominate: e.g., in search tasks [7, 8], strong contextual effects [9], or in free-viewing of images without clearly isolated objects, such as 1 forest scenes or foliage [10]. Here, we test how images containing faces - ecologically highly relevant objects - influence variability of scanpaths across subjects. In a second step, we improve the standard saliency model by adding a “face channel” based on an established face detector algorithm. Although there is an ongoing debate regarding the exact mechanisms which underlie face detection, there is no argument that a normal subject (in contrast to autistic patients) will not interpret a face purely as a reddish blob with four lines, but as a much more significant entity ([11, 12]. In fact, there is mounting evidence of infants’ preference for face-like patterns before they can even consciously perceive the category of faces [13], which is crucial for emotion and social processing ([13, 14, 15, 16]). Face detection is a well investigated area of machine vision. There are numerous computer-vision models for face detection with good results ([17, 18, 19, 20]). One widely used model for face recognition is the Viola & Jones [21] feature-based template matching algorithm (VJ). There have been previous attempts to incorporate face detection into a saliency model. However, they have either relied on biasing a color channel toward skin hue [22] - and thus being ineffective in many cases nor being face-selective per se - or they have suffered from lack of generality [23]. We here propose a system which combines the bottom-up saliency map model of Itti et al. [5] with the Viola & Jones face detector. The contributions of this study are: (1) Experimental data showing that subjects exhibit significantly less variable scanpaths when viewing natural images containing faces, marked by a strong tendency to fixate on faces early. (2) A novel saliency model which combines a face detector with intensity, color, and orientation information. (3) Quantitative results on two versions of this saliency model, including one extended from a recent graph-based approach, which show that, compared to previous approaches, it better predicts subjects’ fixations on images with faces, and predicts as well otherwise. 2 Methods 2.1 Experimental procedures Seven subjects viewed a set of 250 images (1024 × 768 pixels) in a three phase experiment. 200 of the images included frontal faces of various people; 50 images contained no faces but were otherwise identical, allowing a comparison of viewing a particular scene with and without a face. In the first (“free-viewing”) phase of the experiment, 200 of these images (the same subset for each subject) were presented to subjects for 2 s, after which they were instructed to answer “How interesting was the image?” using a scale of 1-9 (9 being the most interesting). Subjects were not instructed to look at anything in particular; their only task was to rate the entire image. In the second (“search”) phase, subjects viewed another 200 image subset in the same setup, only this time they were initially presented with a probe image (either a face, or an object in the scene: banana, cell phone, toy car, etc.) for 600 ms after which one of the 200 images appeared for 2 s. They were then asked to indicate whether that image contained the probe. Half of the trials had the target probe present. In half of those the probe was a face. Early studies suggest that there should be a difference between free-viewing of a scene, and task-dependent viewing of it [2, 4, 6, 7, 24]. We used the second task to test if there are any differences in the fixation orders and viewing patterns between freeviewing and task-dependent viewing of images with faces. In the third phase, subjects performed a 100 images recognition memory task where they had to answer with y/n whether they had seen the image before. 50 of the images were taken from the experimental set and 50 were new. Subjects’ mean performance was 97.5% correct, verifying that they were indeed alert during the experiment. The images were introduced as “regular images that one can expect to find in an everyday personal photo album”. Scenes were indoors and outdoors still images (see examples in Fig. 1). Images included faces in various skin colors, age groups, and positions (no image had the face at the center as this was the starting fixation location in all trials). A few images had face-like objects (see balloon in Fig. 1, panel 3), animal faces, and objects that had irregular faces in them (masks, the Egyptian sphinx face, etc.). Faces also vary in size (percentage of the entire image). The average face was 5% ± 1% (mean ± s.d.) of the entire image - between 1◦to 5◦of the visual field; we also varied the number of faces in the image between 1-6, with a mean of 1.1 ± 0.48. Image order was randomized throughout, and subjects were na¨ıve to the purpose of the experiment. Subjects fixated on a cross in the center before each image onset. Eye-position data were acquired at 1000 Hz using an Eyelink 1000 (SR Research, Osgoode, Canada) eye-tracking device. The images were presented on a CRT 2 screen (120 Hz), using Matlab’s Psychophysics and eyelink toolbox extensions ([25, 26]). Stimulus luminance was linear in pixel values. The distance between the screen and the subject was 80 cm, giving a total visual angle for each image of 28◦× 21◦. Subjects used a chin-rest to stabilize their head. Data were acquired from the right eye alone. All subjects had uncorrected normal eyesight. Figure 1: Examples of stimuli during the “free-viewinng” phase. Notice that faces have neutral expressions. Upper 3 panels include scanpaths of one individual. The red triangle marks the first and the red square the last fixation, the yellow line the scanpath, and the red circles the subsequent fixations. Lower panels show scanpaths of all 7 subjects. The trend of visiting the faces first - typically within the 1st or 2nd fixation - is evident. All images are available at http://www.klab.caltech.edu/˜moran/db/faces/. 2.2 Combining face detection with various saliency algorithms We tried to predict the attentional allocation via fixation patterns of the subjects using various saliency maps. In particular, we computed four different saliency maps for each of the images in our data set: (1) a saliency map based on the model of [5] (SM), (2) a graph-based saliency map according to [27] (GBSM), (3) a map which combines SM with face-detection via VJ (SM+VJ), and (4) a saliency map combining the outputs of GBSM and VJ (GBSM+VJ). Each saliency map was represented as a positive valued heat map over the image plane. SM is based on computing feature maps, followed by center-surround operations which highlight local gradients, followed by a normalization step prior to combining the feature channels. We used the “Maxnorm” normalization scheme which is a spatial competition mechanism based on the squared ratio of global maximum over average local maximum. This promotes feature maps with one conspicuous location to the detriment of maps presenting numerous conspicuous locations. The graphbased saliency map model (GBSM) employs spectral techniques in lieu of center surround subtraction and “Maxnorm” normalization, using only local computations. GBSM has shown more robust correlation with human fixation data compared with standard SM [27]. For face detection, we used the Intel Open Source Computer Vision Library (“OpenCV”) [28] implementation of [21]. This implementation rapidly processes images while achieving high detection rates. An efficient classifier built using the Ada-Boost learning algorithm is used to select a small number of critical visual features from a large set of potential candidates. Combining classifiers in a cascade allows background regions of the image to be quickly discarded, so that more cycles process promising face-like regions using a template matching scheme. The detection is done by applying a classifier to a sliding search window of 24x24 pixels. The detectors are made of three joined black and white rectangles, either up-right or rotated by 45◦. The values at each point are calculated as a weighted sum of two components: the pixel sum over the black rectangles and the sum over the whole detector area. The classifiers are combined to make a boosted cascade with classifiers going from simple to more complex, each possibly rejecting the candidate window as “not a face” [28]. This implementation of the facedetect module was used with the standard default training set of the original model. We used it to form a “Faces conspicuity map”, or “Face channel” 3 by convolving delta functions at the (x,y) detected facial centers with 2D Gaussians having standard deviation equal to estimated facial radius. The values of this map were normalized to a fixed range. For both SM and GBSM, we computed the combined saliency map as the mean of the normalized color (C), orientation (O), and intensity (I) maps [5]: 1 3(N(I) + N(C) + N(O)) And for SM+VJ and GBSM+VJ, we incorporated the normalized face conspicuity map (F) into this mean (see Fig 2): 1 4(N(I) + N(C) + N(O) + N(F)) This is our combined face detector/saliency model. Although we could have explored the space of combinations which would optimize predictive performance, we chose to use this simplest possible combination, since it is the least complicated to analyze, and also provides us with first intuition for further studies. Face detection Color Intensity Orientation False False positive Saliency Map Saliency Map with face detection Figure 2: Modified saliency model. An image is processed through standard [5] color, orientation and intensity multi-scale channels, as well as through a trained template-matching face detection mechanism. Face coordinates and radius from the face detector are used to form a face conspicuity map (F), with peaks at facial centers. All four maps are normalized to the same dynamic range, and added with equal weights to a final saliency map (SM+VJ, or GBSM+VJ). This is compared to a saliency map which only uses the three bottom-up features maps (SM or GBSM). 3 Results 3.1 Psychophysical results To evaluate the results of the 7 subjects’ viewing of the images, we manually defined minimally sized rectangular regions-of-interest (ROIs) around each face in the entire image collection. We first assessed, in the “free-viewing” phase, how many of the first fixations went to a face, how many of the second, third fixations and so forth. In 972 out of the 1050 (7 subjects x 150 images with faces) trials (92.6%), the subject fixated on a face at least once. In 645/1050 (61.4%) trials, a 4 face was fixated on within the first fixation, and of the remaining 405 trials, a face was fixated on in the second fixation in 71.1% (288/405), i.e. after two fixations a face was fixated on in 88.9% (933/1050) of trials (Fig. 3). Given that the face ROIs were chosen very conservatively (i.e. fixations just next to a face do not count as fixations on the face), this shows that faces, if present, are typically fixated on within the first two fixations (327 ms ± 95 ms on average). Furthermore, in addition to finding early fixations on faces, we found that inter-subject scanpath consistency on images with faces was higher. For the free-viewing task, the mean minimum distance to another’s subject’s fixation (averaged over fixations and subjects) was 29.47 pixels on images with faces, and a greater 34.24 pixels on images without faces (different with p < 10−6). We found similar results using a variety of different metrics (ROC, Earth Mover’s Distance, Normalized Scanpath Saliency, etc.). To verify that the double spatial bias of photographer and observer ([29] for discussion of this issue) did not artificially result in high fractions of early fixations on faces, we compared our results to an unbiased baseline: for each subject, the fraction of fixations from all images which fell in the ROIs of one particular image. The null hypothesis that we would see the same fraction of first fixations on a face at random is rejected at p < 10−20 (t-test). To test for the hypothesis that face saliency is not due to top-down preference for faces in the absence of other interesting things, we examined the results of the “search” task, in which subjects were presented with a non-face target probe in 50% of the trials. Provided the short amount of time for the search (2 s), subjects should have attempted to tune their internal saliency weights to adjust color, intensity, and orientation optimally for the searched target [30]. Nevertheless, subjects still tended to fixate on the faces early. A face was fixated on within the first fixation in 24% of trials, within the first two fixations in 52% of trials, and within the three fixations in 77% of the trials. While this is weaker than in free-viewing, where 88.9% was achieved after just two fixations, the difference from what would be expected for random fixation selection (unbiased baseline as above) is still highly significant (p < 10−8). Overall, we found that in both experimental conditions (“free-viewing” and “search”), faces were powerful attractors of attention, accounting for a strong majority of early fixations when present. This trend allowed us to easily improve standard saliency models, as discussed below. Figure 3: Extent of fixation on face regions-of-interest (ROIs) during the “free-viewing” phase . Left: image with all fixations (7 subjects) superimposed. First fixation marked in blue, second in cyan, remaining fixations in red. Right: Bars depict percentage of trials, which reach a face the first time in the first, second, third, . . . fixation. The solid curve depicts the integral, i.e. the fraction of trials in which faces were fixated on at least once up to and including the nth fixation. 3.2 Assessing the saliency map models We ran VJ on each of the 200 images used in the free viewing task, and found at least one face detection on 176 of these images, 148 of which actually contained faces (only two images with faces were missed). For each of these 176 images, we computed four saliency maps (SM, GBSM, SM+VJ, GBSM+VJ) as discussed above, and quantified the compatibility of each with our scanpath recordings, in particular fixations, using the area under an ROC curve. The ROC curves were generated by sweeping over saliency value thresholds, and treating the fraction of non-fixated pixels 5 on a map above threshold as false alarms, and the fraction of fixated pixels above threshold as hits [29, 31]. According to this ROC fixation “prediction” metric, for the example image in Fig. 4, all models predict above chance (50%): SM performs worst, and GBSM+VJ best, since including the face detector substantially improves performance in both cases. Figure 4: Comparison of the area-under-the-curve (AUC) for an image (chosen arbitrarily. Subjects’ scanpaths shown on the left panels of figure 1). Top panel: image with the 49 fixations of the 7 subjects (red). First central fixations for each subject were excluded. From left to right, saliency map model of Itti et al. (SM), saliency map with the VJ face detection map (SM+VJ), the graph-based saliency map (GBSM), and the graph-based saliency map with face detection channel (GBSM+VJ). Red dots correspond to fixations. Lower panels depict ROC curves corresponding to each map. Here, GBSM+VJ predicts fixations best, as quantified by the highest AUC. Across all 176 images, this trend prevails (Fig. 5): first, all models perform better than chance, even over the 28 images without faces. The SM+VJ model performed better than the SM model for 154/176 images. The null hypothesis to get this result by chance can be rejected at p < 10−22 (using a coin-toss sign-test for which model does better, with uniform null-hypothesis, neglecting the size of effects). Similarly, the GBSM+VJ model performed better than the GBSM model for 142/176 images, a comparably vast majority (p < 10−15) (see Fig. 5, right). For the 148/176 images with faces, SM+VJ was better than SM alone for 144/148 images (p < 10−29), whereas VJ alone (equal to the face conspicuity map) was better than SM alone for 83/148 images, a fraction that fails to reach significance. Thus, although the face conspicuity map was surprisingly predictive on its own, fixation predictions were much better when it was combined with the full saliency model. For the 28 images without faces, SM (better than SM+VJ for 18) and SM+VJ (better than SM for 10) did not show a significant difference, nor did GBSM vs. GBSM+VJ (better on 15/28 compared to 13/28, respectively. However, in a recent follow-up study with more non-face images, we found preliminary results indicating that the mean ROC score of VJ-enhanced saliency maps is higher on such non-face images, although the median is slightly lower, i.e. performance is much improved when improved at all indicating that VJ false positives can sometimes enhance saliency maps. In summary, we found that adding a face detector channel improves fixation prediction in images with faces dramatically, while it does not impair prediction in images without faces, even though the face detector has false alarms in those cases. 4 Discussion First, we demonstrated that in natural scenes containing frontal shots of people, faces were fixated on within the first few fixations, whether subjects had to grade an image on interest value or search it for a specific possibly non-face target. This powerful trend motivated the introduction of a new saliency 6 0.5 0.9 0.7 0.6 0.8 AUC (SM) AUC (SM+VJ) 0.5 0.6 0.7 0.8 0.9 image with face image without face 60 0 60 154 22 *** 0 0.5 0.9 0.7 0.6 0.8 AUC (GBSM) AUC (GBSM+VJ) 0.5 0.6 0.7 0.8 0.9 1 1 60 0 *** 142 34 70 0 Figure 5: Performance of SM compared to SM+VJ and GBSM compared to GBSM+VJ. Scatterplots depict the area under ROC curves (AUC) for the 176 images in which VJ found a face. Each point represents a single image. Points above the diagonal indicate better prediction of the model including face detection compared to the models without face channel. Blue markers denote images with faces; red markers images without faces (i.e. false positives of the VJ face detector). Histograms of the SM and SM+VJ (GBSM and GBSM+VJ) are depicted to the top and left (binning: 0.05); colorcode as in scatterplots. model, which combined the “bottom-up” feature channels of color, orientation, and intensity, with a special face-detection channel, based on the Viola & Jones algorithm. The combination was linear in nature with uniform weight distribution for maximum simplicity. In attempting to predict the fixations of human subjects, we found that this additional face channel improved the performance of both a standard and a more recent graph-based saliency model (almost all blue points in Fig. 5 are above the diagonal) in images with faces. In the few images without faces, we found that the false positives represented in the face-detection channel did not significantly alter the performance of the saliency maps – although in a preliminary follow-up on a larger image pool we found that they boost mean performance. Together, these findings point towards a specialized “face channel” in our vision system, which is subject to current debate in the attention literature [11, 12, 32]. In conclusion, inspired by biological understanding of human attentional allocation to meaningful objects - faces - we presented a new model for computing an improved saliency map which is more consistent with gaze deployment in natural images containing faces than previously studied models, even though the face detector was trained on standard sets. This suggests that faces always attract attention and gaze, relatively independent of the task. They should therefore be considered as part of the bottom-up saliency pathway. References [1] G. 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Local Algorithms for Approximate Inference in Minor-Excluded Graphs Kyomin Jung Dept. of Mathematics, MIT kmjung@mit.edu Devavrat Shah Dept. of EECS, MIT devavrat@mit.edu Abstract We present a new local approximation algorithm for computing MAP and logpartition function for arbitrary exponential family distribution represented by a finite-valued pair-wise Markov random field (MRF), say G. Our algorithm is based on decomposing G into appropriately chosen small components; computing estimates locally in each of these components and then producing a good global solution. We prove that the algorithm can provide approximate solution within arbitrary accuracy when G excludes some finite sized graph as its minor and G has bounded degree: all Planar graphs with bounded degree are examples of such graphs. The running time of the algorithm is Θ(n) (n is the number of nodes in G), with constant dependent on accuracy, degree of graph and size of the graph that is excluded as a minor (constant for Planar graphs). Our algorithm for minor-excluded graphs uses the decomposition scheme of Klein, Plotkin and Rao (1993). In general, our algorithm works with any decomposition scheme and provides quantifiable approximation guarantee that depends on the decomposition scheme. 1 Introduction Markov Random Field (MRF) based exponential family of distribution allows for representing distributions in an intuitive parametric form. Therefore, it has been successful for modeling in many applications Specifically, consider an exponential family on n random variables X = (X1, . . . , Xn) represented by a pair-wise (undirected) MRF with graph structure G = (V, E), where vertices V = {1, . . . , n} and edge set E ⊂V × V . Each Xi takes value in a finite set Σ (e.g. Σ = {0, 1}). The joint distribution of X = (Xi): for x = (xi) ∈Σn, Pr[X = x] ∝ exp X i∈V φi(xi) + X (i,j)∈E ψij(xi, xj) . (1) Here, functions φi : Σ →R+ △= {x ∈R : x ≥0}, and ψij : Σ2 →R+ are assumed to be arbitrary non-negative (real-valued) functions.1 The two most important computational questions of interest are: (i) finding maximum a-posteriori (MAP) assignment x∗, where x∗= arg maxx∈Σn Pr[X = x]; and (ii) marginal distributions of variables, i.e. Pr[Xi = x]; for x ∈Σ, 1 ≤i ≤n. MAP is equivalent to a minimal energy assignment (or ground state) where energy, E(x), of state x ∈Σn is defined as E(x) = −H(x) + Constant, where H(x) = P i∈V φi(xi)+P (i,j)∈E ψij(xi, xj). Similarly, computing marginal is equivalent to computing logpartition function, defined as log Z = log P x∈Σn exp P i∈V φi(xi) + P (i,j)∈E ψij(xi, xj) . In this paper, we will find ε-approximation solutions of MAP and log-partition function: that is, ˆx and log ˆZ such that: (1 −ε)H(x∗) ≤H(ˆx) ≤H(x∗), (1 −ε) log Z ≤log ˆZ ≤(1 + ε) log Z. 1Here, we assume the positivity of φi’s and ψij’s for simplicity of analysis. 1 Previous Work. The question of finding MAP (or ground state) comes up in many important application areas such as coding theory, discrete optimization, image denoising.Similarly, log-partition function is used in counting combinatorial objects loss-probability computation in computer networks, etc. Both problems are NP-hard for exact and even (constant) approximate computation for arbitrary graph G. However, applications require solving this problem using very simple algorithms. A plausible approach is as follows. First, identify wide class of graphs that have simple algorithms for computing MAP and log-partition function. Then, try to build system (e.g. codes) so that such good graph structure emerges and use the simple algorithm or else use the algorithm as a heuristic. Such an approach has resulted in many interesting recent results starting the Belief Propagation (BP) algorithm designed for Tree graph [1].Since there a vast literature on this topic, we will recall only few results. Two important algorithms are the generalized belief propagation (BP) [2] and the tree-reweighted algorithm (TRW) [3,4].Key properties of interest for these iterative procedures are the correctness of fixed points and convergence. Many results characterizing properties of the fixed points are known starting from [2]. Various sufficient conditions for their convergence are known starting [5]. However, simultaneous convergence and correctness of such algorithms are established for only specific problems, e.g. [6]. Finally, we discuss two relevant results. The first result is about properties of TRW. The TRW algorithm provides provable upper bound on log-partition function for arbitrary graph [3]However, to the best of authors’ knowledge the error is not quantified. The TRW for MAP estimation has a strong connection to specific Linear Programming (LP) relaxation of the problem [4]. This was made precise in a sequence of work by Kolmogorov [7], Kolmogorov and Wainwright [8] for binary MRF. It is worth noting that LP relaxation can be poor even for simple problems. The second is an approximation algorithm proposed by Globerson and Jaakkola [9] to compute log-partition function using Planar graph decomposition (PDC). PDC uses techniques of [3] in conjunction with known result about exact computation of partition function for binary MRF when G is Planar and the exponential family has specific form. Their algorithm provides provable upper bound for arbitrary graph. However, they do not quantify the error incurred. Further, their algorithm is limited to binary MRF. Contribution. We propose a novel local algorithm for approximate computation of MAP and logpartition function. For any ε > 0, our algorithm can produce an ε-approximate solution for MAP and log-partition function for arbitrary MRF G as long as G excludes a finite graph as a minor (precise definition later). For example, Planar graph excludes K3,3, K5 as a minor. The running time of the algorithm is Θ(n), with constant dependent on ε, the maximum vertex degree of G and the size of the graph that is excluded as minor. Specifically, for a Planar graph with bounded degree, it takes ≤C(ε)n time to find ε-approximate solution with log log C(ε) = O(1/ε). In general, our algorithm works for any G and we can quantify bound on the error incurred by our algorithm. It is worth noting that our algorithm provides a provable lower bound on log-partition function as well unlike many of previous works. The precise results for minor-excluded graphs are stated in Theorems 1 and 2. The result concerning general graphs are stated in the form of Lemmas 2-3-4 for log-partition and Lemmas 5-6-7 for MAP. Techniques. Our algorithm is based on the following idea: First, decompose G into small-size connected components say G1, . . . , Gk by removing few edges of G. Second, compute estimates (either MAP or log-partition) in each of Gi separately. Third, combine these estimates to produce a global estimate while taking care of the effect induced by removed edges. We show that the error in the estimate depends only on the edges removed. This error bound characterization is applicable for arbitrary graph. Klein, Plotkin and Rao [10]introduced a clever and simple decomposition method for minorexcluded graphs to study the gap between max-flow and min-cut for multicommodity flows. We use their method to obtain a good edge-set for decomposing minor-excluded G so that the error induced in our estimate is small (can be made as small as required). In general, as long as G allows for such good edge-set for decomposing G into small components, our algorithm will provide a good estimate. To compute estimates in individual components, we use dynamic programming. Since each component is small, it is not computationally burdensome. 2 However, one may obtain further simpler heuristics by replacing dynamic programming by other method such as BP or TRW for computation in the components. 2 Preliminaries Here we present useful definitions and previous results about decomposition of minor-excluded graphs from [10,11]. Definition 1 (Minor Exclusion) A graph H is called minor of G if we can transform G into H through an arbitrary sequence of the following two operations: (a) removal of an edge; (b) merge two connected vertices u, v: that is, remove edge (u, v) as well as vertices u and v; add a new vertex and make all edges incident on this new vertex that were incident on u or v. Now, if H is not a minor of G then we say that G excludes H as a minor. The explanation of the following statement may help understand the definition: any graph H with r nodes is a minor of Kr, where Kr is a complete graph of r nodes. This is true because one may obtain H by removing edges from Kr that are absent in H. More generally, if G is a subgraph of G′ and G has H as a minor, then G′ has H as its minor. Let Kr,r denote a complete bipartite graph with r nodes in each partition. Then Kr is a minor of Kr,r. An important implication of this is as follows: to prove property P for graph G that excludes H, of size r, as a minor, it is sufficient to prove that any graph that excludes Kr,r as a minor has property P. This fact was cleverly used by Klein et. al. [10] to obtain a good decomposition scheme described next. First, a definition. Definition 2 ((δ, ∆)-decomposition) Given graph G = (V, E), a randomly chosen subset of edges B ⊂E is called (δ, ∆) decomposition of G if the following holds: (a) For any edge e ∈E, Pr(e ∈B) ≤δ. (b) Let S1, . . . , SK be connected components of graph G′ = (V, E\B) obtained by removing edges of B from G. Then, for any such component Sj, 1 ≤j ≤K and any u, v ∈Sj the shortest-path distance between (u, v) in the original graph G is at most ∆with probability 1. The existence of (δ, ∆)-decomposition implies that it is possible to remove δ fraction of edges so that graph decomposes into connected components whose diameter is small. We describe a simple and explicit construction of such a decomposition for minor excluded class of graphs. This scheme was proposed by Klein, Plotkin, Rao [10] and Rao [11]. DeC(G, r, ∆) (0) Input is graph G = (V, E) and r, ∆∈N. Initially, i = 0, G0 = G, B = ∅. (1) For i = 0, . . . , r −1, do the following. (a) Let Si 1, . . . , Si ki be the connected components of Gi. (b) For each Si j, 1 ≤j ≤ki, pick an arbitrary node vj ∈Si j. ◦Create a breadth-first search tree T i j rooted at vj in Si j. ◦Choose a number Li j uniformly at random from {0, . . ., ∆−1}. ◦Let Bi j be the set of edges at level Li j, ∆+ Li j, 2∆+ Li j, . . . in T i j . ◦Update B = B ∪ki j=1 Bi j. (c) set i = i + 1. (3) Output B and graph G′ = (V, E\B). As stated above, the basic idea is to use the following step recursively (upto depth r of recursion): in each connected component, say S, choose a node arbitrarily and create a breadth-first search tree, say T . Choose a number, say L, uniformly at random from {0, . . ., ∆−1}. Remove (add to B) all edges that are at level L + k∆, k ≥0 in T . Clearly, the total running time of such an algorithm is O(r(n + |E|)) for a graph G = (V, E) with |V | = n; with possible parallel implementation across different connected components. The algorithm DeC(G, r, ∆) is designed to provide a good decomposition for class of graphs that exclude Kr,r as a minor. Figure 1 explains the algorithm for a line-graph of n = 9 nodes, which excludes K2,2 as a minor. The example is about a sample run of DeC(G, 2, 3) (Figure 1 shows the first iteration of the algorithm). 3 G0 1 2 3 4 5 6 7 8 9 5 4 3 2 1 6 7 8 9 5 4 6 7 8 9 3 2 1 L1 T1 G1 S1 S2 S3 S5 S4 Figure 1: The first of two iterations in execution of DeC(G, 2, 3) is shown. Lemma 1 If G excludes Kr,r as a minor, then algorithm DeC(G, r, ∆) outputs B which is (r/∆, O(∆))-decomposition of G. It is known that Planar graph excludes K3,3 as a minor. Hence, Lemma 1 implies the following. Corollary 1 Given a planar graph G, the algorithm DeC(G, 3, ∆) produces (3/∆, O(∆))decomposition for any ∆≥1. 3 Approximate log Z Here, we describe algorithm for approximate computation of log Z for any graph G. The algorithm uses a decomposition algorithm as a sub-routine. In what follows, we use term DECOMP for a generic decomposition algorithm. The key point is that our algorithm provides provable upper and lower bound on log Z for any graph; the approximation guarantee and computation time depends on the property of DECOMP. Specifically, for Kr,r minor excluded G (e.g. Planar graph with r = 3), we will use DeC(G, r, ∆) in place of DECOMP. Using Lemma 1, we show that our algorithm based on DeC provides approximation upto arbitrary multiplicative accuracy by tuning parameter ∆. LOG PARTITION(G) (1) Use DECOMP(G) to obtain B ⊂E such that (a) G′ = (V, E\B) is made of connected components S1, . . . , SK. (2) For each connected component Sj, 1 ≤j ≤K, do the following: (a) Compute partition function Zj restricted to Sj by dynamic programming(or exhaustive computation). (3) Let ψL ij = min(x,x′)∈Σ2 ψij(x, x′), ψU ij = max(x,x′)∈Σ2 ψij(x, x′). Then log ˆZLB = K X j=1 log Zj + X (i,j)∈B ψL ij; log ˆZUB = K X j=1 log Zj + X (i,j)∈B ψU ij. (4) Output: lower bound log ˆZLB and upper bound log ˆZUB. In words, LOG PARTITION(G) produces upper and lower bound on log Z of MRF G as follows: decompose graph G into (small) components S1, . . . , SK by removing (few) edges B ⊂E using DECOMP(G). Compute exact log-partition function in each of the components. To produce bounds log ˆZLB, log ˆZUB take the summation of thus computed component-wise log-partition function along with minimal and maximal effect of edges from B. Analysis of LOG PARTITION for General G : Here, we analyze performance of LOG PARTITION for any G. In the next section, we will specialize our analysis for minor excluded G when LOG PARTITION uses DeC as the DECOMP algorithm. Lemma 2 Given an MRF G described by (1), the LOG PARTITION produces log ˆZLB, log ˆZUB such that log ˆZLB ≤log Z ≤log ˆZUB, log ˆZUB −log ˆZLB = X (i,j)∈B ψU ij −ψL ij . 4 It takes O |E|KΣ|S∗| + TDECOMP time to produce this estimate, where |S∗| = maxK j=1 |Sj| with DECOMP producing decomposition of G into S1, . . . , SK in time TDECOMP. Lemma 3 If G has maximum vertex degree D then, log Z ≥ 1 D+1 hP (i,j)∈E ψU ij −ψL ij i . Lemma 4 If G has maximum vertex degree D and the DECOMP(G) produces B that is (δ, ∆)decomposition, then E h log ˆZUB −log ˆZLB i ≤δ(D + 1) log Z, w.r.t. the randomness in B, and LOG PARTITION takes time O(nD|Σ|D∆ ) + TDECOMP. Analysis of LOG PARTITION for Minor-excluded G : Here, we specialize analysis of LOG PARTITIONfor minor exclude graph G. For G that exclude minor Kr,r, we use algorithm DeC(G, r, ∆). Now, we state the main result for log-partition function computation. Theorem 1 Let G exclude Kr,r as minor and have D as maximum vertex degree. Given ε > 0, use LOG PARTITION algorithm with DeC(G, r, ∆) where ∆= ⌈r(D+1) ε ⌉. Then, log ˆZLB ≤log Z ≤log ˆZUB; E h log ˆZUB −log ˆZLB i ≤ε log Z. Further, algorithm takes (nC(D, |Σ|, ε)), where constant C(D, |Σ|, ε) = D|Σ|DO(rD/ε). We obtain the following immediate implication of Theorem 1. Corollary 2 For any ε > 0, the LOG PARTITION algorithm with DeC algorithm for constant degree Planar graph G based MRF, produces log ˆZLB, log ˆZUB so that (1 −ε) log Z ≤log ˆZLB ≤log Z ≤log ˆZUB ≤(1 + ε) log Z, in time O(nC(ε)) where log log C(ε) = O(1/ε). 4 Approximate MAP Now, we describe algorithm to compute MAP approximately. It is very similar to the LOG PARTITION algorithm: given G, decompose it into (small) components S1, . . . , SK by removing (few) edges B ⊂E. Then, compute an approximate MAP assignment by computing exact MAP restricted to the components. As in LOG PARTITION, the computation time and performance of the algorithm depends on property of decomposition scheme. We describe algorithm for any graph G; which will be specialized for Kr,r minor excluded G using DeC(G, r, ∆). MODE(G) (1) Use DECOMP(G) to obtain B ⊂E such that (a) G′ = (V, E\B) is made of connected components S1, . . . , SK. (2) For each connected component Sj, 1 ≤j ≤K, do the following: (a) Through dynamic programming (or exhaustive computation) find exact MAP x∗,j for component Sj, where x∗,j = (x∗,j i )i∈Sj. (3) Produce output c x∗, which is obtained by assigning values to nodes using x∗,j, 1 ≤j ≤K. Analysis of MODE for General G : Here, we analyze performance of MODE for any G. Later, we will specialize our analysis for minor excluded G when it uses DeC as the DECOMP algorithm. Lemma 5 Given an MRF G described by (1), the MODE algorithm produces outputs c x∗such that H(x∗) −P (i,j)∈B ψU ij −ψL ij ≤H(c x∗) ≤H(x∗). It takes O |E|KΣ|S∗| + TDECOMP time to produce this estimate, where |S∗| = maxK j=1 |Sj| with DECOMP producing decomposition of G into S1, . . . , SK in time TDECOMP. Lemma 6 If G has maximum vertex degree D, then H(x∗) ≥ 1 D + 1 X (i,j)∈E ψU ij ≥ 1 D + 1 X (i,j)∈E ψU ij −ψL ij . 5 Lemma 7 If G has maximum vertex degree D and the DECOMP(G) produces B that is (δ, ∆)decomposition, then E h H(x∗) −H(c x∗) i ≤δ(D + 1)H(x∗), where expectation is w.r.t. the randomness in B. Further, MODE takes time O(nD|Σ|D∆ )+TDECOMP. Analysis of MODE for Minor-excluded G : Here, we specialize analysis of MODE for minor exclude graph G. For G that exclude minor Kr,r, we use algorithm DeC(G, r, ∆). Now, we state the main result for MAP computation. Theorem 2 Let G exclude Kr,r as minor and have D as the maximum vertex degree. Given ε > 0, use MODE algorithm with DeC(G, r, ∆) where ∆= ⌈r(D+1) ε ⌉. Then, (1 −ε)H(x∗) ≤H(c x∗) ≤H(x∗). Further, algorithm takes n · C(D, |Σ|, ε) time, where constant C(D, |Σ|, ε) = D|Σ|DO(rD/ε) . We obtain the following immediate implication of Theorem 2. Corollary 3 For any ε > 0, the MODE algorithm with DeC algorithm for constant degree Planar graph G based MRF, produces estimate c x∗so that (1 −ε)H(x∗) ≤H(c x∗) ≤H(x∗), in time O(nC(ε)) where log log C(ε) = O(1/ε). 5 Experiments Our algorithm provides provably good approximation for any MRF with minor excluded graph structure, with planar graph as a special case. In this section, we present experimental evaluation of our algorithm for popular synthetic model. Setup 1.2 Consider binary (i.e. Σ = {0, 1}) MRF on an n × n lattice G = (V, E): Pr(x) ∝exp X i∈V θixi + X (i,j)∈E θijxixj , for x ∈{0, 1}n2. Figure 2 shows a lattice or grid graph with n = 4 (on the left side). There are two scenarios for choosing parameters (with notation U[a, b] being uniform distribution over interval [a, b]): (1) Varying interaction. θi is chosen independently from distribution U[−0.05, 0.05] and θij chosen independent from U[−α, α] with α ∈{0.2, 0.4, . . ., 2}. (2) Varying field. θij is chosen independently from distribution U[−0.5, 0.5] and θi chosen independently from U[−α, α] with α ∈{0.2, 0.4, . . ., 2}. The grid graph is planar. Hence, we run our algorithms LOG PARTITION and MODE, with decomposition scheme DeC(G, 3, ∆), ∆∈{3, 4, 5}. We consider two measures to evaluate performance: error in log Z, defined as 1 n2 | log Zalg −log Z|; and error in H(x∗), defined as 1 n2 |H(xalg −H(x∗)|. We compare our algorithm for error in log Z with the two recently very successful algorithms – Tree re-weighted algorithm (TRW) and planar decomposition algorithm (PDC). The comparison is plotted in Figure 3 where n = 7 and results are averages over 40 trials. The Figure 3(A) plots error with respect to varying interaction while Figure 3(B) plots error with respect to varying field strength. Our algorithm, essentially outperforms TRW for these values of ∆and perform very competitively with respect to PDC. The key feature of our algorithm is scalability. Specifically, running time of our algorithm with a given parameter value ∆scales linearly in n, while keeping the relative error bound exactly the same. To explain this important feature, we plot the theoretically evaluated bound on error in log Z 2Though this setup has φi, ψij taking negative values, they are equivalent to the setup considered in the paper as the function values are lower bounded and hence affine shift will make them non-negative without changing the distribution. 6 in Figure 4 with tags (A), (B) and (C). Note that error bound plot is the same for n = 100 (A) and n = 1000 (B). Clearly, actual error is likely to be smaller than these theoretically plotted bounds. We note that these bounds only depend on the interaction strengths and not on the values of fields strengths (C). Results similar to of LOG PARTITION are expected from MODE. We plot the theoretically evaluated bounds on the error in MAP in Figure 4 with tags (A), (B) and (C). Again, the bound on MAP relative error for given ∆parameter remains the same for all values of n as shown in (A) for n = 100 and (B) for n = 1000. There is no change in error bound with respect to the field strength (C). Setup 2. Everything is exactly the same as the above setup with the only difference that grid graph is replaced by cris-cross graph which is obtained by adding extra four neighboring edges per node (exception of boundary nodes). Figure 2 shows cris-cross graph with n = 4 (on the right side). We again run the same algorithm as above setup on this graph. For cris-cross graph, we obtained its graph decomposition from the decomposition of its grid sub-graph. graph Though the cris-cross graph is not planar, due to the structure of the cris-cross graph it can be shown (proved) that the running time of our algorithm will remain the same (in order) and error bound will become only 3 times weaker than that for the grid graph ! We compute these theoretical error bounds for log Z and MAP which is plotted in Figure 5. This figure is similar to the Figure 4 for grid graph. This clearly exhibits the generality of our algorithm even beyond minor excluded graphs. References [1] J. Pearl, “Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference,” San Francisco, CA: Morgan Kaufmann, 1988. [2] J. Yedidia, W. Freeman and Y. Weiss, “Generalized Belief Propagation,” Mitsubishi Elect. Res. Lab., TR2000-26, 2000. [3] M. J. Wainwright, T. Jaakkola and A. S. Willsky, “Tree-based reparameterization framework for analysis of sum-product and related algorithms,” IEEE Trans. on Info. Theory, 2003. [4] M. J. Wainwright, T. S. Jaakkola and A. S. Willsky, “MAP estimation via agreement on (hyper)trees: Message-passing and linear-programming approaches,” IEEE Trans. on Info. Theory, 51(11), 2005. [5] S. C. Tatikonda and M. I. Jordan, “Loopy Belief Propagation and Gibbs Measure,” Uncertainty in Artificial Intelligence, 2002. [6] M. Bayati, D. Shah and M. Sharma, “Maximum Weight Matching via Max-Product Belief Propagation,” IEEE ISIT, 2005. [7] V. Kolmogorov, “Convergent Tree-reweighted Message Passing for Energy Minimization,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 2006. [8] V. Kolmogorov and M. Wainwright, “On optimality of tree-reweighted max-product message-passing,” Uncertainty in Artificial Intelligence, 2005. [9] A. Globerson and T. Jaakkola, “Bound on Partition function through Planar Graph Decomposition,” NIPS, 2006. [10] P. Klein, S. Plotkin and S. Rao, “Excluded minors, network decomposition, and multicommodity flow,” ACM STOC, 1993. [11] S. Rao, “Small distortion and volume preserving embeddings for Planar and Euclidian metrics,” ACM SCG, 1999. Grid Cris Figure 2: Example of grid graph (left) and cris-cross graph (right) with n = 4. 7 Z Error Z Error (1-A) Grid, N=7 ͡ ͦ͟͡͡ ͟͢͡ ͦ͟͢͡ ͣ͟͡ ͣͦ͟͡ ͤ͟͡ ͣ͟͡ ͥ͟͡ ͧ͟͡ ͩ͟͡ ͢ ͣ͢͟ ͥ͢͟ ͧ͢͟ ͩ͢͟ ͣ Interaction Strength TRW PDC 3 ' 4 ' 5 ' (1-B) Gird, n=7 ͡ ͦ͟͡͡͡ ͟͢͡͡ ͦ͟͢͡͡ ͣ͟͡͡ ͣͦ͟͡͡ ͤ͟͡͡ ͤͦ͟͡͡ ͣ͟͡ ͥ͟͡ ͧ͟͡ ͩ͟͡ ͢ ͣ͢͟ ͥ͢͟ ͧ͢͟ ͩ͢͟ ͣ Field Strength TRW PDC 3 ' 4 ' 5 ' Figure 3: Comparison of TRW, PDC and our algorithm for grid graph with n = 7 with respect to error in log Z. Our algorithm outperforms TRW and is competitive with respect to PDC. (2-A) Grid, n=100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Interaction Strength (2-B) Grid, n=1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Interaction Strength (2-C) Grid, n=1000 0 0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Field Strength (3-A) Grid, n=100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Interaction Strength (3-B) Grid, n=1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Interaction Strength (3-C) Grid, n=1000 0 0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Field Strength Z Error Bound Z Error Bound MAP Error Bound MAP Error Bound MAP Error Bound Z Error Bound 5 ' 20 ' 10 ' 5 ' 20 ' 10 ' Figure 4: The theoretically computable error bounds for log Z and MAP under our algorithm for grid with n = 100 and n = 1000 under varying interaction and varying field model. This clearly shows scalability of our algorithm. (4-A) Cris Cross, n=100 0 0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Interaction Strength (4-B) Cris Cross, n=1000 0 0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Interaction Strength (4-C) Cris Cross, n=1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Field Strength (5-B) Cris Cross, n=1000 0 0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Interaction Strength (5-C) Cris Cross, n=1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Field Strength Z Error Bound Z Error Bound Z Error Bound MAP Error Bound MAP Error Bound MAP Error Bound 5 ' 20 ' 10 ' (5-A) Criss Cross, n=100 0 0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Interaction Strength 5 ' 20 ' 10 ' Figure 5: The theoretically computable error bounds for log Z and MAP under our algorithm for cris-cross with n = 100 and n = 1000 under varying interaction and varying field model. This clearly shows scalability of our algorithm and robustness to graph structure. 8
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Estimating divergence functionals and the likelihood ratio by penalized convex risk minimization XuanLong Nguyen SAMSI & Duke University Martin J. Wainwright UC Berkeley Michael I. Jordan UC Berkeley Abstract We develop and analyze an algorithm for nonparametric estimation of divergence functionals and the density ratio of two probability distributions. Our method is based on a variational characterization of f-divergences, which turns the estimation into a penalized convex risk minimization problem. We present a derivation of our kernel-based estimation algorithm and an analysis of convergence rates for the estimator. Our simulation results demonstrate the convergence behavior of the method, which compares favorably with existing methods in the literature. 1 Introduction An important class of “distances” between multivariate probability distributions P and Q are the AliSilvey or f-divergences [1, 6]. These divergences, to be defined formally in the sequel, are all of the form Dφ(P, Q) = R φ(dQ/dP)dP, where φ is a convex function of the likelihood ratio. This family, including the Kullback-Leibler (KL) divergence and the variational distance as special cases, plays an important role in various learning problems, including classification, dimensionality reduction, feature selection and independent component analysis. For all of these problems, if f-divergences are to be used as criteria of merit, one has to be able to estimate them efficiently from data. With this motivation, the focus of paper is the problem of estimating an f-divergence based on i.i.d. samples from each of the distributions P and Q. Our starting point is a variational characterization of f-divergences, which allows our problem to be tackled via an M-estimation procedure. Specifically, the likelihood ratio function dP/dQ and the divergence functional Dφ(P, Q) can be estimated by solving a convex minimization problem over a function class. In this paper, we estimate the likelihood ratio and the KL divergence by optimizing a penalized convex risk. In particular, we restrict the estimate to a bounded subset of a reproducing kernel Hilbert Space (RKHS) [17]. The RKHS is sufficiently rich for many applications, and also allows for computationally efficient optimization procedures. The resulting estimator is nonparametric, in that it entails no strong assumptions on the form of P and Q, except that the likelihood ratio function is assumed to belong to the RKHS. The bulk of this paper is devoted to the derivation of the algorithm, and a theoretical analysis of the performance of our estimator. The key to our analysis is a basic inequality relating a performance metric (the Hellinger distance) of our estimator to the suprema of two empirical processes (with respect to P and Q) defined on a function class of density ratios. Convergence rates are then obtained using techniques for analyzing nonparametric M-estimators from empirical process theory [20]. Related work. The variational representation of divergences has been derived independently and exploited by several authors [5, 11, 14]. Broniatowski and Keziou [5] studied testing and estimation problems based on dual representations of f-divergences, but working in a parametric setting as opposed to the nonparametric framework considered here. Nguyen et al. [14] established a one-to-one correspondence between the family of f-divergences and the family of surrogate loss functions [2], through which the (optimum) “surrogate risk” is equal to the negative of an associated f-divergence. Another link is to the problem of estimating integral functionals of a single density, with the Shannon entropy being a well-known example, which has been studied extensively dating back to early 1 work [9, 13] as well as the more recent work [3, 4, 12]. See also [7, 10, 8] for the problem of (Shannon) entropy functional estimation. In another branch of related work, Wang et al. [22] proposed an algorithm for estimating the KL divergence for continuous distributions, which exploits histogram-based estimation of the likelihood ratio by building data-dependent partitions of equivalent (empirical) Q-measure. The estimator was empirically shown to outperform direct plug-in methods, but no theoretical results on its convergence rate were provided. This paper is organized as follows. Sec. 2 provides a background of f-divergences. In Sec. 3, we describe an estimation procedure based on penalized risk minimization and accompanying convergence rates analysis results. In Sec. 4, we derive and implement efficient algorithms for solving these problems using RKHS. Sec. 5 outlines the proof of the analysis. In Sec. 6, we illustrate the behavior of our estimator and compare it to other methods via simulations. 2 Background We begin by defining f-divergences, and then provide a variational representation of the fdivergence, which we later exploit to develop an M-estimator. Consider two distributions P and Q, both assumed to be absolutely continuous with respect to Lebesgue measure µ, with positive densities p0 and q0, respectively, on some compact domain X ⊂Rd. The class of Ali-Silvey or f-divergences [6, 1] are “distances” of the form: Dφ(P, Q) = Z p0φ(q0/p0) dµ, (1) where φ : R →¯R is a convex function. Different choices of φ result in many divergences that play important roles in information theory and statistics, including the variational distance, Hellinger distance, KL divergence and so on (see, e.g., [19]). As an important example, the Kullback-Leibler (KL) divergence between P and Q is given by DK(P, Q) = R p0 log(p0/q0) dµ, corresponding to the choice φ(t) = −log(t) for t > 0 and +∞otherwise. Variational representation: Since φ is a convex function, by Legendre-Fenchel convex duality [16] we can write φ(u) = supv∈R(uv −φ∗(v)), where φ∗is the convex conjugate of φ. As a result, Dφ(P, Q) = Z p0 sup f (fq0/p0 −φ∗(f)) dµ = sup f µZ f dQ − Z φ∗(f) dP ¶ , where the supremum is taken over all measurable functions f : X →R, and R f dP denotes the expectation of f under distribution P. Denoting by ∂φ the subdifferential [16] of the convex function φ, it can be shown that the supremum will be achieved for functions f such that q0/p0 ∈∂φ∗(f), where q0, p0 and f are evaluated at any x ∈X. By convex duality [16], this is true if f ∈∂φ(q0/p0) for any x ∈X. Thus, we have proved [15, 11]: Lemma 1. Letting F be any function class in X →R, there holds: Dφ(P, Q) ≥sup f∈F Z f dQ −φ∗(f) dP, (2) with equality if F ∩∂φ(q0/p0) ̸= ∅. To illustrate this result in the special case of the KL divergence, here the function φ has the form φ(u) = −log(u) for u > 0 and +∞for u ≤0. The convex dual of φ is φ∗(v) = supu(uv−φ(u)) = −1 −log(−v) if u < 0 and +∞otherwise. By Lemma 1, DK(P, Q) = sup f<0 Z f dQ − Z (−1 −log(−f)) dP = sup g>0 Z log g dP − Z gdQ + 1. (3) In addition, the supremum is attained at g = p0/q0. 3 Penalized M-estimation of KL divergence and the density ratio Let X1, . . . , Xn be a collection of n i.i.d. samples from the distribution Q, and let Y1, . . . , Yn be n i.i.d. samples drawn from the distribution P. Our goal is to develop an estimator of the KL divergence and the density ratio g0 = p0/q0 based on the samples {Xi}n i=1 and {Yi}n i=1. 2 The variational representation in Lemma 1 motivates the following estimator of the KL divergence. First, let G be a function class of X →R+. We then compute ˆDK = sup g∈G Z log g dPn − Z gdQn + 1, (4) where R dPn and R dQn denote the expectation under empirical measures Pn and Qn, respectively. If the supremum is attained at ˆgn, then ˆgn serves as an estimator of the density ratio g0 = p0/q0. In practice, the “true” size of G is not known. Accordingly, our approach in this paper is an alternative approach based on controlling the size of G by using penalties. More precisely, let I(g) be a non-negative measure of complexity for g such that I(g0) < ∞. We decompose the function class G as follows: G = ∪1≤M≤∞GM, (5) where GM := {g | I(g) ≤M} is a ball determined by I(·). The estimation procedure involves solving the following program: ˆgn = argming∈G Z gdQn − Z log g dPn + λn 2 I2(g), (6) where λn > 0 is a regularization parameter. The minimizing argument ˆgn is plugged into (4) to obtain an estimate of the KL divergence DK. For the KL divergence, the difference | ˆDK −DK(P, Q)| is a natural performance measure. For estimating the density ratio, various metrics are possible. Viewing g0 = p0/q0 as a density function with respect to Q measure, one useful metric is the (generalized) Hellinger distance: h2 Q(g0, g) := 1 2 Z (g1/2 0 −g1/2)2 dQ. (7) For the analysis, several assumptions are in order. First, assume that g0 (not all of G) is bounded from above and below: 0 < η0 ≤g0 ≤η1 for some constants η0, η1. (8) Next, the uniform norm of GM is Lipchitz with respect to the penalty measure I(g), i.e.: sup g∈GM |g|∞≤cM for any M ≥1. (9) Finally, on the bracket entropy of G [21]: For some 0 < γ < 2, HB δ (GM, L2(Q)) = O(M/δ)γ for any δ > 0. (10) The following is our main theoretical result, whose proof is given in Section 5: Theorem 2. (a) Under assumptions (8), (9) and (10), and letting λn →0 so that: λ−1 n = OP(n2/(2+γ))(1 + I(g0)), then under P: hQ(g0, ˆgn) = OP(λ1/2 n )(1 + I(g0)), I(ˆgn) = OP(1 + I(g0)). (b) If, in addition to (8), (9) and (10), there holds infg∈G g(x) ≥η0 for any x ∈X, then | ˆDK −DK(P, Q)| = OP(λ1/2 n )(1 + I(g0)). (11) 4 Algorithm: Optimization and dual formulation G is an RKHS. Our algorithm involves solving program (6), for some choice of function class G. In our implementation, relevant function classes are taken to be a reproducing kernel Hilbert space induced by a Gaussian kernel. The RKHS’s are chosen because they are sufficiently rich [17], and as in many learning tasks they are quite amenable to efficient optimization procedures [18]. 3 Let K : X × X →R be a Mercer kernel function [17]. Thus, K is associated with a feature map Φ : X →H, where H is a Hilbert space with inner product ⟨., .⟩and for all x, x′ ∈X, K(x, x′) = ⟨Φ(x), Φ(x′)⟩. As a reproducing kernel Hilbert space, any function g ∈H can be expressed as an inner product g(x) = ⟨w, Φ(x)⟩, where ∥g∥H = ∥w∥H. A kernel used in our simulation is the Gaussian kernel: K(x, y) := e−∥x−y∥2/σ, where ∥.∥is the Euclidean metric in Rd, and σ > 0 is a parameter for the function class. Let G := H, and let the complexity measure be I(g) = ∥g∥H. Thus, Eq. (6) becomes: min w J := min w 1 n n X i=1 ⟨w, Φ(xi)⟩−1 n n X j=1 log⟨w, Φ(yj)⟩+ λn 2 ∥w∥2 H, (12) where {xi} and {yj} are realizations of empirical data drawn from Q and P, respectively. The log function is extended take value −∞for negative arguments. Lemma 3. minw J has the following dual form: −min α>0 n X j=1 −1 n−1 n log nαj+ 1 2λn X i,j αiαjK(yi, yj)+ 1 2λnn2 X i,j K(xi, xj)−1 λnn X i,j αjK(xi, yj). Proof. Let ψi(w) := 1 n⟨w, Φ(xi)⟩, ϕj(w) := −1 n log⟨w, Φ(yj)⟩, and Ω(w) = λn 2 ∥w∥2 H. We have min w J = −max w (⟨0, w⟩−J(w)) = −J∗(0) = −min ui,vj n X i=1 ψ∗ i (ui) + n X j=1 ϕ∗ j(vj) + Ω∗(− n X i=1 ui − n X j=1 vj), where the last line is due to the inf-convolution theorem [16]. Simple calculations yield: ϕ∗ j(v) = −1 n −1 n log nαj if v = −αjΦ(yj) and + ∞otherwise ψ∗ i (u) = 0 if u = 1 nΦ(xi) and + ∞otherwise Ω∗(v) = 1 2λn ∥v∥2 H. So, minw J = −minαi Pn j=1(−1 n −1 n log nαj)+ 1 2λn ∥Pn j=1 αjΦ(yj)−1 n Pn i=1 Φ(xi)∥2 H, which implies the lemma immediately. If ˆα is solution of the dual formulation, it is not difficult to show that the optimal ˆw is attained at ˆw = 1 λn (Pn j=1 ˆαjΦ(yj) −1 n Pn i=1 Φ(xi)). For an RKHS based on a Gaussian kernel, the entropy condition (10) holds for any γ > 0 [23]. Furthermore, (9) trivially holds via the Cauchy-Schwarz inequality: |g(x)| = |⟨w, Φ(x)⟩| ≤ ∥w∥H∥Φ(x)∥H ≤I(g) p K(x, x) ≤I(g). Thus, by Theorem 2(a), ∥ˆw∥H = ∥ˆgn∥H = OP(∥g0∥H), so the penalty term λn∥ˆw∥2 vanishes at the same rate as λn. We have arrived at the following estimator for the KL divergence: ˆDK = 1 + n X j=1 (−1 n −1 n log nˆαj) = n X j=1 −1 n log nˆαj. log G is an RKHS. Alternatively, we could set log G to be the RKHS, letting g(x) = exp⟨w, Φ(x)⟩, and letting I(g) = ∥log g∥H = ∥w∥H. Theorem 2 is not applicable in this case, because condition (9) no longer holds, but this choice nonetheless seems reasonable and worth investigating, because in effect we have a far richer function class which might improve the bias of our estimator when the true density ratio is not very smooth. 4 A derivation similar to the previous case yields the following convex program: min w J := min w 1 n n X i=1 e⟨w, Φ(xi)⟩−1 n n X j=1 ⟨w, Φ(yj)⟩+ λn 2 ∥w∥2 H = −min α>0 n X i=1 αi log(nαi) −αi + 1 2λn ∥ n X i=1 αiΦ(xi) −1 n n X j=1 Φ(yj)∥2 H. Letting ˆα be the solution of the above convex program, the KL divergence can be estimated by: ˆDK = 1 + n X i=1 ˆαi log ˆαi + ˆαi log n e . 5 Proof of Theorem 2 We now sketch out the proof of the main theorem. The key to our analysis is the following lemma: Lemma 4. If ˆgn is an estimate of g using (6), then: 1 4h2 Q(g0, ˆgn) + λn 2 I2(ˆgn) ≤− Z (ˆgn −g0)d(Qn −Q) + Z 2 log ˆgn + g0 2g0 d(Pn −P) + λn 2 I2(g0). Proof. Define dl(g0, g) = R (g −g0)dQ −log g g0 dP. Note that for x > 0, 1 2 log x ≤√x −1. Thus, R log g g0 dP ≤2 R (g1/2g−1/2 0 −1) dP. As a result, for any g, dl is related to hQ as follows: dl(g0, g) ≥ Z (g −g0) dQ −2 Z (g1/2g−1/2 0 −1) dP = Z (g −g0) dQ −2 Z (g1/2g1/2 0 −g0) dQ = Z (g1/2 −g1/2 0 )2dQ = 2h2 Q(g0, g). By the definition (6) of our estimator, we have: Z ˆgndQn − Z log ˆgndPn + λn 2 I2(ˆgn) ≤ Z g0dQn − Z log g0dPn + λn 2 I2(g0). Both sides (modulo the regularization term I2) are convex functionals of g. By Jensen’s inequality, if F is a convex function, then F((u + v)/2) −F(v) ≤(F(u) −F(v))/2. We obtain: Z ˆgn + g0 2 dQn − Z log ˆgn + g0 2 dPn + λn 4 I2(ˆgn) ≤ Z g0dQn − Z log g0dPn + λn 4 I2(g0). Rearranging, R ˆgn−g0 2 d(Qn −Q) − R log ˆgn+g0 2g0 d(Pn −P) + λn 4 I2(ˆgn) ≤ Z log ˆgn + g0 2g0 dP − Z ˆgn −g0 2 dQ + λn 4 I2(g0) = −dl(g0, g0 + ˆgn 2 ) + λn 4 I2(g0) ≤−2h2 Q(g0, g0 + ˆgn 2 ) + λn 4 I2(g0) ≤−1 8h2 Q(g0, ˆgn) + λn 4 I2(g0), where the last inequality is a standard result for the (generalized) Hellinger distance (cf. [20]). Let us now proceed to part (a) of the theorem. Define fg := log g+g0 2g0 , and let FM := {fg|g ∈GM}. Since fg is a Lipschitz function of g, conditions (8) and (10) imply that HB δ (FM, L2(P)) = O(M/δ)γ. (13) Apply Lemma 5.14 of [20] using distance metric d2(g0, g) = ∥g −g0∥L2(Q), the following is true under Q (and so true under P as well, since dP/dQ is bounded from above), sup g∈G | R (g −g0)d(Qn −Q)| n−1/2d2(g0, g)1−γ/2(1 + I(g) + I(g0))γ/2 ∨n− 2 2+γ (1 + I(g) + I(g0)) = OP(1). (14) 5 In the same vein, we obtain that under P measure: sup g∈G | R fgd(Pn −P)| n−1/2d2(g0, g)1−γ/2(1 + I(g) + I(g0))γ/2 ∨n− 2 2+γ (1 + I(g) + I(g0)) = OP(1). (15) By condition (9), we have: d2(g0, g) = ∥g −g0∥L2(Q) ≤2c1/2(1 + I(g) + I(g0))1/2hQ(g0, g). Combining Lemma 4 and Eqs. (15), (14), we obtain the following: 1 4h2 Q(g0, ˆgn) + λn 2 I2(ˆgn) ≤λnI(g0)2/2+ OP µ n−1/2hQ(g0, g)1−γ/2(1 + I(g) + I(g0))1/2+γ/4 ∨n− 2 2+γ (1 + I(g) + I(g0)) ¶ . (16) From this point, the proof involves simple algebraic manipulation of (16). To simplify notation, let ˆh = hQ(g0, ˆgn), ˆI = I(ˆgn), and I0 = I(g0). There are four possibilities: Case a. ˆh ≥n−1/(2+γ)(1 + ˆI + I0)1/2 and ˆI ≥1 + I0. From (16), either ˆh2/4 + λn ˆI2/2 ≤OP(n−1/2)ˆh1−γ/2 ˆI1/2+γ/4 or ˆh2/4 + λn ˆI2/2 ≤λnI2 0/2, which implies, respectively, either ˆh ≤λ−1/2 n OP(n−2/(2+γ)), ˆI ≤λ−1 n OP(n−2/(2+γ)) or ˆh ≤OP(λ1/2 n I0), ˆI ≤OP(I0). Both scenarios conclude the proof if we set λ−1 n = OP(n2/(γ+2)(1 + I0)). Case b. ˆh ≥n−1/(2+γ)(1 + ˆI + I0)1/2 and ˆI < 1 + I0. From (16), either ˆh2/4 + λn ˆI2/2 ≤OP(n−1/2)ˆh1−γ/2(1 + I0)1/2+γ/4 or ˆh2/4 + λn ˆI2/2 ≤λnI2 0/2, which implies, respectively, either ˆh ≤(1 + I0)1/2OP(n−1/(γ+2)), ˆI ≤1 + I0 or ˆh ≤OP(λ1/2 n I0), ˆI ≤OP(I0). Both scenarios conclude the proof if we set λ−1 n = OP(n2/(γ+2)(1 + I0)). Case c. ˆh ≤n−1/(2+γ)(1 + ˆI + I0)1/2 and ˆI ≥1 + I0. From (16) ˆh2/4 + λn ˆI2/2 ≤OP(n−2/(2+γ))ˆI, which implies that ˆh ≤OP(n−1/(2+γ))ˆI1/2 and ˆI ≤λ−1 n OP(n−2/(2+γ)). This means that ˆh ≤ OP(λ1/2 n )(1 + I0), ˆI ≤OP(1 + I0) if we set λ−1 n = OP(n2/(2+γ))(1 + I0). Case d. ˆh ≤n−1/(2+γ)(1 + ˆI + I0)1/2 and ˆI ≤1 + I0. Part (a) of the theorem is immediate. Finally, part (b) is a simple consequence of part (a) using the same argument as in Thm. 9 of [15]. 6 Simulation results In this section, we describe the results of various simulations that demonstrate the practical viability of our estimators, as well as their convergence behavior. We experimented with our estimators using various choices of P and Q, including Gaussian, beta, mixture of Gaussians, and multivariate Gaussian distributions. Here we report results in terms of KL estimation error. For each of the eight estimation problems described here, we experiment with increasing sample sizes (the sample size, n, ranges from 100 to 104 or more). Error bars are obtained by replicating each set-up 250 times. For all simulations, we report our estimator’s performance using the simple fixed rate λn ∼1/n, noting that this may be a suboptimal rate. We set the kernel width to be relatively small (σ = .1) for one-dimension data, and larger for higher dimensions. We use M1 to denote the method in which G is the RKHS, and M2 for the method in which log G is the RKHS. Our methods are compared to 6 −0.1 0 0.1 0.2 0.3 0.4 100 200 500 1000 2000 5000 10000 20000 50000 Estimate of KL(Beta(1,2),Unif[0,1]) 0.1931 M1, σ = .1, λ = 1/n M2, σ = .1, λ = .1/n WKV, s = n1/2 WKV, s = n1/3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 100 200 500 1000 2000 5000 10000 Estimate of KL(1/2 Nt(0,1)+ 1/2 Nt(1,1),Unif[−5,5]) 0.414624 M1, σ = .1, λ = 1/n M2, σ = 1, λ = .1/n WKV, s = n1/3 WKV, s = n1/2 WKV, s = n2/3 0 0.5 1 1.5 2 2.5 100 200 500 1000 2000 5000 10000 Estimate of KL(Nt(0,1),Nt(4,2)) 1.9492 M1, σ = .1, λ = 1/n M2, σ = .1, λ = .1/n WKV, s = n1/3 WKV, s = n1/2 WKV, s = n2/3 −2 −1 0 1 2 3 4 5 6 100 200 500 1000 2000 5000 10000 Estimate of KL(Nt(4,2),Nt(0,1)) 4.72006 M1, σ = 1, λ = .1/n M2, σ = 1, λ = .1/n WKV, s = n1/4 WKV, s = n1/3 WKV, s = n1/2 0 0.5 1 1.5 100 200 500 1000 2000 5000 10000 Estimate of KL(Nt(0,I2),Unif[−3,3]2) 0.777712 M1, σ = .5, λ = .1/n M2, σ = .5, λ = .1/n WKV, n1/3 WKV, n1/2 0 0.5 1 1.5 100 200 500 1000 2000 5000 10000 Estimate of KL(Nt(0,I2),Nt(1,I2)) 0.959316 M1, σ = .5, λ = .1/n M2, σ = .5, λ = .1/n WKV, n1/3 WKV, n1/2 0 0.5 1 1.5 2 100 200 500 1000 2000 5000 10000 Estimate of KL(Nt(0,I3),Unif[−3,3]3) 1.16657 M1 σ = 1, λ = .1/n1/2 M2, σ = 1, λ = .1/n M2, σ = 1, λ = .1/n2/3 WKV, n1/3 WKV, n1/2 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 100 200 500 1000 2000 5000 10000 Estimate of KL(Nt(0,I3),Nt(1,I3)) 1.43897 M1, σ = 1, λ = .1/n M2, σ = 1, λ = .1/n WKV, n1/2 WKV, n1/3 Figure 1. Results of estimating KL divergences for various choices of probability distributions. In all plots, the X-axis is the number of data points plotted on a log scale, and the Y-axis is the estimated value. The error bar is obtained by replicating the experiment 250 times. Nt(a, Ik) denotes a truncated normal distribution of k dimensions with mean (a, . . . , a) and identity covariance matrix. 7 algorithm A in Wang et al [22], which was shown empirically to be one of the best methods in the literature. Their method, denoted by WKV, is based on data-dependent partitioning of the covariate space. Naturally, the performance of WKV is critically dependent on the amount s of data allocated to each partition; here we report results with s ∼nγ, where γ = 1/3, 1/2, 2/3. The first four plots present results with univariate distributions. In the first two, our estimators M1 and M2 appear to have faster convergence rate than WKV. The WKV estimator performs very well in the third example, but rather badly in the fourth example. The next four plots present results with two and three dimensional data. Again, M1 has the best convergence rates in all examples. The M2 estimator does not converge in the last example, suggesting that the underlying function class exhibits very strong bias. The WKV methods have weak convergence rates despite different choices of the partition sizes. It is worth noting that as one increases the number of dimensions, histogram based methods such as WKV become increasingly difficult to implement, whereas increasing dimension has only a mild effect on our method. References [1] S. M. Ali and S. D. Silvey. A general class of coefficients of divergence of one distribution from another. J. Royal Stat. Soc. Series B, 28:131–142, 1966. [2] P. L. Bartlett, M. I. Jordan, and J. D. McAuliffe. Convexity, classification, and risk bounds. Journal of the American Statistical Association, 101:138–156, 2006. [3] P. Bickel and Y. Ritov. Estimating integrated squared density derivatives: Sharp best order of convergence estimates. Sankhy¯a Ser. A, 50:381–393, 1988. [4] L. Birg´e and P. Massart. Estimation of integral functionals of a density. Ann. Statist., 23(1):11–29, 1995. [5] M. Broniatowski and A. Keziou. Parametric estimation and tests through divergences. Technical report, LSTA, Universit´e Pierre et Marie Curie, 2004. [6] I. Csisz´ar. Information-type measures of difference of probability distributions and indirect observation. Studia Sci. Math. Hungar, 2:299–318, 1967. [7] L. Gyorfiand E.C. van der Meulen. Density-free convergence properties of various estimators of entropy. Computational Statistics and Data Analysis, 5:425–436, 1987. [8] P. Hall and S. Morton. On estimation of entropy. Ann. Inst. Statist. Math., 45(1):69–88, 1993. [9] I. A. Ibragimov and R. Z. Khasminskii. On the nonparametric estimation of functionals. In Symposium in Asymptotic Statistics, pages 41–52, 1978. [10] H. Joe. Estimation of entropy and other functionals of a multivariate density. Ann. Inst. Statist. Math., 41:683–697, 1989. [11] A. Keziou. Dual representation of φ-divergences and applications. C. R. Acad. Sci. Paris, Ser. I 336, pages 857–862, 2003. [12] B. Laurent. Efficient estimation of integral functionals of a density. Ann. Statist., 24(2):659–681, 1996. [13] B. Ya. Levit. Asymptotically efficient estimation of nonlinear functionals. Problems Inform. Transmission, 14:204–209, 1978. [14] X. Nguyen, M. J. Wainwright, and M. I. Jordan. On divergences, surrogate losses and decentralized detection. Technical Report 695, Dept of Statistics, UC Berkeley, October 2005. [15] X. Nguyen, M. J. Wainwright, and M. I. Jordan. Nonparametric estimation of the likelihood ratio and divergence functionals. In International Symposium on Information Theory (ISIT), 2007. [16] G. Rockafellar. Convex Analysis. Princeton University Press, Princeton, 1970. [17] S. Saitoh. Theory of Reproducing Kernels and its Applications. Longman, Harlow, UK, 1988. [18] B. Sch¨olkopf and A. Smola. Learning with Kernels. MIT Press, Cambridge, MA, 2002. [19] F. Topsoe. Some inequalities for information divergence and related measures of discrimination. IEEE Transactions on Information Theory, 46:1602–1609, 2000. [20] S. van de Geer. Empirical Processes in M-Estimation. Cambridge University Press, 2000. [21] A. W. van der Vaart and J. Wellner. Weak Convergence and Empirical Processes. Springer-Verlag, New York, NY, 1996. [22] Q. Wang, S. R. Kulkarni, and S. Verd´u. Divergence estimation of continuous distributions based on data-dependent partitions. IEEE Transactions on Information Theory, 51(9):3064–3074, 2005. [23] D. X. Zhou. The covering number in learning theory. Journal of Complexity, 18:739–767, 2002. 8
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Learning with Tree-Averaged Densities and Distributions Sergey Kirshner AICML and Dept of Computing Science University of Alberta Edmonton, Alberta, Canada T6G 2E8 sergey@cs.ualberta.ca Abstract We utilize the ensemble of trees framework, a tractable mixture over superexponential number of tree-structured distributions [1], to develop a new model for multivariate density estimation. The model is based on a construction of treestructured copulas – multivariate distributions with uniform on [0, 1] marginals. By averaging over all possible tree structures, the new model can approximate distributions with complex variable dependencies. We propose an EM algorithm to estimate the parameters for these tree-averaged models for both the real-valued and the categorical case. Based on the tree-averaged framework, we propose a new model for joint precipitation amounts data on networks of rain stations. 1 Introduction Multivariate real-valued data appears in many real-world data sets, and a lot of research is being focused on the development of multivariate real-valued distributions. One of the challenges in constructing such distributions is that univariate continuous distributions commonly do not have a clear multivariate generalization. The most studied exception is the multivariate Gaussian distribution owing to properties such as closed form density expression with a convenient generalization to higher dimensions and closure over the set of linear projections. However, not all problems can be addressed fairly with Gaussians (e.g., mixtures, multimodal distributions, heavy-tailed distributions), and new approaches are needed for such problems. While modeling multivariate distributions is in general difficult due to complicated functional forms and the curse of dimensionality, learning models for individual variables (univariate marginals) is often straightforward. Once the univariate marginals are known (or assumed known), the rest can be modeled using copulas, multivariate distributions with all univariate marginals equal to uniform distributions on [0, 1] (e.g., [2, 3]). A large portion of copula research concentrated on bivariate copulas as extensions to higher dimensions are often difficult. Thus if the desired distribution decomposes into its univariate marginals and only bivariate distributions, the machinery of copulas can be effectively utilized. Distributions with undirected tree-structured graphical models (e.g., [4]) have exactly these properties, as probability density functions over the variables with tree-structured conditional independence graphs can be written as a product involving univariate marginals and bivariate marginals corresponding to the edges of the tree. While tree-structured dependence is perhaps too restrictive, a richer variable dependence can be obtained by averaging over a small number of different tree structures [5] or all possible tree structures; the latter can be done analytically for categorical-valued distributions with an ensemble-of-trees model [1]. In this paper, we extend this tree-averaged model to continuous variables with the help of copulas and derive a learning algorithm to estimate the parameters within the maximum likelihood framework with EM [6]. Within this framework, the 1 parameter estimation for tree-structured and tree-averaged models requires optimization over only univariate and bivariate densities potentially avoiding the curse of dimensionality, a property not shared by alternative models that relax the dependence restriction of trees (e.g., vines [7]). The main contributions of the paper are the new tree-averaged model for multivariate copulas, a parameter estimation algorithm for tree-averaged framework (for both categorical and real-valued complete data), and a new model for multi-site daily precipitation amounts, an important application in hydrology. In the process, we introduce previously unexplored tree-structured copula density and an algorithm for estimation of its structure and parameters. The paper is organized as follows. First, we describe copulas, their densities, and some of their useful properties (Section 2). We then construct multivariate copulas with tree-structured dependence from bivariate copulas (Section 3.1) and show how to estimate the parameters of the bivariate copulas and perform the edge selection. To allow more complex dependencies between the variables, we describe a tree-averaged copula, a novel copula object constructed by averaging over all possible spanning trees for tree-structured copulas, and derive a learning algorithm for the estimation of the parameters from data for the treeaveraged copulas (Section 4). We apply our new method to a benchmark data set (Section 5.1); we also develop a new model for multi-site precipitation amounts, a problem involving both binary (rain/no rain) and continuous (how much rain) variables (Section 5.2). 2 Copulas Let X = (X1, . . . , Xd) be a vector random variable with corresponding probability distribution F (cdf) defined on Rd. We denote by V the set of d components (variables) of X and refer to individual variables as Xv for v ∈V. For simplicity, we will refer to assignments to random variables by lower case letters, e.g., Xv = xv will be denoted by xv. Let Fv (xv) = F (Xv = xv, Xu = ∞: u ∈V \ {v}) denote a univariate marginal of F over the variable Xv. Let pv (xv) denote the probability density function (pdf) of Xv. Let av = Fv (xv), and let a = (a1, . . . , ad), so a is a vector of quantiles of components of x with respect to corresponding univariate marginals. Next, we define copula, a multivariate distribution over vectors of quantiles. Definition 1. The copula associated with F is a distribution function C : [0, 1]d →[0, 1] that satisfies F (x) = C (F1 (x1) , . . . , Fd (xd)) , x ∈Rd. (1) If F is a continuous distribution on Rd with univariate marginals F1, . . . , Fd, then C (a) = F F −1 1 (a1) , . . . , F −1 d (ad) is the unique choice for (1). Assuming that F has d-th order partial derivatives, the probability density function (pdf) can be obtained from the distribution function via differentiation and expressed in terms of a derivative of a copula: p (x) = ∂dF (x) ∂x1 . . . ∂xd = ∂dC (a) ∂x1 . . . ∂xd = ∂dC (a) ∂a1 . . . ∂ad Y v∈V ∂av ∂xv = c (a) Y v∈V pv (xv) (2) where c (a) = ∂dC(a) ∂a1...∂ad is referred to as a copula density function. Suppose we are given a complete data set D = x1, . . . , xN of d-component real-valued vectors xn = xn 1, . . . , xd 1 under i.i.d. assumption. A maximum likelihood (ML) estimate for the parameters of c (or p) from data can be obtained my maximizing the log-likelihood of D ln p (D) = X v∈V N X n=1 ln pv (xn v) + N X n=1 ln c (F1 (xn 1) , . . . , Fd (xn d)) . (3) The first term of the log-likelihood corresponds to the total log-likelihood of all univariate marginals of p, and the second term to the log-likelihood of its d-variate copula. These terms are not independent as the second term in the sum is defined in terms of the probability expressions in the first summand; except for a few special cases, a direct optimization of (3) is prohibitively complicated. However a useful (and asymptotically consistent) heuristic is first to maximize the log-likelihood for the marginals (first term only), and then to estimate the parameters for the copula given the solution 2 for the marginals. The univariate marginals can be accurately estimated by either fitting the parameters for some appropriately chosen univariate distributions or by applying non-parametric methods1 as the marginals are estimated independent of each other and do not suffer from the curse of dimensionality. Let ˆpv (xv) be the estimated pdf for component v, and ˆFv be the corresponding cdf. Let A = a1, . . . , aN where an = (an 1, . . . , an d) = ˆF (xn 1) , . . . , ˆF (xn d) be a set of estimated quantiles. Under the above heuristic, ML estimate for copula density c is computed by maximizing ln c (A) = PN n=1 ln c (an). 3 Exploiting Tree-Structured Dependence Joint probability distributions are often modeled with probabilistic graphical models where the structure of the graph captures the conditional independence relations of the variables. The joint distribution is then represented as a product of functions over subsets of variables. We would like to keep the number of variables for each of the functions small as the number of parameters and the number of points needed for parameter estimation often grows exponentially with the number of variables. Thus, we focus on copulas with tree dependence. Trees play an important role in probabilistic graphical models as they allow for efficient exact inference [10] as well as structure and parameter learning [4]. They can also be placed in a fully Bayesian framework with decomposable priors allowing to compute expected values (over all possible spanning trees) of product of functions defined on the edges of the trees [1]. As we will see later in this section, under the tree-structured dependence, a copula density can be computed as products of bivariate copula densities over the edges of the graph. This property allows us to estimate the parameters for the edge copulas independently. 3.1 Tree-Structured Copulas We consider tree-structured Markov networks, i.e., undirected graphs that do not have loops. For a distribution F admitting tree-structured Markov networks (referred from now on as tree-structured distributions), assuming that p (x) > 0 and p (x) < ∞for x ∈R ⊆Rd, the density (for x ∈R) can be rewritten as p (x) = "Y v∈V pv (xv) # Y {u,v}∈E puv (xu, xv) pu (xu) pv (xv). (4) This formulation easily follows from the Hammersley-Clifford theorem [11]. Note that for {u, v} ∈ E, a copula density cuv (au, av) for F (xu, xv) can be computed using Equation 2: cuv (au, av) = puv (xu, xv) pu (xu) pv (xv). (5) Using Equations 2, 4, and 5, cp (a) for F (x) can be computed as cp (a) = p (x) Q v∈V pv (xv) = Y {u,v}∈E puv (xu, xv) pu (xu) pv (xv) = Y {u,v}∈E cp (au, av) . (6) Equation 6 states that a copula density for a tree-structured distribution decomposes as a product of bivariate copulas over its edges. The converse is true as well; a tree-structured copula can be constructed by specifying copulas for the edges of the tree. Theorem 1. Given a tree or a forest G = (V, E) and copula densities cuv (au, av) for {u, v} ∈E, cE (a) = Y {u,v}∈E cuv (au, av) is a valid copula density. For a tree-structured density, the copula log-likelihood can be rewritten as ln c (A) = X {u,v}∈E N X n=1 ln cuv (an u, an v) , 1These approaches for copula estimation are referred to as inference for the margins (IFM) [8] and canonical maximum likelihood (CML) [9] for parametric and non-parametric forms for the marginals, respectively. 3 and the parameters can be fitted by maximizing PN n=1 ln cuv (an u, an v) independently for different pairs {u, v} ∈E. The tree structure can be learned from the data as well, as in the Chow-Liu algorithm [4]. Full algorithm can be found in an extended version of the paper [12]. 4 Tree-Averaged Copulas While the framework from Section 3.1 is computationally efficient and convenient for implementation, the imposed tree-structured dependence is too restrictive for real-world problems. Vines [7], for example, deal with this problem by allowing recursive refinements for the bivariate probabilities over variables not connected by the tree edges. However, vines require estimation of additional characteristics of the distribution (e.g., conditional rank correlations) requiring estimation over large sets of variables, which is not advisable when the amount of available data is not large. Our proposed method would only require optimization of parameters of bivariate copulas from the corresponding two components of weighted data vectors. Using the Bayesian framework for spanning trees from [1], it is possible to construct an object constituting a convex combination over all possible spanning trees allowing a much richer set of conditional independencies than a single tree. Meil˘a and Jaakkola [1] proposed a decomposable prior over all possible spanning tree structures. Let β be a symmetric matrix of non-negative weights for all pairs of distinct variables and zeros on the diagonal. Let E be a set of all possible spanning trees over V. The probability distribution over all spanning tree structures over V is defined as P (E ∈E|β) = 1 Z Y {u,v}∈E βuv where Z = X E∈E Y {u,v}∈E βuv. (7) Even though the sum is over |E| = dd−2 trees, Z can be efficiently computed in closed form using a weighted generalization of Kirchoff’s Matrix Tree Theorem (e.g., [1]). Theorem 2. Let P (E) be a distribution over spanning tree structures defined by (7). Then the normalization constant Z is equal to the determinant |L⋆(β)|, with matrix L⋆(β) representing the first (d −1) rows and columns of the matrix L (β) given by: Luv (β) = Lvu (β) = −βuv u, v ∈V, u ̸= v; P w∈V βvw u, v ∈V, u = v. β is a generalization of an adjacency matrix, and L (β) is a generalization of the Laplacian matrix. The decomposability property of the tree prior (Equation 7) allows us to compute the average of the tree-structured distributions over all dd−2 tree structures. In [1], such averaging was applied to tree-structured distributions over categorical variables. Similarly, we define a tree-averaged copula density as a convex combination of copula densities of the form (6): r (a) = X E∈E P (E|β) c (a) = 1 Z X E∈E Y {u,v}∈E βuv Y {u,v}∈E cuv (au, av) = |L⋆(βc (a))| |L⋆(β)| where entry (uv) of matrix βc (a) denotes βuvcuv (au, av). A finite convex combination of copulas is a copula, so r (a) is a copula density. 4.1 Parameter Estimation Given a set of estimated quantile values A, a suitable parameter values β (edge weight matrix) and θ (parameters for bivariate edge copulas) can be found by maximizing the log-likelihood of A: l (β, θ) = ln r (A|β, θ) = N X n=1 ln r (an|β, θ) = N X n=1 ln |L⋆(βc (an|θ))| −N ln |L⋆(β)| . (8) However, the parameter optimization of l (β, θ) cannot be done analytically. Instead, noticing that we are dealing with a mixture model (granted, one where the number of mixture components is super-exponential), we propose performing the parameter optimization with the EM algorithm [6].2 2A possibility of EM algorithm for ensemble-of-trees with categorical data was mentioned [1], but the idea was abandoned due to the concern about the M-step. 4 Algorithm TREEAVERAGEDCOPULADENSITY(D, c) Inputs: A complete data set D of d-component real-valued vectors; a set of of bivariate parametric copula densities c = {cuv : u, v ∈V} 1. Estimate univariate margins ˆFv (Xv) for all components v ∈V treating all components independently. 2. Replace D with A consisting of vectors an = ˆF1 (xn 1) , . . . , ˆFd (xn d) for each vector xn in D 3. Initialize β and θ 4. Run until convergence (as determined by change in log-likelihood, Equation 8) • E-step: For all vectors an and pairs {u, v}, compute P ({u, v} ∈E|an, β, θ) • M-step: – Update β with gradient ascent – Update θuv for all pairs by setting partial derivative with respect to parameters of θuv (Equation 9) to zero and solving corresponding equations Output: Denoting au = ˆF (xu) and av = ˆF (xv), ˆp (x) = Q v∈V ˆpv (xv) |L⋆(βc(a))| |L⋆(β)| Figure 1: Algorithm for estimation of a pdf with tree-averaged copulas. While there are dd−2 possible mixture components (spanning trees), in the E-step, we only need to compute the posterior probabilities for d (d −1) /2 edges. Each step of EM consists of finding parameters β′, θ′ maximizing the expected joint log-likelihood M β′, θ′; β, θ given current parameter values β, θ where M β′, θ′; β, θ = N X n=1 X En∈E P (En|an, β, θ) ln P E|β′ c an|E, θ′ = X {u,v} N X n=1 sn ({u, v}) (ln β′ uv + ln cuv (an u, an v|θ′ uv)) −N ln L⋆ β′ ; sn ({u, v}) = X E∈E {u,v}∈E P (En|an, β, θ) = X E∈E {u,v}∈E Q {u,v}∈E (βuvcuv (an u, an v|θuv)) |L⋆(βc (an))| . The probability distribution P (En|an, β, θ) is of the same form as the tree prior, so to compute sn ({u, v}) one needs to compute the sum of probabilities of all trees containing edge {u, v}. Theorem 3. Let P (E|β) be a tree prior defined in Equation 7. Let Q (β) = (L⋆(β))−1 where L⋆ is obtained by removing row and column w from L. Then X E∈E: {u,v}∈E P (E|β) = ( βuv (Quu (β) + Qvv (β) −2Quv (β)) : u ̸= v, u ̸= w, v ̸= w, βuwQuu (β) : v = w, βwvQvv (β) : u = w. As a consequence of Theorem 3, for each an, all d (d −1) /2 edge probabilities sn ({u, v}) can be computed simultaneously with time complexity of a single (d −1) × (d −1) matrix inversion, O d3 . Assuming a candidate bivariate copula cuv has one free parameter θuv, θuv can be optimized by setting ∂M β′, θ′; β, θ ∂θ′uv = N X n=1 sn ({u, v}) ∂ln cuv (an u, an v; θ′ uv) ∂θ′uv , (9) to 0. (See [12] for more details.) The parameters of the tree prior can be updated by maximizing X {u,v} 1 N N X n=1 sn ({u, v}) ! ln β′ uv −ln |L⋆(β)| , 5 an expression concave in ln βuv ∀{u, v}. β′ can be updated using a gradient ascent algorithm on ln βuv ∀{u, v}, with time complexity O d3 per iteration. The outline of the EM algorithm is shown in Figure 1. Assuming the complexity of each bivariate copula update is O (N), the time complexity of each EM iteration is O Nd3 . The EM algorithm can be easily transferred to tree averaging for categorical data. The E-step does not change, and in the M-step, the parameters for the univariate marginals are updated ignoring bivariate terms. Then, the parameters for the bivariate distributions for each edge are updated constrained on the new values of the parameters for the univariate distributions. While the algorithm does not guarantee a maximization of the expected log-likelihood, it nonetheless worked well in our experiments. 5 Experiments 5.1 MAGIC Gamma Telescope Data Set First, we tested our tree-averaged density estimator on a MAGIC Gamma Telescope Data Set from the UCI Machine Learning Repository [13]. We considered only the examples from class gamma (signal); this set consists of 12332 vectors of d = 10 real-valued components. The univariate marginals are not Gaussian (some are bounded; some have multiple modes). Fig. 2 shows an average log-likelihood of models trained on training sets with N = 50, 100, 200, 500, 1000, 2000, 5000, 10000 and evaluated on 2000-example test sets (averaged over 10 training and test sets). The marginals were estimated using Gaussian kernel density estimators (KDE) with Rule-of-Thumb bandwidth selection. All of the models except for full Gaussian have the same marginals, differ only in the multivariate dependence (copula). As expected from the curse of dimensionality, product KDE improves logarithmically with the amount of data. Not only the marginals are not Gaussian (evidenced by a Gaussian copula with KDE marginals outperforming a Gaussian distribution), the multivariate dependence is also not Gaussian, evidenced by a tree-structured Frank copula outperforming a tree-structured and a full Gaussian copula. However, model averaging even with the wrong dependence model (tree-averaged Gaussian copula) yields superior performance. 5.2 Multi-Site Precipitation Modeling We applied the tree-averaged framework to the problem of modeling daily rainfall amounts for a regional spatial network of stations. The task is to build a generative model capturing the spatial and temporal properties of the data. This model can be used in at least two ways: first, to sample sequences from it and to use them as inputs for other models, e.g., crop models; and second, as a descriptive model of the data. Hidden Markov models (possible with non-homogeneous transitions) are being frequently used for this task (e.g., [14]) with the transition distribution responsible for modeling of temporal dependence, and the emission distributions capturing most of the spatial dependence. Additionally, HMMs can be viewed as assigning rainfall daily patterns to “weather states” (or corresponding emission components), and both these states (as described by either their parameters or the statistics of the patterns associated with it) and their temporal evolution often offer useful synoptic insight. We will use HMMs as the wrapper model with tree-averaged (and tree-structured) distributions to model the emission components. The distribution of daily rainfall amounts for any given station can be viewed as a non-overlapping mixture with one component corresponding to zero precipitation, and the other component to positive precipitation. For a station v, let rv be the precipitation amount, πv be a probability of positive precipitation, and let fv (rv|λv) be a probability density function for amounts given positive precipitation: p (rv|πv, λv) = 1 −πv : rv = 0, πvfv (rv|λv) : rv > 0. For a pair of stations {u, v}, let πuv denote the probability of simultaneous positive amounts and cuv (Fu (ru|λu) , Fv (rv|λv) |θuv) denote the copula density for simultaneous positive amounts; 6 then p (ru, rv|πu, πv, πuv, λu, λv) = 1 −πu −πv + πuv : ru = 0, rv = 0, (πv −πuv) fv (rv|λv) : ru = 0, rv > 0, (πu −πuv) fu (ru|λu) : ru > 0, rv = 0, πuvfu (ru) fv (rv) c (Fu (ru) , Fv (rv)) : ru > 0, rv > 0. We can now define a tree-structured and tree-averaged probability distributions, pt (r) and pta (r), respectively, over the amounts: ωuv (r) = p (ru, rv|πu, πv, πuv, λu, λv) p (ru|πu, λu) p (rv|πv, λv) , pt (r|π, λ, θ, E) = "Y v∈V p (rv|πv) # Y {u,v}∈E ωuv (r) , pta (r|π, λ, θ, β) = X E∈E P (E|β) pt (r|π, λ, θ, E) = "Y v∈V p (rv|πv) # |L⋆(βω (r))| |L⋆(β)| . We employ univariate exponential distributions fv (rv) = λve−λvrv and bivariate Gaussian copulas cuv (au, av) = 1 √ 1−θ2 uv e − θ2 uvΦ−1(au)2+θ2 uvΦ−1(av)2−2θuvΦ−1(au)Φ−1(av) 2(1−θ2uv) . We applied the models to a data set collected from 30 stations from a region in Southeastern Australia (Fig. 3) 1986-2005, April-October, (20 sequences 214 30-dimensional vectors each). We used a 5-state HMM with three different types of emission distributions: tree-averaged (pta), treestructured (pt), and conditionally independent (first term of pt and pta). We will refer to these models HMM-TA, HMM-Tree, and HMM-CI, respectively. For HMM-TA, we reduced the number of free parameters by only allowing edges for stations adjacent to each other as determined by the the Delaunay triangulation (Fig. 3). We also did not learn the edge weights (β) setting them to 1 for selected edges and to 0 for the rest. To make sure that the models do not overfit, we computed their out-of-sample log-likelihood with cross-validation, leaving out one year at a time (not shown). (5 states were chosen because the leave-one-year out log-likelihood starts to flatten out for HMM-TA at 5 states.) The resulting log-likelihoods divided by the number of days and stations are −0.9392, −0.9522, and −1.0222 for HMM-TA, HMM-Tree, and HMM-CI, respectively. To see how well the models capture the properties of the data, we trained each model on the whole data set (with 50 restarts of EM), and then simulated 500 sequences of length 214. We are particularly interested in how well they measure pairwise dependence; we concentrate on two measures: log-odds ratio for occurrence and Kendall’s τ measure of concordance for pairs when both stations had positive amounts. Both are shown in Fig. 4. Both plots suggest that HMM-CI underestimates the pairwise dependence for strongly dependent pairs (as indicated by its trend to predict lower absolute values for log-odds and concordance); HMM-Tree estimating the dependence correctly mostly for strongly dependent pairs (as indicated by good prediction for high values), but underestimating it for moderate dependence; and HMM-TA performing the best for most pairs except for the ones with very strong dependence. Acknowledgements This work has been supported by the Alberta Ingenuity Fund through the AICML. We thank Stephen Charles (CSIRO, Australia) for providing us with precipitation data. References [1] M. Meil˘a and T. Jaakkola. Tractable Bayesian learning of tree belief networks. Statistics and Computing, 16(1):77–92, 2006. [2] H. Joe. Multivariate Models and Dependence Concepts, volume 73 of Monographs on Statistics and Applied Probability. Chapman & Hall/CRC, 1997. [3] R. B. Nelsen. An Introduction to Copulas. Springer Series in Statistics. Springer, 2nd edition, 2006. [4] C. K. Chow and C. N. Liu. Approximating discrete probability distributions with dependence trees. IEEE Transactions on Information Theory, IT-14(3):462–467, May 1968. [5] M. Meil˘a and M. I. Jordan. Learning with mixtures of trees. Journal of Machine Learning Research, 1(1):1–48, October 2000. 7 50 100 200 500 1000 2000 5000 10000 −3.2 −3.1 −3 −2.9 −2.8 −2.7 −2.6 Training set size Log−likelihood per feature Independent KDE Product KDE Gaussian Gaussian Copula Gaussian TCopula Frank TCopula Gaussian TACopula Figure 2: Averaged test set per-feature loglikelihood for MAGIC data: independent KDE (black solid ), product KDE (blue dashed ◦), Gaussian (brown solid ♦), Gaussian copula (orange solid +), Gaussian tree-copula (magenta dashed x), Frank tree-copula (blue dashed □), Gaussian tree-averaged copula (red solid x). 143 144 145 146 147 148 149 150 −38 −37 −36 −35 −34 −33 Longitude Latitude Coastline Stations Selected pairs Figure 3: Station map with station locations (red dots), coastline, and the pairs of stations selected according to Delaunay triangulation (dotted lines) 1 1.5 2 2.5 3 3.5 4 4.5 5 1 1.5 2 2.5 3 3.5 4 4.5 5 Log−odds from the historical data Log−odds from the simulated data HMM−TA HMM−Tree HMM−CI y=x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Kendall’s τ from the historical data Kendall’s τ from the simulated data HMM−TA HMM−Tree HMM−CI y=x Figure 4: Scatter-plots of log-odds ratios for occurrence (left) and Kendall’s τ measure of concordance (right) for all pairs of stations for the historical data vs HMM-TA (red o), HMM-Tree (blue x), and HMM-CI (green ·). [6] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via EM algorithm. Journal of the Royal Statistical Society Series B-Methodological, 39(1):1–38, 1977. [7] T. Bedford and R. M. Cooke. Vines – a new graphical model for dependent random variables. The Annals of Statistics, 30(4):1031–1068, 2002. [8] H. Joe and J.J. Xu. The estimation method of inference functions for margins for multivariate models. Technical report, Department of Statistics, University of British Columbia, 1996. [9] C. Genest, K. Ghoudi, and L.-P. Rivest. A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika, 82:543–552, 1995. [10] J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann Publishers, Inc., San Francisco, California, 1988. [11] J. Besag. Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society Series B-Methodological, 36(2):192–236, 1974. [12] S. Kirshner. Learning with tree-averaged densities and distributions. Technical Report TR 08-01, Department of Computing Science, University of Alberta, 2008. [13] A. Asuncion and D.J. Newman. UCI machine learning repository, 2007. [14] E. Bellone. Nonhomogeneous Hidden Markov Models for Downscaling Synoptic Atmospheric Patterns to Precipitation Amounts. PhD thesis, Department of Statistics, University of Washington, 2000. 8
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Variational inference for Markov jump processes Manfred Opper Department of Computer Science Technische Universit¨at Berlin D-10587 Berlin, Germany opperm@cs.tu-berlin.de Guido Sanguinetti Department of Computer Science University of Sheffield, U.K. guido@dcs.shef.ac.uk Abstract Markov jump processes play an important role in a large number of application domains. However, realistic systems are analytically intractable and they have traditionally been analysed using simulation based techniques, which do not provide a framework for statistical inference. We propose a mean field approximation to perform posterior inference and parameter estimation. The approximation allows a practical solution to the inference problem, while still retaining a good degree of accuracy. We illustrate our approach on two biologically motivated systems. Introduction Markov jump processes (MJPs) underpin our understanding of many important systems in science and technology. They provide a rigorous probabilistic framework to model the joint dynamics of groups (species) of interacting individuals, with applications ranging from information packets in a telecommunications network to epidemiology and population levels in the environment. These processes are usually non-linear and highly coupled, giving rise to non-trivial steady states (often referred to as emerging properties). Unfortunately, this also means that exact statistical inference is unfeasible and approximations must be made in the analysis of these systems. A traditional approach, which has been very successful throughout the past century, is to ignore the discrete nature of the processes and to approximate the stochastic process with a deterministic process whose behaviour is described by a system of non-linear, coupled ODEs. This approximation relies on the stochastic fluctuations being negligible compared to the average population counts. There are many important situations where this assumption is untenable: for example, stochastic fluctuations are reputed to be responsible for a number of important biological phenomena, from cell differentiation to pathogen virulence [1]. Researchers are now able to obtain accurate estimates of the number of macromolecules of a certain species within a cell [2, 3], prompting a need for practical statistical tools to handle discrete data. Sampling approaches have been extensively used to simulate the behaviour of MJPs. Gillespie’s algorithm and its generalisations [4, 5] form the basis of many simulators used in systems biology studies. The simulations can be viewed as individual samples taken from a completely specified MJP, and can be very useful to reveal possible steady states. However, it is not clear how observed data can be incorporated in a principled way, which renders this approach of limited use for posterior inference and parameter estimation. A Markov chain Monte Carlo (MCMC) approach to incorporate observations has been recently proposed by Boys et al. [6]. While this approach holds a lot of promise, it is computationally very intensive. Despite several simplifying approximations, the correlations between samples mean that several millions of MCMC iterations are needed even in simple examples. In this paper we present an alternative, deterministic approach to posterior inference and parameter estimation in MJPs. We extend the mean-field (MF) variational approach ([cf. e.g. 7]) to approximate a probability distribution over an (infinite dimensional) space of discrete paths, representing the time-evolving state of the system. In this way, we replace the couplings between the 1 different species by their average, mean-field (MF) effect. The result is an iterative algorithm that allows parameter estimation and prediction with reasonable accuracy and very contained computational costs. The rest of this paper is organised as follows: in sections 1 and 2 we review the theory of Markov jump processes and introduce our general strategy to obtain a MF approximation. In section 3 we introduce the Lotka-Volterra model which we use as an example to describe how our approach works. In section 4 we present experimental results on simulated data from the Lotka-Volterra model and from a simple gene regulatory network. Finally, we discuss the relationship of our study to other stochastic models, as well as further extensions and developments of our approach. 1 Markov jump processes We start off by establishing some notation and basic definitions. A D-dimensional discrete stochastic process is a family of D-dimensional discrete random variables x(t) indexed by the continuous time t. In our examples, the values taken by x(t) will be restricted to the non-negative integers ND 0 . The dimensionality D represents the number of (molecular) species present in the system; the components of the vector x (t) then represent the number of individuals of each species present at time t. Furthermore, the stochastic processes we will consider will always be Markovian, i.e. given any sequence of observations for the state of the system (xt1, . . . , xtN ), the conditional probability of the state of the system at a subsequent time xtN+1 depends only on the last of the previous observations. A discrete stochastic process which exhibits the Markov property is called a Markov jump process (MJP). A MJP is characterised by its process rates f (x′|x), defined ∀x′ ̸= x; in an infinitesimal time interval δt, the quantity f (x′|x) δt represents the infinitesimal probability that the system will make a transition from state x at time t to state x′ at time t + δt. Explicitly, p (x′|x) ≃δx′x + δtf (x′|x) (1) where δx′x is the Kronecker delta and the equation becomes exact in the limit δt →0. Equation (1) implies by normalisation that f (x|x) = −P x′̸=x f (x′|x). The interpretation of the process rates as infinitesimal transition probabilities highlights the simple relationship between the marginal distribution pt (x) and the process rates. The probability of finding the system in state x at time t + δt will be given by the probability that the system was already in state x at time t, minus the probability that the system was in state x at time t and jumped to state x′, plus the probability that the system was in a different state x ′′ at time t and then jumped to state x. In formulae, this is given by pt+δt (x) = pt (x) 1 − X x′̸=x f (x′|x) δt + X x′̸=x pt (x′) f (x|x′) δt. Taking the limit for δt →0 we obtain the (forward) Master equation for the marginal probabilities dpt (x) dt = X x′̸=x [−pt (x) f (x′|x) + pt (x′) f (x|x′)] . (2) 2 Variational approximate inference Let us assume that we have noisy observations yl l = 1, . . . , N of the state of the system at a discrete number of time points; the noise model is specified by a likelihood function ˆp (yl|x (tl)). We can combine this likelihood with the prior process to obtain a posterior process. As the observations happen at discrete time points, the posterior process is clearly still a Markov jump process. Given the Markovian nature of the processes, one could hope to obtain the posterior rate functions g(x′|x) by a forward-backward procedure similar to the one used for Hidden Markov Models. While this is possible in principle, the computations would require simultaneously solving a very large system of coupled linear ODEs (the number of equations is of order SD, S being the number of states accessible to the system), which is not feasible even in simple systems. 2 In the following, we will use the variational mean field (MF) approach to approximate the posterior process by a factorizing process, minimising the Kullback - Leibler (KL) divergence between processes. The inference process is then reduced to the solution of D one - dimensional Master and backward equations of size S. This is still nontrivial because the KL divergence requires the joint probabilities of variables x(t) at infinitely many different times t, i.e. probabilities over entire paths of a process rather than the simpler marginals pt(x). We will circumvent this problem by working with time discretised trajectories and then passing on to the continuum time limit. We denote such a trajectory as x0:K = (x (t0) , . . . , x (t0 + Kδt)) where δt is a small time interval and K is very large. Hence, we write the joint posterior probability as ppost(x0:K) = 1 Z pprior(x0:K) × N Y l=1 ˆp (yl|x (tl)) with pprior(x0:K) = p(x0) K−1 Y k=0 p(xk+1|xk) with Z = p(y1, . . . , yN). Note that x (tl) ∈x0:K. In the rest of this section, we will show how to compute the posterior rates and marginals by minimising the KL divergence. We notice in passing that a similar framework for continuous stochastic processes was proposed recently in [8]. 2.1 KL divergence between MJPs The KL divergence between two MJPs defined by their path probabilities p(x0:K) and q(x0:K) is KL [q, p] = X x0:K q(x0:K) ln q(x0:K) p(x0:K) = K−1 X k=0 X xk q(xk) X xk+1 q(xk+1|xk) ln q(xk+1|xk) p(xk+1|xk) + K0 and where K0 = P x0 q (x0) log q(x0) p(x0) will be set to zero in the following. We can now use equation (1) for the conditional probabilities; letting δt →0 and simultaneously K →∞so that Kδt →T, we obtain KL [q, p] = Z T 0 dt X x qt(x) X x′:x′̸=x g(x′|x) ln g(x′|x) f(x′|x) + f(x′|x) −g(x′|x) (3) where f(x′|x) and g(x′|x) are the rates of the p and q process respectively. Notice that we have swapped from the path probabilities framework to an expression that depends solely on the process rates and marginals. 2.2 MF approximation to posterior MJPs We will now consider the case where p is a posterior MJP and q is an approximating process. The prior process will be denoted as pprior and its rates will be denoted by f. The KL divergence then is KL(q, ppost) = ln Z + KL(q, pprior) − N X l=1 Eq [ln ˆp (yl|x (tl))] . To obtain a tractable inference problem, we will assume that, in the approximating process q, the joint path probability for all the species factorises into the product of path probabilities for individual species. This gives the following equations for the species probabilities and transition rates qt (x) = D Y i=1 qit (xi) gt (x′|x) = D X i=1 Y j̸=i δx′ j,xjgit (x′ i|xi) . (4) Notice that we have emphasised that the process rates for the approximating process may depend explicitly on time, even if the process rates of the original process do not. Exploiting these assumptions, we obtain that the KL divergence between the approximating process and the posterior process is given by KL [q, ppost] = ln Z − N X l=1 Eq [ln ˆp (yl|x (tl))] + Z T 0 dt X i X x qit(x) X x′:x′̸=x ( git (x′|x) ln git (x′|x) ˆfi (x′|x) + ˜fi (x′|x) −git (x′|x) ) (5) 3 where we have defined ˆfi (x′|x) = exp Ex\i[ln fi x′|x : x′ j = xj, ∀j ̸= i ] ˜fi (x′|x) = Ex\i[fi x′|x : x′ j = xj, ∀j ̸= i ] (6) and Ex\i[. . .] denotes an expectation over all components of x except xi (using the measure q). In order to find the MF approximation to the posterior process we must optimise the KL divergence (5) with respect to the marginals qit(x) and the rates git (x′|x). These, however, are not independent but fulfill the Master equation (2). We will take care of this constraint by using a Lagrange multiplier function λi(x, t) and compute the stationary values of the Lagrangian L =KL (q, ppost) − X i Z T 0 dt X x λi (x, t) ∂tqit (x) − X x′̸=x {git (x|x′) qit (x′) −git (x′|x) qit (x)} . (7) We can now compute functional derivatives of (7) to obtain δL δqit(x) = X x′̸=x " git (x′|x) ln git (x′|x) ˆfi (x′|x) −git (x′|x) + ˜fi (x′|x) # + ∂tλi (x, t) + X x′ git (x′|x) {λi (x′, t) −λi (x, t)} − X l ln ˆp (yl|x (t)) δ (t −tl) = 0 (8) δL δgit (x′|x) = qit (x) ln git (x′|x) ˆfi (x′|x) + λi (x′, t) −λi (x, t) ! = 0 (9) Defining ri(x, t) = e−λi(x,t) and inserting (9) into (8), we arrive at the linear differential equation dri(x, t) dt = X x′̸=x ˜fi (x′|x) ri (x, t) −ˆfi (x′|x) ri (x′, t) (10) valid for all times outside of the observations. To include the observations, we assume for simplicity that the noise model factorises across the species, so that ˆp (yl|x(t)) = Q i ˆpi (yil|xi(tl)) ∀l. Then equation (8) yields lim t→t− l ri (x, t) = ˆpi (yil|xi(tl)) lim t→t+ l ri (x, t) . We can then optimise the Lagrangian (7) using an iterative strategy. Starting with an initial guess for qt(x) and selecting a species i, we can compute ˆfi (x′|x) and ˜fi (x′|x). Using these, we can solve equation (10) backwards starting from the condition ri (x, T) = 1∀x (i.e., the constraint becomes void at the end of the time under consideration). This allows us to update our estimate of the rates git (x′|x) using equation (9), which can then be used to solve the master equation (2) and update our guess of qit(x). This procedure can be followed sequentially for all the species; as each step leads to a decrease in the value of the Lagrangian, this guarantees that the algorithm will converge to a (local) minimum. 2.3 Parameter estimation Since KL [q, ppost] ≥0, we obtain as useful by-product of the MF approximation a tractable variational lower bound on the log - likelihood of the data log Z = log p(y1, . . . , yN) from (5). As usual [e.g 7] such a bound can be used in order to optimise model parameters using a variational E-M algorithm. 4 3 Example: the Lotka-Volterra process The Lotka-Volterra (LV) process is often used as perhaps the simplest non-trivial MJP [6, 4]. Introduced independently by Alfred J. Lotka in 1925 and Vito Volterra in 1926, it describe the dynamics of a population composed of two interacting species, traditionally referred to as predator and prey. The process rates for the LV system are given by fprey (x + 1|x, y) = αx fprey (x −1|x, y) = βxy fpredator (y + 1|x, y) = δxy fprey (y −1|x, y) = γy (11) where x is the number of preys and y is the number of predators. All other rates are zero: individuals can only be created or destroyed one at the time. Rate sparsity is a characteristic of very many processes, including all chemical kinetic processes (indeed, the LV model can be interpreted as a chemical kinetic model). An immediate difficulty in implementing our strategy is that some of the process rates are identically zero when one of the species is extinct (i.e. its numbers have reached zero); this will lead to infinities when computing the expectation of the logarithm of the rates in equation (6). To avoid this, we will “regularise” the process by adding a small constant to the f(1|0); it can be proved that on average over the data generating process the variational approximation to the regularised process still optimises a bound analogous to (3) on the original process [9]. The variational estimates for the parameters of the LV process are obtained by inserting the process rates (11) into the MF bound and taking derivatives w.r.t. the parameters. Setting them to zero, we obtain a set of fixed point equations α = R T 0 ⟨gpreyt (x + 1|x)⟩preyt R T 0 dt ⟨x⟩preyt , β = R T 0 ⟨gpreyt (x −1|x)⟩preyt R T 0 dt ⟨x⟩preyt ⟨y⟩predatort , γ = R T 0 ⟨gpredatort (y −1|y)⟩predatort R T 0 dt ⟨y⟩predatort , δ = R T 0 ⟨gpredatort (y + 1|y)⟩predatort R T 0 dt ⟨y⟩predatort ⟨x⟩preyt . (12) Equations (12) have an appealing intuitive meaning in terms of the physics of the process: for example, α is given by the average total increase rate of the approximating process divided by the average total number of preys. We generated 15 counts of predator and prey numbers at regular intervals from a LV process with parameters α = 5 × 10−4, β = 1 × 10−4, γ = 5 × 10−4 and δ = 1 × 10−4, starting from initial population levels of seven predators and nineteen preys. These counts were then corrupted according to the following noise model ˆpi (yil|xi (tl)) ∝ 1 2|yil−xi(tl)| + 10−6 , (13) where xi (tl) is the (discrete) count for species i at time tl before the addition of noise. Notice that, since population numbers are constrained to be positive, the noise model is not symmetric. The original count is placed at the mode, rather than the mean, of the noise model. This asymmetry is unavoidable when dealing with quantities that are constrained positive. While in theory each species can have an arbitrarily large number of individuals, in order to solve the differential equations (2) and (10) we have to truncate the process. While the truncation threshold could be viewed as another parameter and optimised variationally, in these experiments we took a more heuristic approach and limited the maximum number of individuals of each species to 200. This was justified by considering that an exponential growth pattern fitted to the available data led to an estimate of approximately 90 individuals in the most abundant species, well below the truncation threshold. The results of the inference are shown in Figure 1. The solid line is the mean of the approximating distribution, the dashed lines are the 90% confidence intervals, the dotted line is the true path from which the data was obtained. The diamonds are the noisy observations. The parameter values inferred are reasonably close to the real parameter values: α = 1.35 × 10−3, β = 2.32 × 10−4, 5 0 500 1000 1500 2000 2500 3000 0 5 10 15 20 25 (a) t y 0 500 1000 1500 2000 2500 3000 0 5 10 15 20 25 x t (b) Figure 1: MF approximation to posterior LV process: (a) predator population and (b) prey population. Diamonds are the (noisy) observed data points, solid line the mean, dashed lines 90% confidence intervals, dotted lines the true path from which the data was sampled. γ = 1.57 × 10−3 and δ = 1.78 × 10−4. While the process is well approximated in the area where data is present, the free-form prediction is less good, especially for the predator population. This might be due to the inaccuracies in the estimates of the parameters. The approximate posterior displays nontrivial emerging properties: for example, we predict that there is a 10% chance that the prey population will become extinct at the end of the period of interest. These results were obtained in approximately fifteen minutes on an Intel Pentium M 1.7GHz laptop computer. To check the reliability of our inference results and the rate with which the estimated parameter values converge to the true values, we repeated our experiments for 5, 10, 15 and 20 available data points. For each sample size, we drew five independent samples from the same LV process. Figure 2(a) shows the average and standard deviation of the mean squared error (MSE) in the estimate of the parameters as a function of the number of observations N; as expected, this decreases uniformly with the sample size. 4 Example: gene autoregulatory network As a second example we consider a gene autoregulatory network. This simple network motif is one of the most important building blocks of the transcriptional regulatory networks found in cells because of its ability to increase robustness in the face of fluctuation in external signals [10]. Because of this, it is one of the best studied systems, both at the experimental and at the modelling level [11, 3]. The system consists again of two species, mRNA and protein; the process rates are given by fRNA(x + 1|x, y) = α (1 −0.99 × Θ (y −yc)) fRNA (x −1|x, y) = βx fp (y + 1|x, y) = γx fp (y −1|x, y) = δy (14) where Θ is the Heavyside step function, y the protein number and x the mRNA number. The intuitive meaning of these equations is simple: both protein and mRNA decay exponentially. Proteins are produced through translation of mRNA with a rate proportional to the mRNA abundance. On the other hand, mRNA production depends on protein concentration levels through a logical function: as soon as protein numbers increase beyond a certain critical parameter yc, mRNA production drops dramatically by a factor 100. The optimisation of the variational bound w.r.t. the parameters α, β, γ and δ is straightforward and yields fixed point equations similar to the ones for the LV process. The dependence of the MF bound on the critical parameter yc is less straightforward and is given by Lyc = const + ( 2 Z T 0 dt¯gh (yc) + log " 1 −0.99 1 T Z T 0 h (yc) dt # Z T 0 dt¯g ) (15) where ¯g = ⟨gRNA (x + 1|x)⟩qRNA and h (yc) = P y≥yc qp (y). A plot of this function obtained during the inference task below can be seen in Figure 2(b). We can determine the minimum of (15) by searching over the possible (discrete) values of yc. 6 4 6 8 10 12 14 16 18 20 22 0.5 1 1.5 2 x 10 −4 (a) N MSE 0 20 40 60 80 100 120 140 160 180 200 −0.5 0 0.5 1 1.5 2 2.5 L yc (b) Figure 2: (a) Mean squared error (MSE) in the estimate of the parameters as a function of the number of observations N for the LV process. (b) Negative variational likelihood bound for the gene autoregulatory network as a function of the critical parameter yc. 0 500 1000 1500 2000 14 16 18 20 22 24 26 28 30 (a) t y 0 500 1000 1500 2000 9 10 11 12 13 14 15 16 17 18 x t (b) Figure 3: MF approximation to posterior autoregulatory network process: (a) protein population and (b) mRNA population. Diamonds are the (noisy) observed data points, solid line the mean, dashed lines 90% confidence intervals, dotted lines the true path from which the data was taken. Again, we generated data by simulating the process with parameter values yc = 20, α = 2 × 10−3, β = 6 × 10−5, γ = 5 × 10−4 and δ = 7 × 10−5. Fifteen counts were generated for both mRNA and proteins, with initial count of 17 protein and 12 mRNA molecules. These were then corrupted with noise generated from the distribution shown in equation (13). The results of the approximate posterior inference are shown in Figure 3. The inferred parameter values are in good agreement with the true values: yc = 19, α = 2.20 × 10−3, β = 1.84 × 10−5, γ = 4.01 × 10−4 and δ = 1.54 × 10−4. Interestingly, if the data is such that the protein count never exceeds the critical parameter yc, this becomes unidentifiable (the likelihood bound is optimised by yc = ∞or yc = 0), as may be expected. The likelihood bound loses its sharp optimum evident from Figure 2(b) (results not shown). 5 Discussion In this contribution we have shown how a MF approximation can be used to perform posterior inference in MJPs from discretely observed noisy data. The MF approximation has been shown to perform well and to retain much of the richness of these complex systems. The proposed approach is conceptually very different from existing MCMC approaches [6]. While these focus on sampling from the distribution of reactions happening in a small interval in time, we compute an approximation to the probability distribution over possible paths of the system. This allows us to easily factorise across species; by contrast, sampling the number of reactions happening in a certain time 7 interval is difficult, and not amenable to simple techniques such as Gibbs sampling. While it is possible that future developments will lead to more efficient sampling strategies, our approach outstrips current MCMC based methods in terms of computational efficiency, A further strength of our approach is the ease with which it can be scaled to more complex systems involving larger numbers of species. The factorisation assumption implies that the computational complexity grows linearly in the number of species D; it is unclear how MCMC would scale to larger systems. An alternative suggestion, proposed in [11], was somehow to seek a middle way between a MJP and a deterministic, ODE based approach by approximating the MJP with a continuous stochastic process, i.e. by using a diffusion approximation. While these authors show that this approximation works reasonably well for inference purposes, it is worth pointing out that the population sizes in their experimental results were approximately one order of magnitude larger than in ours. It is arguable that a diffusion approximation might be suitable for population sizes as low as a few hundreds, but it cannot be expected to be reasonable for population sizes of the order of 10. The availability of a practical tool for statistical inference in MJPs opens a number of important possible developments for modelling. It would be of interest, for example, to develop mixed models where one species with low counts interacts with another species with high counts that can be modelled using a deterministic or diffusion approximation. This situation would be of particular importance for biological applications, where different proteins can have very different copy numbers in a cell but still be equally important. Another interesting extension is the possibility of introducing a spatial dimension which influences how likely interactions are. Such an extension would be very important, for example, in an epidemiological study. All of these extensions rely centrally on the possibility of estimating posterior probabilities, and we expect that the availability of a practical tool for the inference task will be very useful to facilitate this. References [1] Harley H. McAdams and Adam Arkin. Stochastic mechanisms in gene expression. Proceedings of the National Academy of Sciences USA, 94:814–819, 1997. [2] Long Cai, Nir Friedman, and X. Sunney Xie. Stochastic protein expression in individual cells at the single molecule level. Nature, 440:580–586, 2006. [3] Yoshito Masamizu, Toshiyuki Ohtsuka, Yoshiki Takashima, Hiroki Nagahara, Yoshiko Takenaka, Kenichi Yoshikawa, Hitoshi Okamura, and Ryoichiro Kageyama. Real-time imaging of the somite segmentation clock: revelation of unstable oscillators in the individual presomitic mesoderm cells. Proceedings of the National Academy of Sciences USA, 103:1313–1318, 2006. [4] Daniel T. Gillespie. Exact stochastic simulation of coupled chemical reactions. Journal of Physical Chemistry, 81(25):2340–2361, 1977. [5] Eric Mjolsness and Guy Yosiphon. Stochastic process semantics for dynamical grammars. to appear in Annals of Mathematics and Artificial Intelligence, 2006. [6] Richard J. Boys, Darren J. Wilkinson, and Thomas B. L. Kirkwood. Bayesian inference for a discretely observed stochastic kinetic model. available from http://www.staff.ncl.ac.uk/d.j.wilkinson/pub.html, 2004. [7] Manfred Opper and David Saad (editors). Advanced Mean Field Methods. MIT press, Cambridge,MA, 2001. [8] Cedric Archambeau, Dan Cornford, Manfred Opper, and John Shawe-Taylor. Gaussian process approximations of stochastic differential equations. Journal of Machine Learning Research Workshop and Conference Proceedings, 1(1):1–16, 2007. [9] Manfred Opper and David Haussler. Bounds for predictive errors in the statistical mechanics of supervised learning. Physical Review Letters, 75:3772–3775, 1995. [10] Uri Alon. An introduction to systems biology. Chapman and Hall, London, 2006. [11] Andrew Golightly and Darren J. Wilkinson. Bayesian inference for stochastic kinetic models using a diffusion approximation. Biometrics, 61(3):781–788, 2005. 8
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Expectation Maximization and Posterior Constraints Jo˜ao V. Grac¸a L2F INESC-ID INESC-ID Lisboa, Portugal Kuzman Ganchev Computer & Information Science University of Pennsylvania Philadelphia, PA Ben Taskar Computer & Information Science University of Pennsylvania Philadelphia, PA Abstract The expectation maximization (EM) algorithm is a widely used maximum likelihood estimation procedure for statistical models when the values of some of the variables in the model are not observed. Very often, however, our aim is primarily to find a model that assigns values to the latent variables that have intended meaning for our data and maximizing expected likelihood only sometimes accomplishes this. Unfortunately, it is typically difficult to add even simple a-priori information about latent variables in graphical models without making the models overly complex or intractable. In this paper, we present an efficient, principled way to inject rich constraints on the posteriors of latent variables into the EM algorithm. Our method can be used to learn tractable graphical models that satisfy additional, otherwise intractable constraints. Focusing on clustering and the alignment problem for statistical machine translation, we show that simple, intuitive posterior constraints can greatly improve the performance over standard baselines and be competitive with more complex, intractable models. 1 Introduction In unsupervised problems where observed data has sequential, recursive, spatial, relational, or other kinds of structure, we often employ statistical models with latent variables to tease apart the underlying dependencies and induce meaningful semantic parts. Part-of-speech and grammar induction, word and phrase alignment for statistical machine translation in natural language processing are examples of such aims. Generative models (graphical models, grammars, etc.) estimated via EM [6] are one of the primary tools for such tasks. The EM algorithm attempts to maximize the likelihood of the observed data marginalizing over the hidden variables. A pernicious problem with most models is that the data likelihood is not convex in the model parameters and EM can get stuck in local optima with very different latent variable posteriors. Another problem is that data likelihood may not guide the model towards the intended meaning for the latent variables, instead focusing on explaining irrelevant but common correlations in the data. Very indirect methods such as clever initialization and feature design (as well as ad-hoc procedural modifications) are often used to affect the posteriors of latent variables in a desired manner. By allowing to specify prior information directly about posteriors of hidden variables, we can help avoid these difficulties. A somewhat similar in spirit approach is evident in work on multivariate information bottleneck [8], where extra conditional independence assumptions between latent variables can be imposed to control their “meaning”. Similarly, in many semisupervised approaches, assumptions about smoothness or other properties of the posteriors are often used as regularization [18, 13, 4]. In [17], deterministic annealing was used to to explicitly control a particular feature of the posteriors of a grammar induction model. In this paper, we present an approach that effectively incorporates rich constraints on posterior distributions of a graphical model into a simple and efficient EM scheme. An important advantage of our approach is that the E-step remains tractable in a large class of problems even though incorporating the desired constraints directly into the model would make it intractable. We test our approach on synthetic clustering data as well as statistical 1 word alignment and show that we can significantly improve the performance of simple, tractable models, as evaluated on hand-annotated alignments for two pairs of languages, by introducing intuitive constraints such as limited fertility and the agreement of two models. Our method is attractive in its simplicity and efficiency and is competitive with more complex, intractable models. 2 Expectation Maximization and posterior constraints We are interested in estimating the parameters θ of a model pθ(x, z) over observed variables X taking values x ∈X and latent variables Z taking values z ∈Z. We are often even more interested in the induced posterior distribution over the latent variables, pθ(z | x), as we ascribe domainspecific semantics to these variables. We typically represent pθ(x, z) as a directed or undirected graphical model (although the discussion below also applies to context free grammars and other probabilistic models). We assume that computing the joint and the marginals is tractable and that the model factors across cliques as follows: pθ(x, z) ∝Q α φθ(xα, zα), where φθ(xα, zα) are clique potentials or conditional probability distributions. Given a sample S = {x1, . . . , xn}, EM maximizes the average log likelihood function LS(θ) via an auxiliary lower bound F(q, θ) (cf. [14]): LS(θ) = ES[log pθ(x)] = ES " log X z pθ(x, z) # = ES " log X z q(z | x)pθ(x, z) q(z | x) # (1) ≥ ES "X z q(z | x) log pθ(x, z) q(z | x) # = F(q, θ), (2) where ES[f(x)] = 1 n P i f(xi) denotes the sample average and q(z | x) is non-negative and sums to 1 over z for each x. The lower bound above is a simple consequence of Jensen’s inequality for the log function. It can be shown that the lower bound can be made tight for a given value of θ by maximizing over q and under mild continuity conditions on pθ(x, z), local maxima (q∗, θ∗) of F(q, θ) correspond to local maxima θ∗of LS(θ) [14]. Standard EM iteration performs coordinate ascent on F(q, θ) as follows: E: qt+1(z | x) = arg max q(z|x) F(q, θt) = arg min q(z|x) KL(q(z | x) || pθt(z | x)) = pθt(z | x); (3) M: θt+1 = arg max θ F(qt+1, θ) = arg max θ ES "X z qt+1(z | x) log pθ(x, z) # , (4) where KL(q||p) = Eq[log q(·) p(·)] is Kullback-Leibler divergence. The E step computes the posteriors of the latent variables given the observed variables and current parameters. The M step uses q to “fill in” the values of latent variables z and estimate parameters θ as if the data was complete. This step is particularly easy for exponential models, where θ is a simple function of the (expected) sufficient statistics. This modular split into two intuitive and straightforward steps accounts for the vast popularity of EM. In the following, we build on this simple scheme while incorporating desired constraints on the posteriors over latent variables. 2.1 Constraining the posteriors Our goal is to allow for finer-level control over posteriors, bypassing the likelihood function. We propose an intuitive way to modify EM to accomplish this and discuss the implications of the new procedure below in terms of the objective it attempts to optimize. We can express our desired constraints on the posteriors as the requirement that pθ(z | x) ∈Q(x). For example, in dependency grammar induction, constraining the average length of dependency attachments is desired [17]; in statistical word alignment, the constraint might involve the expected degree of each node in the alignment [3]. Instead of restricting p directly, which might not be feasible, we can penalize the distance of p to the constraint set Q. As it turns out, we can accomplish this by restricting q to be constrained to Q instead. This results in a very simple modification to the E step of EM, by constraining the set of q over which F(q, θ) is optimized (M step is unchanged): E: qt+1(z | x) = arg max q(z|x)∈Q(x) F(q, θt) = arg min q(z|x)∈Q(x) KL(q(z | x) || pθt(z | x)) (5) 2 Note that in variational EM, the set Q(x) is usually a simpler inner bound (as in mean field) or outer bound (as in loopy belief propagation) on the intractable original space of posteriors [9]. The situation here is the opposite: we assume the original posterior space is tractable but we add constraints to enforce intended semantics not captured by the simple model. Of course to make this practical, the set Q(x) needs to be well-behaved. We assume that Q(x) is convex and non-empty for every x so that the problem in Eq. (5) becomes a strictly convex minimization over a non-empty convex set, guaranteed to have a unique minimizer [1]. A natural and general way to specify constraints on q is by bounding expectations of given functions: Eq[f(x, z)] ≤b (equality can be achieved by adding Eq[−f(x, z)] ≤−b). Stacking functions f() into a vector f() and constants b into a vector b, the minimization problem in Eq. (5) becomes: arg min q KL(q(z | x) || pθt(z | x)) s.t. Eq[f(x, z)] ≤b. (6) In the next section, we discuss how to solve this optimization problem (also called I-projection in information geometry), but before we move on, it is interesting to consider what this new procedure in Eq. (5) converges to. The new scheme alternately maximizes F(q, θ), but over a subspace of the original space of q, hence using a looser lower-bound than original EM. We are no longer guaranteed that the local maxima of the constrained problem are local maxima of the log-likelihood. However, we can characterize the objective maximized at local maxima as log-likelihood penalized by average KL divergence of posteriors from Q: Proposition 2.1 The local maxima of F(q, θ) such that q(z | x) ∈Q(x), ∀x ∈S are local maxima of ES[log pθ(x)] −ES[KL(Q(x) || pθ(z | x)], where KL(Q(x) || pθ(z | x) = minq(z|x))∈Q(x) KL(q(z | x) || pθ(z | x)). Proof: By adding and subtracting ES[P z q(z | x) log pθ(z | x)] from F(q, θ), we get: F(q, θ) = ES "X z q(z | x) log pθ(x, z) q(z | x) # (7) = ES "X z q(z | x) log pθ(x, z) pθ(z | x) # −ES "X z q(z | x) log q(z | x) pθ(z | x) # (8) = ES "X z q(z | x) log pθ(x) # −ES[KL(q(z | x)||pθ(z | x)] (9) = ES[log pθ(x)] −ES[KL(q(z | x) || pθ(z | x)]. (10) Since the first term does not depend on q, the second term is minimized by q∗(z | x) = minq(z|x))∈Q(x) KL(q(z | x) || pθ(z | x)) at local maxima. This proposition implies that our procedure trades off likelihood and distance to the desired posterior subspace (modulo getting stuck in local maxima) and provides an effective method of controlling the posteriors. 2.2 Computing I-projections onto Q(x) The KL-projection onto Q(x) in Eq. (6) is easily solved via the dual (cf. [5, 1]): arg max λ≥0 λ⊤b −log X z pθt(z | x) exp{λ⊤f(x, z)} ! (11) Define qλ(z | x) ∝pθt(z | x) exp{λ⊤f(x, z)}, then at the dual optimum λ∗, the primal solution is given by qλ∗(z | x). Such projections become particularly efficient when we assume the constraint functions decompose the same way as the graphical model: f(x, z) = P α f(xα, zα). Then qλ(z | x) ∝ Q α φθt(xα, zα) exp{λ⊤f(xα, zα)}, which factorizes the same way as pθ(x, z). In case the constraint functions do not decompose over the model cliques but require additional cliques, the resulting qλ will factorize over the union of the original cliques and the constraint function cliques, 3 Initial Configuration Output of EM Output of Constrained EM Figure 1: Synthetic data results. The dataset consists of 9 points drawn as dots and there are three clusters represented by ovals centered at their mean with dimensions proportional to their standard deviation. The EM algorithm clusters each column of points together, but if we introduce the constraint that each column should have at least one of the clusters, we get the clustering to the right. 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 · · · · · · · · · 0 · · · · · · · · · 0 · · · · · · · · · it 1 · · · · · · · · · 1 · · · · · · · · · 1 · · · · · · · · · was 2 · · • · · · · · · 2 · · • · · · · · · 2 · · • · · · · · · an 3 · · · · • · · · · 3 • · · • • · · · · 3 • · · · • · · · · animated 4 · · · · · • · · · 4 · · · · · • · · · 4 · · · · · • · · · , 5 · · • · · · • · · 5 · · · · · · • · · 5 · · · · · · • · · very 6 • · · • • • · • · 6 · · · · · · · • · 6 · · · · · · · • · convivial 7 • · · · · · · · · 7 • · · · · · · · · 7 • · · · · · · · · game 8 · · · · · · · · • 8 · · · · · · · · • 8 · · · · · · · · • . jugaban de una manera animada y muy cordial . jugaban de una manera animada y muy cordial . jugaban de una manera animada y muy cordial . Figure 2: An example of the output of HMM trained on 100k the EPPS data. Left: Baseline model. Middle: Substochastic constraints. Right Agreement constraints. potentially making inference more expensive. In our experiments, we used constraint functions that decompose with the original model. Note that even in this case, the graphical model pθ(x, z) can not in general satisfy the expectation constraints for every setting of θ and x. Instead, the constrained EM procedure is tuning θ to the distribution of x to satisfy these constraints in expectation. The dual of the projection problem can be solved using a variety of optimization methods; perhaps the simplest of them is projected gradient (since λ is non-negative, we need to simply truncate negative values as we perform gradient ascent). The gradient of the objective in Eq. (11) is given by: b −Eqλ[f(x, z)] = b −P α Eqλ(zα|x)[f(xα, zα)]. Every gradient computation thus involves computing marginals of qλ(z | x), which is of the same complexity as computing marginals of pθ(z | x) if no new cliques are added by the constraint functions. In practice, we do not need to solve the dual to a very high precision in every round of EM, so several (about 5-10) gradient steps suffice. When the number of constraints is small, alternating projections are also a good option. 3 Clustering A simple but common problem that employs EM is clustering a group of points using a mixture of Gaussians. In practice, the data points and Gaussian clusters have some meaning not captured by the model. For example, the data points could correspond to descriptors of image parts and the clusters could be “words” used for later processing of the image. In that case, we often have special knowledge about the clusters that we expect to see that is difficult to express in the original model. For example, we might know that within each image two features that are of different scales should not be clustered together. As another example, we might know that each image has at least one copy of each cluster. Both of these constraints are easy to capture and implement in our framework. Let zij = 1 represent the event that data point i is assigned to cluster j. If we want to ensure that data point i is not assigned to the same cluster as data point i′ then we need to enforce the constraint E [zij + zi′j] ≤1, ∀j. To ensure the constraint that each cluster has at least one data point assigned to it from an instance I we need to enforce the constraint E P i∈I zij ≤1, ∀j. We implemented this constraint in a mixture of Gaussians clustering algorithm. Figure 1 compares clustering of synthetic data using unconstrained EM as well as our method with the constraint that each column of data points has at least one copy of each cluster in expectation. 4 4 Statistical word alignment Statistical word alignment, used primarily for machine translation, is a task where the latent variables are intended to have a meaning: whether a word in one language translates into a word in another language in the context of the given sentence pair. The input to an alignment systems is a sentence aligned bilingual corpus, consisting of pairs of sentences in two languages. Figure 2 shows three machine-generated alignments of a sentence pair. The black dots represent the machine alignments and the shading represents the human annotation. Darkly shaded squares with a border represent a sure alignments that the system is required to produce while lightly shaded squares without a border represent possible alignments that the system is optionally allowed to produce. We denote one language the “source” language and use s for its sentences and one language the “target” language and use t for its sentences. It will also be useful to talk about an alignment for a particular sentence pair as a binary matrix z, with zij = 1 representing “source word i generates target word j.” The generative models we consider generate target word j from only one source word, and so an alignment is only valid from the point of view of the model when P i zij = 1, so we can equivalently represent an alignment as an array a of indices, with aj = i ⇔zij = 1. Figure 2 shows three alignments performed by a baseline model as well as our two modifications. We see that the rare word “convivial” acts as a garbage collector[2], aligning to words that do not have a simple translation in the target sentence. Both of the constraints we suggest repair this problem to different degrees. We now introduce the baseline models and the constraints we impose on them. 4.1 Baseline models We consider three models below: IBM Model 1, IBM Model 2 [3] and the HMM model proposed by [20]. The three models can be expressed as: p(t, a | s) = Y j pd(aj|j, aj−1)pt(tj|saj), (12) with the three models differing in their definition of the distortion probability pd(aj|j, aj−1). Model 1 assumes that the positions of the words are not important and assigns uniform distortion probability. Model 2 allows a dependence on the positions pd(aj|j, aj−1) = pd(aj|j) and the HMM model assumes that the only the distance between the current and previous source word are important pd(aj|j, aj−1) = pd(aj|aj −aj−1). All the models are augmented by adding a special “null” word to the source sentence. The likelihood of the corpus, marginalized over possible alignments is concave for Model 1, but not for the other models [3]. 4.1.1 Substochastic Constraints A common error for our baseline models is to use rare source words as garbage collectors [2]. The models align target words that do not match any of the source words to rare source words rather than to the null word. While this results in higher data likelihood, the resulting alignments are not desirable, since they cannot be interpreted as translations. Figure 2 shows an example. One might consider augmenting the models to disallow this, for example by restricting that the alignments are at most one-to-one. Unfortunately computing the normalization for such a model is a ♯P complete problem [19]. Our approach is to instead constrain the posterior distribution over alignments during the E-step. More concretely we enforce the constraint Eq[zij] ≤1. Another way of thinking of this constraint is that we require the expected fertility of each source word to be at most one. For our hand-aligned corpora Hansards [15] and EPPS [11, 10], the average fertility is around 1 and 1.2, respectively, with standard deviation of 0.01. We will see that these constraints improve alignment accuracy. 4.1.2 Agreement Constraints Another weakness of our baseline models is that they are asymmetric. Usually, a model is trained in each direction and then they are heuristically combined. [12] introduce an objective to train the two models concurrently and encourage them to agree. Unfortunately their objective leads to an intractable E-step and they are forced to use a heuristic approximation. In our framework, we can 5 Hansards 447 sentences EPPS 400 sentences Language Max Avg. Fertility Avg. F. Language Max Avg. Fertility Avg. F. English 30 15.7 6 1.02 English 90 29 218 1.20 French 30 17.4 3 1.00 Spanish 99 31.2 165 1.17 Table 1: Test Corpus statistics. Max and Avg. refer to sentence length. Fertility is the number of words that occur at least twice and have on average at least 1.5 sure alignment when they have any. Avg. F. is the average word fertility. All average fertilities have a standard deviation of 0.01. also enforce agreement in expectation without approximating. Denote one direction the “forward” direction and the other the “backward” direction. Denote the forward model −→p with hidden variables −→z ∈−→ Z and backward model ←−p with hidden variables ←−z ∈←− Z and note −→p (←−z ) = 0 and ←−p (−→z ) = 0. Define a mixture p(z) = 1 2−→p (z)+ 1 2←−p (z) for z ∈←− Z ∪−→ Z. The constraints that enforce agreement in this setup are Eq[f(x, z)] = 0 with fij(x, z) = 1 z ∈−→ Z and zij = 1 −1 z ∈←− Z and zij = 1 0 otherwise . 5 Evaluation We evaluated our augmented models on two corpora: the Hansards corpus [15] of English/French and the Europarl corpus [10] with EPPS annotation [11]. Table 1 presents some statistics for the two corpora. Notably, Hansards is a much easier corpus than EPPS. Hansards test sentences are on average only half as long as those of EPPS and only 21% of alignments in Hansards are sure and hence required compared with 69% for EPPS. Additionally, more words in EPPS are aligned to multiple words in the other language. Since our models cannot model this “fertility” we expect their performance to be worse on EPPS data. Despite these differences, the corpora are also similar in some ways. Both are alignments of a Romance language to English and the average distance of an alignment to the diagonal is around 2 for both corpora. The error metrics we use are precision, recall and alignment error rate (AER), which is a weighted combination of precision and recall. Although AER is the standard metric in word alignment is has been shown [7] that it has a weak correlation with the standard MT metric, Bleu, when the alignments are used in a phrase-based translation system. [7] suggest weighted F-Measure1 as an alternative that correlates well with Bleu, so we also report precision and recall numbers. Following prior work [16], we initialize Model 1 translation table with uniform probabilities over word pairs that occur together in same sentence. Model 2 and Model HMM were initialized with the translation probabilities from Model 1 and with uniform distortion probabilities. All models were trained for 5 iterations. We used a maximum length cutoff for training sentences of 40. For the Hansards corpus this leaves 87.3% of the sentences, while for EPPS this leaves 74.5%. Following common practice, we included the unlabeled test and development data during training. We report results for the model with English as the “source” language when using posterior decoding [12]. Figures 3 shows alignment results for the baselines models as well as the models with additional constraints. We show precision, recall and AER for the HMM model as well as precision and recall for Model 2. We note that both constraints improve all measures of performance for all dataset sizes, with most improvement for smaller dataset sizes. We performed additional experiments to verify that our model is not unfairly aided by the standard but arbitrary choice of 5 iterations of EM. Figure 4 shows AER and data likelihood as a function of the number of EM iterations. We see that the performance gap between the model with and without agreement constraints is preserved as the number of EM iterations increases. Note also that likelihood increases monotonically for all the models and that the baseline model always achieves higher likelihood as expected. 1defined as ( α P recision + 1−α Recall)−1 with 0.1 ≤α ≤0.4 showing good correlation with Bleu [7]. 6 60 65 70 75 80 85 90 95 100 0 20 40 60 80 100 Precision Recall Agreement Substochastic Baseline 60 65 70 75 80 85 90 95 100 10 100 1000 Precision Recall Agreement Substochastic Baseline 0 5 10 15 20 25 30 10 100 1000 Baseline Substochastic Agreement 30 40 50 60 70 80 90 100 0 20 40 60 80 100 Precision Recall Agreement Substochastic Baseline 30 40 50 60 70 80 90 100 10 100 1000 Precision Recall Agreement Substochastic Baseline 0 5 10 15 20 25 30 35 40 45 10 100 1000 Baseline Substochastic Agreement Figure 3: Effect of posterior constraints on learning curves for IBM Model 2 and HMM. From left to right: Precision/Recall for IBM Model 2, Precision/Recall for HMM Model and AER for HMM Model. Top: Hansards Bottom: EPPS. Both types of constraints improve all accuracy measures across both datasets and models. 2 4 6 8 10 12 14 Baseline Substochastic Agreement (a) Hansards negative log Likelihood 10 12 14 16 18 20 2 4 6 8 10 12 14 Baseline Substochastic Agreement (b) Hansards AER Figure 4: Data likelihood and AER vs. EM iteration using HMM on 100k Hansards. 6 Conclusions In this paper we described a general and principled way to introduce prior knowledge to guide the EM algorithm. Intuitively, we can view our method as a way to exert flexible control during the execution of EM. More formally, our method can be viewed as a regularization of the expectations of the hidden variables during EM. Alternatively, it can be viewed as an augmentation of the EM objective function with KL divergence from a set of feasible models. We implemented our method on two different problems: probabilistic clustering using mixtures of Gaussians and statistical word alignment and tested it on synthetic and real data. We observed improved performance by introducing simple and intuitive prior knowledge into the learning process. Our method is widely applicable to other problems where the EM algorithm is used but prior knowledge about the problem is hard to introduce directly into the model. 7 Acknowledgments J. V. Grac¸a was supported by a fellowship from Fundac¸˜ao para a Ciˆencia e Tecnologia (SFRH/ BD/ 27528/ 2006). K. Ganchev was partially supported by NSF ITR EIA 0205448. References [1] D. Bertsekas. Nonlinear Programming. Athena Scientific, Belmont, MA, 1999. [2] P. F. Brown, S. A. Della Pietra, V. J. Della Pietra, M. J. Goldsmith, J. Hajic, R. L. Mercer, and S. Mohanty. But dictionaries are data too. In Proc. HLT, 1993. [3] Peter F. Brown, Stephen Della Pietra, Vincent J. Della Pietra, and Robert L. Mercer. The mathematic of statistical machine translation: Parameter estimation. Computational Linguistics, 19(2):263–311, 1994. [4] O. Chapelle, B. Sch¨olkopf, and A. Zien, editors. Semi-Supervised Learning. MIT Press, Cambridge, MA, 2006. [5] I. Csiszar. I-divergence geometry of probability distributions and minimization problems. The Annals of Probability, 3, 1975. [6] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the em algorithm. Journal of the Royal Statistical Society. Series B (Methodological), 39(1):1–38, 1977. [7] Alexander Fraser and Daniel Marcu. Measuring word alignment quality for statistical machine translation. Comput. Linguist., 33(3):293–303, 2007. [8] Nir Friedman, Ori Mosenzon, Noam Slonim, and Naftali Tishby. Multivariate information bottleneck. In UAI, 2001. [9] Michael I. Jordan, Zoubin Ghahramani, Tommi Jaakkola, and Lawrence K. Saul. An introduction to variational methods for graphical models. Machine Learning, 37(2):183–233, 1999. [10] Philipp Koehn. Europarl: A multilingual corpus for evaluation of machine translation, 2002. [11] P. Lambert, A.De Gispert, R. Banchs, and J. B. Mari˜no. Guidelines for word alignment evaluation and manual alignment. In Language Resources and Evaluation, Volume 39, Number 4, pages 267–285, 2005. [12] Percy Liang, Ben Taskar, and Dan Klein. Alignment by agreement. In Proc. HLT-NAACL, 2006. [13] Gideon S. Mann and Andrew McCallum. Simple, robust, scalable semi-supervised learning via expectation regularization. In Proc. ICML, 2007. [14] R. M. Neal and G. E. Hinton. A new view of the EM algorithm that justifies incremental, sparse and other variants. In M. I. Jordan, editor, Learning in Graphical Models, pages 355– 368. Kluwer, 1998. [15] Franz Josef Och and Hermann Ney. Improved statistical alignment models. In ACL, 2000. [16] Franz Josef Och and Hermann Ney. A systematic comparison of various statistical alignment models. Comput. Linguist., 29(1):19–51, 2003. [17] Noah A. Smith and Jason Eisner. Annealing structural bias in multilingual weighted grammar induction. In Proc. ACL, pages 569–576, 2006. [18] Martin Szummer and Tommi Jaakkola. Information regularization with partially labeled data. In Proc. NIPS, pages 1025–1032, 2003. [19] L. G. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8:189–201, 1979. [20] Stephan Vogel, Hermann Ney, and Christoph Tillmann. Hmm-based word alignment in statistical translation. In Proc. COLING, 1996. 8
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Unsupervised Feature Selection for Accurate Recommendation of High-Dimensional Image Data Sabri Boutemedjet DI, Universite de Sherbrooke 2500 boulevard de l’Universit´e Sherbrooke, QC J1K 2R1, Canada sabri.boutemedjet@usherbrooke.ca Djemel Ziou DI, Universite de Sherbrooke 2500 boulevard de l’Universit´e Sherbrooke, QC J1K 2R1, Canada djemel.ziou@usherbrooke.ca Nizar Bouguila CIISE, Concordia University 1515 Ste-Catherine Street West Montreal, QC H3G 1T7, Canada bouguila@ciise.concordia.ca Abstract Content-based image suggestion (CBIS) targets the recommendation of products based on user preferences on the visual content of images. In this paper, we motivate both feature selection and model order identification as two key issues for a successful CBIS. We propose a generative model in which the visual features and users are clustered into separate classes. We identify the number of both user and image classes with the simultaneous selection of relevant visual features using the message length approach. The goal is to ensure an accurate prediction of ratings for multidimensional non-Gaussian and continuous image descriptors. Experiments on a collected data have demonstrated the merits of our approach. 1 Introduction Products in today’s e-market are described using both visual and textual information. From consumer psychology, the visual information has been recognized as an important factor that influences the consumer’s decision making and has an important power of persuasion [4]. Furthermore, it is well recognized that the consumer choice is also influenced by the external environment or context such as the time and location [4]. For example, a consumer could express an information need during a travel that is different from the situation when she or he is working or even at home. “Content-Based Image Suggestion” (CBIS) [4] motivates the modeling of user preferences with respect to visual information under the influence of the context. Therefore, CBIS aims at the suggestion of products whose relevance is inferred from the history of users in different contexts on images of the previously consumed products. The domains considered by CBIS are a set of users U = {1, 2, . . ., Nu}, a set of visual documents V = {⃗v1,⃗v2, . . . ,⃗vNv}, and a set of possible contexts E = {1, 2, . . ., Ne}. Each ⃗vk is an arbitrary descriptor (visual, textual, or categorical) used to represent images or products. In this work, we consider an image as a D-dimensional vector ⃗v = (v1, v2, . . . , vD). The visual features may be local such as interest points or global such as color, texture, or shape. The relevance is expressed explicitly on an ordered voting (or rating) scale defined as R = {r1, r2, . . . , rNr}. For example, the five star scale (i.e. Nr = 5) used by Amazon allows consumers to give different degrees of appreciation. The history of each user u ∈U, is defined as Du = {< u, e(j),⃗v(j), r(j) > |e(j) ∈E,⃗v(j) ∈V, r(j) ∈R, j = 1, . . . , |Du|}. Figure 1: The VCC-FMM identifies like-mindedness from similar appreciations on similar images represented in 3-dimensional space. Notice the inter-relation between the number of image clusters and the considered feature subset. In literature, the modeling of user preferences has been addressed mainly within collaborative filtering (CF) and content-based filtering (CBF) communities. On the one hand, CBF approaches [12] build a separate model of “liked” and “disliked” discrete data (word features) from each D u taken individually. On the other hand, CF approaches predict the relevance of a given product for a given user based on the preferences provided by a set of “like-minded” (similar tastes) users. The data set used by CF is the user-product matrix (∪Nu u=1Du) which is discrete since each product is represented by a categorical index. The Aspect model [7] and the flexible mixture model (FMM) [15] are examples of some model-based CF approaches. Recently, the authors in [4] have proposed a statistical model for CBIS which uses both visual and contextual information in modeling user preferences with respect to multidimensional non Gaussian and continuous data. Users with similar preferences are considered in [4] as those who appreciated with similar degrees similar images. Therefore, instead of considering products as categorical variables (CF), visual documents are represented by a richer visual information in the form of a vector of visual features (texture, shape, and interest points). The similarity between images and between user preferences is modeled in [4] through a single graphical model which clusters users and images separately into homogeneous groups in a similar way to the flexible mixture model (FMM) [15]. In addition, since image data are generally non-Gaussian [1], class-conditional distributions of visual features are assumed Dirichlet densities. By this way, the like-mindedness in user preferences is captured at the level of visual features. Statistical models for CBIS are useful tools in modeling for many reasons. First, once the model is learned from training data (union of user histories), it can be used to “suggest” unknown (possibly unrated) images efficiently i.e. few effort is required at the prediction phase. Second, the model can be updated from new data (images or ratings) in an online fashion in order to handle the changes in either image clusters and/or user preferences. Third, model selection approaches can be employed to identify “without supervision” both numbers of user preferences and image clusters (i.e. model order) from the statistical properties of the data. It should be stressed that the unsupervised selection of the model order was not addressed in CF/CBF literature. Indeed, the model order in many wellfounded statistical models such as the Aspect model [7] or FMM [15] was set “empirically” as a compromise between the model’s complexity and the accuracy of prediction, but not from the data. From an “image collection modeling” point of view, the work in [4] has focused on modeling user preferences with respect to non-Gaussian image data. However, since CBIS employs generally highdimensional image descriptors, then the problem of modeling accurately image collections needs to be addressed in order to overcome the curse of dimensionality and provide accurate suggestions. Indeed, the presence of many irrelevant features degrades substantially the performance of the modeling and prediction [6] in addition to the increase of the computational complexity. To achieve a better modeling, we consider feature selection and extraction as another “key issue” for CBIS. In literature [6], the process of feature selection in mixture models have not received as much attention as in supervised learning. The main reason is the absence of class labels that may guide the selection process [6]. In this paper, we address the issue of feature selection in CBIS through a new generative model which we call Visual Content Context-aware Flexible Mixture Model (VCC-FMM). Due to the problem of the inter-relation between feature subsets and the model order i.e. different feature subsets correspond to different natural groupings of images, we propose to learn the VCC-FMM from unlabeled data using the Minimum Message Length (MML) approach [16]. The next Section details the VCC-FMM model with an integrated feature selection. After that, we discuss the identification of the model order using the MML approach in Section 3. Experimental results are presented in Section 4. Finally, we conclude this paper by a summary of the work. 2 The Visual Content Context Flexible Mixture Model The data set D used to learn a CBIS system is the union of all user histories i.e. D = ∪u∈UDu. From this data set we model both like-mindedness shared by user groups as well as the visual and semantic similarity between images [4]. For that end, we introduce two latent variables z and c to label each observation < u, e,⃗v, r > with information about user classes and image classes, respectively. In order to make predictions on unseen images, we need to model the joint event p(⃗v, r, u, e) = z,c p(⃗v, r, u, e, z, c). Then, the rating r for a given user u, context e and a visual document ⃗v can be predicted on the basis of probabilities p(r|u, e, v) that can be derived by conditioning the generative model p(u, e, v, r). We notice that the full factorization of p(⃗v, r, u, e, z, c) using the chain rule leads to quantities with a huge number of parameters which are difficult to interpret in terms of the data [4]. To overcome this problem, we make use of some conditional independence assumptions that constitute our statistical approximation of the joint event p(⃗v, r, u, e). These assumptions are illustrated by the graphical representation of the model in figure 2. Let K and M be the number of user classes and images classes respectively, an initial model for CBIS can be derived as [4]: p(⃗v, r, u, e) = K z=1 M c=1 p(z)p(c)p(u|z)p(e|z)p(⃗v|c)p(r|z, c) (1) The quantities p(z) and p(c) denote the a priori weights of user and image classes. p(u|z) and p(e|z) denote the likelihood of a user and context to belong respectively to the user’s class z. p(r|z, c) is the probability to sample a rating for a given user class and image class. All these quantities are modeled from discrete data. On the other hand, image descriptors are high-dimensional, continuous and generally non Gaussian data [1]. Thus, the distribution of class-conditional densities p(⃗v|c) should be modeled carefully in order to capture efficiently the added-value of the visual information. In this work, we assume that p(⃗v|c) is a Generalized Dirichlet distribution (GDD) which is more appropriate than other distributions such as the Gaussian or Dirichlet distributions in modeling image collections [1]. This distribution has a more general covariance structure and provides multiple shapes. The distribution of the c-th component ⃗Θ∗ c is given by equation (2). The ∗superscript is used to denote the unknown true GDD distribution. p(⃗v|⃗Θ∗ c) = D l=1 Γ(α∗ cl + β∗ cl) Γ(α∗ cl)Γ(β∗ cl)v α∗ cl−1 l (1 − l k=1 vk)γ∗ cl (2) where D l=l vl < 1 and 0 < vl < 1 for l = 1, . . . , D. γ∗ cl = β∗ cl−α∗ cl+1−β∗ cl+1 for l = 1, . . . , D−1 and γ∗ D = β∗ D −1. In equation (2) we have set ⃗Θ∗ c = (α∗ c1, β∗ c1, . . . , α∗ cD, β∗ cD). From the mathematical properties of the GDD, we can transform using a geometric transformation the data point Figure 2: Graphical representation of VCC-FMM. ⃗v into another data point ⃗x = (x1, . . . , xD) with independent features without loss of information [1]. In addition, each xl of ⃗x generated by the c-th component, follows a Beta distribution p b(.|θ∗ cl) with parameters θ∗ cl = (α∗ cl, β∗ cl) which leads to the fact p(⃗x|⃗Θ∗ c) = D l=1 pb(xl|θ∗ cl). The independence between xl makes the estimation of a GDD very efficient i.e. D estimations of univariate Beta distributions without loss of accuracy. However, even with independent features, the unsupervised identification of image clusters based on high-dimensional descriptors remains a hard problem due to the omnipresence of noisy, redundant and uninformative features [6] that degrade the accuracy of the modeling and prediction. We consider feature selection and extraction as a “key” methodology in order to remove that kind of features in our modeling. Since x l are independent, then we can extract “relevant” features in the representation space X. However, we need some definition of feature’s relevance. From figure 1, four well-separated image clusters can be identified from only two relevant features 1 and 2 which are multimodal and influenced by class labels. On the other hand, feature 3 is unimodal (i.e. irrelevant) and can be approximated by a single Beta distribution pb(.|ξl) common to all components. This definition of feature’s relevance has been motivated in unsupervised learning [2][9]. Let ⃗φ = (φ1, . . . , φD) be a set of missing binary variables denoting the relevance of all features. φl is set to 1 when the l-th feature is relevant and 0 otherwise. The “true” Beta distribution θ∗ cl can be approximated as [2][9]: p(xl|θ∗ cl, φl) ≃ pb(xl|θcl) φl pb(xl|ξl) 1−φl (3) By considering each φl as Bernoulli variable with parameters p(φl = 1) = ϵl1 and p(φl = 0) = ϵl2 (ϵl1 + ϵl2 = 1) then, the distribution p(xl|θ∗ cl) can be obtained after marginalizing over φl [9] as: p(xl|θ∗ cl) ≃ϵl1pb(xl|θcl) + ϵl2pb(xl|ξl). The VCC-FMM model is given by equation (4). We notice that both models [3] [4] are special cases of VCC-FMM. p(⃗x, r, u, e) = K z=1 M c=1 p(z)p(u|z)p(e|z)p(c)p(r|z, c) D l=1 [ϵl1pb(xl|θcl) + ϵl2pb(xl|ξl)] (4) 3 A Unified Objective for Model and Feature Selection using MML We denote by ⃗θA π the parameter vector of the multinomial distribution of any discrete variable A conditioned on its parent Π of VCC-FMM (see figure 2). We have A| Π=π ∼Multi(1;⃗θA π ) where θA πa = p(A = a|Π = π) and a θA πa = 1. Also, we employ the superscripts θ and ξ to denote the parameters of the Beta distribution of relevant and irrelevant components, respectively i.e. θcl = (αθ cl, βθ cl) and ξl = (αξ l , βξ l ) . The set Θ of all VCC-FMM parameters is defined by ⃗θU z , ⃗θE z , ⃗θR zc, ⃗θφl, ⃗θZ, ⃗θC and θcl, ξl. The log-likelihood of a data set of N independent and identically distributed observations D = {< u(i), e(i), ⃗x(i), r(i) > |i = 1, . . . , N, u(i) ∈U, e(i) ∈E, ⃗x(i) ∈X, r(i) ∈R} is given by: log p(D|Θ) = N i=1 log K z=1 M c=1 p(z)p(c)p(u(i)|z)p(e(i)|z)p(r(i)|z, c) D l=1 [ϵl1pb(x(i) l |θcl) + ϵl2pb(x(i) l |ξl)] (5) The maximum likelihood (ML) approach which optimizes equation (5) w.r.t Θ is not appropriate for learning VCC-FMM since both K and M are unknown. In addition, the likelihood increases monotonically with the number of components and favors lower dimensions [5]. To overcome these problems, we define a message length objective [16] for both the estimation of Θ and identification of K and M using MML [9][2]. This objective incorporates in addition to the log-likelihood, a penalty term which encodes the data to penalize complex models as: MML(K, M) = −log p(Θ) + 1 2 log |I(Θ)| + s 2(1 + log 1 12) −log p(D|Θ) (6) In equation (6), |I(Θ)|, p(Θ), and s denote the Fisher information, prior distribution and the total number of parameters, respectively. The Fisher information of a parameter is the expectation of the second derivatives with respect to the parameter of the minus log-likelihood. It is common sense to assume an independence among the different groups of parameters which factorizes both |I(Θ)| and p(Θ) over the Fisher and prior distribution of different groups of parameters, respectively. We approximate the Fisher information of the VCC-FMM from the complete likelihood which assumes the knowledge about the values of hidden variables for each observation < u(i), e(i), ⃗x(i), r(i) >∈D. The Fisher information of θcl and ξl can be computed by following a similar methodology of [1]. Also, we use the result found in [8] in computing the Fisher information of ⃗θA π of a discrete variable A with NA different values in a data set of N observations. |I(⃗θA π )| is given by |I(⃗θA π )| = Np(Π = π) NA−1/ NA a=1 θA πa [8], where p(Π = π) is the marginal probability of the parent Π. The graphical representation of of VCC-FMM does not involve variable ancestors (parents of parents). Therefore, the marginal probabilities p(Π = π) are simply the parameters of the multinomial distribution of the parent variable. For example, |I( ⃗θR zc)| is computed as: |I(⃗θR zc)| = N Nr−1(θC c θZ z )Nr−1 / Nr r=1 θR zcr. In case of complete ignorance, it is common to employ the Jeffrey’s prior for different groups of parameters. Replacing p(Θ) and I(Θ) in (6), and after discarding the first order terms, the MML objective is given by: MML(K, M) = Np 2 log N + M D l=1 log ϵl1 + D l=1 log ϵl2 + 1 2N Z p K z=1 log θZ z + 1 2(Nr −1) M c=1 log θC c −log p(D|Θ) (7) with Np = 2D(M + 1) + K(Nu + Ne −2) + MK(Nr −1) and N Z p = Nr + Nu + Ne −3. For fixed values of K, M and D, the minimization of MML objective with respect to Θ is equivalent to a maximum a posteriori (MAP) estimate with the following improper Dirichlet priors [9]: p(⃗θC) ∝ M c=1 (θC c )−Nr−1 2 , p(⃗θZ) ∝ K z=1 (θZ z )− NZ p 2 , p(ϵ1, . . . , ϵD) ∝ D l=1 ϵ−M l1 ϵ−1 l2 (8) 3.1 Estimation of parameters We optimize the MML of the data set using the Expectation-Maximization (EM) algorithm in order to estimate the parameters. In the E-step, the joint posterior probabilities of the latent variables given the observations are computed as Qzci = p(z, c|u(i), e(i), ⃗x(i), r(i), ˆΘ): Qzci = ˆθZ z ˆθC c ˆθU zu(i) ˆθE ze(i) ˆθR zcr(i) l(ϵl1p(x(i) l |ˆθcl) + ϵl2p(x(i) l |ˆξl)) z,c ˆθZ z ˆθC c ˆθU zu(i) ˆθE ze(i) ˆθR zcr(i) l(ϵl1p(x(i) l |ˆθcl) + ϵl2p(x(i) l |ˆξl)) (9) In the M-step, the parameters are updated using the following equations: ˆθZ z = max i c Qzci − NZ p 2 , 0 z max i c Qzci − NZ p 2 , 0 , ˆθC c = max i z Qzci −Nr−1 2 , 0 c max i z Qzci −Nr−1 2 , 0 (10) ˆθU zu = i:u(i)=u c Qzci N ˆθZ z , ˆ θE ze = i:e(i)=e c Qzci N ˆθZ z ˆθR zcr = i:r(i)=r Qzci i Qzci (11) 1 ϵl1 = 1 + max z,c,i Qzciϵl2pb(x(i) l |ξl) ϵl1pb(x(i) l |θcl)+ϵl2pb(x(i) l |ξl) −1, 0 max z,c,i Qzciϵl1pb(Xil|θcl) ϵl1pb(x(i) l |θcl)+ϵl2pb(x(i) l |ξl) −M, 0 (12) The parameters of Beta distributions θcl and ξl are updated using the Fisher scoring method based on the first and second order derivatives of the MML objective [1]. 4 Experiments The benefits of using feature selection and the contextual information are evaluated by considering two variants: V-FMM and V-GD-FMM in addition the original VCC-FMM given by equation (4). V-FMM does not handle the contextual information and assumes θ E ze constant for all e ∈E. On the other hand, feature selection is not considered for V-GD-FMM by setting ϵ l1 = 1 and pruning the uninformative components ξl for l = 1, . . . , D. 4.1 Data Set We have collected ratings from 27 subjects who participated in the experiment (i.e. N u = 27) during a period of three months. The participating subjects are graduate students in faculty of science. Subjects received periodically (twice a day) a list of three images on which they assign relevance degrees expressed on a five star rating scale (i.e. Nr = 5). We define the context as a combination of two attributes: location L = {in−campus, out−campus} inferred from the Internet Protocol (IP) address of the subject, and time as T = (weekday, weekend) i.e Ne = 4. A data set D of 13446 ratings is collected (N = 13446). We have used a collection of 4775 (i.e. Nv = 4775) images collected from Washington University [10] and collections of free photographs which we categorized manually into 41 categories. For visual content characterization, we have employed both local and global descriptors. For local descriptors, we use the 128-dimensional Scale Invariant Feature Transform (SIFT) [11] to represent image patches. We employ vector quantization to SIFT descriptors and we build a histogram for each image (“bag of visual words”). The size of the visual vocabulary is 500. For global descriptors, we used the color correlogram for image texture representation, and the edge histogram descriptor. Therefore, a visual feature vector is represented in a 540-dimensional space (D = 540). We measure the accuracy of the prediction by the Mean Absolute Error (MAE) which is the average of the absolute deviation between the actual and predicted ratings. 4.2 First Experiment: Evaluating the influence of model order on the prediction accuracy This experiment tries to investigate the relationship between the assumed model order defined by K and M on the prediction accuracy of VCC-FMM. It should be noticed that the ground truth number of user classes K∗is not known for our data set D. We run this experiment on a ground truth (artificial) data with known K and M. DGT is sampled from the preferences P1 and P2 of two most dissimilar subjects according to Pearson correlation coefficients [14]. We sample ratings for 100 simulated users from the preferences P1 and P2 only on images of four image classes. For each user, we generate 80 ratings (∼20 ratings per context). Therefore, the ground truth model order is K∗= 2 and M ∗= 4. The choice of M ∗is purely motivated by convenience of presentation since similar performance was reported for higher values of M ∗. We learn the VCC-FMM model using one half of DGT for different choices of training and validation data. The model order defined by M = 15 and K = 15 is used to initialize EM algorithm. Figure 3(a) shows that both K and M have been identified correctly on D GT since the lowest MML was reported for the model order defined by M = 4 and K = 2. The selection of the best model order is important since it influences the accuracy of the prediction (MAE) as illustrated by Figure 3(b). It should be noticed that the over-estimation of M (M > M ∗) leads to more errors than the over-estimation of K (K > K ∗). 4.3 Second Experiment: Comparison with state-of-the-art The aim of this experiment is to measure the contribution of the visual information and the user’s context in making accurate predictions comparatively with some existing CF approaches. We make comparisons with the Aspect model [7], Pearson Correlation (PCC)[14], Flexible Mixture Model (FMM) [15], and User Rating Profile (URP) [13]. For accurate estimators, we learn the URP model using Gibs sampling. We retained for the previous algorithms, the model order that ensured the lowest MAE. (a) MML (b) MAE Figure 3: MML and MAE curves for different model orders on D GT . Table 1: Averaged MAE over 10 runs of the different algorithms on D PCC(baseline) Aspect FMM URP V-FMM V-GD-FMM VCC-FMM Avg MAE 1.327 1.201 1.145 1.116 0.890 0.754 0.646 Deviation 0.040 0.051 0.036 0.042 0.034 0.027 0.014 Improvement 0.00% 9.49% 13.71% 15.90% 32.94% 43.18% 55.84% The first five columns of table 1 show the added value provided by the visual information comparatively with pure CF techniques. For example, the improvement in the rating’s prediction reported by V-FMM is 3.52% and 1.97% comparatively with FMM and URP, respectively. The algorithms (with context information) shown in the last two columns have also improved the accuracy of the prediction comparatively with the others (at least 15.28%). This explains the importance of the contextual information on user preferences. Feature selection is also important since VCC-FMM has reported a better accuracy (14.45%) than V-GD-FMM. Furthermore, it is reported in figure 4(a) that VCCFMM is less sensitive to data sparsity (number of ratings per user) than pure CF techniques. Finally, the evolution of the average MAE provided VCC-FMM for different proportions of unrated images remains under < 25% for up to 30% of unrated images as shown in Figure 4(b). We explain the stability of the accuracy of VCC-FMM for data sparsity and new images by the visual information since only cluster representatives need to be rated. (a) Data sparsity (b) new images Figure 4: MAE curves with error bars on the data set D. 5 Conclusions This paper has motivated theoretically and empirically the importance of both feature selection and model order identification from unlabeled data as important issues in content-based image suggestion. Experiments on collected data showed also the importance of the visual information and the user’s context in making accurate suggestions. Acknowledgements The completion of this research was made possible thanks to Natural Sciences and Engineering Research Council of Canada (NSERC), Bell Canada’s support through its Bell University Laboratories R&D program and a start-up grant from Concordia University. References [1] N. Bouguila and D. Ziou. High-Dimensional Unsupervised Selection and Estimation of a Finite Generalized Dirichlet Mixture Model Based on Minimum Message Length. IEEE Transactions on Pattern Analysis and Machine Intelligence, 29(10):1716–1731, 2007. [2] S. Boutemedjet, N. Bouguila, and D. Ziou. Unsupervised Feature and Model Selection for Generalized Dirichlet Mixture Models. In Proc. of International Conference on Image Analysis and Recognition (ICIAR), pages 330–341. LNCS 4633, 2007. [3] S. Boutemedjet and D. Ziou. Content-based Collaborative Filtering Model for Scalable Visual Document Recommendation. In Proc. of IJCAI-2007 Workshop on Multimodal Information Retrieval, pages 11–18, 2007. [4] S. Boutemedjet and D. Ziou. A Graphical Model for Context-Aware Visual Content Recommendation. IEEE Transactions on Multimedia, 10(1):52–62, 2008. [5] J. G. Dy and C. E. Brodley. Feature Selection for Unsupervised Learning. Journal of Machine Learning Research, 5:845–889, 2004. [6] I. Guyon and A. Elisseeff. An Introduction to Variable and Feature Selection. Journal of Machine Learning Research, 3:1157–1182, 2003. [7] T. Hofmann. Latent Semantic Models for Collaborative Filtering. ACM Transactions on Information Systems, 22(1):89–115, 2004. [8] P. Kontkanen, P. Myllymki, T. Silander, H. Tirri, and P. Grnwald. On Predictive Distributions and Bayesian Networks. Statistics and Computing, 10(1):39–54, 2000. [9] M. H. C. Law, M.A.T. Figueiredo, and A. K. Jain. Simultaneous Feature Selection and Clustering Using Mixture Models. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(9), 2004. [10] J. Li and J. Z. Wang. Automatic Linguistic Indexing of Pictures by a Statistical Modeling Approach. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(9):49–68, 2003. [11] D.G. Lowe. Distinctive Image Features From Scale-Invariant Keypoints. International Journal of Computer Vision, 60(2):91–110, 2004. [12] J. Muramastsu M. Pazzani and D. Billsus. Syskill and Webert:Identifying Interesting Web Sites. In In Proc. of the 13th National Conference on Artificial Intelligence (AAAI), 1996. [13] B. Marlin. Modeling User Rating Profiles For Collaborative Filtering. In Proc. of Advances in Neural Information Processing Systems 16 (NIPS), 2003. [14] P. Resnick, N. Iacovou, M. Suchak, P. Bergstrom, and J. Riedl. Grouplens: An Open Architecture for Collaborative Filtering of Netnews. In Proc. of ACM Conference on Computer Supported Cooperative Work, 1994. [15] L. Si and R. Jin. Flexible Mixture Model for Collaborative Filtering. In Proc. of 20th International Conference on Machine Learning (ICML), pages 704–711, 2003. [16] C. Wallace. Statistical and Inductive Inference by Minimum Message Length. Information Science and Statistics. Springer, 2005.
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Anytime Induction of Cost-sensitive Trees Saher Esmeir Computer Science Department Technion—Israel Institute of Technology Haifa 32000, Israel esaher@cs.technion.ac.il Shaul Markovitch Computer Science Department Technion—Israel Institute of Technology Haifa 32000, Israel shaulm@cs.technion.ac.il Abstract Machine learning techniques are increasingly being used to produce a wide-range of classifiers for complex real-world applications that involve nonuniform testing costs and misclassification costs. As the complexity of these applications grows, the management of resources during the learning and classification processes becomes a challenging task. In this work we introduce ACT (Anytime Cost-sensitive Trees), a novel framework for operating in such environments. ACT is an anytime algorithm that allows trading computation time for lower classification costs. It builds a tree top-down and exploits additional time resources to obtain better estimations for the utility of the different candidate splits. Using sampling techniques ACT approximates for each candidate split the cost of the subtree under it and favors the one with a minimal cost. Due to its stochastic nature ACT is expected to be able to escape local minima, into which greedy methods may be trapped. Experiments with a variety of datasets were conducted to compare the performance of ACT to that of the state of the art cost-sensitive tree learners. The results show that for most domains ACT produces trees of significantly lower costs. ACT is also shown to exhibit good anytime behavior with diminishing returns. 1 Introduction Suppose that a medical center has decided to use machine learning techniques to induce a diagnostic tool from records of previous patients. The center aims to obtain a comprehensible model, with low expected test costs (the costs of testing attribute values) and high expected accuracy. Moreover, in many cases there are costs associated with the predictive errors. In such a scenario, the task of the inducer is to produce a model with low expected test costs and low expected misclassification costs. A good candidate for achieving the goals of comprehensibility and reduced costs is a decision tree model. Decision trees are easily interpretable because they mimic the way doctors think [13][chap. 9]. In the context of cost-sensitive classification, decision trees are the natural form of representation: they ask only for the values of the features along a single path from the root to a leaf. Indeed, cost-sensitive trees have been the subject of many research efforts. Several works proposed learners that consider different misclassification costs [7, 18, 6, 9, 10, 14, 1]. These methods, however, do not consider test costs. Other authors designed tree learners that take into account test costs, such as IDX [16], CSID3 [22], and EG2 [17]. These methods, however, do not consider misclassification costs. The medical center scenario exemplifies the need for considering both types of cost together: doctors do not perform a test before considering both its cost and its importance to the diagnosis. Minimal Cost trees, a method that attempts to minimize both types of costs simultaneously has been proposed in [21]. A tree is built top-down. The immediate reduction in total cost each split results in is estimated, and a split with the maximal reduction is selected. Although efficient, the Minimal Cost approach can be trapped into a local minimum and produce trees that are not globally optimal. 1 a1 a7 a9 a9 0 1 1 0 a6 a4 a4 0 1 1 0 a9 a10 1 0 cost(a1-8) = $$ cost(a9,10) = $$$$$$ cost(a1-10) = $$ a10 0 1 Figure 1: A difficulty for greedy learners (left). Importance of context-based evaluation (right). For example, consider a problem with 10 attributes a1−10, of which only a9 and a10 are relevant. The cost of a9 and a10, however, is significantly higher than the others but lower than the cost of misclassification. This may hide their usefulness, and mislead the learner to fit a large expensive tree. The problem is intensified if a9 and a10 were interdependent with a low immediate information gain (e.g., a9 ⊕a10), as illustrated in Figure 1 (left). In such a case, even if the costs were uniform, local measures would fail in recognizing the relevance of a9 and a10 and other attributes might be preferred. The Minimal Cost method is appealing when resources are very limited. However, it requires a fixed runtime and cannot exploit additional resources. In many real-life applications, we are willing to wait longer if a better tree can be induced. For example, due to the importance of the model, the medical center is ready to allocate 1 week to learn it. Algorithms that can exploit more time to produce solutions of better quality are called anytime algorithms [5]. One way to exploit additional time when searching for a tree of lower costs is to widen the search space. In [2] the cost-sensitive learning problem is formulated as a Markov Decision Process (MDP) and a systematic search is used to solve the MDP. Although the algorithm searches for an optimal strategy, the time and memory limits prevent it from always finding optimal solutions. The ICET algorithm [24] was a pioneer in searching non-greedily for a tree that minimizes both costs together. ICET uses genetic search to produce a new set of costs that reflects both the original costs and the contribution each attribute can make to reduce misclassification costs. Then it builds a tree using the greedy EG2 algorithm but with the evolved costs instead of the original ones. ICET was shown to produce trees of lower total cost. It can use additional time resources to produce more generations and hence to widen its search in the space of costs. Nevertheless, it is limited in the way it can exploit extra time. Firstly, it builds the final tree using EG2. EG2 prefers attributes with high information gain (and low test cost). Therefore, when the concept to learn hides interdependency between attributes, the greedy measure may underestimate the usefulness of highly relevant attributes, resulting in more expensive trees. Secondly, even if ICET may overcome the above problem by reweighting the attributes, it searches the space of parameters globally, regardless of the context. This imposes a problem if an attribute is important in one subtree but useless in another. To illustrate the above consider the concept in Figure 1 (right). There are 10 attributes of similar costs. Depending on the value of a1, the target concept is a7 ⊕a9 or a4 ⊕a6. Due to interdependencies, all attributes will have a low gain. Because ICET assigns costs globally, they will have similar costs as well. Therefore, ICET will not be able to recognize which attribute is relevant in what context. Recently, we have introduced LSID3, a cost-insensitive algorithm, which can induce more accurate trees when given more time [11]. The algorithm uses stochastic sampling techniques to evaluate candidate splits. It is not designed, however, to minimize test and misclassification costs. In this work we build on LSID3 and propose ACT, an Anytime Cost-sensitive Tree learner that can exploit additional time to produce trees of lower costs. Applying the sampling mechanism to the costsensitive setup, however, is not trivial and imposes several challenges which we address in Section 2. Extensive set of experiments that compares ACT to EG2 and to ICET is reported in Section 3. The results show that ACT is significantly better for the majority of problems. In addition ACT is shown to exhibit good anytime behavior with diminishing returns. The major contributions of this paper are: (1) a non-greedy algorithm for learning trees of lower costs that allows handling complex cost structures, (2) an anytime framework that allows learning time to be traded for reduced classification costs, and (3) a parameterized method for automatic assigning of costs for existing datasets. Note that costs may also be involved during example acquisition [12, 15]. In this work, however, we assume that the full training examples are in hand. Moreover, we assume that during the test phase, all tests in the relevant path will be taken. Several test strategies that determine which values to query for and at what order have been recently studied [21]. These strategies are orthogonal to our work because they assume a given tree. 2 2 The ACT Algorithm Offline concept learning consists of two stages: learning from labelled examples; and using the induced model to classify unlabelled instances. These two stages involve different types of cost [23]. Our primary goal in this work is to trade the learning time for reduced test and misclassification costs. To make the problem well defined, we need to specify how to: (1) represent misclassification costs, (2) calculate test costs, and (3) combine both types of cost. To answer these questions, we adopt the model described by Turney [24]. In a problem with |C| different classes, a classification cost matrix M is a |C| × |C| matrix whose Mi,j entry defines the penalty of assigning the class ci to an instance that actually belongs to the class cj. To calculate the test costs of a particular case, we sum the cost of the tests along the path from the root to the appropriate leaf. For tests that appear several times we charge only for the first occurrence. The model handles two special test types, namely grouped and delayed. Grouped tests share a common cost that is charged only once per group. Each test also has an extra cost charged when the test is actually made. For example, consider a tree path with tests like cholesterol level and glucose level. For both values to be measured, a blood test is needed. Clearly, once blood samples are taken to measure the cholesterol level, the cost for measuring the glucose level is lower. Delayed tests are tests whose outcome cannot be obtained immediately, e.g., lab test results. Such tests force us to wait until the outcome is available. Alternatively, we can take into account all possible outcomes and follow several paths in the tree simultaneously (and pay for their costs). Once the result of the delayed test is available, the prediction is in hand. Note that we might be charged for tests that we would not perform if the outcome of the delayed tests were available. In this work we do not handle delayed costs but we do explain how to adapt our framework to scenarios that involve them. Having measured the test costs and misclassification costs, an important question is how to combine them. Following [24] we assume that both types of cost are given in the same scale. Alternatively, Qin et. al. [19] presented a method to handle the two kinds of cost scales by setting a maximal budget for one kind and minimizing the other. ACT, our proposed anytime framework for induction of cost-sensitive trees, builds on the recently introduced LSID3 algorithm [11]. LSID3 adopts the general top-down induction of decision trees scheme (TDIDT): it starts from the entire set of training examples, partitions it into subsets by testing the value of an attribute, and then recursively builds subtrees. Unlike greedy inducers, LSID3 invests more time resources for making better split decisions. For every candidate split, LSID3 attempts to estimate the size of the resulting subtree were the split to take place and following Occam’s razor [4] it favors the one with the smallest expected size. The estimation is based on a biased sample of the space of trees rooted at the evaluated attribute. The sample is obtained using a stochastic version of ID3, called SID3 [11]. In SID3, rather than choosing an attribute that maximizes the information gain ∆I (as in ID3), the splitting attribute is chosen semi-randomly. The likelihood that an attribute will be chosen is proportional to its information gain. LSID3 is a contract algorithm parameterized by r, the sample size. When r is larger, the resulting estimations are expected to be more accurate, therefore improving the final tree. Let m = |E| be the number of examples and n = |A| be the number of attributes. The runtime complexity of LSID3 is O(rmn3) [11]. LSID3 was shown to exhibit a good anytime behavior with diminishing returns. When applied to hard concepts, it produced significantly better trees than ID3 and C4.5. ACT takes the same sampling approach as in LSID3. However, three major components of LSID3 need to be replaced for the cost-sensitive setup: (1) sampling the space of trees, (2) evaluating a tree, and (3) pruning. Obtaining the Sample. LISD3 uses SID3 to bias the samples towards small trees. In ACT, however, we would like to bias our sample towards low cost trees. For this purpose, we designed a stochastic version of the EG2 algorithm, that attempts to build low cost trees greedily. In EG2, a tree is built top-down, and the attribute that maximizes ICF (Information Cost Function) is chosen for splitting a node, where, ICF (a) = 2∆I(a) −1 / ((cost (a) + 1)w). In Stochastic EG2 (SEG2), we choose splitting attributes semi-randomly, proportionallyto their ICF. Due to the stochastic nature of SEG2 we expect to be able to escape local minima for at least some of the trees in the sample. To obtain a sample of size r, ACT uses EG2 once and SEG2 r −1 times. Unlike ICET, we give EG2 and SEG2 a direct access to context-based costs, i.e., if an attribute has already been tested its cost would be zero and if another attribute that belongs to the same group has been tested, a group discount is applied. The parameter w controls the bias towards lower cost 3 attributes. While ICET tunes this parameter using genetic search, we set w inverse proportionally to the misclassification cost: a high misclassification cost results in a smaller w, reducing the effect of attribute costs. One direction for future work would be to tune w a priori. Evaluating a Subtree. As a cost insensitive learner, the main goal of LSID3 is to maximize the expected accuracy of the learned tree. Following Occam’s razor, it uses the tree size as a preference bias and favors splits that are expected to reduce the final tree size. In a cost-sensitive setup, our goal is to minimize the expected cost of classification. Following the same lookahead strategy as LSID3, we sample the space of trees under each candidate split. However, instead of choosing an attribute that minimizes the size, we would like to choose one that minimizes costs. Therefore, given a tree, we need to come up with a procedure that estimates the expected costs when classifying a future case. This cost consists of two components: the test cost and misclassification cost. Assuming that the distribution of future cases would be similar to that of the learning examples, we can estimate the test costs using the training data. Given a tree, we calculate the average test cost of the training examples and use it to approximate the test cost of new cases. For a tree T and a set of training examples E, we denote the average cost of traversing T for an example from E (average testing cost) by tst-cost(T, E). Note that group discounts and delayed cost penalties do not need a special care because they will be incorporated when calculating the average test costs. Estimating the cost of errors is not obvious. One can no longer use the tree size as a heuristic for predictive errors. Occam’s razor allows to compare two consistent trees but does not provide a mean to estimate accuracy. Moreover, tree size is measured in a different currency than accuracy and hence cannot be easily incorporated in the cost function. Instead, we propose using a different estimator: the expected error [20]. For a leaf with m training examples, of which e are misclassified the expected error is defined as the upper limit on the probability for error, i.e., EE(m, e, cf) = Ucf(e, m) where cf is the confidence level and U is the confidence interval for binomial distribution. The expected error of a tree is the sum of the expected errors in its leafs. Originally, the expected error was used by C4.5 to predict whether a subtree performs better than a leaf. Although it lacks theoretical basis, it was shown experimentally to be a good heuristic. In ACT we use the expected error to approximate the misclassification cost. Assume a problem with |C| classes and a misclassification cost matrix M. Let c be the class label in a leaf l. Let m be the total number of examples in l and mi be the number of examples in l that belong to class i. The expected misclassification cost in l is (the right most expression assumes uniform misclassification cost Mi,j = mc) mc-cost(l) = EE(m, m −mc, cf) · 1 |C| −1 X i̸=c Mc,i = EE(m, m −mc, cf) · mc The expected error of a tree is the sum of the expected errors in its leafs. In our experiments we use cf = 0.25, as in C4.5. In the future, we intend to tune cf if the allocated time allows. Alternatively, we also plan to estimate the error using a set-aside validation set, when the training set size allows. To conclude, let E be the set of examples used to learn a tree T , and let m be the size of E. Let L be the set of leafs in T . The expected total cost of T when classifying an instance is: tst-cost(T, E) + 1 m · X l∈L mc-cost (l). Having decided about the sampler and the tree utility function we are ready to formalize the tree growing phase in ACT. A tree is built top-down. The procedure for selecting splitting test at each node is listed in Figure 2 (left), and exemplified in Figure 2 (right). The selection procedure, as formalized is Figure 2 (left) needs to be slightly modified when an attribute is numeric: instead of iterating over the values the attribute can take, we examine r cutting points, each is evaluated with a single invocation of EG2. This guarantees that numeric and nominal attributes get the same resources. The r points are chosen dynamically, according to their information gain. Costs-sensitive Pruning. Pruning plays an important role in decision tree induction. In costinsensitive environments, the main goal of pruning is to simplify the tree in order to avoid overfitting. A subtree is pruned if the resulting tree is expected to yield a lower error. When test costs are taken into account, pruning has another important role: reducing costs. It is worthwhile to keep a subtree only if its expected reduction to the misclassification cost is larger that the cost of its tests. If the misclassification cost was zero, it makes no sense to keep any split in the tree. If, on the other hand, 4 Procedure ACT-CHOOSE-ATTRIBUTE(E, A, r) If r = 0 Return EG2-CHOOSE-ATTRIBUTE(E, A) Foreach a ∈A Foreach vi ∈domain(a) Ei ←{e ∈E | a(e) = vi} T ←EG2(a, Ei, A −{a}) mini ←COST(T, Ei) Repeat r −1 times T ←SEG2(a, Ei, A −{a}) mini ←min (mini, COST(T, Ei)) totala ←COST(a) + P|domain(a)| i=1 mini Return a for which totala is minimal a cost(EG2) =4.1 cost(EG2) =8.9 cost(SEG2) =5.1 cost(SEG2) =4.9 Figure 2: Attribute selection (left) and evaluation (right) in ACT (left). Assume that the cost of a in the current context is 1. The estimated cost of a subtree rooted at a is therefore 1 + min(4.1, 5.1) + min(8.9, 4.9) = 9. the misclassification cost was very large, we would expect similar behavior to the cost-insensitive setup. To handle this challenge, we propose a novel approach for cost-sensitive pruning. Similarly to error-based pruning [20], we scan the tree bottom-up. For each subtree, we compare its expected total cost to that of a leaf. Formally, assume that e examples in E do not belong to the default class.1 We prune a subtree T into a leaf if: 1 m · mc-cost(l) ≤tst-cost(T, E) + 1 m · X l∈L mc-cost(l). 3 Empirical Evaluation A variety of experiments were conducted to test the performance and behavior of ACT. First we describe and motivate our experimental methodology. We then present and discuss our results. 3.1 Methodology We start our experimental evaluation by comparing ACT, given a fixed resource allocation, with EG2 and ICET. EG2 was selected as a representative for greedy learners. We also tested the performance of CSID3 and IDX but found the results very similar to EG2, confirming the report in [24]. Our second set of experiments compares the anytime behavior of ACT to that of ICET. Because the code of EG2 and ICET is not publicly available we have reimplemented them. To verify the reimplementation results, we compared them with those reported in literature. We followed the same experimental setup and used the same 5 datasets. The results are indeed similar with the basic version of ICET achieving an average cost of 49.9 in our reimplementation vs. 49 in Turney’s paper [24]. One possible reason for the slight difference may be the randomization involved in the genetic search as well as in data partitioning into training, validating, and testing sets. Datasets. Typically, machine learning researchers use datasets from the UCI repository [3]. Only five UCI datasets, however, have assigned test costs [24]. To gain a wider perspective, we developed an automatic method that assigns costs to existing datasets randomly. The method is parameterized with: (1) cr the cost range, (2) g the number of desired groups as a percentage of the number of attributes, and (3) sc the group shared cost as a percentage of the maximal marginal cost in the group. Using this method we assigned costs to 25 datasets: 21 arbitrarily chosen UCI datasets2 and 4 datasets that represent hard concept and have been used in previous research. The online appendix 3 gives detailed descriptions of these datasets. Two versions of each dataset have been created, both with cost range of 1-100. In the first g and sc were set to 20% and in the second they were set to 80%. These parameters were chosen arbitrarily, in attempt to cover different types of costs. In total we have 55 datasets: 5 with costs assigned as in [24] and 50 with random costs. Cost-insensitive learning algorithms focus on accuracy and therefore are expected to perform well 1The default class is the one that minimizes the misclassification cost in the node. 2The chosen UCI datasets vary in their size, type of attributes and dimension. 3http://www.cs.technion.ac.il/∼esaher/publications/nips07 5 Table 1: Average cost of classification as a percentage of the standard cost of classification. The table also lists for each of ACT and ICET the number of significant wins they had using t-test. The last row shows the winner, if any, as implied by a Wilcoxon test over all datasets with α = 5%. mc = 10 mc = 100 mc = 1000 mc = 10000 EG2 ICET ACT EG2 ICET ACT EG2 ICET ACT EG2 ICET ACT AVERAGE 22.37 10.23 2.21 25.93 17.15 11.86 38.69 35.28 34.38 54.22 47.47 41.62 BETTER 0 34 0 25 3 11 10 12 WILCOXON √ √ √ 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 Figure 3: Illustration of the differences in performance between ACT and ICET for misclassification costs (from left to right: 10, 100, 1000, and 10000). Each point represents a dataset. The x-axis represents the cost of ICET while the y-axis represents that of ACT. The dashed line indicates equality. Points are below it if ACT performs better and above it if ICET is better. when testing costs are negligible relative to misclassification costs. On the other hand, when testing costs are significant, ignoring them would result in expensive classifiers. Therefore, to evaluate a cost-sensitive learner a wide spectrum of misclassification costs is needed. For each problem out of the 55, we created 4 instances, with uniform misclassification costs mc = 10, 100, 1000, 10000. Normalized Cost. As pointed out by Turney [24], using the average cost is problematic because: (1) the differences in costs among the algorithms become small as misclassification cost increases, (2) it is difficult to combine the results for the multiple datasets, and (3) it is difficult to combine average costs for different misclassification costs. To overcome these problems, Turney suggests to normalize the average cost of classification by dividing it by the standard cost, defined as (T C + mini (1 −fi) · maxi,j (Mi,j)), The standard cost is an approximation for the maximal cost in a given problem. It consists of two components: (1) TC, the cost if we take all tests, and (2) the misclassification cost if the classifier achieves only the base-line accuracy. fi denotes the frequency of class i in the data and hence (1 −fi) would be the error if the response would always be class i. Statistical Significance. For each problem, one 10 fold cross-validation experiment has been conducted. The same partition to train-test sets was used for all compared algorithms. To test the statistical significance of the differences between ACT and ICET we used two tests. The first is t-test with a α = 5% confidence: for each method we counted how many times it was a significant winner. The second is Wilcoxon test [8], which compares classifiers over multiple datasets and states whether one method is significantly better than the other (α = 5%). 3.2 Fixed-time Comparison For each of the 55 × 4 problem instances, we run the seeded version of ICET with its default parameters (20 generations),4 EG2, and ACT with r = 5. We choose r = 5 so the average runtime of ACT would be shorter than ICET for all problems. EG2 and ICET use the same post-pruning mechanism as in C4.5. In EG2 the default confidence factor is used (0.25) while in ICET this value is tuned using the genetic search. Table 1 lists the average results, Figure 3 illustrates the differences between ICET and ACT, and Figure 4 (left) plots the average cost for the different values of mc. The full results are available in the online appendix. Similarly to the results reported in [24] ICET is clearly better than EG2, because the latter does not consider misclassification costs. When mc is set to 10 and to 100 ACT significantly outperforms ICET for most datasets. In these cases ACT was able to produce very small trees, sometimes consist of one node, neglecting the accuracy of the learned model. For mc set to 1000 and 10000 there are fewer significant wins, yet it is clear that ACT is dominating: the 4Seeded ICET includes the true costs in the initial population and was reported to perform better [24]. 6 0 10 20 30 40 50 10 100 1000 10000 Average Cost Misclassification Cost EG2 ICET ACT 50 55 60 65 70 75 80 85 10 100 1000 10000 Average Accuracy Misclassification Cost C4.5 ICET ACT 40 42 44 46 48 50 0 1 2 3 4 5 Average Cost Time [sec] EG2 ICET ACT 20 25 30 35 40 45 0 1 2 3 4 5 6 Average Cost Time [sec] EG2 ICET ACT Figure 4: Average cost (left most) and accuracy (mid-left) as a function of misclassification cost. Average cost as a function of time for Breast-cancer-20 (mid-right) and Multi-XOR-80 (right most). number of ACT wins is higher and the average results indicate that ACT trees are cheaper. The Wilcoxon test, states that for mc = 10, 100, 10000, ACT is significantly better than ICET, and that for mc = 1000 no significant winner was found. When misclassification costs are low, an optimal algorithm would produce a very shallow tree. When misclassification costs are dominant, an optimal algorithm would produce a highly accurate tree. Some concepts, however, are not easily learnable and even cost-insensitive algorithms fail to achieve perfect accuracy on them. Hence, with the increase in the importance of accuracy the normalized cost increases: the predictive errors affect the cost more dramatically. To learn more about the effect of accuracy, we compared the accuracy of ACT to that of C4.5 and ICET mc values. Figure 4 (mid-left) shows the results. An important property of both ICET and ACT is their ability to compromise on accuracy when needed. ACT’s flexibility, however, is more noteworthy: from the least accurate method it becomes the most accurate one. Interestingly, when accuracy is extremely important both ICET and ACT achieves even better accuracy than C4.5. The reason is their non-greedy nature. ICET performs an implicit lookahead by reweighting attributes according to their importance. ACT performs lookahead by sampling the space of subtrees under every split. Among the two, the results indicates that ACT’s lookahead is more efficient in terms of accuracy. We also compared ACT to LSID3. As expected, ACT was significantly better for mc ≤1000. For mc = 10000 their performance was similar. In addition, we compared the studied methods on nonuniform misclassification costs and found ACT’s advantage to be consistent. 3.3 Anytime Comparison Both ICET and ACT are anytime algorithms that improve their performance with time. ICET is expected to exploit extra time by producing more generations and hence better tuning the parameters for the final invocation of EG2. ACT can use additional time to acquire larger samples and hence achieve better cost estimations. A typical anytime algorithm would produce improved results with the increase in resources. The improvements diminish with time, reaching a stable performance. To examine the anytime behavior of ICET and ACT, we run each of them on 2 problems, namely Breast-cancer-20 and Multi-XOR-80, with exponentially increasing time allocation. ICET was run with 2, 4, 8 . . . generations and ACT with a sample size of 1, 2, 4, . . .. Figure 4 plots the results. The results show a good anytime behavior of both ICET and ACT. For both algorithms, it is worthwhile to allocate more time. ACT dominates ICET for both domains and is able to produce trees of lower costs in shorter time. The Multi-XOR dataset is an example for a concept with attributes being important only in one sub-concept. As we expected, ACT outperforms ICET significantly because the latter cannot assign context-based costs. Allowing ICET to produce more and more generations (up to 128) does not result in trees comparable to those obtained by ACT. 4 Conclusions Machine learning techniques are increasingly being used to produce a wide-range of classifiers for real-world applications that involve nonuniform testing costs and misclassification costs. As the complexity of these applications grows, the management of resources during the learning and classification processes becomes a challenging task. In this work we introduced a novel framework for operating in such environments. Our framework has 4 major advantages: (1) it uses a non-greedy approach to build a decision tree and therefore is able to overcome local minima problems, (2) it evaluates entire trees and therefore can be adjusted to any cost scheme that is defined over trees. (3) it exhibits good anytime behavior and produces significantly better trees when more time is available, and (4) it can be easily parallelized and hence can benefit from distributed computer power. 7 To evaluate ACT we have designed an extensive set of experiments with a wide range of costs. The experimental results show that ACT is superior over ICET and EG2. Significance tests found the differences to be statistically strong. ACT also exhibited good anytime behavior: with the increase in time allocation, there was a decrease in the cost of the learned models. ACT is a contract anytime algorithm that requires its sample size to be pre-determined. In the future we intend to convert ACT into an interruptible anytime algorithm, by adopting the IIDT general framework [11]. In addition, we plan to apply monitoring techniques for optimal scheduling of ACT and to examine other strategies for evaluating subtrees. References [1] N. Abe, B. Zadrozny, and J. Langford. An iterative method for multi-class cost-sensitive learning. In KDD, 2004. [2] V. Bayer-Zubek and Dietterich. Integrating learning from examples into the search for diagnostic policies. Artificial Intelligence, 24:263–303, 2005. [3] C. L. Blake and C. J. Merz. UCI repository of machine learning databases, 1998. [4] A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth. Occam’s Razor. Information Processing Letters, 24(6):377–380, 1987. [5] M. Boddy and T. L. Dean. Deliberation scheduling for problem solving in time constrained environments. Artificial Intelligence, 67(2):245–285, 1994. [6] J. Bradford, C. Kunz, R. Kohavi, C. Brunk, and C. Brodley. Pruning decision trees with misclassification costs. In ECML, 1998. [7] L. Breiman, J. Friedman, R. Olshen, and C. Stone. Classification and Regression Trees. Wadsworth and Brooks, Monterey, CA, 1984. [8] J. Demsar. Statistical comparisons of classifiers over multiple data sets. Journal of Machine Learning Research, 7:1–30, 2006. [9] P. Domingos. Metacost: A general method for making classifiers cost-sensitive. In KDD, 1999. [10] C. Elkan. The foundations of cost-sensitive learning. In IJCAI, 2001. [11] S. Esmeir and S. Markovitch. Anytime learning of decision trees. Journal of Machine Learning Research, 8, 2007. [12] R. Greiner, A. J. Grove, and D. Roth. Learning cost-sensitive active classifiers. Artificial Intelligence, 139(2):137–174, 2002. [13] T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. New York: Springer-Verlag, 2001. [14] D. Margineantu. Active cost-sensitive learning. In IJCAI, 2005. [15] P. Melville, M. Saar-Tsechansky, F. Provost, and R. J. Mooney. Active feature acquisition for classifier induction. In ICDM, 2004. [16] S. W. Norton. Generating better decision trees. In IJCAI, 1989. [17] M. Nunez. The use of background knowledge in decision tree induction. Machine Learning, 6:231–250, 1991. [18] F. Provost and B. Buchanan. Inductive policy: The pragmatics of bias selection. Machine Learning, 20(1-2):35–61, 1995. [19] Z. Qin, S. Zhang, and C. Zhang. Cost-sensitive decision trees with multiple cost scales. Lecture Notes in Computer Science, AI, Volume 3339/2004:380–390, 2004. [20] J. R. Quinlan. C4.5: Programs for Machine Learning. Morgan Kaufmann, 1993. [21] S. Sheng, C. X. Ling, A. Ni, and S. Zhang. Cost-sensitive test strategies. In AAAI, 2006. [22] M. Tan and J. C. Schlimmer. Cost-sensitive concept learning of sensor use in approach and recognition. In Proceedings of the 6th international workshop on Machine Learning, 1989. [23] P. Turney. 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Optimal ROC Curve for a Combination of Classifiers Marco Barreno Alvaro A. C´ardenas J. D. Tygar Computer Science Division University of California at Berkeley Berkeley, California 94720 {barreno,cardenas,tygar}@cs.berkeley.edu Abstract We present a new analysis for the combination of binary classifiers. Our analysis makes use of the Neyman-Pearson lemma as a theoretical basis to analyze combinations of classifiers. We give a method for finding the optimal decision rule for a combination of classifiers and prove that it has the optimal ROC curve. We show how our method generalizes and improves previous work on combining classifiers and generating ROC curves. 1 Introduction We present an optimal way to combine binary classifiers in the Neyman-Pearson sense: for a given upper bound on false alarms (false positives), we find the set of combination rules maximizing the detection rate (true positives). This forms the optimal ROC curve of a combination of classifiers. This paper makes the following original contributions: (1) We present a new method for finding the meta-classifier with the optimal ROC curve. (2) We show how our framework can be used to interpret, generalize, and improve previous work by Provost and Fawcett [1] and Flach and Wu [2]. (3) We present experimental results that show our method is practical and performs well, even when we must estimate the distributions with insufficient data. In addition, we prove the following results: (1) We show that the optimal ROC curve is composed in general of 2n + 1 different decision rules and of the interpolation between these rules (over the space of 22n possible Boolean rules). (2) We prove that our method is optimal in this space. (3) We prove that the Boolean AND and OR rules are always part of the optimal set for the special case of independent classifiers (though in general we make no independence assumptions). (4) We prove a sufficient condition for Provost and Fawcett’s method to be optimal. 2 Background Consider classification problems where examples from a space of inputs X are associated with binary labels {0, 1} and there is a fixed but unknown probability distribution P(x, c) over examples (x, c) ∈X × {0, 1}. H0 and H1 denote the events that c = 0 and c = 1, respectively. A binary classifier is a function f : X →{0, 1} that predicts labels on new inputs. When we use the term “classifier” in this paper we mean binary classifier. We address the problem of combining results from n base classifiers f1, f2, . . . , fn. Let Yi = fi(X) be a random variable indicating the output of classifier fi and Y ∈{0, 1}n = (Y1, Y2, . . . , Yn). We can characterize the performance of classifier fi by its detection rate (also true positives, or power) PDi = Pr[Yi = 1|H1] and its false alarm rate (also false positives, or test size) PF i = Pr[Yi = 1|H0]. In this paper we are concerned with proper classifiers, that is, classifiers where PDi > PF i. We sometimes omit the subscript i. 1 The Receiver Operating Characteristic (ROC) curve plots PF on the x-axis and PD on the y-axis (ROC space). The point (0, 0) represents always classifying as 0, the point (1, 1) represents always classifying as 1, and the point (0, 1) represents perfect classification. If one classifier’s curve has no points below another, it weakly dominates the latter. If no points are below and at least one point is strictly above, it dominates it. The line y = x describes a classifier that is no better than chance, and every proper classifier dominates this line. When an ROC curve consists of a single point, we connect it with straight lines to (0, 0) and (1, 1) in order to compare it with others (see Lemma 1). In this paper, we focus on base classifiers that occupy a single point in ROC space. Many classifiers have tunable parameters and can produce a continuous ROC curve; our analysis can apply to these cases by choosing representative points and treating each one as a separate classifier. 2.1 The ROC convex hull Provost and Fawcett [1] give a seminal result on the use of ROC curves for combining classifiers. They suggest taking the convex hull of all points of the ROC curves of the classifiers. This ROC convex hull (ROCCH) combination rule interpolates between base classifiers f1, f2, . . . , fn, selecting (1) a single best classifier or (2) a randomization between the decisions of two classifiers for every false alarm rate [1]. This approach, however, is not optimal: as pointed out in later work by Fawcett, the Boolean AND and OR rules over classifiers can perform better than the ROCCH [3]. AND and OR are only 2 of 22n possible Boolean rules over the outputs of n base classifiers (n classifiers ⇒2n possible outcomes ⇒22n rules over outcomes). We address finding optimal rules. 2.2 The Neyman-Pearson lemma In this section we introduce Neyman-Pearson theory from the framework of statistical hypothesis testing [4, 5], which forms the basis of our analysis. We test a null hypothesis H0 against an alternative H1. Let the random variable Y have probability distributions P(Y|H0) under H0 and P(Y|H1) under H1, and define the likelihood ratio ℓ(Y) = P(Y|H1)/P(Y|H0). The Neyman-Pearson lemma states that the likelihood ratio test D(Y) = ( 1 if ℓ(Y) > τ γ if ℓ(Y) = τ 0 if ℓ(Y) < τ , (1) for some τ ∈(0, ∞) and γ ∈[0, 1], is a most powerful test for its size: no other test has higher PD = Pr[D(Y) = 1|H1] for the same bound on PF = Pr[D(Y) = 1|H0]. (When ℓ(Y) = τ, D = 1 with probability γ and 0 otherwise.) Given a test size α, we maximize PD subject to PF ≤α by choosing τ and γ as follows. First we find the smallest value τ ∗such that Pr[ℓ(Y) > τ ∗|H0] ≤ α. To maximize PD, which is monotonically nondecreasing with PF , we choose the highest value γ∗that satisfies Pr[D(Y) = 1|H0] = Pr[ℓ(Y) > τ ∗|H0] + γ∗Pr[ℓ(Y) = τ ∗|H0] ≤α, finding γ∗= (α −Pr[ℓ(Y) > τ ∗|H0])/ Pr[ℓ(Y) = τ ∗|H0]. 3 The optimal ROC curve for a combination of classifiers We characterize the optimal ROC curve for a decision based on a combination of arbitrary classifiers—for any given bound α on PF , we maximize PD. We frame this problem as a NeymanPearson hypothesis test parameterized by the choice of α. We assume nothing about the classifiers except that each produces an output in {0, 1}. In particular, we do not assume the classifiers are independent or related in any way. Before introducing our method we analyze the one-classifier case (n = 1). Lemma 1 Let f1 be a classifier with performance probabilities PD1 and PF 1. Its optimal ROC curve is a piecewise linear function parameterized by a free parameter α bounding PF : for α < PF 1, PD(α) = (PD1/PF 1)α, and for α > PF 1, PD(α) = [(1−PD1)/(1−PF 1)](α−PF 1)+PD1. Proof. When α < PF 1, we can obtain a likelihood ratio test by setting τ ∗= ℓ(1) and γ∗= α/PF 1, and for α > PF 1, we set τ ∗= ℓ(0) and γ∗= (α −PF 1)/(1 −PF 1). 2 2 The intuitive interpretation of this result is that to decrease or increase the false alarm rate of the classifier, we randomize between using its predictions and always choosing 1 or 0. In ROC space, this forms lines interpolating between (PF 1, PD1) and (1, 1) or (0, 0), respectively. To generalize this result for the combination of n classifiers, we require the distributions P(Y|H0) and P(Y|H1). With this information we then compute and sort the likelihood ratios ℓ(y) for all outcomes y ∈{0, 1}n. Let L be the list of likelihood ratios ranked from low to high. Lemma 2 Given any 0 ≤α ≤1, the ordering L determines parameters τ ∗and γ∗for a likelihood ratio test of size α. Lemma 2 sets up a classification rule for each interval between likelihoods in L and interpolates between them to create a test with size exactly α. Our meta-classifier does this for any given bound on its false positive rate, then makes predictions according to Equation 1. To find the ROC curve for our meta-classifier, we plot PD against PF for all 0 ≤α ≤1. In particular, for each y ∈{0, 1}n we can compute Pr[ℓ(Y) > ℓ(y)|H0], which gives us one value for τ ∗and a point in ROC space (PF and PD follow directly from L and P). Each τ ∗will turn out to be the slope of a line segment between adjacent vertices, and varying γ∗interpolates between the vertices. We call the ROC curve obtained in this way the LR-ROC. Theorem 1 The LR-ROC weakly dominates the ROC curve of any possible combination of Boolean functions g : {0, 1}n →{0, 1} over the outputs of n classifiers. Proof. Let α′ be the probability of false alarm PF for g. Let τ ∗and γ∗be chosen for a test of size α′. Then our meta-classifier’s decision rule is a likelihood ratio test. By the Neyman-Pearson lemma, no other test has higher power for any given size. Since ROC space plots power on the y-axis and size on the x-axis, this means that the PD for g at PF = α′ cannot be higher than that of the LR-ROC. Since this is true at any α′, the LR-ROC weakly dominates the ROC curve for g. 2 3.1 Practical considerations To compute all likelihood ratios for the classifier outcomes we need to know the probability distributions P(Y|H0) and P(Y|H1). In practice these distributions need to be estimated. The simplest method is to run the base classifiers on a training set and count occurrences of each outcome. It is likely that some outcomes will not occur in the training, or will occur only a small number of times. Our initial approach to deal with small or zero counts when estimating was to use add-one smoothing. In our experiments, however, simple special-case treatment of zero counts always produced better results than smoothing, both on the training set and on the test set. See Section 5 for details. Furthermore, the optimal ROC curve may have a different likelihood ratio for each possible outcome from the n classifiers, and therefore a different point in ROC space, so optimal ROC curves in general have up to 2n points. This implies an exponential (in the number of classifiers) lower bound on the running time of any algorithm to compute the optimal ROC curve for a combination of classifiers. For a handful of classifiers, such a bound is not problematic, but it is impractical to compute the optimal ROC curve for dozens or hundreds of classifiers. (However, by computing and sorting the likelihood ratios we avoid a 22n-time search over all possible classification functions.) 4 Analysis 4.1 The independent case In this section we take an in-depth look at the case of two binary classifiers f1 and f2 that are conditionally independent given the input’s class, so that P(Y1, Y2|Hc) = P(Y1|Hc)P(Y2|Hc) for c ∈{0, 1} (this section is the only part of the paper in which we make any independence assumptions). Since Y1 and Y2 are conditionally independent, we do not need the full joint distribution; we need only the probabilities PD1, PF 1, PD2, and PF 2 to find the combined PD and PF . For example, ℓ(01) = ((1 −PD1)PD2)/((1 −PF 1)PF 2). The assumption that f1 and f2 are conditionally independent and proper defines a partial ordering on the likelihood ratio: ℓ(00) < ℓ(10) < ℓ(11) and ℓ(00) < ℓ(01) < ℓ(11). Without loss of 3 Table 1: Two probability distributions. Class 1 (H1) Class 0 (H0) Y1 Y2 0 1 0 0.2 0.375 1 0.1 0.325 Y1 Y2 0 1 0 0.5 0.1 1 0.3 0.1 (a) Class 1 (H1) Class 0 (H0) Y1 Y2 0 1 0 0.2 0.1 1 0.2 0.5 Y1 Y2 0 1 0 0.1 0.3 1 0.5 0.1 (b) generality, we assume ℓ(00) < ℓ(01) < ℓ(10) < ℓ(11). This ordering breaks the likelihood ratio’s range (0, ∞) into five regions; choosing τ in each region defines a different decision rule. The trivial cases 0 ≤τ < ℓ(00) and ℓ(11) < τ < ∞correspond to always classifying as 1 and 0, respectively. PD and PF are therefore both equal to 1 and both equal to 0, respectively. For the case ℓ(00) ≤τ < ℓ(01), Pr [ℓ(Y) > τ] = Pr [Y = 01 ∨Y = 10 ∨Y = 11] = Pr [Y1 = 1 ∨Y2 = 1] . Thresholds in this range define an OR rule for the classifiers, with PD = PD1 + PD2 −PD1PD2 and PF = PF 1 + PF 2 −PF 1PF 2. For the case ℓ(01) ≤τ < ℓ(10), we have Pr [ℓ(Y) > τ] = Pr [Y = 10 ∨Y = 11] = Pr [Y1 = 1] . Therefore the performance probabilities are simply PD = PD1 and PF = PF 1. Finally, the case ℓ(10) ≤τ < ℓ(11) implies that Pr [ℓ(Y) > τ] = Pr [Y = 11] , and therefore thresholds in this range define an AND rule, with PD = PD1PD2 and PF = PF 1PF 2. Figure 1a illustrates this analysis with an example. The assumption of conditional independence is a sufficient condition for ensuring that the AND and OR rules improve on the ROCCH for n classifiers, as the following result shows. Theorem 2 If the distributions of the outputs of n proper binary classifiers Y1, Y2, . . . , Yn are conditionally independent given the instance class, then the points in ROC space for the rules AND (Y1 ∧Y2 ∧· · · ∧Yn) and OR (Y1 ∨Y2 ∨· · · ∨Yn) are strictly above the convex hull of the ROC curves of the base classifiers f1, . . . , fn. Furthermore, these Boolean rules belong to the LR-ROC. Proof. The likelihood ratio of the case when AND outputs 1 is given by ℓ(11 · · · 1) = (PD1PD2 · · · PDn)/(PF 1PF 2 · · · PF n). The likelihood ratio of the case when OR does not output 1 is given by ℓ(00 · · · 0) = [(1 −PD1)(1 −PD2) · · · (1 −PDn)]/[(1 −PF 1)(1 −PF 2) · · · (1 −PF n)]. Now recall that for proper classifiers fi, PDi > PF i and thus (1−PDi)/(1−PF i) < 1 < PDi/PF i. It is now clear that ℓ(00 · · · 0) is the smallest likelihood ratio and ℓ(11 · · · 1) is the largest likelihood ratio, since others are obtained only by swapping P(F,D)i and (1 −P(F,D)i), and therefore the OR and AND rules will always be part of the optimal set of decisions for conditionally independent classifiers. These rules are strictly above the ROCCH: because ℓ(11 · · · 1) > PD1/PD2, and PD1/PD2 is the slope of the line from (0, 0) to the first point in the ROCCH (f1), the AND point must be above the ROCCH. A similar argument holds for OR since ℓ(00 · · · 0) < (1 −PDn)/(1 −PF n). 2 4.2 Two examples We return now to the general case with no independence assumptions. We present two example distributions for the two-classifier case that demonstrate interesting results. The first distribution appears in Table 1a. The likelihood ratio values are ℓ(00) = 0.4, ℓ(10) = 3.75, ℓ(01) = 1/3, and ℓ(11) = 3.25, giving us ℓ(01) < ℓ(00) < ℓ(11) < ℓ(10). The three non-trivial rules correspond to the Boolean functions Y1 ∨¬Y2, Y1, and Y1 ∧¬Y2. Note that Y2 appears only negatively despite being a proper classifier, and both the AND and OR rules are sub-optimal. The distribution for the second example appears in Table 1b. The likelihood ratios of the outcomes are ℓ(00) = 2.0, ℓ(10) = 1/3, ℓ(01) = 0.4, and ℓ(11) = 5, so ℓ(10) < ℓ(01) < ℓ(00) < ℓ(11) and the three points defining the optimal ROC curve are ¬Y1 ∨Y2, ¬(Y1 ⊕Y2), and Y1 ∧Y2 (see Figure 1b). In this case, an XOR rule emerges from the likelihood ratio analysis. These examples show that for true optimal results it is not sufficient to use weighted voting rules w1Y1 + w2Y2 + · · · + wnYn ≥τ, where w ∈(0, ∞) (like some ensemble methods). Weighted voting always has AND and OR rules in its ROC curve, so it cannot always express optimal rules. 4 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 PF PD ROC of f1 ROC of f2 LR−ROC Y1 ∧Y2 Y2 Y1 Y1 ∨Y2 (a) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 PF PD ROC of f1 ROC of f2 LR−ROC Y1 ∧Y2 ¬Y1 ∨Y2 Y1 Y2 ¬(Y1 ⊕Y2) (b) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 PF PD Original ROC LR−ROC f ′ 2 f1 f ′ 1 f2 f3 f ′ 3 (c) Figure 1: (a) ROC for two conditionally independent classifiers. (b) ROC curve for the distributions in Table 1b. (c) Original ROC curve and optimal ROC curve for example in Section 4.4. 4.3 Optimality of the ROCCH We have seen that in some cases, rules exist with points strictly above the ROCCH. As the following result shows, however, there are conditions under which the ROCCH is optimal. Theorem 3 Consider n classifiers f1, . . . , fn. The convex hull of points (PF i, PDi) with (0, 0) and (1, 1) (the ROCCH) is an optimal ROC curve for the combination if (Yi = 1) ⇒(Yj = 1) for i < j and the following ordering holds: ℓ(00 · · · 0) < ℓ(00 · · · 01) < ℓ(00 · · · 011) < · · · < ℓ(1 · · · 1). Proof. The condition (Yi = 1) ⇒(Yj = 1) for i < j implies that we only need to consider n + 2 points in the ROC space (the two extra points are (0, 0) and (1, 1)) rather than 2n. It also implies the following conditions on the joint distribution: Pr[Y1 = 0 ∧· · · ∧Yi = 0 ∧Yi+1 = 1 ∧· · · ∧Yn = 1|H0] = PF i+1 −PF i, and Pr[Y1 = 1 ∧· · · ∧Yn = 1|H0] = PF 1. With these conditions and the ordering condition on the likelihood ratios, we have Pr[ℓ(Y) > ℓ(1 · · · 1)|H0] = 0, and Pr[ℓ(Y) > ℓ(0 · · · 0 | {z } i 1 · · · 1)|H0] = PF i. Therefore, finding the optimal threshold of the likelihood ratio test for PF i−1 ≤α < PF i, we get τ ∗= ℓ(0 · · · 0 | {z } i−1 1 · · · 1), and for PF i ≤α < PF i+1, τ ∗= ℓ(0 · · · 0 | {z } i 1 · · · 1). This change in τ ∗implies that the point PF i is part of the LR-ROC. Setting α = PF i (thus τ ∗= ℓ(0 · · · 0 | {z } i 1 · · · 1) and γ∗=0) implies Pr[ℓ(Y) > τ ∗|H1] = PDi. 2 The condition Yi = 1 ⇒Yj = 1 for i < j is the same inclusion condition Flach and Wu use for repairing an ROC curve [2]. It intuitively represents the performance in ROC space of a single classifier with different operating points. The next section explores this relationship further. 4.4 Repairing an ROC curve Flach and Wu give a voting technique to repair concavities in an ROC curve that generates operating points above the ROCCH [2]. Their intuition is that points underneath the convex hull can be mirrored to appear above the convex hull in much the same way as an improper classifier can be negated to obtain a proper classifier. Although their algorithm produces better ROC curves, their solution will often yield curves with new concavities (see for example Flach and Wu’s Figure 4 [2]). Their algorithm has a similar purpose to ours, but theirs is a local greedy optimization technique, while our method performs a global search in order to find the best ROC curve. Figure 1c shows an example comparing their method to ours. Consider the following probability distribution on a random variable Y ∈{0, 1}2: P((00, 10, 01, 11)|H1) = (0.1, 0.3, 0.0, 0.6), P((00, 10, 01, 11)|H0) = (0.5, 0.001, 0.4, 0.099). Flach and Wu’s method assumes the original ROC curve to be repaired has three models, or operating points: f1 predicts 1 when Y ∈{11}, f2 predicts 1 when Y ∈{11, 01}, and f3 predicts 1 when Y ∈{11, 01, 10}. If we apply Flach and Wu’s repair algorithm, the point f2 is corrected to the point f ′ 2; however, the operating points of f1 and f3 remain the same. 5 0.00 0.05 0.10 0.15 0.20 0.0 0.2 0.4 0.6 0.8 1.0 Pfa Pd Meta (train) Base (train) Meta (test) Base (test) PART (a) adult 0.000 0.005 0.010 0.015 0.0 0.2 0.4 0.6 0.8 1.0 Pfa Pd Meta (train) Base (train) Meta (test) Base (test) PART (b) hypothyroid 0.00 0.05 0.10 0.15 0.0 0.2 0.4 0.6 0.8 1.0 Pfa Pd Meta (train) Base (train) Meta (test) Base (test) PART (c) sick-euthyroid 0.00 0.02 0.04 0.06 0.08 0.10 0.0 0.2 0.4 0.6 0.8 1.0 Pfa Pd Meta (train) Base (train) Meta (test) Base (test) PART (d) sick Figure 2: Empirical ROC curves for experimental results on four UCI datasets. Our method improves on this result by ordering the likelihood ratios ℓ(01) < ℓ(00) < ℓ(11) < ℓ(10) and using that ordering to make three different rules: f ′ 1 predicts 1 when Y ∈{10}, f ′ 2 predicts 1 when Y ∈{10, 11}, and f ′ 3 predicts 1 when Y ∈{10, 11, 00}. 5 Experiments We ran experiments to test the performance of our combining method on the adult, hypothyroid, sick-euthyroid, and sick datasets from the UCI machine learning repository [6]. We chose five base classifiers from the YALE machine learning platform [7]: PART (a decision list algorithm), SMO (Sequential Minimal Optimization), SimpleLogistic, VotedPerceptron, and Y-NaiveBayes. We used default settings for all classifiers. The adult dataset has around 30,000 training points and 15,000 test points and the sick dataset has around 2000 training points and 700 test points. The others each have around 2000 points that we split randomly into 1000 training and 1000 test. For each experiment, we estimate the joint distribution by training the base classifiers on a training set and counting the outcomes. We compute likelihood ratios for all outcomes and order them. When outcomes have no examples, we treat ·/0 as near-infinite and 0/· as near-zero and define 0/0 = 1. 6 We derive a sequence of decision rules from the likelihood ratios computed on the training set. We can compute an optimal ROC curve for the combination by counting the number of true positives and false positives each rule achieves. In the test set we use the rules learned on the training set. 5.1 Results The ROC graphs for our four experiments appear in Figure 2. The ROC curves in these experiments all rise very quickly and then flatten out, so we show only the range of PF 1 for which the values are interesting. We can draw some general conclusions from these graphs. First, PART clearly outperforms the other base classifiers in three out of four experiments, though it seems to overfit on the hypothyroid dataset. The LR-ROC dominates the ROC curves of the base classifiers on both training and test sets. The ROC curves for the base classifiers are all strictly below the LR-ROC in results on the test sets. The results on training sets seem to imply that the LR-ROC is primarily classifying like PART with a small boost from the other classifiers; however, the test set results (in particular, Figure 2b) demonstrate that the LR-ROC generalizes better than the base classifiers. The robustness of our method to estimation errors is uncertain. In our experiments we found that smoothing did not improve generalization, but undoubtedly our method would benefit from better estimation of the outcome distribution and increased robustness. We ran separate experiments to test how many classifiers our method could support in practice. Estimation of the joint distribution and computation of the ROC curve finished within one minute for 20 classifiers (not including time to train the individual classifiers). Unfortunately, the inherent exponential structure of the optimal ROC curve means we cannot expect to do significantly better (at the same rate, 30 classifiers would take over 12 hours and 40 classifiers almost a year and a half). 6 Related work Our work is loosely related to ensemble methods such as bagging [8] and boosting [9] because it finds meta-classification rules over a set of base classifiers. However, bagging and boosting each take one base classifier and train many times, resampling or reweighting the training data to generate classifier diversity [10] or increase the classification margin [11]. The decision rules applied to the generated classifiers are (weighted) majority voting. In contrast, our method takes any binary classifiers and finds optimal combination rules from the more general space of all binary functions. Ranking algorithms, such as RankBoost [12], approach the problem of ranking points by score or preference. Although we present our methods in a different light, our decision rule can be interpreted as a ranking algorithm. RankBoost, however, is a boosting algorithm and therefore fundamentally different from our approach. Ranking can be used for classification by choosing a cutoff or threshold, and in fact ranking algorithms tend to optimize the common Area Under the ROC Curve (AUC) metric. Although our method may have the side effect of maximizing the AUC, its formulation is different in that instead of optimizing a single global metric, it is a constrained optimization problem, maximizing PD for each PF . Another more similar method for combining classifiers is stacking [13]. Stacking trains a metalearner to combine the predictions of several base classifiers; in fact, our method might be considered a stacking method with a particular meta-classifier. It can be difficult to show the improvement of stacking in general over selecting the best base-level classifier [14]; however, stacking has a useful interpretation as generalized cross-validation that makes it practical. Our analysis shows that our combination method is the optimal meta-learner in the Neyman-Pearson sense, but incorporating the model validation aspect of stacking would make an interesting extension to our work. 7 Conclusion In this paper we introduce a new way to analyze a combination of classifiers and their ROC curves. We give a method for combining classifiers and prove that it is optimal in the Neyman-Pearson sense. This work raises several interesting questions. Although the algorithm presented in this paper avoids checking the whole doubly exponential number of rules, the exponential factor in running time limits the number of classifiers that can be 7 combined in practice. Can a good approximation algorithm approach optimality while having lower time complexity? Though in general we make no assumptions about independence, Theorem 2 shows that certain simple rules are optimal when we do know that the classifiers are independent. Theorem 3 proves that the ROCCH can be optimal when only n output combinations are possible. Perhaps other restrictions on the distribution of outcomes can lead to useful special cases. Acknowledgments This work was supported in part by TRUST (Team for Research in Ubiquitous Secure Technology), which receives support from the National Science Foundation (NSF award number CCF-0424422) and the following organizations: AFOSR (#FA9550-06-1-0244), Cisco, British Telecom, ESCHER, HP, IBM, iCAST, Intel, Microsoft, ORNL, Pirelli, Qualcomm, Sun, Symantec, Telecom Italia, and United Technologies; and in part by the UC Berkeley-Taiwan International Collaboration in Advanced Security Technologies (iCAST) program. The opinions expressed in this paper are solely those of the authors and do not necessarily reflect the opinions of any funding agency or the U.S. or Taiwanese governments. References [1] Foster Provost and Tom Fawcett. Robust classification for imprecise environments. Machine Learning Journal, 42(3):203–231, March 2001. [2] Peter A. Flach and Shaomin Wu. Repairing concavities in ROC curves. In Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI’05), pages 702–707, August 2005. [3] Tom Fawcett. ROC graphs: Notes and practical considerations for data mining researchers. Technical Report HPL-2003-4, HP Laboratories, Palo Alto, CA, January 2003. Updated March 2004. [4] J. Neyman and E. S. Pearson. On the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society of London, Series A, Containing Papers of a Mathematical or Physical Character, 231:289–337, 1933. [5] Vincent H. Poor. An Introduction to Signal Detection and Estimation. Springer-Verlag, second edition, 1988. [6] D. J. Newman, S. Hettich, C. L. Blake, and C. J. Merz. UCI repository of machine learning databases, 1998. http://www.ics.uci.edu/∼mlearn/MLRepository.html. [7] I. Mierswa, M. Wurst, R. Klinkenberg, M. Scholz, and T. Euler. YALE: Rapid prototyping for complex data mining tasks. In Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD), 2006. [8] L. Breiman. Bagging predictors. Machine Learning, 24(2):123–140, 1996. [9] Y. Freund and R. E. Schapire. Experiments with a new boosting algorithm. In Thirteenth International Conference on Machine Learning, pages 148–156, Bari, Italy, 1996. Morgan Kaufmann. [10] Thomas G. Dietterich. Ensemble methods in machine learning. Lecture Notes in Computer Science, 1857:1–15, 2000. [11] Robert E. Schapire, Yoav Freund, Peter Bartlett, and Wee Sun Lee. Boosting the margin: A new explanation for the effectiveness of voting methods. The Annals of Statistics, 26(5):1651–1686, October 1998. [12] Yoav Freund, Raj Iyer, Robert E. Schapire, and Yoram Singer. An efficient boosting algorithm for combining preferences. Journal of Machine Learning Research (JMLR), 4:933–969, 2003. [13] D. H. Wolpert. Stacked generalization. Neural Networks, 5:241–259, 1992. [14] Saso D˘zeroski and Bernard ˘Zenko. Is combining classifiers with stacking better than selecting the best one? Machine Learning, 54:255–273, 2004. 8
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Modeling homophily and stochastic equivalence in symmetric relational data Peter D. Hoff Departments of Statistics and Biostatistics University of Washington Seattle, WA 98195-4322. hoff@stat.washington.edu Abstract This article discusses a latent variable model for inference and prediction of symmetric relational data. The model, based on the idea of the eigenvalue decomposition, represents the relationship between two nodes as the weighted inner-product of node-specific vectors of latent characteristics. This “eigenmodel” generalizes other popular latent variable models, such as latent class and distance models: It is shown mathematically that any latent class or distance model has a representation as an eigenmodel, but not vice-versa. The practical implications of this are examined in the context of three real datasets, for which the eigenmodel has as good or better out-of-sample predictive performance than the other two models. 1 Introduction Let {yi,j : 1 ≤i < j ≤n} denote data measured on pairs of a set of n objects or nodes. The examples considered in this article include friendships among people, associations among words and interactions among proteins. Such measurements are often represented by a sociomatrix Y , which is a symmetric n × n matrix with an undefined diagonal. One of the goals of relational data analysis is to describe the variation among the entries of Y , as well as any potential covariation of Y with observed explanatory variables X = {xi,j, 1 ≤i < j ≤n}. To this end, a variety of statistical models have been developed that describe yi,j as some function of node-specific latent variables ui and uj and a linear predictor βT xi,j. In such formulations, {u1, . . . , un} represent across-node variation in the yi,j’s and β represents covariation of the yi,j’s with the xi,j’s. For example, Nowicki and Snijders [2001] present a model in which each node i is assumed to belong to an unobserved latent class ui, and a probability distribution describes the relationships between each pair of classes (see Kemp et al. [2004] and Airoldi et al. [2005] for recent extensions of this approach). Such a model captures stochastic equivalence, a type of pattern often seen in network data in which the nodes can be divided into groups such that members of the same group have similar patterns of relationships. An alternative approach to representing across-node variation is based on the idea of homophily, in which the relationships between nodes with similar characteristics are stronger than the relationships between nodes having different characteristics. Homophily provides an explanation to data patterns often seen in social networks, such as transitivity (“a friend of a friend is a friend”), balance (“the enemy of my friend is an enemy”) and the existence of cohesive subgroups of nodes. In order to represent such patterns, Hoff et al. [2002] present a model in which the conditional mean of yi,j is a function of β′xi,j −|ui −uj|, where {u1, . . . , un} are vectors of unobserved, latent characteristics in a Euclidean space. In the context of binary relational data, such a model predicts the existence of more transitive triples, or “triangles,” than would be seen under a random allocation of edges among pairs of nodes. An important assumption of this model is that two nodes with a strong 1 G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G Figure 1: Networks exhibiting homophily (left panel) and stochastic equivalence (right panel). relationship between them are also similar to each other in terms of how they relate to other nodes: A strong relationship between i and j suggests |ui −uj| is small, but this further implies that |ui −uk| ≈|uj −uk|, and so nodes i and j are assumed to have similar relationships to other nodes. The latent class model of Nowicki and Snijders [2001] and the latent distance model of Hoff et al. [2002] are able to identify, respectively, classes of nodes with similar roles, and the locational properties of the nodes. These two items are perhaps the two primary features of interest in social network and relational data analysis. For example, discussion of these concepts makes up more than half of the 734 pages of main text in Wasserman and Faust [1994]. However, a model that can represent one feature may not be able to represent the other: Consider the two graphs in Figure 1. The graph on the left displays a large degree of transitivity, and can be well-represented by the latent distance model with a set of vectors {u1, . . . , un} in two-dimensional space, in which the probability of an edge between i and j is decreasing in |ui −uj|. In contrast, representation of the graph by a latent class model would require a large number of classes, none of which would be particularly cohesive or distinguishable from the others. The second panel of Figure 1 displays a network involving three classes of stochastically equivalent nodes, two of which (say A and B) have only across-class ties, and one (C) that has both within- and across-class ties. This graph is well-represented by a latent class model in which edges occur with high probability between pairs having one member in each of A and B or in B and C, and among pairs having both members in C (in models of stochastic equivalence, nodes within each class are not differentiated). In contrast, representation of this type of graph with a latent distance model would require the dimension of the latent characteristics to be on the order of the class membership sizes. Many real networks exhibit combinations of structural equivalence and homophily in varying degrees. In these situations, use of either the latent class or distance model would only be representing part of the network structure. The goal of this paper is to show that a simple statistical model based on the eigenvalue decomposition can generalize the latent class and distance models: Just as any symmetric matrix can be approximated with a subset of its largest eigenvalues and corresponding eigenvectors, the variation in a sociomatrix can be represented by modeling yi,j as a function of β′xi,j + uT i Λuj, where {u1, . . . , un} are node-specific factors and Λ is a diagonal matrix. In this article, we show mathematically and by example how this eigenmodel can represent both stochastic equivalence and homophily in symmetric relational data, and thus is more general than the other two latent variable models. The next section motivates the use of latent variables models for relational data, and shows mathematically that the eigenmodel generalizes the latent class and distance models in the sense that it can compactly represent the same network features as these other models but not vice-versa. Section 3 compares the out-of-sample predictive performance of these three models on three different datasets: a social network of 12th graders; a relational dataset on word association counts from the first chapter of Genesis; and a dataset on protein-protein interactions. The first two networks exhibit latent homophily and stochastic equivalence respectively, whereas the third shows both to some degree. 2 In support of the theoretical results of Section 2, the latent distance and class models perform well for the first and second datasets respectively, whereas the eigenmodel performs well for all three. Section 4 summarizes the results and discusses some extensions. 2 Latent variable modeling of relational data 2.1 Justification of latent variable modeling The use of probabilistic latent variable models for the representation of relational data can be motivated in a natural way: For undirected data without covariate information, symmetry suggests that any probability model we consider should treat the nodes as being exchangeable, so that Pr({yi,j : 1 ≤i < j ≤n} ∈A) = Pr({yπi,πj : 1 ≤i < j ≤n} ∈A) for any permutation π of the integers {1, . . . , n} and any set of sociomatrices A. Results of Hoover [1982] and Aldous [1985, chap. 14] show that if a model satisfies the above exchangeability condition for each integer n, then it can be written as a latent variable model of the form yi,j = h(µ, ui, uj, ϵi,j) (1) for i.i.d. latent variables {u1, . . . , un}, i.i.d. pair-specific effects {ϵi,j : 1 ≤i < j ≤n} and some function h that is symmetric in its second and third arguments. This result is very general - it says that any statistical model for a sociomatrix in which the nodes are exchangeable can be written as a latent variable model. Difference choices of h lead to different models for y. A general probit model for binary network data can be put in the form of (1) as follows: {ϵi,j : 1 ≤i < j ≤n} ∼ i.i.d. normal(0, 1) {u1, . . . , un} ∼ i.i.d. f(u|ψ) yi,j = h(µ, ui, uj, ϵi,j) = δ(0,∞)(µ + α(ui, uj) + ϵi,j), where µ and ψ are parameters to be estimated, and α is a symmetric function, also potentially involving parameters to be estimated. Covariation between Y and an array of predictor variables X can be represented by adding a linear predictor βT xi,j to µ. Finally, integrating over ϵi,j we obtain Pr(yi,j = 1|xi,j, ui, uj) = Φ[µ + βT xi,j + α(ui, uj)]. Since the ϵi,j’s can be assumed to be independent, the conditional probability of Y given X and {u1, . . . , un} can be expressed as Pr(yi,j = 1|xi,j, ui, uj) ≡θi,j = Φ[µ + βT xi,j + α(ui, uj)] (2) Pr(Y |X, u1, . . . , un) = Y i<j θyi,j i,j (1 −θi,j)yi,j Many relational datasets have ordinal, non-binary measurements (for example, the word association data in Section 3.2). Rather than “thresholding” the data to force it to be binary, we can make use of the full information in the data with an ordered probit version of (2): Pr(yi,j = y|xi,j, ui, uj) ≡θ(y) i,j = Φ[µy + βT xi,j + α(ui, uj)] −Φ[µy+1 + βT xi,j + α(ui, uj)] Pr(Y |X, u1, . . . , un) = Y i<j θ(yi,j) i,j , where {µy} are parameters to be estimated for all but the lowest value y in the sample space. 2.2 Effects of nodal variation The latent variable models described in the Introduction correspond to different choices for the symmetric function α: Latent class model: α(ui, uj) = mui,uj ui ∈{1, . . . , K}, i ∈{1, . . . , n} 3 M a K × K symmetric matrix Latent distance model: α(ui, uj) = −|ui −uj| ui ∈RK, i ∈{1, . . . , n} Latent eigenmodel: α(ui, uj) = uT i Λuj ui ∈RK, i ∈{1, . . . , n} Λ a K × K diagonal matrix. Interpretations of the latent class and distance models were given in the Introduction. An interpretation of the latent eigenmodel is that each node i has a vector of unobserved characteristics ui = {ui,1, . . . , ui,K}, and that similar values of ui,k and uj,k will contribute positively or negatively to the relationship between i and j, depending on whether λk > 0 or λk < 0. In this way, the model can represent both positive or negative homophily in varying degrees, and stochastically equivalent nodes (nodes with the same or similar latent vectors) may or may not have strong relationships with one another. We now show that the eigenmodel generalizes the latent class and distance models: Let Sn be the set of n × n sociomatrices, and let CK = {C ∈Sn : ci,j = mui,uj, ui ∈{1, . . . , K}, M a K × K symmetric matrix}; DK = {D ∈Sn : di,j = −|ui −uj|, ui ∈RK}; EK = {E ∈Sn : ei,j = uT i Λuj, ui ∈RK, Λ a K × K diagonal matrix}. In other words, CK is the set of possible values of {α(ui, uj), 1 ≤i < j ≤n} under a Kdimensional latent class model, and similarly for DK and EK. EK generalizes CK: Let C ∈CK and let ˜C be a completion of C obtained by setting ci,i = mui,ui. There are at most K unique rows of ˜C and so ˜C is of rank K at most. Since the set EK contains all sociomatrices that can be completed as a rank-K matrix, we have CK ⊆EK. Since EK includes matrices with n unique rows, CK ⊂EK unless K ≥n in which case the two sets are equal. EK+1 weakly generalizes DK: Let D ∈DK. Such a (negative) distance matrix will generally be of full rank, in which case it cannot be represented exactly by an E ∈EK for K < n. However, what is critical from a modeling perspective is whether or not the order of the entries of each D can be matched by the order of the entries of an E. This is because the probit and ordered probit model we are considering include threshold variables {µy : y ∈Y} which can be adjusted to accommodate monotone transformations of α(ui, uj). With this in mind, note that the matrix of squared distances among a set of K-dimensional vectors {z1, . . . , zn} is a monotonic transformation of the distances, is of rank K + 2 or less (as D2 = [z′ 1z1, . . . , z′ nzn]T 1T + 1[z′ 1z1, . . . , z′ nzn] −2ZZT ) and so is in EK+2. Furthermore, letting ui = (zi, p r2 −zT i zi) ∈RK+1 for each i ∈{1, . . . , n}, we have u′ iuj = z′ izj + p (r2 −|ui|2)(r2 −|uj|2). For large r this is approximately r2 −|zi −zj|2/2, which is an increasing function of the negative distance di,j. For large enough r the numerical order of the entries of this E ∈EK+1 is the same as that of D ∈DK. DK does not weakly generalize E1: Consider E ∈E1 generated by Λ = 1, u1 = 1 and ui = r < 1 for i > 1. Then r = e1,i1 = e1,i2 > ei1,i2 = r2 for all i1, i2 ̸= 1. For which K is such an ordering of the elements of D ∈DK possible? If K = 1 then such an ordering is possible only if n = 3. For K = 2 such an ordering is possible for n ≤6. This is because the kissing number in R2, or the number of non-overlapping spheres of unit radius that can simultaneously touch a central sphere of unit radius, is 6. If we put node 1 at the center of the central sphere, and 6 nodes at the centers of the 6 kissing spheres, then we have d1,i1 = d1,i2 = di1,i2 for all i1, i2 ̸= 1. We can only have d1,i1 = d1,i2 > di1,i2 if we remove one of the non-central spheres to allow for more room between those remaining, leaving one central sphere plus five kissing spheres for a total of n = 6. Increasing n increases the necessary dimension of the Euclidean space, and so for any K there are n and E ∈E1 that have entry orderings that cannot be matched by those of any D ∈DK. 4 A less general positive semi-definite version of the eigenmodel has been studied by Hoff [2005], in which Λ was taken to be the identity matrix. Such a model can weakly generalize a distance model, but cannot generalize a latent class model, as the eigenvalues of a latent class model could be negative. 3 Model comparison on three different datasets 3.1 Parameter estimation Bayesian parameter estimation for the three models under consideration can be achieved via Markov chain Monte Carlo (MCMC) algorithms, in which posterior distributions for the unknown quantities are approximated with empirical distributions of samples from a Markov chain. For these algorithms, it is useful to formulate the probit models described in Section 2.1 in terms of an additional latent variable zi,j ∼normal[β′xi,j + α(ui, uj)], for which yi,j = y if µy < zi,j < µy+1. Using conjugate prior distributions where possible, the MCMC algorithms proceed by generating a new state φ(s+1) = {Z(s+1), µ(s+1), β(s+1), u(s+1) 1 , . . . , u(s+1) n } from a current state φ(s) as follows: 1. For each {i, j}, sample zi,j from its (constrained normal) full conditional distribution. 2. For each y ∈Y, sample µy from its (normal) full conditional distribution. 3. Sample β from its (multivariate normal) full conditional distribution. 4. Sample u1, . . . , un and their associated parameters: • For the latent distance model, propose and accept or reject new values of the ui’s with the Metropolis algorithm, and then sample the population variances of the ui’s from their (inverse-gamma) full conditional distributions. • For the latent class model, update each class variable ui from its (multinomial) conditional distribution given current values of Z, {uj : j ̸= i} and the variance of the elements of M (but marginally over M to improve mixing). Then sample the elements of M from their (normal) full conditional distributions and the variance of the entries of M from its (inverse-gamma) full conditional distribution. • For the latent vector model, sample each ui from its (multivariate normal) full conditional distribution, sample the mean of the ui’s from their (normal) full conditional distributions, and then sample Λ from its (multivariate normal) full conditional distribution. To facilitate comparison across models, we used prior distributions in which the level of prior variability in α(ui, uj) was similar across the three different models (further details and code to implement these algorithms are available at my website). 3.2 Cross validation To compare the performance of these three different models we evaluated their out-of-sample predictive performance under a range of dimensions (K ∈{3, 5, 10}) and on three different datasets exhibiting varying combinations of homophily and stochastic equivalence. For each combination of dataset, dimension and model we performed a five-fold cross validation experiment as follows: 1. Randomly divide the n 2 data values into 5 sets of roughly equal size, letting si,j be the set to which pair {i, j} is assigned. 2. For each s ∈{1, . . . , 5}: (a) Obtain posterior distributions of the model parameter conditional on {yi,j : si,j ̸= s}, the data on pairs not in set s. (b) For pairs {k, l} in set s, let ˆyk,l = E[yk,l|{yi,j : si,j ̸= s}], the posterior predictive mean of yk,l obtained using data not in set s. This procedure generates a sociomatrix ˆY , in which each entry ˆyi,j represents a predicted value obtained from using a subset of the data that does not include yi,j. Thus ˆY is a sociomatrix of out-of-sample predictions of the observed data Y . 5 Table 1: Cross validation results and area under the ROC curves. K Add health Genesis Protein interaction dist class eigen dist class eigen dist class eigen 3 0.82 0.64 0.75 0.62 0.82 0.82 0.83 0.79 0.88 5 0.81 0.70 0.78 0.66 0.82 0.82 0.84 0.84 0.90 10 0.76 0.69 0.80 0.74 0.82 0.82 0.85 0.86 0.90 G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G 0 5000 15000 25000 0 100 200 300 400 false positives true positives distance class vector Figure 2: Social network data and unscaled ROC curves for the K = 3 models. 3.3 Adolescent Health social network The first dataset records friendship ties among 247 12th-graders, obtained from the National Longitudinal Study of Adolescent Health (www.cpc.unc.edu/projects/addhealth). For these data, yi,j = 1 or 0 depending on whether or not there is a close friendship tie between student i and j (as reported by either i or j). These data are represented as an undirected graph in the first panel of Figure 2. Like many social networks, these data exhibit a good deal of transitivity. It is therefore not surprising that the best performing models considered (in terms of area under the ROC curve, given in Table 1) are the distance models, with the eigenmodels close behind. In contrast, the latent class models perform poorly, and the results suggest that increasing K for this model would not improve its performance. 3.4 Word neighbors in Genesis The second dataset we consider is derived from word and punctuation counts in the first chapter of the King James version of Genesis (www.gutenberg.org/dirs/etext05/bib0110.txt). There are 158 unique words and punctuation marks in this chapter, and for our example we take yi,j to be the number of times that word i and word j appear next to each other (a model extension, appropriate for an asymmetric version of this dataset, is discussed in the next section). These data can be viewed as a graph with weighted edges, the unweighted version of which is shown in the first panel of Figure 3. The lack of a clear spatial representation of these data is not unexpected, as text data such as these do not have groups of words with strong within-group connections, nor do they display much homophily: a given noun may appear quite frequently next to two different verbs, but these verbs will not appear next to each other. A better description of these data might be that there are classes of words, and connections occur between words of different classes. The cross validation results support this claim, in that the latent class model performs much better than the distance model on these data, as seen in the second panel of Figure 3 and in Table 1. As discussed in the previous section, the eigenmodel generalizes the latent class model and performs equally well. 6 , ; : . a above abundantly after air all also and appear be bearing beast beginning behold blessed bring brought calledcattle created creature creepeth creeping darkness day days deep divide divided dominion dry earth evening every face female fifth fill finished firmament first fish fly for form forth fourth fowl from fruit fruitful gathered gathering give given god good grass great greater green had hath have he heaven heavens herb him his host i image in is it itself kind land lesser let life light lights likeness living made make male man may meat midst morning moved moveth moving multiply night of one open our over own place replenish rule said saw saying seaseas seasons second seed set shall signs sixth so spirit stars subdue that the their them there thing third thus to together tree two under unto upon us very void was waters were whales wherein which whose winged without years yielding you 0 4000 8000 12000 0 100 200 300 400 false positives true positives distance class vector Figure 3: Relational text data from Genesis and unscaled ROC curves for the K = 3 models. We note that parameter estimates for these data were obtained using the ordered probit versions of the models (as the data are not binary), but the out-of-sample predictive performance was evaluated based on each model’s ability to predict a non-zero relationship. 3.5 Protein-protein interaction data Our last example is the protein-protein interaction data of Butland et al. [2005], in which yi,j = 1 if proteins i and j bind and yi,j = 0 otherwise. We analyze the large connected component of this graph, which includes 230 proteins and is displayed in the first panel of 4. This graph indicates patterns of both stochastic equivalence and homophily: Some nodes could be described as “hubs”, connecting to many other nodes which in turn do not connect to each other. Such structure is better represented by a latent class model than a distance model. However, most nodes connecting to hubs generally connect to only one hub, which is a feature that is hard to represent with a small number of latent classes. To represent this structure well, we would need two latent classes per hub, one for the hub itself and one for the nodes connecting to the hub. Furthermore, the core of the network (the nodes with more than two connections) displays a good degree of homophily in the form of transitive triads, a feature which is easiest to represent with a distance model. The eigenmodel is able to capture both of these data features and performs better than the other two models in terms of out-of-sample predictive performance. In fact, the K = 3 eigenmodel performs better than the other two models for any value of K considered. 4 Discussion Latent distance and latent class models provide concise, easily interpreted descriptions of social networks and relational data. However, neither of these models will provide a complete picture of relational data that exhibit degrees of both homophily and stochastic equivalence. In contrast, we have shown that a latent eigenmodel is able to represent datasets with either or both of these data patterns. This is due to the fact that the eigenmodel provides an unrestricted low-rank approximation to the sociomatrix, and is therefore able to represent a wide array of patterns in the data. The concept behind the eigenmodel is the familiar eigenvalue decomposition of a symmetric matrix. The analogue for directed networks or rectangular matrix data would be a model based on the singular value decomposition, in which data yi,j could be modeled as depending on uT i Dvj, where ui and vj represent vectors of latent row and column effects respectively. Statistical inference using the singular value decomposition for Gaussian data is straightforward. A model-based version of 7 G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G 0 5000 15000 25000 0 100 300 500 700 false positives true positives distance class vector Figure 4: Protein-protein interaction data and unscaled ROC curves for the K = 3 models. the approach for binary and other non-Gaussian relational datasets could be implemented using the ordered probit model discussed in this paper. Acknowledgment This work was partially funded by NSF grant number 0631531. References Edoardo Airoldi, David Blei, Eric Xing, and Stephen Fienberg. A latent mixed membership model for relational data. In LinkKDD ’05: Proceedings of the 3rd international workshop on Link discovery, pages 82–89, New York, NY, USA, 2005. ACM Press. ISBN 1-59593-215-1. doi: http://doi.acm.org/10.1145/1134271.1134283. David J. Aldous. Exchangeability and related topics. In ´Ecole d’´et´e de probabilit´es de Saint-Flour, XIII—1983, volume 1117 of Lecture Notes in Math., pages 1–198. Springer, Berlin, 1985. G. Butland, J. M. Peregrin-Alvarez, J. Li, W. Yang, X. Yang, V. Canadien, A. Starostine, D. Richards, B. Beattie, N. Krogan, M. Davey, J. Parkinson, J. Greenblatt, and A. Emili. Interaction network containing conserved and essential protein complexes in escherichia coli. Nature, 433:531–537, 2005. Peter D. Hoff. Bilinear mixed-effects models for dyadic data. J. Amer. Statist. Assoc., 100(469): 286–295, 2005. ISSN 0162-1459. Peter D. Hoff, Adrian E. Raftery, and Mark S. Handcock. Latent space approaches to social network analysis. J. Amer. Statist. Assoc., 97(460):1090–1098, 2002. ISSN 0162-1459. D. N. Hoover. Row-column exchangeability and a generalized model for probability. In Exchangeability in probability and statistics (Rome, 1981), pages 281–291. North-Holland, Amsterdam, 1982. Charles Kemp, Thomas L. Griffiths, and Joshua B. Tenenbaum. Discovering latent classes in relational data. AI Memo 2004-019, Massachusetts Institute of Technology, 2004. Krzysztof Nowicki and Tom A. B. Snijders. Estimation and prediction for stochastic blockstructures. J. Amer. Statist. Assoc., 96(455):1077–1087, 2001. ISSN 0162-1459. Stanley Wasserman and Katherine Faust. Social Network Analysis: Methods and Applications. Cambridge University Press, Cambridge, 1994. 8
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On Sparsity and Overcompleteness in Image Models Pietro Berkes, Richard Turner, and Maneesh Sahani Gatsby Computational Neuroscience Unit, UCL Alexandra House, 17 Queen Square, London WC1N 3AR Abstract Computational models of visual cortex, and in particular those based on sparse coding, have enjoyed much recent attention. Despite this currency, the question of how sparse or how over-complete a sparse representation should be, has gone without principled answer. Here, we use Bayesian model-selection methods to address these questions for a sparse-coding model based on a Student-t prior. Having validated our methods on toy data, we find that natural images are indeed best modelled by extremely sparse distributions; although for the Student-t prior, the associated optimal basis size is only modestly over-complete. 1 Introduction Computational models of visual cortex, and in particular those based on sparse coding, have recently enjoyed much attention. The basic assumption behind sparse coding is that natural scenes are composed of structural primitives (edges or lines, for example) and, although there are a potentially large number of these primitives, typically only a few are active in a single natural scene (hence the term sparse, [1, 2]). The claim is that cortical processing uses these statistical regularities to shape a representation of natural scenes, and in particular converts the pixel-based representation at the retina to a higher-level representation in terms of these structural primitives. Traditionally, research has focused on determining the characteristics of the structural primitives and comparing their representational properties with those of V1. This has been a successful enterprise, but as a consequence other important questions have been neglected. The two we focus on here are: How large is the set of structural primitives best suited to describe all natural scenes (how over-complete), and how many primitives are active in a single scene (how sparse)? We will also be interested in the coupling between sparseness and over-completeness. The intuition is that, if there are a great number of structural primitives, they can be very specific and only a small number will be active in a visual scene. Conversely if there are a small number they have to be more general and a larger number will be active on average. We attempt to map this coupling by evaluating models with different over-completenesses and sparsenesses and discover where natural scenes live along this trade-off (see Fig. 1). In order to test the sparse coding hypothesis it is necessary to build algorithms that both learn the primitives and decompose natural scenes in terms of them. There have been many ways to derive such algorithms, but one of the more successful is to regard the task of building a representation of natural scenes as one of probabilistic inference. More specifically, the unknown activities of the structural primitives are viewed as latent variables that must be inferred from the natural scene data. Commonly the inference is carried out by writing down a generative model (although see [3] for an alternative), which formalises the assumptions made about the data and latent variables. The rules of probability are then used to derive inference and learning algorithms. Unfortunately the assumption that natural scenes are composed of a small number of structural primitives is not sufficient to build a meaningful generative model. Other assumptions must therefore be made and typically these are that the primitives occur independently, and combine linearly. These 1 0 2 4 6 8 10 sparsity overcompleteness Figure 1: Schematic showing the space of possible sparse coding models in terms of sparseness (increasing in the direction of the arrow) and over-completeness. For reference, complete models lie along the dashed black line. Ideally every model could be evaluated (e.g. via their marginal likelihood or cross-validation) and the grey contours illustrate what we might expect to discover if this were possible: The solid black line illustrates the hypothesised trade-off between over-completeness and sparsity, whilst the star shows the optimal point in this trade-off. are drastic approximations and it is an open question to what extent this affects the results of sparse coding. The distribution over the latent variables xt,k is chosen to be sparse and typical choices are Student-t, a Mixture of Gaussians (with zero means), and the Generalised Gaussian (which includes the Laplace distribution). The output yt is then given by a linear combination of the K, D-dimensional structural primitives gk, weighted by their activities, plus some additive Gaussian noise (the model reduces to independent components analysis in the absence of this noise [4]), p(xt,k|α) = psparse(α) (1) p(yt|xt, G) = Nyt(Gxt, Σy) . (2) The goal of this paper will be to learn the optimal dimensionality of the latent variables (K) and the optimal sparseness of the prior (α). In order to do this a notion of optimality has to be defined. One option is to train many different sparse-coding models and find the one which is most “similar” to visual processing. (Indeed this might be a fair characterisation of much of the current activity in field.) However, this is fraught with difficulty not least as it is unclear how recognition models map to neural processes. We believe the more consistent approach is, once again, to use the Bayesian framework and view this as a problem of probabilistic inference. In fact, if the hypothesis is that the visual system is implementing an optimal generative model, then questions of over-completeness and sparsity should be addressed in this context. Unfortunately, this is not a simple task and quite sophisticated machine-learning algorithms have to be harnessed in order to answer these seemingly simple questions. In the first part of this paper we describe these algorithms and then validate them using artificial data. Finally, we present results concerning the optimal sparseness and over-completeness for natural image patches in the case that the prior is a Student-t distribution. 2 Model As discussed earlier, there are many variants of sparse-coding. Here, we focus on the Student-t prior for the latent variables xt,k: p(xt,k|α, λ) = Γ α+1 2 λ√απ Γ α 2 1 + 1 α xt,k λ 2−α+1 2 (3) There are two main reasons for this choice: The first is that this is a widely used model [1]. The second is that by implementing the Student-t prior using an auxiliary variable, all the distributions in the generative model become members of the exponential family [5]. This means it is easy to derive efficient approximate inference schemes like variational Bayes and Gibbs sampling. The auxiliary variable method is based on the observation that a Student-t distribution is a continuous mixture of zero-mean Gaussians, whose mixing proportions are given by a Gamma distribution over 2 the precisions. This indicates that we can exchange the Student-t prior for a two-step prior in which we first draw a precision from a Gamma distribution and then draw an activation from a Gaussian with that precision, p(ut,k|α, λ) = Gut,k α 2 , 2 αλ2 , (4) p(xt,k|ut,k) = Nxt,k 0, u−1 t,k , (5) p(yt|xt, G) = Nyt(Gxt, Σy) , (6) Σy := diag σ2 y . (7) This model produces data which are often near zero, but occasionally highly non-zero. These nonzero elements form star-like patterns, where the points of the star are determined by the direction of the weights (e.g., Fig. 2). One of the major technical difficulties posed by sparse-coding is that, in the over-complete regime, the posterior distribution of the latent variables p(X|Y, θ) is often complex and multi-modal. Approximation schemes are therefore required, but we must be careful to ensure that the scheme we choose does not bias the conclusions we are trying to draw. This is true for any application of sparse coding, but is particularly pertinent for our problem as we will be quantitatively comparing different sparse-coding models. 3 Bayesian Model Comparison A possible strategy for investigating the sparseness/over-completeness coupling would be to tile the space with models and learn the parameters at each point (as schematised in Fig. 1). A model comparison criterion could then be used to rank the models, and to find the optimal sparseness/overcompleteness. One such criterion would be to use cross validation and evaluate the likelihoods on some held-out test data. Another is to use (approximate) Bayesian Model Comparison, and it is on this method that we focus. To evaluate the plausibility of two alternative versions of a model M, each with a different setting of the hyperparameters Ξ1 and Ξ2, in the light of some data Y , we compute the evidence [6]: p(M, Ξ1|Y ) p(M, Ξ2|Y ) = p(Y |M, Ξ1) P(M, Ξ1) p(Y |M, Ξ2) P(M, Ξ2) . (8) Since we do not have any reason a priori to prefer one particular configuration of hyperparameters to another, we take the prior terms P(M, Ξi) to be equal, which leaves us with the ratio of the marginal-likelihoods (or Bayes Factor), P(Y |M, Ξ1) P(Y |M, Ξ2) , (9) The marginal-likelihoods themselves are hard to compute, being formed from high dimensional integrals over the latent variables V and parameters Θ, p(Y |M, Ξi) = Z dV dΘ p(Y, V, Θ|M, Ξi) (10) = Z dV dΘ p(Y, V |Θ, M, Ξi)p(Θ|M, Ξi) . (11) One concern in model comparison might be that the more complex models (those which are more over-complete) have a larger number parameters and therefore ‘fit’ any data set better. However, the Bayes factor (Eq. 9) implicitly implements a probabilistic version of Occam’s razor that penalises more complex models and mitigates this effect [6]. This makes the Bayesian method appealing for determining the over-completeness of a sparse-coding model. Unfortunately computing the marginal-likelihood is computationally intensive, and this precludes tiling the sparseness/over-completeness space. However, an alternative is to learn the optimal overcompleteness at a given sparseness using automatic relevance determination (ARD) [7, 8]. The 3 advantage of ARD is that it changes a hard and lengthy model comparison problem (i.e., computing the marginal-likelihood for many models of differing dimensionalities) into a much simpler inference problem. In a nutshell, the idea is to equip the model with many more components than are believed to be present in the data, and to let it prune out the weights which are unnecessary. Practically this involves placing a (Gaussian) prior over the components which favours small weights, and then inferring the scale of this prior. In this way the scale of the superfluous weights is driven to zero, removing them from the model. The necessary ARD hyper-priors are p(gk|γk) = Ngk 0, γ−1 k , (12) p(γk) = Gγk(θk, lk) . (13) 4 Determining the over-completeness: Variational Bayes In the previous two sections we described a generative model for sparse coding that is theoretically able to learn the optimal over-completeness of natural scenes. We have two distinct uses for this model: The first, and computationally more demanding task, is to learn the over-completeness at a variety of different, fixed, sparsenesses (that is, to find the optimal over-completeness in a vertical slice through Fig. 1); The second is to determine the optimal point on this trade-off by evaluating the (approximate) marginal-likelihood (that is, evaluating points along the trade-off line in Fig. 1 to find the optimal model - the star). It turns out that no single method is able to solve both these tasks, but that it is possible to develop a pair of approximate algorithms to solve them separately. The first approximation scheme is Variational Bayes (VB), and it excels at the first task, but is severely biased in the case of the second. The second scheme is Annealed Importance Sampling (AIS) which is prohibitively slow for the first task, but much more accurate on the second. We describe them in turn, starting with VB. The quantity required for learning is the marginal-likelihood, log p(Y |M, Ξ) = log Z dV dΘ p(Y, V, Θ|M, Ξ). (14) Computing this integral is intractable (for reasons similar to those given in Sec. 2), but a lowerbound can be constructed by introducing any distribution over the latent variables and parameters, q(V, Θ), and using Jensen’s inequality, log p(Y |M, Ξ) ≥ Z dV dΘ q(V, Θ) log p(Y, V, Θ|M, Ξ) q(V, Θ) =: F(q(V, Θ)) (15) = log p(Y |M, Ξ) −KL(q(V, Θ)||p(V, Θ|Y )) (16) This lower-bound is called the free-energy, and the idea is to repeatedly optimise it with respect to the distribution q(V, Θ) so that it becomes as close to the true marginal likelihood as possible. Clearly the optimal choice for q(V, Θ) is the (intractable) true posterior. However, by constraining this distribution headway can be made. In particular if we assume that the set of parameters and set of latent variables are independent in the posterior, so that q(V, Θ) = q(V )q(Θ) then we can sequentially optimise the free-energy with respect to each of these distributions. For large hierarchical models, including the one described in this paper, it is often necessary to introduce further factorisations within these two distributions in order to derive the updates. Their general form is, q(Vi) ∝exp ⟨log p(V, Θ)⟩q(Θ) Q j̸=i q(Vi) (17) q(Θi) ∝exp ⟨log p(V, Θ)⟩q(V ) Q j̸=i q(Θi) . (18) As the Bayesian Sparse Coding model is composed of distributions from the exponential family, the functional form of these updates is the same as the corresponding priors. So, for example the latent variables have the following form: q(xt) is Gaussian and q(ut,k) is Gamma distributed. Although this approximation is good at discovering the over-completeness of data at fixed sparsities, it provides an estimate of the marginal-likelihood (the free-energy) which is biased toward regions of low sparsity. The reason is simple to understand. The difference between the free energy and the true likelihood is given by the KL divergence between the approximate and true posterior. Thus, the freeenergy bound is tightest in regions where q(V, Θ) is a good match to the true posterior, and loosest in 4 regions where it is a poor match. At high sparsities, the true posterior is multimodal and highly nonGaussian. In this regime q(V, Θ) – which is always uni-modal – is a poor approximation. At lowsparsities the prior becomes Gaussian-like and the posterior also becomes a uni-modal Gaussian. In this regime q(V, Θ) is an excellent approximation. This leads to a consistent bias in the peak of the free-energy toward regions of low sparsity. One might also be concerned with another potential source of bias: The number of modes in the posterior increases with the number of components in the model, which gives a worse match to the variational approximation for more over-complete models. However, because of the sparseness of the prior distribution, most of the modes are going to be very shallow for typical inputs, so that this effect should be small. We verify this claim on artificial data in Section 6.2. 5 Determining the sparsity: AIS An approximation scheme is required to estimate the marginal-likelihood, but without a sparsitydependent bias. Any scheme which uses a uni-modal approximation to the posterior will inevitably fall victim to such biases. This rules out many alternate variational schemes, as well as methods like the Laplace approximation, or Expectation Propagation. One alternative might be to use a variational method which has a multi-modal approximating distribution (e.g. a mixture model). The approach taken here is to use Annealed Importance Sampling (AIS) [9] which is one of the few methods for evaluating normalising constants of intractable distributions. The basic idea behind AIS is to estimate the marginal-likelihood using importance sampling. The twist is that the proposal distribution for the importance sampler is itself generated using an MCMC method. Briefly, this inner loop starts by drawing samples from the model’s prior distribution and continues to sample as the prior is deformed into the posterior, according to an annealing schedule. Both the details of this schedule, and having a quick-mixing MCMC method, are critical for good results. In fact it is simple to derive a quick-mixing Gibbs sampler for our application and this makes AIS particularly appealing. 6 Results Before tackling natural images, it is necessary to verify that the approximations can discover the correct degree of over-completeness and sparsity in the case where the data are drawn from the forward model. This is done in two stages: Firstly we focus on a very simple, low-dimensional example that is easy to visualise and which helps explicate the learning algorithms, allowing them to be tuned; Secondly, we turn to a larger scale example designed to be as similar to the tests on natural data as possible. 6.1 Verification using simple artifical data In the first experiment the training data are produced as follows: Two-dimensional observations are generated by three Student-t sources with degree of freedom chosen to be 2.5. The generative weights are fixed to be 60 degrees apart from one another, as shown in Figure 2. A series of VB simulations were then run, differing only in the sparseness level (as measured by the degrees of freedom of the Student-t distribution over xt). Each simulation consisted of 500 VB iterations performed on a set of 3000 data points randomly generated from the model. We initialised the simulations with K = 7 components. To improve convergence, we started the simulations with weights near the origin (drawn from a normal distribution with mean 0 and standard deviation 10−8) and a relatively large input noise variance, and annealed the noise variance between the iterations of VBEM. The annealing schedule was as following: we started with σ2 y = 0.3 for 100 iterations, reduced this linearly down to σ2 y = 0.1 in 100 iterations, and finally to σ2 y = 0.01 in a further 50 iterations. During the annealing process, the weights typically grew from the origin and spread in all directions to cover the input space. After an initial growth period, where the representation usually became as over-complete as allowed by the model, some of the weights rapidly shrank again and collapsed to the origin. At the same time, the corresponding precision hyperparameters grew and effectively pruned the unnecessary components. We performed 7 blocks of simulations at different sparseness levels. In every block we performed 3 runs of the algorithm and retained the result with the highest free energy. 5 4 2 0 2 4 4 2 0 2 4 2.1 2.2 2.4 2.5 3.0 3.5 −9600 −9400 −9200 −9000 −8800 −8600 −8400 −8200 −8000 free energy (NATS) α 2.1 2.2 2.4 2.5 3.0 3.5 −6850 −6800 −6750 −6700 −6650 −6600 −6550 −6500 log P(Y) (NATS) α Figure 2: Left: Test data drawn from the simple artificial model. Centre: Free energy of the models learned by VBEM in the artificial data case. Right: Estimated log marginal likelihood. Error bars are 3 times the estimated standard deviation. The marginal likelihoods of the selected results were then estimated using AIS. We derived the importance weights using a fixed data set with 2500 data points, 250 samples, and 300 intermediate distributions. Following the recommendations in [9], the annealing schedule was chosen to be linear initially (with 50 inverse temperatures spaced uniformly from 0 to 0.01), followed by a geometric section (250 inverse temperatures spaced geometrically from 0.01 to 1). This mean that there were a total of 300 distributions between the prior and posterior. The results indicate that the combination of the two methods is successful at learning both the overcompleteness and sparseness. In particular the VBEM algorithm was able to recover the correct dimensionality for all sparseness levels, except for the sparsest case α = 2.1, where it preferred a model with 5 significant components. As expected, however, figure 2 shows that the maximum free energy is biased toward the more Gaussian models. In contrast to this, the marginal likelihood estimated by AIS (Fig. 2), which is strictly greater than the free-energy as expected, favours sparseness levels close to the true value. 6.2 Verification using complex artificial data Although it is necessary that the inference scheme should pass simple tests like that in the previous section, they are not sufficient to give us confidence that it will perform successfully on natural data. One pertinent criticism is that the regime in which we tested the algorithms in the previous section (two dimensional observations, and three hidden latents) is quite different from that required to model natural data. To that end, in this section we first learn a sparse model for natural images with fixed over-completeness levels using a Maximum A Posteriori (MAP) algorithm [2] (degree of freedom 2.5). These solutions are then used to generate artificial data as in the previous section. The goal is to validate the model on data which has a content and scale similar to the natural images case, but with a controlled number of generative components. The image data comprised patches of size 9 × 9 pixels, taken at random positions from 36 natural images randomly selected from the van Hateren database (preprocessed as described in [10]). The patches were whitened and their dimensionality reduced from 81 to 36 by principal component analysis. The MAP solution was trained for 500 iterations, with every iteration performed on a new batch of 1440 patches (100 patches per image). The model was initialised with a 3-times over-complete number of components (K = 108). As above, the weights were initialised near the origin, and the input noise was annealed linearly from σd = 0.5 to σd = 0.2 in the first 300 iterations, remaining constant thereafter. Every run consisted of 500 VBEM iterations, with every iteration performed on 3600 patches generated from the MAP solution. We performed several simulations for over-completeness levels between 0.5 and 4.5, and retained the solutions with the highest free energy. The results are summarised in Figure 3: The model is able to recover the underlying dimensionality for data between 0.5 and 2 times over-complete, and correctly saturates to 3 times over-complete (the maximum attainable level here) when the data over-completeness exceeds 3. In the regime between 2.5 and 3 times over-complete data, the model returns solutions with a smaller number of components, which is possibly due to the bias described at the end of Section 5. However, these 6 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.5 1 1.5 2 2.5 3 True Overcompleteness Inferred Overcompleteness Figure 3: True versus inferred over-completeness from data drawn from the forward model trained on natural images. If inference was perfect, the true over-completeness would be recovered (black line). This straight line saturates when we hit the number of latent variables with which ARD was initialised (three times overcomplete). The results using multiple runs of ARD are close to this line (open circles, simulations with the highest free-energy are shown as closed circles). The maximal and best over-completeness inferred from natural scenes is shown by the dotted line, and lies well below the over-completenesses we are able to infer. values are still far above the highest over-completeness learned from natural images (see section 6.3), so that we believe that the bias does not invalidate our conclusions. 6.3 Natural images Having established that the model performs as expected, at least when the data is drawn from the forward model, we now turn to natural image data and examine the optimal over-completeness ratio and sparseness degree for natural scene statistics. The image data for this simulation and the model initialisation and annealing procedure are identical to the ones in the experiments in the preceeding section. We performed 20 simulations with different sparseness levels, especially concentrated on the more sparse values. Every run comprised 500 VBEM iterations, with every iteration performed on a new batch of 3600 patches. As shown in Figure 4, the free energy increased almost monotonically until α = 5 and then stabilised and started to decrease for more Gaussian models. The algorithm learnt models that were only slightly over-complete: the over-completeness ratio was distributed between 1 and 1.3, with a trend for being more over-complete at high sparseness levels (Fig. 4). Although this general trend accords with the intuition that sparseness and over-completeness are coupled, both the magnitude of the effect and the degree of over-completeness is smaller than might have been anticipated. Indeed, this result suggests that highly over-complete models with a Student-t prior may very well be overfitting the data. Finally we performed AIS using the same annealing schedule as in Section 6.1, using 250 samples for the first 6 sparseness levels and 50 for the successive 14. The estimates obtained for the log marginal likelihood, shown in Figure 4, were monotonically increasing with increasing sparseness (decreasing α). This indicates that sparse models are indeed optimal for natural scenes. Note that this is exactly the opposite trend to that of the free energy, indicating that it is also biased for natural scenes. Figure 4 shows the basis vectors learned in the simulation with α = 2.09, which had maximal marginal likelihood. The weights resemble the Gabor wavelets, typical of sparse codes for natural images [1]. 7 Discussion Our results suggest that the optimal sparse-coding model for natural scenes is indeed one which is very sparse, but only modestly over-complete. The anticipated coupling between the degree of sparsity and the over-completeness in the model is visible, but is weak. One crucial question is how far these results will generalise to other prior distributions; and indeed, which of the various possible sparse-coding priors is best able to capture the structure of natural scenes. One indication that the Student-t might not be optimal, is its behaviour as the degree-of7 2 4 6 8 10 −9.2 −9.1 −9 −8.9 −8.8x 10 4 free energy (NATS) α 2 3 4 5 6 7 8 9 −8.6 −8.5 −8.4 −8.3 −8.2 −8.1 x 10 4 log P(Y) (NATS) α d) b) 2 4 6 8 1 1.1 1.2 1.3 1.4 overcompleteness ratio α c) a) Figure 4: Natural images results. a) Free energy b) Marginal likelihood c) Estimated over-completeness d) Basis vectors freedom parameter moves towards sparser values. The distribution puts a very small amount of mass at a very great distance from the mean (for example, the kurtosis is undefined for α < 4). It is not clear that data with such extreme values will be encountered in typical data sets, and so the model may become distorted at high sparseness values. Future work will be directed towards more general prior distributions. The formulation of the Student-t in terms of a random precision Gaussian is computationally helpful. While no longer within the exponential family, other distributions on the precision (such as a uniform one) may be approximated using a similar approach. Acknowledgements This work has been supported by the Gatsby Charitable Foundation. We thank Yee Whye Teh, Iain Murray, and David McKay for fruitful discussions. References [1] B.A. Olshausen and D.J. Field. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 381(6583):607–609, 1996. [2] B.A. Olshausen and D.J. Field. Sparse coding with an overcomplete basis set: A strategy employed by V1? Vision Research, 37:3311–3325, 1997. [3] Y.W Teh, M. Welling, S. Osindero, and G.E. Hinton. Energy-based models for sparse overcomplete representations. Journal of Machine Learning Research, 4:1235–1260, 2003. [4] A.J. Bell and T.J. Sejnowski. The ‘independent components’ of natural scenes are edge filters. Vision Research, 37(23):3327–3338, 1997. [5] S. Osindero, M. Welling, and G.E. Hinton. Topographic product models applied to natural scene statistics. Neural Computation, 18:381–344, 2006. [6] D.J.C. McKay. Bayesian interpolation. Neural Comput, 4(3):415–447, 1992. [7] C.M. Bishop. Variational principal components. In ICANN 1999 Proceedings, pages 509–514, 1999. [8] M.J. Beal. Variational Algorithms for Approximate Bayesian Inference. PhD thesis, Gatsby Computational Neuroscience Unit, University College London, 2003. [9] R.M. Neal. Annealed importance sampling. Statistics and Computing, 11:125–139, 2001. [10] J.H. van Hateren and A. van der Schaaf. Independent component filters of natural images compared with simple cells in primary visual cortex. Proc. R. Soc. Lond. B, 265:359–366, 1998. 8
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A Probabilistic Approach to Language Change Alexandre Bouchard-Cˆot´e∗ Percy Liang∗ Thomas L. Griffiths† Dan Klein∗ ∗Computer Science Division †Department of Psychology University of California at Berkeley Berkeley, CA 94720 Abstract We present a probabilistic approach to language change in which word forms are represented by phoneme sequences that undergo stochastic edits along the branches of a phylogenetic tree. This framework combines the advantages of the classical comparative method with the robustness of corpus-based probabilistic models. We use this framework to explore the consequences of two different schemes for defining probabilistic models of phonological change, evaluating these schemes by reconstructing ancient word forms of Romance languages. The result is an efficient inference procedure for automatically inferring ancient word forms from modern languages, which can be generalized to support inferences about linguistic phylogenies. 1 Introduction Languages evolve over time, with words changing in form, meaning, and the ways in which they can be combined into sentences. Several centuries of linguistic analysis have shed light on some of the key properties of this evolutionary process, but many open questions remain. A classical example is the hypothetical Proto-Indo-European language, the reconstructed common ancestor of the modern Indo-European languages. While the existence and general characteristics of this proto-language are widely accepted, there is still debate regarding its precise phonology, the original homeland of its speakers, and the date of various events in its evolution. The study of how languages change over time is known as diachronic (or historical) linguistics (e.g., [4]). Most of what we know about language change comes from the comparative method, in which words from different languages are compared in order to identify their relationships. The goal is to identify regular sound correspondences between languages and use these correspondences to infer the forms of proto-languages and the phylogenetic relationships between languages. The motivation for basing the analysis on sounds is that phonological changes are generally more systematic than syntactic or morphological changes. Comparisons of words from different languages are traditionally carried out by hand, introducing an element of subjectivity into diachronic linguistics. Early attempts to quantify the similarity between languages (e.g., [15]) made drastic simplifying assumptions that drew strong criticism from diachronic linguists. In particular, many of these approaches simply represent the appearance of a word in two languages with a single bit, rather than allowing for gradations based on correspondences between sequences of phonemes. We take a quantitative approach to diachronic linguistics that alleviates this problem by operating at the phoneme level. Our approach combines the advantages of the classical, phoneme-based, comparative method with the robustness of corpus-based probabilistic models. We focus on the case where the words are etymological cognates across languages, e.g. French faire and Spanish hacer from Latin facere (to do). Following [3], we use this information to estimate a contextualized model of phonological change expressed as a probability distribution over rules applied to individual phonemes. The model is fully generative, and thus can be used to solve a variety of problems. For example, we can reconstruct ancestral word forms or inspect the rules learned along each branch of 1 a phylogeny to identify sound laws. Alternatively, we can observe a word in one or more modern languages, say French and Spanish, and query the corresponding word form in another language, say Italian. Finally, models of this kind can potentially be used as a building block in a system for inferring the topology of phylogenetic trees [3]. In this paper, we use this general approach to evaluate the performance of two different schemes for defining probability distributions over rules. The first scheme, used in [3], treats these distributions as simple multinomials and uses a Dirichlet prior on these multinomials. This approach makes it difficult to capture rules that apply at different levels of granularity. Inspired by the prevalence of multi-scale rules in diachronic phonology and modern phonological theory, we develop a new scheme in which rules possess a set of features, and a distribution over rules is defined using a loglinear model. We evaluate both schemes in reconstructing ancient word forms, showing that the new linguistically-motivated change can improve performance significantly. 2 Background and previous work Most previous computational approaches to diachronic linguistics have focused on the reconstruction of phylogenetic trees from a Boolean matrix indicating the properties of words in different languages [10, 6, 14, 13]. These approaches descend from glottochronology [15], which measures the similarity between languages (and the time since they diverged) using the number of words in those languages that belong to the same cognate set. This information is obtained from manually curated cognate lists such as the data of [5]. The modern instantiations of this approach rely on sophisticated techniques for inferring phylogenies borrowed from evolutionary biology (e.g., [11, 7]). However, they still generally use cognate sets as the basic data for evaluating the similarity between languages (although some approaches incorporate additional manually constructed features [14]). As an example of a cognate set encoding, consider the meaning “eat”. There would be one column for the cognate set which appears in French as manger and Italian as mangiare since both descend from the Latin mandere (to chew). There would be another column for the cognate set which appears in both Spanish and Portuguese as comer, descending from the Latin comedere (to consume). If these were the only data, algorithms based on this data would tend to conclude that French and Italian were closely related and that Spanish and Portuguese were equally related. However, the cognate set representation has several disadvantages: it does not capture the fact that the cognate is closer between Spanish and Portuguese than between French and Spanish, nor do the resulting models let us conclude anything about the regular processes which caused these languages to diverge. Also, curating cognate data can be expensive. In contrast, each word in our work is tracked using an automatically obtained cognate list. While these cognates may be noisier, we compensate for this by modeling phonological changes rather than Boolean mutations in cognate sets. Another line of computational work has explored using phonological models as a way to capture the differences between languages. [16] describes an information theoretic measure of the distance between two dialects of Chinese. They use a probabilistic edit model, but do not consider the reconstruction of ancient word forms, nor do they present a learning algorithm for such models. There have also been several approaches to the problem of cognate prediction in machine translation (essentially transliteration), e.g., [12]. Compared to our work, the phenomena of interest, and therefore the models, are different. [12] presents a model for learning “sound laws,” general phonological changes governing two completely observed aligned cognate lists. This model can be viewed as a special case of ours using a simple two-node topology. 3 A generative model of phonological change In this section, we outline the framework for modeling phonological change that we will use throughout the paper. Assume we have a fixed set of word types (cognate sets) in our vocabulary V and a set of languages L. Each word type i has a word form wil in each language l ∈L, which is represented as a sequence of phonemes which might or might not be observed. The languages are arranged according to some tree topology T (see Figure 2(a) for examples). It is possible to also induce the topology or cognate set assignments, but in this paper we assume that the topology is fixed and cognates have already been identified. 2 For each word i ∈V : wiROOT ∼LanguageModel For each branch (k →l) ∈T: θk→l ∼Rules(σ2) [choose edit parameters] For each word i ∈V : wil ∼Edit(wik, θk→l) [sample word form] (a) Generative description # C V C V C # # f o k u s # # f w O k o # # C V V C V # f →f / # V o →w O / C C k →k / V V u →o / C C s → / V # Edits applied Rules used (b) Example of edits · · · wiA wiB wiC wiD · · · · · · word type i = 1 . . . |V | eiA→B θA→B eiB→C θB→C eiB→D θB→D (c) Graphical model Figure 1: (a) A description of the generative model. (b) An example of edits that were used to transform the Latin word focus (/fokus/) into the Italian word fuoco (/fwOko/) (fire) along with the context-specific rules that were applied. (c) The graphical model representation of our model: θ are the parameters specifying the stochastic edits e, which govern how the words w evolve. The probabilistic model specifies a distribution over the word forms {wil} for each word type i ∈V and each language l ∈L via a simple generative process (Figure 1(a)). The generative process starts at the root language and generates all the word forms in each language in a top-down manner. The w ∼LanguageModel distribution is a simple bigram phoneme model. A root word form w consisting of n phonemes x1 · · · xn is generated with probability plm(x1) = Qn j=2 plm(xj | xj−1), where plm is the distribution of the language model. The stochastic edit model w′ ∼Edit(w, θ) describes how a single old word form w = x1 · · · xn changes along one branch of the phylogeny with parameters θ to produce a new word form w′. This process is parametrized by rule probabilities θk→l, which are specific to branch (k →l). The generative process used in the edit model is as follows: for each phoneme xi in the old word form, walking from left to right, choose a rule to apply. There are three types of rules: (1) deletion of the phoneme, (2) substitution with some phoneme (possibly the same one), or (3) insertion of another phoneme, either before or after the existing one. The probability of applying a rule depends on the context (xi−1, xi+1). Context-dependent rules are often used to characterize phonological changes in diachronic linguistics [4]. Figure 1(b) shows an example of the rules being applied. The context-dependent form of these rules allows us to represent phenomena such as the likely deletion of s in word-final positions. 4 Defining distributions over rules In the model defined in the previous section, each branch (k →l) ∈T has a collection of contextdependent rule probabilities θk→l. Specifically, θk→l specifies a collection of multinomial distributions, one for each C = (cl, x, cr), where cl is left phoneme, x is the old phoneme, cr is the right phoneme. Each multinomial distribution is over possible right-hand sides α of the rule, which could consist of 0, 1, or 2 phonemes. We write θk→l(C, α) for the probability of rule x →α / c1 c2. Previous work using this probabilistic framework simply placed independent Dirichlet priors on each of the multinomial distributions [3]. While this choice results in a simple estimation procedure, it has some severe limitations. Sound changes happen at many granularities. For example, from Latin to Vulgar Latin, u →o occurs in many contexts while s →∅occurs only in word-final contexts. Using independent Dirichlets forces us to commit to a single context granularity for C. Since the different multinomial distributions are not tied together, generalization becomes very difficult, especially as data is limited. It is also difficult to interpret the learned rules, since the evidence for a coarse phenomenon such as u →o would be unnecessarily fragmented across many different 3 context-dependent rules. We would like to ideally capture a phenomenon using a single rule or feature. We could relate the rule probabilities via a simple hierarchical Bayesian model, but we would still have to define a single hierarchy of contexts. This restriction might be inappropriate given that sound changes often depend on different contexts that are not necessarily nested. For these reasons, we propose using a feature-based distribution over the rule probabilities. Let F(C, α) be a feature vector that depends on the context-dependent rule (C, α), and λk→l be the log-linear weights for branch (k →l). We use a Normal prior on the log-linear weights, λk→l ∼ N(0, σ2I). The rule probabilities are then deterministically related to the weights via the softmax function: θk→l(C, α; λk→l) = eλT k→lF (C,α) P α′ eλT k→lF (C,α′) . (1) For each rule x →α / cl cr, we defined features based on whether x = α (i.e. self-substitution), and whether |α| = n for each n = 0, 1, 2 (corresponding to deletion, substitution, and insertion). We also defined sets of features using three partitions of phonemes c into “natural classes”. These correspond to looking at the place of articulation (denoted A2(c)), testing whether c is a vowel, consonant, or boundary symbol (A1(c)), and the trivial wildcard partition (A0(c)), which allows rules to be insensitive to c. Using these partitions, the final set of features corresponded to whether Akl(cl) = al and Akr(cr) = ar for each type of partitioning kl, kr ∈{0, 1, 2} and natural classes al, ar. The move towards using a feature-based scheme for defining rule probabilities is not just motivated by the greater expressive capacity of this scheme. It also provides a connection with contemporary phonological theory. Recent work in computational linguistics on probabilistic forms of optimality theory has begun to use a similar approach, characterizing the distribution over word forms within a language using a log-linear model applied to features of the words [17, 9]. Using similar features to define a distribution over phonological changes thus provides a connection between synchronic and diachronic linguistics in addition to a linguistically-motivated method for improving reconstruction. 5 Learning and inference We use a Monte Carlo EM algorithm to fit the parameters of both models. The algorithm iterates between a stochastic E-step, which computes reconstructions based on the current edit parameters, and an M-step, which updates the edit parameters based on the reconstructions. 5.1 Monte Carlo E-step: sampling the edits The E-step computes the expected sufficient statistics required for the M-step, which in our case is the expected number of times each edit (such as o →O) was used in each context. Note that the sufficient statistics do not depend on the prior over rule probabilities; in particular, both the model based on independent Dirichlet priors and the one based on a log-linear prior require the same E-step computation. An exact E-step would require summing over all possible edits involving all languages in the phylogeny (all unobserved {e}, {w} variables in Figure 1(c)), which does not permit a tractable dynamic program. Therefore, we resort to a Monte Carlo E-step, where many samples of the edit variables are collected, and counts are computed based on these samples. Samples are drawn using Gibbs sampling [8]: for each word form of a particular language wil, we fix all other variables in the model and sample wil along with its corresponding edits. Consider the simple four-language topology in Figure 1(c). Suppose that the words in languages A, C and D are fixed, and we wish to sample the word at language B along with the three corresponding sets of edits (remember that the edits fully determine the words). While there are an exponential number of possible words/edits, we can exploit the Markov structure in the edit model to consider all such words/edits using dynamic programming, in a way broadly similar to the forward-backward algorithm for HMMs. See [3] for details of the dynamic program. 4 la es it la vl ib es pt it Topology 1 Topology 2 (a) Topologies Experiment Topology Model Heldout Latin reconstruction (6.1) 1 Dirichlet la:293 1 Log-linear la:293 Sound changes (6.2) 2 Log-linear None (b) Experimental conditions Figure 2: Conditions under which each of the experiments presented in this section were performed. The topology indices correspond to those displayed at the left. The heldout column indicates how many words, if any, were held out for edit distance evaluation, and from which language. All the experiments were run on a data set of 582 cognates from [3]. 5.2 M-step: updating the parameters In the M-step, we estimate the distribution over rules for each branch (k →l). In the Dirichlet model, this can be done in closed form [3]. In the log-linear model, we need to optimize the feature weights λk→l. Let us fix a single branch and drop the subscript. Let N(C, α) be the expected number of times the rule (C, α) was used in the E-step. Given these sufficient statistics, the estimate of λ is given by optimizing the expected complete log-likelihood plus the regularization penalty from the prior on λ, O(λ) = X C,α N(C, α) h λT F(C, α) −log X α′ eλT F (C,α′)i −||λ||2 2σ2 . (2) We use L-BFGS to optimize this convex objective. which only requires the partial derivatives: ∂O(λ) ∂λj = X C,α N(C, α) h Fj(C, α) − X α′ θ(C, α′; λ)Fj(C, α′) i −λj σ2 (3) = ˆFj − X C,α′ N(C, ·)θ(C, α′; λ)Fj(C, α′) −λj σ2 , (4) where ˆFj def = P C,α N(C, α)Fj(C, α) is the empirical feature vector and N(C, ·) def = P α N(C, α) is the number of times context C was used. ˆFj and N(C, ·) do not depend on λ and thus can be precomputed at the beginning of the M-step, thereby speeding up each L-BFGS iteration. 6 Experiments In this section, we summarize the results of the experiments testing our different probabilistic models of phonological change. The experimental conditions are summarized in Table 2. Training and test data sets were taken from [3]. 6.1 Reconstruction of ancient word forms We ran the two models using Topology 1 in Figure 2 to assess the relative performance of Dirichletparametrized versus log-linear-parametrized models. Half of the Latin words at the root of the tree were held out, and the (uniform cost) Levenshtein edit distance from the predicted reconstruction to the truth was computed. While the uniform-cost edit distance misses important aspects of phonology (all phoneme substitutions are not equal, for instance), it is parameter-free and still seems to correlate to a large extent with linguistic quality of reconstruction. It is also superior to held-out log-likelihood, which fails to penalize errors in the modeling assumptions, and to measuring the percentage of perfect reconstructions, which ignores the degree of correctness of each reconstructed word. 5 Model Baseline Model Improvement Dirichlet 3.59 3.33 7% Log-linear (0) 3.59 3.21 11% Log-linear (0,1) 3.59 3.14 12% Log-linear (0,1,2) 3.59 3.10 14% Table 1: Results of the edit distance experiment. The language column corresponds to the language held out for evaluation. We show the mean edit distance across the evaluation examples. Improvement rate is computed by comparing the score of the algorithm against the baseline described in Section 6.1. The numbers in parentheses for the log-linear model indicate which levels of granularity were used to construct the features (see Section 4). /dEntis/ /djEntes/ /dEnti/ i →E E →j E s → Figure 3: An example of the proper Latin reconstruction given the Spanish and Italian word forms. Our model produces /dEntes/, which is nearly correct, capturing two out of three of the phenomena. We ran EM for 10 iterations for each model, and evaluated performance via a Viterbi derivation produced using these parameters. Our baseline for comparison was picking randomly, for each heldout node in the tree, an observed neighboring word (i.e., copy one of the modern forms). Both models outperformed this baseline (see Figure 3), and the log-linear model outperformed the Dirichlet model, suggesting that the featurized system better captures the phonological changes. Moreover, adding more features further improved the performance, indicating that being able to express rules at multiple levels of granularity allows the model to capture the underlying phonological changes more accurately. To give a qualitative feel for the operation of the system (good and bad), consider the example in Figure 3, taken from the Dirichlet-parametrized experiment. The Latin dentis /dEntis/ (teeth) is nearly correctly reconstructed as /dEntes/, reconciling the appearance of the /j/ in the Spanish and the disappearance of the final /s/ in the Italian. Note that the /is/ vs. /es/ ending is difficult to predict in this context (indeed, it was one of the early distinctions to be eroded in Vulgar Latin). 6.2 Inference of phonological changes Another use of this model is to automatically recover the phonological drift processes between known or partially-known languages. To facilitate evaluation, we continued in the well-studied Romance evolutionary tree. Again, the root is Latin, but we now add an additional modern language, Portuguese, and two additional hidden nodes. One of the nodes characterizes the least common ancestor of modern Spanish and Portuguese; the other, the least common ancestor of all three modern languages. In Figure 2, Topology 2, these two nodes are labeled vl (Vulgar Latin) and ib (ProtoIbero Romance), respectively. Since we are omitting many other branches, these names should not be understood as referring to actual historical proto-languages, but, at best, to collapsed points representing several centuries of evolution. Nonetheless, the major reconstructed rules still correspond to well-known phenomena and the learned model generally places them on reasonable branches. Figure 4 shows the top four general rules for each of the evolutionary branches recovered by the log-linear model. The rules are ranked by the number of times they were used in the derivations during the last iteration of EM. The la, es, pt, and it forms are fully observed while the vl and ib forms are automatically reconstructed. Figure 4 also shows a specific example of the evolution of the Latin VERBUM (word), along with the specific edits employed by the model. For this particular example, both the Dirichlet and the log-linear models produced the same reconstruction in the internal nodes. However, the log-linear parametrization makes inspection of sound laws easier. Indeed, with the Dirichlet model, since the natural classes are of fixed granularity, some 6 r →R / * * e → / ALV # t →d / * * Ù →s / * * u →o / * * o →o s / C # v →b / * * t →t e / * * /werbum/ (la) /verbo/ (vl) /veRbo/ (ib) /beRbo/ (es) /veRbu/ (pt) /vErbo/ (it) s → / * # m → / * # i → / * V ï →n / * VELAR u →o / * * e →E / * * i → / C V a →j a / * * n →m / * * a →5 / * * o →u / * * e →1 / * * m → u →o w →v r →R v →b o →u e →E Figure 4: The tree shows the system’s hypothesized transformation of a selected Latin word form, VERBUM (word) into the modern Spanish, Italian, and Portuguese pronunciations. The Latin root and modern leaves were observed while the hidden nodes as well as all the derivations were obtained using the parameters computed by our model after 10 iterations of EM. Nontrivial rules (i.e. rules that are not identities) used at each stage are shown along the corresponding edge. The boxes display the top four nontrivial rules corresponding to each of these evolutionary branches, ordered by the number of times they were applied during the last E step. These are grouped and labeled by their active feature of highest weight. ALV stands for alveolar consonant. rules must be redundantly discovered, which tends to flood the top of the rule lists with duplicates. In contrast, the log-linear model groups rules with features of the appropriate degree of generality. While quantitative evaluation such as measuring edit distance is helpful for comparing results, it is also illuminating to consider the plausibility of the learned parameters in a historical light, which we do here briefly. In particular, we consider rules on the branch between la and vl, for which we have historical evidence. For example, documents such as the Appendix Probi [2] provide indications of orthographic confusions which resulted from the growing gap between Classical Latin and Vulgar Latin phonology around the 3rd and 4th centuries AD. The Appendix lists common misspellings of Latin words, from which phonological changes can be inferred. On the la to vl branch, rules for word-final deletion of classical case markers dominate the list. It is indeed likely that these were generally eliminated in Vulgar Latin. For the deletion of the /m/, the Appendix Probi contains pairs such as PASSIM NON PASSI and OLIM NON OLI. For the deletion of final /s/, this was observed in early inscriptions, e.g. CORNELIO for CORNELIOS [1]. The frequent leveling of the distinction between /o/ and /u/ (which was ranked 5, but was not included for space reasons) can be also be found in the Appendix Probi: COLUBER NON COLOBER. Note that in the specific example shown, the model lowers the original /u/ and then re-raises it in the pt branch due to a later process along that branch. Similarly, major canonical rules were discovered in other branches as well, for example, /v/ to /b/ fortition in Spanish, palatalization along several branches, and so on. Of course, the recovered words and rules are not perfect. For example, reconstructed Ibero /trinta/ to Spanish /treinta/ (thirty) is generated in an odd fashion using rules /e/ to /i/ and /n/ to /in/. In the Dirichlet model, even when otherwise reasonable systematic sound changes are captured, the crudeness of the fixed-granularity contexts can prevent the true context from being captured, resulting in either rules applying with low probability in overly coarse environments or rules being learned redundantly in overly fine environments. The featurized model alleviates this problem. 7 Conclusion Probabilistic models have the potential to replace traditional methods used for comparing languages in diachronic linguistics with quantitative methods for reconstructing word forms and inferring phylogenies. In this paper, we presented a novel probabilistic model of phonological change, in which the rules governing changes in the sound of words are parametrized using the features of the phonemes involved. This model goes beyond previous work in this area, providing more accurate reconstructions of ancient word forms and connections to current work on phonology in synchronic linguistics. Using a log-linear model to define the probability of a rule being applied results in a 7 straightforward inference procedure which can be used to both produce accurate reconstructions as measured by edit distance and identify linguistically plausible rules that account for phonological changes. We believe that this probabilistic approach has the potential to support quantitative analysis of the history of languages in a way that can scale to large datasets while remaining sensitive to the concerns that have traditionally motivated diachronic linguistics. Acknowledgments We would like to thank Bonnie Chantarotwong for her help with the IPA converter and our reviewers for their comments. This work was supported by a FQRNT fellowship to the first author, a NDSEG fellowship to the second author, NSF grant number BCS-0631518 to the third author, and a Microsoft Research New Faculty Fellowship to the fourth author. References [1] W. Sidney Allen. Vox Latina: The Pronunciation of Classical Latin. Cambridge University Press, 1989. [2] W.A. Baehrens. Sprachlicher Kommentar zur vulg¨arlateinischen Appendix Probi. Halle (Saale) M. Niemeyer, 1922. [3] A. Bouchard-Cˆot´e, P. Liang, T. Griffiths, and D. Klein. A Probabilistic Approach to Diachronic Phonology. In Empirical Methods in Natural Language Processing and Computational Natural Language Learning (EMNLP/CoNLL), 2007. [4] L. Campbell. Historical Linguistics. The MIT Press, 1998. [5] I. Dyen, J.B. Kruskal, and P. Black. FILE IE-DATA1. Available at http://www.ntu.edu.au/education/langs/ielex/IE-DATA1, 1997. [6] S. N. Evans, D. Ringe, and T. Warnow. Inference of divergence times as a statistical inverse problem. In P. Forster and C. Renfrew, editors, Phylogenetic Methods and the Prehistory of Languages. McDonald Institute Monographs, 2004. [7] J. Felsenstein. Inferring Phylogenies. Sinauer Associates, 2003. [8] S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6:721–741, 1984. [9] S. Goldwater and M. Johnson. Learning ot constraint rankings using a maximum entropy model. Proceedings of the Workshop on Variation within Optimality Theory, 2003. [10] R. D. Gray and Q. Atkinson. Language-tree divergence times support the Anatolian theory of Indo-European origins. Nature, 2003. [11] J. P. Huelsenbeck, F. Ronquist, R. Nielsen, and J. P. Bollback. Bayesian inference of phylogeny and its impact on evolutionary biology. Science, 2001. [12] G. Kondrak. Algorithms for Language Reconstruction. PhD thesis, University of Toronto, 2002. [13] L. Nakhleh, D. Ringe, and T. Warnow. Perfect phylogenetic networks: A new methodology for reconstructing the evolutionary history of natural languages. Language, 81:382–420, 2005. [14] D. Ringe, T. Warnow, and A. Taylor. Indo-european and computational cladistics. Transactions of the Philological Society, 100:59–129, 2002. [15] M. Swadesh. Towards greater accuracy in lexicostatistic dating. Journal of American Linguistics, 21:121–137, 1955. [16] A. Venkataraman, J. Newman, and J.D. Patrick. A complexity measure for diachronic chinese phonology. In J. Coleman, editor, Computational Phonology. Association for Computational Linguistics, 1997. [17] C. Wilson and B. Hayes. A maximum entropy model of phonotactics and phonotactic learning. Linguistic Inquiry, 2007. 8
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Learning the 2-D Topology of Images Nicolas Le Roux University of Montreal nicolas.le.roux@umontreal.ca Yoshua Bengio University of Montreal yoshua.bengio@umontreal.ca Pascal Lamblin University of Montreal lamblinp@umontreal.ca Marc Joliveau ´Ecole Centrale Paris marc.joliveau@ecp.fr Bal´azs K´egl LAL/LRI, University of Paris-Sud, CNRS 91898 Orsay, France kegl@lal.in2p3.fr Abstract We study the following question: is the two-dimensional structure of images a very strong prior or is it something that can be learned with a few examples of natural images? If someone gave us a learning task involving images for which the two-dimensional topology of pixels was not known, could we discover it automatically and exploit it? For example suppose that the pixels had been permuted in a fixed but unknown way, could we recover the relative two-dimensional location of pixels on images? The surprising result presented here is that not only the answer is yes, but that about as few as a thousand images are enough to approximately recover the relative locations of about a thousand pixels. This is achieved using a manifold learning algorithm applied to pixels associated with a measure of distributional similarity between pixel intensities. We compare different topologyextraction approaches and show how having the two-dimensional topology can be exploited. 1 Introduction Machine learning has been applied to a number of tasks involving an input domain with a special topology: one-dimensional for sequences, two-dimensional for images, three-dimensional for videos and for 3-D capture. Some learning algorithms are generic, e.g., working on arbitrary unstructured vectors in d, such as ordinary SVMs, decision trees, neural networks, and boosting applied to generic learning algorithms. On the other hand, other learning algorithms successfully exploit the specific topology of their input, e.g., SIFT-based machine vision [10], convolutional neural networks [6, 7], time-delay neural networks [5, 16]. It has been conjectured [8, 2] that the two-dimensional structure of natural images is a very strong prior that would require a huge number of bits to specify, if starting from the completely uniform prior over all possible permutations. The question studied here is the following: is the two-dimensional structure of natural images a very strong prior or is it something that can be learned with a few examples? If a small number of examples is enough to discover that structure, then the conjecture in [8] about the image topology was probably incorrect. To answer that question we consider a hypothetical learning task involving images whose pixels have been permuted in a fixed but unknown way. Could we recover the 1 two-dimensional relations between pixels automatically? Could we exploit it to obtain better generalization? A related study performed in the context of ICA can be found in [1]. The basic idea of the paper is that the two-dimensional topology of pixels can be recovered by looking for a two-dimensional manifold embedding pixels (each pixel is a point in that space), such that nearby pixels have similar distributions of intensity (and possibly color) values. We explore a number of manifold techniques with this goal in mind, and explain how we have adapted these techniques in order to obtain the positive and surprising result: the two-dimensional structure of pixels can be recovered from a rather small number of training images. On images we find that the first 2 dimensions are dominant, meaning that even the knowledge that 2 dimensions are most appropriate could probably be inferred from the data. 2 Manifold Learning Techniques Used In this paper we have explored the question raised in the introduction for the particular case of images, i.e., with 2-dimensional structures, and our experiments have been performed with images of size 27 × 27 to 30 × 30, i.e., with about a thousand pixels. It means that we have to look for the embedding of about a thousand points (the pixels) on a two-dimensional manifold. Metric Multi-Dimensional Scaling MDS is a linear embedding technique (analogous to PCA but starting from distances and yielding coordinates on the principal directions, of maximum variance). Nonparametric techniques such as Isomap [13], Local Linear Embedding (LLE) [12], or Semidefinite Embedding (SDE, also known as MVU for Maximum Variance Unfolding) [17] have computation time that scale polynomially in the number of examples n. With n around a thousand, all of these are feasible, and we experimented with MDS, Isomap, LLE, and MVU. Since we found Isomap to work best to recover the pixel topology even on small sets of images, we review the basic elements of Isomap. It applies the metric multidimensional scaling (MDS) algorithm to geodesic distances in the neighborhood graph. The neighborhood graph is obtained by connecting the k nearest neighbors of each point. Each arc of the graph is associated with a distance (the user-provided distance between points), and is used to compute an approximation of the geodesic distance on the manifold with the length of the shortest path between two points. The metric MDS algorithm then transforms these distances into d-dimensional coordinates as follows. It first computes the dot-product (or Gram) n × n matrix M using the “double-centering” formula, yielding entries Mij = −1 2(D2 ij −1 n P i D2 ij −1 n P j D2 ij + 1 n2 P i,j D2 ij). The d principal eigenvectors vk and eigenvalues λk (k = 1, . . . , d) of M are then computed. This yields the coordinates: xik = vki √λk is the k-th embedding coordinate of point i. 3 Topology-Discovery Algorithms In order to apply a manifold learning algorithm, we must generally have a notion of similarity or distance between the points to embed. Here each point corresponds to a pixel, and the data we have about the pixels provide an empirical distribution of intensities for all pixels. Therefore we want to compare two estimate the statistical dependency between two pixels, in order to determine if they should be “neighbors” on the manifold. A simple and natural dependency statistic is the correlation between pixel intensities, and it works very well. The empirical correlation ρij between the intensity of pixel i and pixel j is in the interval [−1, 1]. However, two pixels highly anti-correlated are much more likely to be close than pixels not correlated (think of edges in an image). We should thus consider the absolute value of the correlations. If we assume them to be the value of a Gaussian kernel |ρij| = K(xi, xj) = e−1 2 ∥xi−xj∥2 , then by defining Dij = ∥xi −xj∥and solving the above for Dij we obtain a “distance” formula that can be used with the manifold learning algorithms: Dij = q −log |ρij| . (1) Note that scaling the distances in the Gaussian kernel by a variance parameter would only scale the resulting embedding, so it is unnecessary. 2 Many other measures of distance would probably work as well. However, we found the absolute correlation to be simple and easy to understand while yielding nice embeddings. 3.1 Dealing With Low-Variance Pixels A difficulty we observed in experimenting with different manifold learning algorithms on data sets such as MNIST is the influence of low-variance pixels. On MNIST digit images the border pixels may have 0 or very small variance. This makes them all want to be close to each other, which tends to fold the manifold on itself. To handle this problem we have simply ignored pixels with very low variance. When these represent a fixed background (as in MNIST images), this strategy works fine. In the experiments with MNIST we removed pixels with standard deviation less than 15% of the maximum standard deviation (maximum over all pixels). On the NORB dataset, which has varied backgrounds, this step does not remove any of the pixels (so it is unnecessary). 4 Converting Back to a Grid Image Once we have obtained an embedding for the pixels, the next thing we would like to do is to transform the data vectors back into images. For this purpose we have performed the following two steps: 1. Choosing horizontal and vertical axes (since the coordinates on the manifold can be arbitrarily rotated), and rotating the embedding coordinates accordingly, and 2. Transforming the input vector of intensity values (along with the pixel coordinates) into an ordinary discrete image on a grid. This should be done so that the resulting intensity at position (i, j) is close to the intensity values associated with input pixels whose embedding coordinates are (i, j). Such a mapping of pixels to a grid has already been done in [4], where a grid topology is defined by the connections in a graphical model, which is then trained by maximizing the approximate likelihood. However, they are not starting from a continuous embedding, but from the original data. Let pk (k = 1 . . . N) be the embedding coordinates found by the dimensionality reduction algorithm for the k-th input variable. We select the horizontal axis as the direction of smaller spread, the vertical axis being in the orthogonal direction, and perform the appropriate rotation. Once we have a coordinate system that assigns a 2-dimensional position pk to the k-th input pixel, placed at irregular locations inside a rectangular grid, we can map the input intensities xk into intensities Mi,j, so as to obtain a regular image that can be processed by standard image-processing and machine vision learning algorithms. The output image pixel intensity Mi,j at coordinates (i, j) is obtained through a convex average Mi,j = X k wi,j,kxk (2) where the weights are non-negative and sum to one, and are chosen as follows. wi,j,k = vi,j,k P k vi,j,k with an exponential of the L1 distance to give less weight to farther points: vi,j,k = exp (γ∥(i, j) −pk∥1) N(i,j,k) (3) where N(i, j, k) is true if ∥(i, j) −pk∥1 < 2 (or inferior to a larger radius to make sure that at least one input pixel k is associated with output grid position (i, j)). We used γ = 3 in the experiments, after trying only 1, 3 and 10. Large values of γ correspond to using only the nearest neighbor of (i, j) among the pks. Smaller values smooth the intensities and make the output look better if the embedding is not perfect. Too small values result in a loss of effective resolution. 3 Algorithm 1 Pseudo-code of the topology-learning learning that recovers the 2-D structure of inputs provided in an arbitrary but fixed order. Input: X {Raw input n × N data matrix, one row per example, with elements in fixed but arbitrary order} Input: δ = 0.15 (default value){Minimum relative standard deviation threshold, to remove too low-variance pixels} Input: k = 4 (default value){Number of neighbors used to build Isomap neighborhood graph} Input: L = √ N, W = √ N (default values) {Dimensions (length L, width W of output image)} Input: γ = 3 (default value) {Smoothing coefficient to recover images} Output: p {N × 2 matrix of embedding coordinates (one per row) for each input variable} Output: w {Convolution weights to recover an image from a raw input vector} n = number of examples (rows of X) for all column X.i do µi ←1 n P t Xti {Compute means} σ2 i ←1 n P t(Xti −µi)2 {Compute variances} end for Remove columns of X for which σi maxj σj < δ for all column X.i do for all column X.j do empirical correlation ρij = (X.i−µi)′(X.j−µj) σiσj {Compute all pair-wise empirical correlations} pseudo-distances Dij = p −log |ρij| end for end for {Compute the 2-D embeddings (pk1, pk2) of each input variable k through Isomap} p = Isomap(D, k, 2) {Rotate the coordinates p to try to align them to a vertical-horizontal grid (see text)} {Invert the axes if L < W} {Compute the convolution weights that will map raw values to output image pixel intensities} for all grid position (i, j) in output image (i in 1 . . . L, j in 1 . . . W) do r = 1 repeat neighbors ←{k : ||pk −(i, j)||1 < r} r ←r + 1 until neighbors not empty for all k in neighbors do vk ←eγ||pk−(i,j)||1 end for wi,j,. ←0 for all k in neighbors do wi,j,k = vi,j,k P k vi,j,k {Compute convolution weights} end for end for Algorithm 2 Convolve a raw input vector into a regular grid image, using the already discovered embedding for each input variable. Input: x {Raw input N-vector (in same format as a row of X above)} Input: p {N × 2 matrix of embedding coordinates (one per row) for each input variable} Input: w {Convolution weights to recover an image from a raw input vector} Output: Y {L × W output image} for all grid position (i, j) in output image (i in 1 . . . L, j in 1 . . . W) do Yi,j ←P k wi,j,kxk {Perform the convolution} end for 4 5 Experimental Results We performed experiments on two sets of images: MNIST digits dataset and NORB object classification dataset 1. We used the “jittered objects and cluttered background” image set from NORB. The MNIST images are particular in that they have a white background, whereas the NORB images have more varying backgrounds. The NORB images are originally of dimension 108 × 108; we subsampled them by 4 × 4 averaging into 27 × 27 images. The experiments have been performed with k = 4 neighbors for the Isomap embedding. Smaller values of k often led to unconnected neighborhood graphs, which Isomap cannot deal with. (a) Isomap embedding (b) LLE embedding (c) MDS embedding (d) MVU embedding Figure 1: Examples of embeddings discovered by Isomap, LLE, MDS and MVU with 250 training images from NORB. Each of the original pixel is placed at the location discovered by the algorithm. Size of the circle and gray level indicate the original true location of the pixel. Manifold learning produces coordinates with an arbitrary rotation. Isomap appears most robust, and MDS the worst method, for this task. In Figure 1 we compare four different manifold learning algorithms on the NORB images: Isomap, LLE, MDS and MVU. Figure 2 explains why Isomap is giving good results, especially in comparison with MDS. One the one hand, MDS is using the pseudo-distance defined in equation 1, whose relationship with the real distance between two pixels in the original image is linear only in a small neighborhood. On the other hand, Isomap uses the geodesic distances in the neighborhood graph, whose relationship with the real distance is really close to linear. (a) (b) (c) (d) Figure 2: (a) and (c): Pseudo-distance Dij (using formula 1) vs. the true distance on the grid. (b) and (d): Geodesic distance in neighborhood graph vs. the true distance on the grid. The true distance is on the horizontal axis for all figures. (a) and (b) are for a point in the upper-left corner, (c) and (d) for a point in the center. Figure 3 shows the embeddings obtained on the NORB data using different numbers of examples. In order to quantitatively evaluate the reconstruction, we applied on each embedding the similarity transformation that minimizes the Root of the Mean Squared Error (RMSE) between the coordinates of each pixel on the embedding, and their coordinates on the original grid, before measuring the residual error. This minimization is justified because the discovered embedding could be arbitrarily rotated, isotropically scaled, and mirrored. 100 examples are enough to get a reasonable embedding, and with 2000 or more a very good embedding is obtained: the RMSE for 2000 examples is 1.13, meaning that in expectation, each pixel is off by slightly more than one. 1Both can be obtained from Yann Le Cun’s web site: http://yann.lecun.com/. 5 9.25 2.43 1.68 1.21 1.13 10 examples 50 examples 100 examples 1000 examples 2000 examples Figure 3: Embedding discovered by Isomap on the NORB dataset, with different numbers of training samples (top row). Second row shows the same embeddings aligned (by a similarity transformation) on the original grid, third row shows the residual error (RMSE) after the alignment. Figure 4 shows the whole process of transforming an original image (with pixels possibly permuted) into an embedded image and finally into a reconstructed image as per algorithms 1 and 2. Figure 4: Example of the process of transforming an MNIST image (top) from which pixel order is unknown (second row) into its embedding (third row) and finally reconstructed as an image after rotation and convolution (bottom). In the third row, we show the intensity associated to each original pixel by the grey level in a circle located at the pixel coordinates discovered by Isomap. We also performed experiments with acoustic spectral data to see if the time-frequency topology can be recovered. The acoustic data come from the first 100 blues pieces of a publically available genre classification dataset [14]. The FFT is computed for each frame and there are 86 frames per second. The first 30 frequency bands are kept, each covering 21.51 Hz. We used examples formed by 30-frame spectrograms, i.e., just like images of size 30 × 30. Using the first 600,000 audio samples from each recording yielded 2600 30-frames images, on which we applied our technique. Figure 5 shows the resulting embedding when we removed the 30 coordinates of lowest standard deviation (δ = .15). 6 (a) Blues embedding 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 3.5 4 Eigenvalues Ratio of consecutive eigenvalues (b) Spectrum Figure 5: Embedding and spectrum decay for sequences of blues music. 6 Discussion Although [8] argue that learning the right permutation of pixels with a flat prior might be too difficult (either in a lifetime or through evolution), our results suggest otherwise. How do we interpret that apparent contradiction? The main element of explanation that we see is that the space of permutations of d numbers is not such a large class of functions. There are approximately N = √ 2πd d e d permutations (Stirling approximation) of d numbers. Since this is a finite class of functions, its VC-dimension [15] is h = log N ≈d log d −d. Hence if we had a bounded criterion (say taking values in [0, 1]) to compare different permutations and we used n examples (i.e., n images, here), we would expect the difference between generalization error and test error to be bounded [15] by 1 2 r 2 log N/η n with probability 1−η. Hence, with n a multiple of d log d, we would expect that one could approximately learn a good permutation. When d = 400 (the number of pixels with non-negligible variance in MNIST images), d log d−d ≈2000. This is more than what we have found necessary to recover a “good” representation of the images, but on the other hand there are equivalent classes within the set of permutations that give as good results as far as our objective and subjective criteria are concerned: we do not care about image symmetries, rotations, and small errors in pixel placement. What is the selection criterion that we have used to recover the image structure? Mainly we have used an additional prior which gives a preference to an order for which nearby pixels have similar distributions. How specific to natural images and how strong is that prior? This may be an application of a more general principle that could be advantageous to learning algorithms as well as to brains. When we are trying to compute useful functions from raw data, it is important to discover dependencies between the input random variables. If we are going to perform computations on subsets of variables at a time (which would seem necessary when the number of inputs is very large, to reduce the amount of connecting hardware), it would seem wiser that these computations combine variables that have dependencies with each other. That directly gives rise to the notion of local connectivity between neurons associated to nearby spatial locations, in the case of brains, the same notion that is exploited in convolutional neural networks. The fact that nearby pixels are more correlated is true at many scales in natural images. This is well known and explains why Gabor-like filters often emerge when trying to learn good filters for images, e.g., by ICA [9] or Products of Experts [3, 11]. In addition to the above arguments, there is another important consideration to keep in mind. The way in which we score permutations is not the way that one would score functions in an ordinary learning experiment. Indeed, by using the distributional similarity between pairs of pixels, we get not just a scalar score but d(d−1)/2 scores. Since our “scoring function” is much more informative, it is not surprising that it allows us to generalize from many fewer examples. 7 7 Conclusion and Future Work We proved here that, even with a small number of examples, we are able to recover almost perfectly the 2-D topology of images. This allows us to use image-specific learning algorithms without specifying any prior other than the dimensionnality of the coordinates. We also showed that this algorithm performed well on sound data, even though the topology might be less obvious in that case. However, in this paper, we only considered the simple case where we knew in advance the dimensionnality of the coordinates. One could easily apply this algorithm to data whose intrinsic dimensionality of the coordinates is unknown. In that case, one would not convert the embedding to a grid image but rather keep it and connect only the inputs associated to close coordinates (performing a k nearest neighbor for instance). It is not known if such an embedding might be useful for other types of data than the ones discussed above. Acknowledgements The authors would like to thank James Bergstra for helping with the audio data. They also want to acknowledge the support from several funding agencies: NSERC, the Canada Research Chairs, and the MITACS network. References [1] S. Abdallah and M. Plumbley. Geometry dependency analysis. Technical Report C4DM-TR06-05, Center for Digital Music, Queen Mary, University of London, 2006. [2] Y. Bengio and Y. Le Cun. Scaling learning algorithms towards AI. In L. Bottou, O. Chapelle, D. DeCoste, and J. Weston, editors, Large Scale Kernel Machines. MIT Press, 2007. [3] G. Hinton, M. Welling, Y. Teh, and S. Osindero. A new view of ica. In Proceedings of ICA-2001, San Diego, CA, 2001. [4] A. Hyv¨arinen, P. O. Hoyer, and M. Inki. Topographic independent component analysis. Neural Computation, 13(7):1527–1558, 2001. [5] K. J. Lang and G. E. Hinton. The development of the time-delay neural network architecture for speech recognition. Technical Report CMU-CS-88-152, Carnegie-Mellon University, 1988. [6] Y. LeCun, B. Boser, J. Denker, D. Henderson, R. Howard, W. Hubbard, and L. Jackel. Backpropagation applied to handwritten zip code recognition. Neural Computation, 1(4):541–551, 1989. [7] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, November 1998. [8] Y. LeCun and J. S. Denker. Natural versus universal probability complexity, and entropy. In IEEE Workshop on the Physics of Computation, pages 122–127. IEEE, 1992. [9] T.-W. Lee and M. S. Lewicki. Unsupervised classification segmentation and enhancement of images using ica mixture models. IEEE Trans. Image Proc., 11(3):270–279, 2002. [10] D. Lowe. Distinctive image features from scale-invariant keypoints. International Journal of Computer Vision, 60(2):91–110, 2004. [11] S. Osindero, M. Welling, and G. Hinton. Topographic product models applied to natural scene statistics. Neural Computation, 18:381–344, 2005. [12] S. Roweis and L. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500):2323–2326, Dec. 2000. [13] J. Tenenbaum, V. de Silva, and J. Langford. A global geometric framework for nonlinear dimensionality reduction. Science, 290(5500):2319–2323, Dec. 2000. [14] G. Tzanetakis and P. Cook. Musical genre classification of audio signals. IEEE Transactions on Speech and Audio Processing, 10(5):293–302, Jul 2002. [15] V. Vapnik. Estimation of Dependences Based on Empirical Data. Springer-Verlag, Berlin, 1982. [16] A. Waibel. Modular construction of time-delay neural networks for speech recognition. Neural Computation, 1:39–46, 1989. [17] K. Q. 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A Bayesian LDA-based model for semi-supervised part-of-speech tagging Kristina Toutanova Microsoft Research Redmond, WA kristout@microsoft.com Mark Johnson Brown University Providence, RI Mark Johnson@brown.edu Abstract We present a novel Bayesian model for semi-supervised part-of-speech tagging. Our model extends the Latent Dirichlet Allocation model and incorporates the intuition that words’ distributions over tags, p(t|w), are sparse. In addition we introduce a model for determining the set of possible tags of a word which captures important dependencies in the ambiguity classes of words. Our model outperforms the best previously proposed model for this task on a standard dataset. 1 Introduction Part-of-speech tagging is a basic problem in natural language processing and a building block for many components. Even though supervised part-of-speech taggers have reached performance of over 97% on in-domain data [1, 2], the performance on unknown in-domain words is below 90% and the performance on unknown out-of-domain words can be below 70% [3]. Additionally, few languages have a large amount of data labeled for part-of-speech. Thus it is important to develop methods that can use unlabeled data to learn part-of-speech. Research on unsupervised or partially supervised part-of-speech tagging has a long history [4, 5]. Recent work includes [6, 7, 8, 9, 10]. As in most previous work on partially supervised part-of-speech tagging, our model takes as input a (possibly incomplete) tagging dictionary, specifying, for some words, all of their possible parts of speech, as well as a corpus of unlabeled text. Our model departs from recent work on semisupervised part-of-speech induction using sequence HMM-based models, and uses solely observed context features to predict the tags of words. We show that using this representation of context gives our model substantial advantage over standard HMM-based models. There are two main innovations of our approach. The first is that we incorporate a sparse prior on the distribution over tags for each word, p(t|w), and employ a Bayesian approach that maintains a distribution over parameters, rather than committing to a single parameter value. Previous approaches to part-of-speech tagging ([9, 10]) also use sparse priors and Bayesian inference, but do not incorporate sparse priors directly on the p(t|w) distribution. Our results demonstrate that encoding this sparse prior and employing a Bayesian approach contributes significantly to performance. The second innovation of our approach is that we explicitly model ambiguity class (the set of part-ofspeech tags a word type can appear with). We show that this also results in substantial performance improvement. Our model outperforms the best-performing previously proposed model for this task [7], with an error reduction of up to 57% when the amount of supervision is small. The task setting is more formally as follows. Assume we are given a finite set of possible part-ofspeech tags (labels) T = {t1, t2, . . . , tnT }. The set of part-of-speech tags for English we experiment with has the 17 tags defined by Smith & Eisner [7], and is a coarse-grained version of the 45-tag set in the English Penn Treebank. We are also given a dictionary which specifies the ambiguity classes s ⊆T for a subset of the word types w. The ambiguity class of a word type is the set of all of its 1 α β θ t m4 u s ξ w m1 . . . ω M 4 c1 c4 . . . ϕ1 ϕ4 . . . T γ ψ4 . . . ψ1 T L W si | ξ ∼ MULTI(ξ), i = 1, . . . , L ui | si ∼ UNIFORM(si) i = 1, . . . , L mj,i | ui, ψj ∼ MULTI(ψj,ui), i = 1, . . . , L, j = 1, . . . , 4 wi | mi, ω ∼ MULTI(ωmi), i = 1, . . . , L βi | α, si = SUBSET(α, si), i = 1, . . . , L θi | βi ∼ DIR(βi), i = 1, . . . , L ti,j | θi ∼ MULTI(θi), i = 1, . . . , L, j = 1, . . . , Wi ϕk,ℓ | γ ∼ DIR(γ), k = 1, . . . , 4, ℓ= 1, . . . , T ck,i,j | ti,j, ϕk ∼ MULTI(ϕk,ti,j), i = 1, . . . , L, j = 1, . . . , Wi, k = 1, . . . , 4 Figure 1: A graphical model for the tagging model. In this model, each word type w is associated with a set s of possible parts-of-speech (ambiguity class), and each of its tokens is associated with a part-of-speech tag t, which generates the context words c surrounding that token. The ambiguity class s also generates the morphological features m of the word type w via a hidden tag u ∈s. The dotted line divides the model into the ambiguity class model (on the left) and the word context model (on the right). possible tags. For example, the dictionary might specify that walks has the ambiguity class {N, V } which means that walks can never have a tag which is not an N or a V. Additionally, we are given a large amount of unlabeled natural language text. The task is to label each word token with its correct part-of-speech tag in the corresponding context. This task formulation corresponds to a problem in computational linguistics that frequently arises in practice, because the only available resources for many languages consist of a manually constructed dictionary and a text corpus. Note that it differs from the standard semi-supervised learning setting, where we are given a small amount of labeled data and a large amount of unlabeled data. In the setting we study, we are never given labeled data, but are given instead constraints on possible tags of some words (in the form of a dictionary).1 2 Graphical model Our model is shown in Figure 1. In the figure, T is the set of part-of-speech tags, L is the set of word types (i.e., the set of different orthographic forms), W is the set of tokens (i.e., occurrences) of the word type w, and M 4 is the set of four-element morphological feature vectors described below. This is a generative model for a sequence of word tokens in a text corpus along with part-of-speech tags for all tokens, ambiguity classes for word types and other hidden variables. To generate the text corpus, the model generates the instances of every word type together with their contexts in 1For some words, the dictionary specifies only one possible tag, e.g. information →{N}, in which case all instances of information can be assumed labeled with the tag N. However these constraints are not sufficient to result in fully labeled sentences. 2 turn. The generation of a word type and all of its occurrences can be decomposed into two steps, corresponding to the left and right parts of the model: the ambiguity class model, and the word context model (separated by a dotted line in the figure). For every word type wi ∈L (plate L in the figure), in the first step the model generates an ambiguity class si ⊆T of possible parts of speech. The ambiguity class si is the set of parts-of-speech that tokens of wi can be labeled with. Our dictionary specifies si for some but not all word types wi. The ambiguity class si is generated by a multinomial over 2T with parameters ξ, with support on the different values for s observed in the dictionary. The ambiguity class si for wi generates four different morphological features m1,i, . . . , m4,i of wi representing the suffixes, capitalization, etc., of the orthographic form of wi. These are generated by multinomials with parameters ψ1,u, . . . , ψ4,u respectively, where u ∈s is a hidden variable generated by a uniform distribution over the members of s. For completeness we generate the full surface form of the word type wi from a multinomial distribution selected by its morphology features m1,i, . . . , m4,i. But since the morphology features are always observed (they are determined by wi’s orthographic form), we ignore this part of the model. We discuss the ambiguity class model in detail in Section 3.1. In the second step the word context model generates all instances wi,j of wi together with their part-of-speech tags ti,j and context words (plate W in the figure). This is done by first choosing a multinomial distribution θi over the tags in the set si, which is drawn from a Dirichlet with parameters βi and support si, where βi,t = αt for t ∈s. That is, si identifies the subset of T to receive support in βi, but the value of βi,t for t ∈si is specified by αt. Given these variables, all tokens wi,j of the word wi together with their contexts are generated by first choosing a part-of-speech tag ti,j from θi and then choosing context words ck,i,j preceding and following the word token wi,j according to tag-specific (depending on ti,j) multinomial distributions. The context of a word token c1,i,j . . . , c4,i,j consists of the two preceding and two following words. For example, for the sentence He often walks to school, the context words of that instance of walks are c1=He, c2=often, c3=to, and c4=school. This representation of the context has been used previously by unsupervised models for part-of-speech tagging in different ways [4, 8]. Each context word ck,i,j is generated by a multinomial with parameters ϕk,ti,j, where each ϕk,t is in turn generated by a Dirichlet with parameters γ. The parameters ϕk,t are generated once for the whole corpus as indicated in the figure. A sparse Dirichlet prior on θi with parameter α < 1 allows us to exploit the fact that most words have a very frequent predominant tag, and their distribution over tags p(t|w) is sparse. To verify this, we examined the distribution of the 17-label tag set in the WSJ Penn Treebank. A classifier that always chooses the most frequent tag for every word type, without looking at context, is 90.9% accurate on ambiguous words, indicating that the distribution is heavily skewed. Our model builds upon the Latent Dirichlet Allocation (LDA) model [11] by extending it in several ways. If we assume that the only possible ambiguity class s for all words is the set of all tags (and thus remove the ambiguity class model because it becomes irrelevant), and if we simplify our word context model to generate only one context word (say the word in position −1), we would end up with the LDA model. In this simplified model, we could say that for every word type wi we have a document consisting of all word tokens that occur in position −1 of the word type wi in the corpus. Each context word ci,j in wi’s document is generated by first choosing a tag (topic) from a word (document) specific distribution θi and then generating the word ci,j from a tag (topic) specific multinomial. The LDA model incorporates the same kind of Dirichlet priors on θ and ϕ that our model uses. The additional power of our model stems from the model of ambiguity classes si which can take advantage of the information provided by the dictionary, and from the incorporation of multiple context features. Finally, we note that our model is deficient, because the same word token in the corpus is independently generated multiple times (e.g., each token will appear in the context of four other words and will be generated four times). Even though this is a theoretical drawback of the model, it remains to be seen whether correcting for this deficiency (e.g., by re-normalization) would improve tagging performance. Models with similar deficiencies have been successful in other applications (e.g. the model described in [12], which achieved substantial improvements over the previous state-of-the-art in unsupervised parsing). 3 3 Parameter estimation and tag prediction Here we discuss our method of estimating the parameters of our model and making predictions, given an (incomplete) tagging dictionary and a set of natural language sentences. We train the parameters of the ambiguity class model, ξ, ψ, and ω, separately from the parameters of the word context model: α,θ,γ, and ϕ. This is because the two parts of the model are connected only via the variables si (the ambiguity classes of words), and when these ambiguity classes are given the two sets of parameters are completely decoupled. The dictionary gives us labeled training examples for the ambiguity class model, and we train the parameters of the ambiguity class model only from this data (i.e., the word types in the dictionary). After training the ambiguity class model from the dictionary we fix its parameters and estimate the word context model given these parameters. 3.1 Ambiguity class model: details and parameter estimation Our ambiguity class model captures the strong regularities governing the possible tags of a word type. Empirically we observe that the number of occurring ambiguity classes is very small relative to the number of possible ambiguity classes. For example, in the WSJ Penn Treebank data, the 49, 206 word types belong to 118 ambiguity classes. Modeling these (rather than POS tags directly) constrains the model to avoid assignments of tags to word tokens which would result in improbable ambiguity classes for word types. A related intuition has been used in other contexts before, e.g. [13, 14], but without directly modeling ambiguity classes. The ambiguity class model contributes to biasing p(t|w) toward sparse distributions as well, because most ambiguity classes have very few elements. For example, the top ten most frequent ambiguity classes in the complete dictionary consist of one or two elements. The ambiguity class of a word type can be predicted from its surface morphological features. For example the suffix -s of walks indicates that an ambiguity class of {N, V } is likely for this word. The four morphological features which we used for the ambiguity class model were: a binary feature indicating whether the word is capitalized, a binary feature indicating whether the word contains a hyphen, a binary feature indicating whether the word contains a digit character, and a nominal feature indicating the suffix of a word. We define the suffix of a word to be the longest character suffix (up to three letters) which occurs as a suffix of sufficiently many word types.2 We train the ambiguity class model on the set of word types present in the dictionary. We set the multinomial parameters ψk,l and ξ to maximize the joint likelihood of these word types and their morphological features. Maximum likelihood estimation for ψ is complicated by the hidden variable ui which selects a tag form the ambiguity class with uniform distribution. P(s, m1, m2, m3, m4|ψ, ξ) = P(s|ξ) P u∈s P(u|s) Q4 j=1 P(mj|ψj,l). We fix the probability P(u|s) = 1/|s| to the uniform distribution over tags in s. We estimate the ξ parameters using maximum likelihood estimation with add-1 (Laplace) smoothing and we train the ψ parameters using EM (with add-1 smoothing in the M-step). 3.2 Parameter estimation for the word context model and prediction given complete dictionary We restrict our attention at first to the setting where a complete tagging dictionary is given. The incomplete dictionary generalization is discussed in Section 3.3. When every word is in the dictionary, the ambiguity class si for each word type wi is specified by the tagging dictionary, and the ambiguity class model becomes irrelevant. The relevant parameters of the model in this setting are α,θ,γ, and ϕ. The contexts of word instances ck,i,j and the ambiguity classes si are observed. We integrate over all hidden variables except the uniform Dirichlet parameters α and γ. We set γ = 1 and we use Empirical Bayes to estimate α by maximizing the likelihood of the observed data given α and the ambiguity classes si. Note that if the ambiguity classes si and α are given, βi is fixed. Below we use c to denote the vector of all contexts of all word instances, and s the vector of ambiguity classes for all word types. We use ϕ to denote the vector of all multinomials ϕk,l, θ to 2A suffix occurs with sufficiently many word types if its type-frequency rank is below 100. 4 denote the vector of all θi and t to denote the vector of all tag sequences ti for word types wi. The likelihood we would like to maximize is: L(c|s, α, γ) = R P(ϕ|γ) QL i=1 R P(θi|βi) QWi j=1 PT l=1 θi,l Q4 k=1 P(ck,i,j|ϕk,l) dθidϕ P(ϕ|γ) = Q4 k=1 QT l=1 DIR(ϕk,l|γ) Since exact inference is intractable, we use a variational approximation to the posterior distribution of the hidden variables given the data and maximize instead of the exact log-likelihood, a lower bound given by the variational approximation. This variational approximation is also used for finding the most likely assignment of the part-of-speech tag variables to instances of words. More specifically, the variational approximation has analogous form to the approximation used for the LDA model [11]. It depends on variational parameters λk,l, ηi, and υi,j. Q(ϕ, θ, t|λ, η, υ) = Q4 k=1 QT l=1 DIR(ϕk,l|λk,l) QL i=1 DIR(θi|ηi) QWi j=1 P(ti,j|υi,j) This distribution is an approximation to the posterior distribution of the hidden variables: P(ϕ, θ, t|c, s, α, γ). As we can see, according to the Q distribution, the variables ϕ, θ, and t are independent. Each ϕk,l is distributed according to a Dirichlet distribution with variational parameters λk,l, each θi is also Dirichlet with parameters ηi and each tag ti,j is distributed according to a multinomial υi,j. We obtain the variational parameters by maximizing the following lower bound on the log-likelihood of the data (the dependence of Q on the variational parameters is not shown below for simplicity): EQ [log P(ϕ, θ, t, c|s, α, γ)] −EQ [log Q(ϕ, θ, t)] We use an iterative maximization algorithm for finding the values of the variational parameters. We do not describe it here due to space limitations, but it is analogous to the one used in [11]. Given fixed variational parameters λk,l we maximize with respect to the variational parameters ηi and υi,j corresponding to word types and their instances. Then keeping the latter parameters fixed, we maximize with respect to λk,l. We repeat until the change in the variational bound falls below a threshold. On our dataset, about 100 iterations of the outer loop for maximizing with respect to λk,l were necessary. Given a variational distribution Q we can maximize the lower bound on the loglikelihood with respect to α. Since α is determined by a single real-valued parameter, we maximized with respect to α using a simple grid search. For predicting the tags ti,j of word tokens we use the same approximate posterior distribution Q. Since according to Q all tags ti,j are independent given the variational parameters: Q(ti|υi) = QWi j=1(ti,j|υi,j), finding the most likely assignment is straightforward. 3.3 Parameter estimation for the word context model and prediction with incomplete dictionary So far we have described the training of the parameters of the word context model in the setting where for all words, the ambiguity classes si are known and these variables are observed. When the ambiguity classes si are unknown for some words in the dataset, they become additional hidden variables, and the hidden variables in the word context model become dependent on the morphological features mi and the parameters of the ambiguity class model. Denote the vector of ambiguity classes for the known (in the dictionary) word types by sd and the ambiguity classes for the unknown word types by su. The posterior distribution over the hidden variables of interest given the observations becomes: P(ϕ, θ, t, su|sd, mu, c, α, γ), where mu are the morphological features of the unknown word types. To perform inference in this setting we extend the variational approximation to account for the additional hidden variables. Before we had, for every word type, a variational distribution over the hidden variables corresponding to that word type: Q(θi, ti|ηi, υi,j) = DIR(θi|ηi) QWi j=1 P(ti,j|υi,j) We now introduce a variational distribution including new hidden variables si for unknown words. Q(θi, ti, si|mi, ηi,s, υi,j,s) = P(si|mi)DIR(θi|ηi,si) QWi j=1 P(ti,j|υi,j,si) 5 That is, for each possible ambiguity class si of an unknown word wi we introduce variational parameters specific to that ambiguity class. Instead of single variational parameters θi and υi,j for a word with known si, we now have variational parameters {θi,s} and {υi,j,s} for all possible values s of si. For simplicity, we use the probability P(si|mi) = P(si|mi, ξ, ψ) from the morphology-based ambiguity class model in the approximating distribution rather than introducing new variational parameters and learning this distribution.3 We adapt the algorithm to estimate the variational parameters. The derivation is slightly complicated by the fact that si and θi are not independent according to Q (this makes sense because si determines the dimensionality of θi), but the derived iterative algorithm is essentially the same as for our basic model, if we imagine that an unknown word type wi occurs with each of its possible ambiguity classes si a fractional p(si|mi) number of times. For predicting tag assignments for words according to this extended model, we use the same algorithm as described in Section 3.2, for word types whose ambiguity classes si are known. For words with unknown ambiguity classes, we need to maximize over ambiguity classes as well as tag assignments. We use the following algorithm to obtain a slightly better approximation than the one given by the variational distribution Q. For each possible tag set si, we find the most likely assignment of tags given that ambiguity class t∗(si), using the variational distribution as in the case of known ambiguity classes. We then choose an ambiguity class and an assignment of tags according to: s∗= arg maxsiP(si|mi, ψ, ξ)P(t∗(si), ci|si, D, α, γ) and t = t∗(s∗). We compute P(t∗(si), ci|si, D, α, γ) by integrating with respect to the word context distributions ϕ whose approximate posterior given the data is Dirichlet with parameters λk,l, and by integrating with respect to θi which are Dirichlet with parameters α and dimensionality given by si. 4 Experimental Evaluation We evaluate the performance of our model in comparison with other related models. We train and evaluate the model in three different settings. In the first setting, a complete tagging dictionary is available, and in the other two settings the coverage of the dictionary is greatly reduced. The tagging dictionary was constructed by collecting for each word type, the set of parts-of-speech with which it occurs in the annotated WSJ Penn Treebank, including the test set. This method of constructing a tag dictionary is arguably unrealistic but has been used in previous research [7, 9, 6] and provides a reproducible framework for comparing different models. In the complete dictionary setting, we use the ambiguity class information for all words, and in the second and third setting we remove from the dictionary all word types that have occurred with frequency less than 2 and less than 3, respectively, in the test set of 1,005 sentences. The complete tagging dictionary contains entries for 49, 206 words. The dictionary obtained with cutoff of 2 contains 2,141 words, and the one with cutoff of 3 contains 1,249 words. We train the model on the whole (unlabeled) WSJ Penn Treebank, consisting of 49,208 sentences. We evaluate performance on a set of 1,005 sentences, which is a subset of the training data and is the same test set used by [7, 9]. To see how much removing information from the dictionary impacts the hardness of the problem we can look at the accuracy of a classifier choosing a tag at random from the possible tags of words, shown in the column Random of Table 1. Results for the three settings are shown in the three rows of Table 1. In addition to the Random baseline, we include the results of a frequency baseline, Freq, in which for each word, we choose the most frequent tag from its set of possible tags.4 This baseline uses the same amount of partial supervision as our models. If labeled corpus data were available, a model which assigns the most frequent tag to each word by using ˆp(t|w) would do much better. The models in the table are: LDA is the model proposed in this paper, excluding the ambiguity class model. The ambiguity class model is irrelevant when a compete dictionary is available because all si are observed. In the other two settings for the LDA model we assume that si is the complete ambiguity class (all 17 tags) 3We also limit the number of possible ambiguity classes per word to the three most likely ones and renormalize the probability mass among them. 4Frequency of tags is unigram frequency of tags ˆp(t) by token in the unlabeled data. Since the tokens in the corpus are not actually labeled we compute the frequency by giving fractional counts to each possible tag of words in the dictionary. Only the words present in the dictionary were used for computing ˆp(t). 6 Dictionary LDA LDA PLSA PLSA CE (S&E) Bayesian ML Random Freq coverage + AC +AC +spelling HMM (G&G)HMM (G&G) complete 93.4 93.4 89.7 89.7 88.7 (91.9) 87.3 83.2 69.5 64.8 count ≥2 87.4 91.2 83.4 87.8 79.5 (90.3) 79.6 70.6 56.6 64.8 count ≥3 85.0 89.7 80.2 85.9 78.4 (89.5) 71.0 65.5 51.0 62.9 Table 1: Results from minimally supervised POS-tagging models. for words which are not in the dictionary and do not attempt to predict a more specific ambiguity class. The estimated parameter α for the tag prior was 0.5 for the complete dictionary setting, and 0.2 for the other two settings, encouraging sparse distributions. For this model we estimate the variational parameters λk,l and the Dirichlet parameter α to maximize the variational bound on the log-likelihood of the word types which are in the dictionary only. We found that including unknown word types was detrimental to performance. LDA+AC is our full model including the model of ambiguity classes of words given their morphological features. As mentioned above, this augmented model differs from LDA only when the dictionary is incomplete. We trained this model on all word types as discussed in Section 3.3. The estimated α parameters for this model in the three dictionary settings were 0.5, 0.1, and 0.1, respectively. PLSA is the model analogous to LDA, which has the same structure as our word context model, but excludes the Bayesian components. We include this model in the comparison in order to evaluate the effect on performance of the sparse prior and the integration over model parameters. This model is similar to the PLSA model for text documents [15]. The PLSA model does not have a prior on the word-specific distributions over tags θi = p(t|wi) and it does not have a prior distribution on the topic-specific multinomials for context words ϕk,l. For this model we find maximum likelihood estimates for these parameters by applying an EM algorithm. We do add-1 smoothing for ϕk,l in the M step, because even though this is not theoretically justified for this mixture model, it is frequently used in practice and helps prevent probabilities of zero for possible events. PLSA does not include the ambiguity class model for si and as in the LDA model, word types not in the dictionary were assumed to have ambiguity classes containing all 17 tags. PLSA+AC extends the PLSA model by the inclusion of the ambiguity class model. CE+spelling (S&E) is the sequence model for semi-supervised part-of-speech tagging proposed in [7], based on an HMM-structured model estimated using contrastive estimation. This is the stateof-the-art model for semi-supervised tagging using an incomplete dictionary. In the table we show actual performance and oracle performance for this model (oracle performance is in brackets).The oracle is obtained by testing models with different values of a smoothing hyper-parameter on the test set and choosing the model with the best accuracy. Even though there is only one real-valued hyper-parameter, the accuracies of models using different values can vary by nearly ten accuracy points and it is thus more fair to compare our results to the non-oracle result, until a better criterion for setting the hyper-parameters using only the partial supervision is found. The results shown in the table are for a model which incorporates morphological features. Bayesian HMM (G&G) is a fully Bayesian HMM model for semi-supervised part-of-speech tagging proposed in [9], which incorporates sparse Dirichlet priors on p(w|t) of word tokens given part of speech tags and p(ti|ti−1, ti−2) of transition probabilities in the HMM. We include this model in the comparison, because it uses sparse priors and Bayesian inference as our LDA model, but using a different structure of the model. [9] showed that this model outperforms significantly a non-Bayesian HMM model, whose results we show as well. ML HMM (G&G) is the maximum likelihood version of a trigram HMM for semi-supervised partof-speech tagging. Results for this model have been reported by other researchers as well [7, 6]. We use the performance numbers reported in [9] because they have used the same data sets for testing. The last two models do not use spelling (morphological) features. We should note that even though the same amount of supervision in the form of a tagging dictionary is used by all compared models, the HMM and CE models whose results are shown in the Table have been trained on less unsupervised natural language text: they have been trained using only the test set of 1,005 sentences. However, there is no reason one should limit the amount of unlabeled data used and in addition, 7 other results reported in [7] and [9] show that accuracy does not seem to improve when more unlabeled data are used with these models. There are several points to note about the experimental results. First, the fact that PLSA substantially outperforms ML HMM (and even the Bayesian HMM) models shows that predicting the tags of words from a window of neighboring word tokens and modeling the P(t|w) distribution directly results in an advantage over HMMs with maximum likelihood or Bayesian estimation. This is consistent with the success of other models that used word context for part-of-speech prediction in different ways [4, 8]. Second, the Bayesian and sparse-prior components of our model do indeed contribute substantially to performance, as illustrated by the performance of LDA compared to that of PLSA. LDA achieves an error reduction of up to 36% over PLSA. Third, our ambiguity class model results in a significant improvement as well; LDA+AC reduces the error of LDA by up to 31%. PLSA+AC similarly reduces the error of PLSA. Finally, our complete model outperforms the state-of-the-art model CE+spelling. It reduces the error of the non-oracle models by up to 57% and also outperforms the oracle models. We compared the performance of our model to that of state-of-the-art models applied in the same setting. It will also be interesting to compare our model to the one proposed in [8], which was applied in a different partial supervision setting. In their setting a small set of example word types (which they call prototypes) are provided for each possible tag (only three prototypes per tag were specified). Their model achieved an accuracy of 82.2% on a similar dataset. We can not directly compare the performance of our model to theirs, because our model would need prototypes for every ambiguity class rather than for every tag. In future work we will explore whether a very small set of prototypical ambiguity classes and corresponding word types can achieve the performance we obtained with an incomplete tagging dictionary. Another interesting direction for future work is applying our model to other NLP disambiguation tasks, such as named entity recognition and induction of deeper syntactic or semantic structure, which could benefit from both our ambiguity class model and our word context model. References [1] Kristina Toutanova, Dan Klein, and Christopher D. Manning. Feature-rich part-of-speech tagging with a cyclic dependency network. In Proceedings of HLT-NAACL 03, 2003. [2] Michael Collins. Discriminative training methods for hidden markov models: Theory and experiments with perceptron algorithms. In EMNLP, 2002. [3] John Blitzer, Ryan McDonald, and Fernando Pereira. Domain adaptation with structural correspondence learning. In EMNLP, 2006. [4] Hinrich Sch¨utze. Distributional part-of-speech tagging. In EACL, 1995. [5] Bernard Merialdo. Tagging english text with a probabilistic model. In ICASSP, 1991. [6] Michele Banko and Robert C. Moore. Part of Speech tagging in context. In COLING, 2004. [7] Noah A. Smith and Jason Eisner. Contrastive estimation: Training log-linear models on unlabeled data. In ACL, 2005. [8] Aria Haghighi and Dan Klein. Prototype-driven learning for sequence models. In HLT-NAACL, 2006. [9] Sharon Goldwater and Thomas L. Griffiths. A fully Bayesian approach to unsupervised Part-of-Speech tagging. In ACL, 2007. [10] Mark Johnson. Why doesn’t EM find good HMM POS-taggers. In EMNLP, 2007. [11] David Blei, Andrew Ng, and Michael Jordan. Latent dirichlet allocation. Journal of Machine Learning Research, 3:993–1022, 2003. [12] Dan Klein and Christopher D. Manning. Natural language grammar induction using a constituent-context model. In NIPS 14, 2002. [13] Jenny Rose Finkel, Trond Grenager, and Christopher Manning. Incorporating non-local information into information extraction systems by Gibbs sampling. In ACL, 2005. [14] Tetsuji Nakagawa and Yuji Matsumoto. Guessing parts-of-speech of unknown words using global information. In ACL, 2006. [15] Thomas Hofmann. Probabilistic latent semantic analysis. In UAI, 1999. 8
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Cluster Stability for Finite Samples Ohad Shamir† and Naftali Tishby†‡ † School of Computer Science and Engineering ‡ Interdisciplinary Center for Neural Computation The Hebrew University Jerusalem 91904, Israel {ohadsh,tishby}@cs.huji.ac.il Abstract Over the past few years, the notion of stability in data clustering has received growing attention as a cluster validation criterion in a sample-based framework. However, recent work has shown that as the sample size increases, any clustering model will usually become asymptotically stable. This led to the conclusion that stability is lacking as a theoretical and practical tool. The discrepancy between this conclusion and the success of stability in practice has remained an open question, which we attempt to address. Our theoretical approach is that stability, as used by cluster validation algorithms, is similar in certain respects to measures of generalization in a model-selection framework. In such cases, the model chosen governs the convergence rate of generalization bounds. By arguing that these rates are more important than the sample size, we are led to the prediction that stability-based cluster validation algorithms should not degrade with increasing sample size, despite the asymptotic universal stability. This prediction is substantiated by a theoretical analysis as well as some empirical results. We conclude that stability remains a meaningful cluster validation criterion over finite samples. 1 Introduction Clustering is one of the most common tools of unsupervised data analysis. Despite its widespread use and an immense amount of literature, distressingly little is known about its theoretical foundations [14]. In this paper, we focus on sample based clustering, where it is assumed that the data to be clustered are actually a sample from some underlying distribution. A major problem in such a setting is assessing cluster validity. In other words, we might wish to know whether the clustering we have found actually corresponds to a meaningful clustering of the underlying distribution, and is not just an artifact of the sampling process. This problem relates to the issue of model selection, such as determining the number of clusters in the data or tuning parameters of the clustering algorithm. In the past few years, cluster stability has received growing attention as a criterion for addressing this problem. Informally, this criterion states that if the clustering algorithm is repeatedly applied over independent samples, resulting in ’similar’ clusterings, then these clusterings are statistically significant. Based on this idea, several cluster validity methods have been proposed (see [9] and references therein), and were shown to be relatively successful for various data sets in practice. However, in recent work, it was proven that under mild conditions, stability is asymptotically fully determined by the behavior of the objective function which the clustering algorithm attempts to optimize. In particular, the existence of a unique optimal solution for some model choice implies stability as sample size increase to infinity. This will happen regardless of the model fit to the data. From this, it was concluded that stability is not a well-suited tool for model selection in clustering. This left open, however, the question of why stability is observed to be useful in practice. 1 In this paper, we attempt to explain why stability measures should have much wider relevance than what might be concluded from these results. Our underlying approach is to view stability as a measure of generalization, in a learning-theoretic sense. When we have a ’good’ model, which is stable over independent samples, then inferring its fit to the underlying distribution should be easy. In other words, stability should ’work’ because stable models generalize better, and models which generalize better should fit the underlying distribution better. We emphasize that this idea in itself is not novel, appearing explicitly and under various guises in many aspects of machine learning. The novelty in this paper lies mainly in the predictions that are drawn from it for clustering stability. The viewpoint above places emphasis on the nature of stability for finite samples. Since generalization is meaningless when the sample is infinite, it should come as no surprise that stability displays similar behavior. On finite samples, the generalization uncertainty is virtually always strictly positive, with different model choices leading to different convergence rates towards zero for increasing sample size. Based on the link between stability and generalization, we predict that on realistic data, all risk-minimizing models asymptotically become stable, but the rates of convergence to this ultimate stability differ. In other words, an appropriate scaling of the stability measures will make them independent of the actual sample size used. Using this intuition, we characterize and prove a mild set of conditions, applicable in principle to a wide class of clustering settings, which ensure the relevance of cluster stability for arbitrarily large sample sizes. We then prove that the stability measure used in previous work to show negative asymptotic results on stability, actually allows us to discern the ’correct’ model, regardless of how large is the sample, for a certain simple setting. Our results are further validated by some experiments on synthetic and real world data. 2 Definitions and notation We assume that the data sample to be clustered, S = {x1, .., xm}, is produced by sampling instances i.i.d from an underlying distribution D, supported on a subset X of Rn. A clustering CD for some D ⊆X is a function from D × D to {0, 1}, defining an equivalence relation on D with a finite number of equivalence classes (namely, CD(xi, xj) = 1 if xi and xj belong to the same cluster, and 0 otherwise). For a clustering CX of the instance space, and a finite sample S, let CX |S denote the functional restriction of CX on S × S. A clustering algorithm A is a function from any finite sample S ⊆X, to some clustering CX of the instance space1. We assume the algorithm is driven by optimizing an objective function, and has some user-defined parameters Θ. In particular, Ak denotes the algorithm A with the number of clusters chosen to be k. Following [2], we define the stability of a clustering algorithm A on finite samples of size m as: stab(A, D, m) = ES1,S2dD(A(S1), A(S2)), (1) where S1 and S2 are samples of size m, drawn i.i.d from D, and dD is some ’dissimilarity’ function between clusterings of X, to be specified later. Let ℓdenote a loss function from any clustering CS of a finite set S ⊆X to [0, 1]. ℓmay or may not correspond to the objective function the clustering algorithm attempts to optimize, and may involve a global quality measure rather than some average over individual instances. For a fixed sample size, we say that ℓobeys the bounded differences property (see [11]), if for any clustering CS it holds that |ℓ(CS) −ℓ(CS′)| ≤a, where a is a constant, and CS′ is obtained from CS by replacing at most one instance of S by any other instance from X, and clustering it arbitrarily. A hypothesis class H is defined as some set of clusterings of X. The empirical risk of a clustering CX ∈H on a sample S of size m is ℓ(CX |S). The expected risk of CX , with respect to samples S of size m, will be defined as ESℓ(CX |S). The problem of generalization is how to estimate the expected risk, based on the empirical data. 1Many clustering algorithms, such as spectral clustering, do not induce a natural clustering on X based on a clustering of a sample. In that case, we view the algorithm as a two-stage process, in which the clustering of the sample is extended to X through some uniform extension operator (such as assigning instances to the ’nearest’ cluster in some appropriate sense). 2 3 A Bayesian framework for relating stability and generalization The relationship between generalization and various notions of stability is long known, but has been dealt with mostly in a supervised learning setting (see [3][5] [8] and references therein). In the context of unsupervised data clustering, several papers have explored the relevance of statistical stability and generalization, separately and together (such as [1][4][14][12]). However, there are not many theoretical results quantitatively characterizing the relationship between the two in this setting. The aim of this section is to informally motivate our approach, of viewing stability and generalization in clustering as closely related. Relating the two is very natural in a Bayesian setting, where clustering stability implies an ’unsurprising’ posterior given a prior, which is based on clustering another sample. Under this paradigm, we might consider ’soft clustering’ algorithms which return a distribution over a measurable hypothesis class H, rather than a specific clustering. This distribution typically reflects the likelihood of a clustering hypothesis, given the data and prior assumptions. Extending our notation, we have that for any sample S, A(S) is now a distribution over H. The empirical risk of such a distribution, with respect to sample S′, is defined as ℓ(A(S)|S′) = ECX ∼A(S)ℓ(CX |S′). In this setting, consider for example the following simple procedure to derive a clustering hypothesis distribution, as well as a generalization bound: Given a sample of size 2m drawn i.i.d from D, we randomly split it into two samples S1,S2 each of size m, and use A to cluster each of them separately. Then we have the following: Theorem 1. For the procedure defined above, assume ℓobeys the bounded differences property with parameter 1/m. Define the clustering distance dD(P, Q) in Eq. (1), between two distributions P,Q over the hypothesis class H, as the Kullback-Leibler divergence DKL[Q||P]2. Then for a fixed confidence parameter δ ∈(0, 1), it holds with probability at least 1 −δ over the draw of samples S1 and S2 of size m, that ESℓ(A(S2)|S) −ℓ(A(S2)|S2) ≤ r dD(A(S1), A(S2)) + ln(m/δ) + 2 2m −1 . The theorem is a straightforward variant of the PAC-Bayesian theorem [10]. Since the loss function is not necessarily an empirical average, we need to utilize McDiarmid’s bound for random variables with bounded differences, instead of Hoeffding’s bound. Other than that, the proof is identical, and is therefore ommited. This theorem implies that the more stable is the Bayesian algorithm, the tighter the expected generalization bounds we can achieve. In fact, the ’expected’ magnitude of the high-probability bound we will get (over drawing S1 and S2 and performing the procedure described above) is: ES1,S2 r dD(A(S1), A(S2)) + ln(m/δ) + 2 2m −1 ≤ r ES1,S2dD(A(S1), A(S2)) + ln(m/δ) + 2 2m −1 = r stab(A, D, m) + ln(m/δ) + 2 2m −1 . Note that the only model-dependent quantity in the expression above is stab(A, D, m). Therefore, carrying out model selection by attempting to minimize these types of generalization bounds is closely related to minimizing stab(A, D, m). In general, the generalization bound might converge to 0 as m →∞, but this is immaterial for the purpose of model selection. The important factor is the relative values of the measure, over different choices of the algorithm parameters Θ. In other words, the important quantity is the relative convergence rates of this bound for different choices of Θ, governed by stab(A, D, m). This informal discussion only exemplifies the relationship between generalization and stability, since the setting and the definition of dD here differs from the one we will focus on later in the paper. Although these ideas can be generalized, they go beyond the scope of this paper, and we leave it for future work. 2Where we define DKL[Q||P] = R X Q(X) ln(Q(X)/P(X)), and DKL[q||p] for q, p ∈[0, 1] is defined as the divergence of Bernoulli distributions with parameters q and p. 3 4 Effective model selection for arbitrarily large sample sizes From now on, following [2], we will define the clustering distance function dD of Eq. (1) as: dD(A(S1), A(S2)) = Pr x1,x2∼D (A(S1)(x1, x2) ̸= A(S2)(x1, x2)) . (2) In other words, the clustering distance is the probability that two independently drawn instances from D will be in the same cluster under one clustering, and in different clusters under another clustering. In [2], it is essentially proven that if there exists a unique optimizer to the clustering algorithm’s objective function, to which the algorithm converges for asymptotically large samples, then stab(A, D, m) converges to 0 as m →∞, regardless of the parameters of A. From this, it was concluded that using stability as a tool for cluster validity is problematic, since for large enough samples it would always be approximately zero, for any algorithm parameters chosen. However, using the intuition gleaned from the results of the previous section, the different convergence rates of the stability measure (for different algorithm parameters) should be more important than their absolute values or the sample size. The key technical result needed to substantiate this intuition is the following theorem: Theorem 2. Let X, Y be two random variables bounded in [0, 1], and with strictly positive expected values. Assume E[X]/E[Y ] ≥1 + c for some positive constant c. Letting X1, . . . , Xm and Y1, . . . , Ym be m identical independent copies of X and Y respectively, define ˆX = 1 m Pm i=1 Xi and ˆY = 1 m Pm i=1 Yi. Then it holds that: Pr( ˆX ≤ˆY ) ≤exp à −1 8mE[X] µ c 1 + c ¶4! + exp à −1 4mE[X] µ c 1 + c ¶2! . The importance of this theorem becomes apparent when ˆX, ˆY are taken to be empirical estimators of stab(A, D, m) for two different algorithm parameter sets Θ, Θ′. For example, suppose that according to our stability measure (see Eq. (1)), a cluster model with k clusters is more stable than a model with k′ clusters, where k ̸= k′, for sample size m (e.g., stab(Ak, D, m) < stab(Ak′, D, m)). These stability measures might be arbitrarily close to zero. Assume that with high probability over the choice of samples S1 and S2 of size m, we can show that dD(Ak(S1), Ak(S2)) ≤1/√m, while dD(Ak′(S1), Ak′(S2)) ≥1.01/√m. We cannot compute these exactly, since the definition of dD involves an expectation over the unknown distribution D (see Eq. (2)). However, we can estimate them by drawing another sample S3 of m instance pairs, and computing a sample mean to estimate Eq. (2). According to Thm. 2, since dD(Ak(S1), Ak(S2)) and dD(Ak′(S1), Ak′(S2)) have slightly different convergence rates (c ≥0.01), which are slower than Θ(1/m), then we can discern which number of clusters is more stable, with a high probability which actually improves as m increases. Therefore, we can use Thm. 2 as a guideline for when a stability estimator might be useful for arbitrarily large sample sizes. Namely, we need to show it is an expected value of some random variable, with at least slightly different convergence rates for different model selections, and with at least some of them dominating Θ(1/m). We would expect these conditions to hold under quite general settings, since most stability measures are based on empirically estimating the mean of some random variable. Moreover, a central-limit theorem argument leads us to expect an asymptotic form of Ω(1/√m), with the exact constants dependent on the model. This convergence rate is slow enough for the theorem to apply. The difficult step, however, is showing that the differing convergence rates can be detected empirically, without knowledge of D. In the example above, this reduces to showing that with high probability over S1 and S2, dD(Ak(S1), Ak(S2)) and dD(Ak′(S1), Ak′(S2)) will indeed differ by some constant ratio independent of m. Proof of Thm. 2. Using a relative entropy variant of Hoeffding’s bound [7], we have that for any 1 > b > 0 and 1/E[Y ] > a > 1, it holds that: Pr ³ ˆX ≤bE[X] ´ ≤exp (−m DKL [bE [X] || E [X]]) , Pr ³ ˆY ≥aE[Y ] ´ ≤exp (−m DKL [aE [Y ] || E [Y ]]) . 4 By substituting the bound DKL[p||q] ≥(p −q)2/2 max{p, q} in the two inequalities, we get: Pr ³ ˆX ≤bE[X] ´ ≤exp µ −1 2mE [X] (1 −b)2 ¶ (3) Pr ³ ˆY ≥aE[Y ] ´ ≤exp µ −1 2mE [Y ] µ a + 1 a −2 ¶¶ , (4) which hold whenever 1 > b > 0 and a > 1. Let b = 1 −(1 −E[Y ]/E[X])2 /2, and a = bE[X]/E[Y ]. It is easily verified that b < 1 and a > 1. Substituting these values into the r.h.s of Eq. (3), and to both sides of Eq. (4), and after some algebra, we get: Pr( ˆX ≤bE[X]) ≤exp à −1 8mE[X] µ c 1 + c ¶4! , Pr( ˆY ≥bE[X]) ≤exp à −1 4mE[X] µ c 1 + c ¶2! . As a result, by the union bound, we have that Pr( ˆX ≤ˆY ) is at most the sum of the r.h.s of the last two inequalities, hence proving the theorem. As a proof of concept, we show that for a certain setting, the stability measure used by [2], as defined above, is meaningful for arbitrarily large sample sizes, even when this measure converges to zero for any choice of the required number of clusters. The result is a simple counter-example to the claim that this phenomenon makes cluster stability a problematic tool. The setting we analyze is a mixture distribution of three well-separated unequal Gaussians in R, where an empirical estimate of stability, using a centroid-based clustering algorithm, is utilized to discern whether the data contain 2, 3 or 4 clusters. We prove that with high probability, this empirical estimation process will discern k = 3 as much more stable than both k = 2 and k = 4 (by an amount depending on the separation between the Gaussians). The result is robust enough to hold even if in addition one performs normalization procedures to account for the fact that higher number of clusters entail more degrees of freedom for the clustering algorithm (see [9]). We emphasize that the simplicity of this setting is merely for the sake of analytical convenience. The proof itself relies on a general and intuitive characteristic of what constitutes a ’wrong’ model (namely, having cluster boundaries in areas of high density), rather than any specific feature of this setting. We are currently working on generalizing this result, using a more involved analysis. In this setting, by the results of [2], stab(Ak, D, m) will converge to 0 as m →∞for k = 2, 3, 4. The next two lemmas, however, show that the stability measure for k = 3 (the ’correct’ model order) is smaller than the other two, by a substantial ratio independent of m, and that this will be discerned, with high probability, based on the empirical estimates of dD(Ak(S1), Ak(S2)). The proofs are technical, and appear in the supplementary material to this paper. Lemma 1. For some µ > 0, let D be a Gaussian mixture distribution on R, with density function p(x) = 2 3 √ 2π exp µ −(x + µ)2 2 ¶ + 1 6 √ 2π exp µ −x2 2 ¶ + 1 6 √ 2π exp µ −(x −µ)2 2 ¶ . Assume µ ≫1, so that the Gaussians are well separated. Let Ak be a centroid-based clustering algorithm, which is given a sample and required number of clusters k, and returns a set of k centroids, minimizing the k-means objective function (sum of squared Euclidean distances between each instance and its nearest centroid). Then the following holds, with o(1) signifying factors which converge to 0 as m →∞: stab(A2, D, m) ≥1 −o(1) 7√m exp µ −µ2 32 ¶ , stab(A4, D, m) ≥0.4 −o(1) √m stab(A3, D, m) ≤1.1 + o(1) √m exp µ −µ2 8 ¶ . 5 Lemma 2. For the setting described in Lemma 1, it holds that over the draw of independent sample pairs (S1, S2), (S′ 1, S′ 2), (S′′ 1 , S′′ 2 ) (each of size m from D), the ratio between dD(A2(S′ 1), A2(S′ 2)) and dD(A3(S1), A3(S2)), as well as the ratio between dD(A4(S′′ 1 ), A4(S′′ 2 )) and dD(A3(S1), A3(S2)), is larger than 2 with probability of at least: 1 −(4 + o(1)) µ exp µ −µ2 16 ¶ + exp µ −µ2 32 ¶¶ . It should be noted that the asymptotic notation is merely to get rid of second-order terms, and is not an essential feature. Also, the constants are by no means the tightest possible. With these lemmas, we can prove that a direct estimation of stab(A, D, m), based on a random sample, allows us to discern the more stable model with high probability, for arbitrarily large sample sizes. Theorem 3. For the setting described in Lemma 1, define the following unbiased estimator ˆθk,4m of stab(Ak, D, m): Given a sample of size 4m, split it randomly into 3 disjoint subsets S1,S2,S3 of size m,m and 2m respectively. Estimate dD(Ak(S1), Ak(S2)) by computing 1 m X xi,xm+i∈S3 1 ³ Ak(S1)(xi, xm+i) ̸= Ak(S2)(xi, xm+i) ´ , where (x1, .., xm) is a random permutation of S3, and return this value as an estimate of stab(Ak, D, m). If three samples of size 4m each are drawn i.i.d from D, and are used to calculate ˆθ2,4m, ˆθ3,4m, ˆθ4,4m, then Pr ³ ˆθ3,4m ≥min n ˆθ2,4m, ˆθ4,4m o´ ≤exp ¡ −Ω(µ2) ¢ + exp ¡ −Ω ¡√m ¢¢ . Proof. Using Lemma 2, we have that: Pr µmin {dD(A2(S′ 1), A2(S′ 2)), dD(A4(S′′ 1 ), A4(S′′ 2 ))} dD(A3(S1), A3(S2)) ≤2 ¶ < exp ¡ −Ω(µ2) ¢ . (5) Denoting the event above as B, and assuming it does not occur, we have that the estimators ˆθ2,4m, ˆθ3,4m, ˆθ4,4m are each an empirical average over an additional sample of size m, and the expected value of ˆθ3,4m is at least twice smaller than the expected values of the other two. Moreover, by Lemma 1, the expected value of dD(A3(S1), A3(S2)) is Ω(1/√m). Invoking Thm. 2, we have that: Pr ³ ˆθ3,4m ≥min n ˆθ2,4m, ˆθ4,4m o ¯¯ B∁´ ≤exp ¡ −Ω ¡√m ¢¢ (6) Combining Eq. (5) and Eq. (6) yield the required result. 5 Experiments In order to further substantiate our analysis above, some experiments were run on synthetic and real world data, with the goal of performing model selection over the number of clusters k. Our first experiment simulated the setting discussed in section 4 (see figure 1). We tested 3 different Gaussian mixture distributions (with µ = 5, 7, 8), and sample sizes m ranging from 25 to 222. For each distribution and sample size, we empirically estimated ˆθ2, ˆθ3 and ˆθ4 as described in section 4, using the k-means algorithm, and repeated this procedure over 1000 trials. Our results show that although these empirical estimators converge towards zero, their convergence rates differ, with approximately constant ratios between them. Scaling the graphs by √m results in approximately constant and differing stability measures for each µ. Moreover, the failure rate does not increase with sample size, and decreases rapidly to negligible size as the Gaussians become more well separated - exactly in line with Thm. 3. Notice that although in the previous section we assumed a large separation between the Gaussians for analytical convenience, good results are obtained even when this separation is quite small. For the other experiments, we used the stability-based cluster validation algorithm proposed in [9], which was found to compare favorably with similar algorithms, and has the desirable property of 6 −10 −5 0 5 10 10 0 0.1 0.2 0.3 Distribution p(x) 10 1 10 3 10 5 10 7 10 −8 10 −6 10 −4 10 −2 Values of ˆθ2, ˆθ3, ˆθ4 10 1 10 3 10 5 10 7 0 0.1 0.2 0.3 0.4 0.5 0.5 Failure Rate −10 −5 0 5 10 0 0.1 0.2 0.3 p(x) 10 1 10 3 10 5 10 7 10 −8 10 −6 10 −4 10 −2 10 1 10 3 10 5 10 7 0 0.1 0.2 0.3 0.4 0.5 −10 −5 0 5 10 0 0.1 0.2 0.3 p(x) 10 1 10 3 10 5 10 7 10 −8 10 −6 10 −4 10 −2 m 10 1 10 3 10 5 10 7 0 0.1 0.2 0.3 0.4 0.5 m k=2 k=3 k=4 k=2 k=3 k=4 k=2 k=3 k=4 Figure 1: Empirical validation of results in section 4. In each row, the leftmost sub-figure is the actual distribution, the middle sub-figure is a log-log plot of the estimators ˆθ2, ˆθ3, ˆθ4 (averaged over 1000 trials), as a function of the sample size, and on the right is the failure rate as a function of the sample size (percentage of trials where ˆθ3 was not the smallest of the three). −10 −5 0 5 −10 −5 0 5 Random Sample 10 2 10 3 10 4 10 5 10 6 10 −4 10 −3 10 −2 10 −1 k=3 k=4 k=5 k=6 k=7 Values of stability method index 10 2 10 3 10 4 10 5 0 0.1 0.2 0.3 0.4 0.5 Failure Rate −2 0 2 −2 0 2 2000 4000 8000 10 −3 10 −2 10 −1 k=2 k=3 k=4 k=5 2000 4000 8000 0 0.1 0.2 0.3 0.4 0.5 −100 0 100 200 300 −50 0 50 sh iy dcl aa ao 500 1000 5000 10 −3 10 −2 10 −1 k=3 k=4 k=5 k=6 m 500 1000 5000 0 0.1 0.2 0.3 0.4 0.5 m Figure 2: Performance of stability based algorithm in [9] on 3 data sets. In each row, the leftmost sub-figure is a sample representing the distribution, the middle sub-figure is a log-log plot of the computed stability indices (averaged over 100 trials), and on the right is the failure rate (in detecting the most stable model over repeated trials). In the phoneme data set, the algorithm selects 3 clusters as the most stable models, since the vowels tend to group into a single cluster. The ’failures’ are all due to trials when k = 4 was deemed more stable. 7 producing a clear quantitative stability measure, bounded in [0, 1]. Lower values match models with higher stability. The synthetic data sets selected (see figure 2) were a mixture of 5 Gaussians, and segmented 2 rings. We also experimented on the Phoneme data set [6], which consists of 4, 500 log-periodograms of 5 phonemes uttered by English speakers, to which we applied PCA projection on 3 principal components as a pre-processing step. The advantage of this data set is its clear low-dimensional representation relative to its size, allowing us to get nearer to the asymptotic convergence rates of the stability measures. All experiments used the k-means algorithm, except for the ring data set which used the spectral clustering algorithm proposed in [13]. Complementing our theoretical analysis, the experiments clearly demonstrate that regardless of the actual stability measures per fixed sample size, they seem to eventually follow roughly constant and differing convergence rates, with no substantial degradation in performance. In other words, when stability works well for small sample sizes, it should also work at least as well for larger sample sizes. The universal asymptotic convergence to zero does not seem to be a problem in that regard. 6 Conclusions In this paper, we propose a principled approach for analyzing the utility of stability for cluster validation in large finite samples. This approach stems from viewing stability as a measure of generalization in a statistical setting. It leads us to predict that in contrast to what might be concluded from previous work, cluster stability does not necessarily degrade with increasing sample size. This prediction is substantiated both theoretically and empirically. The results also provide some guidelines (via Thm. 2) for when a stability measure might be relevant for arbitrarily large sample size, despite asymptotic universal stability. They also suggest that by appropriate scaling, stability measures would become insensitive to the actual sample size used. These guidelines do not presume a specific clustering framework. However, we have proven their fulfillment rigorously only for a certain stability measure and clustering setting. The proof can be generalized in principle, but only at the cost of a more involved analysis. We are currently working on deriving more general theorems on when these guidelines apply. Acknowledgements: This work has been partially supported by the NATO SfP Programme and the PASCAL Network of excellence. References [1] Shai Ben-David. A framework for statistical clustering with a constant time approximation algorithms for k-median clustering. In Proceedings of COLT 2004, pages 415–426. [2] Shai Ben-David, Ulrike von Luxburg, and D´avid P´al. A sober look at clustering stability. In Proceedings of COLT 2006, pages 5–19. [3] Olivier Bousquet and Andr´e Elisseeff. Stability and generalization. Journal of Machine Learning Research, 2:499–526, 2002. [4] Joachim M. Buhmann and Marcus Held. Model selection in clustering by uniform convergence bounds. In Advances in Neural Information Processing Systems 12, pages 216–222, 1999. [5] Andrea Caponnetto and Alexander Rakhlin. Stability properties of empirical risk minimization over donsker classes. Journal of Machine Learning Research, 6:2565–2583, 2006. [6] Trevor Hastie, Robert Tibshirani, Jerome Friedman. The Elements of Statistical Learning. Springer, 2001. [7] Wassily Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58(301):13–30, March 1963. [8] Samuel Kutin and Partha Niyogi. Almost-everywhere algorithmic stability and generalization error. In Proceeding of the 18th confrence on Uncertainty in Artificial Intelligence (UAI), pages 275–282, 2002. [9] Tilman Lange, Volker Roth, Mikio L. Braun, and Joachim M. Buhmann. Stability-based validation of clustering solutions. Neural Computation, 16(6):1299–1323, June 2004. [10] D.A. McAllester. Pac-bayesian stochastic model selection. Machine Learning Journal, 51(1):5–21, 2003. [11] C. McDiarmid. On the method of bounded differences. In Surveys in Combinatorics, volume 141 of London Mathematical Society Lecture Note Series, pages 148–188. Cambridge University Press, 1989. [12] Alexander Rakhlin and Andrea Caponnetto. Stability of k-means clustering. In Advances in Neural Information Processing Systems 19. MIT Press, Cambridge, MA, 2007. [13] Jianbo Shi and Jitendra Malik. Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(8):888–905, 2000. [14] Ulrike von Luxburg and Shai Ben-David. Towards a statistical theory of clustering. Technical report, PASCAL workshop on clustering, London, 2005. 8
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Variational Inference for Diffusion Processes C´edric Archambeau University College London c.archambeau@cs.ucl.ac.uk Manfred Opper Technical University Berlin opperm@cs.tu-berlin.de Yuan Shen Aston University y.shen2@aston.ac.uk Dan Cornford Aston University d.cornford@aston.ac.uk John Shawe-Taylor University College London jst@cs.ucl.ac.uk Abstract Diffusion processes are a family of continuous-time continuous-state stochastic processes that are in general only partially observed. The joint estimation of the forcing parameters and the system noise (volatility) in these dynamical systems is a crucial, but non-trivial task, especially when the system is nonlinear and multimodal. We propose a variational treatment of diffusion processes, which allows us to compute type II maximum likelihood estimates of the parameters by simple gradient techniques and which is computationally less demanding than most MCMC approaches. We also show how a cheap estimate of the posterior over the parameters can be constructed based on the variational free energy. 1 Introduction Continuous-time diffusion processes, described by stochastic differential equations (SDEs), arise naturally in a range of applications from environmental modelling to mathematical finance [13]. In statistics the problem of Bayesian inference for both the state and parameters, within partially observed, non-linear diffusion processes has been tackled using Markov Chain Monte Carlo (MCMC) approaches based on data augmentation [17, 11], Monte Carlo exact simulation methods [6], or Langevin / hybrid Monte Carlo methods [1, 3]. Within the signal processing community solutions to the so called Zakai equation [12] based on particle filters [8], a variety of extensions to the Kalman filter/smoother [2, 5] and mean field analysis of the SDE together with moment closure methods [10] have also been proposed. In this work we develop a novel variational approach to the problem of approximate inference in continuous-time diffusion processes, including a marginal likelihood (evidence) based inference technique for the forcing parameters. In general, joint parameter and state inference using naive methods is complicated due to dependencies between state and system noise parameters. We work in continuous time, computing distributions over sample paths1, and discretise only in our posterior approximation, which has advantages over methods based on discretising the SDE directly [3]. The approximate inference approach we describe is more computationally efficient than competing Monte Carlo algorithms and could be further improved in speed by defining a variety of sub-optimal approximations. The approximation is also more accurate than existing Kalman smoothing methods applied to non-linear systems [4]. Ultimately, we are motivated by the critical requirement to estimate parameters within large environmental models, where at present only a small number of Kalman filter/smoother based estimation algorithms have been attempted [2], and there have been no likelihood based attempts to estimate the system noise forcing parameters. 1A sample path is a continuous-time realisation of a stochatic process in a certain time interval. Hence, a sample path is an infinite dimensional object. 1 In Section 2 and 3, we introduce the formalism for a variational treatment of partially observed diffusion processes with measurement noise and we provide the tools to estimate the optimal variational posterior process [4]. Section 4 deals with the estimation of the drift and the system noise parameter, as well as the estimation of the optimal initial conditions. Finally, the approach is validated on a bi-stable nonlinear system in Section 5. In this context, we also discuss how to construct an estimate of the posterior distribution over parameters based on the variational free energy. 2 Diffusion processes with measurement error Consider the continuous-time continuous-state stochastic process X = {Xt, t0 ≤t ≤tf}. We assume this process is a d-dimensional diffusion process. Its time evolution is described by the following SDE (to be interpreted as an Ito stochastic integral): dXt = fθ(t, Xt) dt + Σ1/2 dWt, dWt ∼N(0, dtI). (1) The nonlinear vector function fθ defines the deterministic drift and the positive semi-definite matrix Σ ∈Rd×d is the system noise covariance. The diffusion is modelled by a d-dimensional Wiener process W = {Wt, t0 ≤t ≤tf} (see e.g. [13] for a formal definition). Eq. (1) defines a process with additive system noise. This might seem restrictive at first sight. However, it can be shown [13, 17, 6] that a range of state dependent stochastic forcings can be transformed into this form. It is further assumed that only a small number of discrete-time latent states are observed and that the observations are subject to measurement error. We denote the set of observations at the discrete times {tn}N n=1 by Y = {yn}N n=1 and the corresponding latent states by {xn}N n=1, with xn = Xt=tn.For simplicity, the measurement noise is modelled by a zero-mean multivariate Gaussian density,with covariance matrix R ∈Rd×d. 3 Approximate inference for diffusion processes Our approximate inference scheme builds on [4] and is based on a variational inference approach (see for example [7]). The aim is to minimise the variational free energy, which is defined as follows: FΣ(q, θ) = − ln p(Y, X|θ, Σ) q(X|Σ) q , X = {Xt, t0 ≤t ≤tf}, (2) where q(X|Σ) is an approximate posterior process over sample paths in the interval [t0, tf] and θ are the parameters, excluding the stochastic forcing covariance matrix Σ. Hence, this quantity is an upper bound to the negative log-marginal likelihood: −ln p(Y |θ, Σ) = FΣ(q, θ) −KL [q(X|Σ)∥p(X|Y, θ, Σ)] ≤FΣ(q, θ). (3) As noted in Appendix A, this bound is finite if the approximate process is another diffusion process with a system noise covariance chosen to be identical to that of the prior process induced by (1). The standard approach for learning the parameters in presence of latent variables is to use an EM type algorithm [9]. However, since the variational distribution is restricted to have the same system noise covariance (see Appendix A) as the true posterior, the EM algorithm would leave this covariance completely unchanged in the M step and cannot be used for learning this crucial parameter. Therefore, we adopt a different approach, which is based on a conjugate gradient method. 3.1 Optimal approximate posterior process We consider an approximate time-varying linear process with the same diffusion term, that is the same system noise covariance: dXt = g(t, Xt) dt + Σ1/2 dWt, dWt ∼N(0, dtI), (4) where g(t, x) = −A(t)x+b(t), with A(t) ∈Rd×d and b(t) ∈Rd. In other words, the approximate posterior process q(X|Σ) is restricted to be a Gaussian process [4]. The Gaussian marginal at time t is defined as follows: q(Xt|Σ) = N(Xt|m(t), S(t)), t0 ≤t ≤tf, (5) 2 where m(t) ∈Rd and S(t) ∈Rd×d are respectively the marginal mean and the marginal covariance at time t. In the rest of the paper, we denote q(Xt|Σ) by the shorthand notation qt. For fixed parameters θ and assuming that there is no observation at the initial time t0, the optimal approximate posterior process q(X|Σ) is the one minimizing the variational free energy, which is given by (see Appendix A) FΣ(q, θ) = Z tf t0 Esde(t) dt + Z tf t0 Eobs(t) X n δ(t −tn) dt + KL [q0∥p0] . (6) The function δ(t) is Dirac’s delta function. The energy functions Esde(t) and Eobs(t) are defined as follows: Esde(t) = 1 2 (fθ(t, Xt) −g(t, Xt))⊤Σ−1(fθ(t, Xt) −g(t, Xt)) qt , (7) Eobs(t) = 1 2 (Yt −Xt)⊤R−1(Yt −Xt) qt + d 2 ln 2π + 1 2 ln |R|. (8) where {Yt, t0 ≤t ≤tf} is the underlying continuous-time observable process. 3.2 Smoothing algorithm The variational parameters to optimise in order to find the optimal Gaussian process approximation are A(t), b(t), m(t) and S(t). For a linear SDE with additive system noise, it can be shown that the time evolution of the means and the covariances are described by a set of ordinary differential equations [13, 4]: ˙m(t) = −A(t)m(t) + b(t), (9) ˙S(t) = −A(t)S(t) −S(t)A⊤(t) + Σ, (10) where ˙ denotes the time derivtive. These equations provide us with consistency constraints for the marginal means and the marginal covariances along sample paths. To enforce these constraints we formulate the Lagrangian Lθ,Σ = FΣ(q, θ) − Z tf t0 λ⊤(t) ˙m(t) + A(t)m(t) −b(t) dt − Z tf t0 tr n Ψ(t) ˙S(t) + 2A(t)S(t) −Σ o dt, (11) where λ(t) ∈Rd and Ψ(t) ∈Rd×d are time dependent Lagrange multipliers, with Ψ(t) symmetric. First, taking the functional derivatives of Lθ,Σ with respect to A(t) and b(t) results in the following gradient functions: ∇ALθ,Σ(t) = ∇AEsde(t) −λ(t)m⊤(t) −2Ψ(t)S(t), (12) ∇bLθ,Σ(t) = ∇bEsde(t) + λ(t). (13) The gradients ∇AEsde(t) and ∇bEsde(t) are derived in Appendix B. Secondly, taking the functional derivatives of Lθ,Σ with respect to m(t) and S(t), setting to zero and rearranging leads to a set of ordinary differential equations, which describe the time evolution of the Lagrange multipliers, along with jump conditions when there are observations: ˙λ(t) = −∇mEsde(t) + A⊤(t)λ(t), λ+ n = λ− n −∇mEobs(t)|t=tn, (14) ˙Ψ(t) = −∇SEsde(t) + 2Ψ(t)A(t), Ψ+ n = Ψ− n −∇SEobs(t)|t=tn. (15) The optimal variational functions can be computed by means of a gradient descent technique, such as the conjugate gradient (see e.g., [16]). The explicit gradients with respect to A(t) and b(t) are given by (12) and (13). Since m(t), S(t), λ(t) and Ψ(t) are dependent on these parameters, one needs also to take the corresponding implicit derivatives into account. However, these implicit gradients vanish if the consistency constraints for the means (9) and the covariances (10), as well as the ones for the Lagrange multipliers (14-15), are satisfied. One way to achieve this is to perform a forward propagation of the means and the covariances, followed by a backward propagation of the Lagrange multipliers, and then to take a gradient step. The resulting algorithm for computing the optimal posterior q(X|Σ) over sample paths is detailed in Algorithm 1. 3 Algorithm 1 Compute the optimal q(X|Σ). 1: input(m0, S0, θ, Σ, t0, tf, ∆t, ω) 2: K ←(tf −t0)/∆t 3: initialise {Ak, bk}k≥0 4: repeat 5: for k = 0 to K −1 do 6: mk+1 ←mk −(Akmk −bk)∆t 7: Sk+1 ←Sk −(AkSk + SkA⊤ k −Σ)∆t 8: end for{forward propagation} 9: for k = K to 1 do 10: λk−1 ←λk + (∇mEsde|t=tk −A⊤ k λk)∆t 11: Ψk−1 ←Ψk + (∇SEsde|t=tk −2ΨkAk)∆t 12: if observation at tk−1 then 13: λk−1 ←λk−1 + ∇mEobs|t=tk−1 14: Ψk−1 ←Ψk−1 + ∇SEobs|t=tk−1 15: end if{jumps} 16: end for{backward sweep (adjoint operation)} 17: update {Ak, bk}k≥0 using the gradient functions (12) and (13) 18: until minimum of Lθ,Σ is attained {optimisation loop} 19: return {Ak, bk, mk, Sk, λk, Ψk}k≥0 4 Parameter estimation The parameters to optimise include the parameters of the prior over the initial state, the drift function parameters and the system noise covariance. The estimation of the parameters related to the observable process are not discussed in this work, although it is a straightforward extension. The smoothing algorithm described in the previous section computes the optimal posterior process by providing us with the stationary solution functions A(t) and b(t). Therefore, when subsequently optimising the parameters we only need to compute their explicit derivatives; the implicit ones vanish since ∇ALθ,Σ = 0 and ∇bLθ,Σ = 0. Before computing the gradients, we integrate (11) by parts to make the boundary conditions explicit. This leads to Lθ,Σ = FΣ(q, θ) − Z tf t0 n λ⊤(t) A(t)m(t) −b(t) −˙λ ⊤(t)m(t) o dt −λ⊤ f mf + λ⊤ 0 m0 − Z tf t0 tr n Ψ(t) 2A(t)S(t) −Σ −˙Ψ(t)S(t) o dt −tr {ΨfSf} + tr {Ψ0S0} , (16) At the final time tf, there are no consistency constraints, that is λf and Ψf are both equal to zero. 4.1 Initial state The initial variational posterior q(x0) is chosen equal to N(x0|m0, S0) to ensure that the approximate process is a Gaussian one. Taking the derivatives of (16) with respect to m0 and S0 results in the following expressions: ∇m0Lθ,Σ = λ0 + τ −1 0 (m0 −µ0), ∇S0Lθ,Σ = Ψ0 + 1 2 τ −1 0 I −S−1 0 , (17) where the prior p(x0) is assumed to be an isotropic Gaussian density with mean µ0. Its variance τ0 is taken sufficiently large to give a broad prior. 4.2 Drift The gradients for the drift function parameters θf only depend on the total energy associated to the SDE. Their general expression is given by ∇θf Lθ,Σ = Z tf t0 ∇θf Esde(t) dt, (18) 4 where ∇θf Esde(t) = D (fθ(t, Xt) −g(t, Xt))⊤Σ−1∇θf fθ(t, Xt) E qt. Note that the observations do play a role in this gradient as they enter through g(t, Xt) and the expectation w.r.t. q(Xt|Σ). 4.3 System noise Estimating the system noise covariance (or volatility) is essential as the system noise, together with the drift function, determines the dynamics. In general, this parameter is difficult to estimate using an MCMC approach because the efficiency is strongly dependent on the discrete approximation of the SDE and most methods break down when the time step ∆t gets too small [11, 6]. For example in a Bayesian MCMC approach, which alternates between sampling paths and parameters, the latent paths imputed between observations must have a system noise parameter which is arbitrarily close to its previous value in order to be accepted by a Metropolis sampler. Hence, the algorithm becomes extremely slow. Note, that for the same reason, a naive EM algorithm within our approach breaks down. However, in our method, we can simply compute approximations to the marginal likelihood and its gradient directly. In the next section, we will compare our results to a direct MCMC estimate of the marginal likelihood which is a time consuming method. The gradient of (16) with respect to Σ is given by ∇ΣLθ,Σ = Z tf t0 ∇ΣEsde(t) dt + Z tf t0 Ψ(t) dt, (19) where ∇ΣEsde(t) = −1 2Σ−1 D (fθ(t, Xt) −g(t, Xt)) (fθ(t, Xt) −g(t, Xt))⊤E qt Σ−1. 5 Experimental validation on a bi-stable system In order to validate the approach, we consider the 1 dimensional double-well system: fθ(t, x) = 4x θ −x2 , θ > 0, (20) where fθ(t, x) is the drift function. This dynamical system is highly nonlinear and its stationary distribution is multi-modal. It has two stable states, one in x = −θ and one in x = +θ. The system is driven by the system noise, which makes it occasionally flip from one well to the other. In the experiments, we set the drift parameter θ to 1, the system noise standard deviation σ to 0.5 and the measurement error standard deviation r to 0.2. The time step for the variational approximation is set to ∆t = 0.01, which is identical to the time resolution used to generate the original sample path. In this setting, the exit time from one of the wells is 4000 time units [15]. In other words, the transition from one well to the other is highly unlikely in the window of roughly 8 time units that we consider and where a transition occurs. Figure 1(a) compares the variational solution to the outcomes of a hybrid MCMC simulation of the posterior process using the true parameter values. The hybrid MCMC approach was proposed in [1]. At each step of the sampling process, an entire sample path is generated. In order to keep the acceptance of new paths sufficiently high, the basic MCMC algorithm is combined with ideas from Molecular Dynamics, such that the MCMC sampler moves towards regions of high probability in the state space. An important drawback of MCMC approaches is that it might be extremely difficult to monitor their convergence and that they may require a very large number of samples before actually converging. In particular, over 100, 000 sample paths were necessary to reach convergence in the case of the double-well system. The solution provided by the hybrid MCMC is here considered as the base line solution. One can observe that the variational solution underestimates the uncertainty (smaller error bars). Nevertheless, the time of the transition is correctly located. Convergence of the smoothing algorithm was achieved in approximately 180 conjugate gradient steps, each one involving a forward and backward sweep. The optimal parameters and the optimal initial conditions for the variational solution are given by ˆσ = 0.72, ˆθf = 0.85, ˆm0 = 0.88, ˆs0 = 0.45. (21) Convergence of the outer optimization loop is typically reached after less then 10 conjugate gradient steps. While the estimated value for the drift parameter is within 15% percent from its true value, 5 0 1 2 3 4 5 6 7 8 !2 !1.5 !1 !0.5 0 0.5 1 1.5 2 t x (a) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 !2 (b) Figure 1: (a) Variational solution (solid) compared to the hybrid MCMC solution (dashed), using the true parameter values. The curves denote the mean paths and the shaded regions are the twostandard deviation noise tubes. (b) Posterior of the system noise variance (diffusion term). The plain curve and the dashed curve are respectively the approximations of the posterior shape based on the variational free energy and MCMC. the deviation of the system noise is worse. Deviations may be explained by the fact that the number of observations is relatively small. Furthermore, we have chosen a sample path which contains a transition between the two wells within a small time interval and is thus highly untypical with respect to the prior distribution. This fact was experimentally assessed by estimating the parameters on a sample path without transition, in a time window of the same size. In this case, we obtained estimate roughly within 5% of the true parameter values: ˆσ = 0.46 and ˆθf = 0.92. Finally, it turns out that our estimate for ˆσ is close to the one obtained from the MCMC approach as discussed next. Posterior distribution over the parameters Interestingly, minimizing the free energy Fσ2 for different values of σ provides us with much more information than a single point estimate for the parameters [14]. Using a suitable prior over p(σ), we can approximate the posterior over the system noise variance via p(σ2|Y ) ∝e−Fσ2p(σ2), (22) where we take e−Fσ2 (at its minimum) as an approximation to the marginal likelihood of the observations p(Y |σ2). To illustrate this point, we assume a non-informative Gamma prior p(σ2) = G(α, β), with α = 10−3 and β = 10−3. A comparison with preliminary MCMC estimates for p(Y |σ2) for θ = 1 and a set of system noise variances indicates that the shape of our approximation is a reasonable indicator of the shape of the posterior. Figure 1(b) shows that at least the mean and the variance of the density come out fairly well. 6 Conclusion We have presented a variational approach to the approximate inference of stochastic differential equations from a finite set of noisy observations. So far, we have tested the method on a one dimensional bi-stable system only. Comparison with a Monte Carlo approach suggests that our method can reproduce the posterior mean fairly well but underestimates the variance in the region of the transition. Parameter estimates also agree well with the MC predictions. In the future, we will extend our method in various directions. Although our approach is based on a Gaussian approximation of the posterior process, we expect that one can improve on it and obtain non-Gaussian predictions at least for various marginal posterior distributions, including that of the latent variable Xt at a fixed time t. This should be possible by generalising our method for the computation of a non-Gaussian shaped probability density for the system noise parameter using the free energy. An important extension of our method will be to systems with many degrees of freedom. 6 We hope that the possibility of using simpler suboptimal parametrisations of the approximating Gaussian process will allow us to obtain a tractable inference method that scales well to higher dimensions. Acknowledgments This work has been funded by the EPSRC as part of the Variational Inference for Stochastic Dynamic Environmental Models (VISDEM) project (EP/C005848/1). A The Kullback-Leibler divergence interpreted as a path integral In this section, we show that the Kullback-Leibler divergence between the posterior process p(Xt|Y, θ, Σ) and its approximation q(X|Σ) can be interpreted as a path integral over time. It is an average over all possible realisations, called sample paths, of the continuous-time (i.e., infinite dimensional) random variable described by the SDE in the time interval under consideration. Consider the Euler-Muryama discrete approximation (see for example [13]) of the SDE (1) and its linear approximation (4): ∆xk = fk∆t + Σ1/2∆wk, (23) ∆xk = gk∆t + Σ1/2∆wk, (24) where ∆xk ≡xk+1 −xk and wk ∼N(0, ∆tI). The vectors fk and gk are shorthand notations for fθ(tk, xk) and g(tk, xk). Hence, the joint distributions of discrete sample paths {xk}k≥0 for the true process and its approximation follow from the Markov property: p(x0, . . . , xK|Σ) = p(x0) Y k>0 N(xk+1|xk + fk∆t, Σ∆t), (25) q(x0, . . . , xK|Σ) = q(x0) Y k>0 N(xk+1|xk + gk∆t, Σ∆t), (26) where p(x0) is the prior on the intial state x0 and q(x0) is assumed to be Gaussian. Note thate we do not restrict the variational posterior to factorise over the latent states. The Kullback-Leibler divergence between the two discretized prior processes is given by KL [q∥p] = KL [q(x0)∥p(x0)] − X k>0 Z q(xk) ⟨ln p(xk+1|xk)⟩q(xk+1|xk) dxk = KL [q(x0)∥p(x0)] + 1 2 X k>0 (fk −gk)⊤Σ−1(fk −gk) q(xk) ∆t, where we omitted the conditional dependency on Σ for simplicity. The second term on the right hand side is a sum in ∆t. As a result, taking limits for ∆t →0 leads to a proper Riemann integral, which defines an integral over the average sample path: KL [q(X|Σ)∥p(X|θ, Σ)] = KL [q0∥p0] + 1 2 Z tf t0 (ft −gt)⊤Σ−1(ft −gt) qt dt, (27) where X = {Xt, t0 ≤t ≤tf} denotes the stochastic process in the interval [t0, tf]. The distribution qt = q(Xt|Σ) is the marginal at time t for a given system noise covariance Σ. It is important to realise that the KL between the induced prior process and its approximation is finite because the system noise covariances are chosen to be identical. If this was not the case, the normalizing constants of p(xk+1|xk) and q(xk+1|xk) would not cancel. This would result in KL →∞when ∆t →0. If we assume that the observations are i.i.d., it follows also that FΣ(q, θ) = − X n ⟨ln p(yn|xn)⟩q(xn) + KL [q(X|Σ)∥p(X|θ, Σ)] . Clearly, minimising this expression with respect to the variational parameters for a given system noise Σ and for a fixed parameter vector θ is equivalent to minimising the KL between the variational posterior q(X|Σ) and the true posterior p(X|Y, θ, Σ), since the normalizing constant is independent of sample paths. 7 B The gradient functions The general expressions for the gradients of Esde(t) with respect to the variational functions are given by ∇AEsde(t) = Σ−1 n ⟨∇xfθ(t, Xt)⟩qt + A(t) o S(t) −∇bEsde(t)m⊤(t), (28) ∇bEsde(t) = Σ−1 n −⟨fθ(t, Xt)⟩qt −A(t)m(t) + b(t) o , (29) where ⟨∇xfθ(t, Xt)⟩qt S(t) = D fθ(t, Xt) (Xt −m(t))⊤E qt is invoked in order to obtain (28). References [1] F. J. Alexander, G. L. Eyink, and J. M. Restrepo. Accelerated Monte Carlo for optimal estimation of time series. Journal of Statistical Physics, 119:1331–1345, 2005. [2] J. D. Annan, J. C. Hargreaves, N. R. Edwards, and R. Marsh. Parameter estimation in an intermediate complexity earth system model using an ensemble Kalman filter. Ocean Modelling, 8:135–154, 2005. [3] A. Apte, M. Hairer, A. Stuart, and J. Voss. Sampling the posterior: An approach to non-Gaussian data assimilation. Physica D, 230:50–64, 2007. [4] C. Archambeau, D. Cornford, M. Opper, and J. Shawe-Taylor. Gaussian process approximation of stochastic differential equations. Journal of Machine Learning Research: Workshop and Conference Proceedings, 1:1–16, 2007. [5] D. Barber. Expectation correction for smoothed inference in switching linear dynamical systems. Journal of Machine Learning Research, 7:2515–2540, 2006. [6] A. Beskos, O. Papaspiliopoulos, G. Roberts, and P. Fearnhead. Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes (with discussion). Journal of the Royal Statistical Society B, 68(3):333–382, 2006. [7] Christopher M. Bishop. Pattern Recognition and Machine Learning. Springer, New York, 2006. [8] D. Crisan and T. Lyons. A particle approximation of the solution of the Kushner-Stratonovitch equation. Probability Theory and Related Fields, 115(4):549–578, 1999. [9] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via EM algorithm. Journal of the Royal Statistical Society B, 39(1):1–38, 1977. [10] G. L. Eyink, J. L. Restrepo, and F. J. Alexander. A mean field approximation in data assimilation for nonlinear dynamics. Physica D, 194:347–368, 2004. [11] A. Golightly and D. J. Wilkinson. Bayesian inference for nonlinear multivariate diffusion models observed with error. Computational Statistics and Data Analysis, 2007. Accepted. [12] A. H. Jazwinski. Stochastic Processes and Filtering Theory. Academic Press, New York, 1970. [13] Peter E. Kloeden and Eckhard Platen. Numerical Solution of Stochastic Differential Equations. Springer, Berlin, 1999. [14] H. Lappalainen and J. W. Miskin. Ensemble learning. In M. Girolami, editor, Advances in Independent Component Analysis, pages 76–92. Springer-Verlag, 2000. [15] R. N. Miller, M. Ghil, and F. Gauthiez. Advanced data assimilation in strongly nonlinear dynamical systems. Journal of the Atmospheric Sciences, 51:1037–1056, 1994. [16] Jorge Nocedal and Stephen J. Wright. Numerical Optimization. Springer, 2000. [17] G. Roberts and O. Stramer. On inference for partially observed non-linear diffusion models using the Metropolis-Hastings algorithm. Biometirka, 88:603–621, 2001. 8
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Augmented Functional Time Series Representation and Forecasting with Gaussian Processes Nicolas Chapados and Yoshua Bengio Department of Computer Science and Operations Research University of Montr´eal Montr´eal, Qu´ebec, Canada H3C 3J7 {chapados,bengioy}@iro.umontreal.ca Abstract We introduce a functional representation of time series which allows forecasts to be performed over an unspecified horizon with progressively-revealed information sets. By virtue of using Gaussian processes, a complete covariance matrix between forecasts at several time-steps is available. This information is put to use in an application to actively trade price spreads between commodity futures contracts. The approach delivers impressive out-of-sample risk-adjusted returns after transaction costs on a portfolio of 30 spreads. 1 Introduction Classical time-series forecasting models, such as ARMA models [6], assume that forecasting is performed at a fixed horizon, which is implicit in the model. An overlaying deterministic time trend may be fit to the data, but is generally of fixed and relatively simple functional form (e.g. linear, quadratic, or sinusoidal for periodic data). To forecast beyond the fixed horizon, it is necessary to iterate forecasts in a multi-step fashion. These models are good at representing the short-term dynamics of the time series, but degrade rapidly when longer-term forecasts must be made, usually quickly converging to the unconditional expectation of the process after removal of the deterministic time trend. This is a major issue in applications that require a forecast over a complete future trajectory, and not a single (or restricted) horizon. These models are also constrained to deal with regularly-sampled data, and make it difficult to condition the time trend on explanatory variables, especially when iteration of short-term forecasts has to be performed. To a large extent, the same problems are present with non-linear generalizations of such models, such as time-delay or recurrent neural networks [1], which simply allow the short-term dynamics to become nonlinear but leave open the question of forecasting complete future trajectories. Functional Data Analysis (FDA) [10] has been proposed in the statistical literature as an answer to some of these concerns. The central idea is to consider a whole curve as an example (specified by a finite number of samples ⟨t, yt⟩), which can be represented by coefficients in a non-parametric basis expansion such as splines. This implies learning about complete trajectories as a function of time, hence the “functional” designation. Since time is viewed as an independent variable, the approach can forecast at arbitrary horizons and handle irregularly-sampled data. Typically, FDA is used without explanatory time-dependent variables, which are important for the kind of applications we shall be considering. Furthermore, the question remains of how to integrate a progressivelyrevealed information set in order to make increasingly more precise forecasts of the same future trajectory. To incorporate conditioning information, we consider here the output of a prediction to be a whole forecasting curve (as a function of t). The motivation for this work comes from forecasting and actively trading price spreads between commodity futures contracts (see, e.g., [7], for an introduction). Since futures contracts expire and have a finite duration, this problem is characterized by the presence of a large number of separate 1 historical time series, which all can be of relevance in forecasting a new time series. For example, we expect seasonalities to affect similarly all the series. Furthermore, conditioning information, in the form of macroeconomic variables, can be of importance, but exhibit the cumbersome property of being released periodically, with explanatory power that varies across the forecasting horizon. In other words, when making a very long-horizon forecast, the model should not incorporate conditioning information in the same way as when making a short- or medium-term forecast. A possible solution to this problem is to have multiple models for forecasting each time series, one for each time scale. However, this is hard to work with, requires a high degree of skill on the part of the modeler, and is not amenable to robust automation when one wants to process hundreds of time series. In addition, in order to measure risk associated with a particular trade (buying at time t and selling at time t′), we need to estimate the covariance of the price predictions associated with these two points in the trajectory. These considerations motivate the use of Gaussian processes, which naturally provide a covariance matrix between forecasts made at several points. To tackle the challenging task of forecasting and trading spreads between commodity futures, we introduce here a form of functional data analysis in which the function to be forecast is indexed both by the date of availability of the information set and by the forecast horizon. The predicted trajectory is thus represented as a functional object associated with a distribution, a Gaussian process, from which the risk of different trading decisions can readily be estimated. This approach allows incorporating input variables that cannot be assumed to remain constant over the forecast horizon, like statistics of the short-term dynamics. Previous Work Gaussian processes for time-series forecasting have been considered before. Multi-step forecasts are explicitly tackled by [4], wherein uncertainty about the intermediate values is formally incorporated into the predictive distribution to obtain more realistic uncertainty bounds at longer horizons. However, this approach, while well-suited to purely autoregressive processes, does not appear amenable to the explicit handling of exogenous input variables. Furthermore, it suffers from the restriction of only dealing with regularly-sampled data. Our approach is inspired by the CO2 model of [11] as an example of application-specific covariance function engineering. 2 The Model We consider a set of N real time series each of length Mi, {yi t}, i = 1, . . . , N and t = 1, . . . , Mi. In our application each i represents a different year, and the series is the sequence of commodity spread prices during the period where it is traded. The lengths of all series are not necessarily identical, but we shall assume that the time periods spanned by the series are “comparable” (e.g. the same range of days within a year if the series follow an annual cycle) so that knowledge from past series can be transferred to a new one to be forecast. The forecasting problem is that given observations from the complete series i = 1, . . . , N −1 and from a partial last series, {yN t }, t = 1, . . . , MN, we want to extrapolate the last series until a predetermined endpoint, i.e. characterize the joint distribution of {yN τ }, τ = MN +1, . . . , MN +H. We are also given a set of non-stochastic explanatory variables specific to each series, {xi t}, where xi t ∈Rd. Our objective is to find an effective representation of P({yN τ }τ=MN+1,...,MN+H | {xi t, yi t}i=1,...,N t=1,...,Mi), with τ, i and t ranging, respectively over the forecasting horizon, the available series and the observations within a series. Gaussian Processes Assuming that we are willing to accept a normally-distributed posterior, Gaussian processes [8, 11, 14] have proved a general and flexible tool for nonlinear regression in a Bayesian framework. Given a training set of M input–output pairs ⟨X ∈RM×d, y ∈RM⟩, a set of M ′ test point locations X∗∈RM ′×d and a positive semi-definite covariance function k : Rd × Rd 7→R, the joint posterior distribution of the test outputs y∗follows a normal with mean and covariance given by E [y∗| X, X∗, y] = K(X∗, X)Λ−1y, (1) Cov [y∗| X, X∗, y] = K(X∗, X∗) −K(X∗, X)Λ−1K(X, X∗), (2) where we have set Λ = K(X, X) + σ2 nIM, with K the matrix of covariance evaluations, K(U, V)i,j △= k(Ui, Vj), and σ2 n the assumed process noise level. The specific form of the covariance function used in our application is described below, after introducing the representation used for forecasting. 2 Functional Representation for Forecasting In the spirit of functional data analysis, a first attempt at solving the forecasting problem is to set it forth in terms of regression from the input variables to the series values, adding to the inputs an explicit time index t and series identity i, E yi t Ii t0] = f(i, t, xi t|t0) Cov h yi t, yi′ t′ Ii t0 i = g(i, t, xi t|t0, i′, t′, xi′ t′|t0), (3) these expressions being conditioned on the information set Ii t0 containing information up to time t0 of series i (we assume that all prior series i′ < i are also included in their entirety in Ii t0). The notation xi t|t0 denotes a forecast of xi t given information available at t0. Functions f and g result from Gaussian process training, eq. (1) and (2), using information in Ii t0. To extrapolate over the unknown horizon, one simply evaluates f and g with the series identity index i set to N and the time index t within a series ranging over the elements of τ (forecasting period). Owing to the smoothness properties of an adequate covariance function, one can expect the last time series (whose starting portion is present in the training data) to be smoothly extended, with the Gaussian process borrowing from prior series, i < N, to guide the extrapolation as the time index reaches far enough beyond the available data in the last series. The principal difficulty with this method resides in handling the exogenous inputs xN t|t0 over the forecasting period: the realizations of these variables, xN t , are not usually known at the time the forecast is made and must be extrapolated with some reasonableness. For slow-moving variables that represent a “level” (as opposed to a “difference” or a “return”), one can conceivably keep their value constant to the last known realization across the forecasting period. However, this solution is restrictive, problem-dependent, and precludes the incorporation of short-term dynamics variables (e.g. the first differences over the last few time-steps) if desired. Augmenting the Functional Representation We propose in this paper to augment the functional representation with an additional input variable that expresses the time at which the forecast is being made, in addition to the time for which the forecast is made. We shall denote the former the operation time and the latter the target time. The distinction is as follows: operation time represents the time at which the other input variables are observed and the time at which, conceptually, a forecast of the entire future trajectory is performed. In contrast, target time represents time at a point of the predicted target series (beyond operation time), given the information known at the operation time. As previously, the time series index i remains part of the inputs. In this framework, forecasting is performed by holding the time series index constant to N, the operation time constant to the time MN of the last observation, the other input variables constant to their last-observed values xN MN , and varying the target time over the forecasting period τ. Since we are not attempting to extrapolate the inputs beyond their intended range of validity, this approach admits general input variables, without restriction as to their type, and whether they themselves can be forecast. It can be convenient to represent the target time as a positive delta ∆from the operation time t0. In contrast to eq. (3), this yields the representation E yi t0+∆ Ii t0] = f(i, t0, ∆, xi t0) Cov h yi t0+∆, yi′ t′ 0+∆′ Ii t0 i = g(i, t0, ∆, xi t0, i′, t′ 0, ∆′, xi′ t′ 0), (4) where we have assumed the operation time to coincide with the end of the information set. Note that this augmentation allows to dispense with the problematic extrapolation xi t|t0 of the inputs, instead allowing a direct use of the last available values xi t0. Moreover, from a given information set, nothing precludes forecasting the same trajectory from several operation times t′ < t0, which can be used as a means of evaluating the stability of the obtained forecast. The obvious downside to augmentation lies in the greater computational cost it entails. In particular, the training set must contain sufficient information to represent the output variable for many combinations of operation and target times that can be provided as input. In the worst case, this implies that the number of training examples grows quadratically with the length of the training time series. In practice, a downsampling scheme is used wherein only a fixed number of target-time points is sampled for every operation-time point.1 1This number was 15 in our experiments, and these were not regularly spaced, with longer horizons spaced farther apart. Furthermore, the original daily frequency of the data was reduced to keep approximately one operation-time point per week. 3 Covariance Function We used a modified form of the rational quadratic covariance function with hyperparameters for automatic relevance determination [11], which is expressed as kAUG-RQ(u, v; ℓ, α, σf, σTS) = σ2 f 1 + 1 2α d X k=1 (uk −vk)2 ℓ2 k !−α + σ2 TSδiu,iv, (5) where δj,k △= I[j = k] is the Kronecker delta. The variables u and v are values in the augmented representation introduced previously, containing the three variables representing time (current timeseries index or year, operation time, target time) as well as the additional explanatory variables. The notation iu denotes the time-series index component i of input variable u. The last term of the covariance function, the Kronecker delta, is used to induce an increased similarity among points that belong to the same time series (e.g. the same spread trading year). By allowing a series-specific average level to be maintained into the extrapolated portion, the presence of this term was found to bring better forecasting performance. The hyperparameters ℓi, α, σf, σTS, σn are found by maximizing the marginal likelihood on the training set by a standard conjugate gradient optimization [11]. For tractability, we rely on a two-stage training procedure, wherein hyperparameter optimization is performed on a fairly small training set (M = 500) and final training is done on a larger set (M = 2250), keeping hyperparameters fixed. 3 Evaluating Forecasting Performance To establish the benefits of the proposed functional representation for forecasting commodity spread prices, we compared it against other likely models on three common grain and grain-related spreads:2 the January–July Soybeans, May–September Soybean Meal, and March–July Chicago Hard Red Wheat. The forecasting task is to predict the complete future trajectory of each spread (taken individually), from 200 days before maturity until maturity. Methodology Realized prices in the previous trading years are provided from 250 days to maturity, using data going back to 1989. The first test year is 1994. Within a given trading year, the time variables represent the number of calendar days to maturity of the near leg; since no data is observed on week-ends, training examples are sampled on an irregular time scale. Performance evaluation proceeds through a sequential validation procedure [2]: within a trading year, we first train models 200 days before maturity and obtain a first forecast for the future price trajectory. We then retrain models every 25 days, and obtain revised portions of the remainder of the trajectory. Proceeding sequentially, this operation is repeated for succeeding trading years. All forecasts are compared amongst models on squared-error and negative log-likelihood criteria (see “assessing significance”, below). Input variables are subject to minimal preprocessing: we standardize them to zero mean and unit standard deviation. The price targets require additional treatment: since the price level of a spread can vary significantly from year to year, we normalize the price trajectories to start at zero at the start of every trading year, by subtracting the first price. Furthermore, in order to get slightly better behaved optimization, we divide the price targets by their overall standard deviation. Models Compared The “complete” model to be compared against others is based on the augmented-input representation Gaussian process with the modified rational quadratic covariance function eq. (5). In addition to the three variables required for the representation of time, the following inputs were provided to the model: (i) the current spread price and the price of the three nearest futures contracts on the underlying commodity term structure, (ii) economic variables (the stock-to-use ratio and year-over-year difference in total ending stocks) provided on the underlying commodity by the U.S. Department of Agriculture [13]. This model is denoted AugRQ/all-inp. An example of the sequence of forecasts made by this model, repeated every 25 times steps, is shown in the upper panel of Figure 1. To determine the value added by each type of input variable, we include in the comparison two models based on exactly on the same architecture, but providing less inputs: AugRQ/less-inp does 2Our convention is to first give the short leg of the spread, followed by the long leg. Hence, Soybeans 1–7 should be interpreted as taking a short position (i.e. selling) in the January Soybeans contract and taking an offsetting long (i.e. buying) in the July contract. Traditionally, intra-commodity spread positions are taken so as to match the number of contracts on both legs — the number of short contracts equals the number of long ones — not the dollar value of the long and short sides. 4 Figure 1: Top Panel: Illustration of multiple forecasts, repeated every 25 days, of the 1996 March–July Wheat spread (dashed lines); realized price is in gray. Although the first forecast (smooth solid blue, with confidence bands) mistakes the overall price level, it approximately correctly identifies local price maxima and minima, which is sufficient for trading purposes. Bottom Panel: Position taken by the trading model (in red: short, then neutral, then long), and cumulative profit of that trade (gray). not include the economic variables. AugRQ/no-inp further removes the price inputs, leaving only the time-representation inputs. Moreover, to quantify the performance gain of the augmented representation of time, the model StdRQ/no-inp implements a “standard time representation” that would likely be used in a functional data analysis model; as described in eq. (3), this uses a single time variable instead of splitting the representation of time between the operation and target times. Finally, we compare against simpler models: Linear/all-inp uses a dot-product covariance function to implement Bayesian linear regression, using the full set of input variables described above. And AR(1) is a simple linear autoregressive model. The predictive mean and covariance matrix for this last model are established as follows (see, e.g. [6]). We consider the scalar data generating process yt = φ yt−1 + εt, εt iid ∼N(0, σ2), (6) where the process {yt} has an unconditional mean of zero.3 Given information available at time t, It, the h-step ahead forecast from time t under this model, has conditional expectation and covariance (with the h′-step ahead forecast), expressed as E [yt+h | It] = φhyt, Cov yt+h|t, yt+h′|t | It = σ2φh+h′ 1 −φ−2 min(h,h′) φ2 −1 . Assessing Significance of Forecasting Performance Differences For each trajectory forecast, we measure the squared error (SE) made at each time-step along with the negative log-likelihood (NLL) of the realized price under the predictive distribution. To account for differences in target variable distribution throughout the years, we normalize the SE by dividing it by the standard deviation of the test targets in a given year. Similarly, we normalize the NLL by subtracting the likelihood of a univariate Gaussian distribution estimated on the test targets of the year. Due to the serial correlation it exhibits, the time series of performance differences (either SE or NLL) between two models cannot directly be subjected to a standard t-test of the null hypothesis of no difference in forecasting performance. The well-known Diebold-Mariano test [3] corrects for this correlation structure in the case where a single time series of performance differences is available. This test is usually expressed as follows. Let {dt} be the sequence of error differences between two models to be compared. Let ¯d = 1 M P t dt be the mean difference. The sample variance of ¯d is readily shown [3] to be ˆvDM △= Var[ ¯d] = 1 M K X k=−K ˆγk, 3In our experiments, we estimate an independent empirical mean for each trading year, which is subtracted from the prices before proceeding with the analysis. 5 Table 1: Forecast performance difference between AugRQ/all-inp and all other models, for the three spreads studied. For both the Squared Error and NLL criteria, the value of the cross-correlation-corrected statistic is listed (CCC) along with its p-value under the null hypothesis. A negative CCC statistic indicates that AugRQ/all-inp beats the other model on average. Soybeans 1–7 Soybean Meal 5–9 Wheat 3–7 Sq. Error NLL Sq. Error NLL Sq. Error NLL CCC p CCC p CCC p CCC p CCC p CCC p AugRQ/less-inp −0.86 0.39 −0.89 0.37 −1.05 0.29 −0.95 0.34 −0.05 0.96 1.06 0.29 AugRQ/no-inp −1.68 0.09 −1.73 0.08 −1.78 0.08 −2.42 0.02 −2.75 0.01 −2.42 0.02 Linear/all-inp −1.53 0.13 −1.33 0.18 −1.61 0.11 −2.00 0.05 −4.20 10−4 −3.45 10−3 AR(1) −4.24 10−5 −0.44 0.66 −2.53 0.01 0.12 0.90 −6.50 0.00 −6.07 10−9 StdRQ/no-inp −2.44 0.01 −1.04 0.30 −2.69 0.01 −1.08 0.28 −2.67 0.01 −9.36 0.00 where M is the sequence length and ˆγk is an estimator of the lag-k autocovariance of the dts. The maximum lag order K is a parameter of the test and must be determined empirically. Then the statistic DM = ¯d/√ˆvDM is asymptotically distributed as N(0, 1) and a classical test of the null hypothesis ¯d = 0 can be performed. Unfortunately, even the Diebold-Mariano correction for autocorrelation is not sufficient to compare models in the present case. Due to the repeated forecasts made for the same time-step across several iterations of sequential validation, the error sequences are likely to be cross-correlated since they result from models estimated on strongly overlapping training sets. This suggests that an additional correction should be applied to account for this cross-correlation across test sets, expressed as ˆvCCC−DM = 1 M 2 X i Mi K X k=−K ˆγi k + X i X j̸=i Mi ∩j K′ X k=−K′ ˆγi,j k , (7) where Mi is the number of examples in test set i, M = P i Mi is the total number of examples, Mi ∩j is the number of time-steps where test sets i and j overlap, ˆγi k denote the estimated lag-k autocovariances within test set i, and ˆγi,j k denote the estimated lag-k cross-covariances between test sets i and j. The maximum lag order for cross-covariances, K′, is possibly different from K (our experiments used K = K′ = 15). This revised variance estimator was used in place of the usual Diebold-Mariano statistic in the results presented below. Results Results of the forecasting performance difference between AugRQ/all-inp and all other models is shown in Table 1. We observe that AugRQ/all-inp generally beats the others on both the SE and NLL criteria, often statistically significantly so. In particular, the augmented representation of time is shown to be of value (i.e. comparing against StdRQ/no-inp). Moreover, the Gaussian process is capable of making good use of the additional price and economic input variables, although not always with the traditionally accepted levels of significance. 4 Application: Trading a Portfolio of Spreads We applied this forecasting methodology based on an augmented representation of time to trading a portfolio of spreads. Within a given trading year, we apply an information-ratio criterion to greedily determine the best trade into which to enter, based on the entire price forecast (until the end of the year) produced by the Gaussian process. More specifically, let {pt} be the future prices forecast by the model at some operation time (presumably the time of last available element in the training set). The expected forecast dollar profit of buying at t1 and selling at t2 is simply given by pt2 −pt1. Of course, a prudent investor would take trade risk into consideration. A simple approximation of risk is given by the trade profit volatility. This yields the forecast information ratio4 of the trade c IR(t1, t2) = E[pt2 −pt1|It0] p Var[pt2 −pt1|It0] , (8) 4An information ratio is defined as the average return of a portfolio in excess of a benchmark, divided by the standard deviation of the excess return distribution; see [5] for more details. 6 Figure 2: After a price trajectory forecast (in the top and left portions of the figure), all possible pairs of buyday/sell-day are evaluated on a trade information ratio criterion, whose results are shown by the level plot. The best trade is selected, here shorting 235 days before maturity with forecast price at a local maximum, and covering 100 days later at a local minimum. Table 2: Financial performance statistics for the 30-spread portfolio on the 1994–2007 (until April 30) period, and two disjoint sub-periods. All returns are expressed in excess of the riskfree rate. The information ratio statistics are annualized. Skewness and excess kurtosis are on the monthly return distributions. Drawdown duration is expressed in calendar days. The model displays good performance for moderate risk. Full 1994/01 2003/01 Period 2002/12 2007/04 Avg Annual Return 7.3% 5.9% 10.1% Avg Annual Stddev 4.1% 4.0% 4.1% Information Ratio 1.77 1.45 2.44 Skewness 0.68 0.65 0.76 Excess Kurtosis 3.40 4.60 1.26 Best Month 6.0% 6.0% 4.8% Worst Month −3.4% −3.4% −1.8% Percent Months Up 71% 67% 77% Max. Drawdown −7.7% −7.7% −4.0% Drawdown Duration 653 653 23 Drawdown From 1997/02 1997/02 2004/06 Drawdown Until 1998/11 1998/11 2004/07 where Var[pt2 −pt1|It0] can be computed as Var[pt1|It0] + Var[pt1|It0] −2 Cov[pt1, pt2|It0], each quantity being separately obtainable from the Gaussian process forecast, cf. eq. (2). The trade decision is made in one of two ways, depending on whether a position has already been opened: (i) When making a decision at time t0, if a position has not yet been entered for the spread in a given trading year, eq. (8) is maximized with respect to unconstrained t1, t2 ≥t0. An illustration of this criterion is given in Figure 2, which corresponds to the first decision made when trading the spread shown in Figure 1. (ii) In contrast, if a position has already been opened, eq. (8) is only maximized with respect to t2, keeping t1 fixed at t0. This corresponds to revising the exit point of an existing position. Simple additional filters are used to avoid entering marginal trades: we impose a trade duration of at least four days, a minimum forecast IR of 0.25 and a forecast standard deviation of the price sequence of at least 0.075. These thresholds have not been tuned extensively; they were used only to avoid trading on an approximately flat price forecast. We applied these ideas to trading a portfolio of 30 spreads, selected among the following commodities: Cotton (2 spreads), Feeder Cattle (2), Gasoline (1), Lean Hogs (7), Live Cattle (1), Natural Gas (2), Soybean Meal (5), Soybeans (5), Wheat (5). The spreads were selected on the basis of their good performance on the 1994–2002 period. Our simulations were carried on the 1994–2007 period, using historical data (for Gaussian process training) dating back to 1989. Transaction costs were assumed to be 5 basis points per spread leg traded. Spreads were never traded later than 25 calendar days before maturity of the near leg. Relative returns are computed using as a notional amount half the total exposure incurred by both legs of the spread.5 Financial performance results on the complete test period and two disjoint sub-periods (which correspond, until end-2002 to the model selection period, and after 2003 to a true out-of-sample evaluation) are shown in Table 2. In all sub-periods, but particularly since 2003, the portfolio exhibits a very favorable risk-return profile, including positive skewness and acceptable excess kurtosis.6 A plot of cumulative returns, number of open positions and monthly returns appears in Figure 3. 5This is a conservative assumption, since most exchanges impose considerably reduced margin requirements on recognized spreads. 6By way of comparison, over the period 1 Jan. 1994–30 Apr. 2007, the S&P 500 index has an information ratio of approximately 0.37 against the U.S. three-month treasury bills. 7 Figure 3: Top Panel: cumulative excess return after transaction costs of a portfolio of 30 spreads traded according to the maximum information-ratio criterion; the bottom part plots the number of positions open at a time (right axis). Bottom Panel: monthly portfolio relative excess returns; we observe the significant positive skewness in the distribution. 5 Future Work and Conclusions We introduced a flexible functional representation of time series, capable of making long-term forecasts from progressively-revealed information sets and of handling multiple irregularly-sampled series as training examples. We demonstrated the approach on a challenging commodity spread trading application, making use of a Gaussian process’ ability to compute a complete covariance matrix between several test outputs. Future work includes making more systematic use of approximation methods for Gaussian processes (see [9] for a survey). The specific usage pattern of the Gaussian process may guide the approximation: in particular, since we know in advance the test inputs, the problem is intrinsically one of transduction, and the Bayesian Committee Machine [12] could prove beneficial. References [1] C. Bishop. Neural Networks for Pattern Recognition. Oxford University Press, 1995. [2] N. Chapados and Y. Bengio. Cost functions and model combination for VaR-based asset allocation using neural networks. IEEE Transactions on Neural Networks, 12(4):890–906, July 2001. [3] F. X. Diebold and R. S. Mariano. Comparing predictive accuracy. Journal of Business & Economic Statistics, 13(3):253–263, July 1995. [4] A. Girard, C. E. Rasmussen, J. Q. Candela, and R. Murray-Smith. Gaussian process priors with uncertain inputs – application to multiple-step ahead time series forecasting. In S. T. S. Becker and K. Obermayer, editors, Advances in Neural Information Processing Systems 15, pages 529–536. MIT Press, 2003. [5] R. C. Grinold and R. N. Kahn. Active Portfolio Management. McGraw Hill, 1999. [6] J. D. Hamilton. Time Series Analysis. Princeton University Press, 1994. [7] J. C. Hull. Options, Futures and Other Derivatives. Prentice Hall, Englewood Cliffs, NJ, sixth edition, 2005. [8] A. O’Hagan. Curve fitting and optimal design for prediction. Journal of the Royal Statistical Society B, 40:1–42, 1978. (With discussion). [9] J. Quionero-Candela and C. E. Rasmussen. A unifying view of sparse approximate gaussian process regression. Journal of Machine Learning Research, 6:1939–1959, 2005. [10] J. O. Ramsay and B. W. Silverman. Functional Data Analysis. Springer, second edition, 2005. [11] C. E. Rasmussen and C. K. I. Williams. Gaussian Processes for Machine Learning. MIT Press, 2006. [12] V. Tresp. A bayesian committee machine. Neural Computation, 12:2719–2741, 2000. [13] U.S. Department of Agriculture. Economic research service data sets. WWW publication. Available at http://www.ers.usda.gov/Data/. [14] C. K. I. Williams and C. E. Rasmussen. Gaussian processes for regression. In D. S. Touretzky, M. C. Mozer, and M. E. Hasselmo, editors, Advances in Neural Information Processing Systems 8, pages 514– 520. MIT Press, 1996. 8
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Near-Minimax Recursive Density Estimation on the Binary Hypercube Maxim Raginsky Duke University Durham, NC 27708 m.raginsky@duke.edu Svetlana Lazebnik UNC Chapel Hill Chapel Hill, NC 27599 lazebnik@cs.unc.edu Rebecca Willett Duke University Durham, NC 27708 willett@duke.edu Jorge Silva Duke University Durham, NC 27708 jg.silva@duke.edu Abstract This paper describes a recursive estimation procedure for multivariate binary densities using orthogonal expansions. For d covariates, there are 2d basis coefficients to estimate, which renders conventional approaches computationally prohibitive when d is large. However, for a wide class of densities that satisfy a certain sparsity condition, our estimator runs in probabilistic polynomial time and adapts to the unknown sparsity of the underlying density in two key ways: (1) it attains near-minimax mean-squared error, and (2) the computational complexity is lower for sparser densities. Our method also allows for flexible control of the trade-off between mean-squared error and computational complexity. 1 Introduction Multivariate binary data arise in a variety of fields, such as biostatistics [1], econometrics [2] or artificial intelligence [3]. In these and other settings, it is often necessary to estimate a probability density from a number of independent observations. Formally, we have n i.i.d. samples from a probability density f (with respect to the counting measure) on the d-dimensional binary hypercube Bd, B △= {0, 1}, and seek an estimate bf of f with a small mean-squared error MSE(f, bf) = E P x∈Bd(f(x) −bf(x))2 . In many cases of practical interest, the number of covariates d is much larger than log n, so direct estimation of f as a multinomial density with 2d parameters is both unreliable and impractical. Thus, one has to resort to “nonparametric” methods and search for good estimators in a suitably defined class whose complexity grows with n. Some nonparametric methods proposed in the literature, such as kernels [4] and orthogonal expansions [5, 6], either have very slow rates of MSE convergence or are computationally prohibitive for large d. For example, the kernel method [4] requires O(n2d) operations to compute the estimate at any x ∈Bd, yet its MSE decays as O(n−4/(4+d)) [7], which is extremely slow when d is large. In contrast, orthogonal function methods generally have much better MSE decay rates, but rely on estimating 2d coefficients in a fixed basis, which requires enormous computational resources for large d. For instance, using the Fast Hadamard Transform to estimate the coefficients in the so-called Walsh basis using n samples requires O(nd2d) operations [5]. In this paper we take up the problem of accurate, computationally tractable estimation of a density on the binary hypercube. We take the minimax point of view, where we assume that f comes from a particular function class F and seek an estimator that approximately attains the minimax MSE R∗ n(F) △= inf b f sup f∈F MSE(f, bf), where the infimum is over all estimators based on n i.i.d. samples. We will define our function class to reflect another feature often encountered in situations involving multivariate binary data: namely, that the shape of the underlying density is strongly influenced by small constellations of the d covariates. For example, when working with panel data [2], it may be the case that the answers to some specific subset of questions are highly correlated among a particular group of the panel participants, and the responses of these participants to other questions are nearly random; moreover, there may be several such distinct groups in the panel. To model such “constellation effects” mathematically, we will consider classes of densities that satisfy a particular sparsity condition. Our contribution consists in developing a thresholding density estimator that adapts to the unknown sparsity of the underlying density in two key ways: (1) it is near-minimax optimal, with the error decay rate depending upon the sparsity, and (2) it can be implemented using a recursive algorithm that runs in probabilistic polynomial time and whose computational complexity is lower for sparser densities. The algorithm entails recursively examining empirical estimates of whole blocks of the 2d basis coefficients. At each stage of the algorithm, the weights of the coefficients estimated at previous stages are used to decide which remaining coefficients are most likely to be significant, and computing resources are allocated accordingly. We show that this decision is accurate with high probability. An additional attractive feature of our approach is that it gives us a principled way of trading off MSE against computational complexity by controlling the decay of the threshold as a function of the recursion depth. 2 Preliminaries We first list some definitions and results needed in the sequel. Throughout the paper, C and c denote generic constants whose values may change from line to line. For two real numbers a and b, a ∧b and a ∨b denote, respectively, the smaller and the larger of the two. Biased Walsh bases. Let µd denote the counting measure on the d-dimensional binary hypercube Bd. Then the space of all real-valued functions on Bd is the real Hilbert space L2(µd) with the standard inner product ⟨f, g⟩ △= P x∈Bd f(x)g(x). Given any η ∈(0, 1), we can construct an orthonormal system Φd,η in L2(µd) as follows. Define two functions ϕ0,η, ϕ1,η : B →R by ϕ0,η(x) △= (1 −η)x/2η(1−x)/2 and ϕ1,η(x) △= (−1)xηx/2(1 −η)(1−x)/2, x ∈{0, 1}. (1) Now, for any s = (s(1), . . . , s(d)) ∈Bd define the function ϕs,η : Bd →R by ϕs,η(x) △= d Y i=1 ϕs(i),η(x(i)), ∀x = (x(1), . . . , x(d)) ∈Bd (2) (this is written more succinctly as ϕs,η = ϕs(1),η ⊗. . . ⊗ϕs(d),η, where ⊗is the tensor product). The set Φd,η = {ϕs,η : s ∈Bd} is an orthonormal system in L2(µd), which is referred to as the Walsh system with bias η [8, 9]. Any function f ∈L2(µd) can be uniquely represented as f = X s∈Bd θs,ηϕs,η, where θs,η = ⟨f, ϕs,η⟩. When η = 1/2, we get the standard Walsh system used in [5, 6]; in that case, we shall omit the index η = 1/2 for simplicity. The product structure of the biased Walsh bases makes them especially convenient for statistical applications as it allows for a computationally efficient recursive method for computing accurate estimates of squared coefficients in certain hierarchically structured sets. Sparsity and weak-ℓp balls. We are interested in densities whose representations in some biased Walsh basis satisfy a certain sparsity constraint. Given η ∈(0, 1) and a function f ∈L2(µd), let θ(f) denote the list of its coefficients in Φd,η. We are interested in cases when the components of θ(f) decay according to a power law. Formally, let θ(1), . . . , θ(M), where M = 2d, be the components of θ(f) arranged in decreasing order of magnitude: |θ(1)| ≥|θ(2)| ≥. . . ≥|θ(M)|. Given some 0 < p < ∞, we say that θ(f) belongs to the weak-ℓp ball of radius R [10], and write θ(f) ∈wℓp(R), if |θ(m)| ≤R · m−1/p, 1 ≤m ≤M. (3) It is not hard to show that the coefficients of any probability density on Bd in Φd,η are bounded by R(η) = [η ∨(1 −η)]d/2. With this in mind, let us define the class Fd(p, η) of all functions f on Bd satisfying θ(f) ∈wℓp(R(η)) in RM. We are particularly interested in the case 0 < p < 2. When η = 1/2, with R(η) = 2−d/2, we shall write simply Fd(p). We will need approximation properties of weak-ℓp balls as listed, e.g., in [11]. The basic fact is that the power-law condition (3) is equivalent to the concentration estimate s ∈Bd : |θs| ≥λ ≤(R/λ)p, ∀λ > 0. (4) For any 1 ≤k ≤M, let θk(f) denote the vector θ(f) with θ(k+1), . . . , θ(M) set to zero. Then it follows from (3) that ∥θ(f) −θk(f)∥ℓ2 M ≤CRk−r, where r △= 1/p −1/2, and C is some constant that depends only on p. Given any f ∈Fd(p, η) and denoting by fk the function obtained from it by retaining only the k largest coefficients, we get from Parseval’s identity that ∥f −fk∥L2(µd) ≤CRk−r. (5) To get a feeling for what the classes Fd(p, η) could model in practice, we note that, for a fixed η ∈(0, 1), the product of d Bernoulli(η∗) densities with η∗ △= √η/(√η + √1 −η) is the unique sparsest density in the entire scale of Fd(p, η) spaces with 0 < p < 2: all of its coefficients in Fd,η are zero, except for θs,η with s = (0, . . . , 0), which is equal to (η∗/√η)d. Other densities in {Φd(p, η) : 0 < p < 2} include, for example, mixtures of components that, up to a permutation of {1, . . . , d}, can be written as a tensor product of a large number of Bernoulli(η∗) densities and some other density. The parameter η can be interpreted either as the default noise level in measuring an individual covariate or as a smoothness parameter that interpolates between the point masses δ(0,...,0) and δ(1,...,1). We assume that η is known (e.g., from some preliminary exploration of the data or from domain-specific prior information) and fixed. In the following, we limit ourselves to the “noisiest” case η = 1/2 with R(1/2) = 2−d/2. Our theory can be easily modified to cover any other η ∈(0, 1): one would need to replace R = 2−d/2 with the corresponding R(η) and use the bound ∥ϕs,η∥∞≤R(η) instead of ∥ϕs∥∞≤2−d/2 when estimating variances and higher moments. 3 Density estimation via recursive Walsh thresholding We now turn to our problem of estimating a density f on Bd from a sample {Xi}n i=1 when f ∈Fd(p) for some unknown 0 < p < 2. The minimax theory for weak-ℓp balls [10] says that R∗ n(Fd(p)) ≥CM −p/2n−2r/(2r+1), r = 1/p −1/2 where M = 2d. We shall construct an estimator that adapts to unknown sparsity of f in the sense that it achieves this minimax rate up to a logarithmic factor without prior knowledge of p and that its computational complexity improves as p →0. Our method is based on the thresholding of empirical Walsh coefficients. A thresholding estimator is any estimator of the form bf = X s∈Bd I{T (bθs)≥λn}bθsϕs, where bθs = (1/n) Pn i=1 ϕs(Xi) are empirical estimates of the Walsh coefficients of f, T (·) is some statistic, and I{·} is an indicator function. The threshold λn depends on the sample size. For example, in [5, 6] the statistic T (bθs) = bθ2 s was used with the threshold λn = 1/M(n + 1). This choice was motivated by the considerations of bias-variance trade-off for each individual coefficient. The main disadvantage of such direct methods is the need to estimate all M = 2d Walsh coefficients. While this is not an issue when d ≍log n, it is clearly impractical when d ≫log n. To deal with this issue, we will consider a recursive thresholding approach that will allow us to reject whole groups of coefficients based on efficiently computable statistics. This approach is motivated as follows. For any 1 ≤k ≤d, we can write any f ∈L2(µd) with the Walsh coefficients θ(f) as f = X u∈Bk X v∈Bd−k θuvϕuv = X u∈Bk fu ⊗ϕu, where uv denotes the concatenation of u ∈Bk and v ∈Bd−k and, for each u ∈Bk, fu △= P v∈Bd−k θuvϕv lies in L2(µd−k). By Parseval’s identity, Wu △= ∥fu∥2 L2(µd−k) = P v∈Bd−k θ2 uv. This means that if Wu < λ for some u ∈Bk, then θ2 uv < λ for every v ∈Bd−k. Thus, we could start at u = 0 and u = 1 and check whether Wu ≥λ. If not, then we would discard all θuv with v ∈Bd−1; otherwise, we would proceed on to u0 and u1. At the end of this process, we will be left only with those s ∈Bd for which θ2 s ≥λ. Let fλ denote the resulting function. If f ∈Fd(p) for some p, then we will have ∥f −fλ∥2 L2(µd) ≤CM −1(Mλ)−2r/(2r+1). We will follow this reasoning in constructing our estimator. We begin by developing an estimator for Wu. We will use the following fact, easily proved using the definitions (1) and (2) of the Walsh functions: for any density f on Bd, any k and u ∈Bk, we have fu(y) = Ef ϕu(πk(X))I{σk(X)=y} , ∀y ∈Bd−k and Wu = Ef {ϕu(πk(X))fu(σk(X))} , where πk(x) △= (x(1), . . . , x(k)) and σk(x) △= (x(k + 1), . . . , x(d)) for any x ∈Bd. This suggests that we can estimate Wu by c Wu = 1 n2 n X i1=1 n X i2=1 ϕu(πk(Xi1))ϕu(πk(Xi2))I{σk(Xi1 )=σk(Xi2 )}. (6) Using induction and Eqs. (1) and (2), we can prove that c Wu = P v∈Bd−k bθ2 uv. An advantage of computing c Wu indirectly via (6) rather than as a sum of bθ2 uv, v ∈Bd−k, is that, while the latter has O(2d−kn) complexity, the former has only O(n2d) complexity. This can lead to significant computational savings for small k. When k ≥d −log(nd), it becomes more efficient to use the direct estimator. Now we can define our density estimation procedure. Instead of using a single threshold for all 1 ≤k ≤d, we consider a more flexible strategy: for every k, we shall compare each c Wu to a threshold that depends not only on n, but also on k. Specifically, we will let λk,n = αk log n n , 1 ≤k ≤d (7) where α = {αk}d k=1 satisfies α1 ≥αk ≥αd > 0. (This k-dependent scaling will allow us to trade off MSE and computational complexity.) Given λ = {λk,n}d k=1, define the set A(λ) △= {s ∈Bd : c Wπk(s) ≥λk,n, ∀1 ≤k ≤d} and the corresponding estimator bfRWT △= X s∈Bd I{s∈A(λ)}bθsϕs, (8) where RWT stands for “recursive Walsh thresholding.” To implement bfRWT on a computer, we adapt the algorithm of Goldreich and Levin [12], originally developed for cryptography and later applied to the problem of learning Boolean functions from membership queries [13]: we call the routine RECURSIVEWALSH, shown in Algorithm 1, with u = ∅(the empty string) and with λ from (7). Analysis of the estimator. We now turn to the asymptotic analysis of the MSE and the computational complexity of bfRWT. We first prove that bfRWT adapts to unknown sparsity of f: Theorem 3.1 Suppose the threshold sequence λ = {λk}d k=1 is such that αd ≥(20d + 25)2/2d. Then for all 0 < p < 2 the estimator (8) satisfies sup f∈Fd(p) MSE(f, bfRWT) = sup f∈Fd(p) Ef ∥f −bfRWT∥2 L2(µd) ≤C 2d 2dα1 log n n 2r/(2r+1) , (9) where the constant C depends only on p. Proof: Let us decompose the squared L2 error of bfRWT as ∥f −bfRWT∥2 L2(µd) = X s I{s∈A(λ)}(θs −bθs)2 + X s I{s∈A(λ)c}θ2 s ≡T1 + T2. Algorithm 1 RECURSIVEWALSH(u, λ) k ←length(u) if k = d then compute bθu ←1 n nP i=1 ϕu(Xi); if bθ2 u ≥λd,n then output u, bθu; return end if compute c Wu0 ← 1 n2 nP i1=1 nP i2=1 ϕu0(πk+1(Xi1))ϕu0(πk+1(Xi2))I{σk+1(Xi1 )=σk+1(Xi2 )} compute c Wu1 ← 1 n2 nP i1=1 nP i2=1 ϕu1(πk+1(Xi1))ϕu1(πk+1(Xi2))I{σk+1(Xi1 )=σk+1(Xi2 )} if c Wu0 ≤λk+1,n then return else RECURSIVEWALSH(u0, λ); end if if c Wu1 ≤λk+1,n then return else RECURSIVEWALSH(u1, λ); end if We start by observing that s ∈A(λ) only if bθ2 s ≥λd,n, while for any s ∈A(λ)c there exists some 1 ≤k ≤d such that bθ2 s < λk,n ≤λ1,n. Defining the sets A1 = {s ∈Bd : bθ2 s ≥λd,n} and A2 = {s ∈Bd : bθ2 s < λ1,n}, we get T1 ≤P s I{s∈A1}(θs −bθs)2 and T2 ≤P s I{s∈A2}θ2 s. Further, defining B = {s ∈Bd : θ2 s < λd,n/2} and S = {s ∈Bd : θ2 s ≥3λ1,n/2}, we can write T1 = X s I{s∈A1∩B}(θs −bθs)2 + X s I{s∈A1∩Bc}(θs −bθs)2 ≡T11 + T12, T2 = X s I{s∈A2∩S}θ2 s + X s I{s∈A2∩Sc}θ2 s ≡T21 + T22. First we deal with the easy terms T12, T22. Applying (4), (5) and a bit of algebra, we get E T12 ≤ 1 Mn s : θ2 s ≥λd,n/2 ≤ 1 Mn 2 Mλd,n p/2 ≤1 M n−2r/(2r+1), (10) E T22 ≤ X s∈Bd I{θ2s<(3α1/2) log n/n}θ2 s ≤C M Mα1 log n n 2r/(2r+1) . (11) Next we deal with the large-deviation terms T11 and T21. Using Cauchy–Schwarz, we get E T11 ≤ X s h E(θs −bθs)4 · P(s ∈A1 ∩B) i1/2 . (12) To estimate the fourth moment in (12), we use Rosenthal’s inequality [14] to get E(θs −bθs)4 ≤ c/M 2n2. To bound the probability that s ∈A1 ∩B, we observe that s ∈A1 ∩B implies that |bθs −θs| ≥(1/5) p λd,n, and then use Bernstein’s inequality [14] to get P |bθs −θs| ≥(1/5) p λd,n ≤2 exp − β2 log n 2(1 + 2β/3) = 2n−β2/[2(1+2β/3)] ≤2n−(β−1)/2 with β = (1/5)√Mαd ≥4d + 5. Since n−(β−1)/2 ≤n−2(d+1), we have E T11 ≤Cn−(d+1) ≤C/(Mn). (13) Finally, E T21 ≤P s P(s ∈A2 ∩S)θ2 s. Using the same argument as above, we get P(s ∈A2 ∩S) ≤ 2n−(γ−1)/2, where γ = (1/5)√Mα1. Since θ2 s ≤1/M for all s ∈Bd and since γ ≥β, this gives E T21 ≤2n−2(d+1) ≤2/(Mn). (14) Putting together Eqs. (10), (11), (13), and (14), we get (9), and the theorem is proved. ■ Our second result concerns the running time of Algorithm 1. Let K(α, p) △= Pd k=1 α−p/2 k . Theorem 3.2 Given any δ ∈(0, 1), provided each αk is chosen so that p 2kαkn log n ≥5 C2 √n + (log(d/δ) + k)/ log e , (15) Algorithm 1 runs in O(n2d(n/M log n)p/2K(α, p)) time with probability at least 1 −δ. Proof: The complexity is determined by the number of calls to RECURSIVEWALSH. For each k, a call to RECURSIVEWALSH is made at every u ∈Bk with c Wu ≥λk,n. Let us say that a call to RECURSIVEWALSH(u, λ) is correct if Wu ≥λk,n/2. We will show that, with probability at least 1 −δ, only the correct calls are made. The probability of making at least one incorrect call is P d[ k=1 [ u∈Bk {c Wu ≥λk,n, Wu < λk,n/2} ! ≤ d X k=1 X u∈Bk P c Wu ≥λk,n, Wu < λk,n/2 . For a given u ∈Bk, c Wu ≥λk,n and Wu < λk,n/2 together imply that ∥fu −bfu∥2 L2(µd−k) ≥ (1/5) p λk,n, where bfu △= P v∈Bd−k bθuvϕv. Now, it can be shown that, for every u ∈Bk, the norm ∥fu −bfu∥L2(µd−k) can be expressed as a supremum of an empirical process [15] over a certain function class that depends on k (details are omitted for lack of space). We can then use Talagrand’s concentration-of-measure inequality for empirical processes [16] to get P(c Wu ≥λk,n, Wu < λk,n/2) ≤exp −nC1(2ka2 k,n ∧2k/2ak,n) , where ak,n = (1/5) p αk log n/n−C2/ √ 2kn, and C1, C2 are the absolute constants in Talagrand’s bound. If we choose αk as in (15), then P(c Wu ≥λk,n, Wu < λk,n/2) ≤δ/(d2d−k) for all u ∈Bk. Summing over k, u ∈Bk, we see that, with probability ≥1 −δ, only the correct calls will be made. It remains to bound the number of the correct calls. For each k, Wu ≥λk,n/2 implies that there exists at least one v ∈Bd−k such that θ2 uv ≥λk,n/2. Since for every 1 ≤k ≤d each θs contributes to exactly one Wu, we have by the pigeonhole principle that u ∈Bk : Wu ≥λk,n/2 ≤ s ∈Bd : θ2 s ≥λk,n/2 ≤(2/Mλk,n)p/2, where in the second inequality we used (4) with R = 1/ √ M. Hence, the number of correct recursive calls is bounded by N = Pd k=1(2/Mλk,n)p/2 = (2n/M log n)p/2K(α, p). At each call, we compute an estimate of the corresponding Wu0 and Wu1, which requires O(n2d) operations. Therefore, with probability at least 1 −δ, the time complexity will be as stated in the theorem. ■ MSE vs. complexity. By controlling the rate at which the sequence αk decays with k, we can trade off MSE against complexity. Consider the following two extreme cases: (1) α1 = . . . = αd ∼1/M and (2) αk ∼2d−k/M. The first case, which reduces to term-by-term thresholding, achieves the best bias-variance trade-off with the MSE O((log n/n)2r/(2r+1)(1/M)). However, it has K(α, p) = O(M p/2d), resulting in O(d2n2(n/ log n)p/2) complexity. The second case, which leads to a very severe estimator that will tend to reject a lot of coefficients, has MSE of O((log n/n)2r/(2r+1)M −1/(2r+1)), but K(α, p) = O(M p/2), leading to a considerably better O(dn2(n/ log n)p/2) complexity. From the computational viewpoint, it is preferable to use rapidly decaying thresholds. However, this reduction in complexity will be offset by a corresponding increase in MSE. In fact, using exponentially decaying αk’s in practice is not advisable as its low complexity is mainly due to the fact that it will tend to reject even the big coefficients very early on, especially when d is large. To achieve a good balance between complexity and MSE, a moderately decaying threshold sequence might be best, e.g., αk ∼(d−k+1)m/M for some m ≥1. As p →0, the effect of λ on complexity becomes negligible, and the complexity tends to O(n2d). Positivity and normalization issues. As is the case with orthogonal series estimators, bfRWT may not necessarily be a bona fide density. In particular, there may be some x ∈Bd such that bfRWT(x) < 0, and it may happen that R bfRWTdµd ̸= 1. In principle, this can be handled by clipping the negative values at zero and renormalizing, which can only improve the MSE. In practice renormalization may be computationally expensive when d is very large. If the estimate is suitably sparse, however, the renormalization can be carried out approximately using Monte-Carlo methods. 4 Simulations The focus of our work is theoretical, consisting in the derivation of a recursive thresholding procedure for estimating multivariate binary densities (Algorithm 1), with a proof of its near-minimaxity and an asymptotic analysis of its complexity. Although an extensive empirical evaluation is outside the scope of this paper, we have implemented the proposed estimator, and now present some simulation results to demonstrate its small-sample performance. We generated synthetic observations from a mixture density f on a 15-dimensional binary hypercube. The mixture has 10 components, where each component is a product density with 12 randomly chosen covariates having Bernoulli(1/2) distributions, and the other three having Bernoulli(0.9) distributions. For d = 15, it is still feasible to quickly compute the ground truth, consisting of 32768 values of f and its Walsh coefficients. These values are shown in Fig. 1 (left). As can be seen from the coefficient profile in the bottom of the figure, this density is clearly sparse. Fig. 1 also shows the estimated probabilities and the Walsh coefficients for sample sizes n = 5000 (middle) and n = 10000 (right). Ground truth (f) bfRWT, n = 5000 bfRWT, n = 10000 Figure 1: Ground truth (left) and estimated density for n = 5000 (middle) and n = 10000 (right) with constant thresholding. Top: true and estimated probabilities (clipped at zero and renormalized) arranged in lexicographic order. Bottom: absolute values of true and estimated Walsh coefficients arranged in lexicographic order. For the estimated densities, the coefficient plots also show the threshold level (dotted line) and absolute values of the rejected coefficients (lighter color). 2000 4000 6000 8000 10000 0.1 0.2 0.3 0.4 0.5 0.6 Sample size (n) MSE (× 2d) constant log linear 2000 4000 6000 8000 10000 200 400 600 800 1000 1200 1400 Sample size (n) Time (s) 2000 4000 6000 8000 10000 500 1000 1500 2000 2500 3000 3500 Sample size (n) Recursive calls 2000 4000 6000 8000 10000 10 15 20 25 30 35 40 Sample size (n) Coeffs. estimated (a) (b) (c) (d) Figure 2: Small-sample performance of bfRWT in estimating f wth three different thresholding schemes: (a) MSE; (b) running time (in seconds); (c) number of recursive calls; (d) number of coefficients retained by the algorithm. All results are averaged over five independent runs for each sample size (the error bars show the standard deviations). To study the trade-off between MSE and complexity, we implemented three different thresholding schemes: (1) constant, λk,n = 2 log n/(2dn), (2) logarithmic, λk,n = 2 log(d−k +2) log n/(2dn), and (3) linear, λk,n = 2(d −k + 1) log n/(2dn). Up to the log n factor (dictated by the theory), the thresholds at k = d are set to twice the variance of the empirical estimate of any coefficient whose value is zero; this forces the estimator to reject empirical coefficients whose values cannot be reliably distinguished from zero. Occasionally, spurious coefficients get retained, as can be seen in Fig. 1 (middle) for the estimate for n = 5000. Fig 2 shows the performance of bfRWT. Fig. 2(a) is a plot of MSE vs. sample size. In agreement with the theory, MSE is the smallest for the constant thresholding scheme [which is simply an efficient recursive implementation of a term-by-term thresholding estimator with λn ∼log n/(Mn)], and then it increases for the logarithmic and for the linear schemes. Fig. 2(b,c) shows the running time (in seconds) and the number of recursive calls made to RECURSIVEWALSH vs. sample size. The number of recursive calls is a platformindependent way of gauging the computational complexity of the algorithm, although it should be kept in mind that each recursive call has O(n2d) overhead. The running time increases polynomially with n, and is the largest for the constant scheme, followed by the logarithmic and the linear schemes. We see that, while the MSE of the logarithmic scheme is fairly close to that of the constant scheme, its complexity is considerably lower, in terms of both the number of recursive calls and the running time. In all three cases, the number of recursive calls decreases with n due to the fact that weight estimates become increasingly accurate with n, which causes the expected number of false discoveries (i.e., making a recursive call at an internal node of the tree only to reject its descendants later) to decrease. Finally, Fig. 2(d) shows the number of coefficients retained in the estimate. This number grows with n as a consequence of the fact that the threshold decreases with n, while the number of accurately estimated coefficients increases. The true density f has 40 parameters: 9 to specify the weights of the components, 3 per component to locate the indices of the nonuniform covariates, and the single Bernoulli parameter of the nonuniform covariates. It is interesting to note that the maximal number of coefficients returned by our algorithm approaches 40. Overall, these preliminary simulation results show that our implemented estimator behaves in accordance with the theory even in the small-sample regime. The performance of the logarithmic thresholding scheme is especially encouraging, suggesting that it may be possible to trade off MSE against complexity in a way that will scale to large values of d. In the future, we plan to test our method on high-dimensional real data sets. Our particular interest is in social network data, e.g., records of meetings among large groups of individuals. These are represented by binary strings most of whose entries are zero (i.e., only a very small number of people are present at any given meeting). To model their densities, we plan to experiment with Walsh bases with η biased toward unity. Acknowledgments This work was supported by NSF CAREER Award No. CCF-06-43947 and DARPA Grant No. HR0011-07-1003. References [1] I. Shmulevich and W. Zhang. Binary analysis and optimization-based normalization of gene expression data. Bioinformatics 18(4):555–565, 2002. [2] J.M. Carro. Estimating dynamic panel data discrete choice models with fixed effects. J. Econometrics 140:503–528, 2007. [3] Z. Ghahramani and K. Heller. Bayesian sets. NIPS 18:435–442, 2006. [4] J. Aitchison and C.G.G. Aitken. Multivariate binary discrimination by the kernel method. Biometrika 63(3):413–420, 1976. [5] J. Ott and R.A. Kronmal. Some classification procedures for multivariate binary data using orthogonal functions. J. Amer. Stat. Assoc. 71(354):391–399, 1976. [6] W.-Q. Liang and P.R. Krishnaiah. Nonparametric iterative estimation of multivariate binary density. J. Multivariate Anal. 16:162–172, 1985. [7] J.S. Simonoff. Smoothing categorical data. J. Statist. Planning and Inference 47:41–60, 1995. [8] M. Talagrand. On Russo’s approximate zero-one law. Ann. Probab. 22:1576–1587, 1994. [9] I. Dinur, E. Friedgut, G. Kindler and R. O’Donnell. On the Fourier tails of bounded functions over the discrete cube. Israel J. Math. 160:389–421, 2007. [10] I.M. Johnstone. Minimax Bayes, asymptotic minimax and sparse wavelet priors. In S.S. Gupta and J.O. Berger, eds., Statistical Decision Theory and Related Topics V, pp. 303–326, Springer, 1994. [11] E.J. Cand`es and T. Tao. Near-optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inf. Theory 52(12):5406–5425, 2006. [12] O. Goldreich and L. Levin. A hard-core predicate for all one-way functions. STOC, pp. 25–32, 1989. [13] E. Kushilevitz and Y. Mansour. Learning decision trees using the Fourier spectrum. SIAM J. Comput. 22(6):1331-1348, 1993. [14] W. H¨ardle, G. Kerkyacharian, D. Picard and A.B. Tsybakov. Wavelets, Approximation, and Statistical Applications, Springer, 1998. [15] S.A. van de Geer. Empirical Processes in M-Estimation, Cambridge Univ. Press, 2000. [16] M. Talagrand. Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22:28–76, 1994.
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Asynchronous Distributed Learning of Topic Models Arthur Asuncion, Padhraic Smyth, Max Welling Department of Computer Science University of California, Irvine {asuncion,smyth,welling}@ics.uci.edu Abstract Distributed learning is a problem of fundamental interest in machine learning and cognitive science. In this paper, we present asynchronous distributed learning algorithms for two well-known unsupervised learning frameworks: Latent Dirichlet Allocation (LDA) and Hierarchical Dirichlet Processes (HDP). In the proposed approach, the data are distributed across P processors, and processors independently perform Gibbs sampling on their local data and communicate their information in a local asynchronous manner with other processors. We demonstrate that our asynchronous algorithms are able to learn global topic models that are statistically as accurate as those learned by the standard LDA and HDP samplers, but with significant improvements in computation time and memory. We show speedup results on a 730-million-word text corpus using 32 processors, and we provide perplexity results for up to 1500 virtual processors. As a stepping stone in the development of asynchronous HDP, a parallel HDP sampler is also introduced. 1 Introduction Learning algorithms that can perform in a distributed asynchronous manner are of interest for several different reasons. The increasing availability of multi-processor and grid computing technology provides an immediate and practical motivation to develop learning algorithms that are able take advantage of such computational resources. Similarly, the increasing proliferation of networks of low-cost devices motivates the investigation of distributed learning in the context of sensor networks. On a deeper level, there are fundamental questions about distributed learning from the viewpoints of artificial intelligence and cognitive science. In this paper, we focus on the specific problem of developing asynchronous distributed learning algorithms for a class of unsupervised learning techniques, specifically LDA [1] and HDP [2] with learning via Gibbs sampling. The frameworks of LDA and HDP have recently become popular due to their effectiveness at extracting low-dimensional representations from sparse high-dimensional data, with multiple applications in areas such as text analysis and computer vision. A promising approach to scaling these algorithms to large data sets is to distribute the data across multiple processors and develop appropriate distributed topic-modeling algorithms [3, 4, 5]. There are two somewhat distinct motivations for distributed computation in this context: (1) to address the memory issue when the original data and count matrices used by the algorithm exceed the main memory capacity of a single machine; and (2) using multiple processors to significantly speed up topic-learning, e.g., learning a topic model in near real-time for tens of thousands of documents returned by a search-engine. While synchronous distributed algorithms for topic models have been proposed in earlier work, here we investigate asynchronous distributed learning of topic models. Asynchronous algorithms provide several computational advantages over their synchronous counterparts: (1) no global synchronization step is required; (2) the system is extremely fault-tolerant due to its decentralized nature; (3) heterogeneous machines with different processor speeds and memory capacities can be used; (4) new processors and new data can be incorporated into the system at any time. Our primary novel contribution is the introduction of new asynchronous distributed algorithms for LDA and HDP, based on local collapsed Gibbs sampling on each processor. We assume an asyn1 chronous “gossip-based” framework [6] which only allows pairwise interactions between random processors. Our distributed framework can provide substantial memory and time savings over singleprocessor computation, since each processor only needs to store and perform Gibbs sweeps over 1 P th of the data, where P is the number of processors. Furthermore, the asynchronous approach can scale to large corpora and large numbers of processors, since no global synchronization steps are required. While building towards an asynchronous algorithm for HDP, we also introduce a novel synchronous distributed inference algorithm for HDP, again based on collapsed Gibbs sampling. In the proposed framework, individual processors perform Gibbs sampling locally on each processor based on a noisy inexact view of the global topics. As a result, our algorithms are not necessarily sampling from the proper global posterior distribution. Nonetheless, as we will show in our experiments, these algorithms are empirically very robust and converge rapidly to high-quality solutions. We first review collapsed Gibbs sampling for LDA and HDP. Then we describe the details of our distributed algorithms. We present perplexity and speedup results for our algorithms when applied to text data sets. We conclude with a discussion of related work and future extensions of our work. 2 A brief review of topic models α ij Z ij X wk Φ kj θ k β ij Z ij X kj θ wk Φ K J Nj ∞ J η η γ α Nj Figure 1: Graphical models for LDA (left) and HDP (right). Before delving into the details of our distributed algorithms, we first describe the LDA and HDP topic models. In LDA, each document j is modeled as a mixture over K topics, and each topic k is a multinomial distribution, φwk, over a vocabulary of W words1. Each document’s mixture over topics, θkj, is drawn from a Dirichlet distribution with parameter η. In order to generate a new document, θkj is first sampled from a Dirichlet distribution with parameter α. For each token i in that document, a topic assignment zij is sampled from θkj, and the specific word xij is drawn from φwzij. The graphical model for LDA is shown in Figure 1, and the generative process is below: θk,j ∼D[α] φw,k ∼D[η] zij ∼θk,j xij ∼φw,zij . Given observed data, it is possible to infer the posterior distribution of the latent variables. One can perform collapsed Gibbs sampling [7] by integrating out θkj and φwk and sampling the topic assignments in the following manner: P(zij = k|z¬ij, w) ∝ N ¬ij wk + η P w N ¬ij wk + Wη N ¬ij jk + α . (1) Nwk denotes the number of word tokens of type w assigned to topic k, while Njk denotes the number of tokens in document j assigned to topic k. N ¬ij denotes the count with token ij removed. The HDP mixture model is composed of a hierarchy of Dirichlet processes. HDP is similar to LDA and can be viewed as the model that results from taking the infinite limit of the following finite mixture model. Let L be the number of mixture components, and βk be top level Dirichlet variables drawn from a Dirichlet distribution with parameter γ/L. The mixture for each document, θkj, is generated from a Dirichlet with parameter αβk. The multinomial topic distributions, φwk are drawn from a base Dirichlet distribution with parameter η. As in LDA, zij is sampled from θkj, and word xij is sampled from φwzij. If we take the limit of this model as L goes to infinity, we obtain HDP: βk ∼D[γ/L] θk,j ∼D[αβk] φw,k ∼D[η] zij ∼θk,j xij ∼φw,zij. To sample from the posterior, we follow the details of the direct assignment sampler for HDP [2]. Both θkj and φwk are integrated out, and zij is sampled from a conditional distribution that is almost identical to that of LDA, except that a small amount of probability mass is reserved for the instantiation of a new topic. Note that although HDP is defined to have an infinite number of topics, the only topics that are instantiated are those that are actually used. 3 Asynchronous distributed learning for the LDA model We consider the problem of learning an LDA model with K topics in a distributed fashion where documents are distributed across P processors. Each processor p stores the following local variables: 1To avoid clutter, we write φwk or θkj to denote the set of all components, i.e. {φwk} or {θkj}. Similarly, when sampling from a Dirichlet, we write θkj ∼D[αβk] instead of [θ1,j, ..θK,j] ∼D[αβ1, .., αβK]. 2 wp ij contains the word type for each token i in document j in the processor, and zp ij contains the assigned topic for each token. N ¬p wk is the global word-topic count matrix stored at the processor— this matrix stores counts of other processors gathered during the communication step and does not include the processor’s local counts. N p kj is the local document-topic count matrix (derived from zp), N p w is the simple word count on a processor (derived from wp), and N p wk is the local word-topic count matrix (derived from zp and wp) which only contains the counts of data on the processor. Newman et al. [5] introduced a parallel version of LDA based on collapsed Gibbs sampling (which we will call Parallel-LDA). In Parallel-LDA, each processor receives 1 P of the documents in the corpus and the z’s are globally initialized. Each iteration of the algorithm is composed of two steps: a Gibbs sampling step and a synchronization step. In the sampling step, each processor samples its local zp by using the global topics of the previous iteration. In the synchronization step, the local counts N p wk on each processor are aggregated to produce a global set of word-topic counts Nwk. This process is repeated for either a fixed number of iterations or until the algorithm has converged. Parallel-LDA can provide substantial memory and time savings. However, it is a fully synchronous algorithm since it requires global synchronization at each iteration. In some applications, a global synchronization step may not be feasible, e.g. some processors may be unavailable, while other processors may be in the middle of a long Gibbs sweep, due to differences in processor speeds. To gain the benefits of asynchronous computing, we introduce an asynchronous distributed version of LDA (Async-LDA) that follows a similar two-step process to that above. Each processor performs a local Gibbs sampling step followed by a step of communicating with another random processor. For Async-LDA, during each iteration, the processors perform a full sweep of collapsed Gibbs sampling over their local topic assignment variables zp according to the following conditional distribution, in a manner directly analogous to Equation 1, P(zpij = k|z¬ij p , wp) ∝ (N ¬p + N p)¬ij wk + η P w(N ¬p + N p)¬ij wk + Wη N ¬ij pjk + α . (2) The combination of N ¬p wk and N p wk is used in the sampling equation. Recall that N ¬p wk represents processor p’s belief of the counts of all the other processors with which it has already communicated (not including processor p’s local counts), while N p wk is the processor’s local word-topic counts. Thus, the sampling of the zp’s is based on the processor’s “noisy view” of the global set of topics. Algorithm 1 Async-LDA for each processor p in parallel do repeat Sample zp locally (Equation 2) Receive N g wk from random proc g Send N p wk to proc g if p has met g before then N ¬p wk ←N ¬p wk −˜ N g wk + N g wk else N ¬p wk ←N ¬p wk + N g wk end if until convergence end for Once the inference of zp is complete (and N p wk is updated), the processor finds another finished processor and initiates communication2. We are generally interested in the case where memory and communication bandwidth are both limited. We also assume in the simplified gossip scheme that a processor can establish communication with every other processor – later in the paper we also discuss scenarios that relax these assumptions. In the communication step, let us consider the case where two processors, p and g have never met before. In this case, processors simply exchange their local N p wk’s (their local contribution to the global topic set), and processor p simply adds N g wk to its N ¬p wk , and vice versa. Consider the case where two processors meet again. The processors should not simply swap and add their local counts again; rather, each processor should first remove from N ¬p wk the previous influence of the other processor during their previous encounter, in order to prevent processors that frequently meet from over-influencing each other. We assume in the general case that a processor does not store in memory the previous counts of all the other processors that processor p has already met. Since the previous local counts of the other processor were already absorbed into N ¬p wk and are thus not retrievable, we must take a different approach. In Async-LDA, the processors exchange their N p wk’s, from which the count of words on each processor, N p w can be derived. Using processor g’s N g w, processor p creates ˜N g wk by sampling N g w topic values randomly without replacement from 2We don’t discuss in general the details of how processors might identify other processors that have finished their iteration, but we imagine that a standard protocol could be used, like P2P. 3 collection {N ¬p wk}. We can imagine that there are P k N ¬p wk colored balls, with N ¬p wk balls of color k, from which we pick N g w balls uniformly at random without replacement. This process is equivalent to sampling from a multivariate hypergeometric distribution. ˜N g wk acts as a substitute for the N g wk that processor p received during their previous encounter. Since all knowledge of the previous N g wk is lost, this method can be justified by Laplace’s principle of indifference (or the principle of maximum entropy). Finally, we update N ¬p wk by subtracting ˜N g wk and adding the current N g wk: N ¬p wk ←N ¬p wk −˜N g wk + N g wk where ˜N g w,k ∼MH [N g w; N ¬p w,1, .., N ¬p w,K] . (3) Pseudocode for Async-LDA is provided in the display box for Algorithm 1. The assumption of limited memory can be relaxed by allowing processors to cache previous counts of other processors – the cached N g wk would replace ˜N g wk. We can also relax the assumption of limited bandwidth. Processor p could forward its individual cached counts (from other processors) to g, and vice versa, to quicken the dissemination of information. In fixed topologies where the network is not fully connected, forwarding is necessary to propagate the counts across the network. Our approach can be applied to a wide variety of scenarios with varying memory, bandwidth, and topology constraints. 4 Synchronous and asynchronous distributed learning for the HDP model Inference for HDP can be performed in a distributed manner as well. Before discussing our asynchronous HDP algorithm, we first describe a synchronous parallel inference algorithm for HDP. We begin with necessary notation for HDPs: γ is the concentration parameter for the top level Dirichlet Process (DP), α is the concentration parameter for the document level DP, βk’s are toplevel topic probabilities, and η is the Dirichlet parameter for the base distribution. The graphical model for HDP is shown in Figure 1. We introduce Parallel-HDP, which is analogous to Parallel-LDA except that new topics may be added during the Gibbs sweep. Documents are again distributed across the processors. Each processor maintains local βp k parameters which are augmented when a new topic is locally created. During the Gibbs sampling step, each processor locally samples the zp topic assignments. In the synchronization step, the local word-topic counts N p wk are aggregated into a single matrix of global counts Nwk, and the local βp k’s are averaged to form a global βk. The α, βk and γ hyperparameters are also globally resampled during the synchronization step – see Teh et al. [2] for details. We fix η to be a small constant. While α and γ can also be fixed, sampling these parameters improves the rate of convergence. To facilitate sampling, relatively flat gamma priors are placed on α and γ. Finally, these parameters and the global count matrix are distributed back to the processors. Algorithm 2 Parallel-HDP repeat for each processor p in parallel do Sample zp locally Send N p wk, βp k to master node end for Nwk ←P p N p wk βk ←(P p βp k) / P Resample α, βk, γ globally Distribute Nwk, α, βk, γ to all processors until convergence Algorithm 3 Async-HDP for each processor p in parallel do repeat Sample zp and then αp, βp k, γp locally Receive N g wk, αg, βg k from random proc g Send N p wk, αp, βp k to proc g if p has met g before then N ¬p wk ←N ¬p wk −˜ N g wk + N g wk else N ¬p wk ←N ¬p wk + N g wk end if αp ←(αp + αg) / 2 and βp k ←(βp k + βg k) / 2 until convergence end for Motivated again by the advantages of local asynchronous communication between processors, we propose an Async-HDP algorithm. It is very similar in spirit to Async-LDA, and so we focus on the differences in our description. First, the sampling equation for zp is different to that of Async-LDA, since some probability mass is reserved for new topics: P(zpij = k|z¬ij p , wp) ∝ (N¬p+Np)¬ij wk +η P w(N¬p+Np)¬ij wk +W η N ¬ij pjk + αpβp k , if k ≤Kp αpβp new W , if k is new. 4 KOS NIPS NYT PUBMED Total number of documents in training set 3,000 1,500 300,000 8,200,000 Size of vocabulary 6,906 12,419 102,660 141,043 Total number of words 410,595 1,932,365 99,542,125 737,869,083 Total number of documents in test set 430 184 – – Table 1: Data sets used for perplexity and speedup experiments We resample the hyperparameters αp, βp k, γp locally3 during the inference step, and keep η fixed. In Async-HDP, a processor can add new topics to its collection during the inference step. Thus, when two processors communicate, the number of topics on each processor might be different. One way to merge topics is to perform bipartite matching across the two topic sets, using the Hungarian algorithm. However, performing this topic matching step imposes a computational penalty as the number of topics increases. In our experiments for Async-LDA, Parallel-HDP, and Async-HDP, we do not perform topic matching, but we simply combine the topics on different processors based their topic IDs and (somewhat surprisingly) the topics gradually self-organize and align. Newman et al. [5] also observed this same behavior occurring in Parallel-LDA. During the communication step, the counts N p wk and the parameters αp and βp k values are exchanged and merged. Async-HDP removes a processor’s previous influence through the same MH technique used in Async-LDA. Pseudocode for Async-HDP is provided in the display box for Algorithm 3. 5 Experiments We use four text data sets for evaluation: KOS, a data set derived from blog entries (dailykos.com); NIPS, a data set derived from NIPS papers (books.nips.cc); NYT, a collection of news articles from the New York Times (nytimes.com); and PUBMED, a large collection of PubMed abstracts (ncbi.nlm.nih.gov/pubmed/). The characteristics of these four data sets are summarized in Table 1. For our perplexity experiments, parallel processors were simulated in software and run on smaller data sets (KOS, NIPS), to enable us to test the statistical limits of our algorithms. Actual parallel hardware is used to measure speedup on larger data sets (NYT, PUBMED). Our simulation features a gossip scheme over a fully connected network that lets each processor communicate with one other random processor at the end of every iteration, e.g., with P=100, there are 50 pairs at each iteration. In our perplexity experiments, the data set is separated into a training set and a test set. We learn our models on the training set, and then we measure the performance of our algorithms on the test set using perplexity, a widely-used metric in the topic modeling community. We briefly describe how perplexity is computed for our models. Perplexity is simply the exponentiated average per-word log-likelihood. For each of our experiments, we perform S = 5 different Gibbs runs, with each run lasting 1500 iterations (unless otherwise noted), and we obtain a sample at the end of each of those runs. The 5 samples are then averaged when computing perplexity. For Parallel-HDP, perplexity is calculated in the same way as in standard HDP: log p(xtest) = X jw log 1 S X s X k ˆθs jk ˆφs wk where ˆθs jk = αβk + N s jk P k (αβk) + N s j , ˆφs wk = η + N s wk Wη + N s k . (4) After the model is run on the training data, ˆφs wk is available in sample s. To obtain ˆθs jk, one must resample the topic assignments on the first half of each document in the test set while holding ˆφs wk fixed. Perplexity is evaluated on the second half of each document in the test set, given ˆφs wk and ˆθs jk. The perplexity calculation for Async-LDA and Async-HDP uses the same formula. Since each processor effectively learns a separate local topic model, we can directly compute the perplexity for each processor’s local model. In our experiments, we report the average perplexity among processors, and we show error bars denoting the minimum and maximum perplexity among all processors. The variance of perplexities between processors is usually quite small, which suggests that the local topic models learned on each processor are equally accurate. For KOS and NIPS, we used the same settings for priors and hyperpriors: α = 0.1, η = 0.01 for LDA and Async-LDA, and η = 0.01, γ ∼Gam(10, 1), and α ∼Gam(2, 1) for the HDP algorithms. 3Sampling αp, βp k, γp requires a global view of variables like m·k, the total number of “tables” serving “dish” k [2]. These values can be asynchronously propagated in the same way that the counts are propagated. 5 1 10 100 1400 1500 1600 1700 1800 Perplexity Processors KOS K=8 K=16 K=32 K=64 1 10 100 1400 1600 1800 2000 Perplexity Processors NIPS K=10 K=20 K=40 K=80 1 10 100 500 10001500 1400 1600 1800 2000 Processors Perplexity KOS K=16 LDA Async−LDA Figure 2: (a) Left: Async-LDA perplexities on KOS. (b) Middle: Async-LDA perplexities on NIPS. (c) Right: Async-LDA perplexities on KOS with many procs. Cache=5 when P≥100. 3000 iterations run when P≥500. 5.1 Async-LDA perplexity and speedup results Figures 2(a,b) show the perplexities for Async-LDA on KOS and NIPS data sets for varying numbers of topics. The variation in perplexities between LDA and Async-LDA is slight and is significantly less than the variation in perplexities as the number of topics K is changed. These numbers suggest that Async-LDA converges to solutions of the same quality as standard LDA. While these results are based on a single test/train split of the corpus, we have also performed cross-validation experiments (results not shown) which give essentially the same results across different test/train splits. We also stretched the limits of our algorithm by increasing P (e.g. for P=1500, there are only two documents on each processor), and we found that performance was virtually unchanged (figure 2(c)). As a baseline we ran an experiment where processors never communicate. As the number of processors P was increased from 10 to 1500 the corresponding perplexities increased from 2600 to 5700, dramatically higher than our Async-LDA algorithm, indicating (unsurprisingly) that processor communication is essential to obtain good quality models. Figure 3(a) shows the rate of convergence of Async-LDA. As the number of processors increases, the rate of convergence slows, since it takes more iterations for information to propagate to all the processors. However, it is important to note that one iteration in real time of Async-LDA is up to P times faster than one iteration of LDA. We show the same curve in terms of estimated real time in figure 3(b) , assuming a parallel efficiency of 0.5, and one can see that Async-LDA converges much more quickly than LDA. Figure 3(c) shows actual speedup results for Async-LDA on NYT and PUBMED, and the speedups are competitive to those reported for Parallel-LDA [5]. As the data set size grows, the parallel efficiency increases, since communication overhead is dwarfed by the sampling time. In Figure 3(a), we also show the performance of a baseline asynchronous averaging scheme, where global counts are averaged together: N ¬p wk ←(N ¬p wk + N ¬g wk)/d + N g wk. To prevent unbounded count growth, d must be greater than 2, and so we arbitrarily set d to 2.5. While this averaging scheme initially converges quickly, it converges to a final solution that is worse than Async-LDA, regardless of the setting for d. The rate of convergence for Async-LDA P=100 can be dramatically improved by letting each processor maintain a cache of previous N g wk counts of other processors. Figures 3(a,b), C=5, show the improvement made by letting each processor cache the five most recently seen N g wk’s. Note that we still assume a limited bandwidth – processors do not forward individual cached counts, but instead share a single matrix of combined cache counts that helps the processors to achieve faster burn-in time. In this manner, one can elegantly make a tradeoff between time and memory. 0 500 1000 1500 2000 2500 Iteration Perplexity LDA Async−LDA P=10 Async−LDA P=100 Async−LDA P=100 C=5 Averaging P=100 0 50 100 1500 2000 2500 Relative Time Perplexity LDA Async−LDA P=10 Async−LDA P=100 Async−LDA P=100 C=5 1 8 16 24 32 5 10 15 20 25 30 Processors (MPI) Speedup Perfect Async−LDA (PUBMED) Async−LDA (NYT) Figure 3: (a) Left: Convergence plot for Async-LDA on KOS, K=16. (b) Middle: Same plot with x-axis as relative time. (c) Right: Speedup results for NYT and PUBMED on a cluster, using Message Passing Interface. 6 1 10 100 1200 1300 1400 1300 1400 1500 Processors Perplexity KOS NIPS HDP Parallel−HDP Async−HDP 0 500 1000 0 1000 2000 3000 No. of Topics Perplexity Iteration No. of Topics || Perplexity HDP Parallel−HDP P=10 Parallel−HDP P=100 0 500 1000 0 1000 2000 3000 Iteration No. of Topics || Perplexity No. of Topics Perplexity HDP Async−HDP P=10 Async−HDP P=100 C=5 Figure 4: (a) Left: Perplexities for Parallel-HDP and Async-HDP. Cache=5 used for Async-HDP P=100. (b) Middle: Convergence plot for Parallel-HDP on KOS. (c) Right: Convergence plot for Async-HDP on KOS. 5.2 Parallel-HDP and Async-HDP results Perplexities for Parallel-HDP after 1500 iterations are shown in figure 4(a), and they suggest that the model generated by Parallel-HDP has nearly the same predictive power as standard HDP. Figure 4(b) shows that Parallel-HDP converges at essentially the same rate as standard HDP on the KOS data set, even though topics are generated at a slower rate. Topics grow at a slower rate in ParallelHDP since new topics that are generated locally on each processor are merged together during each synchronization step. In this experiment, while the number of topics is still growing, the perplexity has converged, because the newest topics are smaller and do not significantly affect the predictive power of the model. The number of topics does stabilize after thousands of iterations. Perplexities for Async-HDP are shown in figures 4(a,c) as well. On the NIPS data set, there is a slight perplexity degradation, which is partially due to non-optimal parameter settings for α and γ. Topics are generated at a slightly faster rate for Async-HDP than for Parallel-HDP because AsyncHDP take a less aggressive approach on pruning small topics, since processors need to be careful when pruning topics locally. Like Parallel-HDP, Async-HDP converges rapidly to a good solution. 5.3 Extended experiments for realistic scenarios In certain applications, it is desirable to learn a topic model incrementally as new data arrives. In our framework, if new data arrives, we simply assign the new data to a new processor, and then let that new processor enter the “world” of processors with which it can begin to communicate. Our asynchronous approach requires no global initialization or global synchronization step. We do assume a fixed global vocabulary, but one can imagine schemes which allow the vocabulary to grow as well. We performed an experiment for Async-LDA where we introduced 10 new processors (each carrying new data) every 100 iterations. In the first 100 iterations, only 10% of the KOS data is known, and every 100 iterations, an additional 10% of the data is added to the system through new processors. Figure 5(a) shows that perplexity decreases as more processors and data are added. After 1000 iterations, the perplexity of Async-LDA has converged to the standard LDA perplexity. Thus, in this experiment, learning in an online fashion does not adversely affect the final model. In the experiments previously described, documents were randomly distributed across processors. In reality, a processor may have a document set specialized to only a few topics. We investigated Async-LDA’s behavior on a non-random distribution of documents over processors. After running LDA (K=20) on NIPS, we used the inferred mixtures θjk to separate the corpus into 20 different sets of documents corresponding to the 20 topics. We assigned 2 sets of documents to each of 10 processors, so that each processor had a document set that was specialized to 2 topics. Figure 5(b) shows that Async-LDA performs just as well on this non-random distribution of documents. 0 200 400 600 800 1000 1500 2000 2500 3000 Iteration Perplexity 10% 20% 30% 40% of data seen, etc. LDA Async−LDA P=100 Async−LDA P=100 (Online) 0 100 200 300 400 500 1500 2000 2500 3000 Iteration Perplexity LDA Async−LDA P=10 Random Async−LDA P=10 Non−Random 0 200 400 600 800 1000 1500 2000 2500 3000 Relative Time Perplexity LDA Async−LDA P=10 (Balanced) Async−LDA P=10 (Imbalanced) Figure 5: (a) Left: Online learning for Async-LDA on KOS, K=16. (b) Middle: Comparing random vs. nonrandom distribution of documents for Async-LDA on NIPS, K=20. (c) Right: Async-LDA on KOS, K=16, where processors have varying amounts of data. In all 3 cases, Async-LDA converges to a good solution. 7 Another situation of interest is the case where the amount of data on each processor varies. KOS was divided into 30 blocks of 100 documents and these blocks were assigned to 10 processors according to a distribution: {7, 6, 4, 3, 3, 2, 2, 1, 1, 1}. We assume that if a processor has k blocks, then it will take k units of time to complete one sampling sweep. Figure 5(c) shows that this load imbalance does not significantly affect the final perplexity achieved. More generally, the time T p that each processor p takes to perform Gibbs sampling dictates the communication graph that will ensue. There exist pathological cases where the graph may be disconnected due to phase-locking (e.g. 5 processors with times T = {10, 12, 14, 19, 20} where P1, P2, P3 enter the network at time 0 and P4, P5 enter the network at time 34). However, the graph is guaranteed to be connected over time if Tp has a stochastic component (e.g. due to network delays), a reasonable assumption in practice. In our experiments, we assumed a fully connected network of processors and did not focus on other network topologies. After running Async-LDA on both a 10x10 fixed grid network and a 100 node chain network on KOS K=16, we have verified that Async-LDA achieves the same perplexity as LDA as long as caching and forwarding of cached counts occurs between processors. 6 Discussion and conclusions The work that is most closely related to that in this paper is that of Mimno and McCallum [3] and Newman et al. [5], who each propose parallel algorithms for the collapsed sampler for LDA. In other work, Nallapati et al. [4] parallelize the variational EM algorithm for LDA, and Wolfe et al. [8] examine asynchronous EM algorithms for LDA. The primary distinctions between our work and other work on distributed LDA based on Gibbs sampling are that (a) our algorithms use purely asynchronous communication rather than a global synchronous scheme, and (b) we have also extended these ideas (synchronous and asynchronous) to HDP. More generally, exact parallel Gibbs sampling is difficult to perform due to the sequential nature of MCMC. Brockwell [9] presents a prefetching parallel algorithm for MCMC, but this technique is not applicable to the collapsed sampler for LDA. There is also a large body of prior work on gossip algorithms (e.g., [6]), such as Newscast EM, a gossip algorithm for performing EM on Gaussian mixture learning [10]. Although processors perform local Gibbs sampling based on inexact global counts, our algorithms nonetheless produce solutions that are nearly the same as that of standard single-processor samplers. Providing a theoretical justification for these distributed algorithms is still an open area of research. We have proposed a new set of algorithms for distributed learning of LDA and HDP models. Our perplexity and speedup results suggest that topic models can be learned in a scalable asynchronous fashion for a wide variety of situations. One can imagine our algorithms being performed by a large network of idle processors, in an effort to mine the terabytes of information available on the Internet. Acknowledgments This material is based upon work supported in part by NSF under Award IIS-0083489 (PS, AA), IIS0447903 and IIS-0535278 (MW), and an NSF graduate fellowship (AA). MW was also supported by ONR under Grant 00014-06-1-073, and PS was also supported by a Google Research Award. References [1] D. Blei, A. Ng, and M. Jordan. Latent Dirichlet allocation. JMLR, 3:993–1022, 2003. [2] Y. Teh, M. Jordan, M. Beal, and D. Blei. Hierarchical Dirichlet processes. JASA, 101(476), 2006. [3] D. Mimno and A. McCallum. Organizing the OCA: learning faceted subjects from a library of digital books. In JCDL ’07, pages 376–385, New York, NY, USA, 2007. ACM. [4] R. Nallapati, W. Cohen, and J. Lafferty. Parallelized variational EM for latent Dirichlet allocation: An experimental evaluation of speed and scalability. In ICDM Workshop On High Perf. Data Mining, 2007. [5] D. Newman, A. Asuncion, P. Smyth, and M. Welling. Distributed inference for latent Dirichlet allocation. In NIPS 20. MIT Press, Cambridge, MA, 2008. [6] S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah. Gossip algorithms: design, analysis and applications. In INFOCOM, pages 1653–1664, 2005. [7] T. L. Griffiths and M. Steyvers. Finding scientific topics. PNAS, 101 Suppl 1:5228–5235, April 2004. [8] J. Wolfe, A. Haghighi, and D. Klein. Fully distributed EM for very large datasets. In ICML ’08, pages 1184–1191, New York, NY, USA, 2008. ACM. [9] A. Brockwell. Parallel Markov chain Monte Carlo simulation by pre-fetching. JCGS, 15, No. 1, 2006. [10] W. Kowalczyk and N. Vlassis. Newscast EM. In NIPS 17. MIT Press, Cambridge, MA, 2005. 8
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Exact Convex Confidence-Weighted Learning Koby Crammer Mark Dredze Fernando Pereira∗ Department of Computer and Information Science , University of Pennsylvania Philadelphia, PA 19104 {crammer,mdredze,pereira}@cis.upenn.edu Abstract Confidence-weighted (CW) learning [6], an online learning method for linear classifiers, maintains a Gaussian distributions over weight vectors, with a covariance matrix that represents uncertainty about weights and correlations. Confidence constraints ensure that a weight vector drawn from the hypothesis distribution correctly classifies examples with a specified probability. Within this framework, we derive a new convex form of the constraint and analyze it in the mistake bound model. Empirical evaluation with both synthetic and text data shows our version of CW learning achieves lower cumulative and out-of-sample errors than commonly used first-order and second-order online methods. 1 Introduction Online learning methods for linear classifiers, such as the perceptron and passive-aggressive (PA) algorithms [4], have been thoroughly analyzed and are widely used. However, these methods do not model the strength of evidence for different weights arising from differences in the use of features in the data, which can be a serious issue in text classification, where weights of rare features should be trusted less than weights of frequent features. Confidence-weighted (CW) learning [6], motivated by PA learning, explicitly models classifier weight uncertainty with a full multivariate Gaussian distribution over weight vectors. The PA geometrical margin constraint is replaced by the probabilistic constraint that a classifier drawn from the distribution should, with high probability, classify correctly the next example. While Dredze et al. [6] explained CW learning in terms of the standard deviation of the margin induced by the hypothesis Gaussian, in practice they used the margin variance to make the problem convex. In this work, we use their original constraint but maintain convexity, yielding experimental improvements. Our primary contributions are a mistake-bound analysis [11] and comparison with related methods. We emphasize that this work focuses on the question of uncertainty about feature weights, not on confidence in predictions. In large-margin classification, the margin’s magnitude for an instance is sometimes taken as a proxy for prediction confidence for that instance, but that quantity is not calibrated nor is it connected precisely to a measure of weight uncertainty. Bayesian approaches to linear classification, such as Bayesian logistic regression [9], use a simple mathematical relationship between weight uncertainty and prediction uncertainty, which unfortunately cannot be computed exactly. CW learning preserves the convenient computational properties of PA algorithms while providing a precise connection between weight uncertainty and prediction confidence that has led to weight updates that are more effective in practice [6, 5]. We begin with a review of the CW approach, then show that the constraint can be expressed in a convex form, and solve it to obtain a new CW algorithm. We also examine a dual representation that supports kernelization. Our analysis provides a mistake bound and indicates that the algorithm is invariant to initialization. Simulations show that our algorithm improves over first-order methods ∗Current affiliation: Google, Mountain View, CA 94043, USA. 1 (perceptron and PA) as well as other second order methods (second-order perceptron). We conclude with a review of related work. 2 Confidence-Weighted Linear Classification The CW binary-classifier learner works in rounds. On round i, the algorithm applies its current linear classification rule hw(x) = sign(w · x) to an instance xi ∈Rd to produce a prediction ˆyi ∈{−1, +1}, receives a true label yi ∈{−1, +1} and suffers a loss ℓ(yi, ˆyi). The rule hw can be identified with w up to a scaling, and we will do so in what follows since our algorithm will turn out to be scale-invariant. As usual, we define the margin of an example on round i as mi = yi(wi · xi), where positive sign corresponds to a correct prediction. CW classification captures the notion of confidence in the weights of a linear classifier with a probability density on classifier weight vectors, specifically a Gaussian distribution with mean µ ∈Rd and covariance matrix Σ ∈Rd×d. The values µp and Σp,p represent knowledge of and confidence in the weight for feature p. The smaller Σp,p, the more confidence we have in the mean weight value µp. Each covariance term Σp,q captures our knowledge of the interaction between features p and q. In the CW model, the traditional signed margin is the mean of the induced univariate Gaussian random variable M ∼N ! y(µ · x), x⊤Σx " . (1) This probabilistic model can be used for prediction in different ways. Here, we use the average weight vector E [w] = µ, analogous to Bayes point machines [8]. The information captured by the covariance Σ is then used just to adjust training updates. 3 Update Rule The CW update rule of Dredze et al. [6] makes the smallest adjustment to the distribution that ensures the probability of correct prediction on instance i is no smaller than the confidence hyperparameter η ∈[0, 1]: Pr [yi (w · xi) ≥0] ≥η. The magnitude of the update is measured by its KL divergence to the previous distribution, yielding the following constrained optimization: (µi+1, Σi+1) = arg min µ,Σ DKL (N (µ, Σ) ∥N (µi, Σi)) s.t. Pr [yi (w · xi) ≥0] ≥η . (2) They rewrite the above optimization in terms of the standard deviation as: min 1 2 # log $detΣ i detΣ % + Tr ! Σ−1 i Σ " + (µi −µ)⊤Σ−1 i (µi −µ) & s.t. yi(µ · xi) ≥φ ' x⊤ i Σxi . (3) Unfortunately, while the constraint of this problem is linear in µ, it is not convex in Σ. Dredze et al. [6, eq. (7)] circumvented that lack of convexity by removing the square root from the right-hand-size of the constraint, which yields the variance. However, we found that the original optimization can be preserved while maintaining convexity with a change of variable. Since Σ is positive semidefinite (PSD), it can be written as Σ =Υ 2 with Υ = Qdiag(λ1/2 1 , . . . ,λ 1/2 d )Q⊤ where Q is orthonormal and λ1, . . . ,λ d are the eigenvalues of Σ; Υ is thus also PSD. This change yields the following convex optimization with a convex constraint in µ and Υ simultaneously: (µi+1, Υi+1) = arg min 1 2 log $detΥ 2 i detΥ 2 % + 1 2Tr ! Υ−2 i Υ2" + 1 2 (µi −µ)⊤Υ−2 i (µi −µ) s.t. yi (µ · xi) ≥φ∥Υxi∥ , Υ is PSD . (4) We call our algorithm CW-Stdev and the original algorithm of Dredze et al. CW-Var. 3.1 Closed-Form Update While standard optimization techniques can solve the convex program (4), we favor a closed-form solution. Omitting the PSD constraint for now, we obtain the Lagrangian for (4), L = 1 2 ( log $detΥ 2 i detΥ 2 % + Tr ! Υ−2 i Υ2" + (µi −µ)⊤Υ−2 i (µi −µ) ) +α (−yi (µ · xi) + φ∥Υxi∥) (5) 2 Input parameters a > 0 ; η ∈[0.5, 1] Initialize µ1 = 0 , Σ1 = aI ,φ = Φ−1(η) , ψ = 1 + φ2/2 , ξ = 1 + φ2 . For i = 1, . . . , n • Receive a training example xi ∈Rd • Compute Gaussian margin distribution Mi ∼N ` (µi · xi) , ` x⊤ i Σixi ´ ´ • Receive true label yi and compute vi = x⊤ i Σixi , mi = yi (µi · xi) (11) , ui = 1 4 „ −αviφ + q α2v2 i φ2 + 4vi «2 (12) αi = max ( 0, 1 viξ −miψ + r m2 i φ4 4 + viφ2ξ !) (14) , βi = αiφ √ui + viαiφ (22) • Update µi+1 = µi + αiyiΣixi Σi+1 = Σi −βiΣixix⊤ i Σi (full) (10) Σi+1 = „ Σ−1 i + αiφu −1 2 i diag2 (xi) «−1 (diag) (15) Output Gaussian distribution N ` µn+1, Σn+1 ´ . Figure 1: The CW-Stdev algorithm. The numbers in parentheses refer to equations in the text. At the optimum, it must be that ∂ ∂µL = Υ−2 i (µ −µi) −αyixi = 0 ⇒ µi+1 = µi + αyiΥ2 i xi , (6) where we assumed that Υi is non-singular (PSD). At the optimum, we must also have, ∂ ∂ΥL = −Υ−1 + 1 2Υ−2 i Υ + 1 2ΥΥ−2 i + αφ xix⊤ i Υ 2 * x⊤ i Υ2xi + αφ Υxix⊤ i 2 * x⊤ i Υ2xi = 0 , (7) from which we obtain the implicit-form update Υ−2 i+1 = Υ−2 i + αφ xix⊤ i ' x⊤ i Υ2 i+1xi . (8) Conveniently, these updates can be expressed in terms of the covariance matrix 1 : µi+1 = µi + αyiΣixi , Σ−1 i+1 = Σ−1 i + αφ xix⊤ i * x⊤ i Σi+1xi . (9) We observe that (9) computes Σ−1 i+1 as the sum of a rank-one PSD matrix and Σ−1 i . Thus, if Σ−1 i has strictly positive eigenvalues, so do Σ−1 i+1 and Σi+1. Thus, Σi and Υi are indeed PSD non-singular, as assumed above. 3.2 Solving for the Lagrange Multiplier α We now determine the value of the Lagrange multiplier α and make the covariance update explicit. We start by computing the inverse of (9) using the Woodbury identity [14, Eq. 135] to get Σi+1 = + Σ−1 i + αφ xix⊤ i * x⊤ i Σi+1xi ,−1 =Σi −Σixi + αφ * x⊤ i Σi+1xi + x⊤ i Σixiαφ , x⊤ i Σi . (10) Let ui = x⊤ i Σi+1xi , vi = x⊤ i Σixi , mi = yi (µi · xi) . (11) 1Furthermore, writing the Lagrangian of (3) and solving it would yield the same solution as Eqns. (9). Thus the optimal solution of both (3) and (4) are the same. 3 Multiplying (10) by x⊤ i (left) and xi (right) we get ui = vi −vi αφ √ui+viαφ . vi , which can be solved for ui to obtain √ui = −αviφ + * α2v2 i φ2 + 4vi 2 . (12) The KKT conditions for the optimization imply that either α = 0 and no update is needed, or the constraint (4) is an equality after the update. Using the equality version of (4) and Eqs. (9,10,11,12) we obtain mi + αvi = φ −αviφ+√ α2v2 i φ2+4vi 2 , which can be rearranged into a quadratic equation in α: α2v2 i ! 1 + φ2" + 2αmivi 1 + φ2 2 . + ! m2 i −viφ2" = 0 . The smaller root of this equation is always negative and thus not a valid Lagrange multiplier. We use the following abbreviations for writing the larger root γi: ψ = 1 + φ2/2 ; ξ = 1 + φ2 . The larger root is then γi = −miviψ + * m2 i v2 i ψ2 −v2 i ψ (m2 i −viφ2) v2 i ψ . (13) The constraint (4) is satisfied before the update if mi −φ√vi ≥0. If mi ≤0, then mi ≤φ√vi and from (13) we have that γi > 0. If, instead, mi ≥0, then, again by (13), we have γi > 0 ⇔miviψ < ' m2 i v2 i ψ2 −v2 i ψ (m2 i −viφ2) ⇔mi <φv i . From the KKT conditions, either αi = 0 or (3) is satisfied as an equality and αi = γi > 0. We summarize the discussion in the following lemma: Lemma 1 The solution of (13) satisfies the KKT conditions, that is either αi ≥0 or the constraint of (3) is satisfied before the update with the parameters µi and Σi. We obtain the final form of αi by simplifying (13) together with Lemma 1, max 0, 1 vi −miψ + ' m2 i φ4 4 + viφ2ξ ξ . (14) To summarize, after receiving the correct label yi the algorithm checks whether the probability of a correct prediction under the current parameters is greater than a confidence threshold η =Φ( φ). If so, it does nothing. Otherwise it performs an update as described above. We initialize µ1 = 0 and Σ1 = aI for some a > 0. The algorithm is summarized in Fig. 1. Two comments are in order. First, if η = 0.5, then from Eq. (9) we see that only µ will be updated, not Σ, because φ = 0 ⇔η = 0.5. In this case the covariance Σ parameter does not influence the decision, only the mean µ. Furthermore, for length-one input vectors, at the first round we have Σ1 = aI, so the first-round constraint is yi (wi · xi) ≥a ∥xi∥2 = a, which is equivalent to the original PA update. Second, the update described above yields full covariance matrices. However, sometimes we may prefer diagonal covariance matrices, which can be achieved by projecting the matrix Σi+1 that results from the update onto the set of diagonal matrices. In practice it requires setting all the off-diagonal elements to zero, leaving only the diagonal elements. In fact, if Σi is diagonal then we only need to project xix⊤ i to a diagonal matrix. We thus replace (9) with the following update, Σ−1 i+1 = Σ−1 i + φ αi √ui diag2 (xi) , (15) where diag2 (xi) is a diagonal matrix made from the squares of the elements of xi on the diagonal. Note that for diagonal matrices there is no need to use the Woodbury equation to compute the inverse, as it can be computed directly element-wise. We use CW-Stdev (or CW-Stdev-full) to refer to the full-covariance algorithm, and CW-Stdev-diag to refer to the diagonal-covariance algorithm. Finally, the following property of our algorithm shows that it can be used with Mercer kernels: 4 Theorem 2 (Representer Theorem) The mean µi and covariance Σi parameters computed by the algorithm in Fig. 1 can be written as linear combinations of the input vectors with coefficients that depend only on inner products of input vectors: Σi = i−1 5 p,q=1 π(i) p,qxpx⊤ q + aI , µi = i−1 5 p ν(i) p xp . (16) The proof, given in the appendix, is a simple induction. 4 Analysis We analyze CW-Stdev in two steps. First, we show that performance does not depend on initialization and then we compute a bound on the number of mistakes that the algorithm makes. 4.1 Invariance to Initialization The algorithm in Fig. 1 uses a predefined parameter a to initialize the covariance matrix. Since the decision to update depends on the covariance matrix, which implicitly depends on a through αi and vi, one may assume that a effects performance. In fact the number of mistakes is independent of a, i.e. the constraint of (3) is invariant to scaling. Specifically, if it holds for mean and covariance parameters µ and Σ, it holds also for the scaled parameters cµ and c2Σ for any c > 0. The following lemma states that the scaling is controlled by a. Thus, we can always initialize the algorithm with a value of a = 1. If, in addition to predictions, we also need the distribution over weight vectors, the scale parameter a should be calibrated. Lemma 3 Fix a sequence of examples (x1, y1) . . . (xn, yn). Let Σi, µi, mi, vi, αi, ui be the quantities obtained throughout the execution of the algorithm described in Fig. 1 initialized with (0, I) (a = 1). Let also ˜Σi, ˜µi, ˜mi, ˜vi, ˜αi, ˜ui be the corresponding quantities obtained throughout the execution of the algorithm, with an alternative initialization of (0, aI) (for some a > 0). The following relations between the two set of quantities hold: ˜mi = √ami , ˜vi = avi , ˜αi = 1 √aαi , ˜µi = √aµi , ˜ui = aui , ˜Σi = aΣi . (17) Proof sketch: The proof proceeds by induction. The initial values of these quantities clearly satisfy the required equalities. For the induction step we assume that (17) holds for some i and show that these identities also hold for i + 1 using Eqs. (9,14,11,12) . From the lemma we see that the quantity ˜mi/√˜vi = mi/√vi is invariant to a. Therefore, the behavior of the algorithm in general, and its updates and mistakes in particular, are independent to the choice of a. Therefore, we assume a = 1 in what follows. 4.2 Analysis in the Mistake Bound Model The main theorem of the paper bounds the number of mistakes made by CW-Stdev. Theorem 4 Let (x1, y1) . . . (xn, yn) be an input sequence for the algorithm of Fig. 1, initialized with (0, I), with xi ∈Rd and yi ∈{−1, +1} . Assume there exist µ∗and Σ∗such that for all i for which the algorithm made an update (αi > 0), µ∗⊤xiyi ≥µ⊤ i+1xiyi and x⊤ i Σ∗xi ≤x⊤ i Σi+1xi . (18) Then the following holds: no. mistakes ≤ 5 i α2 i vi ≤1 + φ2 φ2 −log detΣ ∗+ Tr (Σ∗) + µ∗⊤Σ−1 n+1µ∗−d . (19) 5 100 200 300 400 500 600 700 800 900 1000 20 40 60 80 100 120 140 160 180 200 Round Cumulative Loss Perceptron PA 2nd Ord Std−diag Std−full Var−diag Var−full Perceptron PA 2nd OrderStd−diag Std−full Var−diag Var−full 0 1 2 3 4 5 6 7 8 9 Test Error 0.80 0.85 0.90 0.95 1.00 Variance Accuracy 0.80 0.85 0.90 0.95 1.00 Stdev Accuracy Reuters Sentiment 20 Newsgroups (a) (b) (c) Figure 2: (a) The average and standard deviation of the cumulative number of mistakes for seven algorithms. (b) The average and standard deviation of test error (%) over unseen data for the seven algorithms. (c) Comparison between CW-Stdev-diag and CW-Var-diag on text classification. The proof is given in the appendix. The above bound depends on an output of the algorithm, Σn+1, similar to the bound for the secondorder perceptron [3]. The two conditions (18) imply linear separability of the input sequence by µ∗: µ∗⊤xiyi (18) ≥µ⊤ i+1xiyi (4) ≥φ ' x⊤ i Σi+1xi (18) ≥x⊤ i Σ∗xi ≥min i x⊤ i Σ∗xi > 0 , where the superscripts in parentheses refer to the inequalities used. From (10), we observe that Σi+1 ⪯Σi for all i, so Σn+1 ⪯Σi+1 ⪯Σ1 = I for all i. Therefore, the conditions on Σ∗in (18) are satisfied by Σ∗= Σn+1. Furthermore, if µ∗satisfies the stronger conditions yi(µ∗·xi) ≥∥xi∥, from Σi+1 ⪯I above it follows that (φµ∗)⊤xiyi ≥φ∥xi∥= φ ' x⊤ i Ixi ≥φ ' x⊤ i Σi+1xi = µ⊤ i+1xiyi , where the last equality holds since we assumed that an update was made for the ith example. In this situation, the bound becomes φ2 + 1 φ2 (−log detΣ n+1 + Tr (Σn+1) −d) + (φ2 + 1) µ∗⊤Σ−1 n+1µ∗. . The quantity µ∗⊤Σ−1 n+1µ∗in this bound is analogous to the quantity R2 ∥µ∗∥2 in the perceptron bound [13], except that the norm of the examples does not come in explicitly as the radius R of the enclosing ball, but implicitly through the fact that Σ−1 n+1 is a sum of example outer products (9). In addition, in this version of the bound we impose a margin of 1 under the condition that examples have unit norm, whereas in the perceptron bound, the margin of 1 is for examples with arbitrary norm. This follows from the fact that (4) is invariant to the norm of xi. 5 Empirical Evaluation We illustrate the benefits of CW-Stdev with synthetic data experiments. We generated 1, 000 points in R20 where the first two coordinates were drawn from a 45◦rotated Gaussian distribution with standard deviation 1. The remaining 18 coordinates were drawn from independent Gaussian distributions N (0, 2). Each point’s label depended on the first two coordinates using a separator parallel to the long axis of the ellipsoid, yielding a linearly separable set (Fig. 3(top)). We evaluated five online learning algorithms: the perceptron [16] , the passive-aggressive (PA) algorithm [4], the secondorder perceptron (SOP) [3], CW-Var-diag, CW-Var-full [6], CW-Stdev-diag and CW-Stdev-full. All algorithm parameters were tuned over 1, 000 runs. Fig. 2(a) shows the average cumulative mistakes for each algorithm; error bars indicate one unit of standard deviation. Clearly, second-order algorithms, which all made fewer than 80 mistakes, outperform the first-order ones, which made at least 129 mistakes. Additionally, CW-Var makes more mistakes than CW-Stdev: 8% more in the diagonal case and 17% more in the full. The diagonal methods performed better than the first order methods, indicating that while they do not use any 6 second-order information, they capture additional information for single features. For each repetition, we evaluated the resulting classifiers on 10, 000 unseen test examples (Fig. 2(b)). Averaging improved the first-order methods. The second-order methods outperform the first-order methods, and CW-Stdev outperforms all the other methods. Also, the full case is less sensitive across runs. The Gaussian distribution over weight vectors after 50 rounds is represented in Fig. 3(bot). The 20 dimensions of the version space are grouped into 10 pairs, the first containing the two meaningful features. The dotted segment represents the first two coordinates of possible representations of the true hyperplane in the positive quadrant. Clearly, the corresponding vectors are orthogonal to the hyperplane shown in Fig. 3(top). The solid black ellipsoid represents the first two significant feature weights; it does not yet lie of the dotted segment because the algorithm has not converged. Nevertheless, the long axis is already parallel to the true set of possible weight vectors. The axis perpendicular to the weight-vector set is very small, showing that there is little freedom in that direction. The remaining nine ellipsoids represent the covariance of pairs of noise features. Those ellipsoids are close to circular and have centers close to the origin, indicating that the corresponding feature weights should be near zero but without much confidence. NLP Evaluation: We compared CW-Stdev-diag with CW-Var-diag, which beat many state of the art algorithms on 12 NLP datasets [6]. We followed the same evaluation setting using 10-fold cross validation and the same splits for both algorithms. Fig. 2(c) compares the accuracy on test data of each algorithm; points above the line represent improvements of CW-Stdev over CW-Var. Stdev improved on eight of the twelve datasets and, while the improvements are not significant, they show the effectiveness of our algorithm on real world data. 6 Related Work −25 −20 −15 −10 −5 0 5 10 15 20 25 −30 −20 −10 0 10 20 30 −1 −0.5 0 0.5 1 1.5 2 2.5 3 −0.5 0 0.5 1 1.5 2 2.5 Figure 3: Top : Plot of the two informative features of the synthetic data. Bottom: Feature weight distributions of CW-Stdev-full after 50 examples. Online additive algorithms have a long history, from with the perceptron [16] to more recent methods [10, 4]. Our update has a more general form, in which the input vector xi is linearly transformed using the covariance matrix, both rotating the input and assigning weight specific learning rates. Weightspecific learning rates appear in neural-network learning [18], although they do not model confidence based on feature variance. The second order perceptron (SOP) [3] demonstrated that second-order information can improve on first-order methods. Both SOP and CW maintain second-order information. SOP is mistake driven while CW is passive-aggressive. SOP uses the current instance in the correlation matrix for prediction while CW updates after prediction. A variant of CW-Stdev similar to SOP follows from our derivation if we fix the Lagrange multiplier in (5) to a predefined value αi = α, omit the square root, and use a gradient-descent optimization step. Fundamentally, CW algorithms have a probabilistic motivation, while the SOP is geometric: replace the ball around an example with a refined ellipsoid. Shivaswamy and Jebara [17] used a similar motivation in batch learning. Ensemble learning shares the idea of combining multiple classifiers. Gaussian process classification (GPC) maintains a Gaussian distribution over weight vectors (primal) or over regressor values (dual). Our algorithm uses a different update criterion than the standard GPC Bayesian updates [15, Ch.3], avoiding the challenge of approximating posteriors. Bayes point machines [8] maintain a collection of weight vectors consistent with the training data, and use the single linear classifier which best represents the collection. Conceptually, the collection is a non-parametric distribution over the weight vectors. Its online version [7] maintains a finite number of weight-vectors which are updated simultaneously. The rele7 vance vector machine [19] incorporates probabilistic models into the dual formulation of SVMs. As in our work, the dual parameters are random variables distributed according to a diagonal Gaussian with example specific variance. The weighted-majority [12] algorithm and later improvements [2] combine the output of multiple arbitrary classifiers, maintaining a multinomial distribution over the experts. We assume linear classifiers as experts and maintain a Gaussian distribution over their weight vectors. 7 Conclusion We presented a new confidence-weighted learning method for linear classifier based on the standard deviation. We have shown that the algorithm is invariant to scaling and we provided a mistake-bound analysis. Based on both synthetic and NLP experiments, we have shown that our method improves upon recent first and second order methods. Our method also improves on previous CW algorithms. We are now investigating special cases of CW-Stdev for problems with very large numbers of features, multi-class classification, and batch training. References [1] Y. Censor and S.A. Zenios. Parallel Optimization: Theory, Algorithms, and Applications. Oxford University Press, New York, NY, USA, 1997. [2] N. Cesa-Bianchi, Y. Freund, D. Haussler, D. P. Helmbold, R. E. Schapire, and M. K. Warmuth. How to use expert advice. Journal of the Association for Computing Machinery, 44(3):427–485, May 1997. [3] Nicol´o Cesa-Bianchi, Alex Conconi, and Claudio Gentile. A second-order perceptron algorithm. Siam Journal of Commutation, 34(3):640–668, 2005. [4] K. Crammer, O. Dekel, J. Keshet, S. Shalev-Shwartz, and Y. Singer. Online passive-aggressive algorithms. Journal of Machine Learning Research, 7:551–585, 2006. [5] Mark Dredze and Koby Crammer. Active learning with confidence. In ACL, 2008. [6] Mark Dredze, Koby Crammer, and Fernando Pereira. Confidence-weighted linear classification. In International Conference on Machine Learning, 2008. [7] E. Harrington, R. Herbrich, J. Kivinen, J. Platt, and R.C. Williamson. Online bayes point machines. In 7th Pacific-Asia Conference on Knowledge Discovery and Data Mining (PAKDD), 2003. [8] R. Herbrich, T. Graepel, and C. Campbell. Bayes point machines. JMLR, 1:245–279, 2001. [9] T. Jaakkola and M. Jordan. A variational approach to bayesian logistic regression models and their extensions. In Workshop on Artificial Intelligence and Statistics, 1997. [10] J. Kivinen and M. K. Warmuth. Exponentiated gradient versus gradient descent for linear predictors. Information and Computation, 132(1):1–64, January 1997. [11] N. Littlestone. Learning when irrelevant attributes abound: A new linear-threshold algorithm. Machine Learning, 2:285–318, 1988. [12] N. Littlestone and M. K. Warmuth. The weighted majority algorithm. Information and Computation, 108:212–261, 1994. [13] A. B. J. Novikoff. On convergence proofs on perceptrons. In Proceedings of the Symposium on the Mathematical Theory of Automata, volume XII, pages 615–622, 1962. [14] K. B. Petersen and M. S. Pedersen. The matrix cookbook, 2007. [15] C. E. Rasmussen and C. K. I. Williams. Gaussian Processes for Machine Learning. The MIT Press, 2006. [16] F. Rosenblatt. The perceptron: A probabilistic model for information storage and organization in the brain. Psychological Review, 65:386–407, 1958. (Reprinted in Neurocomputing (MIT Press, 1988).). [17] P. Shivaswamy and T. Jebara. Ellipsoidal kernel machines. In AISTATS, 2007. [18] Richard S. Sutton. Adapting bias by gradient descent: an incremental version of delta-bar-delta. In Proceedings of the Tenth National Conference on Artificial Intelligence, pages 171–176. MIT Press, 1992. [19] M. E. Tipping. Sparse bayesian learning and the relevance vector machine. Journal of Machine Learning Research, 1:211–244, 2001. [20] L. Xu, K. Crammer, and D. Schuurmans. Robust support vector machine training via convex outlier ablation. In AAAI-2006, 2006. 8
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The Conjoint Effect of Divisive Normalization and Orientation Selectivity on Redundancy Reduction in Natural Images Fabian Sinz MPI for Biological Cybernetics 72076 T¨ubingen, Germany fabee@tuebingen.mpg.de Matthias Bethge MPI for Biological Cybernetics 72076 T¨ubingen, Germany mbethge@tuebingen.mpg.de Abstract Bandpass filtering, orientation selectivity, and contrast gain control are prominent features of sensory coding at the level of V1 simple cells. While the effect of bandpass filtering and orientation selectivity can be assessed within a linear model, contrast gain control is an inherently nonlinear computation. Here we employ the class of Lp elliptically contoured distributions to investigate the extent to which the two features—orientation selectivity and contrast gain control—are suited to model the statistics of natural images. Within this framework we find that contrast gain control can play a significant role for the removal of redundancies in natural images. Orientation selectivity, in contrast, has only a very limited potential for redundancy reduction. 1 Introduction It is a long standing hypothesis that sensory systems are adapted to the statistics of their inputs. These natural signals are by no means random, but exhibit plenty of regularities. Motivated by information theoretic principles, Attneave and Barlow suggested that one important purpose of this adaptation in sensory coding is to model and reduce the redundancies [4; 3] by transforming the signal into a statistically independent representation. The problem of redundancy reduction can be split into two parts: (i) finding a good statistical model of the natural signals and (ii) a way to map them into a factorial representation. The first part is relevant not only to the study of biological systems, but also to technical applications such as compression and denoising. The second part offers a way to link neural response properties to computational principles, since neural representations of natural signals must be advantageous in terms of redundancy reduction if the hypothesis were true. Both aspects have been extensively studied for natural images [2; 5; 8; 19; 20; 21; 24]. In particular, it has been shown that applying Independent Component Analysis (ICA) to natural images consistently and robustly yields filters that are localized, oriented and show bandpass characteristics [19; 5]. Since those features are also ascribed to the receptive fields of neurons in the primary visual cortex (V1), it has been suggested that the receptive fields of V1 neurons are shaped to form a minimally redundant representation of natural images [5; 19]. From a redundancy reduction point of view, ICA offers a small but significant advantage over other linear representations [6]. In terms of density estimation, however, it is a poor model for natural images since already a simple non-factorial spherically symmetric model yields a much better fit to the data [10]. Recently, Lyu and Simoncelli proposed a method that converts any spherically symmetric distribution into a (factorial) Gaussian (or Normal distribution) by using a non-linear transformation of the 1 norm of the image patches [17]. This yields a non-linear redundancy reduction mechanism, which exploits the superiority of the spherically symmetric model over ICA. Interestingly, the non-linearity of this Radial Gaussianization method closely resembles another feature of the early visual system, known as contrast gain control [13] or divisive normalization [20]. However, since spherically symmetric models are invariant under orthogonal transformations, they are agnostic to the particular choice of basis in the whitened space. Thus, there is no role for the shape of the filters in this model. Combining the observations from the two models of natural images, we can draw two conclusions: On the one hand, ICA is not a good model for natural images, because a simple spherically symmetric model yields a much better fit [10]. On the other hand, the spherically symmetric model in Radial Gaussianization cannot capture that ICA filters do yield a higher redundancy reduction than other linear transformations. This leaves us with the questions whether we can understand the emergence of oriented filters in a more general redundancy reduction framework, which also includes a mechanism for contrast gain control. In this work we address this question by using the more general class of Lp-spherically symmetric models [23; 12; 15]. These models are quite similar to spherically symmetric models, but do depend on the particular shape of the linear filters. Just like spherically symmetric models can be nonlinearly transformed into isotropic Gaussians, Lp-spherically symmetric models can be mapped into a unique class of factorial distributions, called p-generalized Normal distributions [11]. Thus, we are able to quantify the influence of orientation selective filters and contrast gain control on the redundancy reduction of natural images in a joint model. 2 Models and Methods 2.1 Decorrelation and Filters All probabilistic models in this paper are defined on whitened natural images. Let C be the covariance matrix of the pixel intensities for an ensemble x1, ..., xm of image patches, then C−1 2 constitutes the symmetric whitening transform. Note that all vectors y = V C−1 2 x, with V being an orthogonal matrix, have unit covariance. V C−1 2 yield the linear filters that are applied to the raw image patches before feeding them in the probabilistic models described below. Since any decorrelation transform can be written as V C−1 2 , the choice of V determines the shape of the linear filters. In our experiments, we use three different kinds of V : SYM The simplest choice is VSYM = I, i. e. y = C−1 2 x contains the coefficients in the symmetric whitening basis. From a biological perspective, this case is interesting as the filters resemble receptive fields of retinal ganglion cells with center-surround properties. ICA The filters VICA of ICA are determined by maximizing the non-Gaussanity of the marginal distributions. For natural image patches, ICA is known to yield orientation selective filters in resemblance to V1 simple cells. While other orientation selective bases are possible, the filters defined by VICA correspond to the optimal choice for redundancy reduction under the restriction to linear models. HAD The coefficients in the basis VHAD = 1 √mHVICA, with H denoting an arbitrary Hadamard matrix, correspond to a sum over the different ICA coefficients, each possibly having a flipped sign. Hadamard matrices are defined by the two properties Hij = ±1 and HH⊤= mI. This case can be seen as the opposite extreme to the case of ICA. Instead of running an independent search for the most Gaussian marginals, the central limit theorem is used to produce the most Gaussian components by using the Hadamard transformation to mix all ICA coefficients with equal weight resorting to the independence assumption underlying ICA. 2.2 Lp-spherically Symmetric Distributions The contour lines of spherically symmetric distributions have constant Euclidean norm. Similarly, the contour lines of Lp-spherically symmetric distributions have constant p-norm1 ||y||p := 1Note that ||y||p is only a norm in the strict sense if p ≥1. However, since the following considerations also hold for 0 < p < 1, we will employ the term “p-norm” and the notation “||y||p” for notational convenience. 2 ppPn i=1 |yi|p The set of vectors with constant p-norm Sn−1 p (r) := {y ∈Rn : ||y||p = r, p > 0, r > 0} is called p-sphere of radius r. Different examples of p-spheres are shown along the coordinate axis of Figure 1. For p ̸= 2 the distribution is not invariant under arbitrary orthogonal transformations, which means that the choice of the basis V can make a difference in the likelihood of the data. Factorial Distributions Lp Spherically Symmetric Distributions Normal Distribution p ICA cICA SYM cSYM HAD cHAD p-generalized Normal Distributions p=2: Spherically Symmetric Distributions Figure 1: The spherically symmetric distributions are a subset of the Lp-spherical symmetric distributions. The right shapes indicate the iso-density lines for the different distributions. The Gaussian is the only L2-spherically symmetric distribution with independent marginals. Like the Gaussian distribution, all p-generalized Normal distributions have independent marginals. ICA, SYM, ... denote the models used in the experiments below. A multivariate random variable Y is called Lp-spherically symmetric distributed if it can be written as a product Y = RU, where U is uniformly distributed on Sn−1 p (1) and R is a univariate nonnegative random variable with an arbitrary distribution [23; 12]. Intuitively, R corresponds to the radial component, i. e. the length ||y||p measured with the p-norm. U describes the directional components in a polar-like coordinate system (see Extra Material). It can be shown that this definition is equivalent to the density ϱ(y) of Y having the form ϱ(y) = f(||y||p p) [12]. This immediately suggests two ways of constructing an Lp-spherically symmetric distribution. Most obviously, one can specify a density ϱ(y) that has the form ϱ(y) = f(||y||p p). An example is the p-generalized Normal distribution (gN) [11] ϱ(y) = pn Γn 1 p (2σ2) n p 2n exp − Pn i=1 |yi|p 2σ2 = f(||y||p p). (1) Analogous to the Gaussian being the only factorial spherically symmetric distribution [1], this distribution is the only Lp-spherically symmetric distribution with independent marginals [22]. For the p-generalized Normal, the marginals are members of the exponential power family. In our experiments, we will use the p-generalized Normal to model linear marginal independence by fitting it to the coefficients of the various bases in whitened space. Since this distribution is sensitive to the particular filter shapes for p ̸= 2, we can assess how well the distribution of the linearly transformed image patches is matched by a factorial model. An alternative way of constructing an Lp-spherically symmetric distribution is to specify the radial distribution ϱr. One example, which will be used later, is obtained by choosing a mixture of LogNormal distributions (RMixLogN). In Cartesian coordinates, this yields the density ϱ(y) = pn−1Γ n p 2nΓn 1 p K X k=1 ηk ||y||npσk √ 2π exp −(log ||y||p −µk)2 2σ2 k . (2) 3 An immediate consequence of any Lp-spherically symmetric distribution being specified by its radial density is the possibility to change between any two of those distributions by transforming the radial component with (F−1 2 ◦F1)(||y||p), where F1 and F2 are cumulative distribution functions (cdf) of the source and the target density, respectively. In particular, for a fixed p, any Lp-spherically symmetric distribution can be transformed into a factorial one by the transform z = g(y) · y = (F−1 2 ◦F1)(||y||p) ||y||p y. This transform closely resembles contrast gain control models for primary visual cortex [13; 20], which use a different gain function having the form ˜g(y) = 1 c+r with r = ||y||2 2 [17]. We will use the distribution of equation (2) to describe the joint model consisting of a linear filtering step followed by a contrast gain control mechanism. Once, the linear filter responses in whitened space are fitted with this distribution, we non-linearly transform it into a the factorial p-generalized Normal by the transformation g(y) · y = (F−1 gN ◦FRMixLogN)(||y||p)/||y||p · y. Finally, note that because a Lp-spherically symmetric distribution is specified by its univariate radial distribution, fitting it to data boils down to estimating the univariate density for R, which can be done efficiently and robustly. 3 Experiments and Results 3.1 Dataset We use the dataset from the Bristol Hyperspectral Images Database [7], which was already used in previous studies [25; 16]. All images had a resolution of 256×256 pixels and were converted to gray level by averaging over the channels. From each image circa 5000 patches of size 15×15 pixels were drawn at random locations for training (circa 40000 patches in total) as well as circa 6250 patches per image for testing (circa 50000 patches in total). In total, we sampled ten pairs of training and test sets in that way. All results below are averaged over those. Before computing the linear filters, the DC component was projected out with an orthogonal transformation using a QR decomposition. Afterwards, the data was rescaled in order to make whitening a volume conserving transformation (a transformation with determinant one) since those transformations leave the entropy unchanged. 3.2 Evaluation Measure In all our experiments, we used the Average Log Loss (ALL) to assess the quality of the fit and the redundancy reduction achieved. The ALL = 1 nEϱ[−log2 ˆϱ(y)] ≈ 1 mn Pm k=1 −log2 ˆϱ(y) is the negative mean log-likelihood of the model distribution under the true distribution. If the model distribution matches the true one, the ALL equals the entropy. Otherwise, the difference between the ALL and the entropy of the true distribution is exactly the Kullback-Leiber divergence between the two. The difference between the ALLs of two models equals the reduction in multi-information (see Extra Material) and can therefore be used to quantify the amount of redundancy reduction. 3.3 Experiments We fitted the Lp-spherically symmetric distributions from equations (1) and (2) to the image patches in the bases HAD, SYM, and ICA by a maximum likelihood fit on the radial component. For the mixture of Log-Normal distributions, we used EM for a mixture of Gaussians on the logarithm of the p-norm of the image patches. For each model, we computed the maximum likelihood estimate of the model parameters and determined the best value for p according to the ALL in bits per component on a training set. The final ALL was computed on a separate test set. For ICA, we performed a gradient descent over the orthogonal group on the log-likelihood of a product of independent exponential power distributions, where we used the result of the FastICA algorithm by Hyv¨arinen et al. as initial starting point [14]. All transforms were computed separately for each training set. 4 HAD SYM ICA cHAD cSYM cICA Figure 2: ALL in bits per component as a function of p. The linewidth corresponds to the standard deviation over ten pairs of training and test sets. Left: ALL for the bases HAD, SYM and ICA under the p-generalized Normal (HAD, SYM, ICA) and the factorial Lp-spherically symmetric model with the radial component modeled by a mixture of Log-Normal distributions (cHAD, cSYM, cICA). Right: Bar plot for the different ALL indicated by horizontal lines in the left plot. In order to compare the redundancy reduction of the different transforms with respect to the pixel basis (PIX), we computed a non-parametric estimate of the marginal entropies of the patches before the DC component was projected out [6]. Since the estimation is not bound to a particular parametric model, we used the mean of the marginal entropies as an estimate of the average log-loss in the pixel representation. 3.4 Results Figure 2 and Table 1 show the ALL for the bases HAD, SYM, and ICA as a function of p. The upper curve bundle represents the factorial p-generalized Normal model, the lower bundle the nonfactorial model with the radial component modeled by a mixture of Log-Normal distributions with five mixtures. The ALL for the factorial models always exceeds the ALL for the non-factorial models. At p = 2, all curves intersect, because all models are invariant under a change of basis for that value. Note that the smaller ALL of the non-factorial model cannot be attributed to the mixture of Log-Normal distributions having more degrees of freedom. As mentioned in the introduction, the p-generalized Normal is the only factorial Lp-spherically symmetric distribution [22]. Therefore, marginal independence is such a rigid assumption that the output scale is the only degree of freedom left. From the left plot in Figure 2, we can assess the influence of the different filter shapes and contrast gain control on the redundancy reduction of natural images. We used the best ALL of the HAD basis under the p-generalized Normal as a baseline for a whitening transformation without contrast gain control (HAD). Analogously, we used the best ALL of the HAD basis under the non-factorial model as a baseline for a pure contrast gain control model (cHAD). We compared these values to the best ALL obtained by using the SYM and the ICA basis under both models. Because the filters of SYM and ICA resemble receptive field properties of retinal ganglion cells and V1 simple cells, respectively, we can assess their possible influence on the redundancy reduction with and without contrast gain control. The factorial model corresponds to the case without contrast gain control (SYM and ICA). Since we have shown that the non-factorial model can be transformed into a factorial one by a p-norm based divisive normalization operation, these scores correspond to the cases with contrast gain control (cSYM and cICA). The different cases are depicted by the horizontal lines in Figure 2. As already reported in other works, plain orientation selectivity adds only very little to the redundancy reduction achieved by decorrelation and is less effective than the baseline contrast gain control model [10; 6; 17]. If both orientation selectivity and contrast gain control are combined (cICA) it is possible to achieve about 9% extra redundancy reduction in addition to baseline whitening 5 Absolute Difference [Bits/Comp.] Relative Difference [% wrt. cICA] HAD - PIX −3.2947 ± 0.0018 91.0016 ± 0.0832 SYM- PIX −3.3638 ± 0.0022 92.9087 ± 0.0782 ICA - PIX −3.4110 ± 0.0024 94.2135 ± 0.0747 cHAD - PIX −3.5692 ± 0.0045 98.5839 ± 0.0134 cSYM - PIX −3.5945 ± 0.0047 99.2815 ± 0.0098 cICA - PIX −3.6205 ± 0.0049 100.0000 ± 0.0000 Table 1: Difference in ALL for gray value images with standard deviation over ten training and test set pairs. The column on the left displays the absolute difference to the PIX representation. The column on the right shows the relative difference with respect to the largest reduction achieved by ICA with non-factorial model. 10 −1 10 0 10 1 10 2 10 3 10 −1 10 0 10 1 10 2 HAD SYM ICA Figure 3: The curve in the upper right corner depicts the transformation ||z||p = (F−1 gN ◦ FRMixLogN)(||y||p) of the radial component in the ICA basis for gray scale images. The resulting radial distribution over ||z||p corresponds to the radial distribution of the p-generalized Normal. The inset shows the gain function g(||y||p) = FRMixLogN(||y||p) ||y||p in loglog coordinates. The scale parameter of the p-generalized normal was chosen such that the marginal had unit variance. (HAD). By setting the other models in relation to the best joint model (cICA:= 100%), we are able to tell apart the relative contributions of bandpass filtering (HAD= 91%), particular filter shapes (SYM= 93%, ICA= 94%), contrast gain control (cHAD= 98.6%) as well as combined models (cSYM= 99%, cICA := 100%) to redundancy reduction (see Table 1). Thus, orientation selectivity (ICA) contributes less to the overall redundancy reduction than any model with contrast gain control (cHAD, cSYM, cICA). Additionally, the relative difference between the joint model (cICA) and plain contrast gain control (cHAD) is only about 1.4%. For cSYM it is even less, about 0.7%. The difference in redundancy reduction between center-surround filters and orientation selective filters becomes even smaller in combination with contrast gain control (1.3% for ICA vs. SYM, 0.7% for cICA vs. cSYM). However, it is still significant (t-test, p = 5.5217 · 10−9). When examining the gain functions g(||y||p) = (F−1 gN ◦FRMixLogN)(||y||p) ||y||p resulting from the transformation of the radial components, we find that they approximately exhibit the form g(||y||p) = c ||y||κ p . The inset in Figure 3 shows the gain control function g(||y||p) in a log-log plot. While standard contrast gain control models assume p = 2 and κ = 2, we find that κ between 0.90 and 0.93 to be optimal for redundancy reduction. p depends on the shape of the linear filters and ranges from approximately 1.2 to 2. In addition, existing contrast gain models assume the form g(||y||2) = 1 σ+||y||2 2 , while we find that σ must be approximately zero. In the results above, the ICA filters always achieve the lowest ALL under both p-spherically symmetric models. For examining whether these filters really represent the best choice, we also optimized the filter shapes under the model of equation (2) via maximum likelihood estimation on the orthogonal group in whitened space [9; 18]. Figure 4 shows the filter shapes for ICA and the ones obtained from the optimization, where we used either the ICA solution or a random orthogonal matrix as starting point. Qualitatively, the filters look exactly the same. The ALL also changed just 6 Figure 4: Filters optimized for ICA (left) and for the p-spherically symmetric model with radial mixture of Log-Normal distributions starting from the ICA solution (middle) and from a random basis (right). The first filter corresponds to the DC component, the others to the filter shapes under the respective model. Qualitatively the filter shapes are very similar. The ALL for the ICA basis under the mixture of Log-Normal model is 1.6748±0.0058 bits/component (left), the ALL with the optimized filters is 1.6716 ± 0.0056 (middle) and 1.6841 ± 0.0068 (right). marginally from 1.6748 ± 0.0058 to 1.6716 ± 0.0056 or 1.6841 ± 0.0068, respectively. Thus, the ICA filters are a stable and optimal solution under the model with contrast gain control, too. 4 Summary In this report, we studied the conjoint effect of contrast gain control and orientation selectivity on redundancy reduction for natural images. In particular, we showed how the Lp-spherically distribution can be used to tune a nonlinearity of contrast gain control to remove higher-order redundancies in natural images. The idea of using an Lp-spherically symmetric model for natural images has already been brought up by Hyv¨arinen and K¨oster in the context of Independent Subspace Analysis [15]. However, they do not use the Lp-distribution for contrast gain control, but apply a global contrast gain control filter on the images before fitting their model. They also use a less flexible Lp-distribution since their goal is to fit an ISA model to natural images and not to carry out a quantitative comparison as we did. In our work, we find that the gain control function turns out to follow a power law, which parallels the classical model of contrast gain control. In addition, we find that edge filters also emerge in the non-linear model which includes contrast gain control. The relevance of orientation selectivity for redundancy reduction, however, is further reduced. In the linear framework (possibly endowed with a point-wise nonlinearity for each neuron) the contribution of orientation selectivity to redundancy reduction has been shown to be smaller than 5% relative to whitening (i. e. bandpass filtering) alone [6; 10]. Here, we found that the contribution of orientation selectivity is even smaller than two percent relative to whitening plus gain control. Thus, this quantitative model comparison provides further evidence that orientation selectivity is not critical for redundancy reduction, while contrast gain control may play a more important role. Acknowledgements The authors would like to thank Reshad Hosseini, Sebastian Gerwinn and Philipp Berens for fruitful discussions. This work is supported by the German Ministry of Education, Science, Research and Technology through the Bernstein award to MB (BMBF; FKZ: 01GQ0601), a scholarship of the German National Academic Foundation to FS, and the Max Planck Society. References [1] S. F. Arnold and J. Lynch. On Ali’s characterization of the spherical normal distribution. Journal of the Royal Statistical Society. Series B (Methodological), 44(1):49–51, 1982. 7 [2] J. J. Atick. Could information theory provide an ecological theory of sensory processing? Network, 3:213–251, 1992. [3] F. Attneave. Informational aspects of visual perception. Psychological Review, 61:183–193, 1954. [4] H. B. Barlow. Sensory mechanisms, the reduction of redundancy, and intelligence. In The Mechanisation of Thought Processes, pages 535–539, London: Her Majesty’s Stationery Office, 1959. [5] A. J. Bell and T. 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Regularized Co-Clustering with Dual Supervision Vikas Sindhwani Jianying Hu Aleksandra Mojsilovic IBM Research, Yorktown Heights, NY 10598 {vsindhw, jyhu, aleksand}@us.ibm.com Abstract By attempting to simultaneously partition both the rows (examples) and columns (features) of a data matrix, Co-clustering algorithms often demonstrate surprisingly impressive performance improvements over traditional one-sided row clustering techniques. A good clustering of features may be seen as a combinatorial transformation of the data matrix, effectively enforcing a form of regularization that may lead to a better clustering of examples (and vice-versa). In many applications, partial supervision in the form of a few row labels as well as column labels may be available to potentially assist co-clustering. In this paper, we develop two novel semi-supervised multi-class classification algorithms motivated respectively by spectral bipartite graph partitioning and matrix approximation formulations for co-clustering. These algorithms (i) support dual supervision in the form of labels for both examples and/or features, (ii) provide principled predictive capability on out-of-sample test data, and (iii) arise naturally from the classical Representer theorem applied to regularization problems posed on a collection of Reproducing Kernel Hilbert Spaces. Empirical results demonstrate the effectiveness and utility of our algorithms. 1 Introduction Consider the setting where we are given large amounts of unlabeled data together with dual supervision in the form of a few labeled examples as well as a few labeled features, and the goal is to estimate an unknown classification function. This setting arises naturally in numerous applications. Imagine, for example, the problem of inferring sentiment (“positive” versus “negative”) associated with presidential candidates from online political blog posts represented as word vectors, given the following: (a) a vast collection of blog posts easily downloadable from the web (unlabeled examples), (b) a few blog posts whose sentiment for a candidate is manually identified (labeled examples), and (c) prior knowledge of words that reflect positive (e.g., ’superb’) and negative (e.g, ’awful’) sentiment (labeled features). Most existing semi-supervised algorithms do not explicitly incorporate feature supervision. They typically implement the cluster assumption [3] by learning decision boundaries such that unlabeled points belonging to the same cluster are given the same label, and empirical loss over labeled examples is concurrently minimized. In situations where the classes are predominantly supported on unknown subsets of similar features, it is clear that feature supervision can potentially illuminate the true cluster structure inherent in the unlabeled examples over which the cluster assumption ought to be enforced. Even when feature supervision is not available, there is ample empirical evidence in numerous recent papers in the co-clustering literature (see e.g., [5, 1] and references therein), suggesting that the clustering of columns (features) of a data matrix can lead to massive improvements in the quality of row (examples) clustering. An intuitive explanation is that column clustering enforces a form of dimensional reduction or implicit regularization that is responsible for performance enhancements observed in many applications such as text clustering, microarray data analysis and video content mining [1]. In this paper, we utilize data-dependent co-clustering regularizers for semi-supervised learning in the presence of partial dual supervision. 1 Our starting point is the spectral bipartite graph partitioning approach of [5] which we briefly review in Section 2.1. This approach effectively applies spectral clustering on a graph representation of the data matrix and is also intimately related to Singular Value Decomposition. In Section 2.2 we review an equivalence between this approach and a matrix approximation objective function that is minimized under orthogonality constraints [6]. By dropping the orthogonality constraints but imposing non-negativity constraints, one is led to a large family of co-clustering algorithms that arise from the non-negative matrix factorization literature. Based on the algorithmic intuitions embodied in the algorithms above, we develop two semisupervised classification algorithms that extend the spectral bipartite graph partitioning approach and the matrix approximation approach respectively. We start with Reproducing Kernel Hilbert Spaces (RKHSs) defined over both row and column spaces. These RKHSs are then coupled through co-clustering regularizers. In the first algorithm, we directly adopt graph Laplacian regularizers constructed from the bipartite graph of [5] and include it as a row and column smoothing term in the standard regularization objective function. The solution is obtained by solving a convex optimization problem. This approach may be viewed as a modification of the Manifold Regularization framework [2] where we now jointly learn row and column classification functions. In the second algorithm proposed in this paper, we instead add a (non-convex) matrix approximation term to the objective function, which is then minimized using a block-coordinate descent procedure. Unlike, their unsupervised counterparts, our methods support dual supervison and naturally possess out-of-sample extension. In Section 4, we provide experimental results where we compare against various baseline approaches, and highlight the performance benefits of feature supervision. 2 Co-Clustering Algorithms Let X denote the data matrix with n data points and d features. The methods that we discuss in this section output a row partition function πr : {i}n i=1 7→{j}mr j=1 and a column partition function πc : {i}d i=1 7→{j}mc j=1 that give cluster assignments to row and column indices respectively. Here, mr is the desired number of row clusters and mc is the desired number of column clusters. Below, by xi we mean the ith example (row) and by fj we mean the jth column (feature) in the data matrix. 2.1 Bipartite Graph Partitioning In the co-clustering technique introduced by [5], the data matrix is modeled as a bipartite graph with examples (rows) as one set of nodes and features (columns) as another. An edge (i, j) exists if feature fj assumes a non-zero value in example xi, in which case the edge is given a weight of Xij. This bi-partite graph is undirected and there are no inter-example or inter-feature edges. The adjacency matrix, W, and the normalized Laplacian [4], M, of this graph are given by, W = 0 X XT 0 , M = I −D−1 2 WD−1 2 (1) where D is the diagonal degree matrix defined by Dii = P i Wij and I is the (n + d) × (n + d) identity matrix. Guided by the premise that column clustering induces row clustering while row clustering induces column clustering, [5] propose to find an optimal partitioning of the nodes of the bipartite graph. This method is retricted to obtaining co-clusterings where mr = mc = m. The mpartitioning is obtained by minimizing the relaxation of the normalized cut objective function using standard spectral clustering techniques. This reduces to first constructing a spectral representation of rows and columns given by the smallest eigenvectors of M, and then performing standard k-means clustering on this representation, to finally obtain the partition functions πr, πc. Due to the special structure of Eqn. 1, it can be shown that the spectral representation used in this algorithm is related to the singular vectors of a normalized version of X. 2.2 Matrix Approximation Formulation In [6] it is shown that the bipartite spectral graph partitioning is closely related to solving the following matrix approximation problem, (Fr ⋆, Fc ⋆) = argminFrT Fr=I,FcT Fc=I ∥X −FrFc T ∥fro where Fr is an n × m matrix and Fc is a d × m matrix. Once the minimization is performed, 2 πr(i) = argmaxj Fr ⋆ ij and πc(i) = argmaxj Fc ⋆ ij. In a non-negative matrix factorization approach, the orthogonality constraints are dropped to make the optimization easier while non-negativity constraints Fr, Fc ≥0 are introduced with the goal of lending better interpretability to the solutions. There are numerous multiplicative update algorithms for NMF which essentially have the flavor of alternating non-convex optimization. In our empirical comparisons in Section 4, we use the Alternating Constrained Least Squares (ACLS) approach of [12]. In Section 3.2 we consider a 3-factor non-negative matrix approximation to incorporate unequal values of mr and mc, and to improve the quality of the approximation. See [7, 13] for more details on matrix tri-factorization based formulations for co-clustering. 3 Objective Functions for Regularized Co-clustering with Dual Supervision Let us consider examples x to be elements of R ⊂ℜd. We consider column values f for each feature to be a data point in C ⊂ℜn. Our goal is to learn partition functions defined over the entire row and column spaces (as opposed to matrix indices), i.e., πr : R 7→{i}mr i=1 and πc : C 7→{i}mc i=1. For this purpose, let us introduce kr : R × R →ℜto be the row kernel that defines an associated RKHS Hr. Similarly, kc : C × C →ℜdenotes the column kernel whose associated RKHS is Hc. Below, we define πr, πc using these real valued function spaces. Consider a simultaneous assignment of rows into mr classes and columns into mc classes. For any data point x, denote Fr(x) = [f 1 r (x) · · · f mr r (x)]T ∈ℜmr to be a vector whose elements are soft class assignments where f j r ∈Hr for all j. For the given n data points, denote Fr to be the n × mr class assignment matrix. Correspondingly, Fc(f) is defined for any feature f ∈C, and Fc denotes the associated column class assignment matrix. Additionally, we are given dual supervision in the form of label matrices Yr ∈ℜn×mr and Yc ∈ℜm×mc where Yrij = 1 specifies that the ith example is labeled with class j (simlarly for the feature labels matrix Yc). The associated row sum for a labeled point is 1. Unlabeled points have all-zero rows, and the row sums are therefore 0. Let Jr (Jc) denote a diagonal matrix of size n×n (d×d) whose diagonal entry is 1 for labeled examples (features) and 0 otherwise. By Is we will denote an identity matrix of size s×s. We use the notation tr(A) to mean the trace of the matrix A. 3.1 Manifold Regularization with Bipartite Graph Laplacian (MR) In this approach, we setup the following optimization problem, argmin Fr∈Hmr r ,Fc∈Hmc c γr 2 mr X i=1 ∥f i r∥2 Hr + γc 2 mc X i=1 ∥f i c∥2 Hc + 1 2tr (Fr −Yr)T Jr(Fr −Yr) +1 2tr (Fc −Yc)T Jc(Fc −Yc) + µ 2 tr Fr T Fc T M Fr Fc (2) The first two terms impose the usual RKHS norm on the class indicator functions for rows and columns. The middle two terms measure squared loss on labeled data. The final term measure smoothness of the row and column indicator functions with respect to the bipartite graph introduced in Section 2.1. This term also incorporates unlabeled examples and features. γr, γc, µ are real-valued parameters that tradeoff various regularization terms. Clearly, by Representer Theorem the solution is has the form, f j r (x) = n X i=1 αijkr(x, xi), 1 ≤j ≤mr, f j c (f) = d X i=1 βijkc(f, fi), 1 ≤j ≤mc (3) Let α, β denote the corresponding optimal expansion coefficient matrices. Then, plugging in Eqn. 3 and solving the optimization problem, the solution is easily seen to be given by, γrIn 0 0 γcId + µM Kr 0 0 Kc + JrKr 0 0 JcKc α β = Yr Yc (4) 3 where Kr, Kc are gram matrices over datapoints and features respectively. The partition functions are then defined by πr(x) = argmax 1≤j≤m n X i=1 αijkr(x, xi), πc(f) = argmax 1≤j≤m d X i=1 βijkc(f, fi) (5) As in Section 2.1, we assume mr = mc = m. If the linear system above is solved by explicitly computing the matrix inverse, the computational cost is O((n + d)3 + (n + d)2m). This approach is closely related to the Manifold Regularization framework of [2], and may be viewed as an modification of the Laplacian Regularized Least Squares (LAPRLS) algorithm, which uses a euclidean nearest neighbor row similarity graph to capture the manifold structure in the data. Instead of using the squared loss, one can develop variants using the SVM Hinge loss or the logistic loss function. One can also use a large family of graph regularizers derived from the graph Laplacian [3]. In particular, we use the iterated Laplacian of the form M p where p is an integer. 3.2 Matrix Approximation under Dual Supervision (MA) We now consider an alternative objective function where instead of the graph Laplacian regularizer, we add a penalty term that measures how well the data matrix is approximated by a trifactorization FrQFc T , argmin Fr∈Hmr r ,Fc∈Hmc c Q∈ℜmr×mc γr 2 mr X i=1 ∥f i r∥2 Hr + γc 2 mc X i=1 ∥f i c∥2 Hc + 1 2tr (Fr −Yr)T Jr(Fr −Yr) +1 2tr (Fc −Yc)T Jc(Fc −Yc) + µ 2 ∥X −FrQFc T ∥2 fro (6) As before, the first two terms above enforce smoothness, the third and fourth terms measure squared loss over labels and the final term enforces co-clustering. The classical Representer Theorem (Eqn. 3) can again be applied since the above objective function only depends on point evaluations and RKHS norms of functions in Hr, Hc. The optimal expansion coefficient matrices, α, β, in this case are obtained by solving, argmin α,β,Q J (α, β, Q) = γr 2 tr αT Krα + γc 2 tr βT Kcβ + 1 2tr (Krα −Yr)T Jr(Krα −Yr) + 1 2tr (Kcβ −Yc)T Jc(Kcβ −Yc) + µ 2 ∥X −KrαQβT Kc∥2 fro (7) This problem is not convex in α, β, Q. We propose a block coordinate descent algorithm for the problem above. Keeping two variables fixed, the optimization over the other is a convex problem with a unique solution. This guarantees monotonic decrease of the objective function and convergence to a stationary point. We get the simple update equations given below, ∂J ∂Q = 0 =⇒ Q = (αT K2 rα)−1(αT KrXKcβ)(βT K2 cβ)−1 (8) ∂J ∂α = 0 =⇒ [γrIn + JrKr] α + µKrαZc = JrYr + µXKcβQT (9) ∂J ∂β = 0 =⇒ [γcId + JcKc] β + µKcβZr = JcYc + µXT KrαQ (10) where Zc = QβT K2 cβQT , Zr = QT αT K2 rαQ (11) In Eqn 8, we assume that the appropriate matrix inverses exist. Eqns 9 and 10 are generalized Sylvester matrix equations of the form AXB⊤+ CXD⊤= E whose unique solution X under certain regularity conditions can be exactly obtained by an extended version of the classical BartelsStewart method [9] whose complexity is O((p+q)3) for p×q-sized matrix variable X. Alternatively, one can solve the linear system [10]: B⊤⊗A + D⊤⊗C vec(X) = vec(E) where ⊗denotes 4 Kronecker product and vec(X) vectorizes X in a column oriented way (it behaves as the matlab operator X(:)). Thus, the solution to Eqns (9,10) are as follows, [Imr ⊗(γrIn + JrKr) + µZc ⊗Kr] vec(α) = vec(JrYr + µXKcβQT ) (12) [Imc ⊗(γrId + JcKc) + µZr ⊗Kc] vec(β) = vec(JcYc + µXT KrαQ) (13) These linear systems are of size nmr × nmr and dmc × dmc respectively. It is computationally prohibitive to solve these systems by direct matrix inversion. We use an iterative conjugate gradients (CG) technique instead, which can exploit hot-starts from the previous solution, and the fact that the matrix vector products can be computed relatively efficiently as follows, [Imr ⊗(γrIn + JrKr) + µZc ⊗Kr] vec(α) = vec(µKrαZ⊤ c ) + γrvec(α) + vec(JrKrα) To optimize α (β) given fixed Q and β (α), we run CG with a stringent tolerance of 10−10 and maximum of 200 iterations starting from the α(β) from the previous iteration. In an outer loop, we monitor the relative decrease in the objective function and terminate when the relative improvement falls below 0.0001. We use a maximum of 40 outer iterations where each iteration performs one round of α, β, Q optimization. Empirically, we find that the block coordinate descent approach often converges surprisingly quickly (see Section 4.2). The final classification is given by Eqn. 5. 4 Empirical Study In this section, we present an empirical study aimed at comparing the proposed algorithms with several baselines: (i) Unsupervised co-clustering with spectral bipartite graph partitioning (BIPARTITE) and non-negative matrix factorization (NMF), (ii) supervised performance of standard regularized least squares classification (RLS) that ignores unlabeled data, and (iii) one-sided semi-supervised performance obtained with Laplacian RLS (LAPRLS) which uses a euclidean nearest-neighbor row similarity graph. The goal is to observe whether dual supervision particularly along features can help improve classification performance, and whether joint RKHS regularization as formulated in our algorithms (abbreviated MR for the manifold regularization based method of Section 3.1 and MA for the matrix approximation method of Section 3.2) along both rows and columns leads to good quality out-of-sample prediction. In the experiments below, the performance of RLS and LAPRLS is optimized for best performance on the unlabeled set over a grid of hyperparameters. We use Gaussian kernels with width σr for rows and σc for columns. These were set to 2kσ0r, 2kσ0c respectively where σ0r, σ0c are (1/m)-quantile of pairwise euclidean distances among rows and columns respectively for an m class problem, and k is tuned over {−2, −1, 0, 1, 2} to optimize 3fold cross-validation performance of fully supervised RLS. The values γr, γc, µ are loosely tuned for MA,MR with respect to a single random split of the data into training and validation set; more careful hyperparameter tuning may further improve the results presented below. We focus on performance in predicting row labels. To enable comparison with the unsupervised coclustering methods, we use the popularly used F-measure defined on pairs of examples as follows: Precision = Number of Pairs Correctly Predicted Number of Pairs Predicted to be In Same Cluster or Class Recall = Number of Pairs Correctly Predicted Number of Pairs in the Same Cluster or Class F-measure = (2 ∗Precision ∗Recall)/(Precision + Recall) (14) 4.1 A Toy Dataset We generated a toy 2-class dataset with 200 examples per class and 100 features to demonstrate the main observations. The feature vector for a positive example is of the form [2u−0.1 2u+0.1], and for a negative example is of the form [2u+0.1 2u−0.1], where u is a 50-dimensional random vector whose entries are uniformly distributed over the unit interval. It is clear that there is substantial overlap between the two classes. Given a column partitioning πc, consider the following transformation: T(x) = P i:πc(i)=1 xi |i:πc(i)=1| , P i:πc(i)=−1 xi |i:πc(i)=−1| that maps examples in ℜ100 to the plane ℜ2 by composing a single feature whose value equals the mean of all features in the same partition. For the correct column partitioning, πc(i) = 1, 1 ≤i ≤50, πc(i) = −1, 50 < i ≤100, the examples under the 5 action of T are shown in Figure 1 (left). It is clear that T renders the data to be almost separable. It is therefore natural to attempt to (effectively) learn T in a semi-supervised manner. In Figure 1 (right), we plot the learning curves of various algorithms with respect to increasing number of row and column labels. On this dataset, co-clustering techniques (BIPARTITE, NMF) perform fairly well, and even significantly better than RLS, which has an optimized F-measure of 67% with 25 row labels. With increasing amounts of column labels, the learning curves of MR and MA steadily lift eventually outperforming the unsupervised techniques. The hyperparameters used in this experiment are: σr = 2.1, σc = 4.1, γr = γc = 0.001, µ = 10 for MR and 0.001 for MA. Figure 1: left: Examples in the toy dataset under the transformation defined by the correct column partitioning. right: Performance comparison – the number of column labels used are marked. 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 5 10 15 20 25 0.82 0.84 0.86 0.88 0.9 0.92 Number of Row Labels F−measure MR,5 MR,10 bipartite nmf MR,25 MR,50 MA,50 MA,5 MA,10,25 4.2 Text Categorization We performed experiments on document-word matrices drawn from the 20-newgroups dataset preprocessed as in [15]. The preprocessed data has been made publicly available by the authors of [15]1. For each word w and class c, we computed a score as follows: score(w, c) = −P(Y = c) log P(Y = c) −P(W = w)P(Y = c|W = w) log P(Y = c|W = w) −P(W ̸= w)P(Y = c|W ̸= w) log P(Y = c|W ̸= w), where P(Y = c) is the fraction of documents whose category is c, P(W = w) is the fraction of times word w is encountered, and P(Y = c|W = w) (P(Y = c|W ̸= w)) is the fraction of documents with class c when w is present (absent). It is easy to see that the mutual information between the indicator random variable for w and the class variable is P c score(w, c). We simulated manual labeling of words by associating w with the class argmaxc score(w, c). Finally, we restricted attention to 631 words with highest overall mutual information and 2000 documents that belong to the following 5 classes: comp.graphics, rec.motorcycles, rec.sport.baseball, sci.space, talk.politics.mideast. Since words of talk.politics.mideast accounted for more than half the vocabulary, we used the class normalization prescribed in [11] to handle the imbalance in the labeled data. Results presented in Table 1 are averaged over 10 runs. In each run, we randomly split the documents into training and test sets, in the ratio 1 : 3. The training set is then further split into labeled and unlabeled sets by randomly selecting 75 labeled documents. We experimented with increasing number of randomly chosen word labels. The hyperparameters are as follows: σr = 0.43, σc = 0.69, γr = γc = µ = 1 for MR and γr = γc = 0.0001, µ = 0.01 for MA. We observe that even without any word supervision MR outperforms all the baseline approaches: unsupervised co-clustering with BIPARTITE and NMF, standard RLS that only uses labeled documents, and also LAPRLS which uses a graph Laplacian based on document similarity for semisupervised learning. This validates the effectiveness of the bipartite document and word graph regularizer. As the amount of word supervision increases, the performance of both MR and MA improves gracefully. The out-of-sample extension to test data is of good quality, considering that our test sets are much larger than our training sets. We also observed that the mean number of (outer) iterations required for convergence of MA decreases as labels are increased from 0 to 500: 28.7(0), 12.2(100), 12.7(200), 9.3(350), 7.8(500). In, Figure 2 we show the top unlabeled words 1At http://www.princeton.edu/∼nslonim/data/20NG data 74000.mat.gz 6 Table 1: Performance on 5-Newsgroups Dataset with 75 row labels (a) F-measure on Unlabeled Set BIPARTITE NMF RLS LAPRLS 54.8 (7.8) 54.4 (6.2) 62.2 (3.1) 62.5 (3.0) (b) F-measure on Test Set RLS LAPRLS 61.2 (1.7) 61.9 (1.4) (c) F-measure on Unlabeled Set # col labs MR MA 0 64.7 (1.3) 60.4 (5.6) 100 72.3 (2.2) 59.6 (5.7) 200 77.0 (2.5) 69.2 (7.1) 350 78.6 (2.1) 75.1 (4.1) 500 79.3 (1.6) 77.1 (5.8) (d) F-measure on Test Set # col labs MR MA 0 57.1 (2.1) 60.3 (7.0) 100 60.9 (2.4) 60.9 (5.0) 200 66.2 (2.8) 66.2 (6.2) 350 68.1 (1.9) 70.3 (4.4) 500 69.1 (2.4) 71.0 (6.0) for each class sorted by the real-valued prediction score assigned by MR (in one run trained with 100 labeled words). Intuitvely, the main words associated with the class are retrieved. Figure 2: Top unlabeled words categorized by MR COMP.GRAPHICS: polygon, gifs, conversion, shareware, graphics, rgb, vesa, viewers, gif, format, viewer, amiga, raster, ftp, jpeg, manipulation REC.MOTORCYCLES: biker, archive, dogs, yamaha, plo, wheel, riders, motorcycle, probes, ama, rockies, neighbors, saudi, kilometers REC.SPORT.BASEBALL: clemens, morris, pitched, hr, batters, dodgers, offense, reds, rbi, wins, mets, innings, ted, defensive, sox, inning SCI.SPACE: oo, servicing, solar, scispace, scheduled, atmosphere, missions, telescope, bursts, orbiting, energy, observatory, island, hst, dark TALK.POLITICS.MIDEAST:turkish, greek, turkey, hezbollah, armenia, territory, ohanus, appressian, sahak, melkonian, civilians, greeks 4.3 Project Categorization We also considered a problem that arises in a real business-intelligence setting. The dataset is composed of 1169 projects tracked by the Integrated Technology Services division of IBM. These projects need to be categorized into 8 predefined product categories within IBM’s Server Services product line, with the eventual goal of performing various follow-up business analyses at the granularity of categories. Each project is represented as a 112-dimensional vector specifying the distribution of skills required for its delivery. Therefore, each feature is associated with a particular job role/skill set (JR/SS) combination, e.g., “data-specialist (oracle database)”. Domain experts validated project (row) labels and additionally provided category labels for 25 features deemed to be important skills for delivering projects in the corresponding category. By demonstrating our algorithms on this dataset, we are able to validate a general methodology with which to approach project categorization across all service product lines (SPLs) on a regular basis. The amount of dual supervision available in other SPLs is indeed severely limited as both the project categories and skill definitions are constantly evolving due to the highly dynamic business environment. Results presented in Table 2 are averaged over 10 runs. In each run, we randomly split the projects into training and test sets, in the ratio 3 : 1. The training set is then further split into labeled and unlabeled sets by randomly selecting 30 labeled projects. We experimented with increasing number of randomly chosen column labels, from none to all 25 available labels. The hyperparameters are as follows: γr = γc = 0.0001, σr = 0.69, σc = 0.27 chosen as described earlier. Results in Tables 2(c),2(d) are obtained with µ = 10 for MR, µ = 0.001 for MA. We observe that BIPARTITE performs significantly better than NMF on this dataset, and is competitve with supervised RLS performance that relies only on labeled data. By using LAPRLS , performance can be slightly boosted. We find that MR outperforms all approaches significantly even with very few column labels. We conjecture that the comparatively lower mean and high variance in the performance of MA on this dataset is due to suboptimal local minima issues, which may be alleviated using annealing techniques or multiple random starts, commonly used for Transductive SVMs [3]. From Tables 2(c),2(d) we also observe that both methods give high quality out-of-sample extension on this problem. 7 Table 2: Performance on IBM Project Categorization Dataset with 30 row labels (a) F-measure on Unlabeled Set BIPARTITE NMF RLS LAPRLS 89.1 (2.7) 56.5 (1.1) 88.1 (7.3) 90.20 (5.8) (b) F-measure on Test Set RLS LAPRLS 87.8 (8.4) 90.2 (6.0) (c) F-measure on Unlabeled Set # col labs MR MA 0 92.7 (4.6) 90.7 (4.8) 5 94.9 (1.8) 87.8 (6.4) 10 93.0 (4.2) 89.0 (8.0) 15 92.3 (7.0) 89.1 (7.4) 25 98.0 (0.5) 92.2 (6.0) (d) F-measure on Test Set # col labs MR MA 0 89.2 (5.5) 90.0 (5.5) 5 93.3 (1.7) 87.4 (6.6) 10 91.9 (4.2) 89.1 (8.3) 15 92.2 (5.2) 89.2 (8.8) 25 96.4 (1.6) 92.1 (6.8) 5 Conclusion We have developed semi-supervised kernel methods that support partial supervision along both dimensions of the data. Empirical studies show promising results and highlight the previously untapped benefits of feature supervision in semi-supervised settings. For an application of closely related algorithms to blog sentiment classification, we point the reader to [14]. For recent work on text categorization with labeled features instead of labeled examples, see [8]. References [1] A. Banerjee, I. Dhillon, J. Ghosh, S.Merugu, and D.S. Modha. A generalized maximum entropy approach to bregman co-clustering and matrix approximation. JMLR, 8:1919–1986, 2007. [2] M. Belkin, P. Niyogi, and V. Sindhwani. Manifold regularization: A geometric framework for learning from labeled and unlabeled examples. JMLR, 7:2399–2434, 2006. [3] O. Chapelle, B. Sch¨olkopf, and A. Zien, editors. Semi-Supervised Learning. MIT Press, 2006. [4] F. Chung, editor. Spectral Graph Theory. AMS, 1997. [5] I. Dhillon. Co-clustering documents and words using bipartite spectral graph partitioning. In KDD, 2001. [6] C. Ding, X. He, and H.D. Simon. On the equivalence of nonnegative matrix factorization and spectral clustering. In SDM, 2005. [7] C. Ding, T. Li, W. Peng, and H. Park. Orthogonal nonnegative matrix tri-factorizations for clustering. In KDD, 2006. [8] G. Druck, G. Mann, and A. McCallum. Learning from labeled features using generalized expectation criteria. In SIGIR, 2008. [9] J. Gardiner, Laub A.J, Amato J.J, and Moler C.B. Solution of the Sylvester matrix equation AXBT + CXDT = E. ACM Transactions on Mathematical Software, 18(2):223–231, 1992. [10] D. Harville. Matrix Algebra From a Statistician’s Perspective. Springer, New York, 1997. [11] T.M. Huang and V. Kecman. Semi-supervised learning from unbalanced labeled data an improvement. Lecture Notes in Computer Science, 3215:765–771, 2004. [12] A. Langville, C. Meyer, and R. Albright. Initializations for the non-negative matrix factorization. In KDD, 2006. [13] T. Li and C. Ding. The relationships among various nonnegative matrix factorization methods for clustering. In ICDM, 2006. [14] V. Sindhwani and P. Melville. Document-word co-regularization for semi-supervised sentiment analysis. 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Tighter Bounds for Structured Estimation Chuong B. Do, Quoc Le Stanford University {chuongdo,quocle}@cs.stanford.edu Choon Hui Teo Australian National University and NICTA choonhui.teo@anu.edu.au Olivier Chapelle, Alex Smola Yahoo! Research chap@yahoo-inc.com,alex@smola.org Abstract Large-margin structured estimation methods minimize a convex upper bound of loss functions. While they allow for efficient optimization algorithms, these convex formulations are not tight and sacrifice the ability to accurately model the true loss. We present tighter non-convex bounds based on generalizing the notion of a ramp loss from binary classification to structured estimation. We show that a small modification of existing optimization algorithms suffices to solve this modified problem. On structured prediction tasks such as protein sequence alignment and web page ranking, our algorithm leads to improved accuracy. 1 Introduction Structured estimation [18, 20] and related techniques has proven very successful in many areas ranging from collaborative filtering to optimal path planning, sequence alignment, graph matching and named entity tagging. At the heart of those methods is an inverse optimization problem, namely that of finding a function f(x, y) such that the prediction y∗which maximizes f(x, y∗) for a given x, minimizes some loss ∆(y, y∗) on a training set. Typically x ∈X is referred to as a pattern, whereas y ∈Y is a corresponding label. Y can represent a rich class of possible data structures, ranging from binary sequences (tagging), to permutations (matching and ranking), to alignments (sequence matching), to path plans [15]. To make such inherently discontinuous and nonconvex optimization problems tractable, one applies a convex upper bound on the incurred loss. This has two benefits: firstly, the problem has no local minima, and secondly, the optimization problem is continuous and piecewise differentiable, which allows for effective optimization [17, 19, 20]. This setting, however, exhibits a significant problem: the looseness of the convex upper bounds can sometimes lead to poor accuracy. For binary classification, [2] proposed to switch from the hinge loss, a convex upper bound, to a tighter nonconvex upper bound, namely the ramp loss. Their motivation was not the accuracy though, but the faster optimization due to the decreased number of support vectors. The resulting optimization uses the convex-concave procedure of [22], which is well known in optimization as the DC-programming method [9]. We extend the notion of ramp loss to structured estimation. We show that with some minor modifications, the DC algorithms used in the binary case carry over to the structured setting. Unlike the binary case, however, we observe that for structured prediction problems with noisy data, DC programming can lead to improved accuracy in practice. This is due to increased robustness. Effectively, the algorithm discards observations which it labels incorrectly if the error is too large. This ensures that one ends up with a lower-complexity solution while ensuring that the “correctable” errors are taken care of. 1 2 Structured Estimation Denote by X the set of patterns and let Y be the set of labels. We will denote by X := {x1, . . . , xm} the observations and by Y := {y1, . . . , ym} the corresponding set of labels. Here the pairs (xi, yi) are assumed to be drawn from some distribution Pr on X × Y. Let f : X × Y →R be a function defined on the product space. Finally, denote by ∆: Y × Y →R+ 0 a loss function which maps pairs of labels to nonnegative numbers. This could be, for instance, the number of bits in which y and y′ differ, i.e. ∆(y, y′) = ∥y −y′∥1 or considerably more complicated loss functions, e.g., for ranking and retrieval [21]. We want to find f such that for y∗(x, f) := argmax y′ f(x, y′) (1) the loss ∆(y, y∗(x, f)) is minimized: given X and Y we want to minimize the regularized risk, Rreg[f, X, Y ] := 1 m m X i=1 ∆(yi, y∗(xi, f)) + λΩ[f]. (2) Here Ω[f] is a regularizer, such as an RKHS norm Ω[f] = ∥f∥2 H and λ > 0 is the associated regularization constant, which safeguards us against overfitting. Since (2) is notoriously hard to minimize several convex upper bounds have been proposed to make ∆(yi, y∗(xi, f)) tractable in f. The following lemma, which is a generalization of a result of [20] provides a strategy for convexification: Lemma 1 Denote by Γ : R+ 0 →R+ 0 a monotonically increasing nonnegative function. Then l(x, y, y′′, f) := sup y′ Γ(∆(y, y′)) [f(x, y′) −f(x, y′′)] + ∆(y, y′) ≥∆(y, y∗(x, f)) for all y, y′′ ∈Y. Moreover, l(x, y, y′′, f) is convex in f. Proof Convexity follows immediately from the fact that l is the supremum over linear functions in f. To see the inequality, plug y′ = y∗(x, f) into the LHS of the inequality: by construction f(x, y∗(x, f)) ≥f(x, y′′) for all y′′ ∈Y. In regular convex structured estimation, l(x, y, y, f) is used. Methods in [18] choose the constant function Γ(η) = 1, whereas methods in [20] choose margin rescaling by means of Γ(η) = η. This also shows why both formulations lead to convex upper bounds of the loss. It depends very much on the form of f and ∆which choice of Γ is easier to handle. Note that the inequality holds for all y′′ rather than only for the “correct” label y′′ = y. We will exploit this later. 3 A Tighter Bound For convenience denote by β(x, y, y′, f) the relative margin between y and y′ induced by f via β(x, y, y′, f) := Γ(∆(y, y′))[f(x, y′) −f(x, y)]. (3) The loss bound of Lemma 1 suffers from a significant problem: for large values of f the loss may grow without bound, provided that the estimate is incorrect. This is not desirable since in this setting even a single observation may completely ruin the quality of the convex upper bound on the misclassification error. Another case where the convex upper bound is not desirable is the following: imagine that there are a lot of y which are as good as the label in the training set; this happens frequently in ranking where there are ties between the optimal permutations. Let us denote by Yopt := {y′′ such that ∆(y, y′) = ∆(y′′, y′), ∀y′} this set of equally good labels. Then one can replace y by any element of Yopt in the bound of Lemma 1. Minimization over y′′ ∈Yopt leads to a tighter non-convex upper bound: l(x, y, y, f) ≥ inf y′′∈Yopt sup y′ β(x, y′′, y′, f) + ∆(y′′, y′) ≥∆(y, y∗(x, f)) . In the case of binary classification, [2] proposed the following non-convex loss that can be minimized using DC programming: l(x, y, f) := min(1, max(0, 1 −yf(x))) = max(0, 1 −yf(x)) −max(0, −yf(x)). (4) 2 We see that (4) is the difference between a soft-margin loss and a hinge loss. That is, the difference between a loss using a large margin related quantity and one using simply the violation of the margin. This difference ensures that l cannot increase without bound, since in the limit the derivative of l with respect to f vanishes. The intuition for extending this to structured losses is that the generalized hinge loss underestimates the actual loss whereas the soft margin loss overestimates the actual loss. Taking the difference removes linear scaling behavior while retaining the continuous properties. Lemma 2 Denote as follows the rescaled estimate and the margin violator ˜y(x, y, f) := argmax y′ β(x, y, y′, f) and ¯y(x, y, f) := argmax y′ β(x, y, y′, f) + ∆(y, y′) (5) Moreover, denote by l(x, y, f) the following loss function l(x, y, f) := sup y′ [β(x, y, y′, f) + ∆(y, y′)] −sup y′ β(x, y, y′, f). (6) Then under the assumptions of Lemma 1 the following bound holds ∆(y, ¯y(x, y, f)) ≥l(x, y, f) ≥∆(y, y∗(x, f)) (7) This loss is a difference between two convex functions, hence it may be (approximately) minimized by a DC programming procedure. Moreover, it is easy to see that for Γ(η) = 1 and f(x, y) = 1 2yf(x) and y ∈{±1} we recover the ramp loss of (4). Proof Since ¯y(x, y, f) maximizes the first term in (6), replacing y′ by ¯y(x, y, f) in both terms yields l(x, y, f) ≤β(x, y, ¯y, f) + ∆(y, ¯y) −β(x, y, ¯y, f) = ∆(y, ¯y). To show the lower bound, we distinguish the following two cases: Case 1: y∗is a maximizer of supy′ β(x, y, y′, f) Replacing y′ by y∗in both terms of (6) leads to l(x, y, f) ≥∆(y, y∗). Case 2: y∗is not a maximizer of supy′ β(x, y, y′, f) Let ˜y be any maximizer. Because f(x, y∗) ≥f(x, ˜y), we have Γ(∆(y, ˜y)) [f(x, y∗) −f(x, y)] > Γ(∆(y, ˜y)) [f(x, ˜y) −f(x, y)] > Γ(∆(y, y∗)) [f(x, y∗) −f(x, y)] and thus Γ(∆(y, ˜y)) > Γ(∆(y, y∗)). Since Γ is non-decreasing this implies ∆(y, ˜y) > ∆(y, y∗). On the other hand, plugging ˜y in (6) gives l(x, y, f) ≥∆(y, ˜y). Combining both inequalities proves the claim. Note that the main difference between the cases of constant Γ and monotonic Γ is that in the latter case the bounds are not quite as tight as they could potentially be, since we still have some slack with respect to ∆(y, ˜y). Monotonic Γ tend to overscale the margin such that more emphasis is placed on avoiding large deviations from the correct estimate rather than restricting small deviations. Note that this nonconvex upper bound is not likely to be Bayes consistent. However, it will generate solutions which have a smaller model complexity since it is never larger than the convex upper bound on the loss, hence the regularizer on f plays a more important role in regularized risk minimization. As a consequence one can expect better statistical concentration properties. 4 DC Programming We briefly review the basic template of DC programming, as described in [22]. For a function f(x) = fcave(x) + fvex(x) which can be expressed as the sum of a convex fvex and a concave fcave function, we can find a convex upper bound by fcave(x0)+⟨x −x0, f ′ cave(x0)⟩+fvex(x). This follows from the first-order Taylor expansion of the concave part fcave at the current value of x. Subsequently, this upper bound is minimized, a new Taylor approximation is computed, and the procedure is repeated. This will lead to a local minimum, as shown in [22]. We now proceed to deriving an explicit instantiation for structured estimation. To keep things simple, in particular the representation of the functional subgradients of l(x, y, f) with respect to f, we assume that f is drawn from a Reproducing Kernel Hilbert Space H. 3 Algorithm 1 Structured Estimation with Tighter Bounds Using the loss of Lemma 1 initialize f = argminf ′ Pm i=1 l(xi, yi, yi, f ′) + λΩ[f ′] repeat Compute ˜yi := ˜y(xi, yi, f) for all i. Using the tightened loss bound recompute f = argminf ′ Pm i=1 ˜l(xi, yi, ˜yi, f ′) + λΩ[f ′] until converged Denote by k the kernel associated with H, defined on (X × Y) × (X × Y). In this case for f ∈H we have by the reproducing property that f(x, y) = ⟨f, k((x, y), ·)⟩and the functional derivative is given by ∂ff(x, y) = k((x, y), ·). Likewise we may perform the linearization in (6) as follows: −sup y′ β(x, y, y′, f) ≤−β(x, y, ˜y, f) In other words, we use the rescaled estimate ˜y to provide an upper bound on the concave part of the loss function. This leads to the following instantiation of standard convex-concave procedure: instead of the structured estimation loss it uses the loss bound ˜l(x, y, ˜y, f) ˜l(x, y, ˜y, f) := sup y′∈Y [β(x, y, y′, f) + ∆(y, y′)] −β(x, y, ˜y, f) In the case of Γ(η) = 1 this can be simplified significantly: the terms in f(x, y) cancel and ˜l becomes ˜l(x, y, ˜y, f) = sup y′∈Y [f(x, y′) −f(x, ˜y)] + ∆(y, y′). In other words, we replace the correct label y by the rescaled estimate ˜y. Such modifications can be easily implemented in bundle method solvers and related algorithms which only require access to the gradient information (and the function value). In fact, the above strategy follows directly from Lemma 1 when replacing y′′ by the rescaled estimate ˜y. 5 Experiments 5.1 Multiclass Classification In this experiment, we investigate the performance of convex and ramp loss versions of the WinnerTakes-All multiclass classification [1] when the training data is noisy. We performed the experiments on some UCI/Statlog datasets: DNA, LETTER, SATIMAGE, SEGMENT, SHUTTLE, and USPS, with some fixed percentages of the labels shuffled, respectively. Note that we reshuffled the labels in a stratified fashion. That is, we chose a fixed fraction from each class and we permuted the label assignment subsequently. Table 1 shows the results (average accuracy ± standard deviation) on several datasets with different percentages of labels shuffled. We used nested 10-fold crossvalidation to adjust the regularization constant and to compute the accuracy. A linear kernel was used. It can be seen that ramp loss outperforms the convex upper bound when the datasets are noisy. For clean data the convex upper bound is slightly superior, albeit not in a statistically significant fashion. This supports our conjecture that, compared to the convex upper bound, the ramp loss is more robust on noisy datasets. 5.2 Ranking with Normalized Discounted Cumulative Gains Recently, [12] proposed a method for learning to rank for web search. They compared several methods showing that optimizing the Normalized Discounted Cumulative Gains (NDCG) score using a form of structured estimation yields best performance. The algorithm used a linear assignment problem to deal with ranking. In this experiment, we perform ranking experiments with the OHSUMED dataset which is publicly available [13]. The dataset is already preprocessed and split into 5 folds. We first carried out the structured output training algorithm which optimizes the convex upper bound of NDCG as described in [21]. Unfortunately, the returned solution was f = 0. The convex upper bounds led to the 4 Dataset Methods 0% 10% 20% DNA convex 95.2 ± 1.1 88.9 ± 1.5 83.1 ± 2.4 ramp loss 95.1 ± 0.8 89.1 ± 1.3 83.5 ± 2.2 LETTER convex 76.8 ± 0.9 64.6 ± 0.7 50.1 ± 1.4 ramp loss 78.6 ± 0.8 70.8 ± 0.8 63.0 ± 1.5 SATIMAGE convex 85.1 ± 0.9 77.0 ± 1.6 66.4 ± 1.3 ramp loss 85.4 ± 1.2 78.1 ± 1.6 70.7 ± 1.0 SEGMENT convex 95.4 ± 0.9 84.8 ± 2.3 73.8 ± 2.1 ramp loss 95.2 ± 1.0 85.9 ± 2.1 77.5 ± 2.0 SHUTTLE convex 97.4 ± 0.2 89.5 ± 0.2 83.8 ± 0.2 ramp loss 97.1 ± 0.2 90.6 ± 0.8 88.1 ± 0.3 USPS convex 95.1 ± 0.7 85.3 ± 1.3 76.5 ± 1.4 ramp loss 95.1 ± 0.9 86.1 ± 1.6 77.6 ± 1.1 Table 1: Average accuracy for multiclass classification using the convex upper bound and the ramp loss. The third through fifth columns represent results for datasets with none, 10%, and 20% of the labels randomly shuffled, respectively. 1 2 3 4 5 6 7 8 9 10 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 truncation level NDCG@k svmrank rankboost ndcg optimization Figure 1: NDCG comparison against ranking SVM and RankBoost. We report the NDCG computed at various truncation levels. Our non-convex upper bound consistently outperforms other rankers. In the context of web page ranking an improvement of 0.01 −0.02 in the NDCG score is considered substantial. undesirable situation where no nonzero solution would yield any improvement, since the linear function class was too simple. This problem is related to the fact that there are a lot of rankings which are equally good because of the ties in the editorial judgments (see beginning of section 3). As a result, there is no w that learns the data well, and for each w the associated maxy′ f(x, y′) −f(x, y) + ∆(y, y′) causes either the first part or the second part of the loss to be big such that the total value of the loss function always exceeds max ∆(y, y′). When using the non-convex formulation the problem can be resolved because we do not entirely rely on the y given in the training set, but instead find the y that minimizes the loss. We compared the results of our method and two standard methods for ranking: ranking SVM [10, 8] and RankBoost [6] (the baselines for OHSUMED are shown in [13]) and used NDCG as the performance criterion. We report the aggregate performance in Figure 1. As can be seen from the figure, the results from the new formulation are better than standard methods for ranking. It is worth emphasizing that the most important contribution is not only that the new formulation can give comparable results to the state-of-the-art algorithms for ranking but also that it provides useful solutions when the convex structured estimation setting provides only useless results (obviously f = 0 is highly undesirable). 5.3 Structured classification We also assessed the performance of the algorithm on two different structured classification tasks for computational biology, namely protein sequence alignment and RNA secondary structure prediction. Protein sequence alignment is the problem of comparing the amino acid sequences corresponding to two different proteins in order to identify regions of the sequences which have common ancestry or biological function. In the pairwise sequence alignment task, the elements of the input space X consist of pairs of amino acid sequences, represented as strings of approximately 100-1000 char5 0-10% 11-20% 21-30% 31-40% Overall Method (324) (793) (429) (239) (1785) CRF 0.111 0.316 0.634 0.877 0.430 convex 0.116 0.369 0.699 0.891 0.472 ramp loss 0.138 0.387 0.708 0.905 0.488 Table 2: Protein pairwise sequence alignment results, stratified by reference alignment percentage identity. The second through fifth columns refer to the four non-overlapping reference alignment percentage identity ranges described in the text, and the sixth column corresponds to overall results, pooled across all four subsets. Each non-bolded value represents the average test set recall for a particular algorithm on alignment from the corresponding subset. The numbers in parentheses indicate the total number of sequences in each subset. 1-50 51-100 101-200 201+ Overall Method (118) (489) (478) (274) (1359) CRF 0.546 / 0.862 0.586 / 0.727 0.467 / 0.523 0.414 / 0.472 0.505 / 0.614 convex 0.690 / 0.755 0.664 / 0.629 0.571 / 0.501 0.542 / 0.484 0.608 / 0.565 ramp loss 0.725 / 0.708 0.705 / 0.602 0.612 / 0.489 0.569 / 0.461 0.646 / 0.542 Table 3: RNA secondary structure prediction results. The second through fifth columns represent subsets of the data stratified by sequence length. The last column presents overall results, pooled across all four subsets. Each pair of non-bolded numbers indicates the sensitivity / selectivity for structures in the two-fold cross-validation. The numbers in parentheses indicate the total number of sequences in each subset. acters in length. The output space Y contains candidate alignments which identify the corresponding positions in the two sequences which are hypothesized to be evolutionarily related. We developed a structured prediction model for pairwise protein sequence alignment, using the types of features described in [3, 11] For the loss function, we used ∆(y, y′) = 1 −recall (where recall is the proportion of aligned amino acid matches in the true alignment y that appear in the predicted alignment y′. For each inner optimization step, we used a fast-converging subgradientbased optimization algorithm with an adaptive Polyak-like step size [23]. We performed two-fold cross-validation over a collection of 1785 pairs of structurally aligned protein domains [14]. All hyperparameters were selected via holdout cross validation on the training set, and we pooled the results from the two folds. For evaluation, we used recall, as described previously, and compared the performance of our algorithm to a standard conditional random field (CRF) model and max-margin model using the same features. The percentage identity of a reference alignment is defined as the proportion of aligned residue pairs corresponding to identical amino acids. We partitioned the alignments in the testing collection into four subsets based on percent identity (0-10%, 11-20%, 21-30%, and 31+%), showed the recall of the algorithm for each subset in addition to overall recall (see Table 2). Here, it is clear that our method obtains better accuracy than both the CRF and max-margin models.1 We note that the accuracy differences are most pronounced at the low percentage identity ranges, the ‘twilight zone’ regime where better alignment accuracy has far reaching consequences in many other computational biology applications [16]. RNA secondary structure prediction Ribonucleic acid (RNA) refers to a class of long linear polymers composed of four different types of nucleotides (A, C, G, U). Nucleotides within a single RNA molecule base-pair with each other, giving rise to a pattern of base-pairing known as the RNA’s secondary structure. In the RNA secondary structure prediction problem, we are given an RNA sequence (a string of approximately 20-500 characters) and are asked to predict the secondary structure that the RNA molecule will form in vivo. Conceptually, an RNA secondary structure can be thought of as a set of unordered pairs of nucleotide indices, where each pair designates two 1We note that the results here are based on using the Viterbi algorithm for parsing, which differs from the inference method used in [3]. In practice this is preferable to posterior decoding as it is significantly faster which is crucial applications to large amounts of data. 6 (a) (b) (c) Figure 2: Tightness of the nonconvex bound. Figures (a) and (b) show the value of the nonconvex loss, the convex loss and the actual loss as a function of the number of iterations when minimizing the nonconvex upper bound. At each relinearization, which occurs every 1000 iterations, the nonconvex upper bound decreases. Note that the convex upper bound increases in the process as convex and nonconvex bound diverge further from each other. We chose λ = 2−6 in Figure (a) and λ = 27 for Figure (b). Figure (c) shows the tightness of the final nonconvex bound at the end of optimization for different values of the regularization parameter λ. nucleotides in the RNA molecule which base-pair with each other. Following convention, we take the structured output space Y to be the set of all possible pseudoknot-free structures. We used a max-margin model for secondary structure prediction. The features of the model were chosen to match the energetic terms in standard thermodynamic models for RNA folding [4]. As our loss function, we used ∆(y, y′) = 1−recall (where recall is the proportion of base-pairs in the reference structure y that are recovered in the predicted structure y′). We again used the subgradient algorithm for optimization. To test the algorithm, we performed two-fold cross-validation over a large collection of 1359 RNA sequences with known secondary structures from the RFAM database (release 8.1) [7]. We evaluated the methods using two standard metrics for RNA secondary structure prediction accuracy known as sensitivity and selectivity (which are the equivalent of recall and precision, respectively, for this domain). For reporting, we binned the sequences in the test collection by length into four ranges (150, 51-100, 101-200, 201+ nucleotides), and evaluated the sensitivity and selectivity of the algorithm for each subset in addition to overall accuracy (see Table 3). Again, our algorithm consistently outperforms an equivalently parameterized CRF and max-margin model in terms of sensitivity.2 The selectivity of the predictions from our algorithm is often worse than that of the other two models. This is likely because we opted for a loss function that penalizes for “false negative” base-pairings but not “false-positives” since our main interest is in identifying correct base-pairings (a harder task than predicting only a small number of high-confidence basepairings). An alternative loss function that chooses a different balance between penalizing false positives and false negatives would achieve a different trade-off of sensitivity and selectivity. Tightness of the bound: We generated plots of the convex, nonconvex, and actual losses (which correspond to l(x, y, y, f), l(x, y, f), and ∆(y, y∗(x, f)), respectively, from Lemma 2) over the course of optimization for our RNA folding task (see Figure 2). From Figures 2a and 2b, we see that the nonconvex loss provides a much tighter upper bound on the actual loss function. Figure 2c shows that the tightness of the bound decreases for increasing regularization parameters λ. In summary, our bound leads to improvements whenever there is a large number of instances (x, y) which cannot be classified perfectly. This is not surprising as for “clean” datasets even the convex upper bound vanishes when no margin errors are encountered. Hence noticeable improvements can be gained mainly in the structured output setting rather than in binary classification. 2Note that the results here are based on using the CYK algorithm for parsing, which differs from the inference method used in [4]. 7 6 Summary and Discussion We proposed a simple modification of the convex upper bound of the loss in structured estimation which can be used to obtain tighter bounds on sophisticated loss functions. The advantage of our approach is that it requires next to no modification of existing optimization algorithms but rather repeated invocation of a structured estimation solver such as SVMStruct, BMRM, or Pegasos. In several applications our approach outperforms the convex upper bounds. This can be seen both for multiclass classification, for ranking where we encountered underfitting and undesirable trivial solutions for the convex upper bound, and in the context of sequence alignment where in particular for the hard-to-align observations significant gains can be found. From this experimental study, it seems that the tighter non-convex upper bound is useful in two scenarios: when the labels are noisy and when for each example there is a large set of labels which are (almost) as good as the label in the training set. Future work includes studying other types of structured estimation problems such as the ones encountered in NLP to check if our new upper bound can also be useful for these problems. References [1] K. Crammer, and Y. Singer. On the Learnability and Design of Output Codes for Multiclass Problems. In COLT 2000, pages 35–46. Morgan Kaufmann, 2000. [2] R. Collobert, F.H. Sinz, J. Weston, and L. Bottou. Trading convexity for scalability. In ICML 2006, pages 201–208. ACM, 2006. [3] C. B. Do, S. S. Gross, and S. Batzoglou. CONTRAlign: discriminative training for protein sequence alignment. In RECOMB, pages 160–174, 2006. [4] C. B. Do, D. A. Woods, and S. Batzoglou. CONTRAfold: RNA secondary structure prediction without physics-based models. Bioinformatics, 22(14):e90–e98, 2006. [5] S. R. Eddy. Non-coding RNA genes and the modern RNA world. Nature Reviews Genetics, 2(12):919– 929, 2001. [6] Y. Freund, R. Iyer, R.E. Schapire, and Y. Singer. An efficient boosting algorithm for combining preferences. In ICML 1998, pages 170–178., 1998. [7] S. Griffiths-Jones, S. Moxon, M. Marshall, A. Khanna, S. R. Eddy, and A. Bateman. Rfam: annotating non-coding RNAs in complete genomes. Nucl. Acids Res., 33:D121–D124, 2005. [8] R. Herbrich, T. Graepel, and K. Obermayer. Large margin rank boundaries for ordinal regression. In Advances in Large Margin Classifiers, pages 115–132, 2000. MIT Press. [9] T. Hoang. DC optimization: Theory, methods, and applications. In R. Horst and P. Pardalos, editors, Handbook of Global Optimization, Kluwer. [10] T. Joachims. Optimizing search engines using clickthrough data. In KDD. ACM, 2002. [11] T. Joachims, T. Galor, and R. Elber. Learning to align sequences: A maximum-margin approach. In New Algorithms for Macromolecular Simulation, LNCS 49, 57–68. Springer, 2005. [12] Q. Le and A.J. Smola. Direct optimization of ranking measures. NICTA-TR, 2007. [13] T.-Y. Liu, J. Xu, T. Qin, W. Xiong, and H. Li. Letor: Benchmark dataset for research on learning to rank for information retrieval. In LR4IR, 2007. [14] J. Pei and N. V. Grishin. MUMMALS: multiple sequence alignment improved by using hidden Markov models with local structural information. Nucl. Acids Res., 34(16):4364–4374, 2006. [15] N. Ratliff, J. Bagnell, and M. Zinkevich. (online) subgradient methods for structured prediction. In AISTATS, 2007. [16] B. Rost. Twilight zone of protein sequence alignments. Protein Eng., 12(2):85–94, 1999. [17] S. Shalev-Shwartz, Y. Singer, and N. Srebro. Pegasos: Primal estimated sub-gradient solver for svm. In Proc. Intl. Conf. Machine Learning, 2007. [18] B. Taskar, C. Guestrin, and D. Koller. Max-margin Markov networks. In NIPS 16, pages 25–32, 2004. MIT Press. [19] C.H. Teo, Q. Le, A.J. Smola, and S.V.N. Vishwanathan. A scalable modular convex solver for regularized risk minimization. In KDD. ACM, 2007. [20] I. Tsochantaridis, T. Joachims, T. Hofmann, and Y. Altun. Large margin methods for structured and interdependent output variables. J. Mach. Learn. Res., 6:1453–1484, 2005. [21] M. Weimer, A. Karatzoglou, Q. Le, and A. Smola. Cofirank - maximum margin matrix factorization for collaborative ranking. In NIPS 20. MIT Press, 2008. [22] A.L. Yuille and A. Rangarajan. The concave-convex procedure. Neural Computation, 15:915–936, 2003. [23] A. Nedic and D. P. Bertsekas. Incremental subgradient methods for nondifferentiable optimization. Siam J. on Optimization, 12:109–138, 2001. 8
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Multi-Agent Filtering with Infinitely Nested Beliefs Luke S. Zettlemoyer MIT CSAIL Cambridge, MA 02139 lsz@csai.mit.edu Brian Milch∗ Google Inc. Mountain View, CA 94043 brian@google.com Leslie Pack Kaelbling MIT CSAIL Cambridge, MA 02139 lpk@csail.mit.edu Abstract In partially observable worlds with many agents, nested beliefs are formed when agents simultaneously reason about the unknown state of the world and the beliefs of the other agents. The multi-agent filtering problem is to efficiently represent and update these beliefs through time as the agents act in the world. In this paper, we formally define an infinite sequence of nested beliefs about the state of the world at the current time t, and present a filtering algorithm that maintains a finite representation which can be used to generate these beliefs. In some cases, this representation can be updated exactly in constant time; we also present a simple approximation scheme to compact beliefs if they become too complex. In experiments, we demonstrate efficient filtering in a range of multi-agent domains. 1 Introduction The existence of nested beliefs is one of the defining characteristics of a multi-agent world. As an agent acts, it often needs to reason about what other agents believe. For instance, a teacher must consider what a student knows to decide how to explain important concepts. A poker agent must think about what cards other players might have — and what cards they might think it has — in order to bet effectively. In this paper, we assume a cooperative setting where all the agents have predetermined, commonly-known policies expressed as functions of their beliefs; we focus on the problem of efficient belief update, or filtering. We consider the nested filtering problem in multi-agent, partially-observable worlds [6, 1, 9]. In this setting, agents receive separate observations and independently execute actions, which jointly change the hidden state of the world. Since each agent does not get to see the others’ observations and actions, there is a natural notion of nested beliefs. Given its observations and actions, an agent can reason not only about the state of the external world, but also about the other agents’ observations and actions. It can also condition on what others might have seen and done to compute their beliefs at the next level of nesting. This pattern can be repeated to arbitrary depth. The multi-agent filtering problem is to efficiently represent and update these nested beliefs through time. In general, an agent’s beliefs depend on its entire history of actions and observations. One approach to computing these beliefs would be to remember the entire history, and perform inference to compute whatever probabilities are needed at each time step. But the time required for this computation would grow with the history length. Instead, we maintain a belief state that is sufficient for predicting future beliefs and can be approximated to achieve constant-time belief updates. We begin by defining an infinite sequence of nested beliefs about the current state st, and showing that it is sufficient for predicting future beliefs. We then present a multi-agent filtering algorithm that maintains a compact representation sufficient for generating this sequence. Although in the worst case this representation grows exponentially in the history length, we show that its size remains constant for several interesting problems. We also describe an approximate algorithm that always ∗This work was done while the second author was at MIT CSAIL. maintains a constant representation size (and constant-time updates), possibly at the cost of accuracy. In experiments, we demonstrate efficient and accurate filtering in a range of multi-agent domains. 2 Related Work In existing research on partially observable stochastic games (POSGs) and Decentralized POMDPs (DEC-POMDPs) [6, 1, 9], policies are represented as direct mappings from observation histories to actions. That approach removes the need for the agents to perform any kind of filtering, but requires the specification of some particular class of policies that return actions for arbitrarily long histories. In contrast, many successful algorithms for single-agent POMDPs represent policies as functions on belief states [7], which abstract over the specifics of particular observation histories. Gmytrasiewicz and Doshi [5] consider filtering in interactive POMDPs. Their approach maintains finitely nested beliefs that are derived from a world model as well as hand-specified models of how each agent reasons about the other agents. In this paper, all of the nested reasoning is derived from a single world model, which eliminates the need for any agent-specific models. To the best of our knowledge, our work is the first to focus on filtering of infinitely nested beliefs. There has been significant work on infinitely nested beliefs in game theory, where Brandenburger and Dekel [2] introduced the notion of an infinite sequence of finitely nested beliefs. However, they do not describe any method for computing these beliefs from a world model or updating them over time. Another long-standing line of related work is in the epistemic logic community. Fagin and Halpern [3] define labeled graphs called probabilistic Kripke structures, and show how a graph with finitely many nodes can define an infinite sequence of nested beliefs. Building on this idea, algorithms have been proposed for answering queries on probabilistic Kripke structures [10] and on influence diagrams that define such structures [8]. However, these algorithms have not addressed the fact that as agents interact with the world over time, the set of observation sequences they could have received (and possibly the set of beliefs they could arrive at) grows exponentially. 3 Nested Filtering In this section, we describe the world model and define the multi-agent filtering problem. We then present a detailed example where a simple problem leads to a complex pattern of nested reasoning. 3.1 Partially observable worlds with many agents We will perform filtering given a multi-agent, decision-theoretic model for acting in a partially observable world.1 Agents receive separate observations and independently execute actions, which jointly change the state of the world. There is a finite set of states S, but the current state s ∈S cannot be observed directly by any of the agents. Each agent j has a finite set of observations Oj that it can receive and a finite set of actions Aj that it can execute. Throughout this paper, we will use superscripts and vector notation to name agents and subscripts to indicate time. For example, aj t ∈Aj is the action for agent j at time t; ⃗at = ⟨ai t, . . . , aj t⟩is a vector with actions for each of the agents; and aj 0:t = (aj 0, . . . , aj t) is a sequence of actions for agent j at time steps 0 . . . t. The state dynamics is defined by a distribution p0(s) over initial states and a transition distribution p(st|st−1,⃗at−1) that is conditioned on the previous state st−1 and the action vector ⃗at−1. For each agent j, observations are generated from a distribution p(oj t|st,⃗at−1) conditioned on the current state and the previous joint action. Each agent j sees only its own actions and observations. To record this information, it is useful to define a history hj 0:t = (aj 0:t−1, oj 1:t) for agent j at time t. A policy is a distribution πj(aj t|hj 0:t) over the actions agent j will take given this history. Together, these distributions define the joint world model: p(s0:t,⃗h0:t) = p0(s0) t−1 Y i=0 ⃗π(⃗ai|⃗h0:i)p(si+1|si,⃗ai)p(⃗oi+1|si+1,⃗ai) (1) where ⃗π(⃗at|⃗h0:t) = Q j πj(aj t|hj 0:t) and p(⃗ot+1|st+1,⃗at) = Q j p(oj t+1|st+1,⃗at). 1This is the same type of world model that is used to define POSGs and DEC-POMDPs. Since we focus on filtering instead of planning, we do not need to define reward functions for the agents. 3.2 The nested filtering problem In this section, we describe how to compute infinitely nested beliefs about the state at time t. We then define a class of policies that are functions of these beliefs. Finally, we show that the current nested belief for an agent i contains all of the information required to compute future beliefs. Throughout the rest of this paper, we use a minus notation to define tuples indexed by all but one agent. For example, h−i 0:t and π−i are tuples of histories and policies for all agents k ̸= i. We define infinitely nested beliefs by presenting an infinite sequence of finitely nested beliefs. For each agent i and nesting level n, the belief function Bi,n : hi 0:t →bi,n t maps the agent’s history to its nth-level beliefs at time t. The agent’s zeroth-level belief function Bi,0(hi 0:t) returns the posterior distribution bi,0 t = p(st|hi 0:t) over states given the input history, which can be computed from Eq. 1: Bi,0(hi 0:t) = p(st|hi 0:t) ∝ P s0:t−1,h−i 0:t p(s0:t,⃗h0:t). Agent i’s first-level belief function Bi,1(hi 0:t) returns a joint distribution on st and the zeroth-level beliefs of all the other agents (what the other agents believe about the state of the world). We can compute the tuple of zeroth-level beliefs b−i,0 t for all agents k ̸= i by summing the probabilities of all histories h−i 0:t that lead to these beliefs (that is, such that b−i,0 t = B−i,0(h−i 0:t)): Bi,1(hi 0:t) = p(st, b−i,0 t |hi 0:t) ∝ P s0:t−1,h−i 0:t p(s0:t,⃗h0:t)δ(b−i,0 t , B−i,0(h−i 0:t)). The delta function δ(·, ·) returns one when its arguments are equal and zero otherwise. For level n, Bi,n(hi 0:t) returns a distribution over states and level n −1 beliefs for the other agents. For example, at level 2, the function returns a joint distribution over: the state, what the other agents believe about the state, and what they believe others believe. Again, these beliefs are computed by summing over histories for the other agents that lead to the appropriate level n −1 beliefs: Bi,n(hi 0:t) = p(st, b−i,n−1 t |hi 0:t) ∝ P s0:t−1,h−i 0:t p(s0:t,⃗h0:t)δ(b−i,n−1 t , B−i,n−1(h−i 0:t)). Note that for all nesting levels n, Bi,n(hi 0:t) is a discrete distribution. There are only finitely many beliefs each agent k could hold at time t — each arising from one of the possible histories hk 0:t. Define bi,∗ t = Bi,∗(hi 0:t) to be the infinite sequence of nested beliefs generated by computing Bi,n(hi 0:t) for n = 0, 1, . . .. We can think of bi,∗ t as a belief state for agent i, although not one that can be used directly by a filtering algorithm. We will assume that the policies πi are represented as functions of these belief states: that is, πi(ai t|bi,∗ t ) can be thought of as a procedure that looks at arbitrary parts of the infinite sequence bi,∗ t and returns a distribution over actions. We will see examples of this type of policy in the next section. Under this assumption, bi,∗ t is a sufficient statistic for predicting future beliefs in the following sense: Proposition 1 In a model with policies πj(aj t|bj,∗ t ) for each agent j, there exists a belief estimation function BE s.t. ∀ai 0:t−1, oi 1:t, ai t, oi t+1 . Bi,∗(ai 0:t, oi 1:t+1) = BE(Bi,∗(ai 0:t−1, oi 1:t), ai t, oi t+1). To prove this result, we need to demonstrate a procedure that correctly computes the new belief given only the old belief and the new action and observation. The filtering algorithm we will present in Sec. 4 achieves this goal by representing the nested belief with a finite structure that can be used to generate the infinite sequence, and showing how these structures are updated over time. 3.3 Extended Example: The Tiger Communication World We now describe a simple two-agent “tiger world” where the optimal policies require the agents to coordinate their actions. In this world there are two doors: behind one randomly chosen door is a hungry tiger, and behind the other is a pile of gold. Each agent has unique abilities. Agent l (the tiger listener) can hear the tiger roar, which is a noisy indication of its current location, but cannot open the doors. Agent d (the door opener) can open doors but cannot hear the roars. To facilitate communication, agent l has two actions, signal left and signal right, which each produce a unique observation for agent d. When a door is opened, the world resets and the tiger is placed behind a randomly chosen door. To act optimally, agent l must listen to the tiger’s roars until it is confident about the tiger’s location and then send the appropriate signal to agent d. Agent d must wait for this bl,∗ al πl(al|bl,∗) bl,0(T L) > 0.8 SL 1.0 bl,0(T L) > 0.8 SR 1.0 otherwise L 1.0 bd,∗ ad πd(ad|bd,∗) bd,0(T L) > 0.8 OR 1.0 bd,0(T R) > 0.8 OL 1.0 otherwise L 1.0 Figure 1: Deterministic policies for the tiger world that depend on each agent’s beliefs about the physical state, where the tiger can be on the left (TL) or the right (TR). The tiger listener, agent l, will signal left (SL) or right (SR) if it confident of the tiger’s location. The door opener, agent d, will open the appropriate door when it is confident about the tiger’s location. Otherwise both agents listen (to the tiger or for a signal). signal and then open the appropriate door. Fig. 1 shows a pair of policies that achieve this desired interaction and depend only on each agent’s level-zero beliefs about the state of the world. However, as we will see, the agents cannot maintain their level-zero beliefs in isolation. To correctly update these beliefs, each agent must reason about the unseen actions and observations of the other agent. Consider the beliefs that each agent must maintain to execute its policies during a typical scenario. Assume the tiger starts behind the left door. Initially, both agents have uniform beliefs about the location of the tiger. As agent d waits for a signal, it does not gain any information about the tiger’s location. However, it maintains a representation of the possible beliefs for agent l and knows that l is receiving observations that correlate with the state of the tiger. In this case, the most likely outcome is that agent l will hear enough roars on the left to do a “signal left” action. This action produces an observation for agent d which allows it to gain information about l’s beliefs. Because agent d has maintained the correspondence between the true state and agent l’s beliefs, it can now infer that the tiger is more likely to be on the left (it is unlikely that l could have come to believe the tiger was on the left if that were not true). This inference makes agent d confident enough about the tiger’s location to open the right door and reset the world. Agent l must also represent agent d’s beliefs, because it never receives any observations that indicate what actions agent d is taking. It must track agent d’s belief updates to know that d will wait for a signal and then immediately open a door. Without this information, l cannot predict when the world will be reset, and thus when it should disregard past observations about the location of the tiger. Even in this simple tiger world, we see a complicated reasoning pattern: the agents must track each others’ beliefs. To update its belief about the external world, each agent must infer what actions the other agent has taken, which requires maintaining that agent’s beliefs about the world. Moreover, updating the other agent’s beliefs requires maintaining what it believes you believe. Continuing this reasoning to deeper levels leads to the infinitely nested beliefs defined in Sec. 3.2. However, we will never explicitly construct these infinite beliefs. Instead, we maintain a finite structure that is sufficient to recreate them to arbitrary depth, and only expand as necessary to compute action probabilities. 4 Efficient Filtering In this section, we present an algorithm for performing belief updates bi,∗ t = BE(bi,∗ t−1, ai t−1, oi t) on nested beliefs. This algorithm is applicable in the cooperative setting where there are commonly known policies πj(aj t|bj,∗ t ) for each agent j. The approach, which we call the SDS filter, maintains a set of Sparse Distributions over Sequences of past states, actions, and observations. Sequence distributions. The SDS filter deals with two kinds of sequences: histories hj 0:t = (aj 0:t−1, oj 1:t) and trajectories x0:t = (s0:t,⃗a0:t−1). A history represents what agent j knows before acting at time t; a trajectory is a trace of the states and joint actions through time t. The filter for agent i maintains the following sequence sets: a set X of trajectories that might have occurred so far, and for each agent j (including i itself), a set Hj of possible histories. One of the elements of Hi is marked as being the history that i has actually experienced. The SDS filter maintains belief information in the form of sequence distributions αj(x0:t|hj 0:t) = p(x0:t|hj 0:t) and βj(hj 0:t|x0:t) = p(hj 0:t|x0:t) for all agents j, histories hj 0:t ∈Hj, and trajectories x0:t ∈X.2 The αj distributions represent what agent j would believe about the possible sequences of states and other agents’ actions given hj 0:t. The βj distributions represent the probability of j receiving the observations in hj 0:t if the trajectory x0:t had actually happened. 2Actions are included in both histories and trajectories; when x0:t and hj 0:t specify different actions, both αj(x0:t|hj 0:t) and βj(hj 0:t|x0:t) are zero. The insight behind the SDS filter is that these sequence distributions can be used to compute the nested belief functions Bi,n(hi 0:t) from Sec. 3.2 to arbitrary depth. The main challenge is that sets of possible histories and trajectories grow exponentially with the time t. To avoid this blow-up, the SDS filter does not maintain the complete set of possible sequences. We will see that some sequences can be discarded without affecting the results of the belief computations. If this pruning is insufficient, the SDS filter can drop low-probability sequences and perform approximate filtering. A second challenge is that if we represent each sequence explicitly, the space required grows linearly with t. However, the belief computations do not require the details of each trajectory and history. To compute beliefs about current and future states, it suffices to maintain the sequence distributions αj and βj defined above, along with the final state st in each trajectory. The SDS filter maintains only this information.3 For clarity, we will continue to use full sequence notation in the paper. In the rest of this section, we first show how the sequence distributions can be used to compute nested beliefs of arbitrary depth. Then, we show how to maintain the sequence distributions. Finally, we present an algorithm that computes these distributions while maintaining small sequence sets. The nested beliefs from Sec. 2.2 can be written in terms of the sequence distributions as follows: Bj,0(hj 0:t)(s) = X x0:t∈X : xt=s αj(x0:t|hj 0:t) (2) Bj,n(hj 0:t)(s, b−j,n−1) = X x0:t∈X : xt=s αj(x0:t|hj 0:t) Y k̸=j X hk 0:t∈Hk βk(hk 0:t|x0:t)δ(bk,n−1, Bk,n−1(hk 0:t)) (3) At level zero, we sum over the probabilities according to agent j of all trajectories with the correct final state. At level n, we perform the same outer sum, but for each trajectory we sum the probabilities of the histories for agents k ̸= j that would lead to the beliefs we are interested in. Thus, the sequence distributions at time t are sufficient for computing any desired element of the infinite belief sequence Bj,∗(hj 0:t) for any agent j and history hj 0:t. Updating the distributions. The sequence distributions are updated at each time step t as follows. For each agent j, trajectory x0:t = (s0:t,⃗a0:t−1) and history hj 0:t = (aj 0:t−1, oj 1:t): βj(hj 0:t|x0:t) = βj(hj 0:t−1|x0:t−1)p(oj t|st,⃗at−1) (4) αj(x0:t|hj 0:t) = αj(x0:t−1|hj 0:t−1)p(⃗at−1|x0:t−1)p(st|st−1, oj t,⃗at−1) (5) The values of βj on length-t histories are computed from existing βj values by multiplying in the probability of the most recent observation. To extend αj to length-t trajectories, we multiply in the probability of the state transition and the probability of the agents’ actions given the past trajectory: p(⃗at−1|x0:t−1) = Y k X hk 0:t−1 βk(hk 0:t−1|x0:t−1)πk(ak t−1|Bk,∗(hk 0:t−1)) (6) Here, to predict the actions for agent k, we take an expectation over its possible histories hk 0:t−1 (according to the βk distribution from the previous time step) of the probability of each action ak t−1 given the beliefs Bk,∗(hk 0:t−1) induced by the history. In practice, only some of the entries in Bk,∗(hk 0:t−1) will be needed to compute k’s action; for example, in the tiger world, the policies are functions of the zero-level beliefs. The necessary entries are computed from the the previous α and β distributions as described in Eqs. 2 and 3. This computation is not prohibitive because, as we will see later, we only consider a small subset of the possible histories. Returning to the example tiger world, we can see that maintaining these sequence distributions will allow us to achieve the desired interactions described in Sec. 3.3. For example, when the door opener receives a “signal left” observation, it will infer that the tiger is on the left because it has done the reasoning in Eq. 6 and determined that, with high probability, the trajectories that would have led the tiger listener to take this action are the ones where the tiger is actually on the left. 3This data structure is closely related to probabilistic Kripke structures [3] which are known to be sufficient for recreating nested beliefs. We are not aware of previous work that guarantees compactness through time. Initialization. Input: Distribution p(s) over states. 1. Initialize trajectories and histories: X = {((s), ())|s ∈S}, Hj = {((), ())} 2. Initialize distributions: ∀x = ((s), ()) ∈X, j, hj ∈Hj: αj(x|hj) = p(s) and βj(hj|x) = 1. Filtering. Input: Action ai t−1 and observation oi t. 1. Compute new sequence sets X and Hj, for all agents j, by adding all possible states, actions, and observations to sequences in the previous sets. Compute new sequence distributions αj and βj, for all agents j, as described in Eqs. 5, 4, and 6. Mark the observed history hi 0:t ∈Hi. 2. Merge and drop sequences: (a) Drop trajectories and histories that are commonly known to be impossible: • ∀x0:t ∈X s.t. ∀j, hj 0:t ∈Hj . αj(x0:t|hj 0:t) = 0: Set X = X \ {x0:t}. • ∀j, hj 0:t ∈Hj s.t. ∀x0:t ∈X . βj(hj 0:t|x0:t) = 0: Set Hj = Hj \ {hj 0:t}. (b) Merge histories that lead to the same beliefs: • ∀j, hj 0:t ∈Hj, h′j 0:t ∈Hj s.t. ∀x0:t ∈X . αj(x0:t|hj 0:t) = αj(x0:t|h′j 0:t): Set Hj = Hj \ {h′j 0:t} and βj(hj 0:t|x0:t) = βj(hj 0:t|x0:t) + βj(h′j 0:t|x0:t) for all x0:t. (c) Reset when marginal of st is common knowledge: • If ∀j, k, hj 0:t ∈Hj, hk 0:t, ∈Hk, st . αj(st|hj 0:t) = αk(st|hk 0:t): Reinitialize the filter using the distribution αj(st|hj 0:t) instead of the prior p0(s). 3. Prune: For all αj or βj with m ≥N non-zero entries: Remove the m −N lowest-probability sequences and renormalize. Figure 2: The SDS filter for agent i. At all times t, the filter maintains sequence sets X and Hj, for all agents j, along with the sequence distributions αj and βj for all agents j. Agent i’s actual observed history is marked as a distinguished element hi 0:t ∈Hi and used to compute its beliefs Bi,∗(hi 0:t). Filtering algorithm. We now consider the challenge of maintaining small sequence sets. Fig. 2 provides a detailed description of the SDS filtering algorithm for agent i. The filter is initialized with empty histories for each agent and trajectories with single states that are distributed according to the prior. At each time t, Step 1 extends the sequence sets, computes the sequence distributions, and records agent i’s history. Running a filter with only this step would generate all possible sequences. Step 2 introduces three operations that reduce the size of the sequence sets while guaranteeing that Eqs. 2 and 3 still produce the correct nested beliefs at time t. Step 2(a) removes trajectories and histories when all the agents agree that they are impossible; there is no reason to track them. For example, in the tiger communication world, the policies are such that for the first few time steps each agent will always listen (to the tiger or for signals). During this period all the trajectories where other actions are taken are known to be impossible and can be ignored. Step 2(b) merges histories for an agent j that lead to the same beliefs. This is achieved by arbitrarily selecting one history to be deleted and adding its βj probability to the other’s βj. For example, as the tiger listener hears roars, any two observation sequences with the same numbers of roars on the left and right provide the same information about the tiger and can be merged. Step 2(c) resets the filter if the marginal over states at time t has become commonly known to all the agents. For example, when both agents know that a door has been opened, this implies that the world has reset and all previous trajectories and histories can be discarded. This type of agreement is not limited to cases where the state of the world is reset. It occurs with any distribution over states that the agents agree on, for example when they localize and both know the true state, even if they disagree about the trajectory of past states. Together, these three operators can significantly reduce the size of the sequence sets. We will see in the experiments (Sec. 5) that they enable the SDS filter to exactly track the tiger communication world extremely efficiently. However, in general, there is no guarantee that these operators will be enough to maintain small sets of trajectories and histories. Step 3 introduces an approximation by removing low-probability sequences and normalizing the belief distributions. This does guarantee that we will maintain small sequence sets, possibly at the cost of accuracy. In many domains we can ignore unlikely histories and trajectories without significantly changing the current beliefs. 5 Evaluation In this section, we describe the performance of the SDS algorithm on three nested filtering problems. 0 2 4 6 8 10 0 5 10 15 20 25 30 Running Time (seconds) Time Step SDS -a,-b,-c SDS -b,-c SDS -c SDS (a) Tiger world: time. 0 2 4 6 8 10 12 14 0 5 10 15 20 Running Time (seconds) Time Step SDS N=10 SDS N=50 SDS N=100 SDS N=∞ (b) Box pushing: time. 0 0.05 0.1 0.15 0.2 0.25 0 5 10 15 20 Empirical Variational Distance Time Step SDS N=10 SDS N=50 SDS N=100 (c) Box pushing: error. Figure 3: Time per filtering step, and error, for the SDS algorithm on two domains. Tiger Communication World. The tiger communication world was described in detail in Sec. 3.3. Fig. 3(a) shows the average computation time used for filtering at each time step. The full algorithm (SDS) maintains a compact, exact representation without any pruning and takes only a fraction of a second to do each update. The graph also shows the results of disabling different parts of Step 2(a-c) of the algorithm (for example, SDS -a,-b,-c does not do any simplifications from Step 2). Without these steps, the algorithm runs in exponential time. Each simplification allows the algorithm to perform better, but all are required for constant-time performance. Since the SDS filter runs without the pruning in Step 3, we know that it computes the correct beliefs; there is no approximation error.4 Box Pushing. The DEC-POMDP literature includes several multi-agent domains; we evaluate SDS on the largest of them, known as the box-pushing domain [9]. In this scenario, two agents interact in a 3x4 grid world where they must coordinate their actions to move a large box and then independently push two small boxes. The state encodes the positions and orientations of the robots, as well as the locations of the three boxes. The agents can move forward, rotate left and right, or stay still. These actions fail with probability 0.1, leaving the state unchanged. Each agent receives deterministic observations about what is in the location in front of it (empty space, a robot, etc.). We implemented policies for each agent that consist of a set of 20 rules specifying actions given its zeroth-level beliefs about the world state. While executing their policies, the agents first coordinate to move the large box and then independently move the two small boxes. The policies are such that, with high probability, the agents will always move the boxes. There is uncertainty about when this will happen, since actions can fail. We observed, in practice, that it rarely took more than 20 steps. Fig. 3(b) shows the running time of the SDS filter on this domain, with various pruning parameters (N = 10, 50, 100, ∞in Step 3). Without pruning (N = ∞), the costs are too high for the filter to move beyond time step five. With pruning, however, the cost remains reasonable. Fig. 3(c) shows the error incurred with various degrees of pruning, in terms of the difference between the estimated zeroth-level beliefs for the agents and the true posterior over physical states given their observations.5 Note that in order to accurately maintain each agent’s beliefs about the physical state—which includes the position of the other robot—the filter must assign accurate probabilities to unobserved actions by the other agent , which depend on its beliefs. This is the same reasoning pattern we saw in the tiger world where we are required to maintain infinitely nested beliefs. As expected, we see that more pruning leads to faster running time but decreased accuracy. We also find that the problem is most challenging around time step ten and becomes easier in the limit, as the world moves towards the absorbing state where both agents have finished their tasks. With N = 100, we get high-quality estimates in an acceptable amount of time. Noisy Muddy Children. The muddy children problem is a classic puzzle often discussed by researchers in epistemic logic [4]. There are n agents and 2n possible states. Each agent’s forehead can be either muddy or clean, but it does not get any direct observations about this fact. Initially, it is commonly known that at least one agent has a muddy forehead. As time progresses, the agents follow a policy of raising their hand if they know that their forehead is muddy; they must come to this conclusion given only observations about the cleanliness of the other agents’ foreheads and who has 4The exact version of SDS also runs in constant time on the broadcast channel domain of Hansen et al. [6]. 5Because the box-pushing problem is too large for beliefs to be computed exactly, we compare the filter’s performance to empirical distributions obtained by generating 10,000 sequences of trajectories and histories. We group the runs by the history hi 0:t; for all histories that appear at least ten times, we compare the empirical distribution ˆbt of states occurring after that history to the filter’s computed beliefs ˜bi,0 t , using the variational distance V D(ˆbt,˜bi,0 t ) = P s |ˆbt(s) −˜bi,0 t (s)|. raised their hands (this yields 22n possible observations for each agent). This puzzle is represented in our framework as follows. The initial knowledge is encoded with a prior that is uniform over all states with in which at least one agent is muddy. The state of the world never changes. Observations about the muddiness of the other agents are only correct with probability ν, and each agent raises its hand if it assigns probability at least 0.8 to being muddy. When there is no noise, ν = 1.0, the agents behave as follows. With m ≤n muddy agents, everyone waits m time steps and then all of the muddy agents simultaneously raise their hands.6 The SDS filter exhibits exactly this behavior and runs in reasonable time, using only a few seconds per filtering step, for problem instances with up to 10 agents without pruning. We also ran the filter on instances with noise (ν = 0.9) and up to 5 agents. This required pruning histories to cope with the extremely large number of possible but unlikely observation sequences. The observed behavior is similar to the deterministic case: eventually, all of the m muddy agents raise their hands. In expectation, this happens at a time step greater than m, since the agents must receive multiple observations before they are confident about each other’s cleanliness. If one agent raises its hand before the others, this provides more information to the uncertain agents, who usually raise their hands soon after. 6 Conclusions We have considered the problem of efficient belief update in multi-agent scenarios. We introduced the SDS algorithm, which maintains a finite belief representation that can be used to compute an infinite sequence of nested beliefs about the physical world and the beliefs of other agents. We demonstrated that on some problems, SDS can maintain this representation exactly in constant time per filtering step. On more difficult examples, SDS maintains constant-time filtering by pruning low-probability trajectories, yielding acceptable levels of approximation error. These results show that efficient filtering is possible in multi-agent scenarios where the agents’ policies are expressed as functions of their beliefs, rather than their entire observation histories. These belief-based policies are independent of the current time step, and have the potential to be more compact than history-based policies. In the single-agent setting, many successful POMDP planning algorithms construct belief-based policies; we plan to investigate how to do similar beliefbased planning in the multi-agent case. References [1] D. S. Bernstein, E. Hansen, and S. Zilberstein. Bounded policy iteration for decentralized POMDPs. In Proc. of the 19th International Joint Conference on Artificial Intelligence (IJCAI), 2005. [2] A. Brandenburger and E. Dekel. Hierarchies of beliefs and common knowledge. Journal of Economic Theory, 59:189–198, 1993. [3] R. Fagin and J. Y. Halpern. Reasoning about knowledge and probability. Journal of the ACM, 41(2):340– 367, 1994. [4] R. Fagin, J. Y. Halpern, Y. Moses, and M. Y. Vardi. Reasoning About Knowledge. The MIT Press, 1995. [5] P. J. Gmytrasiewicz and P. Doshi. A framework for sequential planning in multi-agent settings. Journal of Artificial Intelligence Research, 24:49–79, 2005. [6] E. A. Hansen, D. S. Bernstein, and S. Zilberstein. Dynamic programming for partially observable stochastic games. In Proc. of the 19th National Conf, on Artificial Intelligence (AAAI), 2004. [7] L. P. Kaelbling, M. L. Littman, and A. R. Cassandra. Planning and acting in partially observable stochastic domains. Artificial Intelligence, 101:99–134, 1998. [8] B. Milch and D. Koller. Probabilistic models for agents’ beliefs and decisions. In Proc. 16th Conference on Uncertainty in Artificial Intelligence (UAI), 2000. [9] S. Seuken and S. Zilberstein. Improved memory-bounded dynamic programming for decentralized POMDPs. In Proc. of the 23rd Conference on Uncertainty in Artificial Intelligences (UAI), 2007. [10] A. Shirazi and E. Amir. Probabilistic modal logic. In Proc. of the 22nd National Conference on Artificial Intelligence (AAAI), 2007. 6This behavior can be verified by induction. If there is one muddy agent, it will see that the others are clean and raise its hand immediately. This implies that if no one raises their hand in the first round, there must be at least two muddy agents. At time two, they will both see only one other muddy agent and infer that they are muddy. The pattern follows for larger m.
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Beyond Novelty Detection: Incongruent Events, when General and Specific Classifiers Disagree Abstract Unexpected stimuli are a challenge to any machine learning algorithm. Here we identify distinct types of unexpected events, focusing on ’incongruent events’ when ’general level’ and ’specific level’ classifiers give conflicting predictions. We define a formal framework for the representation and processing of incongruent events: starting from the notion of label hierarchy, we show how partial order on labels can be deduced from such hierarchies. For each event, we compute its probability in different ways, based on adjacent levels (according to the partial order) in the label hierarchy. An incongruent event is an event where the probability computed based on some more specific level (in accordance with the partial order) is much smaller than the probability computed based on some more general level, leading to conflicting predictions. We derive algorithms to detect incongruent events from different types of hierarchies, corresponding to class membership or part membership. Respectively, we show promising results with real data on two specific problems: Out Of Vocabulary words in speech recognition, and the identification of a new sub-class (e.g., the face of a new individual) in audio-visual facial object recognition. 1 Introduction Machine learning builds models of the world using training data from the application domain and prior knowledge about the problem. The models are later applied to future data in order to estimate the current state of the world. An implied assumption is that the future is stochastically similar to the past. The approach fails when the system encounters situations that are not anticipated from the past experience. In contrast, successful natural organisms identify new unanticipated stimuli and situations and frequently generate appropriate responses. By definition, an unexpected event is one whose probability to confront the system is low, based on the data that has been observed previously. In line with this observation, much of the computational work on novelty detection focused on the probabilistic modeling of known classes, identifying outliers of these distributions as novel events (see e.g. [1, 2] for recent reviews). More recently, oneclass classifiers have been proposed and used for novelty detection without the direct modeling of data distribution [3, 4]. There are many studies on novelty detection in biological systems [5], often focusing on regions of the hippocampus [6]. To advance beyond the detection of outliers, we observe that there are many different reasons why some stimuli could appear novel. Our work, presented in Section 2, focuses on unexpected events which are indicated by the incongruence between prediction induced by prior experience (training) and the evidence provided by the sensory data. To identify an item as incongruent, we use two parallel classifiers. One of them is strongly constrained by specific knowledge (both prior and dataderived), the other classifier is more general and less constrained. Both classifiers are assumed to yield class-posterior probability in response to a particular input signal. A sufficiently large discrepancy between posterior probabilities induced by input data in the two classifiers is taken as indication that an item is incongruent. Thus, in comparison with most existing work on novelty detection, one new and important characteristic of our approach is that we look for a level of description where the novel event is highly probable. Rather than simply respond to an event which is rejected by all classifiers, which more often than not requires no special attention (as in pure noise), we construct and exploit a hierarchy of 1 representations. We attend to those events which are recognized (or accepted) at some more abstract levels of description in the hierarchy, while being rejected by the more concrete classifiers. There are various ways to incorporate prior hierarchical knowledge and constraints within different classifier levels, as discussed in Section 3. One approach, used to detect images of unexpected incongruous objects, is to train the more general, less constrained classifier using a larger more diverse set of stimuli, e.g., the facial images of many individuals. The second classifier is trained using a more specific (i.e. smaller) set of specific objects (e.g., the set of Einstein’s facial images). An incongruous item (e.g., a new individual) could then be identified by a smaller posterior probability estimated by the more specific classifiers relative to the probability from the more general classifier. A different approach is used to identify unexpected (out-of-vocabulary) lexical items. The more general classifier is trained to classify sequentially speech sounds (phonemes) from a relatively short segments of the input speech signal (thus yielding an unconstrained sequence of phoneme labels); the more constrained classifier is trained to classify a particular set of words (highly constrained sequences of phoneme labels) from the information available in the whole speech sentence. A word that did not belong to the expected vocabulary of the more constrained recognizer could then be identified by discrepancy in posterior probabilities of phonemes derived from both classifiers. Our second contribution in Section 2 is the presentation of a unifying theoretical framework for these two approaches. Specifically, we consider two kinds of hierarchies: Part membership as in biological taxonomy or speech, and Class membership, as in human categorization (or levels of categorization). We define a notion of partial order on such hierarchies, and identify those events whose probability as computed using different levels of the hierarchy does not agree. In particular, we are interested in those events that receive high probability at more general levels (for example, the system is certain that the new example is a dog), but low probability at more specific levels (in the same example, the system is certain that the new example is not any known dog breed). Such events correspond to many interesting situations that are worthy of special attention, including incongruous scenes and new sub-classes, as shown in Section 3. 2 Incongruent Events - unified approach 2.1 Introducing label hierarchy The set of labels represents the knowledge base about stimuli, which is either given (by a teacher in supervised learning settings) or learned (in unsupervised or semi-supervised settings). In cognitive systems such knowledge is hardly ever a set; often, in fact, labels are given (or can be thought of) as a hierarchy. In general, hierarchies can be represented as directed graphs. The nodes of the graphs may be divided into distinct subsets that correspond to different entities (e.g., all objects that are animals); we call these subsets “levels”. We identify two types of hierarchies: Part membership, as in biological taxonomy or speech. For example, eyes, ears, and nose combine to form a head; head, legs and tail combine to form a dog. Class membership, as in human categorization – where objects can be classified at different levels of generality, from sub-ordinate categories (most specific level), to basic level (intermediate level), to super-ordinate categories (most general level). For example, a Beagle (sub-ordinate category) is also a dog (basic level category), and it is also an animal (super-ordinate category). The two hierarchies defined above induce constraints on the observed features in different ways. In the class-membership hierarchy, a parent class admits higher number of combinations of features than any of its children, i.e., the parent category is less constrained than its children classes. In contrast, a parent node in the part-membership hierarchy imposes stricter constraints on the observed features than a child node. This distinction is illustrated by the simple ”toy” example shown in Fig. 1. Roughly speaking, in the class-membership hierarchy (right panel), the parent node is the disjunction of the child categories. In the part-membership hierarchy (left panel), the parent category represents a conjunction of the children categories. This difference in the effect of constraints between the two representations is, of course, reflected in the dependency of the posterior probability on the class, conditioned on the observations. 2
Figure 1: Examples. Left: part-membership hierarchy, the concept of a dog requires a conjunction of parts a head, legs and tail. Right: class-membership hierarchy, the concept of a dog is defined as the disjunction of more specific concepts - Afghan, Beagle and Collie. In order to treat different hierarchical representations uniformly we invoke the notion of partial order. Intuitively speaking, different levels in each hierarchy are related by a partial order: the more specific concept, which corresponds to a smaller set of events or objects in the world, is always smaller than the more general concept, which corresponds to a larger set of events or objects. To illustrate this point, consider Fig. 1 again. For the part-membership hierarchy example (left panel), the concept of ’dog’ requires a conjunction of parts as in DOG = LEGS ∩HEAD ∩TAIL, and therefore, for example, DOG ⊂LEGS ⇒DOG ⪯LEGS. Thus DOG ⪯LEGS, DOG ⪯HEAD, DOG ⪯TAIL In contrast, for the class-membership hierarchy (right panel), the class of dogs requires the conjunction of the individual members as in DOG = AFGHAN ∪BEAGEL ∪COLLIE, and therefore, for example, DOG ⊃AFGHAN ⇒DOG ⪰AFGHAN . Thus DOG ⪰AFGHAN, DOG ⪰BEAGEL, DOG ⪰COLLIE 2.2 Definition of Incongruent Events Notations We assume that the data is represented as a Graph {G, E} of Partial Orders (GPO). Each node in G is a random variable which corresponds to a class or concept (or event). Each directed link in E corresponds to partial order relationship as defined above, where there is a link from node a to node b iff a ⪯b. For each node (concept) a, define As = {b ∈G, b ⪯a} - the set of all nodes (concepts) b more specific (smaller) than a in accordance with the given partial order; similarly, define Ag = {b ∈ G, a ⪯b} - the set of all nodes (concepts) b more general (larger) than a in accordance with the given partial order. For each concept a and training data T , we train up to 3 probabilistic models which are derived from T in different ways, in order to determine whether the concept a is present in a new data point X: • Qa(X): a probabilistic model of class a, derived from training data T without using the partial order relations in the GPO. • If |As| > 1 Qs a(X): a probabilistic model of class a which is based on the probability of concepts in As, assuming their independence of each other. Typically, the model incorporates some relatively simple conjunctive and/or disjunctive relations among concepts in As. • If |Ag| > 1 Qg a(X): a probabilistic model of class a which is based on the probability of concepts in Ag, assuming their independence of each other. Here too, the model typically incorporates some relatively simple conjunctive and/or disjunctive relations among concepts in Ag. 3 Examples To illustrate, we use the simple examples shown in Fig. 1, where our concept of interest a is the concept ‘dog’: In the part-membership hierarchy (left panel), |Ag| = 3 (head, legs, tail). We can therefore learn 2 models for the class ‘dog’ (Qs dog is not defined): 1. Qdog - obtained using training pictures of ’dogs’ and ’not dogs’ without body part labels. 2. Qg dog - obtained using the outcome of models for head, legs and tail, which were trained on the same training set T with body part labels. For example, if we assume that concept a is the conjunction of its part member concepts as defined above, and assuming that these part concepts are independent of each other, we get Qg dog = Y b∈Ag Qb = QHead · QLegs · QTail (1) In the class-membership hierarchy (right panel), |As| = 3 (Afghan, Beagle, Collie). If we further assume that a class-membership hierarchy is always a tree, then |Ag| = 1. We can therefore learn 2 models for the class ‘dog’ (Qg dog is not defined): 1. Qdog - obtained using training pictures of ’dogs’ and ’not dogs’ without breed labels. 2. Qs dog - obtained using the outcome of models for Afghan, Beagle and Collie, which were trained on the same training set T with only specific dog type labels. For example, if we assume that concept a is the disjunction of its sub-class concepts as defined above, and assuming that these sub-class concepts are independent of each other, we get Qs dog = X b∈As Qb = QAfghan + QBeagle + QCollie Incongruent events In general, we expect the different models to provide roughly the same probability for the presence of concept a in data X. A mismatch between the predictions of the different models should raise the red flag, possibly indicating that something new and interesting had been observed. In particular, we are interested in the following discrepancy: Definition: Observation X is incongruent if there exists a concept ′a′ such that Qg a(X) ≫Qa(X) or Qa(X) ≫Qs a(X). (2) Alternatively, observation X is incongruent if a discrepancy exists between the inference of the two classifiers: either the classifier based on the more general descriptions from level g accepts the X while the direct classier rejects it, or the direct classifier accepts X while the classifier based on the more specific descriptions from level s rejects it. In either case, the concept receives high probability at the more general level (according to the GPO), but much lower probability when relying only on the more specific level. Let us discuss again the examples we have seen before, to illustrate why this definition indeed captures interesting “surprises”: • In the part-membership hierarchy (left panel of Fig. 1), we have Qg dog = QHead · QLegs · QTail ≫Qdog In other words, while the probability of each part is high (since the multiplication of those probabilities is high), the ’dog’ classifier is rather uncertain about the existence of a dog in this data. How can this happen? Maybe the parts are configured in an unusual arrangement for a dog (as in a 3-legged cat), or maybe we encounter a donkey with a cat’s tail (as in Shrek 3). Those are two examples of the kind of unexpected events we are interested in. 4 • In the class-membership hierarchy (right panel of Fig. 1), we have Qs dog = QAfghan + QBeagle + QCollie ≪Qdog In other words, while the probability of each sub-class is low (since the sum of these probabilities is low), the ’dog’ classifier is certain about the existence of a dog in this data. How may such a discrepancy arise? Maybe we are seeing a new type of dog that we haven’t seen before - a Pointer. The dog model, if correctly capturing the notion of ’dogness’, should be able to identify this new object, while models of previously seen dog breeds (Afghan, Beagle and Collie) correctly fail to recognize the new object. 3 Incongruent events: algorithms Our definition for incongruent events in the previous section is indeed unified, but as a result quite abstract. In this section we discuss two different algorithmic implementations, one generative and one discriminative, which were developed for the part membership and class membership hierarchies respectively (see definition in Section 1). In both cases, we use the notation Q(x) for the class probability as defined above, and p(x) for the estimated probability. 3.1 Part membership - a generative algorithm Consider the left panel of Fig. 1. The event in the top node is incongruent if its probability is low, while the probability of all its descendants is high. In many applications, such as speech recognition, one computes the probability of events (sentences) based on a generative model (corresponding to a specific language) which includes a dictionary of parts (words). At the top level the event probability is computed conditional on the model; in which case typically the parts are assumed to be independent, and the event probability is computed as the multiplication of the parts probabilities conditioned on the model. For example, in speech processing and assuming a specific language (e.g., English), the probability of the sentence is typically computed by multiplying the probability of each word using an HMM model trained on sentences from a specific language. At the bottom level, the probability of each part is computed independently of the generative model. More formally, Consider an event u composed of parts wk. Using the generative model of events and assuming the conditional independence of the parts given this model, the prior probability of the event is given by the product of prior probabilities of the parts, p(u|L) = Y k p(wk|L) (3) where L denotes the generative model (e.g., the language). For measurement X, we compute Q(X) as follows Q(X) = p(X|L) = X u p(X|u, L)p(u|L) ≈p(X|¯u, L)p(¯u|L) = p(X|¯u) Y k p(wk|L) (4) using p(X|u, L) = p(X|u) and (3), and where ¯u = arg max u p(u|L) is the most likely interpretation. At the risk of notation abuse, {wk} now denote the parts which compose the most likely event ¯u. We assume that the first sum is dominated by the maximal term. Given a part-membership hierarchy, we can use (1) to compute the probability Qg(X) directly, without using the generative model L. Qg(X) = p(X) = X u p(X|u)p(u) ≥p(X|¯u)p(¯u) = p(X|¯u) Y k p(wk) (5) It follows from (4) and (5) that Q(X) Qg(X) ≤ Y k p(wk|L) p(wk) (6) 5 We can now conclude that X is an incongruent event according to our definition if there exists at least one part k in the final event ¯u, such that p(wk) ≫p(wk|L) (assuming all other parts have roughly the same conditional and unconditional probabilities). In speech processing, a sentence is incongruent if it includes an incongruent word - a word whose probability based on the generative language model is low, but whose direct probability (not constrained by the language model) is high. Example: Out Of Vocabulary (OOV) words For the detection of OOV words, we performed experiments using a Large Vocabulary Continuous Speech Recognition (LVCSR) system on the Wall Street Journal Corpus (WSJ). The evaluation set consists of 2.5 hours. To introduce OOV words, the vocabulary was restricted to the 4968 most frequent words from the language training texts, leaving the remaining words unknown to the model. A more detailed description is given in [7]. In this task, we have shown that the comparison between two parallel classifiers, based on strong and weak posterior streams, is effective for the detection of OOV words, and also for the detection of recognition errors. Specifically, we use the derivation above to detect out of vocabulary words, by comparing their probability when computed based on the language model, and when computed based on mere acoustic modeling. The best performance was obtained by the system when a Neural Network (NN) classifier was used for the direct estimation of frame-based OOV scores. The network was directly fed by posteriors from the strong and the weak systems. For the WSJ task, we achieved performance of around 11% Equal-Error-Rate (EER) (Miss/False Alarm probability), see Fig. 2. Figure 2: Several techniques used to detect OOV: (i) Cmax: Confidence measure computed ONLY from strongly constrained Large Vocabulary Continuous Speech Recognizer (LVCSR), with frame-based posteriors. (ii) LVCSR+weak features: Strongly and weakly constrained recognizers, compared via the KL-divergence metric. (iii) LVCSR+NN posteriors: Combination of strong and weak phoneme posteriors using NN classifier. (iv) all features: fusion of (ii) and (iii) together. 3.2 Class membership - a discriminative algorithm Consider the right panel of Fig. 1. The general class in the top node is incongruent if its probability is high, while the probability of all its sub-classes is low. In other words, the classifier of the parent object accepts the new observation, but all the children object classifiers reject it. Brute force computation of this definition may follow the path taken by traditional approaches to novelty detection, e.g., looking for rejection by all one class classifiers corresponding to sub-class objects. The result we have obtained by this method were mediocre, probably because generative models are not well suited for the task. Instead, it seems like discriminative classifiers, trained to discriminate 6 between objects at the sub-class level, could be more successful. We note that unlike traditional approaches to novelty detection, which must use generative models or one-class classifiers in the absence of appropriate discriminative data, our dependence on object hierarchy provides discriminative data as a by-product. In other words, after the recognition by a parent-node classifier, we may use classifiers trained to discriminate between its children to implement a discriminative novelty detection algorithm. Specifically, we used the approach described in [8] to build a unified representation for all objects in the sub-class level, which is the representation computed for the parent object whose classifier had accepted (positively recognized) the object. In this feature space, we build a classifier for each sub-class based on the majority vote between pairwise discriminative classifiers. Based on these classifiers, each example (accepted by the parent classifier) is assigned to one of the sub-classes, and the average margin over classifiers which agree with the final assignment is calculated. The final classifier then uses a threshold on this average margin to identify each object as known sub-class or new sub-class. Previous research in the area of face identification can be viewed as an implicit use of this propsed framework, see e.g. [9]. Example: new face recognition from audio-visual data We tested our algorithm on audio-visual speaker verification. In this setup, the general parent category level is the ‘speech’ (audio) and ‘face’ (visual), and the different individuals are the offspring (sub-class) levels. The task is to identify an individual as belonging to the trusted group of individuals vs. being unknown, i.e. known sub-class vs. new sub-class in a class membership hierarchy. The unified representation of the visual cues was built using the approach described in [8]. All objects in the sub-class level (different individuals) were represented using the representation learnt for the parent level (’face’). For the audio cues we used the Perceptual linear predictive (PLP) Cepstral features [10] as the unified representation. We used SVM classifiers with RBF kernel as the pairwise discriminative classifiers for each of the different audio/visual representations separately. Data was collected for our experiments using a wearable device, which included stereo panoramic vision sensors and microphone arrays. In the recorded scenario, individuals walked towards the device and then read aloud an identical text; we acquired 30 sequences with 17 speakers (see Fig. 3 for an example). We tested our method by choosing six speakers as members of the trusted group, while the rest were assumed unknown. The method was applied separately using each one of the different modalities, and also in an integrated manner using both modalities. For this fusion the audio signal and visual signal were synchronized, and the winning classification margins of both signals were normalized to the same scale and averaged to obtain a single margin for the combined method. Since the goal is to identify novel incongruent events, true positive and false positive rates were calculated by considering all frames from the unknown test sequences as positive events and the known individual test sequences as negative events. We compared our method to novelty detection based on one-class SVM [3] extended to our multi-class case. Decision was obtained by comparing the maximal margin over all one-class classifiers to a varying threshold. As can be seen in Fig. 3, our method performs substantially better in both modalities as compared to the “standard” one class approach for novelty detection. Performance is further improved by fusing both modalities. 4 Summary Unexpected events are typically identified by their low posterior probability. In this paper we employed label hierarchy to obtain a few probability values for each event, which allowed us to tease apart different types of unexpected events. In general there are 4 possibilities, based on the classifiers’ response at two adjacent levels: Specific level General level possible reason 1 reject reject noisy measurements, or a totally new concept 2 reject accept incongruent concept 3 accept reject inconsistent with partial order, models are wrong 4 accept accept known concept 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 False positive rate True positive rate audio visual audio−visual audio (OC−SVM) visual (OC−SVM) Figure 3: Left: Example: one frame used for the visual verification task. Right: True Positive vs. False Positive rates when detecting unknown vs. trusted individuals. The unknown are regarded as positive events. Results are shown for the proposed method using both modalities separately and the combined method (solid lines). For comparison, we show results with a more traditional novelty detection method using One Class SVM (dashed lines). We focused above on the second type of events - incongruent concepts, which have not been studied previously in isolation. Such events are characterized by some discrepancy between the response of two classifiers, which can occur for a number different reasons: Context: in a given context such as the English language, a sentence containing a Czech word is assigned low probability. In the visual domain, in a given context such as a street scene, otherwise high probability events such as “car” and “elephant” are not likely to appear together. New sub-class: a new object has been encountered, of some known generic type but unknown specifics. We described how our approach can be used to design new algorithms to address these problems, showing promising results on real speech and audio-visual facial datasets. References [1] Markou, M., Singh, S.: Novelty detection: a review-part 1: statistical approaches. Signal Processing 83 (2003) 2499 – 2521 [2] Markou, M., Singh, S.: Novelty detection: a review-part 2: neural network based approaches. Signal Processing 83 (2003) 2481–2497 [3] Scholkopf, B., Williamson, R.C., Smola, A.J., Shawe-Taylor, J., Platt, J.: Support vector method for novelty detection. In: Proc. NIPS. Volume 12. (2000) 582–588 [4] Lanckrietand, G.R.G., Ghaoui, L.E., Jordan, M.I.: Robust novelty detection with single-class mpm. In: Proc. NIPS. Volume 15. (2003) 929–936 [5] Berns, G.S., Cohen, J.D., Mintun, M.A.: Brain regions responsive to novelty in the absence of awareness. Science 276 (1997) 1272 – 1275 [6] Rokers, B., Mercado, E., Allen, M.T., Myers, C.E., Gluck, M.A.: A connectionist model of septohippocampal dynamics during conditioning: Closing the loop. Behavioral Neuroscience 116 (2002) 48–62 [7] Burget, L., Schwarz, P., Matejka, P., Hannemann, M., Rastrow, A., White, C., Khudanpur, S., Hermansky, H., Cernocky, J.: Combination of strongly and weakly constrained recognizers for reliable detection of oovs. In: Proceedings of IEEE Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP). (2008) [8] Bar-Hillel, A., Weinshall, D.: Subordinate class recognition using relational object models. Proc. NIPS 19 (2006) [9] Lanitis, A., Taylor, C.J., Cootes, T.F.: A unified approach to coding and interpreting face images. In: Proc. ICCV. (1995) 368–373 [10] Hermansky, H.: Perceptual linear predictive (PLP) analysis of speech. The Journal of the Acoustical Society of America 87 (1990) 1738 8
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Look Ma, No Hands: Analyzing the Monotonic Feature Abstraction for Text Classification Doug Downey Electrical Engineering and Computer Science Department Northwestern University Evanston, IL 60208 ddowney@eecs.northwestern.edu Oren Etzioni Turing Center, Department of Computer Science and Engineering University of Washington Seattle, WA 98195 etzioni@cs.washington.edu Abstract Is accurate classification possible in the absence of hand-labeled data? This paper introduces the Monotonic Feature (MF) abstraction—where the probability of class membership increases monotonically with the MF’s value. The paper proves that when an MF is given, PAC learning is possible with no hand-labeled data under certain assumptions. We argue that MFs arise naturally in a broad range of textual classification applications. On the classic “20 Newsgroups” data set, a learner given an MF and unlabeled data achieves classification accuracy equal to that of a state-of-the-art semi-supervised learner relying on 160 hand-labeled examples. Even when MFs are not given as input, their presence or absence can be determined from a small amount of hand-labeled data, which yields a new semi-supervised learning method that reduces error by 15% on the 20 Newsgroups data. 1 Introduction Is accurate classification possible in the complete absence of hand-labeled data? A Priori, the answer would seem to be no, unless the learner has knowledge of some additional problem structure. This paper identifies a problem structure, called Monotonic Features (MFs), that enables the learner to automatically assign probabilistic labels to data. A feature is monotonic when the probability of class membership increases monotonically with that feature’s value, all else being equal. MFs occur naturally in a broad range of textual classification tasks. For example, it can be shown that Naive Bayes text classifiers return probability estimates that are monotonic in the frequency of a word—for the class in which the word is most common. Thus, if we are trying to discriminate between documents about New York and Boston, then we expect to find that the Naive Bayes feature measuring the frequency of “Giants” in the corpus is an MF for the class New York, and likewise for “Patriots” and Boston. In document classification, the name of the class is a natural MF—the more times it is repeated in a document, all other things being equal, the more likely it is that the document belongs to the class. We demonstrate this to be the case empirically in Section 4, extending the experiments of [8] and [3]. Similarly, information retrieval systems classify documents into relevant and irrelevant documents based, in part, on the Term-Frequency-Inverse-Document-Frequency (TF-IDF) metric, and then proceed to rank the relevant documents. The term frequency component of this metric is a monotonic feature. The power of MFs is not restricted to textual classification. Consider Information Extraction (IE) where strings are extracted from sentences and classified into categories (e.g. City or Film) based on their proximity to “extraction patterns”. For example, the phrase “cities such as” is an extraction pattern. Any proper noun immediately following this pattern is likely to denote a city, as in the phrase “cities such as Boston, Seattle, and New York” [9]. When classifying a proper noun, the number of times that it follows an extraction pattern in a corpus turns out to be a powerful MF. This observation is implicit in the combinatorial model of unsupervised IE put forth in [6]. Finally, MF-based techniques have been demonstrated to be effective for word-sense disambiguation using a set of manually-specified MFs [13]; this work was later extended to automatically derive MFs from resources like Wordnet [10]. Thus, MFs have been used implicitly in a broad range of textual classification tasks. This paper makes the MF abstraction explicit, provides a formal theory of MFs, an automatic method for explicitly detecting and utilizing MFs, and quantifies the method’s benefits empirically. 1.1 Contribution Typically, MFs cannot serve directly as classifiers. Instead, this paper presents theoretical and empirical results showing that even relatively weak MFs can be used to induce a noisy labeling over examples, and these examples can then be used to train effective classifiers utilizing existing supervised or semi-supervised techniques. Our contributions are as follows: 1. We prove that the Monotonic Feature (MF) structure guarantees PAC learnability using only unlabeled data, and that MFs are distinct from and complementary to standard biases used in semisupervised learning, including the manifold and cluster assumptions. 2. We present a general technique, called MFA, for employing MFs in combination with an arbitrary concept learning algorithm. We demonstrate experimentally that MFA can outperform state-of-theart techniques for semi-supervised document classification, including Naive Bayes with Expectation Maximization (NB-EM), and Label Propagation, on the 20 Newsgroups data set [11]. The remainder of the paper is organized as follows. Section 2 formally defines our problem structure and the properties of monotonic features. Section 3 presents our theoretical results, and formalizes the relationship between the MF approach and previous work. We present experimental results in Section 4, and conclude with directions for future work. 2 Formal Framework We consider a semi-supervised classification task, in which the goal is to produce a mapping from an instance space X consisting of d-tuples x = (x1, . . . , xd), to a binary output space Y = {0, 1}.1 We denote the concept class of mappings f : X →Y as C. We assume the following inputs: • A set of zero or more labeled examples DL = {(xi, yi)|i = 1 . . . n}, drawn i.i.d. from a distribution P(x, y) for x ∈X and y ∈Y. • A set of zero or more unlabeled examples DU = {(xi)|i = 1 . . . u} drawn from the marginal distribution P(x) = P y P(x, y). • A set M ⊂{1, . . . , d} of zero or more monotonic features for the positive class y = 1. The monotonic features have properties specified below. The goal of the classification task is to produce a mapping c ∈C that maximizes classification accuracy evaluated over a set of test examples drawn i.i.d. from P(x, y). 1For convenience, we restrict our formal framework to the binary case, but the techniques and analysis can be extended trivially to the multi-class case. We further define CM ⊂C as the concept class of binary classifiers that use only the monotonic features. Similarly, let C¬M ⊂C indicate the concept class of binary classifiers using only the non-monotonic features. Monotonic features exhibit a monotonically increasing relationship with the probability that an example is a member of the positive class. More formally, we define monotonic features as follows: Definition 1 A monotonic feature for class y is a feature i ∈{1, . . . , d} for which the following three properties hold: • The domain of xi is fully ordered and discrete, and has finite support.2 • The conditional probability that an example is an element of class y = 1 increases strictly monotonically with the value of xi. That is, P(y = 1|xi = r) > P(y = 1|xi = r′) if r > r′. • The monotonicity is non-trivial in that P(xi) has positive probability for more than one feature value. That is, there exists r > r′ and ϵ > 0 such that P(xi = r), P(xi = r′) > ϵ. With this definition, we can state precisely the monotonic feature structure: Definition 2 For a learning problem from the input space X of d-tuples x = (x1, . . . , xd) to the output space Y, the monotonic feature structure (MFS) holds if and only if at least one of the features i ∈{1, . . . , d} is a monotonic feature for the positive class y = 1. When tasked with a learning problem for which the MFS holds, three distinct configurations of the input are possible. First, monotonic features may be known in the absence of labeled data (|M| > 0, DL = ∅). This is the setting considered in previous applications of monotonic features, as discussed in the introduction. Second, monotonic features may be unknown, but labeled data may be provided (M = ∅, |DL| > 0); this corresponds to standard semi-supervised learning. In this case, the MFS can still be exploited by identifying monotonic features using the labeled data. Lastly, both monotonic features and labeled data may be provided (|M|, |DL| > 0). We provide algorithms for each case and evaluate each experimentally in Section 4. 3 Theory of Monotonic Features This section shows that under certain assumptions, knowing the identity of a single monotonic feature is sufficient to PAC learn a target concept from only unlabeled data. Further, we prove that monotonic features become more informative relative to labeled examples as the feature set size increases. Lastly, we discuss and formally establish distinctions between the monotonic feature abstraction and other semi-supervised techniques. We start by introducing the conditional independence assumption, which states that the monotonic features are conditionally independent of the non-monotonic features given the class. Formally, the conditional independence assumption is satisfied iff P({xi : i ∈M}|y, {xj : j /∈M}) = P({xi : i ∈M}|y). While this assumption is clearly an idealization, it is not uncommon in semi-supervised learning (for example, an analogous assumption was introduced to theoretically demonstrate the power of co-training [2]). Further , techniques based upon the idealization of conditional independence are often effective in practice (e.g., Naive Bayes Classifiers). We show that when the concept class C¬M is learnable in the PAC model with classification noise, and the conditional independence assumption holds, then knowledge of a single monotonic feature makes the full concept class C learnable from only unlabeled data. Our result builds on a previous theorem from [2], and requires the following definition: Definition 3 A classifier h ∈CM is weakly-useful iff there exists ϵ > 0 such that P(h(x) = 1) ≥ϵ and P(y = 1|h(x) = 1) ≥P(y = 1) + ϵ. 2For convenience, we present our analysis in terms of discrete and finite monotonic features, but the results can be extended naturally to the continuous case. Theorem 4 If the conditional independence assumption is satisfied and the concept class C¬M is learnable in the PAC model with classification noise, then given a single monotonic feature, C is learnable from unlabeled data only. Proof Sketch. The result follows from Theorem 1 in [2] and an application of Hoeffding bounds to show that the monotonic feature can be used to construct a weakly-useful classifier.3 □ The next theorem demonstrates that monotonic features are relatively more informative than labeled examples as the feature space increases in size. This result suggests that MF-based approaches to text classification may become increasingly valuable over time, as corpora become larger and the number of distinct words and phrases available to serve as features increases. We compare the value of monotonic features and labeled examples in terms of information gain, defined below. For convenience, these results are presented using a feature space XB, in which all features are binaryvalued. Definition 5 The information gain with respect to an unlabeled example’s label y provided by a variable v is defined as the reduction in entropy of y when v is given, that is: P y′=0,1 P(y = y′|v) log P(y = y′|v) −P(y = y′) log P(y = y′). Next, we define the two properties of the classification task that our theorem requires. Informally speaking, the first property states that the feature space does not have fully redundant features, whereas the second states that examples which are far apart have less dependent labels than those which are close together. We would expect these properties to hold for most tasks in practice. Definition 6 A distribution D on (XB, Y) has bounded feature dependence if there exists ϵF > 0 such that the conditional probability PD(xi = r|{xj = rj : j ̸= i}) < 1 −ϵF for all i, r, and sets {xj = rj : j ̸= i} of assignments to one or more xj. Definition 7 A distribution D on (XB, Y) has distance-diminishing information gain if the information gain of an example x with respect to the label of any neighboring example x′ is less than KIδr I for some δI < 1, where r is the Hamming distance between x and x′. The following theorem shows that whenever the above properties hold to a sufficient degree, the expected information gain from a labeled example falls as the size of the feature space increases. Theorem 8 For a learning problem governed by distribution D with bounded feature dependence and distance-dimishing information gain, with ϵF > δI δI+1, as the number of features d increases, the expected information gain provided by a labeled example about unlabeled examples’ labels decreases to zero. However, the information gain from an MF xf with given relationship P(y|xf) remains constant as d increases. The portion of Theorem 8 which concerns information gain of a labeled example is a version of the well-known “curse of dimensionality” [1], which states that the number of examples needed to estimate a function scales exponentially with the number of dimensions under certain assumptions. Theorem 8 differs in detail, however; it states the curse of dimensionality in terms of information gain, making possible a direct comparison with monotonic features. 3.1 Relation to Other Approaches In the introduction, we identified several learning methods that utilized Monotonic Features (MFs) implicitly, which was a key motivation for formalizing MFs. This section explains the ways in which MF-based classification is distinct from previous semi-supervised learning methods. When MFs are provided as input, they can be viewed as a kind of “labeled feature” studied in [7]. However, instead of a generalized expectation criteria, we use the prior to generate noisy labels for examples. Thus, MFs can complement any concept learning algorithm, not just discriminative probabilistic models as in [7]. Moreover, while [7] focuses on a problem setting in which selected 3Proofs of the theorems in this paper can be found in [5], Chapter 2. features are labeled by hand, we show in Section 4 that MFs can either obviate hand-labeled data, or can be estimated automatically from a small set of hand-labeled instances. Co-training [2] is a semi-supervised technique that also considers a partition of the feature space into two distinct “views.” One might ask if monotonic feature classification is equivalent to cotraining with the monotonic features serving as one view, and the other features forming the other. However, co-training requires labeled data to train classifiers for each view, unlike monotonic feature classification which can operate without any labeled data. Thus, there are cases where an MF-based algorithm like MFA is applicable, but co-training is not. Even when labeled data is available, co-training takes the partition of the feature set as input, whereas monotonic features can be detected automatically using the labeled data. Also, co-training is an iterative algorithm in which the most likely examples of a class according to one view are used to train a classifier on the other view in a mutually recursive fashion. For a given set of monotonic features, however, iterating this process is ineffective, because the mostly likely examples of a class according to the monotonic feature view are fixed by the monotonicity property. 3.1.1 Semi-supervised Smoothness Assumptions The MFS is provably distinct from certain smoothness properties typically assumed in previous semi-supervised learning methods, known as the cluster and manifold assumptions. The cluster assumption states that in the target concept, the boundaries between classes occupy relatively lowdensity regions of the distribution P(x). The manifold assumption states that the distribution P(x) is embedded on a manifold of strictly lower dimension than the full input space X. It can be shown that classification tasks with the MFS exist for which neither the cluster assumption nor the manifold assumption holds. Similarly, we can construct classification tasks exhibiting the manifold assumption, the cluster assumption, or their conjunction, but without the MFS. Thus, we state the following theorem. Theorem 9 The monotonic feature structure neither implies nor is implied by the manifold assumption, the cluster assumption, or their conjunction or disjuntion. 4 Experiments This section reports on our experiments in utilizing MFs for text classification. As discussed in the introduction, MFs have been used implicitly by several classification methods in numerous tasks. Here we quantify their impact on the standard “20 newsgroups” dataset [11]. We show that MFs can be employed to perform accurate classification even without labeled examples, extending the results from [8] and [3] to a semi-supervised setting. Further, we also demonstrate that whether or not the identities of MFs are given, exploiting the MF structure by learning MFs can improve performance. 4.1 General Methods for Monotonic Feature Classification Here, we define a set of abstract methods for incorporating monotonic features into any existing learning algorithm. The first method, MFA, is an abstraction of the MF word sense disambiguation algorithm first introduced in [13]. It is applicable when monotonic features are given but labeled examples are not provided. The second, MFA-SSL , applies in the standard semi-supervised learning case when some labeled examples are provided, but the identities of the MFs are unknown and must be learned. Lastly, MFA-BOTH applies when both labeled data and MF identities are given. MFA proceeds as shown in Figure 1. MFA labels the unlabeled examples DU as elements of class y = 1 iff some monotonic feature value xi for i ∈M exceeds a threshold τ. The threshold is set using unlabeled data so as to maximize the minimum probability mass on either side of the threshold.4 This set of bootstrapped examples D′ L is then fed as training data into a supervised or semi-supervised algorithm Φ(D′ L, DU), and MFA outputs the resulting classifier. In general, the MFA schema can be instantiated with any concept learning algorithm Φ. 4This policy is suggested by the proof of Theorem 4, in which the only requirement of the threshold is that sufficient mass lies on each side. MFA(M, DU, Φ) 1. D′ L = Labeled examples (x, y) such that y = 1 iff a xi > τ for some i ∈M 2. Output Φ(D′ L, DU) Figure 1: Pseudocode for MFA. The inputs are M, a set of monotonic features, DU, a set of unlabeled examples, and Φ(L, U), a supervised or semi-supervised machine learning algorithm which outputs a classifier given labeled data L and unlabeled data U. The threshold τ is derived from the unlabeled data and M (see text). MFA-SSL(DL, DU, Φ, ΦM) 1. M = the k strongest monotonic features in DL 2. D′ L = Examples from DU probabilistically labeled with ΦM(M, DL, DU) 3. Output Φ(DL ∪D′ L, DU) Figure 2: Pseudocode for MFA-SSL. The inputs DU and Φ inputs are the same as those of MFA (see Figure 1). The additional inputs include labeled data DL and a machine learning algorithm ΦM(M, L, U) which given labeled data L and unlabeled data U outputs a probabilistic classifier that uses only monotonic features M. k is a parameter of MFA-SSL(see text). When MFs are unknown, but some labeled data is given, the MFA-SSL (Figure 2) algorithm attempts to identify MFs using the labeled training data DL, adding the most strongly monotonic features to the set M. Monotonicity strength can be measured in various ways; in our experiments, we rank each feature xi by the quantity f(y, xi) = P r P(y, xi = r)r for each class y.5 MFA-SSL adds monotonic features to M in descending order of this value, up to a limit of k = 5 per class.6 MFA-SSL then invokes a given machine learning algorithm ΦM(M, DL, DU) to learn a probabilistic classifier that employs only the monotonic features in M. MFA-SSL uses this classifier to probabilistically label the examples in DU to form D′ L. MFA-SSL then returns Φ(D′ L, DU) as in MFA. Note that when no monotonic features are identified, MFA-SSL defaults to the underlying algorithm Φ. When monotonic features are known and labeled examples are available, we run a derivative of MFA-SSL denoted as MFA-BOTH. The algorithm is the same as MFA-SSL, except that any given monotonic features are added to the learned set in Step 1 of Figure 2, and bootstrapped examples using the given monotonic features (from Step 1 in Figure 1) are added to D′ L. 4.2 Experimental Methodology and Baseline Methods The task we investigate is to determine from the text of a newsgroup post the newsgroup in which it appeared. We used bag-of-word features after converting terms to lowercase, discarding the 100 most frequent terms and all terms appearing only once. Below, we present results averaged over four disjoint training sets of variable size, using a disjoint test set of 5,000 documents and an unlabeled set of 10,000 documents. We compared the monotonic feature approach with two alternative algorithms, which represent two distinct points in the space of semi-supervised learning algorithms. The first, NB-EM, is a semisupervised Naive Bayes with Expectation Maximization algorithm [12], employing settings previously shown to be effective on the 20 Newsgroups data. The second, LP, is a semi-supervised graph-based label propagation algorithm recently employed for text classification [4]. We found that on this dataset, the NB-EM algorithm substantially outperformed LP (providing a 41% error reduction in the experiments in Figure 3), so below we compare exclusively with NB-EM. When the identities of monotonic features are given, we obtained one-word monotonic features simply using the newsgroup name, with minor modifications to expand abbreviations. This methodology closely followed that of [8]. For example, the occurrence count of the term “politics” was 5This measure is applicable for features with numeric values. For non-numeric features, alternative measures (e.g. rank correlation) could be employed to detect MFs. 6A sensitivity analysis revealed that varying k by up to 40% in either direction did not decrease performance of MFA-SSL in the experiments in Section 4.3. a monotonic feature for the talk.politics.misc newsgroup. We also expanded the set of monotonic features to include singular/plural variants. We employed Naive Bayes classifiers for both Φ and ΦM. We weighted the set of examples labeled using the monotonic features (D′ L) equally with the original labeled set, increasing the weight by the equivalent of 200 labeled examples when monotonic features are given to the algorithm. 4.3 Experimental Results The first question we investigate is what level of performance MFA can achieve without labeled training data, when monotonic features are given. The results of this experiment are shown in Table 1. MFA achieves accuracy on the 20-way classification task of 0.563. Another way to measure this accuracy is in terms of the number of labeled examples that a baseline semi-supervised technique would require in order to achieve comparable performance. We found that MFA outperformed NBEM with up to 160 labeled examples. This first experiment is similar to that of [8], except that instead of evaluating against only supervised techniques, we use a more comparable semi-supervised baseline (NB-EM). Could the monotonic features, on their own, suffice to directly classify the test data? To address this question, the table also reports the performance of using the given monotonic features exclusively to label the test data (MF Alone), without using the semi-supervised technique Φ. We find that the bootstrapping step provides large benefits to performance; MFA has an effective number of labeled examples eight times more than that of MF Alone. Random Baseline MF Alone MFA Accuracy 5% 24% 56% (2.33x) Labeled Example Equivalent 0 20 160 (8x) Table 1: Performance of MFA when monotonic features are given, and no labeled examples are provided. MFA achieves accuracy of 0.563, which is ten fold that of a Random Baseline classifier that assigns labels randomly, and more than double that of “MF Alone”, which uses only the monotonic features and ignores the other features. MFA’s accuracy exceeds that of the NB-EM baseline with 160 labeled training examples, and is eight fold that of “MF Alone”. The second question we investigate is whether the monotonic feature approach can improve performance even when the class name is not given. MFA-SSL takes the same inputs as the NB-EM technique, without the identities of monotonic features. The performance of MFA-SSL as the size of the labeled data set varies is shown in Figure 3. The graph shows that for small labeled data sets of size 100-400, MFA-SSL outperforms NB-EM by an average error reduction of 15%. These results are statistically significant (p < 0.001, Fisher Exact Test). One important question is whether MFASSL’s performance advantage over NB-EM is in fact due to the presence of monotonic features, or if it instead results from simply utilizing feature selection in Step 2 of Figure 2. We investigated this by replacing MFA-SSL’s monotonic feature measure f(y, xi) with a standard information gain measure, and learning an equal number of features distinct from those selected by MFA-SSL originally. This method has performance essentially equivalent to that of NB-EM, suggesting that MFA-SSL’s performance advantage is not due merely to feature selection. Lastly, when both monotonic features and labeled examples are available, MFA-BOTH reduces error over the NB-EM baseline by an average of 31% across the training set sizes shown in Figure 3. For additional analysis of the above experiments, and results in another domain, see [5]. 5 Conclusions We have presented a general framework for utilizing Monotonic Features (MFs) to perform classification without hand-labeled data, or in a semi-supervised setting where monotonic features can be discovered from small numbers of hand-labeled examples. While our experiments focused on the 20 Newsgroups data set, we have complemented them with both a theoretical analysis, and by enumerating a wide variety of algorithms that have used MFs implicitly. MFA-SSL MFA-both Figure 3: Performance in document classification. MFA-SSL reduces error over the NB-EM baseline by 15% for training sets between 100 and 400 examples, and MFA-BOTH reduces error by 31% overall. Acknowledgements We thank Stanley Kok, Daniel Lowd, Mausam, Hoifung Poon, Alan Ritter, Stefan Schoenmackers, and Dan Weld for helpful comments. This research was supported in part by NSF grants IIS0535284 and IIS-0312988, ONR grant N00014-08-1-0431 as well as gifts from Google, and carried out at the University of Washington’s Turing Center. The first author was supported by a Microsoft Research Graduate Fellowship sponsored by Microsoft Live Labs. References [1] R. Bellman. Adaptive Control Processes: A Guided Tour. Princeton University Press, 1961. [2] A. Blum and T. Mitchell. Combining labeled and unlabeled data with co-training. In COLT: Proceedings of the Workshop on Computational Learning Theory, Morgan Kaufmann Publishers, pages 92–100, 1998. [3] M.-W. Chang, L.-A. Ratinov, D. Roth, and V. Srikumar. Importance of semantic representation: Dataless classification. In D. Fox and C. P. Gomes, editors, AAAI, pages 830–835. AAAI Press, 2008. [4] J. Chen, D.-H. Ji, C. L. Tan, and Z.-Y. Niu. Semi-supervised relation extraction with label propagation. In HLT-NAACL, 2006. [5] D. Downey. Redundancy in Web-scale Information Extraction: Probabilistic Model and Experimental Results. PhD thesis, University of Washington, 2008. [6] D. Downey, O. Etzioni, and S. Soderland. A Probabilistic Model of Redundancy in Information Extraction. In Procs. of IJCAI, 2005. [7] G. Druck, G. Mann, and A. McCallum. Learning from labeled features using generalized expectation criteria. In Proceedings of SIGIR, 2008. [8] A. Gliozzo, C. Strapparava, and I. Dagan. Investigating unsupervised learning for text categorization bootstrapping. In Proceedings of HLT 2005, pages 129–136, Morristown, NJ, USA, 2005. [9] M. Hearst. Automatic Acquisition of Hyponyms from Large Text Corpora. In Procs. of the 14th International Conference on Computational Linguistics, pages 539–545, Nantes, France, 1992. [10] R. Mihalcea and D. I. Moldovan. An automatic method for generating sense tagged corpora. In AAAI/IAAI, pages 461–466, 1999. [11] T. M. Mitchell. Machine Learning. McGraw-Hill, New York, 1997. [12] K. Nigam, A. McCallum, S. Thrun, and T. Mitchell. Text Classification from Labeled and Unlabeled Documents using EM. Machine Learning, 39(2/3):103–134, 2000. [13] D. Yarowsky. Unsupervised word sense disambiguation rivaling supervised methods. In Meeting of the Association for Computational Linguistics, pages 189–196, 1995.
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Nonparametric Regression and Classification with Joint Sparsity Constraints Han Liu John Lafferty Larry Wasserman Carnegie Mellon University Pittsburgh, PA 15213 Abstract We propose new families of models and algorithms for high-dimensional nonparametric learning with joint sparsity constraints. Our approach is based on a regularization method that enforces common sparsity patterns across different function components in a nonparametric additive model. The algorithms employ a coordinate descent approach that is based on a functional soft-thresholding operator. The framework yields several new models, including multi-task sparse additive models, multi-response sparse additive models, and sparse additive multi-category logistic regression. The methods are illustrated with experiments on synthetic data and gene microarray data. 1 Introduction Many learning problems can be naturally formulated in terms of multi-category classification or multi-task regression. In a multi-category classification problem, it is required to discriminate between the different categories using a set of high-dimensional feature vectors—for instance, classifying the type of tumor in a cancer patient from gene expression data. In a multi-task regression problem, it is of interest to form several regression estimators for related data sets that share common types of covariates—for instance, predicting test scores across different school districts. In other areas, such as multi-channel signal processing, it is of interest to simultaneously decompose multiple signals in terms of a large common overcomplete dictionary, which is a multi-response regression problem. In each case, while the details of the estimators vary from instance to instance, across categories, or tasks, they may share a common sparsity pattern of relevant variables selected from a high-dimensional space. How to find this common sparsity pattern is an interesting learning task. In the parametric setting, progress has been recently made on such problems using regularization based on the sum of supremum norms (Turlach et al., 2005; Tropp et al., 2006; Zhang, 2006). For example, consider the K-task linear regression problem y(k) i = β(k) 0 + ∑p j=1 β(k) j x(k) ij + ϵ(k) i where the superscript k indexes the tasks, and the subscript i = 1, . . . , nk indexes the instances within a task. Using quadratic loss, Zhang (2006) suggests the following estimator bβ = arg min β K ∑ k=1 1 2nk nk ∑ i=1 y(k) i −β(k) 0 − p ∑ j=1 β(k) j x(k) ij 2 + λ p ∑ j=1 max k |β(k) j | (1) where maxk |β(k) j | = ∥βj∥∞is the sup-norm of the vector βj ≡(β(1) j , . . . , β(K) j )T of coefficients for the jth feature across different tasks. The sum of sup-norms regularization has the effect of “grouping” the elements in βj such that they can be shrunk towards zero simultaneously. The problems of multi-response (or multivariate) regression and multi-category classification can be viewed as a special case of the multi-task regression problem where tasks share the same design matrix. Turlach et al. (2005) and Fornasier and Rauhut (2008) propose the same sum of sup-norms 1 regularization as in (1) for such problems in the linear model setting. In related work, Zhang et al. (2008) propose the sup-norm support vector machine, demonstrating its effectiveness on gene data. In this paper we develop new methods for nonparametric estimation for such multi-task and multicategory regression and classification problems. Rather than fitting a linear model, we instead estimate smooth functions of the data, and formulate a regularization framework that encourages joint functional sparsity, where the component functions can be different across tasks while sharing a common sparsity pattern. Building on a recently proposed method called sparse additive models, or “SpAM” (Ravikumar et al., 2007), we propose a convex regularization functional that can be viewed as a nonparametric analog of the sum of sup-norms regularization for linear models. Based on this regularization functional, we develop new models for nonparametric multi-task regression and classification, including multi-task sparse additive models (MT-SpAM), multi-response sparse additive models (MR-SpAM), and sparse multi-category additive logistic regression (SMALR). The contributions of this work include (1) an efficient iterative algorithm based on a functional soft-thresholding operator derived from subdifferential calculus, leading to the multi-task and multiresponse SpAM procedures, (2) a penalized local scoring algorithm that corresponds to fitting a sequence of multi-response SpAM estimates for sparse multi-category additive logistic regression, and (3) the successful application of this methodology to multi-category tumor classification and biomarker discovery from gene microarray data. 2 Nonparametric Models for Joint Functional Sparsity We begin by introducing some notation. If X has distribution PX, and f is a function of x, its L2(PX) norm is denoted by ∥f∥2 = ∫ X f 2(x)dPX = E(f 2). If v = (v1, . . . , vn)T is a vector, define ∥v∥2 n = 1 n ∑n j=1 v2 j and ∥v∥∞= maxj |vj|. For a p-dimensional random vector (X1, . . . , Xp), let Hj denote the Hilbert subspace L2(PXj) of PXj-measurable functions fj(xj) of the single scalar variable Xj with zero mean, i.e. E[fj(Xj)] = 0. The inner product on this space is defined as ⟨fj, gj⟩= E [fj(Xj)gj(Xj)]. In this paper, we mainly study multivariate functions f(x1, . . . , xp) that have an additive form, i.e., f(x1, . . . , xp) = α + ∑ j fj(xj), with fj ∈Hj for j = 1, . . . , p. With H ≡{1} ⊕H1 ⊕H2 ⊕. . . ⊕Hp denoting the direct sum Hilbert space, we have that f ∈H. 2.1 Multi-task/Multi-response Sparse Additive Models In a K-task regression problem, we have observations {(x(k) i , y(k) i ), i = 1, . . . , nk, k = 1, . . . , K}, where x(k) i = (x(k) i1 , . . . , x(k) ip )T is a p-dimensional covariate vector, the superscript k indexes tasks and i indexes the i.i.d. samples for each task. In the following, for notational simplicity, we assume that n1 = . . . = nK = n. We also assume different tasks are comparable and each Y (k) and X(k) j has been standardized, i.e., has mean zero and variance one. This is not really a restriction of the model since a straightforward weighting scheme can be adopted to extend our approach to handle noncomparable tasks. We assume the true model is E ( Y (k) | X(k) = x(k)) = f (k)(x(k)) ≡ ∑p j=1 f (k) j (x(k) j ) for k = 1, . . . , K, where, for simplicity, we take all intercepts α(k) to be zero. Let Qf (k)(x, y) = (y −f (k)(x))2 denote the quadratic loss. To encourage common sparsity patterns across different function components, we define the regularization functional ΦK(f) by ΦK(f) = p ∑ j=1 max k=1,...,K ∥f (k) j ∥. (2) The regularization functional ΦK(f) naturally combines the idea of the sum of sup-norms penalty for parametric joint sparsity and the regularization idea of SpAM for nonparametric functional sparsity; if K = 1, then Φ1(f) is just the regularization term introduced for (single-task) sparse additive models by Ravikumar et al. (2007). If each f (k) j is a linear function, then ΦK(f) reduces to the sum of sup-norms regularization term as in (1). We shall employ ΦK(f) to induce joint functional sparsity in nonparametric multi-task inference. 2 Using this regularization functional, the multi-task sparse additive model (MT-SpAM) is formulated as a penalized M-estimator, by framing the following optimization problem bf (1), . . . , bf (K) = arg min f (1),...,f (K) { 1 2n n ∑ i=1 K ∑ k=1 Qf (k)(x(k) i , y(k) i ) + λΦK(f) } (3) where f (k) j ∈H(k) j for j = 1, . . . , p and k = 1, . . . , K, and λ > 0 is a regularization parameter. The multi-response sparse additive model (MR-SpAM) has exactly the same formulation as in (3) except that a common design matrix is used across the K different tasks. 2.2 Sparse Multi-Category Additive Logistic Regression In a K-category classification problem, we are given n examples (x1, y1), . . . , (xn, yn) where xi = (xi1, . . . , xip)T is a p-dimensional predictor vector and yi = (y(1) i , . . . , y(K−1) i )T is a (K −1)dimensional response vector in which at most one element can be one, with all the others being zero. Here, we adopt the common “1-of-K” labeling convention where y(k) i = 1 if xi has category k and y(k) i = 0 otherwise; if all elements of yi are zero, then xi is assigned the K-th category. The multi-category additive logistic regression model is P(Y (k) = 1 | X = x) = exp ( f (k)(x) ) 1 + ∑K−1 k′=1 exp ( f (k′)(x) ), k = 1, . . . , K −1 (4) where f (k)(x) = α(k) +∑p j=1 f (k) j (xj) has an additive form. We define f = (f (1), . . . , f (K−1)) to be a discriminant function and p(k) f (x) = P(Y (k) = 1 | X = x) to be the conditional probability of category k given X = x. The logistic regression classifier hf(·) induced by f, which is a mapping from the sample space to the category labels, is simply given by hf(x) = arg maxk=1,...,K p(k) f (x). If a variable Xj is irrelevant, then all of the component functions f (k) j are identically zero, for each k = 1, 2, . . . , K −1. This motivates the use of the regularization functional ΦK−1(f) to zero out entire vectors fj = (f (1) j , . . . , f (K−1) j ). Denoting ℓf(x, y) = K−1 ∑ k=1 y(k)f (k)(x) −log ( 1 + K−1 ∑ k′=1 exp f (k′)(x) ) as the multinomial log-loss, the sparse multi-category additive logistic regression estimator (SMALR) is thus formulated as the solution to the optimization problem bf (1), . . . , bf (K−1) = arg min f (1),...,f (K−1) { −1 n n ∑ i=1 ℓf(xi, yi) + λΦK−1(f) } (5) where f (k) j ∈H(k) j for j = 1, . . . , p and k = 1, . . . , K −1. 3 Simultaneous Sparse Backfitting We use a blockwise coordinate descent algorithm to minimize the functional defined in (3). We first formulate the population version of the problem by replacing sample averages by their expectations. We then derive stationary conditions for the optimum and obtain a population version algorithm for computing the solution by a series of soft-thresholded univariate conditional expectations. Finally, a finite sample version of the algorithm can be derived by plugging in nonparametric smoothers for these conditional expectations. For the jth block of component functions f (1) j , . . . , f (K) j , let R(k) j = Y (k) −∑ l̸=j f (k) l (X(k) l ) denote the partial residuals. Assuming all but the functions in the jth block to be fixed, the optimization problem is reduced to bf (1) j , . . . , bf (K) j = arg min f (1) j ,...,f (K) j { 1 2E [ K ∑ k=1 ( R(k) j −f (k) j (X(k) j ) )2] + λ max k=1,...,K ∥f (k) j ∥ } . (6) 3 The following result characterizes the solution to (6). Theorem 1. Let P (k) j = E ( R(k) j | X(k) j ) and s(k) j = ∥P (k) j ∥, and order the indices according to s(k1) j ≥s(k2) j ≥. . . ≥s(kK) j . Then the solution to (6) is given by f (ki) j = P (ki) j for i > m∗ 1 m∗ [ m∗ ∑ i′=1 s(ki′) j −λ ] + P (ki) j s(ki) j for i ≤m∗. (7) where m∗= arg maxm 1 m (∑m i′=1 s(ki′) j −λ ) and [·]+ denotes the positive part. Therefore, the optimization problem in (6) is solved by a soft-thresholding operator, given in equation (7), which we shall denote as (f (1) j , . . . , f (K) j ) = Soft(∞) λ [R(1) j , . . . , R(K) j ]. (8) While the proof of this result is lengthy, we sketch the key steps below, which are a functional extension of the subdifferential calculus approach of Fornasier and Rauhut (2008) in the linear setting. First, we formulate an optimality condition in terms of the Gˆateaux derivative as follows. Lemma 2. The functions f (k) j are solutions to (6) if and only if f (k) j −P (k) j + λukvk = 0 (almost surely), for k = 1, . . . , K, where uk are scalars and vk are measurable functions of X(k) j , with (u1, . . . , uK)T ∈∂∥· ∥∞ “ ∥f (1) j ∥,...,∥f (K) j ∥ ”T and vk ∈∂∥f (k) j ∥, k = 1, . . . , K. Here the former one denotes the subdifferential of the convex functional ∥· ∥∞evaluated at (∥f (1) j ∥, . . . , ∥f (K) j ∥)T , it lies in a K-dimensional Euclidean space. And the latter denotes the subdifferential of ∥f (k) j ∥, which is a set of functions. Next, the following proposition from Rockafellar and Wets (1998) is used to characterize the subdifferential of sup-norms. Lemma 3. The subdifferential of ∥· ∥∞on RK is ∂∥· ∥∞ x = {B1(1) if x = 0 conv{sign(xk)ek : |xk| = ∥x∥∞} otherwise. where B1(1) denotes the ℓ1 ball of radius one, conv(A) denotes the convex hull of set A, and ek is the k-th canonical unit vector in RK. Using Lemma 2 and Lemma 3, the proof of Theorem 1 proceeds by considering three cases for the sup-norm subdifferential evaluated at (∥f (1) j ∥, . . . , ∥f (K) j ∥)T : (1) ∥f (k) j ∥= 0 for k = 1, . . . , K; (2) there exists a unique k, such that ∥f (k) j ∥= maxk′=1,...,K ∥f (k′) j ∦= 0; (3) there exists at least two k ̸= k′, such that ∥f (k) j ∥= ∥f (k′) j ∥= maxm=1,...,K ∥f (m) j ∦= 0. The derivations for cases (1) and (2) are relatively straightforward, but for case (3) we prove the following. Lemma 4. The sup-norm is attained precisely at m > 1 entries if only if m is the largest number such that s(km) j ≥ 1 m−1 (∑m−1 i′=1 s(ki′) j −λ ) . The proof of Theorem 1 then follows from the above lemmas and some calculus. Based on this result, the data version of the soft-thresholding operator is obtained by replacing the conditional expectation P (k) j = E(R(k) j | X(k) j ) by S(k) j R(k) j , where S(k) j is a nonparametric smoother for variable X(k) j , e.g., a local linear or spline smoother; see Figure 1. The resulting simultaneous sparse backfitting algorithm for multi-task and multi-response sparse additive models (MT-SpAM and MR-SpAM) is shown in Figure 2. The algorithm for the multi-response case (MR-SpAM) has S(1) j = . . . = S(K) j since there is only a common design matrix. 4 SOFT-THRESHOLDING OPERATOR SOFT(∞) λ [R(1) j , . . . , R(K) j ; S(1) j , . . . , S(K) j ]: DATA VERSION Input: Smoothing matrices S(k) j , residuals R(k) j for k = 1, . . . , K, regularization parameter λ. (1) Estimate P (k) j = E h R(k) j | X(k) j i by smoothing: bP (k) j = S(k) j R(k) j ; (2) Estimate norm: bs(k) j = ∥bPj∥n and order the indices according to bs(k1) j ≥bs(k2) j ≥. . . ≥bs(kK) j ; (3) Find m∗= arg maxm 1 m “Pm i′=1 s (ki′ ) j −λ ” and calculate bf (ki) j = 8 > > < > > : bP (ki) j for i > m∗ 1 m∗ " m∗ X i′=1 bs (ki′ ) j −λ # + bP (ki) j bs(ki) j for i ≤m∗; (4) Center bf (k) j ←bf (k) j −mean( bf (k) j ) for k = 1, . . . , K. Output: Functions bf (k) j for k = 1, . . . , K. Figure 1: Data version of the soft-thresholding operator. MULTI-TASK AND MULTI-RESPONSE SPAM Input: Data (x(k) i , y(k) i ), i = 1, . . . , n, k = 1, . . . , K and regularization parameter λ. Initialize: Set bf (k) j = 0 and compute smoothers S(k) j for j = 1, . . . , p and k = 1, . . . , K; Iterate until convergence: For each j = 1, . . . , p: (1) Compute residuals: R(k) j = y(k) −P k′̸=j bf (k) k′ for k = 1, . . . , K; (2) Threshold: bf (1) j , . . . , bf (K) j ←Soft(∞) λ [R(1) j , . . . , R(K) j ; S(1) j , . . . , S(K) j ]. Output: Functions bf (k) for k = 1, . . . , K. Figure 2: The simultaneous sparse backfitting algorithm for MT-SpAM or MR-SpAM. For the multiresponse case, the same smoothing matrices are used for each k. 3.1 Penalized Local Scoring Algorithm for SMALR We now derive a penalized local scoring algorithm for sparse multi-category additive logistic regression (SMALR), which can be viewed as a variant of Newton’s method in function space. At each iteration, a quadratic approximation to the loss is used as a surrogate functional with the regularization term added to induce joint functional sparsity. However, a technical difficulty is that the approximate quadratic problem in each iteration is weighted by a non-diagonal matrix in function space, thus a trivial extension of the algorithm in (Ravikumar et al., 2007) for sparse binary nonparametric logistic regression does not apply. To tackle this problem, we use an auxiliary function to lower bound the log-loss, as in (Krishnapuram et al., 2005). The population version of the log-loss is L(f) = E[ℓf(X, Y )] with f = (f (1), . . . , f (K−1)). A second-order Lagrange form Taylor expansion to L(f) at bf is then L(f) = L( bf) + E [ ∇L( bf)T (f −bf) ] + 1 2E [ (f −bf)T H( ef)(f −bf) ] (9) for some function ef, where the gradient is ∇L( bf) = Y −p b f(X) with p b f(X) = (p b f(Y (1) = 1 | X), . . . , p b f(Y (K−1) = 1 | X))T , and the Hessian is H( ef) = −diag ( p e f(X) ) + p e f(X)p e f(X)T . Defining B = −(1/4)IK−1, it is straightforward to show that B ≼H( ef), i.e., H( ef) −B is positive-definite. Therefore, we have that L(f) ≥L( bf) + E [ ∇L( bf)T (f −bf) ] + 1 2E [ (f −bf)T B(f −bf) ] . (10) 5 SMALR: SPARSE MULTI-CATEGORY ADDITIVE LOGISTIC REGRESSION Input: Data (xi, yi), i = 1, . . . , n and regularization parameter λ. Initialize: bf (k) j = 0 and bα(k) = log “Pn i=1 y(k) i . “ n −Pn i=1 PK−1 k′=1 y(k′) i ”” , k = 1, . . . , K −1 Iterate until convergence: (1) Compute p(k) b f (xi) ≡P(Y (k) = 1 | X = xi) as in (4) for k = 1, . . . , K −1; (2) Calculate the transformed responses Z(k) i = 4 “ y(k) i −p(k) b f (xi) ” +bα(k)+Pp j=1 bf (k) j (xij) for k = 1, . . . , K −1 and i = 1, . . . , n; (3) Call subroutines ( bf (1), . . . , bf (K−1)) ←MR-SpAM “ (xi, Z(k) i )n i=1, √ 2λ ” ; (4) Adjust the intercepts: α(k) ←1 n n X i=1 Z(k) i ; Output: Functions bf (k) and intercepts bα(k) for k = 1, . . . , K −1. Figure 3: The penalized local scoring algorithm for SMALR. The following lemma results from straightforward calculation. Lemma 5. The solution f that maximizes the righthand side of (10) is equivalent to the solution that minimizes 1 2E ( ∥Z −Af∥2 n ) where A = (−B)1/2 and Z = A−1(Y −p b f) + A bf. Recalling that f (k) = α(k) + ∑p j=1 f (k) j , equation (9) and Lemma 5 then justify the use of the auxiliary functional 1 2 K−1 ∑ k=1 E [( Z′(k) −∑p j=1 f (k)(Xj) )2] + λ′ΦK−1(f) (11) where Z′(k) = 4 ( Y (k) −P b f(Y (k) = 1 | X) ) + bα(k) + ∑p j=1 bf (k) j (Xj) and λ′ = √ 2λ. This is precisely in the form of a multi-response SpAM optimization problem in equation (3). The resulting algorithm, in the finite sample case, is shown in Figure 3. 4 Experiments In this section, we first use simulated data to investigate the performance of the MT-SpAM simultaneous sparse backfitting algorithm. We then apply SMALR to a tumor classification and biomarker identification problem. In all experiments, the data are rescaled to lie in the p-dimensional cube [0, 1]p. We use local linear smoothing with a Gaussian kernel. To choose the regularization parameter λ, we simply use J-fold cross-validation or the GCV score from (Ravikumar et al., 2007) extended to the multi-task setting: GCV(λ) = ∑n i=1 ∑K k=1 Q b f (k)(x(k) i , y(k) i ))/(n2K2−(nK)df(λ))2 where df(λ) = ∑K k=1 ∑p j=1 ν(k) j I ( ∥bf (k) j ∥n ̸= 0 ) , and ν(k) j = trace(S(k) j ) is the effective degrees of freedom for the univariate local linear smoother on the jth variable. 4.1 Synthetic Data We generated n = 100 observations from a 10-dimensional three-task additive model with four relevant variables: y(k) i = ∑4 j=1 f (k) j (x(k) ij ) + ϵ(k) i , k = 1, 2, 3, where ϵ(k) i ∼N(0, 1); the component functions f (k) j are plotted in Figure 4. The 10-dimensional covariates are generated as X(k) j = (W (k) j + tU (k))/(1 + t), j = 1, . . . , 10 where W (k) 1 , . . . , W (k) 10 and U (k) are i.i.d. sampled from Uniform(−2.5, 2.5). Thus, the correlation between Xj and Xj′ is t2/(1 + t2) for j ̸= j′. The results of applying MT-SpAM with the bandwidths h = (0.08, . . . , 0.08) and regularization parameter λ = 0.25 are summarized in Figure 4. The upper 12 figures show the 12 relevant component functions for the three tasks; the estimated function components are plotted as solid black 6 0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 0 1 2 k=1 x1 0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 0 1 2 k=2 x1 0.0 0.2 0.4 0.6 0.8 1.0 −1 0 1 2 3 4 k=3 x1 0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 0 1 2 k=1 x2 0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 0 1 2 k=2 x2 0.0 0.2 0.4 0.6 0.8 1.0 −1 0 1 2 3 4 k=3 x2 0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 0 1 2 k=1 x3 0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 0 1 2 k=2 x3 0.0 0.2 0.4 0.6 0.8 1.0 −1 0 1 2 3 4 k=3 x3 0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 0 1 2 k=1 x4 0.0 0.2 0.4 0.6 0.8 1.0 −2 −1 0 1 2 k=2 x4 0.0 0.2 0.4 0.6 0.8 1.0 −1 0 1 2 3 4 k=3 x4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 t=0 Path Index Empirical sup−L1 norm 10 1 2 3 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 t=2 Path Index Empirical sup−L1 norm 10 6 4 3 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 t=4 Path Index Empirical sup−L1 norm 7 8 1 3 2 Variable selection accuracy Estimation accuracy: MSE (sd) Model t = 0 t = 1 t = 2 t = 3 t = 0 t = 1 t = 2 t = 3 MR−SpAM 89 80 47 37 7.43 (0.71) 5.82 (0.60) 3.83 (0.37) 3.07 (0.30) MARS 0 0 0 0 8.66 (0.78) 7.52 (0.61) 5.36 (0.40) 4.64 (0.35) Figure 4: (Top) Estimated vs. true functions from MT-SpAM; (Middle) Regularization paths using MT-SpAM. (Bottom) Quantitative comparison between MR-SpAM and MARS lines and the true function components are plotted using dashed red lines. For all the other variables (from dimension 5 to dimension 10), both the true and estimated components are zero. The middle three figures show regularization paths as the parameter λ varies; each curve is a plot of the maximum empirical L1 norm of the component functions for each variable, with the red vertical line representing the selected model using the GCV score. As the correlation increases (t increases), the separation between the relevant dimensions and the irrelevant dimensions becomes smaller. Using the same setup but with one common design matrix, we also compare the quantitative performance of MR-SpAM with MARS (Friedman, 1991), which is a popular method for multi-response additive regression. Using 100 simulations, the table illustrates the number of times the models are correctly identified and the mean squared error with the standard deviation in the parentheses. (The MARS simulations are carried out in R, using the default options of the mars function in the mda library.) 4.2 Gene Microarray Data Here we apply the sparse multi-category additive logistic regression model to a microarray dataset for small round blue cell tumors (SRBCT) (Khan et al., 2001). The data consist of expression profiles of 2,308 genes (Khan et al., 2001) with tumors classified into 4 categories: neuroblastoma (NB), rhabdomyosarcoma (RMS), non-Hodgkin lymphoma (NHL), and the Ewing family of tumors (EWS). The dataset includes a training set of size 63 and a test set of size 20. These data have been analyzed by different groups. The main purpose is to identify important biomarkers, which are a small set of genes that can accurately predict the type of tumor of a patient. To achieve 100% accuracy on the test data, Khan et al. (2001) use an artificial neural network approach to identify 96 genes. Tibshirani et al. (2002) identify a set of only 43 genes, using a method called nearest shrunken centroids. Zhang et al. (2008) identify 53 genes using the sup-norm support vector machine. In our experiment, SMALR achieves 100% prediction accuracy on the test data with only 20 genes, which is a much smaller set of predictors than identified in the previous approaches. We follow the same procedure as in (Zhang et al., 2008), and use a very simple screening step based on the marginal correlation to first reduce the number of genes to 500. The SMALR model is then trained using a plugin bandwidth h0 = 0.08, and the regularization parameter λ is tuned using 4-fold cross validation. The results are tabulated in Figure 5. In the left figure, we show a “heat map” of the selected variables on the training set. The rows represent the selected genes, with their cDNA chip image id. The patients are grouped into four categories according to the corresponding tumors, 7 770394 377461 1435862 486110 383188 134748 841620 325182 812105 308231 377048 784224 244618 796258 207274 296448 814526 80649 236282 701751 RMS.T11 RMS.T10 RMS.T3 RMS.T5 RMS.T8 RMS.T7 RMS.T6 RMS.T2 RMS.T4 RMS.T1 RMS.C11 RMS.C10 RMS.C8 RMS.C7 RMS.C6 RMS.C5 RMS.C2 RMS.C9 RMS.C3 RMS.C4 NB.C8 NB.C9 NB.C11 NB.C10 NB.C5 NB.C4 NB.C7 NB.C12 NB.C6 NB.C3 NB.C2 NB.C1 BL.C4 BL.C3 BL.C2 BL.C1 BL.C8 BL.C7 BL.C6 BL.C5 EWS.C10 EWS.C11 EWS.C1 EWS.C7 EWS.C9 EWS.C6 EWS.C4 EWS.C2 EWS.C3 EWS.C8 EWS.T19 EWS.T15 EWS.T14 EWS.T13 EWS.T12 EWS.T11 EWS.T9 EWS.T7 EWS.T6 EWS.T4 EWS.T3 EWS.T2 EWS.T1 0.0 0.2 0.4 0.6 0.8 1.0 −3 −2 −1 0 1 2 3 ID.207274 k=1 0.0 0.2 0.4 0.6 0.8 1.0 −3 −2 −1 0 1 2 3 ID.1435862 k=1 0.0 0.2 0.4 0.6 0.8 1.0 −3 −2 −1 0 1 2 3 ID.207274 k=3 0.0 0.2 0.4 0.6 0.8 1.0 −3 −2 −1 0 1 2 3 ID.770394 k=1 0.0 0.2 0.4 0.6 0.8 1.0 −3 −2 −1 0 1 2 3 ID.377048 k=2 0.0 0.2 0.4 0.6 0.8 1.0 −3 −2 −1 0 1 2 3 ID.377048 k=3 0.1 0.3 0.5 0.7 lambda CV Score 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Figure 5: SMALR results on gene data: heat map (left), marginal fits (center), and CV score (right). as illustrated in the vertical groupings. The genes are ordered by hierarchical clustering of their expression profiles. The heatmap clearly shows four block structures for the four tumor categories. This suggests visually that the 20 genes selected are highly informative of the tumor type. In the middle of Figure 5, we plot the fitted discriminant functions of different genes, with their image ids listed on the plot. The values k = 1, 2, 3 under each subfigure indicate the discriminant function the plot represents. We see that the fitted functions are nonlinear. The right subfigure illustrates the total number of misclassified samples using 4-fold cross validation, the λ values 0.3, 0.4 are both zero, for the purpose of a sparser biomarker set, we choose λ = 0.4. Interestingly, only 10 of the 20 identified genes from our method are among the 43 genes selected using the shrunken centroids approach of Tibshirani et al. (2002). 16 of them are are among the 96 genes selected by neural network approach of Khan et al. (2001). This non-overlap may suggest some further investigation. 5 Discussion and Acknowledgements We have presented new approaches to fitting sparse nonparametric multi-task regression models and sparse multi-category classification models. Due to space constraints, we have not discussed results on the statistical properties of these methods, such as oracle inequalities and risk consistency; these theoretical results will be reported elsewhere. This research was supported in part by NSF grant CCF-0625879. References FORNASIER, M. and RAUHUT, H. (2008). Recovery algorithms for vector valued data with joint sparsity constraints. SIAM Journal of Numerical Analysis 46 577–613. FRIEDMAN, J. H. (1991). Multivariate adaptive regression splines. The Annals of Statistics 19 1–67. KHAN, J., WEI, J. S., RINGNER, M., SAA, L. H., LADANYI, M., WESTERMANN, F., BERTHOLD, F., SCHWAB, M., ANTONESCU, C. R., PETERSON, C. and MELTZER, P. S. (2001). Classification and diagnostic prediction of cancers using gene expression profiling and artificial neural networks. Nature Medicine 7 673 –679. KRISHNAPURAM, B., CARIN, L., FIGUEIREDO, M. and HARTEMINK, A. (2005). Sparse multinomial logistic regression: Fast algorithms and generalization bounds. IEEE Transactions on Pattern Analysis and Machine Intelligence 27 957– 968. RAVIKUMAR, P., LIU, H., LAFFERTY, J. and WASSERMAN, L. (2007). SpAM: Sparse additive models. In Advances in Neural Information Processing Systems 20. MIT Press. ROCKAFELLAR, R. T. and WETS, R. J.-B. (1998). Variational Analysis. Springer-Verlag Inc. TIBSHIRANI, R., HASTIE, T., NARASIMHAN, B., and CHU, G. (2002). Diagnosis of multiple cancer types by shrunken centroids of gene expression. Proc Natl Acad Sci U.S.A. 99 6567–6572. TROPP, J., GILBERT, A. C. and STRAUSS, M. J. (2006). Algorithms for simultaneous sparse approximation. Part II: Convex relaxation. Signal Processing 86 572–588. TURLACH, B., VENABLES, W. N. and WRIGHT, S. J. (2005). Simultaneous variable selection. Technometrics 27 349–363. ZHANG, H. H., LIU, Y., WU, Y. and ZHU, J. (2008). Variable selection for the multicategory SVM via adaptive sup-norm regularization. Electronic Journal of Statistics 2 149–1167. ZHANG, J. (2006). A probabilistic framework for multitask learning. Tech. Rep. CMU-LTI-06-006, Ph.D. thesis, Carnegie Mellon University. 8
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Recursive Segmentation and Recognition Templates for 2D Parsing Long (Leo) Zhu CSAIL MIT leozhu@csail.mit.edu Yuanhao Chen USTC yhchen4@ustc.edu.cn Yuan Lin Shanghai Jiaotong University loirey@sjtu.edu.cn Chenxi Lin Microsoft Research Asia chenxil@microsoft.com Alan Yuille UCLA yuille@stat.ucla.edu Abstract Language and image understanding are two major goals of artificial intelligence which can both be conceptually formulated in terms of parsing the input signal into a hierarchical representation. Natural language researchers have made great progress by exploiting the 1D structure of language to design efficient polynomialtime parsing algorithms. By contrast, the two-dimensional nature of images makes it much harder to design efficient image parsers and the form of the hierarchical representations is also unclear. Attempts to adapt representations and algorithms from natural language have only been partially successful. In this paper, we propose a Hierarchical Image Model (HIM) for 2D image parsing which outputs image segmentation and object recognition. This HIM is represented by recursive segmentation and recognition templates in multiple layers and has advantages for representation, inference, and learning. Firstly, the HIM has a coarse-to-fine representation which is capable of capturing long-range dependency and exploiting different levels of contextual information. Secondly, the structure of the HIM allows us to design a rapid inference algorithm, based on dynamic programming, which enables us to parse the image rapidly in polynomial time. Thirdly, we can learn the HIM efficiently in a discriminative manner from a labeled dataset. We demonstrate that HIM outperforms other state-of-the-art methods by evaluation on the challenging public MSRC image dataset. Finally, we sketch how the HIM architecture can be extended to model more complex image phenomena. 1 Introduction Language and image understanding are two major tasks in artificial intelligence. Natural language researchers have formalized this task in terms of parsing an input signal into a hierarchical representation. They have made great progress in both representation and inference (i.e. parsing). Firstly, they have developed probabilistic grammars (e.g. stochastic context free grammar (SCFG) [1] and beyond [2]) which are capable of representing complex syntactic and semantic language phenomena. For example, speech contains elementary constituents, such as nouns and verbs, that can be recursively composed into a hierarchy of (e.g. noun phrase or verb phrase) of increasing complexity. Secondly, they have exploited the one-dimensional structure of language to obtain efficient polynomial-time parsing algorithms (e.g. the inside-outside algorithm [3]). By contrast, the nature of images makes it much harder to design efficient image parsers which are capable of simultaneously performing segmentation (parsing an image into regions) and recognition (labeling the regions). Firstly, it is unclear what hierarchical representations should be used to model images and there are no direct analogies to the syntactic categories and phrase structures that occur in speech. Secondly, the inference problem is formidable due to the well-known complexity 1 and ambiguity of segmentation and recognition. Unlike most languages (Chinese is an exception), whose constituents are well-separated words, the boundaries between different image regions are usually highly unclear. Exploring all the different image partitions results in combinatorial explosions because of the two-dimensional nature of images (which makes it impossible to order these partitions to enable dynamic programming). Overall it has been hard to adapt methods from natural language parsing and apply them to vision despite the high-level conceptual similarities (except for restricted problems such as text [4]). Attempts at image parsing must make trade-offs between the complexity of the models and the complexity of the computation (for inference and learning). Broadly speaking, recent attempts can be divided into two different styles. The first style emphasizes the modeling problem and develops stochastic grammars [5, 6] capable of representing a rich class of visual relationships and conceptual knowledge about objects, scenes, and images. This style of research pays less attention to the complexity of computation. Learning is usually performed, if at all, only for individual components of the models. Parsing is performed by MCMC sampling and is only efficient provided effective proposal probabilities can be designed [6]. The second style builds on the success of conditional random fields (CRF’s) [7] and emphasizes efficient computation. This yields simpler (discriminative) models which are less capable of representing complex image structures and long range interactions. Efficient inference (e.g. belief propagation and graph-cuts) and learning (e.g. AdaBoost, MLE) are available for basic CRF’s and make these methods attractive. But these inference algorithms become less effective, and can fail, if we attempt to make the CRF models more powerful. For example, TextonBoost [8] requires the parameters of the CRF to be tuned manually. Overall, it seems hard to extend the CRF style methods to include long-range relationships and contextual knowledge without significantly altering the models and the algorithms. In this paper, we introduce Hierarchical Image Models (HIM)’s for image parsing. HIM’s balance the trade-off between model and inference complexity by introducing a hierarchy of hidden states. In particular, we introduce recursive segmentation and recognition templates which represent complex image knowledge and serve as elementary constituents analogous to those used in speech. As in speech, we can recursively compose these constituents at lower levels to form more complex constituents at higher level. Each node of the hierarchy corresponds to an image region (whose size depends on the level in the hierarchy). The state of each node represents both the partitioning of the corresponding region into segments and the labeling of these segments (i.e. in terms of objects). Segmentations at the top levels of the hierarchy give coarse descriptions of the image which are refined by the segmentations at the lower levels. Learning and inference (parsing) are made efficient by exploiting the hierarchical structure (and the absence of loops). In short, this novel architecture offers two advantages: (I) Representation – the hierarchical model using segmentation templates is able to capture long-range dependency and exploiting different levels of contextual information, (II) Computation – the hierarchical tree structure enables rapid inference (polynomial time) and learning by variants of dynamic programming (with pruning) and the use of machine learning (e.g. structured perceptrons [9]). To illustrate the HIM we implement it for parsing images and we evaluate it on the public MSRC image dataset [8]. Our results show that the HIM outperforms the other state-of-the-art approaches. We discuss ways that HIM’s can be extended naturally to model more complex image phenomena. 2 Hierarchical Image Model 2.1 The Model We represent an image by a hierarchical graph defined by parent-child relationships. See figure 1. The hierarchy corresponds to the image pyramid (with 5 layers in this paper). The top node of the hierarchy represents the whole image. The intermediate nodes represent different sub-regions of the image. The leaf nodes represent local image patches (27 × 27 in this paper). We use a to index nodes of the hierarchy. A node a has only one parent node denoted by Pa(a) and four child nodes denoted by Ch(a). Thus, the hierarchy is a quad tree and Ch(a) encodes all its vertical edges. The image region represented by node a is denoted by R(a). A pixel in R(a), indexed by r, corresponds to an image pixel. The set of pairs of neighbor pixels in R(a) is denoted by E(a). A configuration of the hierarchy is an assignment of state variables y = {ya} with ya = (sa, ca) at each node a, where s and c denote region partition and object labeling, respectively and (s, c) is called the “Segmentation and Recognition” pair, which we call an S-R pair. All state variables are 2 Figure 1: The left panel shows the structure of the Hierarchical Image Model. The grey circles are the nodes of the hierarchy. All nodes, except the top node, have only one parent nodes. All nodes except the leafs are connected to four child nodes. The middle panel shows a dictionary of 30 segmentation templates. The color of the sub-parts of each template indicates the object class. Different sub-parts may share the same label. For example, three sub-parts may have only two distinct labels. The last panel shows that the ground truth pixel labels (upper right panel) can be well approximated by composing a set of labeled segmentation templates (bottom right panel). Figure 2: This figure illustrates how the segmentation templates and object labels (S-R pair) represent image regions in a coarse-to-fine way. The left figure is the input image which is followed by global, mid-level and local S-R pairs. The global S-R pair gives a coarse description of the object identity (horse), its background (grass), and its position in the image (central). The mid-level S-R pair corresponds to the region bounded by the black box in the input image. It represents (roughly) the shape of the horse’s leg. The four S-R pairs at the lower level combine to represent the same leg more accurately. unobservable. More precisely, each region R(a) is described by a segmentation templates which is selected from a dictionary DS. Each segmentation template consists of a partition of the region into K non-overlapping sub-parts, see figure 1. In this paper K ≤3, |Ds| = 30, and the segmentation templates are designed by hand to cover the taxonomy of shape segmentations that happen in images, such as T-junctions, Y-junctions, and so on. The variable s refers to the indexes of the segmentation templates in the dictionary, i.e., sa ∈{1..|Ds|}. c gives the object labels of K sub-parts (i.e. labels one sub-part as “horse” another as “dog” and another as “grass”). Hence ca is a K-dimension vector whose components take values 1, ..., M where M is the number of object classes. The labeling of a pixel r in region R(a) is denoted by or a ∈{1..M} and is directly obtained from sa, ca. Any two pixels belonging to the same sub-part share the same label. The labeling or a is defined at the level of node a. In other words, each level of the hierarchy has a separate labeling field. We will show how our model encourages the labelings or a at different levels to be consistent. A novel feature of this hierarchical representation is the multi-level S-R pairs which explicitly model both the segmentation and labeling of its corresponding region, while traditional vision approaches [8, 10, 11] use labeling only. The S-R pairs defined in a hierarchical form provide a coarse-to-fine representation which captures the “gist” (semantical meaning) of image regions. As one can see in figure 2, the global S-R pair gives a coarse description (the identities of objects and their spatial layout) of the whole image which is accurate enough to encode high level image properties in a compact form. The mid-level one represents the leg of a horse roughly. The four templates at the lower level further refine the interpretations. We will show this approximation quality empirically in section 3. The conditional distribution over all the states is given by: p(y|x; α) = 1 Z(x; α) exp{−E1(x, s, c; α1) −E2(x, s, c; α2) −E3(s, c; α3) (1) −E4(c; α4) −E5(s; α5) −E6(s, c; α6)} where x refers to the input image, y is the parse tree, α are the parameters to be estimated, Z(x; α) is the partition function and Ei(x, y) are energy terms. Equivalently, the conditional distribution can be reformulated in a log-linear form: log p(y|x; α) = ψ(x, y) · α −log Z(x; α) (2) 3 Each energy term is of linear form, Ei(x, y) = −ψi(x, y) · αi, where the inner product is calculated on potential functions defined over the hierarchical structure. There are six types of energy terms defined as follows. The first term E1(x, s, c) is an object specific data term which represents image features of regions. We set E1(x, s, c) = −P a α1ψ1(x, sa, ca) where P a is the summation over all nodes at different levels of the hierarchy, and ψ1(x, sa, ca) is of the form: ψ1(x, sa, ca) = 1 |R(a)| X r∈R(a) log p(or a|x) (3) where p(or a|x) = exp{F (xr,or a)} P o′ exp{F (xr,o′)}, xr is a local image region centered at the location of r, and F(·, ·) is a strong classifier output by multi-class boosting [12]. The image features used by the classifier (47 in total) are the greyscale intensity, the color (R,G, B channels), the intensity gradient, the Canny edge, the response of DOG (difference of Gaussians) and DOOG (Difference of Offset Gaussian) filters at different scales (13*13 and 22*22) and orientations (0,30,60,...), and so on. We use 55 types of shape (spatial) filters (similar to [8]) to calculate the responses of 47 image features. There are 2585 = 47 ∗55 features in total. The second term (segmentation specific) E2(x, s, c) = −P a α2ψ2(x, sa, ca) is designed to favor the segmentation templates in which the pixels belonging to the same partitions (i.e., having the same labels) have similar appearance. We define: ψ2(x, sa, ca) = 1 |E(a)| X (q,r)∈E(a) φ(xr, xq|or a, oq a) (4) where E(a) are the set of edges connecting pixels q, r in a neighborhood and φ(xr, xq|or a, oq a) has the form of φ(xr, xq|or a, oq a) = ½ γ(r, q) if or a = oq a 0 if or a ̸= oq a , where γ(r, q) = λ exp{−g2(r,q) 2γ2 } 1 dist(r,q), g(., .) is a distance measure on the colors xr, xq and dist(r, q) measures the spatial distance between r and q. φ(xr, xq|or a, oq a) is so called the contrast sensitive Potts model which is widely used in graph-cut algorithms [13] as edge potentials (only in one level) to favors pixels with similar colour having the same labels. The third term, defined as E3(s, c) = −P a,b=P a(a) α3ψ3(sa, ca, sb, cb) (i.e. the nodes a at all levels are considered and b is the parent of a) is proposed to encourage the consistency between the configurations of every pair of parent-child nodes in two consecutive layers. ψ3(sa, ca, sb, cb) is defined by the Hamming distance: ψ3(sa, ca, sb, cb) = 1 |R(a)| X r∈R(a) δ(or a, or b) (5) where δ(or a, or b) is the Kronecker delta, which equals one whenever or a = or b and zero otherwise. The hamming function ensures to glue the segmentation templates (and their labels) at different levels together in a consistent hierarchical form. This energy term is a generalization of the interaction energy in the Potts model. However, E3(s, c) has a hierarchical form which allows multi-level interactions. The fourth term E4(c) is designed to model the co-occurrence of two object classes (e.g., a cow is unlikely to appear next to an aeroplane): E4(c) = − X a X i,j=1..M α4(i, j)ψ4(i, j, ca, ca) − X a,b=P a(a) X i,j=1..M α4(i, j)ψ4(i, j, ca, cb) (6) where ψ4(i, j, ca, cb) is an indicator function which equals one while i ≡ca and j ≡cb (i ≡ca means i is a component of ca) hold true and zero otherwise. α4 is a matrix where each entry α4(i, j) encodes the compatibility between two classes i and j. The first term on the r.h.s encodes the classes in a single template while the second term encodes the classes in two templates of the parent-child nodes. It is worth noting that class dependency is encoded at all levels to capture both short-range and long-range interactions. 4 The fifth term E5(s) = −P a α5ψ5(sa), where ψ5(sa) = log p(sa) encode the generic prior of the segmentation template. Similarly the sixth term E6(s, c) = −P a P j≡ca α6ψ6(sa, j), where ψ6(sa, j) = log p(sa, j), models the co-occurrence of the segmentation templates and the object classes. ψ5(sa) and ψ6(sa, j) are directly obtained from training data by label counting. The parameters α5 and α6 are both scalars. Justifications. The HIM has several partial similarities with other work. HIM is a coarse-to-fine representation which captures the “gist” of image regions by using the S-R pairs at multiple levels. But the traditional concept of “gist” [14] relies only on image features and does not include segmentation templates. Levin and Weiss [15] use a segmentation mask which is more object-specific than our segmentation templates (and they do not have a hierarchy). It is worth nothing that, in contrast to TextonBoost [8], we do not use “location features” in order to avoid the dangers of overfitting to a restricted set of scene layouts. Our approach has some similarities to some hierarchical models (which have two-layers only) [10],[11] – but these models also lack segmentation templates. The hierarchial model proposed by [16] is an interesting alternative but which does not perform explicit segmentation. 2.2 Parsing by Dynamic Programming Parsing an image is performed as inference of the HIM. More precisely, the task of parsing is to obtain the maximum a posterior (MAP): y∗= arg max y p(y|x; α) = arg max y ψ(x, y) · α (7) The size of the states of each node is O(M K|Ds|) where K = 3, M = 21, |Ds| = 30 in our case. Since the form of y is a tree, Dynamic Programming (DP) can be applied to calculate the best parse tree y∗according to equation 7. Note that the pixel label oa is determined by (s, c), so we only need consider a subset of pixel labelings. It is unlike flat MRF representation where we need to do exhaustive search over all pixel labels o (which would be impractical for DP). The final output of the model for segmentation is the pixel labeling determined by the (s, c) of the lowest level. It is straight forward to see that the computational complexity of DP is O(M 2K|Ds|2H) where H is the number of edges of the hierarchy. Although DP can be performed in polynomial time, the huge number of states make exact DP still impractical. Therefore, we resort to a pruned version of DP similar to the method described in [17]. For brevity we omit the details. 2.3 Learning the Model Since HIM is a conditional model, in principle, estimation of its parameters can be achieved by any discriminative learning approach, such as maximum likelihood learning as used in Conditional Random Field (CRF) [7], max-margin learning [18], and structure-perceptron learning [9]. In this paper, we adopt the structure-perceptron learning which has been applied for learning the recursive deformable template (see paper [19]). Note that structure-perceptron learning is simple to implement and only needs to calculate the most probable configurations (parses) of the model. By contrast, maximum likelihood learning requires calculating the expectation of features which is difficult due to the large states of HIM. Therefore, structure-perceptron learning is more flexible and computationally simpler. Moreover, Collins [9] proved theoretical results for convergence properties, for both separable and non-separable cases, and for generalization. The structure-perceptron learning will not compute the partition function Z(x; α). Therefore we do not have a formal probabilistic interpretation. The goal of structure-perceptron learning is to learn a mapping from inputs x ∈X to output structure y ∈Y . In our case, X is a set of images, with Y being a set of possible parse trees which specify the labels of image regions in a hierarchical form. It seems that the ground truth of parsing trees needs all labels of both segmentation template and pixel labelings. In our experiment, we will show that how to obtain the ground truth directly from the segmentation labels without extra human labeling. We use a set of training examples {(xi, yi) : i = 1...n} and a set of functions ψ which map each (x, y) ∈X × Y to a feature vector ψ(x, y) ∈Rd. The task is to estimate a parameter vector α ∈Rd for the weights of the features. The feature vectors ψ(x, y) can include arbitrary features of parse trees, as we discussed in section 2.1. The loss function used in structure-perceptron learning is usually of form: Loss(α) = ψ(x, y) · α −max y ψ(x, y) · α, (8) 5 Input: A set of training images with ground truth (xi, yi) for i = 1..N. Initialize parameter vector α = 0. For t = 1..T, i = 1..N • find the best state of the model on the i’th training image with current parameter setting, i.e., y∗= arg maxy ψ(xi, y) · α • Update the parameters: α = α + ψ(xi, yi) −ψ(xi, y∗) • Store: αt,i = α Output: Parameters γ = P t,i αt,i/NT Figure 3: Structure-perceptron learning where y is the correct structure for input x, and y is a dummy variable. The basic structure-perceptron algorithm is designed to minimize the loss function. We adapt “the averaged parameters” version whose pseudo-code is given in figure 3. The algorithm proceeds in a simple way (similar to the perceptron algorithm for classification). The parameters are initialized to zero and the algorithm loops over the training examples. If the highest scoring parse tree for input x is not correct, then the parameters α are updated by an additive term. The most difficult step of the method is finding y∗= arg maxy ψ(xi, y) · α. This is precisely the parsing (inference) problem. Hence the practicality of structure-perceptron learning, and its computational efficiency, depends on the inference algorithm. As discussed earlier, see section 2.2, the inference algorithm has polynomial computational complexity for an HIM which makes structure-perceptron learning practical for HIM. The averaged parameters are defined to be γ = PT t=1 PN i=1 αt,i/NT, where T is the number of epochs, NT is the total number of iterations. It is straightforward to store these averaged parameters and output them as the final estimates. 3 Experimental Results Dataset. We use a standard public dataset, the MSRC 21-class Image Dataset [8], to perform experimental evaluations for the HIM. This dataset is designed to evaluate scene labeling including both image segmentation and multi-class object recognition. The ground truth only gives the labeling of the image pixels. To supplement this ground truth (to enable learning), we estimate the true labels (states of the S-R pair ) of the nodes in the five-layer hierarchy of HIM by selecting the S-R pairs which have maximum overlap with the labels of the image pixels. This approximation only results in 2% error in labeling image pixels. There are a total of 591 images. We use the identical splitting as [8], i.e., 45% for training, 10% for validation, and 45% for testing. The parameters learnt from the training set, with the best performance on validation set, are selected. Implementation Details. For a given image x, the parsing result is obtained by estimating the best configuration y∗of the HIM. To evaluate the performance of parsing we use the global accuracy measured in terms of all pixels and the average accuracy over the 21 object classes (global accuracy pays most attention to frequently occurring objects and penalizes infrequent objects). A computer with 8 GB memory and 2.4 GHz CPU was used for training and testing. For each class, there are around 4, 500 weak classifiers selected by multi-class boosting. The boosting learning takes about 35 hours of which 27 hours are spent on I/O processing and 8 hours on computing. The structureperceptron learning takes about 20 hours to converge in 5520(T = 20, N = 276) iterations. In the testing stage, it takes 30 seconds to parse an image with size of 320 × 200 (6s for extracting image features, 9s for computing the strong classifier of boosting and 15s for parsing the HIM). Results. Figure 4 (best viewed in color) shows several parsing results obtained by the HIM and by the classifier by itself (i.e. p(or a|x) learnt by boosting). One can see that the HIM is able to roughly capture different shaped segmentation boundaries (see the legs of the cow and sheep in rows 1 and 3, and the boundary curve between sky and building in row 4). Table 1 shows that HIM improves the results obtained by the classifier by 6.9% for average accuracy and 5.3% for global accuracy. In particular, in rows 6 and 7 in figure 4, one can observe that boosting gives many incorrect labels. It is impossible to correct such large mislabeled regions without the long-range interactions in the HIM, which improves the results by 20% and 32%. Comparisons. In table 1, we compare the performance of our approach with other successful methods [8, 20, 21]. Our approach outperforms those alternatives by 6% in average accuracy and 4% in global accuracy. Our boosting results are better than Textonboost [8] because of image features. Would we get better results if we use a flat CRF with our boosting instead of a hierarchy? We argue that we would not because the CRF only improves TextonBoost’s performance by 3 percent [8], while we gain 5 percent by using the hierarchy (and we start with a higher baseline). Some other 6 Figure 4: This figure is best viewed in color. The colors indicate the labels of 21 object classes as in [8]. The columns (except the fourth “accuracy” column) show the input images, ground truth, the labels obtained by HIM and the boosting classifier respectively. The “accuracy” column shows the global accuracy obtained by HIM (left) and the boosting classifier (right). In these 7 examples, HIM improves boosting by 1%, -1% (an outlier!), 1%, 10%, 18%, 20% and 32% in terms of global accuracy. Textonboost[8] PLSA-MRF [20] Auto-context [21] Classifier only HIM Average 57.7 64.0 68 67.2 74.1 Global 72.2 73.5 77.7 75.9 81.2 Table 1: Performance Comparisons for average accuracy and global accuracy. “Classifier only” are the results where the pixel labels are predicted by the classifier obtained by boosting only. methods [22, 11, 10], which are worse than [20, 21] and evaluated on simpler datasets [10, 11] (less than 10 classes), are not listed here due to lack of space. In summary, our results are significantly better than the state-of-the-art methods. Diagnosis on the function of S-R Pair. Figure 5 shows how the S-R pairs (which include the segmentation templates) can be used to (partially) parse an object into its constituent parts, by the correspondence between S-R pairs and specific parts of objects. We plot the states of a subset of S-R pairs for some images. For example, the S-R pair consisting of two horizontal bars labeled “cow” and “grass” respectively indicates the cow’s stomach consistently across different images. Similarly, the cow’s tail can be located according to the configuration of another S-R pair with vertical bars. In principle, the whole object can be parsed into its constituent parts which are aligned consistently. Developing this idea further is an exciting aspect of our current research. 4 Conclusion This paper describes a novel hierarchical image model (HIM) for 2D image parsing. The hierarchical nature of the model, and the use of recursive segmentation and recognition templates, enables the HIM to represent complex image structures in a coarse-to-fine manner. We can perform inference (parsing) rapidly in polynomial time by exploiting the hierarchical structure. Moreover, we can learn the HIM probability distribution from labeled training data by adapting the structure-perceptron algorithm. We demonstrated the effectiveness of HIM’s by applying them to the challenging task of segmentation and labeling of the public MSRC image database. Our results show that we outperform other state-of-the-art approaches. 7 Figure 5: The S-R pairs can be used to parse the object into parts. The colors indicate the identities of objects. The shapes (spacial layout) of the segmentation templates distinguish the constituent parts of the object. Observe that the same S-R pairs (e.g. stomach above grass, and tail to the left of grass) correspond to the same object part in different images. The design of the HIM was motivated by drawing parallels between language and vision processing. We have attempted to capture the underlying spirit of the successful language processing approaches – the hierarchical representations based on the recursive composition of constituents and efficient inference and learning algorithms. Our current work attempts to extend the HIM’s to improve their representational power while maintaining computational efficiency. 5 Acknowledgments This research was supported by NSF grant 0413214 and the W.M. Keck foundation. References [1] F. Jelinek and J. D. Lafferty, “Computation of the probability of initial substring generation by stochastic context-free grammars,” Computational Linguistics, vol. 17, no. 3, pp. 315–323, 1991. [2] M. Collins, “Head-driven statistical models for natural language parsing,” Ph.D. Thesis, University of Pennsylvania, 1999. [3] K. Lari and S. J. Young, “The estimation of stochastic context-free grammars using the inside-outside algorithm,” in Computer Speech and Languag, 1990. [4] M. Shilman, P. Liang, and P. A. Viola, “Learning non-generative grammatical models for document analysis,” in Proceedings of IEEE International Conference on Computer Vision, 2005, pp. 962–969. [5] Z. Tu and S. C. Zhu, “Image segmentation by data-driven markov chain monte carlo,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 24, no. 5, pp. 657–673, 2002. [6] Z. Tu, X. Chen, A. L. Yuille, and S. C. Zhu, “Image parsing: Unifying segmentation, detection, and recognition,” in Proceedings of IEEE International Conference on Computer Vision, 2003, pp. 18–25. [7] J. D. Lafferty, A. McCallum, and F. C. N. Pereira, “Conditional random fields: Probabilistic models for segmenting and labeling sequence data,” in Proceedings of International Conference on Machine Learning, 2001, pp. 282–289. [8] J. Shotton, J. M. Winn, C. Rother, and A. Criminisi, “TextonBoost: Joint appearance, shape and context modeling for multi-class object recognition and segmentation,” in Proceedings of European Conference on Computer Vision, 2006, pp. 1–15. [9] M. Collins, “Discriminative training methods for hidden markov models: theory and experiments with perceptron algorithms,” in Proceedings of Annual Meeting on Association for Computational Linguistics conference on Empirical methods in natural language processing, 2002, pp. 1–8. [10] X. He, R. S. Zemel, and M. ´A. Carreira-Perpi˜n´an, “Multiscale conditional random fields for image labeling,” in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004, pp. 695–702. [11] S. Kumar and M. Hebert, “A hierarchical field framework for unified context-based classification,” in Proceedings of IEEE International Conference on Computer Vision, 2005, pp. 1284–1291. [12] E. L. Allwein, R. E. Schapire, and Y. Singer, “Reducing multiclass to binary: A unifying approach for margin classifiers,” Journal of Machine Learning Research, vol. 1, pp. 113–141, 2000. [13] Y. Boykov and M.-P. Jolly, “Interactive graph cuts for optimal boundary and region segmentation of objects in n-d images,” in Proceedings of IEEE International Conference on Computer Vision, 2001, pp. 105–112. [14] A. Oliva and A. Torralba, “Building the gist of a scene: the role of global image features in recognition,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 155, pp. 23–36, 2006. [15] A. Levin and Y. Weiss, “Learning to combine bottom-up and top-down segmentation,” in Proceedings of European Conference on Computer Vision, 2006, pp. 581–594. [16] E. B. Sudderth, A. B. Torralba, W. T. Freeman, and A. S. Willsky, “Learning hierarchical models of scenes, objects, and parts,” in Proceedings of IEEE International Conference on Computer Vision, 2005, pp. 1331–1338. [17] Y. Chen, L. Zhu, C. Lin, A. L. Yuille, and H. Zhang, “Rapid inference on a novel and/or graph for object detection, segmentation and parsing,” in Advances in Neural Information Processing Systems, 2007. [18] B. Taskar, D. Klein, M. Collins, D. Koller, and C. Manning, “Max-margin parsing,” in Proceedings of Annual Meeting on Association for Computational Linguistics conference on Empirical methods in natural language processing, 2004. [19] L. Zhu, Y. Chen, X. Ye, and A. L. Yuille, “Structure-perceptron learning of a hierarchical log-linear model,” in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2008. [20] J. Verbeek and B. Triggs, “Region classification with markov field aspect models,” in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2007. [21] Z. Tu, “Auto-context and its application to high-level vision tasks,” in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2008. [22] J. Verbeek and B. Triggs, “Scene segmentation with crfs learned from partially labeled images,” in Advances in Neural Information Processing Systems, vol. 20, 2008. 8
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Predicting the Geometry of Metal Binding Sites from Protein Sequence Paolo Frasconi Universit`a degli Studi di Firenze Via di S. Marta 3, 50139 Firenze, Italy p-f@dsi.unifi.it Andrea Passerini Universit`a degli Studi di Trento Via Sommarive, 14, 38100 Povo, Italy passerini@disi.unitn.it Abstract Metal binding is important for the structural and functional characterization of proteins. Previous prediction efforts have only focused on bonding state, i.e. deciding which protein residues act as metal ligands in some binding site. Identifying the geometry of metal-binding sites, i.e. deciding which residues are jointly involved in the coordination of a metal ion is a new prediction problem that has been never attempted before from protein sequence alone. In this paper, we formulate it in the framework of learning with structured outputs. Our solution relies on the fact that, from a graph theoretical perspective, metal binding has the algebraic properties of a matroid, enabling the application of greedy algorithms for learning structured outputs. On a data set of 199 non-redundant metalloproteins, we obtained precision/recall levels of 75%/46% correct ligand-ion assignments, which improves to 88%/88% in the setting where the metal binding state is known. 1 Introduction Metal ions play important roles in protein function and structure and metalloproteins are involved in a number of diseases for which medicine is still seeking effective treatment, including cancer, Parkinson, dementia, and AIDS [10]. A metal binding site typically consists of an ion bound to one or more protein residues (called ligands). In some cases, the ion is embedded in a prosthetic group (e.g. in the case of heme). Among the 20 amino acids, the four most common ligands are cysteine (C), histidine (H), aspartic acid (D), and glutamic acid (E). Highly conserved residues are more likely to be involved in the coordination of a metal ion, although in the case of cysteines, conservation is also often associated with the presence of a disulfide bridge (a covalent bond between the sulfur atoms of two cysteines) [8]. Predicting metal binding from sequence alone can be very useful in genomic annotation for characterizing the function and the structure of non determined proteins, but also during the experimental determination of new metalloproteins. Current high-throughput experimental technologies only annotate whole proteins as metal binding [13], but cannot determine the involved ligands. Most of the research for understanding metal binding has focused on finding sequence patterns that characterize binding sites [8]. Machine learning techniques have been applied only more recently. The easiest task to formulate in this context is bonding state prediction, which is a binary classification problem: either a residue is involved in the coordination of a metal ion or is free (in the case of cysteines, a third class can also be introduced for disulfide bridges). This prediction task has been addressed in a number of recent works in the case of cysteines only [6], in the case of transition metals (for C and H residues) [12] and for in the special but important case of zinc proteins (for C,H,D, and E residues) [11, 14]. Hovever, classification of individual residues does not provide sufficient information about a binding site. Many proteins bind to several ions in their holo form and a complete characterization requires us to identify the site geometry, i.e. the tuple of residues coordinating each individual ion. This problem has been only studied assuming knowledge of the protein 3D structure (e.g. [5, 1]), limiting its applicability to structurally determined proteins or their close homologs, but not from sequence alone. Abstracting away the biology, this is a structured output prediction problem where the input consists of a string of protein residues and the output is a labeling of each residue with the corresponding ion identifier (specific details are given in the next section). The supervised learning problem with structured outputs has recently received a considerable amount of attention (see [2] for an overview). The common idea behind most methods consists of learning a function F(x, y) on input-output pairs (x, y) and, during prediction, searching the argument y that maximises F when paired with the query input x. The main difficulty is that the search space on which y can take values has usually exponential size (in the length of the query). Different structured output learners deal with this issue by exploiting specific domain properties for the application at hand. Some researchers have proposed probabilistic modeling and efficient dynamic programming algorithms (e.g. [16]). Others have proposed large margin approaches combined with clever algorithmic ideas for reducing the number of constraints (e.g. [15] in the case of graph matching). Another solution is to construct the structured output in a suitable Hilbert space of features and seek the corresponding pre-image for obtaining the desired discrete structure [17]. Yet another is to rely on a state-space search procedure and learn from examples good moves leading to the desired goal [4]. In this paper we develop a large margin solution that does not require a generative model for producing outputs. We borrow ideas from [15] and [4] but specifically take advantage of the fact that, from a graph theoretical perspective, the metal binding problem has the algebraic structure of a matroid, enabling the application of greedy algorithms. 2 A formalization of the metal binding sites prediction problem A protein sequence s is a string in the alphabet of the 20 amino acids. Since only some of the 20 amino acids that exist in nature can act as ligands, we begin by extracting from s the subsequence x obtained by deleting characters corresponding to amino acids that never (or very rarely) act as ligands. By using T = {C, H, D, E} as the set of candidate ligands, we cover 92% ligands of structurally known proteins. A large number of interesting cases (74% in transition metals) is covered by just considering cysteines and histidines, i.e. T = {C, H}. We also introduce the set I of symbols associated with metal ion identifiers. I includes the special nil symbol. The goal is to predict the coordination relation between amino acids in x and metal ions identifiers in I. Amino acids that are not metal-bound are linked to nil. Ideally, it would be also interesting to predict the chemical element of the bound metal ion. However, previous studies suggest that distinguishing the chemical element from sequence alone is a difficult task [12]. Hence, ion identifiers will have no chemical element attribute attached. In practice, we fix a maximum number m of possible ions (m = 4 in the subsequent experiments, covering 93% of structurally known proteins) and let I = {nil, ι1, . . . , ιm}. The number of admissible binding geometries for a given protein chain having n candidate ligands is the multinomial coefficient n! k1!k2!···km!(n−k1−···−km)! being m the number of ions and ki the number of ligands for ion ιi. In practice, each ion is coordinated by a variable number of ligands (typically ranging from 1 to 4, but occasionally more), and each protein chain binds a variable number of ions (typically ranging from 1 to 4). The number of candidate ligands n grows linearly with the protein chain. For example, in the case of PDB chain 1H0Hb (see Figure 1), there are n = 52 candidate ligands and m = 3 ions coordinated by 4 residues each, yielding a set of 7 · 1015 admissible conformations. It is convenient to formulate the problem in a graph theoretical setting. In this view, the string x should be regarded as a set of vertices labeled with the corresponding amino acid in T . The semantic of x will be clear from the context and for simplicity we will avoid additional notation. Definition 2.1 (MBG property). Let x and I be two sets of vertices (associated with candidate ligands and metal ion identifiers, respectively). We say that a bipartite edge set y ⊂x × I satisfies the metal binding geometry (MBG) property if the degree of each vertex in x in the graph (x ∪I, y) is at most 1. For a given x, let Yx denote the set of y that satisfy the MBG property. Let Fx : Yx 7→IR+ be a function that assigns a positive score to each bipartite edge set in Yx. The MBG problem consists of finding arg maxy∈Yx Fx(y). nil … D C C C C H E H D H H E E D D D C H C C D E D H D D C D E D E C D E C D C D C C D E E E D C D D C H H E 1 1 0 2 0 3 0 4 0 5 0 ι1 ι2 ι3 Figure 1: Metal binding structure of PDB entry 1H0Hb. For readability, only a few connections from free residues to the nil symbol are shown. Note that the MBG problem is not a matching problem (such as those studied in [15]) since more than one edge can be incident to vertices belonging to I. As discussed above, we are not interested in distinguishing metal ions based on the element type. Hence, any two label-isomorphic bipartite graphs (obtained by exchanging two non-nil metal ion vertices) should be regarded as equivalent. Outputs y should be therefore regarded as equivalence classes of structures (in the 1H0Hb example above, there are 7 · 1015/3! equivalence classes, each corresponding to a permutation of ι1, ι2, ι3). For simplicity, we will slightly abuse notation and avoid this distinction in the following. We could also look over the MBG problem by analogy with language parsing using formal grammars. In this view, the binding geometry consists of a very shallow “parse tree” for string x, as examplified in Figure 1. A difficulty that is immediately apparent is that the underlying grammar needs to be context sensitive in order to capture the crossing-dependencies between bound amino acids. In real data, when representing metal bonding state in this way, crossing edges are very common. This view enlightens a difficulty that would be encountered by attempting to solve the structured output problem with a generative model as in [16]. 3 A greedy algorithm for constructing structured outputs The core idea of the solution used in this paper is to avoid a generative model as a component of the structured output learner and cast the construction of an output structure into a maximum weight problem that can be solved by a greedy algorithm. Definition 3.1 (Matroid). A matroid (see e.g. [9]) is an algebraic structure M = (S, Y) where S is a finite set and Y a family of subsets of S such that: i) ∅⊆Y; ii) all proper subsets of a set y in Y are in Y; iii) if y and y′ are in Y and |y| < |y′| then there exists e ∈y′ \ y such that y ∪{e} ∈Y. Elements of Y are called independent sets. If y is an independent set, then ext(y) = {e ∈S : y ∪{e} ∈Y} is called the extension set of y. A maximal (having an empty extension set) independent set is called a base. In a weighted matroid, a local weight function v : S 7→IR+ assigns a positive number v(e) to each element e ∈S. The weight function allows us to compare two structures in the following sense. A set y = {e1, . . . , en} is lexicographically greater than set y′ if its monotonically decreasing sequence of weights (v(e1), . . . , v(en)) is lexicographically greater than the corresponding sequence for y′. The following classic result (see e.g. [9]) is the underlying support for many greedy algorithms: Theorem 3.2 (Rado 1957; Edmonds 1971). For any nonnegative weighting over S, a lexicographically maximum base in Y maximizes the global objective function F(y) = P e∈y v(e). Weighted matroids can be seen as a kind of discrete counterparts of concave functions: thanks to the above theorem, if M is a weighted matroid, then the following greedy algorithm is guaranteed to find the optimal structure, i.e. arg maxy∈Y F(y): GREEDYCONSTRUCT(M, F) y ←∅ while ext(y) ̸= ∅ do y ←y ∪ n arg maxe∈ext(y) F(y ∪{e}) o return y This theory shows that if the structured output space being searched satisfies the property of a matroid, learning structured outputs may be cast into the problem of learning the objective function F for the greedy algorithm. When following this strategy, however, we may perceive the additive form of F as a strong limitation as it would prescribe to predict v(e) independently for each part e ∈S, while the whole point of structured output learning is to end-up with a collective decision about which parts should be present in the output structure. But interestingly, the additive form of the objective function as in Theorem 3.2 is not a necessary condition for the greedy optimality of matroids. In facts, Helman et al. [7] show that the classic theory can be generalized to so-called consistent objective functions, i.e. functions that satisfy the following additional constraints: F(y ∪{e}) ≥F(y ∪{e′}) ⇒F(y′ ∪{e}) ≥F(y′ ∪{e′}) (1) for any y ⊂y′ ⊂S and e, e′ ∈S \ y′. Theorem 3.3 (Helman et al. 1993). If F is a consistent objective function then, for each matroid on S, all greedy bases are optimal. Note that the sufficient condition of Theorem 3.3 is also necessary for a slighly more general class of algebraic structures that include matroids, called matroid embeddings [7]. We now show that the MBG problem is a suitable candidate for a greedy algorithmic solution. Theorem 3.4. If each y ∈Yx satisfies the MBG property, then Mx = (Sx, Yx) is a matroid. Proof. Suppose y′ ∈Yx and y ⊆y′. Removing an edge from y′ cannot increase the degree of any vertex in the bipartite graph so y ∈Yx. Also, suppose y ∈Yx, y′ ∈Yx, and |y| < |y′|. Then there must be at least one vertex t in x having no incident edges in y and such that (ι, t) ∈y′ for some ι ∈I. Therefore y ∪{(ι, t)} also satisfies the MBG property and belongs to Yx, showing that Mx is a matroid. We can finally formulate the greedy algorithm for constructing the structured output in the MBG problem. Given the input x, we begin by forming the associated MBG matroid Mx and a corresponding objective function Fx : Yx 7→IR+ (in the next section we will show how to learn the objective function from data). The output structure associated with x is then computed as f(x) = arg max y∈Yx Fx(y) = GREEDYCONSTRUCT(Mx, Fx). (2) The following result immediately follows from Definition 2.1 and Theorem 3.3: Corollary 3.5. Let (x, y) be an MBG instance. If Fx is a consistent objective function and Fx(y′ ∪{e}) > Fx(y′ ∪{e′}) for each y′ ⊂y, e ∈ext(y′) ∩y and e′ ∈ext(y′) \ y, then GREEDYCONSTRUCT((Sx, Yx), Fx) returns y. 4 Learning the greedy objective function A data set for the MBG problem consist of pairs D = {(xi, yi)} where xi is a string in T ∗and yi a bipartite graph. Corollary 3.5 directly suggests the kind of constraints that the objective function needs to satisfy in order to minimize the empirical error of the structured-output problem. For any input string x and (partial) output structure y ∈Y, let Fx(y) = wT φx(y), being w a weight vector and φx(y) a feature vector for (x, y). The corresponding max-margin formulation is min 1 2∥w∥2 (3) subject to: wT φxi(y′ ∪{e}) −φxi(y′ ∪{e′}) ≥1 (4) wT φxi(y′′ ∪{e}) −φxi(y′′ ∪{e′}) ≥1 (5) ∀i = 1, . . . , |D|, ∀y′ ⊂yi, ∀e ∈ext(y′) ∩yi, ∀e′ ∈ext(y′) \ yi, ∀y′′ : y′ ⊂y′′ ⊂Sx. Intuitively, the first set of constraints (Eq. 4) ensures that “correct” extensions (i.e. edges that actually belong to the target output structure yi) receive a higher weight than “wrong” extensions (i.e. edges that do not belong to the target output structure). The purpose of the second set of constraints (Eq. 5) is to force the learned objective function to obey the consistency property of Eq. (1), which in turns ensures the correctness of the greedy algorithm thanks to Theorem 3.3. As usual, a regularized variant with soft constraints can be formulated by introducing positive slack variables and adding their 1-norm times a regularization coefficient to Eq. (3). The number of resulting constraints in the above formulation grows exponentially with the number of edges in each example, hence naively solving problem (3–5) is practically unfeasible. However, we can seek an approximate solution by leveraging the efficiency of the greedy algorithm also during learning. For this purpose, we will use an online active learner that samples constraints chosen by the execution of the greedy construction algorithm. For each epoch, the algorithm maintains the current highest scoring partial correct output y′ i ⊆yi for each example, initialized with the empty MBG structure, where the score is computed by the current objective function F. While there are “unprocessed” examples in D, the algorithm picks a random one and its current best MBG structure y′. If there are no more correct extensions of y′, then y′ = yi and the example is removed from D. Otherwise, the algorithm evaluates each correct extension of y′, updates the current best MBG structure, and invokes the online learner by calling FORCE-CONSTRAINT, which adds a constraint derived from a random incorrect extension (see Eq. 4). It also performs a predefined number L of lookaheads by picking a random superset of y′′ which is included in the target yi, evaluating it and updating the best MBG structure if needed, and adding a corresponding consistency constraint (see Eq. 5). The epoch terminates when all examples are processed. In practice, we found that a single epoch over the data set is sufficient for convergence. Pseudocode for one epoch is given below. GREEDYEPOCH(D, L) for i ←1, . . . , |D| do y′ i ←∅ while D ̸= ∅ do pick a random example (xi, yi) ∈D y′ ←y′ i, y′ i ←∅ if ext(y′) ∩yi = ∅ then D ←D \ (xi, yi) else for each e ∈ext(y′) ∩yi do pick randomly e′ ∈ext(y′) \ yi if F(y′ i) < F(y′ ∪{e}) then y′ i ←y′ ∪{e} FORCE-CONSTRAINT(Fxi(y′ ∪{e}) −Fxi(y′ ∪{e′}) ≥1) for l ←1, . . . , L do randomly choose y′′ : y′ ⊂y′′ ⊂yi ∧e, e′ ∈Sx \ y′′ FORCE-CONSTRAINT(Fxi(y′′ ∪{e}) −Fxi(y′′ ∪{e′}) ≥1) if F(y′ i) < F(y′′ ∪{e}) then y′ i ←y′′ ∪{e} There are several suitable online learners implementing the interface required by the above procedure. Possible candidates include perceptron-like or ALMA-like update rules like those proposed in [4] for structured output learning (in our case the update would depend on the difference between feature vectors of correctly and incorrectly extended structures in the inner loop of GREEDYEPOCH). An alternative online learner is the LaSVM algorithm [3] equipped with obvious modifications for handling constraints between pairs of examples. LaSVM is an SMO-like solver for the dual version of problem (3–5) that optimizes one or two coordinates at a time, alternating process (on newly acquired examples, generated in our case by the FORCE-CONSTRAINT procedure) and reprocess (on previously seen support vectors or patterns) steps. The ability to work efficiently in the dual is the most appealing feature of LaSVM in the present context and advantageous with respect to perceptron-like approaches. Our unsuccessful preliminary experiments with simple feature vectors confirmed the necessity of flexible design choices for developing rich feature spaces. Kernel methods are clearly more attractive in this case. We will therefore rewrite the objective function F using a kernel k(z, z′) = ⟨φx(y), φx′(y′)⟩between two structured instances z = (x, y) and z′ = (x′, y′), so that Fx(y) = F(z) = P i αik(z, zi). Let σi(z) denote the set of edges incident on ion ιi ∈I \ nil and n(z) the number of non-nil ion identifiers that have at least one incident edge. Below is a top-down definition of the kernel used in the subsequent experiments. k(z, z′) = kglob(z, z′) n(z) X i=1 n(z′) X j=1 kmbs(σi(z), σj(z′)) n(z)n(z′) (6) kglob(z, z′) = δ(n(z), n(z′))2 min{|x|, |x′|} |x| + |x′| (7) kmbs(σi(z), σj(z′)) = δ(|σi(z)|, |σj(z′)|) |σi(z)| X ℓ=1 kres(xi(ℓ), x′ j(ℓ)) (8) where δ(a, b) = 1 iff a = b, xi(ℓ) denotes the ℓ-th residue in σi(z), taken in increasing order of sequential position in the protein, and kres(xi(ℓ), x′ j(ℓ)) is simply the dot product between the feature vectors describing residues xi(ℓ) and x′ j(ℓ) (details on these features are given in Section 5). kmbs measures the similarity between individual sites (two sites are orthogonal if have a different number of ligands, a choice that is supported by protein functional considerations). kglob ensures that two structures are orthogonal unless they have the same number of sites and down weights their similarity when their number of candidate ligands differs. 5 Experiments We tested the method on a dataset of non-redundant proteins previously used in [12] for metal bonding state prediction (http://www.dsi.unifi.it/˜passe/datasets/ mbs06/dataset.tgz). Proteins that do not bind metal ions (used in [12] as negative examples) are of no interest in the present case and were removed, resulting in a set of 199 metalloproteins binding transition metals. Following [12], we used T = {C, H} as the set of candidate ligands. Protein sequences were enriched with evolutionary information derived from multiple alignments. Profiles were obtained by running one iteration of PSI-BLAST on the non-redundant (nr) NCBI dataset, with an e-value cutoff of 0.005. Each candidate ligand xi(ℓ) was described by a feature vector of 221 real numbers. The first 220 attributes consist of multiple alignment profiles in the window of 11 amino acids centered around xi(ℓ) (the window was formed from the original protein sequence, not the substring xi of candidate ligands). The last attribute is the normalized sequence separation between xi(ℓ) and xi(ℓ−1), using the N-terminus of the chain for ℓ= 1. A modified version of LaSVM (http://leon.bottou.org/projects/lasvm) was run with constraints produced by the GREEDYEPOCH procedure of Section 4, using a fixed regularization parameter C = 1, and L ∈{0, 5, 10}. All experiments were repeated 30 times, randomly splitting the data into a training and test set in a ratio of 80/20. Two prediction tasks were considered, from unknown and from known metal bonding state (a similar distinction is also customary for the related task of disulfide bonds prediction, see e.g. [15]). In the latter case, the input x only contains actual ligands and no nil symbol is needed. Several measures of performance are reported in Table 1. PE and RE are the precision and recall for the correct assignment between a residue and the metal ion identifier (ratio of correctly predicted coordinations to the number of predicted/actual coordinations); correct links to the nil ion (that would optimistically bias the results) are ignored in these measures. AG is the geometry accuracy, i.e. the fraction of chains that are entirely correctly predicted. PS and RS are the metal binding site precision and recall, respectively (ratio of correctly predicted sites to the number of predicted/actual sites). Finally, PB and RB are precision and recall for metal bonding state prediction (as in binary classification, being “bonded” the positive class). Table 2 reports the breakdown of these performance measures for proteins binding different numbers of metal ions (for L = 10). Results show that enforcing consistency constraints tends to improve recall, especially for the bonding state prediction, i.e. helps the predictor to assign a residue to a metal ion identifier rather than to nil. However, it only marginally improves precision and recall at the site level. Correct prediction of whole sites is very challenging and correct prediction of whole chains even more difficult (given the enormous number of alternatives to be compared). Hence, it is not surprising that some of these performance indicators are low. By comparison, absolute figures are not high even for the much easier task of disulfide bonds prediction [15]. Correct edge assignment, however, appears satisfactory and reasonably good when the bonding state is given. The complete experimental environment can be obtained from http://www.disi.unitn.it/˜passerini/nips08.tgz. Table 1: Experimental results. ab-initio L PE RE AG PS RS PB RB 0 75±5 46±5 12±4 18±6 14±6 81±5 51±6 5 66±5 52±4 14±6 20±7 17±6 79±4 64±6 10 63±5 52±5 13±6 20±7 15±6 78±4 68±5 metal bonding state given L PE RE AG PS RS 0 87±2 87±2 64±6 65±6 65±6 5 87±3 87±3 65±7 66±7 66±7 10 88±3 88±3 67±7 67±7 67±7 Table 2: Breakdown by number of sites each chain. BS= (K)nown/(U)nknown bonding state. # sites = 1 (132 chains) # sites = 2 (48 chains) BS PE RE PS RS AG PE RE PS RS AG U 62±6 57±6 25±9 21±8 19±8 67±9 46±8 14±12 6±8 3±6 K 97±2 97±2 92±6 92±6 92±6 73±5 73±5 21±10 21±10 20±11 # sites = 3 (11 chains) # sites = 4 (8 chains) BS PE RE PS RS AG PE RE PS RS AG U 65±16 33±13 1±5 1±5 0 44±31 24±20 3±11 2±6 0 K 61±12 61±12 8±11 9±13 0 37±25 37±25 1±2 1±2 0 6 Related works As mentioned in the Introduction, methods for structured outputs usually learn a function F on inputoutput pairs (x, y) and construct the predicted output as f(x) = arg maxy F(x, y). Our approach follows the same general principle. There is a notable analogy between the constrained optimization problem (3–5) and the set of constraints derived in [15] for the related problem of disulfide connectivity. As in [15], our method is based on a large-margin approach for solving a structured output prediction problem. The underlying formal problems are however very different and require different algorithmic solutions. Disulfide connectivity is a (perfect) matching problem since each cysteine is bound to exactly one other cysteine (assuming known bonding state, yielding a perfect matching) or can be bound to another cysteine or free (unknown bonding state, yielding a non-perfect matching). The original set of constraints in [15] only focuses on complete structures (non extensible set or bases, in our terminology). It also has exponential size but the matching structure of the problem in that case allows the authors to derive a certificate formulation that reduces it to polynomial size. The MBG problem is not a matching problem but has the structure of a matroid and our formulation allows us to control the number of effectively enforced constraints by taking advantage of a greedy algorithm. The idea of an online learning procedure that receives examples generated by an algorithm which constructs the output structure was inspired from the Learning as Search Optimization (LaSO) approach [4]. LaSO aims to solve a much broader class of structured output problems where good output structures can be generated by AI-style search algorithms such as beam search or A*. The generation of a fresh set of siblings in LaSO when the search is stuck with a frontier of wrong candidates (essentially a backtrack) is costly compared to our greedy selection procedure and (at least in principle) unnecessary when working on matroids. Another general way to deal with the exponential growth of the search space is to introduce a generative model so that arg maxy F(x, y) can be computed efficiently, e.g. by developing an appropriate dynamic programming algorithm. Stochastic grammars and related conditional models have been extensively used for this purpose [2]. These approaches work well if the generative model matches or approximates well the domain at hand. Unfortunately, as discussed in Section 2, the specific application problem we study in this paper cannot be even modeled by a context-free grammar. While we do not claim that it is impossible to devise a suitable generative model for this task (and indeed this is an interesting direction of research), we can argue that handling context-sensitiveness is hard. It is of course possible to approximate context sensitive dependencies using a simplified model. Indeed, an alternative view of the MBG problem is supervised sequence labeling, where the output string consists of symbols in I. A (higher-order) hidden Markov model or chain-structured conditional random field could be used as the underlying generative model for structured output learning. Unfortunately, these approaches are unlikely to be very accurate since models that are structured as linear chains of dependencies cannot easily capture long-ranged interactions such as those occurring in the example. In our preliminary experiments, SVMHMM [16] systematically assigned all bonded residues to the same ion, thus never correctly predicted the geometry except in trivial cases. 7 Conclusions We have reported about the first successful solution to the challenging problem of predicting protein metal binding geometry from sequence alone. The result fills-in an important gap in structural and functional bioinformatics. Learning with structured outputs is a fairly difficult task and in spite of the fact that several methodologies have been proposed, no single general approach can effectively solve every possible application problem. The solution proposed in this paper draws on several previous ideas and specifically leverages the existence of a matroid for the metal binding problem. Other problems that formally exhibit a greedy structure might benefit of similar solutions. Acknowledgments We thank Thomas G¨artner for very fruitful discussions. References [1] M. Babor, S. Gerzon, B. Raveh, V. Sobolev, and M. Edelman. Prediction of transition metal-binding sites from apo protein structures. Proteins, 70(1):208–217, 2008. [2] G. Bakir, T. Hofmann, B. Sch¨olkopf, A. Smola, B. Taskar, and S. Vishwanathan, editors. Predicting Structured Data. The MIT Press, 2007. [3] A. Bordes, S. Ertekin, J. Weston, and L. Bottou. Fast kernel classifiers with online and active learning. Journal of Machine Learning Research, 6:1579–1619, 2005. [4] H. Daume III and D. Marcu. Learning as search optimization: Approximate large margin methods for structured prediction. In Proc. of the 22nd Int. Conf. on Machine Learning (ICML’05), 2005. [5] J. C. Ebert and R. B. Altman. Robust recognition of zinc binding sites in proteins. Protein Sci, 17(1):54– 65, 2008. [6] F. Ferr`e and P. Clote. DiANNA 1.1: an extension of the DiANNA web server for ternary cysteine classification. Nucleic Acids Res, 34:W182–W185, 2006. [7] P. Helman, B. M. E. Moret, and H. D. Shapiro. An exact characterization of greedy structures. SIAM J. Disc. Math., 6(2):274–283, 1993. [8] N. Hulo, A. Bairoch, V. Bulliard, L. Cerutti, B. A. Cuche, E. de Castro, C. Lachaize, P. S. LangendijkGenevaux, and C. J. A. Sigrist. The 20 years of prosite. Nucleic Acids Res, 36:D245–9, 2008. [9] E. L. Lawler. Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, 1976. [10] A. Messerschmidt, R. Huber, K. Wieghardt, and T. Poulos, editors. Handbook of Metalloproteins. John Wiley & Sons, 2004. [11] A. Passerini, C. Andreini, S. Menchetti, A. Rosato, and P. Frasconi. Predicting zinc binding at the proteome level. BMC Bioinformatics, 8:39, 2007. [12] A. Passerini, M. Punta, A. Ceroni, B. Rost, and P. Frasconi. Identifying cysteines and histidines in transition-metal-binding sites using support vector machines and neural networks. Proteins, 65(2):305– 316, 2006. [13] W. Shi, C. Zhan, A. Ignatov, B. A. Manjasetty, N. Marinkovic, M. Sullivan, R. Huang, and M. R. Chance. Metalloproteomics: high-throughput structural and functional annotation of proteins in structural genomics. Structure, 13(10):1473–1486, 2005. [14] N. Shu, T. Zhou, and S. Hovmoller. Prediction of zinc-binding sites in proteins from sequence. Bioinformatics, 24(6):775–782, 2008. [15] B. Taskar, V. Chatalbashev, D. Koller, and C. Guestrin. Learning structured prediction models: a large margin approach. Proc. of the 22nd Int. Conf. on Machine Learning (ICML’05), pages 896–903, 2005. [16] I. Tsochantaridis, T. Joachims, T. Hofmann, and Y. Altun. Large Margin Methods for Structured and Interdependent Output Variables. The Journal of Machine Learning Research, 6:1453–1484, 2005. [17] J. Weston, O. Chapelle, A. Elisseeff, B. Scholkopf, and V. Vapnik. Kernel dependency estimation. Advances in Neural Information Processing Systems, 15:873–880, 2003.
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Cascaded Classification Models: Combining Models for Holistic Scene Understanding Geremy Heitz Stephen Gould Department of Electrical Engineering Stanford University, Stanford, CA 94305 {gaheitz,sgould}@stanford.edu Ashutosh Saxena Daphne Koller Department of Computer Science Stanford University, Stanford, CA 94305 {asaxena,koller}@cs.stanford.edu Abstract One of the original goals of computer vision was to fully understand a natural scene. This requires solving several sub-problems simultaneously, including object detection, region labeling, and geometric reasoning. The last few decades have seen great progress in tackling each of these problems in isolation. Only recently have researchers returned to the difficult task of considering them jointly. In this work, we consider learning a set of related models in such that they both solve their own problem and help each other. We develop a framework called Cascaded Classification Models (CCM), where repeated instantiations of these classifiers are coupled by their input/output variables in a cascade that improves performance at each level. Our method requires only a limited “black box” interface with the models, allowing us to use very sophisticated, state-of-the-art classifiers without having to look under the hood. We demonstrate the effectiveness of our method on a large set of natural images by combining the subtasks of scene categorization, object detection, multiclass image segmentation, and 3d reconstruction. 1 Introduction The problem of “holistic scene understanding” encompasses a number of notoriously difficult computer vision tasks. Presented with an image, scene understanding involves processing the image to answer a number of questions, including: (i) What type of scene is it (e.g., urban, rural, indoor)? (ii) What meaningful regions compose the image? (iii) What objects are in the image? (iv) What is the 3d structure of the scene? (See Figure 1). Many of these questions are coupled—e.g., a car present in the image indicates that the scene is likely to be urban, which in turn makes it more likely to find road or building regions. Indeed, this idea of communicating information between tasks is not new and dates back to some of the earliest work in computer vision (e.g., [1]). In this paper, we present a framework that exploits such dependencies to answer questions about novel images. While our focus will be on image understanding, the goal of combining related classifiers is relevant to many other machine learning domains where several related tasks operate on the same (or related) raw data and provide correlated outputs. In the area of natural language processing, for instance, we might want to process a single document and predict the part of speech of all words, correspond the named entities, and label the semantic roles of verbs. In the area of audio signal processing, we might want to simultaneously do speech recognition, source separation, and speaker recognition. In the problem of scene understanding (as in many others), state-of-the-art models already exist for many of the tasks of interest. However, these carefully engineered models are often tricky to modify, or even simply to re-implement from available descriptions. As a result, it is sometimes desirable to treat these models as “black boxes,” where we have we have access only to a very simple input/output interface. in short, we require only the ability to train on data and produce classifications for each data instance; specifics are given in Section 3 below. In this paper, we present the framework of Cascaded Classification Models (CCMs), where stateof-the-art “black box” classifiers for a set of related tasks are combined to improve performance on 1 (a) Detected Objects (b) Classified Regions (c) 3D Structure (d) CCM Framework Figure 1: (a)-(c) Some properties of a scene required for holistic scene understanding that we seek to unify using a cascade of classifiers. (d) The CCM framework for jointly predicting each of these label types. some or all tasks. Specifically, the CCM framework creates multiple instantiations of each classifier, and organizes them into tiers where models in the first tier learn in isolation, processing the data to produce the best classifications given only the raw instance features. Lower tiers accept as input both the features from the data instance, as well as features computed from the output classifications of the models at the previous tier. While only demonstrated in the computer vision domain, we expect the CCM framework have broad applicability to many applications in machine learning. We apply our model to the scene understanding task by combining scene categorization, object detection, multi-class segmentation, and 3d reconstruction. We show how “black-box” classifiers can be easily integrated into our framework. Importantly, in extensive experiments on large image databases, we show that our combined model yields superior results on all tasks considered. 2 Related Work A number of works in various fields aim to combine classifiers to improve final output accuracy. These works can be divided into two broad groups. The first is the combination of classifiers that predict the same set of random variables. Here the aim is to improved classifications by combining the outputs of the individual models. Boosting [6], in which many weak learners are combined into a highly accurate classifier, is one of the most common and powerful such scemes. In computer vision, this idea has been very successfully applied to the task of face detection using the so-called Cascade of Boosted Ensembles (CoBE) [18, 2] framework. While similar to our work in constructing a cascade of classifiers, their motivation was computational efficiency, rather than a consideration of contextual benefits. Tu [17] learns context cues by cascading models for pixel-level labeling. However, the context is, again, limited to interactions between labels of the same type. The other broad group of works that combine classifiers is aimed at using the classifiers as components in large intelligent systems. Kumar and Hebert [9], for example, develop a large MRF-based probabilistic model linking multiclass segmentation and object detection. Such approaches have also been used in the natural language processing literature. For example, the work of Sutton and McCallum [15] combines a parsing model with a semantic role labeling model into a unified probabilistic framework that solves both simultaneously. While technically-correct probabilistic representations are appealing, it is often painful to fit existing methods into a large, complex, highly interdependent network. By leveraging the idea of cascades, our method provides a simplified approach that requires minimal tuning of the components. The goal of holistic scene understanding dates back to the early days of computer vision, and is highlighted in the “intrinsic images” system proposed by Barrow and Tenenbaum [1], where maps of various image properties (depth, reflectance, color) are computed using information present in other maps. Over the last few decades, however, researchers have instead targeted isolated computer vision tasks, with considerable success in improving the state-of-the-art. For example, in our work, we build on the prior work in scene categorization of Li and Perona [10], object detection of Dalal and Triggs [4], multi-class image segmentation of Gould et al. [7], and 3d reconstruction of Saxena et al. [13]. Recently, however, researchers have returned to the question of how one can benefit from exploiting the dependencies between different classifiers. Torralba et al. [16] use context to significantly boost object detection performance, and Sudderth et al. [14] use object recognition for 3d structure estimation. In independent contemporary work, Hoiem et al. [8] propose an innovative system for integrating the tasks of object recognition, surface orientation estimation, and occlusion boundary detection. Like ours, their system is modular and leverages state-of-the-art components. However, their work has a strong leaning towards 3d scene 2 reconstruction rather than understanding, and their algorithms contain many steps that have been specialized for this purpose. Their training also requires intimate knowledge of the implementation of each module, while ours is more flexible allowing integration of many related vision tasks regardless of their implementation details. Furthermore, we consider a broader class of images and object types, and label regions with specific classes, rather than generic properties. 3 Cascaded Classification Models Our goal is to classify various characteristics of our data using state-of-the-art methods in a way that allows the each model to benefit from the others’ expertise. We are interested in using proven “off-the-shelf” classifiers for each subtask. As such these classifiers will be treated as “black boxes,” each with its own (specialized) data structures, feature sets, and inference and training algorithms. To fit into our framework, we only require that each classifier provides a mechanism for including additional (auxiliary) features from other modules. Many state-of-the-art models lend themselves to the easy addition of new features. In the case of “intrinsic images” [1], the output of each component is converted into an image-sized feature map (e.g., each “pixel” contains the probability that it belongs to a car). These maps can easily be fed into the other components as additional image channels. In cases where this cannot be done, it is trivial to convert the original classifier’s output to a log-odds ratio and use it along with features from their other classifiers in a simple logistic model. A standard setup has, say, two models that predict the variables YD and YS respectively for the same input instance I. For example, I might be an image, and YD could be the locations of all cars in the image, while YS could be a map indicating which pixels are road. Most algorithms begin by processing I to produce a set of features, and then learn a function that maps these features into a predicted label (and in some cases also a confidence estimate). Cascaded Classification Models (CCMs) is a joint classification model that shares information between tasks by linking component classifiers in order to leverage their relatedness. Formally: Definition 3.1: An L-tier Cascaded Classification Model (L-CCM) is a cascade of classifiers of the target labels Y = {Y1, . . . , YK}L (L “copies” of each label) consisting of independent classifiers fk,0(φk(I); θk,0) →ˆY0 k and a series of conditional classifiers fk,ℓ(φk(I, yℓ−1 −k ); θc,ℓ) →ˆYℓ k, indexed by ℓ, indicating the “tier” of the model, where y−k indicates the assignment to all labels other than yk. The labels at the final tier (L −1) represent the final classification outputs. A CCM uses L copies of each component model, stacked into tiers, as depicted in Figure 1(d). One copy of each model lies in the first tier, and learns with only the image features, φk(I), as input. Subsequent tiers of models accepts a feature vector, φk(I, yℓ−1 −k ), containing the original image features and additional features computed from the outputs of models in the preceeding tier. Given a novel test instance, classification is performed by predicting the most likely (MAP) assignment to each of the variables in the final tier. We learn our CCM in a feed-forward manner. That is, we begin from the top level, training the independent (fk,0) classifiers first, in order to maximize the classification performance on the training data. Because we assume a learning interface into each model, we simply supply the subset of data that has ground labels for that model to its learning function. For learning each component k in each subsequent level ℓof the CCM, we first perform classification using the (ℓ−1)-tier CCM that has already been trained. From these output assignments, each classifier can compute a new set of features and perform learning using the algorithm of choice for that classifier. For learning a CCM, we assume that we have a dataset of fully or partially annotated instances. It is not necessary for every instance to have groundtruth labels for every component, and our method works even when the training sets are disjoint. This is appealing since the prevalence of large, volunteer-annotated datasets (e.g., the LabelMe dataset [12] in vision or the Penn Treebank [11] in language processing), is likely to provide large amounts of heterogeneously labeled data. 4 CCM for Holistic Scene Understanding Our scene understanding model uses a CCM to combine various subsets of four computer vision tasks: scene categorization, multi-class image segmentation, object detection, and 3d reconstruction. We first introduce the notation for the target labels and then briefly describe the specifics of each component. Consider an image I. Our scene categorization classifier produces a scene label C from one of a small number of classes. Our multi-class segmentation model produces a class label Sj 3 Figure 2: (left,middle) Two exmaple features used by the “context” aware object detector. (right) Relative location maps showing the relative location of regions (columns) to objects (rows). Each map shows the prevalence of the region relative to the center of object. For example, the top row shows that cars are likely to have road beneath and sky above, while the bottom rows show that cows and sheep are often surrounded by grass. for each of a predefined set of regions j in the image. The base object detectors produce a set of scored windows (Wc,i) that potentially contain an object of type c. We attach a label Dc,i to each window, that indicates whether or not the window contains the object. Our last component module is monocular 3d reconstruction, which produces a depth Zi for every pixel i in the image. Scene Categorization Our scene categorization module is a simple multi-class logistic model that classifies the entire scene into one of a small number of classes. The base model uses a 13 dimensional feature vector φ(I) with elements based on mean and variance of RGB and YCrCb color channels over the entire image, plus a bias term. In the conditional model, we include features that indicate the relative proportions of each region label (a histogram of Sj values) in the image, plus counts of the number of objects of each type detected, producing a final feature vector of length 26. Multiclass Image Segmentation The segmentation module aims to assign a label to each pixel. We base our model on the work of Gould et al. [7] who make use of relative location—the preference for classes to be arranged in a consistent configuration with respect to one another (e.g., cars are often found above roads). Each image is pre-partitioned into a set {S1, . . . , SN} of regions (superpixels) and the pixels are labeled by assigning a class to each region Sj. The method employs a pairwise conditional Markov random field (CRF) constructed over the superpixels with node potentials based on appearance features and edge potentials encoding a preference for smoothness. In our work we wish to model the relative location between detected objects and region labels. This has the advantage of being able to encode scale, which was not possible in [7]. The right side of Figure 2 shows the relative location maps learned by our model. These maps model the spatial location of all classes given the location and scale of detected objects. Because the detection model provides probabilities for each detection, we actually use the relative location maps multiplied by the probability that each detection is a true detection. Preliminary results showed an improvement in using these soft detections over hard (thresholded) detections. Object Detectors Our detection module builds on the HOG detector of Dalal and Triggs [4]. For each class, the HOG detector is trained on a set of images disjoint from our datasets below. This detector is then applied to all images in our dataset with a low threshold that produces an overdetection. For each image I, and each object class c, we typically find 10-100 candidate detection windows Wc,i. Our independent detector model learns a logistic model over a small feature vector φc,i that can be extracted directly from the candidate window. Our conditional classifier seeks to improve the accuracy of the HOG detector by using image segmentation (denoted by Sj for each region j), 3d reconstruction of the scene, with depths (Zj) for each region, and a categorization of the scene as a whole (C), to improve the results of the HOG detector. Thus, the output from other modules and the image are combined into a feature vector φk(I, C, S, Z). A sampling of some features used are shown in Figure 2. This augmented feature vector is used in a logistic model as in the independent case. Both the independent and context aware logistics are regularized with a small ridge term to prevent overfitting. Reconstruction Module Our reconstruction module is based on the work of Saxena et al. [13]. Our Markov Random Field (MRF) approach models the 3d reconstruction (i.e., depths Z at each point in the image) as a function of the image features and also models the relations between depths at 4 CAR PEDES. BIKE BOAT SHEEP COW Mean Segment Category HOG 0.39 0.29 0.13 0.11 0.19 0.28 0.23 N/A N/A Independent 0.55 0.53 0.57 0.31 0.39 0.49 0.47 72.1% 70.6% 2-CCM 0.58 0.55 0.65 0.48 0.45 0.53 0.54 75.0% 77.3% 5-CCM 0.59 0.56 0.63 0.47 0.40 0.54 0.53 75.8% 76.8% Ground 0.49 0.53 0.62 0.35 0.40 0.51 0.48 73.6% 69.9% Ideal Input 0.63 0.64 0.56 0.65 0.45 0.56 0.58 78.4% 86.7% Table 1: Numerical evaluation of our various training regimes for the DS1 dataset. We show average precision (AP) for the six classes, as well as the mean. We also show segmentation and scene categorization accuracy. various points in the image. For example, unless there is occlusion, it is more likely that two nearby regions in the image would have similar depths. More formally, our variables are continuous, i.e., at a point i, the depth Zi ∈R. Our baseline model consists of two types of terms. The first terms model the depth at each point as a linear function of the local image features, and the second type models relationships between neighboring points, encouraging smoothness. Our conditional model includes an additional set of terms that models the depth at each point as a function of the features computed from an image segmentation S in the neighborhood of a point. By including this third term, our model benefits from the segmentation outputs in various ways. For example, a classification of grass implies a horizontal surface, and a classification of sky correlates with distant image points. While detection outputs might also help reconstruction, we found that most of the signal was present in the segmentation maps, and therefore dropped the detection features for simplicity. 5 Experiments We perform experiments on two subsets of images. The first subset DS1 contains 422 fully-labeled images of urban and rural outdoor scenes. Each image is assigned a category (urban, rural, water, other). We hand label each pixel as belonging to one of: tree, road, grass, water, sky, building and foreground. The foreground class captures detectable objects, and a void class (not used during training or evaluation) allows for the small number of regions not fitting into one of these classes (e.g., mountain) to be ignored. This is standard practice for the pixel-labeling task (e.g., see [3]). We also annotate the location of six different object categories (car, pedestrian, motorcycle, boat, sheep, and cow) by drawing a tight bounding box around each object. We use this dataset to demonstrate the combining of three vision tasks: object detection, multi-class segmentation, and scene categorization using the models described above. Our much larger second dataset DS2 was assembled by combining 362 images from the DS1 dataset (including either the segmentation or detection labels, but not both), 296 images from the Microsoft Research Segmentation dataset [3] (labeled with segments), 557 images from the PASCAL VOC 2005 and 2006 challenges [5] (labeled with objects), and 534 images with ground truth depth information. This results in 1749 images with disjoint labelings (no image contains groundtruth labels for more than one task). Combining these datasets results in 534 reconstruction images with groundtruth depths obtained by laser range-finder (split into 400 training and 134 test), 596 images with groundtruth detections (same 6 classes as above, split into 297 train and 299 test), and 615 with groundtruth segmentations (300 train and 315 test). This dataset demonstrates the typical situation in learning related tasks whereby it is difficult to obtain large fully-labeled datasets. We use this dataset to demonstrate the power of our method in leveraging the data from these three tasks to improve performance. 5.1 DS1 Dataset Experiments with the DS1 dataset were performed using 5-fold cross validation, and we report the mean performance results across folds. We compare five training/testing regimes (see Table 1). Independent learns parameters on a 0-Tier (independent) CCM, where no information is exchanged between tasks. We compare two levels of complexity for our method, a 2-CCM and a 5-CCM to test how the depth of the cascade affects performance. The last two training/testing regimes involve using groundtruth information at every stage for training and for both training and testing, respectively. Groundtruth trains a 5-CCM using groundtruth inputs for the feature construction (i.e., as if each tier received perfect inputs from above), but is evaluated with real inputs. The Ideal 5 (a) Cars (b) Pedestrians (c) Motorbikes (d) Categorization (e) Boats (f) Sheep (g) Cows (h) Segmentation Figure 3: Results for the DS1 dataset. (a-c,e-g) show precision-recall curves for the six object classes that we consider, while (d) shows our accuracy on the scene categorization task and (h) our accuracy in labeling regions in one of seven classes. Input experiment uses the Groundtruth model and also uses the groundtruth input to each tier at testing time. We could do this since, for this dataset, we had access to fully labeled groundtruth. Obviously this is not a legitimate operating mode, but does provide an interesting upper bound on what we might hope to achieve. To quantitatively evaluate our method, we consider metrics appropriate to the tasks in question. For scene categorization, we report an overall accuracy for assigning the correct scene label to an image. For segmentation, we compute a per-segment accuracy, where each segment is assigned the groundtruth label that occurs for the majority of pixels in the region. For detection, we consider a particular detection correct if the overlap score is larger than 0.2 (overlap score equals the area of intersection divided by the area of union between the detected bounding box and the groundtruth). We plot precision-recall (PR) curves for detections, and report the average precision of these curves. AP is a more stable version of the area under the PR curve. Our numerical results are shown in Table 1, and the corresponding graphs are given in Figure 3. The PR curves compare the HOG detector results to our Independent results and to our 2-CCM results. It is interesting to note that a large gain was achieved by adding the independent features to the object detectors. While the HOG score looks at only the pixels inside the target window, the other features take into account the size and location of the window, allowing our model to capture the fact that foreground object tend to occur in the middle of the image and at a relatively small range of scales. On top of this, we were able to gain an additional benefit through the use of context in the CCM framework. For the categorization task, we gained 7% using the CCM framework, and for segmentation, CCM afforded a 3% improvement in accuracy. Furthermore, for this task, running an additional three tiers, for a 5-CCM, produced an additional 1% improvement. Interestingly, the Groundtruth method performs little better than Independent for these three tasks. This shows that it is better to train the models using input features that are closer to the features it will see at test time. In this way, the downstream tiers can learn to ignore signals that the upstream tiers are bad at capturing, or even take advantage of consistent upstream bias. Also, the Ideal Input results show that CCMs have made significant progress towards the best we can hope for from these models. 5.2 DS2 Dataset For this dataset we combine the three subtasks of reconstruction, segmentation, and object detection. Furthermore, as described above, the labels for our training data are disjoint. We trained an Independent model and a 2-CCM on this data. Quantitatively, 2-CCM outperformed Independent on segmentation by 2% (75% vs. 73% accuracy), on detection by 0.02 (0.33 vs. 0.31 mean average precision), and on depth reconstruction by 1.3 meters (15.4 vs. 16.7 root mean squared error). 6 Figure 4: (top two rows) three cases where CCM improved results for all tasks. In the first, for instance, the presence of grass allows the CCM to remove the boat detections. The next four rows show four examples where detections are improved and four examples where segmentations are improved. Figure 4 shows example outputs from each component. The first three (top two rows) show images where all components improved over the independent model. In the top left our detectors removed some false boat detections which were out of context and determined that the watery appearance of the bottom of the car was actually foreground. Also by providing a sky segment, our method allowed the 3d reconstruction model to infer that those pixels must be very distant (red). The next two examples show similar improvement for detections of boats and water. The remaining examples show how separate tasks improve by using information from the others. In each example we show results from the independent model for the task in question, the independent contextual task and the 2-CCM output. The first four examples show that our method was able to make correct detections whereas the independent model could not. The last examples show improvements in multi-class image segmentation. 7 6 Discussion In this paper, we have presented the Cascaded Classification Models (CCM) method for combining a collection of state-of-the-art classifiers toward improving the results of each. We demonstrated our method on the task of holistic scene understanding by combining scene categorization, object detection, multi-class segmentation and depth reconstruction, and improving on all. Our results are consistent with other contemporary research, including the work of Hoiem et al. [8], which uses different components and a smaller number of object classes. Importantly, our framework is very general and can be applied to a number of machine learning domains. This result provides hope that we can improve by combining our complex models in a simple way. The simplicity of our method is one of its most appealing aspects. Cascades of classifiers have been used extensively within a particular task, and our results suggest that this should generalize to work between tasks. In addition, we showed that CCMs can benefit from the cascade even with disjoint training data, e.g., no images containing labels for more than one subtask. In our experiments, we passed relatively few features between the tasks. Due to the homogeneity of our data, many of the features carried the same signal (e.g., a high probability of an ocean scene is a surrogate for a large portion of the image containing water regions). For larger, more heterogeneous datasets, including more features may improve performance. In addition, larger datasets will help prevent the overfitting that we experienced when trying to include a large number of features. It is an open question how deep a CCM is appropriate in a given scenario. Overfitting is anticipated for very deep cascades. Furthermore, because of limits in the context signal, we cannot expect to get unlimited improvements. Further exploration of cases where this combination is appropriate is an important future direction. Another exciting avenue is the idea of feeding back information from the later classifiers to the earlier ones. Intuitively, a later classifier might encourage earlier ones to focus its effort on fixing certain error modes, or allow the earlier classifiers to ignore mistakes that do not hurt “downstream.” This also should allow components with little training data to optimize their results to be most beneficial to other modules, while worrying less about their own task. Acknowledgements This work was supported by the DARPA Transfer Learning program under contract number FA8750-05-2-0249 and the Multidisciplinary University Research Initiative (MURI), contract number N000140710747, managed by the Office of Naval Research. References [1] H. G. Barrow and J.M. Tenenbaum. Recovering intrinsic scene characteristics from images. CVS, 1978. [2] S.C. Brubaker, J. Wu, J. Sun, M.D. Mullin, and J.M. Rehg. On the design of cascades of boosted ensembles for face detection. In Tech report GIT-GVU-05-28, 2005. [3] A. Criminisi. Microsoft research cambridge object recognition image database (version 1.0 and 2.0)., 2004. Available Online: http://research.microsoft.com/vision/cambridge/recognition. [4] N. Dalal and B. Triggs. Histograms of oriented gradients for human detection. In CVPR, 2005. [5] M. Everingham et al. The 2005 pascal visual object classes challenge. In MLCW, 2005. [6] Y. Freund and R.E. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. In European Conference on Computational Learning Theory, pages 23–37, 1995. [7] S. Gould, J. Rodgers, D. Cohen, G. Elidan, and D. Koller. Multi-class segmentation with relative location prior. IJCV, 2008. [8] D. Hoiem, A.A. Efros, and M. Hebert. Closing the loop on scene interpretation, 2008. [9] S. Kumar and M. Hebert. A hier. field framework for unified context-based classification. In ICCV, 2005. [10] F. Li and P. Perona. A bayesian hier. model for learning natural scene categories. In CVPR, 2005. [11] M. P. Marcus, M.A. Marcinkiewicz, and B. Santorini. Building a large annotated corpus of english: the penn treebank. Comput. Linguist., 19(2), 1993. [12] B.C. Russell, A.B. Torralba, K.P. Murphy, and W.T. Freeman. Labelme: A database and web-based tool for image annotation. IJCV, 2008. [13] A. Saxena, M. Sun, and A.Y. Ng. Learning 3-d scene structure from a single still image. In PAMI, 2008. [14] E.B. Sudderth, A. Torralba, W.T. Freeman, and A.S. Willsky. Depth from familiar objects: A hierarchical model for 3d scenes. In CVPR, 2006. [15] C. Sutton and A. McCallum. Joint parsing and semantic role labeling. In CoNLL, 2005. [16] Antonio B. Torralba, Kevin P. Murphy, and William T. Freeman. Contextual models for object detection using boosted random fields. In NIPS, 2004. [17] Z. Tu. Auto-context and its application to high-level vision tasks. In CVPR, 2008. [18] P. Viola and M.J. Jones. Robust real-time object detection. IJCV, 2001. 8
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Cell Assemblies in Large Sparse Inhibitory Networks of Biologically Realistic Spiking Neurons Adam Ponzi OIST, Uruma, Okinawa, Japan. adamp@oist.jp Jeff Wickens OIST, Uruma, Okinawa, Japan. wickens@oist.jp Abstract Cell assemblies exhibiting episodes of recurrent coherent activity have been observed in several brain regions including the striatum[1] and hippocampus CA3[2]. Here we address the question of how coherent dynamically switching assemblies appear in large networks of biologically realistic spiking neurons interacting deterministically. We show by numerical simulations of large asymmetric inhibitory networks with fixed external excitatory drive that if the network has intermediate to sparse connectivity, the individual cells are in the vicinity of a bifurcation between a quiescent and firing state and the network inhibition varies slowly on the spiking timescale, then cells form assemblies whose members show strong positive correlation, while members of different assemblies show strong negative correlation. We show that cells and assemblies switch between firing and quiescent states with time durations consistent with a power-law. Our results are in good qualitative agreement with the experimental studies. The deterministic dynamical behaviour is related to winner-less competition[3], shown in small closed loop inhibitory networks with heteroclinic cycles connecting saddle-points. 1 Introduction Cell assemblies exhibiting episodes of recurrent coherent activity have been observed in several brain regions including the striatum[1] and hippocampus CA3[2], but how such correlated activity emerges in neural microcircuits is not well understood. Here we address the question of how coherent assemblies can emerge in large inhibitory neural networks and what this implies for the structure and function of one such network, the striatum. Carrillo-Reid et al.[1] performed calcium imaging of striatal neuronal populations and revealed sporadic and asynchronous activity. They found that burst firing neurons were widespread within the field of observation and that sets of neurons exhibited episodes of recurrent and synchronized bursting. Furthermore dimensionality reduction of network dynamics revealed functional states defined by cell assemblies that alternated their activity and displayed spatiotemporal pattern generation. Recurrent synchronous activity traveled from one cell assembly to the other often returning to the original assembly; suggesting a robust structure. Assemblies were visited non-randomly in sequence and not all state transitions were allowed. Moreover the authors showed that while each cell assembly comprised different cells, a small set of neurons was shared by different assemblies. Although the striatum is an inhibitory network composed of GABAergic projection neurons, similar types of cell assemblies have also been observed in excitatory networks such as the hippocampus. In a related and similar study Sasaki et al.[2] analysed spontaneous CA3 network activity in hippocampal slice cultures using principal component analysis. They found discrete heterogeneous network states defined by active cell ensembles which were stable against external perturbations through synaptic activity. Networks tended to remain in a single state for tens of seconds and then suddenly jump to a new state. Interestingly the authors tried to model the temporal profile of state transitions by a 1 hidden Markov model, but found that the transitions could not be simulated in this way. The authors suggested that state dynamics is non-random and governed by local attractor-like dynamics. We here address the important question of how such assemblies can appear deterministically in biologically realistic cell networks. We focus our modeling on the inhibitory network of the striatum, however similar models can be proposed for networks such as CA3 if the cell assembly activity is controlled by the inhibitory CA3 interneurons. Network synchronization dynamics[4, 5] of random sparse inhibitory networks of CA3 interneurons has been addressed by Wang and Buzsaki[5]. They determined specific conditions for population synchronization including that the ratio between the synaptic decay time constant and the oscillation period be sufficiently large and that a critical minimal average number of synaptic contacts per cell, which was not sensitive to the network size, was required. Here we extend this work focusing on the formation of burst firing cell assemblies The striatum is composed of GABAergic projection neurons with fairly sparse asymmetric inhibitory collaterals which seem quite randomly structured and that receive an excitatory cortical projection[6]. Each striatal medium spiny neuron (MSN) is inhibited by about 500 other MSNs in the vicinity via these inhibitory collaterals and similarly each MSN inhibits about 500 MSNs. However only about 10%−30% of MSNs are actually excited by cortex at any particular time. This implies that each MSN is actively inhibited by about 50−150 cortically excited cells in general. It is important to understand why the striatum has this particular structure, which is incompatible with its putative winner-take-all role. We show by numerical computer simulation that very general random networks of biologically realistic neurons coupled with inhibitory Rall-type synapses[7] and individually driven by excitatory input can show switching assembly dynamics. We commonly observed a switching bursting regime in networks with sparse to intermediate connectivity when the level of network inhibition approximately balanced the external excitation so that the individual cells were near a bifurcation point. In our simulations, cells and assemblies slowly and spontaneously switch between a depolarized firing state and a more hyperpolarized quiescent state. The proportion of switching cells varies with the network connectivity, peaking at low connection probability for fixed total inhibition. The sorted cross correlation matrix of the firing rates time series for switching cells shows a fascinating multiscale clustered structure of cell assemblies similar to observations in[1, 2]. The origin of the deterministic switching dynamics in our model is related to the principle of winnerless competition (WLC) which has previously been observed by Rabinovich and coworkers[3] in small inhibitory networks with closed loops based on heteroclinic cycles connecting saddle points. Rabinovich and coworkers[3] demonstrate that such networks can generate stimulus specific patterns by switching among small and dynamically changing neural ensembles with application to insect olfactory coding[8, 9], sequential decision making[10] and central pattern generation[11]. Networks produce this switching mode of dynamical activity when lateral inhibitory connections are strongly non-symmetric. WLC can represent information dynamically and is reproducible, robust against intrinsic noise and sensitive to changes in the sensory input. A closely related dynamical phenomenon is referred to as chaotic itinerancy,[12]. This is a state that switches between fully developed chaos and ordered behavior. The orbit remains in the vicinity of lower dimensional quasi-stable nearly periodic “attractor ruins” for some time before eventually exiting to a state of high dimensional chaos. This high-dimensional state is also quasi-stable, and after chaotic wandering the orbit is again attracted to one of the attractor ruins. Our study suggests attractor switching may be ubiquitous in biologically realistic large sparse random inhibitory networks. 2 Model The network is composed of biologically realistic model neurons in the vicinity of a bifurcation from a stable fixed point to spiking limit cycle dynamical behaviour. To describe the cells we use the INa,p +Ik model described in Izhikevich[13] although any model near such a bifurcation would be appropriate. The INa,p + Ik cell model is two-dimensional and described by, C dVi dt = Ii(t) −gL(Vi −EL) −gNam∞(Vi)(Vi −ENa) −gkni(Vi −Ek) (1) dni dt = (n∞−ni)/τn (2) having leak current IL, persistent Na+ current INa,p with instantaneous activation kinetic and a relatively slower persistent K+ current IK. Vi(t) is the membrane potential of the i −th cell, C 2 the membrane capacitance, EL,Na,k are the channel reversal potentials and gL,Na,k are the maximal conductances. ni(t) is K+ channel activation variable of the i −th cell. The steady state activation curves m∞and n∞are both described by, x∞(V ) = 1/(1+exp{(V x ∞−V )/kx ∞}) where x denotes m or n and V x ∞and kx ∞are fixed parameters. τn is the fixed timescale of the K+ activation variable. The term Ii(t) is the input current to the i −th cell. The parameters are chosen so that the cell is the vicinity of a saddle-node on invariant circle bifurcation. As the current Ii(t) in Eq.1 increases through the bifurcation point the stable node fixed point and the unstable saddle fixed point annihilate each other and a limit cycle having zero frequency is formed[13]. Increasing current further increases the frequency of the limit cycle. The input current Ii(t) in Eq.1 is composed of both excitatory and inhibitory parts and given by, Ii(t) = Ic i + X j −ksyn,ijgj(t)(Vi(t) −Vsyn). (3) The excitatory part is represented by Ic i and models the effect of the cortico-striatal synapses. It has a fixed magnitude for the duration of a simulation, but varying across cells. In the simulations reported here the Ic i are quenched random variables drawn uniformly randomly from the interval [Ibif, Ibif + 1] where Ibif = 4.51 is the current at the saddle-node bifurcation point. These values of excitatory input current mean that all cells would be on limit cycles and firing with low rates if the network inhibition were not present. In fact the inhibitory network may cause some cells to become quiescent by reducing the total input current to below the bifurcation point. Since the inhibitory current part is provided by the GABAergic collaterals of the striatal network it is dynamically variable. These synapses are described by Rall-type synapses[7] in Eq.3 where the current into postsynaptic neuron i is summed over all inhibitory presynaptic neurons j and Vsyn and ksyn,ij are channel parameters. gj(t) is the quantity of postsynaptically bound neurotransmitter given by, τg dgj dt = Θ(Vj(t) −Vth) −gj(t) (4) for the j −th presynaptic cell. Here Vth is a threshold, and Θ(x) is the Heaviside function. gj is essentially a low-pass filter of presynaptic firing. The timescale τg should be set relatively large so that the postsynaptic conductance follows the exponentially decaying time average of many preceding presynaptic high frequency spikes. The network structure is described by the parameters ksyn,ij = (ksyn/p)ϵijXij where ϵij is another uniform quenched random variable on [0.5, 1.5] independent in i and j. Xij = 1 if cells i and j are connected and zero otherwise. In the simulations reported here we use random networks where cells i and j are connected with probability p, and there are no self-connections, Xii = 0. ksyn is a parameter which is rescaled by the connection probability p so that average total inhibition on each cell is constant independent of p. All simulations were carried out with fourth order Runge-Kutta. 3 Results Figure 1(a) shows a time series segment of membrane potentials Vi(t) for some randomly selected cells from an N = 100 cell network. The switching between firing and quiescent states can clearly be seen. Cells fire with different frequencies and become quiescent for variable periods before starting to fire again apparently randomly. However the model has no stochastic variables and therefore this switching is caused by deterministic chaos. As explained above the firing rate is determined by the proximity of the limit cycle to the saddle-node bifurcation and can therefore be arbitrarily low for this type of bifurcation. Since we have set the unit parameters so that all units are near the bifurcation point even weak network inhibition is able to cause the cells to become quiescent at times. The parameter settings are biologically realistic[13] and MSN cells are known to show irregular quiescent and firing states in vivo[14]. The complex bursting structure is easier to see from raster plots. A segment from a N = 100 cell time series is shown in Fig.1(b). This figure clearly shows attractor switching, or chaotic itinerancy[12], where a quasi-stable nearly-periodic state (an “attractor ruin”) is visited from higher dimensional chaos. To make this plot the cells have been ordered by the k-means algorithm with five clusters (see below). The cells are coloured according to the cluster assigned to them by the algorithm. During the periodic window, most cells are silent however some cells fire continuously 3 Figure 1: (a) Membrane potential Vi(t) time series segment for a few cells from a N = 100 cell network simulation with 20 connections per cell. Each cell time series is a different colour. (b) Spike raster plot from an N = 100 cell network simulation with 20 connections per cell. Each line is a different cell and the 71 cells which fire at least one spike during the period shown are plotted. Cells are ordered by k-means with five clusters and coloured according to their assigned clusters. at fixed frequency and some cells fire in periodic bursts. In fact the cells which fire in bursts have been separated into two clusters, as can be seen in Fig.1(b), the blue and green clusters. These two clusters fire periodic bursts in anti-phase. Cell assemblies can also be seen in the chaotic regions. The cells in the black cluster fire together in a burst around t = 17500 while the cells in the orange cluster fire a burst together around t = 16000. Fig.2(a) shows another example of a spike raster plot from a N = 100 cell network simulation where again the cells have been ordered by the k-means algorithm with five clusters. Now cell assemblies, blue, orange and red coloured, can clearly be seen which appear to switch in alternation. This switching is further interrupted from time to time by the green and black assemblies. Due to the presence of attractor switching where cell assemblies can burst in antiphase we can expect the appearance of strongly positively and strongly negatively correlated cell pairs. Correlation matrices are constructed by dragging a moving window over a long spike time series and counting the spikes to construct the associated firing rate time series. The correlation matrix of the rate time series is then sorted by the k-means method[2], which is equivalent to PCA. Each cell is assigned to one of a fixed number of clusters and the cells indices are reordered accordingly. Fig.2(b) shows the cross-correlation matrix corresponding to the spike raster plot in Fig.2(a) with cells ordered the same way. Within an assembly cells are positively correlated, while cells in different assemblies often show negative correlation. Larger networks with appropriate connectivites also show complex identity-temporal patterns. A patch-work of switching cell assembly clusters can be seen in the spike raster plot and corresponding cross-correlation matrix shown in Figs.2(c) and (d) respectively for a N = 500 cell system where the cells have been ordered by the k-means algorithm, now with 30 clusters. Any particular assembly can seem to be burst firing periodically for a spell before becoming quiescent for long spells. Other cell assemblies burst very occasionally for no apparent reason. Notice from the cross-correlation matrices in Fig.2(b) and (d) that although some cell assemblies are positively correlated with each other, they have different relationships to other cell assemblies, and therefore cannot be combined into a single larger assemblies. Fig.2(c) reveals many cells switching between a firing state and quiescent state. What is the structure of this switching state? To investigate this we analyse inter spike interval (ISI) distributions. Shown in Fig.1(b) are three ISI distributions for three 500 cell network simulations in the sparse to intermediate regime with 30 connections per cell. The distributions are very broad and far from the exponential distribution one would expect from a Poisson process. They are consistent with a scalefree power law behaviour for three orders of magnitude, but exponentially cut off at large ISIs due to finite size effects. It is this distribution which produces the appearance of the complex identitytemporal patterns shown in the 500 cell time series figure in Fig.2(c) with the long ISIs interspersed with the bursts of short ISIs. Power-law distributions are characteristic of systems showing chaotic 4 Figure 2: (a) Spike raster plot from all 69 cells in a 100 cell network with 20 connections per cell which fire at least one spike. The cells are ordered by k-means with five clusters and coloured according to their assigned cluster. (b) Cross-correlation matrix corresponding to (a). The cells are ordered by the k-means algorithm the same way as (a). Red colour means positive correlation, blue means negative correlation, colour intensity matches strength. White is weak or no correlation. (c) Spike raster plot from an N = 500 cell sparse network with 6 connections per cell. The 379 cells which fire at least one spike during the period shown are plotted. The cells are ordered by k-means with 30 clusters. (d) Cross-correlation matrix corresponding to (c) with same conventions as (b). attractor switching and have been studied in connection with deterministic intermittency[15]. Intermittency consists of laminar phases where the system orbits appear to be relatively regular, and bursts phases where the motion is quite violent and irregular. Interestingly a power-law distribution of state sojourn times was also observed in the hippocampal study of by Sasaki et al.[2] described above. Plenz and Thiagarajan[16] discuss cortical cell assemblies in the framework of scale free avalanches which are associated with intermittency[17]. The broad power-law distribution produces the temporal aspect of the complex identity-temporal patterns observed in the time series in Fig.2(c), however the fact that the cells show strong crosscorrelation produces the spatial structure aspect. In the above we have shown how this structure can be revealed using the k-means sorting algorithm. By combining the spikes of cells in a cluster into a “cluster spike train” preserving each spikes’ timing we can study the ISIs of cluster spike time series. However the k-means algorithm produces a different clustering depending on the initial choice of centroids. To control for this we perform the clustering many, here 200, times and combine the ISI time series so generated into a single distribution. The black circles in Fig.3(a) show the cluster ISI distribution after cells have been associated to clusters with the k-means algorithm with 10 clusters. The cluster ISI distribution, like the individual cell ISI distribution, also shows a powerlaw over several orders of magnitude. This implies clusters also burst in a multiple scale way. The slope of the power law is greater than the individual cell result and the cut-off is lower as would be expected when spike trains are combined. Nevertheless the distribution is still very broad. To demonstrate this we perform a bootstrap type test where rather than making each cluster spike train 5 Figure 3: (a) Green, brown, blue: Three cell cumulative ISI distributions from 500 cell network simulations with 30 connections per cell, all cells combined. Log-log scale. The slope of the dashed line is −1.38. Black: ISI distribution for clusters formed by k-means algorithm corresponding to green single cell distribution. The slope of the solid line is −2.35. Red: ISI distribution for clusters formed from cells randomly corresponding to green single cell distribution. (b) Variation of connectivity for 500 cell networks. Inset shows low connectivity detail. Each point calculated from a different network simulation for observation period t = 2000 to t = 12000 msec. Red: Proportion of cells which fire at least one spike during the period. Blue: Proportion of cells firing periodically. Black: Average absolute cross-correlation ⟨|Cij|⟩between all cells in network calculated from rate time series constructed from counting spikes in moving window of size 2000 msec. Green: Coefficient of variation ⟨CV ⟩of ISI distribution averaged across all cells in network rescaled by 1/3. from the cells associated to the cluster we perform the same k-means clustering to obtain correct cluster sizes but then scramble the cell indices, associating the cells to the clusters randomly. Again we do this 200 times and combine all the results into one cluster ISI distribution. The red circles in Fig.3(a) show this random cluster ISI distribution. The distribution is much narrower than the distribution obtained from the non-randomized k-means clustering. This demonstrates further that the time series have a clustered structure which can be revealed by the k-means algorithm and that the clusters produced have a larger periods of quiescence between bursting than would be expected from randomly associating cells, even when the cells themselves have power-law distributed ISIs. This broadened distribution produced by the clustering reflects the complex identity-temporal structure of the ordered spike time series figures such as shown in Fig.2(c). The model has several parameters, in particular the connection probability p. How does the formation of switching assembly dynamics depend on the network connectivity? To study this we perform many numerical simulations while varying p. As described above the synaptic efficacy is rescaled by the connection probability so the total inhibition on each cell is fixed and therefore effects arise purely from variations in connectivity. Fig.3(b) (red) shows the proportion of cells which fire at least one spike versus average connections per cell for 500 cell network simulations. This quantity shows a transition around 5 connections per cell to state where almost all the network is burst firing and then decays off to a plateau region at higher connectivity. Fig.3(b) (blue) shows the proportion of cells firing periodically. This is zero above the transition. Below the transition a large proportion of cells are not inhibited and firing periodically due to the excitatory cortical drive, while another large proportion are not firing at all, inhibited by the periodically firing group. At high connectivities however most cells receive similar inhibition levels which leaves a certain proportion firing. Fig.3(b) (green) shows the coefficient of variation CV of the single cell ISI distribution averaged across all cells and rescaled by 1/3. CV is defined to be the ISI standard deviation normalized by the mean ISI. It is unity for Poisson processes. Below the transition CV is very low due to many periodic firing cells. At high connectivities it is also low and inspection of spike time series shows all cells firing with fairly regular ISIs. In intermediate regions however this quantity can become very large reflecting long periods of quiescent interrupted by high frequency bursting, as also reflected in the single cell ISI distributions in Fig.3(a). Fig.3(b) (black) shows the average absolute cross-correlation ⟨|Cij|⟩where Cij is the cross-correlation co6 efficient between cells i and j firing rate time series’ and its absolute value is averaged across all cells. This quantity also shows the low connectivity transition but peaks around 200 connections per cell, where many cells are substantially cross-correlated (both positively and negatively). This is in accordance with the study of Wang and Buzsaki[5]. Fig.3(b) therefore displays an interesting regime between about 50 and 200 connections per cell where many cells are burst firing with long periods of quiescence but have substantial cross-correlation. It is in this regime that spike time series often show the complex identity-temporal patterns and switching cell assemblies exemplified in Fig.2(c). 4 Discussion We have shown that inhibitory networks of biologically realistic spiking neurons obeying deterministic dynamical equations with sparse to intermediate connectivity can show bursting dynamics, complex identity-temporal patterns and form cell assemblies. The cells should be near a bifurcation point where even weak inhibition can cause them to become quiescent. The synapses should have a slower timescale, τg > 10 in Eq.4, which produces a low pass filter of presynaptic spiking. This slow change in inhibition allows the bursting assembly dynamics since presynaptic cells do not instantly inhibit postsynaptic cells, but inhibition builds up gradually, allowing the formation of assemblies which eventually becoming strong enough to quench the postsynaptic cell activity. At low connectivities sets of cells with sufficiently few and/or sufficiently weak connections between them will exist and these cells will fire together as an assembly due to the cortical excitation, if the rest of the network which inhibits them is sufficiently quiescent for a period. Such a set of weakly connected cells can be inhibited by another such set of weakly connected cells if each member of the first set is inhibited by a sufficient number of cells of the second set. When the second set ceases firing the first set will start to fire. These assemblies can exist in asymmetric closed loops which slowly switch active set. Multiple “frustrated” interlocking loops can exist where the slow switching of one loop will interfere with the dynamical switching of another loop; only when inhibition on one member set is removed will the loop be able to continue slow switching, producing a type of neural computation. Furthermore any given cell can be a member of several such sets of weakly connected cells, as also described by Assisi and Bazhenov[18]. This can explain the findings of Carrillo-Reid et.al.[1] who show some cells firing with only one assembly and other cells firing in multiple assemblies. These cross-coupled switching assemblies with partially shared members produce complex multiple timescale dynamics and identity-temporal patterning for appropriate connectivities. Switching assemblies are most likely to be observed in networks of sparse to intermediate connectivities. This is consistent with WLC based attractor switching. Indeed networks with non-symmetric inhibitory connections which form closed circuits display WLC dynamics[3] and these will be likely to occur in networks with sparse to intermediate connectivities. The spike time series in Figs.2(a) and (c), indicate that cell assemblies switch non-randomly in sequence due to the deterministic attractor switching. This is in good agreement with Carrillo-Reid et al.[1] study of striatal dynamics and also with the Sasaki et al.[2] study of CA3 cell assemblies. Our time series and the crosscorrelation matrices demonstrate that while most cells fire with only one particular assembly, some cells are shared between assemblies, as observed by Carrillo-Reid et al.[1]. We have shown that cells form assemblies of positively correlated cells and assemblies are negatively correlated with each other, in accordance with the similarity matrix results shown in Sasaki et al.[2]. Very interestingly cell assemblies are predominantly found in a connectivity regime appropriate for the striatum[6], where each cell is likely to be connected to about 100 cortically excited cells, suggesting the striatum may have adapted to be in this regime. Studies of spontaneous firing in the striatum also show very variable firing patterns with long periods of quiescence[14], as shown in our simulations at this connectivity. Based on studies of random striatal connectivity[6] we have simulated a random network without real spatial dimension. In support of this assumption Carrillo-Reid et al.[1] find that correlated activity is spatially distributed, noting that neurons firing synchronously could be hundreds of microns apart intermingled with silent cells. Although we leave this point for future work the dynamics can also be affected by the details of the spiking. Detailed inspection of the spike raster plot in Fig.1(b) confirms three cells firing with identical frequency. Since these cells are driven by different levels of cortical excitation, the synchronization can only result from an entrainment produced by the spiking. This is possible in cells with close firing rates since the effect an inhibitory spike has on a post-synaptic cell depends on 7 the post-synaptic membrane potential[13, 19]. In this way the spiking can affect cluster formation dynamics and may prolong the lifetime of visits to quasi-stable periodic states. The coupling of assembly dynamics and spiking may be relevant for coding in the insect olfactory lobe for example[9]. The striatum is the main input structure to the basal ganglia (BG). Correlated activity in cortico-basal ganglia circuits is important in the encoding of movement, associative learning, sequence learning and procedural memory. Aldridge and Berridge[21] demonstrate that the striatum implements action syntax in rats grooming behaviour. BG may contain central pattern generators (CPGs) that activate innate behavioral routines, procedural memories, and learned motor programs[20] and recurrent alternating bursting is characteristic of cell assemblies included in CPGs[20]. WLC has been applied to modeling CPGs[11]. Our modeling suggests that complex switching dynamics based in the sparse striatal inhibitory network may allow the generation of cell assemblies which interface sensory driven cortical patterns to dynamical sequence generation. Further work is underway to demonstrate how these dynamics may be utilized in behavioural tasks recruiting the striatum. References [1] Carrillo-Reid L, Tecuapetla F, Tapia D, Hernandez-Cruz A, Galarraga E, Drucker-Colin R, Bargas J. J.Neurophys. 99, 1435 (2008). [2] Sasaki T, Matsuki N, Ikegaya Y. J.Neurosci. 27(3), 517-528 (2007). Sasaki T, Kimura R, Tsukamoto M, Matsuki N, Ikegaya Y. J.Physiol. 574.1, 195-208 (2006). [3] Rabinovich MI, Huerta R, Volkovskii A, Abarbanel HDI, Stopfer M, Laurent G. J.Physiol. 94, 465 (2000). Rabinovich M, Volkovskii A, Lecanda P, Huerta R, Abarbanel HDI, Laurent G PRL 87,06:U149-U151 (2001). Rabinovich MI, Huerta R, Varona P, Afraimovich V. Biol. Cybern. 95:519-536 (2006). Nowotny T, Rabinovich MI. PRL 98,128106 (2007). [4] Golomb D and Rinzel J. PRE 48, 4810-4814 (1993). Golomb D and Hansel D. Neur.Comp. 12, 10951139 (2000). Tiesinga PHE and Jose VJ. J.Comp.Neuro. 9(1):49-65 (2000). [5] Wang X-J and Buzsaki G. J.Neurosci. 16(20):6402-6413 (1996). [6] Wickens JR, Arbuthnott G, Shindou T. Prog. Brain Res. 160, 316 (2007). [7] Rall WJ. Neurophys. 30, 1138-1168 (1967). [8] Laurent G. Science 286, 723-728 (1999). Laurent G and Davidowitz H. Science 265, 1872-1875 (1994). Wehr M and Laurent G. Nature 384, 162-166 (1996). Laurent G, Stopfer M, Friedrich RW, Rabinovich MI, Abarbanel HDI Annu. Rev. Neurosci. 24:263-297 (2001). [9] Bazhenov M, Stopfer M, Rabinovich MI, Huerta R, Abarbanel HDI, Sejnowski TJ, Laurent G. Neuron, 30, 553-567 (2001). Nowotny T, Huerta R, Abarbanel HDI, Rabinovich MI. Biol. Cybern 93, 436-446 (2005). Huerta R, Nowotny T, Garcia-Sanchez M, Abarbanel HDI. Neur.Comp. 16, 1601-1640 (2004). [10] Rabinovich MI, Huerta R,Afraimovich VS. PRL 97, 188103 (2006). Rabinovich MI, Huerta R, Varona P, Afraimovich VS. PLoS Comput Biol 4(5): e1000072. doi:10.1371 (2008). [11] Selverston A, Rabinovich M, Abarbanel H, Elson R, Szncs A, Pinto R, Huerta R, Varona P. J.Physiol. 94:357-374 (2000). Varona P, Rabinovich MI, Selverston AI, Arshavsky YI. Chaos 12(3) 672, (2003). [12] Kaneko K, Tsuda I. Chaos 13(3), 926-936 (2003). K. Kaneko. Physica D 41, 137 (1990). I. Tsuda. Neural Networks 5, 313 (1992). Tsuda I, Fujii H, Tadokoro S, Yasuoka T, Yamaguti Y. J.Int.Neurosci, 3(2), 159182 (2004). Fujii H and Tsuda I. Neurocomp. 58:151-157 (2004). [13] Izhikevich E.M. Dynamical Systems in Neuroscience: The Geometry of .... MIT press (2005). [14] Wilson CJ and Groves PM. Brain Research 220:67-80 (1981). [15] Gluckenheimer J, Holmes P. Non-linear Oscillations,Dynamical Systems and Bifurcations of Vector Fields, Springer, Berlin (1983). Ott E. Chaos in Dynamical Systems. Cambridge, U.K.: Cambridge Univ. Press (2002). Pomeau Y and Manneville P. Commun. Math. Phys., 74, 189-197 (1980). Pikovsky A, J. Phys. A, 16, L109-L112, (1984). [16] Plenz D, Thiagarajan TC. Trends Neurosci 30: 101-110, (2007). [17] Bak P, Tang C, Wiesenfeld K. PRA 38, 364-374 (1988). Bak P, Sneppen K. PRL 71, 4083-4086 (1993). [18] Assisi CG and Bazhenov MV. SfN 2007 abstract. [19] Ermentrout B, Rep. Prog. Phys. 61, 353-430 (1998). van-Vreeswijk C, Abbott LF, Ermentrout B. J. Comp. Neuro., 1, 313-321 (1994). [20] Grillner S, Hellgren J, Menard A, Saitoh K, Wikstrom MA. Trends Neurosci. 28: 364-370, (2005). Takakusaki K, Oohinata-Sugimoto J, Saitoh K, Habaguchi T. Prog Brain Res. 143: 231-237, (2004). [21] Aldridge JW and Berridge KC. J. Neurosci., 18(7):2777-2787 (1998). 8
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High-dimensional support union recovery in multivariate regression Guillaume Obozinski Department of Statistics UC Berkeley gobo@stat.berkeley.edu Martin J. Wainwright Department of Statistics Dept. of Electrical Engineering and Computer Science UC Berkeley wainwright@stat.berkeley.edu Michael I. Jordan Department of Statistics Department of Electrical Engineering and Computer Science UC Berkeley jordan@stat.berkeley.edu Abstract We study the behavior of block ℓ1/ℓ2 regularization for multivariate regression, where a K-dimensional response vector is regressed upon a fixed set of p covariates. The problem of support union recovery is to recover the subset of covariates that are active in at least one of the regression problems. Studying this problem under high-dimensional scaling (where the problem parameters as well as sample size n tend to infinity simultaneously), our main result is to show that exact recovery is possible once the order parameter given by θℓ1/ℓ2(n, p, s) : = n/[2ψ(B∗) log(p −s)] exceeds a critical threshold. Here n is the sample size, p is the ambient dimension of the regression model, s is the size of the union of supports, and ψ(B∗) is a sparsity-overlap function that measures a combination of the sparsities and overlaps of the K-regression coefficient vectors that constitute the model. This sparsity-overlap function reveals that block ℓ1/ℓ2 regularization for multivariate regression never harms performance relative to a naive ℓ1-approach, and can yield substantial improvements in sample complexity (up to a factor of K) when the regression vectors are suitably orthogonal relative to the design. We complement our theoretical results with simulations that demonstrate the sharpness of the result, even for relatively small problems. 1 Introduction A recent line of research in machine learning has focused on regularization based on block-structured norms. Such structured norms are well motivated in various settings, among them kernel learning [3, 8], grouped variable selection [12], hierarchical model selection [13], simultaneous sparse approximation [10], and simultaneous feature selection in multi-task learning [7]. Block-norms that compose an ℓ1-norm with other norms yield solutions that tend to be sparse like the Lasso, but the structured norm also enforces blockwise sparsity, in the sense that parameters within blocks are more likely to be zero (or non-zero) simultaneously. The focus of this paper is the model selection consistency of block-structured regularization in the setting of multivariate regression. Our goal is to perform model or variable selection, by which we mean extracting the subset of relevant covariates that are active in at least one regression. We refer to this problem as the support union problem. In line with a large body of recent work in statistical machine learning (e.g., [2, 9, 14, 11]), our analysis is high-dimensional in nature, meaning that we allow the model dimension p (as well as other structural parameters) to grow along with the sample size n. A great deal of work has focused on the case of ordinary ℓ1-regularization (Lasso) [2, 11, 14], showing for instance that the Lasso can recover the support of a sparse signal even when p ≫n. 1 Some more recent work has studied consistency issues for block-regularization schemes, including classical analysis (p fixed) of the group Lasso [1], and high-dimensional analysis of the predictive risk of block-regularized logistic regression [5]. Although there have been various empirical demonstrations of the benefits of block regularization, the generalizations of the result of [11] obtained by [6, 4] fail to capture the improvements observed in practice. In this paper, our goal is to understand the following question: under what conditions does block regularization lead to a quantifiable improvement in statistical efficiency, relative to more naive regularization schemes? Here statistical efficiency is assessed in terms of the sample complexity, meaning the minimal sample size n required to recover the support union; we wish to know how this scales as a function of problem parameters. Our main contribution is to provide a function quantifying the benefits of block regularization schemes for the problem of multivariate linear regression, showing in particular that, under suitable structural conditions on the data, the block-norm regularization we consider never harms performance relative to naive ℓ1-regularization and can lead to substantial gains in sample complexity. More specifically, we consider the following problem of multivariate linear regression: a group of K scalar outputs are regressed on the same design matrix X ∈Rn×p. Representing the regression coefficients as a p × K matrix B∗, the regression model takes the form Y = XB∗+ W, (1) where Y ∈Rn×K and W ∈Rn×K are matrices of observations and zero-mean noise respectively and B∗has columns β∗(1), . . . , β∗(K) which are the parameter vectors of each univariate regression. We are interested in recovering the union of the supports of individual regressions, more specifically if Sk = n i ∈{1, . . . , p}, β∗(k) i ̸= 0 o we would like to recover S = ∪kSk. The Lasso is often presented as a relaxation of the so-called ℓ0 regularization, i.e., the count of the number of non-zero parameter coefficients, an intractable non-convex function. More generally, block-norm regularizations can be thought of as the relaxation of a non-convex regularization which counts the number of covariates i for which at least one of the univariate regression parameters β∗(k) i is non-zero. More specifically, let β∗ i denote the ith row of B∗, and define, for q ≥1, ∥B∗∥ℓ0/ℓq = |{i ∈{1, . . . , p}, ∥β∗ i ∥q > 0}| and ∥B∗∥ℓ1/ℓq = p X i=1 ∥β∗ i ∥q All ℓ0/ℓq norms define the same function, but differ conceptually in that they lead to different ℓ1/ℓq relaxations. In particular the ℓ1/ℓ1 regularization is the same as the usual Lasso. The other conceptually most natural block-norms are ℓ1/ℓ2 and ℓ1/ℓ∞. While ℓ1/ℓ∞is of interest, it seems intuitively to be relevant essentially to situations where the support is exactly the same for all regressions, an assumption that we are not willing to make. In the current paper, we focus on the ℓ1/ℓ2 case and consider the estimator bB obtained by solving the following disguised second-order cone program: min B∈Rp×K ½ 1 2n |||Y −XB|||2 F + λn ∥B∥ℓ1/ℓ2 ¾ , (2) where |||M|||F : = (P i,j m2 ij)1/2 denotes the Frobenius norm. We study the support union problem under high-dimensional scaling, meaning that the number of observations n, the ambient dimension p and the size of the union of supports s can all tend to infinity. The main contribution of this paper is to show that under certain technical conditions on the design and noise matrices, the model selection performance of block-regularized ℓ1/ℓ2 regression (2) is governed by the control parameter θℓ1/ℓ2(n, p ; B∗) : = n 2 ψ(B∗,ΣSS) log(p−s), where n is the sample size, p is the ambient dimension, s = |S| is the size of the union of the supports, and ψ(·) is a sparsity-overlap function defined below. More precisely, the probability of correct support union recovery converges to one for all sequences (n, p, s, B∗) such that the control parameter θℓ1/ℓ2(n, p ; B∗) exceeds a fixed critical threshold θcrit < +∞. Note that θℓ1/ℓ2 is a measure of the sample complexity of the problem—that is, the sample size required for exact recovery as a function of the problem parameters. Whereas the ratio (n/ log p) is standard for high-dimensional theory on ℓ1-regularization (essentially due to covering numberings of ℓ1 balls), the function ψ(B∗, ΣSS) is a novel and interesting quantity, which 2 measures both the sparsity of the matrix B∗, as well as the overlap between the different regression tasks (columns of B∗). In Section 2, we introduce the models and assumptions, define key characteristics of the problem and state our main result and its consequences. Section 3 is devoted to the proof of this main result, with most technical results deferred to the appendix. Section 4 illustrates with simulations the sharpness of our analysis and how quickly the asymptotic regime arises. 1.1 Notations For a (possibly random) matrix M ∈Rp×K, and for parameters 1 ≤a ≤b ≤∞, we distinguish the ℓa/ℓb block norms from the (a, b)-operator norms, defined respectively as ∥M∥ℓa/ℓb : = ½ p X i=1 µ K X k=1 |mik|b ¶ a b ¾ 1 a and |||M|||a, b : = sup ∥x∥b=1 ∥Mx∥a, (3) although ℓ∞/ℓp norms belong to both families (see Lemma B.0.1). For brevity, we denote the spectral norm |||M|||2, 2 as |||M|||2, and the ℓ∞-operator norm |||M|||∞, ∞= maxi P j |Mij| as |||M|||∞. 2 Main result and some consequences The analysis of this paper applies to multivariate linear regression problems of the form (1), in which the noise matrix W ∈Rn×K is assumed to consist of i.i.d. elements Wij ∼N(0, σ2). In addition, we assume that the measurement or design matrices X have rows drawn in an i.i.d. manner from a zero-mean Gaussian N(0, Σ), where Σ ≻0 is a p × p covariance matrix. Suppose that we partition the full set of covariates into the support set S and its complement Sc, with |S| = s, |Sc| = p −s. Consider the following block decompositions of the regression coefficient matrix, the design matrix and its covariance matrix: B∗= · B∗ S B∗ Sc ¸ , X = [XS XSc] , and Σ = · ΣSS ΣSSc ΣScS ΣScSc ¸ . We use β∗ i to denote the ith row of B∗, and assume that the sparsity of B∗is assessed as follows: (A0) Sparsity: The matrix B∗has row support S : = {i ∈{1, . . . , p} | β∗ i ̸= 0}, with s = |S|. In addition, we make the following assumptions about the covariance Σ of the design matrix: (A1) Bounded eigenspectrum: There exist a constant Cmin > 0 (resp. Cmax < +∞) such that all eigenvalues of ΣSS (resp. Σ) are greater than Cmin (resp. smaller than Cmax). (A2) Mutual incoherence: There exists γ ∈(0, 1] such that ¯¯¯¯¯¯ΣScS(ΣSS)−1¯¯¯¯¯¯ ∞≤1 −γ. (A3) Self incoherence: There exists a constant Dmax such that ¯¯¯¯¯¯(ΣSS)−1¯¯¯¯¯¯ ∞≤Dmax. Assumption A1 is a standard condition required to prevent excess dependence among elements of the design matrix associated with the support S. The mutual incoherence assumption A2 is also well known from previous work on model selection with the Lasso [10, 14]. These assumptions are trivially satisfied by the standard Gaussian ensemble (Σ = Ip) with Cmin = Cmax = Dmax = γ = 1. More generally, it can be shown that various matrix classes satisfy these conditions [14, 11]. 2.1 Statement of main result With the goal of estimating the union of supports S, our main result is a set of sufficient conditions using the following procedure. Solve the block-regularized problem (2) with regularization parameter λn > 0, thereby obtaining a solution bB = bB(λn). Use this solution to compute an estimate of the support union as bS( bB) : = n i ∈{1, . . . , p} | bβi ̸= 0 o . This estimator is unambiguously defined if the solution bB is unique, and as part of our analysis, we show that the solution bB is indeed unique with high probability in the regime of interest. We study the behavior of this estimator for a 3 sequence of linear regressions indexed by the triplet (n, p, s), for which the data follows the general model presented in the previous section with defining parameters B∗(n) and Σ(n) satisfying A0A3. As (n, p, s) tends to infinity, we give conditions on the triplet and properties of B∗for which bB is unique, and such that P[bS = S] →1. The central objects in our main result are the sparsity-overlap function, and the sample complexity parameter, which we define here. For any vector βi ̸= 0, define ζ(βi) : = βi ∥βi∥2 . We extend the function ζ to any matrix BS ∈Rs×K with non-zero rows by defining the matrix ζ(BS) ∈Rs×K with ith row [ζ(BS)]i = ζ(βi). With this notation, we define the sparsity-overlap function ψ(B) and the sample complexity parameter θℓ1/ℓ2(n, p ; B∗) as ψ(B) : = ¯¯¯¯¯¯ ζ(BS)T (ΣSS)−1ζ(BS) ¯¯¯¯¯¯ 2 and θℓ1/ℓ2(n, p ; B∗): = n 2 ψ(B∗) log(p−s). (4) Finally, we use b∗ min : = mini∈S ∥β∗ i ∥2 to denote the minimal ℓ2 row-norm of the matrix B∗ S. With this notation, we have the following result: Theorem 1. Consider a random design matrix X drawn with i.i.d. N(0, Σ) row vectors, an observation matrix Y specified by model (1), and a regression matrix B∗such that (b∗ min)2 decays strictly more slowly than f(p) n max {s, log(p −s)}, for any function f(p) →+∞. Suppose that we solve the block-regularized program (2) with regularization parameter λn = Θ ³p f(p) log(p)/n ´ . For any sequence (n, p, B∗) such that the ℓ1/ℓ2 control parameter θℓ1/ℓ2(n, p ; B∗) exceeds the critical threshold θcrit(Σ) : = Cmax γ2 , then with probability greater than 1 −exp(−Θ(log p)), (a) the block-regularized program (2) has a unique solution bB, and (b) its support set bS( bB) is equal to the true support union S. Remarks: (i) For the standard Gaussian ensemble (Σ = Ip), the critical threshold is simply θcrit(Σ) = 1. (ii) A technical condition that we require on the regularization parameter is λ2 nn log(p −s) →∞ (5) which is satisfied by the choice given in the statement. 2.2 Some consequences of Theorem 1 It is interesting to consider some special cases of our main result. The simplest special case is the univariate regression problem (K = 1), in which case the function ζ(β∗) outputs an s-dimensional sign vector with elements z∗ i = sign(β∗ i ), so that ψ(β∗) = z∗T (ΣSS)−1z∗= Θ(s). Consequently, the order parameter of block ℓ1/ℓ2-regression for univariate regresion is given by Θ(n/(2s log(p − s)), which matches the scaling established in previous work on the Lasso [11]. More generally, given our assumption (A1) on ΣSS, the sparsity overlap ψ(B∗) always lies in the interval [ s KCmax , s Cmin ]. At the most pessimistic extreme, suppose that B∗: = β∗⃗1 T K—that is, B∗ consists of K copies of the same coefficient vector β∗∈Rp, with support of cardinality |S| = s. We then have [ζ(B∗)]ij = sign(β∗ i )/ √ K, from which we see that ψ(B∗) = z∗T (ΣSS)−1z∗, with z∗again the s-dimensional sign vector with elements z∗ i = sign(β∗ i ), so that there is no benefit in sample complexity relative to the naive strategy of solving separate Lasso problems and constructing the union of individually estimated supports. This might seem a pessimistic result, since under model (1), we essentially have Kn observations of the coefficient vector β∗with the same design matrix but K independent noise realizations. However, the thresholds as well as the rates of convergence in high-dimensional results such as Theorem 1 are not determined by the noise variance, but rather by the number of interfering variables (p −s). At the most optimistic extreme, consider the case where ΣSS = Is and (for s > K) suppose that B∗is constructed such that the columns of the s × K matrix ζ(B∗) are all orthogonal and of equal length. Under this condition, we have 4 Corollary 1 (Orthonormal tasks). If the columns of the matrix ζ(B∗) are all orthogonal with equal length and ΣSS = Is×s then the block-regularized problem (2) succeeds in union support recovery once the sample complexity parameter n/(2 s K log(p −s)) is larger than 1. For the standard Gaussian ensemble, it is known [11] that the Lasso fails with probability one for all sequences such that n < (2 −ν)s log(p −s) for any arbitrarily small ν > 0. Consequently, Corollary 1 shows that under suitable conditions on the regression coefficient matrix B∗, ℓ1/ℓ2 can provides a K-fold reduction in the number of samples required for exact support recovery. As a third illustration, consider, for ΣSS = Is×s, the case where the supports Sk of individual regression problems are all disjoint. The sample complexity parameter for each of the individual Lassos is n/(2sk log(p −sk)) where |Sk| = sk, so that the sample size required to recover the support union from individual Lassos scales as n = Θ(maxk[sk log(p −sk)]). However, if the supports are all disjoint, then the columns of the matrix Z∗ S = ζ(B∗ S) are orthogonal, and Z∗ S TZ∗ S = diag(s1, . . . , sK) so that ψ(B∗) = maxk sk and the sample complexity is the same. In other words, even though there is no sharing of variables at all there is surprisingly no penalty from regularizing jointly with the ℓ1/ℓ2-norm. However, this is not always true if ΣSS ̸= Is×s and in many situations ℓ1/ℓ2-regularization can have higher sample complexity than separate Lassos. 3 Proof of Theorem 1 In addition to previous notations, the proofs use the shorthands: bΣSS= 1 nXT S XS, bΣScS= 1 nXT ScXS and ΠS = XS(bΣSS)−1XT S denotes the orthogonal projection onto the range of XS. High-level proof outline: At a high level, our proof is based on the notion of what we refer to as a primal-dual witness: we first formulate the problem (2) as a second-order cone program (SOCP), with the same primal variable B as in (2) and a dual variable Z whose rows coincide at optimality with the subgradient of the ℓ1/ℓ2 norm. We then construct a primal matrix bB along with a dual matrix bZ such that, under the conditions of Theorem 1, with probability converging to 1: (a) The pair ( bB, bZ) satisfies the Karush-Kuhn-Tucker (KKT) conditions of the SOCP. (b) In spite of the fact that for general high-dimensional problems (with p ≫n), the SOCP need not have a unique solution a priori, a strict feasibility condition satisfied by the dual variables bZ guarantees that bB is the unique optimal solution of (2). (c) The support union ˆS of bB is identical to the support union S of B∗. At the core of our constructive procedure is the following convex-analytic result, which characterizes an optimal primal-dual pair for which the primal solution bB correctly recovers the support set S: Lemma 1. Suppose that there exists a primal-dual pair ( bB, bZ) that satisfy the conditions: bZS = ζ( bBS) (6a) bΣSS( bBS −B∗ S) −1 nXT S W = −λn bZS (6b) λn °°° bZSc °°° ℓ∞/ℓ2 : = °°°°bΣScS( bBS −B∗ S) −1 nXT ScW °°°° ℓ∞/ℓ2 < λn (6c) bBSc = 0. (6d) Then ( bB, bZ) is the unique optimal solution to the block-regularized problem, with bS( bB) = S by construction. Appendix A proves Lemma 1, with the strict feasibility of bZSc given by (6c) to certify uniqueness. 3.1 Construction of primal-dual witness Based on Lemma 1, we construct the primal dual pair ( bB, bZ) as follows. First, we set bBSc = 0, to satisfy condition (6d). Next, we obtain the pair ( bBS, bZS) by solving a restricted version of (2): bBS = arg min BS∈Rs×K ( 1 2n ¯¯¯¯ ¯¯¯¯ ¯¯¯¯Y −X · BS 0Sc ¸¯¯¯¯ ¯¯¯¯ ¯¯¯¯ 2 F + λn∥BS∥ℓ1/ℓ2 ) . (7) 5 Since s < n, the empirical covariance (sub)matrix bΣSS = 1 nXT S XS is strictly positive definite with probability one, which implies that the restricted problem (7) is strictly convex and therefore has a unique optimum bBS. We then choose bZS to be the solution of equation (6b). Since any such matrix bZS is also a dual solution to the SOCP (7), it must be an element of the subdifferential ∂∥bBS∥ℓ1/ℓ2. It remains to show that this construction satisfies conditions (6a) and (6c). In order to satisfy condition (6a), it suffices to show that bβi ̸= 0, i ∈S. From equation (6b) and since bΣSS is invertible, we may solve as follows ( bBS −B∗ S) = ³ bΣSS ´−1 ·XT S W n −λn bZS ¸ = : US. (8) For any row i ∈S, we have ∥bβi∥2 ≥∥β∗ i ∥2 −∥US∥ℓ∞/ℓ2 . Thus, it suffices to show that the following event occurs with high probability E(US) : = ½ ∥US∥ℓ∞/ℓ2 ≤1 2 b∗ min ¾ (9) to show that no row of bBS is identically zero. We establish this result later in this section. Turning to condition (6c), by substituting expression (8) for the difference ( bBS −B∗ S) into equation (6c), we obtain a (p −s) × K random matrix VSc, whose row j ∈Sc is given by Vj : = XT j µ [ΠS −In]W n −λn XS n (bΣSS)−1 bZS ¶ . (10) In order for condition (6c) to hold, it is necessary and sufficient that the probability of the event E(VSc) : = n ∥VSc∥ℓ∞/ℓ2 < λn o (11) converges to one as n tends to infinity. Correct inclusion of supporting covariates: We begin by analyzing the probability of E(US). Lemma 2. Under assumption A3 and conditions (5) of Theorem 1, with probability 1 − exp(−Θ(log s)), we have ∥US∥ℓ∞/ℓ2 ≤O ³p (log s)/n ´ + λn ³ Dmax + O ³p s2/n ´´ . This lemma is proved in in the Appendix. With the assumed scaling n = Ω(s log(p −s)), and the assumed slow decrease of b∗ min, which we write explicitly as (b∗ min)2 ≥ 1 ε2n f(p) max{s,log(p−s)} n for some εn →0, we have ∥US∥ℓ∞/ℓ2 b∗ min ≤ O(εn), (12) so that the conditions of Theorem 1 ensure that E(US) occurs with probability converging to one. Correct exclusion of non-support: Next we analyze the event E(VSc). For simplicity, in the following arguments, we drop the index Sc and write V for VSc. In order to show that ∥V ∥ℓ∞/ℓ2 < λn with probability converging to one, we make use of the decomposition 1 λn ∥V ∥ℓ∞/ℓ2 ≤ 3 X i=1 T ′ i where T ′ 1 : = 1 λn ∥E [V | XS]∥ℓ∞/ℓ2 , T ′ 2 : = 1 λn ∥E [V |XS, W] −E [V |XS]∥ℓ∞/ℓ2 and T ′ 3 : = 1 λn ∥V −E [V |XS, W]∥ℓ∞/ℓ2 . Lemma 3. Under assumption A2, T ′ 1 ≤1 −γ . Under conditions (5) of Theorem 1, T ′ 2 = op(1). Therefore, to show that 1 λn ∥V ∥ℓ∞/ℓ2 < 1 with high probability, it suffices to show that T ′ 3 < γ with high probability. Until now, we haven’t appealed to the sample complexity parameter θℓ1/ℓ2(n, p ; B∗). In the next section, we prove that θℓ1/ℓ2(n, p ; B∗) > θcrit(Σ) implies that T ′ 3 < γ with high probability. 6 Lemma 4. Conditionally on W and XS, we have ¡ ∥Vj −E [Vj | XS, W]∥2 2 | W, XS ¢ d= ¡ ΣSc | S ¢ jj ξT j Mnξj, where ξj ∼N(⃗0K, IK) and where the K × K matrix Mn = Mn(XS, W) is given by Mn : = λ2 n n bZT S (bΣSS)−1 bZS + 1 n2 W T (ΠS −In)W. (13) But the covariance matrix Mn is itself concentrated. Indeed, Lemma 5. Under the conditions (5) of Theorem 1, for any δ > 0, the following event T (δ) has probability converging to 1: T (δ) : = n |||Mn|||2 ≤λ2 n ψ(B∗) n (1 + δ) o . (14) For any fixed δ > 0, we have P[T ′ 3 ≥γ] ≤P[T ′ 3 ≥γ | T (δ)] + P[T (δ)c], but, from lemma 5, P[T (δ)c] →0, so that it suffices to deal with the first term. Given that (ΣSc | S)jj ≤(ΣScSc)jj ≤Cmax for all j, on the event T (δ), we have max j∈Sc (ΣSc | S)jj ξT j Mnξj ≤Cmax |||Mn|||2 max j∈Sc ∥ξj∥2 2 ≤Cmax λ2 n ψ(B∗) n max j∈Sc ∥ξj∥2 2 and P[T ′ 3 ≥γ |T (δ)] ≤P · max j∈Sc ∥ξj∥2 2 ≥2t∗(n, B∗) ¸ with t∗(n, B∗) : = 1 2 γ2 Cmax n ψ(B∗) (1 + δ). Finally using the union bound and a large deviation bound for χ2 variates we get the following condition which is equivalent to the condition of Theorem 1: θℓ1/ℓ2(n, p ; B∗) > θcrit(Σ): Lemma 6. P · max j∈Sc ∥ξj∥2 2 ≥2t∗(n, B∗) ¸ →0 if t∗(n, B∗) > (1 + ν) log(p −s) for some ν > 0. 4 Simulations In this section, we illustrate the sharpness of Theorem 1 and furthermore ascertain how quickly the predicted behavior is observed as n, p, s grow in different regimes, for two regression tasks (i.e., K = 2). In the following simulations, the matrix B∗of regression coefficients is designed with entries β∗ ij in {−1/ √ 2, 1/ √ 2} to yield a desired value of ψ(B∗). The design matrix X is sampled from the standard Gaussian ensemble. Since |β∗ ij| = 1/ √ 2 in this construction, we have B∗ S = ζ(B∗ S), and b∗ min = 1. Moreover, since Σ = Ip, the sparsity-overlap ψ(B∗) is simply ¯¯¯¯¯¯ ζ(B∗)T ζ(B∗) ¯¯¯¯¯¯ 2 . From our analysis, the sample complexity parameter θℓ1/ℓ2 is controlled by the “interference” of irrelevant covariates, and not by the variance of a noise component. We consider linear sparsity with s = αp, for α = 1/8, for various ambient model dimensions p ∈{32, 256, 1024}. For each value of p, we perform simulations varying the sample size n to match corresponding values of the basic Lasso sample complexity parameter, given by θLas : = n/(2s log(p −s)), in the interval [0.25, 1.5]. In each case, we solve the blockregularized problem (2) with sample size n = 2θLass log(p −s) using the regularization parameter λn = p log(p −s) (log s)/n. In all cases, the noise level is set at σ = 0.1. For our construction of matrices B∗, we choose both p and the scalings for the sparsity so that the obtained values for s that are multiples of four, and construct the columns Z(1)∗and Z(2)∗of the matrix B∗= ζ(B∗) from copies of vectors of length 4. Denoting by ⊗the usual matrix tensor product, we consider: Identical regressions: We set Z(1)∗= Z(2)∗= 1 √ 2⃗1s, so that the sparsity-overlap is ψ(B∗) = s. Orthogonal regression: Here B∗is constructed with Z(1)∗⊥Z(2)∗, so that ψ(B∗) = s 2, the most favorable situation. To achieve this, we set Z(1)∗= 1 √ 2⃗1s and Z(2)∗= 1 √ 2⃗1s/2⊗(1, −1)T . Intermediate angles: In this intermediate case, the columns Z(1)∗and Z(2)∗are at a 60◦angle, which leads to ψ(B∗) = 3 4s. We set Z(1)∗= 1 √ 2⃗1s and Z(2)∗= 1 √ 2⃗1s/4 ⊗(1, 1, 1, −1)T . Figure 1 shows plots of all three cases and the reference Lasso case for the three different values of the ambient dimension and the two types of sparsity described above. Note how the curves all undergo a threshold phenomenon, with the location consistent with the predictions of Theorem 1. 7 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 θ P(support correct) p=32 s=p/8=4 L1 Z1=Z2 Ð (Z1,Z2)=60o Z1⊥ Z2 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 θ P(support correct) p=256 s=p/8=32 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 θ P(support correct) p=1024 s=p/8=128 Figure 1. Plots of support recovery probability P[bS = S] versus the basic ℓ1 control parameter θLas=n/[2s log(p −s)] for linear sparsity s=p/8, and for increasing values of p ∈{32, 256, 1024} from left to right. Each graph shows four curves corresponding to the case of independent ℓ1 regularization (pluses), and for ℓ1/ℓ2 regularization, the cases of identical regression (crosses), intermediate angles (nablas), and orthogonal regressions (squares). As plotted in dotted vertical lines, Theorem 1 predicts that identical case should succeed for θLas>1 (same as ordinary Lasso), intermediate case for θLas>0.75, and orthogonal case for θLas>0.50. The shift of these curves confirms this prediction. 5 Discussion We studied support union recovery under high-dimensional scaling with the ℓ1/ℓ2 regularization, and shown that its sample complexity is determined by the function ψ(B∗). The latter integrates the sparsity of each univariate regression with the overlap of all the supports and the discrepancies between each of the vectors of parameter estimated. In favorable cases, for K regressions, the sample complexity for ℓ1/ℓ2 is K times smaller than that of the Lasso. Moreover, this gain is not obtained at the expense of an assumption of shared support over the data. In fact, for standard Gaussian designs, the regularization seems “adaptive” in sense that it doesn’t perform worse than the Lasso for disjoint supports. This is not necessarily the case for more general designs and in some situations, which need to be characterized in future work, it could do worse than the Lasso. References [1] F. Bach. Consistency of the group Lasso and multiple kernel learning. Technical report, INRIA D´epartement d’Informatique, Ecole Normale Sup´erieure, 2008. [2] F. Bach, G. Lanckriet, and M. Jordan. Multiple kernel learning, conic duality, and the SMO algorithm. In Proc. Int. Conf. Machine Learning (ICML). Morgan Kaufmann, 2004. [3] D. Donoho, M. Elad, and V. M. Temlyakov. Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Info Theory, 52(1):6–18, January 2006. [4] H. Liu and J. Zhang. On the ℓ1−ℓq regularized regression. Technical Report arXiv:0802.1517v1, Carnegie Mellon University, 2008. [5] L. Meier, S. van de Geer, and P. B¨uhlmann. The group lasso for logistic regression. Technical report, Mathematics Department, Swiss Federal Institute of Technology Z¨urich, 2007. [6] Y. Nardi and A. Rinaldo. On the asymptotic properties of the group lasso estimator for linear models. Electronic Journal of Statistics, 2:605–633, 2008. [7] G. Obozinski, B. Taskar, and M. Jordan. Joint covariate selection and joint subspace selection for multiple classification problems. Statistics and Computing, 2009. To appear. [8] M. Pontil and C.A. Michelli. Learning the kernel function via regularization. Journal of Machine Learning Research, 6:1099–1125, 2005. [9] P. Ravikumar, J. Lafferty, H. Liu, and L. Wasserman. SpAM: sparse additive models. In Neural Info. Proc. Systems (NIPS) 21, Vancouver, Canada, December 2007. [10] J. A. Tropp. Just relax: Convex programming methods for identifying sparse signals in noise. IEEE Trans. Info Theory, 52(3):1030–1051, March 2006. [11] M. J. Wainwright. Sharp thresholds for high-dimensional and noisy recovery of sparsity using using ℓ1-constrained quadratic programs. Technical Report 709, Department of Statistics, UC Berkeley, 2006. [12] M. Yuan and Y. Lin. Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society B, 1(68):4967, 2006. [13] P. Zhao, G. Rocha, and B. Yu. Grouped and hierarchical model selection through composite absolute penalties. Technical report, Statistics Department, UC Berkeley, 2007. [14] P. Zhao and B. Yu. Model selection with the lasso. J. of Machine Learning Research, pages 2541–2567, 2007. 8
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Convergence and Rate of Convergence of A Manifold-Based Dimension Reduction Algorithm Andrew K. Smith, Xiaoming Huo School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta, GA 30332 andrewsmith81@gmail.com, huo@gatech.edu Hongyuan Zha College of Computing Georgia Institute of Technology Atlanta, GA 30332 zha@cc.gatech.edu Abstract We study the convergence and the rate of convergence of a local manifold learning algorithm: LTSA [13]. The main technical tool is the perturbation analysis on the linear invariant subspace that corresponds to the solution of LTSA. We derive a worst-case upper bound of errors for LTSA which naturally leads to a convergence result. We then derive the rate of convergence for LTSA in a special case. 1 Introduction Manifold learning (ML) methods have attracted substantial attention due to their demonstrated potential. Many algorithms have been proposed and some work has appeared to analyze the performance of these methods. The main contribution of this paper is to establish some asymptotic properties of a local manifold learning algorithm: LTSA [13], as well as a demonstration of some of its limitations. The key idea in the analysis is to treat the solutions computed by LTSA as invariant subspaces of certain matrices, and then carry out a matrix perturbation analysis. Many efficient ML algorithms have been developed including locally linear embedding (LLE) [6], ISOMAP [9], charting [2], local tangent space alignment (LTSA) [13], Laplacian eigenmaps [1], and Hessian eigenmaps [3]. A common feature of many of these manifold learning algorithms is that their solutions correspond to invariant subspaces, typically the eigenspace associated with the smallest eigenvalues of a kernel or alignment matrix. The exact form of this matrix, of course, depends on the details of the particular algorithm. We start with LTSA for several reasons. First of all, in numerical simulations (e.g., using the tools offered by [10]), we find empirically that LTSA performs among the best of the available algorithms. Second, the solution to each step of the LTSA algorithm is an invariant subspace, which makes analysis of its performance more tractable. Third, the similarity between LTSA and several other ML algorithms (e.g., LLE, Laplacian eigenmaps and Hessian eigenmaps) suggests that our results may generalize. Our hope is that this performance analysis will provide a theoretical foundation for the application of ML algorithms. The rest of the paper is organized as follows. The problem formulation and background information are presented in Section 2. Perturbation analysis is carried out, and the main theorem is proved (Theorem 3.7) in Section 3. Rate of convergence under a special case is derived in Section 4. Some discussions related to existing work in this area are included in Section 5. Finally, we present concluding remarks in Section 6. 1 2 Manifold Learning and LTSA We formulate the manifold learning problem as follows. For a positive integer n, let yi ∈IRD, i = 1, 2, . . . , n, denote n observations. We assume that there is a mapping f : IRd →IRD which satisfies a set of regularity conditions (detailed in the next subsection). In addition, we require another set of (possibly multivariate) values xi ∈IRd, d < D, i = 1, 2, . . . , n, such that yi = f(xi) + εi, i = 1, 2, . . . , n, (1) where εi ∈IRD denotes a random error. For example, we may assume εi ∼N(0, σ2ID); i.e., a multivariate normal distribution with mean zero and variance-covariance proportional to the identity matrix. The central questions of manifold learning are: 1) Can we find a set of low-dimensional vectors such that equation (1) holds? 2) What kind of regularity conditions should be imposed on f? 3) Is the model well defined? These questions are the main focus of this paper. 2.1 A Pedagogical Example (a) Embedded Spiral (b) Noisy Observations (c) Learned vs. Truth !1 !0.5 0 0.5 1 !1 !0.5 0 0.5 1 0 0.5 1 1.5 2 2.5 3 !1 !0.5 0 0.5 1 !1 !0.5 0 0.5 1 !0.5 0 0.5 1 1.5 2 2.5 3 3.5 0 0.2 0.4 0.6 0.8 1 !0.05 !0.04 !0.03 !0.02 !0.01 0 0.01 0.02 0.03 0.04 0.05 Figure 1: An illustrative example of LTSA in nonparametric dimension reduction. The straight line pattern in (c) indicates that the underlying parametrization has been approximately recovered. An illustrative example of dimension reduction that makes our formulation more concrete is given in Figure 1. Subfigure (a) shows the true underlying structure of a toy example, a 1-D spiral. The noiseless observations are equally spaced points on this spiral. In subfigure (b), 1024 noisy observations are generated with multivariate noise satisfying εi ∼N(0, 1 100I3). We then apply LTSA to the noisy observations, using k = 10 nearest neighbors. In subfigure (c), the result from LTSA is compared with the true parametrization. When the underlying parameter is faithfully recovered, one should see a straight line, which is observed to hold approximately in subfigure (c). 2.2 Regularity and Uniqueness of the Mapping f If the conditions on the mapping f are too general, the model in equation (1) is not well defined. For example, if the mapping f(·) and point set {xi} satisfy (1), so do f(A−1(· −b)) and {Axi + b}, where A is an invertible d by d matrix and b is a d-dimensional vector. As being common in the manifold-learning literature, we adopt the following condition on f. Condition 2.1 (Local Isometry) The mapping f is locally isometric: For any ε > 0 and x in the domain of f, let Nε(x) = {z : ∥z −x∥2 < ε} denote an ε-neighborhood of x using Euclidean distance. We have ∥f(x) −f(x0)∥2 = ∥x −x0∥2 + o(∥x −x0∥2). The above condition indicates that in a local sense, f preserves Euclidean distance. Let J(f; x0) denote the Jacobian of f at x0. We have J(f; x0) ∈IRD×d, where each column (resp., row) of J(f; x0) corresponds to a coordinate in the feature (resp., data) space. The above in fact implies the following lemma [13]. Lemma 2.2 The matrix J(f; x0) is orthonormal for any x0, i.e., JT (f; x0)J(f; x0) = Id. 2 Given the previous condition, model (1) is still not uniquely defined. For example, for any d by d orthogonal matrix O and any d-dimensional vector b, if f(·) and {xi} satisfy (1) and Condition 2.1, so do f(OT (·−b)) and {Oxi+b}. We can force b to be 0 by imposing the condition that P i xi = 0. In dimension reduction, we can consider the sets {xi} and {Oxi} “invariant,” because one is just a rotation of the other. In fact, the invariance coincides with the concept of “invariant subspace” to be discussed. Condition 2.3 (Local Linear Independence Condition) Let Yi ∈IRD×k, 1 ≤i ≤n, denote a matrix whose columns are made by the ith observation yi and its k −1 nearest neighbors. We choose k −1 neighbors so that the matrix Yi has k columns. It is generally assumed that d < k. For any 1 ≤i ≤n, the rank of YiP k is at least d; in other words, the dth largest singular value of matrix YiP k is greater than 0. In the above, we use the projection matrix P k = Ik−1 k ·1k1T k , where Ik is the k by k identity matrix and 1k is a k-dimensional column vector of ones. The regularity of the manifold can be determined by the Hessians of the mapping. Rewrite f(x) for x ∈IRd as f(x) = (f1(x), f2(x), . . . , fD(x))T . Furthermore, let x = (x1, . . . , xd)T . The Hessian is a D by D matrix, [Hi(f; x)]jk = ∂2fi(x) ∂xj∂xk , 1 ≤i ≤D, 1 ≤j, k ≤d. The following condition ensures that f is locally smooth. We impose a bound on all the components of the Hessians. Condition 2.4 (Regularity of the Manifold) |[Hi(f; x)]jk| ≤C1 for all i, j, and k, where C1 > 0 is a prescribed constant. 2.3 Solutions as Invariant Subspaces and a Related Metric We now give a more detailed discussion of invariant subspaces. Let R(X) denote the subspace spanned by the columns of X. Recall that xi, i = 1, 2, . . . , n, are the true low-dimensional representations of the observations. We treat the xi’s as column vectors. Let X = (x1, x2, · · · , xn)T ; i.e., the ith row of X corresponds to xi, 1 ≤i ≤n. If the set {Oxi}, where O is a d by d orthogonal square matrix, forms another solution to the dimension reduction problem, we have (Ox1, Ox2, · · · , Oxn)T = XOT . It is evident that R(XOT ) = R(X). This justifies the invariance that was mentioned earlier. The goal of our performance analysis is to answer the following question: Letting ∥tan(·, ·)∥2 denote the Euclidean norm of the vector of canonical angles between two invariant subspaces ([8, Section I.5]), and letting X and e X denote the true and estimated parameters, respectively, how do we evaluate ∥tan(R(X), R( e X))∥2? 2.4 LTSA: Local Tangent Space Alignment We now review LTSA. There are two main steps in the LTSA algorithm [13]. 1. The first step is to compute the local representation on the manifold. Recall the projection matrix P k. It is easy to verify that P k = P k · P k, which is a characteristic of projection matrices. We solve the minimization problem: minΛ,V ∥YiP k −ΛV ∥F , where Λ ∈IRD×d, V ∈IRd×k, and V V T = Id. Let Vi denote optimal V . Then the row vectors of Vi are the d right singular vectors of YiP k. 2. The solution to LTSA corresponds to the invariant subspace which is spanned and determined by the eigenvectors associated with the 2nd to the (d + 1)st smallest eigenvalues of the matrix (S1, . . . , Sn)diag(P k −V T 1 V1, . . . , P k −V T n Vn)(S1, . . . , Sn)T . (2) where Si ∈IRn×k is a selection matrix such that Y T Si = Yi, where Y = (y1, y2, . . . , yn)T . 3 As mentioned earlier, the subspace spanned by the eigenvectors associated with the 2nd to the (d + 1)st smallest eigenvalues of the matrix in 2 is an invariant subspace, which will be analyzed using matrix perturbation techniques. We slightly reformulated the original algorithm as presented in [13] for later analysis. 3 Perturbation Analysis We now carry out a perturbation analysis on the reformulated version of LTSA. There are two steps: in the local step (Section 3.1), we characterize the deviation of the null spaces of the matrices P k −V T i Vi, i = 1, 2, . . . , n. In the global step (Section 3.2), we derive the variation of the null space under global alignment. 3.1 Local Coordinates Let X be the matrix of true parameters. We define Xi = XT Si = (x1, x2, · · · , xn)Si; i.e., the columns of Xi are made by xi and those xj’s that correspond to the k −1 nearest neighbors of yi. We require a bound on the size of the local neighborhoods defined by the Xi’s. Condition 3.1 (Universal Bound on the Sizes of Neighborhoods) For all i, 1 ≤i ≤n, we have τi < τ, where τ is a prescribed constant and τi is an upper bound on the distance between two columns of Xi: τi = maxxj,xk ∥xj −xk∥, where the maximum is taken over all columns of Xi. In this paper, we are interested in the case when τ →0. We will need conditions on the local tangent spaces. Let dmin,i (respectively, dmax,i) denote the minimum (respectively, maximum) singular values of XiP k. Let dmin = min 1≤i≤n dmin,i, dmax = max 1≤i≤n dmax,i. We can bound dmax as dmin ≤dmax ≤τ √ k [5]. Condition 3.2 (Local Tangent Space) There exists a constant C2 > 0, such that C2 · τ ≤dmin. (3) The above can roughly be thought of as requiring that the local dimension of the manifold remain constant (i.e., the manifold has no singularities.) The following condition defines a global bound on the errors (εi). Condition 3.3 (Universal Error Bound) There exists σ > 0, such that ∀i, 1 ≤i ≤n, we have ∥yi −f(xi)∥∞< σ. Moreover, we assume σ = o(τ); i.e., we have σ τ →0, as τ →0. It is reasonable to require that the error bound (σ) be smaller than the size of the neighborhood (τ), which is reflected in the above condition. Within each neighborhood, we give a perturbation bound between an invariant subspace spanned by the true parametrization and the invariant subspace spanned by the singular vectors of the matrix of noisy observations. Let XiP k = AiDiBi be the singular value decomposition of the matrix XiP k; here Ai ∈IRd×d is orthogonal (AiAT i = Id), Di ∈IRd×d is diagonal, and the rows of Bi ∈IRd×k are the right singular vectors corresponding to the largest singular values (BiBT i = Id). It is not hard to verify that Bi = BiP k. (4) Let YiP k = eAi eDi eBi be the singular value decomposition of YiP k, and assume that this is the “thin” decomposition of rank d. We may think of this as the perturbed version of J(f; x(0) i )XiP k. The rows of eBi are the eigenvectors of (YiP k)T (YiP k) corresponding to the d largest eigenvalues. Let R(BT i ) (respectively, R( eBT i )) denote the invariant subspace that is spanned by the columns of matrix BT i (respectively, eBT i ). 4 Theorem 3.4 Given invariant subspaces R(BT i ) and R( eBT i )) as defined above, we have lim τ→0 ∥sin(R(BT i ), R( eBT i ))∥2 ≤C3 σ τ + C1τ , where C3 is a constant that depends on k, D and C2. The proof is presented in [5]. The above gives an upper bound on the deviation of the local invariant subspace in step 1 of the modified LTSA. It will be used later to prove a global upper bound. 3.2 Global Alignment Condition 3.5 (No Overuse of One Observation) There exists a constant C4, such that
n X i=1 Si
∞ ≤C4. Note that we must have C4 ≥k. The next condition (Condition 3.6) will implicitly give an upper bound on C4. Recall that the quantity ∥Pn i=1 Si∥∞is the maximum row sum of the absolute values of the entries in Pn i=1 Si. The value of ∥Pn i=1 Si∥∞is equal to the maximum number of nearest neighbor subsets to which a single observation belongs. We will derive an upper bound on the angle between the invariant subspace spanned by the result of LTSA and the space spanned by the true parameters. Given (4), it can be shown that XiP k(P k −BT i Bi)(XiP k)T = 0. Recall X = (x1, x2, . . . , xn)T ∈ IRn×d. It is not hard to verify that the row vectors of (1n, X)T span the (d + 1)-dimensional null space of the matrix: (S1, . . . , Sn)P kdiag(I −BT 1 B1, . . . , I −BT n Bn)P k(S1, . . . , Sn)T . (5) Assume that ( 1n √n, X, (Xc))T is orthogonal, where Xc ∈IRn×(n−1−d). Although in our original problem formulation, we made no assumption about the xi’s, we can still assume that the columns of X are orthonormal because we can transform any set of xi’s into an orthonormal set by rescaling the columns and multiplying by an orthogonal matrix. Based on the previous paragraph, we have 1T n √n XT (Xc)T Mn 1n √n, X, Xc = 0(d+1)×(d+1) 0(d+1)×(n−d−1) 0(n−d−1)×(d+1) L2 (6) where Mn = (S1, . . . , Sn)P kdiag(Ik −BT 1 B1, . . . , Ik −BT n Bn)P k(S1, . . . , Sn)T and L2 = (Xc)T MnXc. Let ℓmin denote the minimum singular value (i.e., eigenvalue) of L2. We will need the following condition on ℓmin. Condition 3.6 (Appropriateness of Global Dimension) ℓmin > 0 and ℓmin goes to 0 at a slower rate than σ τ + 1 2C1τ; i.e., as τ →0, we have σ τ + 1 2C1τ · ∥Pn i=1 Si∥∞ ℓmin →0. As discussed in [12, 11], this condition is actually related to the amount of overlap between the nearest neighbor sets. 5 Theorem 3.7 (Main Theorem) lim τ→0 ∥tan(R( e X), R(X))∥2 ≤C3( σ τ + C1τ) · ∥Pn i=1 Si∥∞ ℓmin . (7) As mentioned in the Introduction, the above theorem gives a worst-case bound on the performance of LTSA. For proofs as well as a discussion of the requirement that σ →0 see [7]. A discussion on when Condition 3.6 is satisfied will be long and beyond the scope of this paper. We leave it to future investigation. We refer to [5] for some simulation results related to the above analysis. 4 A Preliminary Result on the Rate of Convergence We discuss the rate of convergence for LTSA (to the true underlying manifold structure) in the aforementioned framework. We modify the LTSA (mainly on how to choose the size of the nearest neighborhood) for a reason that will become evident later. We assume the following result regarding the relationship between k, ℓmin, and τ (this result can be proved for xi being sampled on a uniform grid, using the properties of biharmonic eigenvalues for partial differential equations) holds: ℓmin ≈C(k) · ν+ min(∆2) · τ 4, (8) where ν+ min(∆2) is a constant, and C(k) ≈k5. We will address such a result in the more general context in the future. So far, we have assumed that k is constant. However, allowing k to be a function of the sample size n, say k = nα, where α ∈[0, 1) allows us to control the asymptotic behavior of ℓmin along with the convergence of the estimated alignment matrix to the true alignment matrix. Consider our original bound on the angle between the true coordinates and the estimated coordinates: lim τ→0 ∥tan(R( e X), R(X))∥2 ≤C3( σ τ + C1τ) · ∥Pn i=1 Si∥∞ ℓmin . Now, set k = nα, where α ∈[0, 1) is an exponent, the value of which will be decided later. We must be careful in disregarding constants, since they may involve k. We have that C3 = √ kD C2 . C1 and C2 are fundamental constants not involving k. Further, it is easy to see that ∥Pn i=1 Si∥∞is O(k) since each point has k neighbors, the maximum number of neighborhoods to which a point belongs is of the same order as k. Now, we can use a simple heuristic to estimate the size of τ, the neighborhood size. For example, suppose we fix ϵ and consider ϵ-neighborhoods. For simplicity, assume that the parameter space is the unit hypercube [0, 1]d, where d is the intrinsic dimension. The law of large numbers tells us that k ≈ϵd · n. Thus we can approximate τ as τ ≈O(n α−1 d ). Plugging all this into the original equation and dropping the constants, we get lim τ→0 ∥tan(R( e X), R(X))∥2 ≤n α−1 d · n 3α 2 ℓmin · Constant. If we conjecture that the relationship in (8) holds in general (i.e., the generating coordinates can follow a more general distribution rather than only lying in a uniform grid), then we have lim τ→0 ∥tan(R( e X), R(X))∥2 ≤n α−1 d · n α 2 · nα n5α · n4· α−1 d · Constant. Now the exponent is a function only of α and the constant d. We can try to solve for α such that the convergence is as fast as possible. Simplifying the exponents, we get lim τ→0 ∥tan(R( e X), R(X))∥2 ≤n −7α 2 −3( α−1 d ) · Constant. As a function of α restricted to the interval [0, 1), there is no minimum—the exponent decreases with α, and we should choose α close to 1. 6 However, in the proof of the convergence of LTSA, it is assumed that the errors in the local step converge to 0. This error is given by ∥sin(R(BT i ), R( eBT i ))∥2 ≤ √ kD · [σ + 1 2C1τ 2] C2 · τ − √ kD · [σ + 1 2C1τ 2] . Thus, our choice of α is restricted by the fact that the RHS of this equation must still converge to 0. Disregarding constants and writing this as a function of n, we get n α 2 · n 2α−2 d n α−1 d −n α 2 · n 2α−2 d . This quantity converges to 0 as n →∞if and only if we have α 2 + 2α −2 d < α −1 d ⇔ α < 2 d + 2. Note that this bound is strictly less than 1 for all positive integers d, so our possible choices of α are restricted further. By the reasoning above, we want the exponent to be as large as possible. Further, it is easy to see that for all d, choosing an exponent roughly equal to 2 d+2 will always yield a bound converging to 0. The following table gives the optimal exponents for selected values of d along with the convergence rate of limτ→0 ∥tan(R( e X), R(X))∥2. In general, using the optimal value of α, the convergence rate will be roughly n −4 d+2 . Table 1: Convergence rates for a few values of the underlying dimension d. d 1 2 3 4 5 Optimal α 0.66 0.5 0.4 0.33 0.29 Convergence rate −1.33 −1 −0.8 −0.66 −0.57 Thesis [7] presents some numerical experiments to illustrate the above results. Associated with each fixed value of k, there seems to be a threshold value of n, above which the performance degrades. This value increases with k, though perhaps at the cost of worse performance for small n. However, we expect from the above analysis that, regardless of the value chosen, the performance will eventually become unacceptable for any fixed k. 5 Discussion To the best of our knowledge, the performance analysis that is based on invariant subspaces is new. Consequently the worst-case upper bound is the first of its kind. There are still open questions to be addressed (Section 5.1). In addition to a discussion on the relation of LTSA to existing dimension reduction methodologies, we will also address relation with known results as well (Section 5.2). 5.1 Open Questions The rate of convergence of ℓmin is determined by the topological structure of f. It is important to estimate this rate of convergence, but this issue has not been addressed here. We did not address the correctness of (8) at all. It turns out the proof of (8) is quite nontrivial and tedious. We assume that τ →0. One can imagine that it is true when the error bound (σ) goes to 0 and when the xi’s are sampled with a sufficient density in the support of f. An open problem is how to derive the rate of convergence of τ →0 as a function of the topology of f and the sampling scheme. After doing so, we may be able to decide where our theorem is applicable. 5.2 Relation to Existing Work The error analysis in the original paper about LTSA is the closest to our result. However, Zhang and Zha [13] do not interpret their solutions as invariant subspaces, and hence their analysis does not yield a worst case bound as we have derived here. 7 Reviewing the original papers on LLE [6], Laplacian eigenmaps [1], and Hessian eigenmaps [3] reveals that their solutions are subspaces spanned by a specific set of eigenvectors. This naturally suggests that results analogous to ours may be derivable as well for these algorithms. A recent book chapter [4] stresses this point. After deriving corresponding upper bounds, we can establish different proofs of consistency than those presented in these papers. ISOMAP, another popular manifold learning algorithm, is an exception. Its solution cannot immediately be rendered as an invariant subspace. However, ISOMAP calls for MDS, which can be associated with an invariant subspace; one may derive an analytical result through this route. 6 Conclusion We derive an upper bound of the distance between two invariant subspaces that are associated with the numerical output of LTSA and an assumed intrinsic parametrization. Such a bound describes the performance of LTSA with errors in the observations, and thus creates a theoretical foundation for its use in real-world applications in which we would naturally expect such errors to be present. Our results can also be used to show other desirable properties, including consistency and rate of convergence. Similar bounds may be derivable for other manifold-based learning algorithms. References [1] M. Belkin and P. Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15(6):1373–1396, 2003. [2] M. Brand. Charting a manifold. In Neural Information Processing Systems, volume 15. Mitsubishi Electric Research Labs, MIT Press, March 2003. [3] D. L. Donoho and C. E. Grimes. Hessian eigenmaps: New locally linear embedding techniques for high-dimensional data. Proceedings of the National Academy of Arts and Sciences, 100:5591–5596, 2003. [4] X. Huo, X. S. Ni, and A. K. Smith. Mining of Enterprise Data, chapter A survey of manifoldbased learning methods. Springer, New York, 2005. Invited book chapter, accepted. [5] X. Huo and A. K. Smith. Performance analysis of a manifold learning algorithm in dimension reduction. Technical report, Georgia Institute of Technology, March 2006. Downloadable at www2.isye.gatech.edu/statistics/papers/06-06.pdf, to appear in Linear Algebra and Its Applications. [6] S. T. Roweis and L. K. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290:2323–2326, 2000. [7] A. K. Smith. New results in dimension reduction and model selection. Ph.D. Thesis. Available at http://etd.gatech.edu, 2008. [8] G. W. Stewart and J.-G. Sun. Matrix Perturbation Theory. Academic Press, Boston, MA, 1990. [9] J. B. Tenenbaum, V. de Silva, and J. C. Langford. A global geometric framework for nonlinear dimensionality reduction. Science, 290:2319–2323, 2000. [10] T. Wittman. MANIfold learning Matlab demo. URL: http://www.math.umn.edu/∼wittman/mani/index.html, April 2005. [11] H. Zha and H. Zhang. Spectral properties of the alignment matrices in manifold learning. SIAM Review, 2008. [12] H. Zha and Z. Zhang. Spectral analysis of alignment in manifold learning. In Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005. [13] Z. Zhang and H. Zha. Principal manifolds and nonlinear dimension reduction via local tangent space alignment. SIAM Journal of Scientific Computing, 26(1):313–338, 2004. 8
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A Convex Upper Bound on the Log-Partition Function for Binary Graphical Models Laurent El Ghaoui Department of Electrical Engineering and Computer Science University of California Berkeley Berkeley, CA 9470 elghaoui@eecs.berkeley.edu Assane Gueye Department of Electrical Engineering and Computer Science University of California Berkeley Berkeley, CA 9470 agueye@eecs.berkeley.edu Abstract We consider the problem of bounding from above the log-partition function corresponding to second-order Ising models for binary distributions. We introduce a new bound, the cardinality bound, which can be computed via convex optimization. The corresponding error on the logpartition function is bounded above by twice the distance, in model parameter space, to a class of “standard” Ising models, for which variable inter-dependence is described via a simple mean field term. In the context of maximum-likelihood, using the new bound instead of the exact log-partition function, while constraining the distance to the class of standard Ising models, leads not only to a good approximation to the log-partition function, but also to a model that is parsimonious, and easily interpretable. We compare our bound with the log-determinant bound introduced by Wainwright and Jordan (2006), and show that when the l1-norm of the model parameter vector is small enough, the latter is outperformed by the new bound. 1 Introduction 1.1 Problem statement This paper is motivated by the problem fitting of binary distributions to experimental data. In the second-order Ising model, PUT REF HERE the fitted distribution p is assumed to have the parametric form p(x; Q, q) = exp(xT Qx + qT x −Z(Q, q)), x ∈{0, 1}n, where Q = QT ∈Rn and q ∈Rn contain the parameters of the model, and Z(Q, q), the normalization constant, is called the log-partition function of the model. Noting that xT Qx + qT x = xT (Q + D(q))x for every x ∈{0, 1}n, we will without loss of generality assume that q = 0, and denote by Z(Q) the corresponding log-partition function Z(Q) := log X x∈{0,1}n exp[xT Qx] . (1) In the Ising model, the maximum-likelihood approach to fitting data leads to the problem min Q∈Q Z(Q) −TrQS, (2) where Q is a subset of the set Sn of symmetric matrices, and S ∈Sn + is the empirical second-moment matrix. When Q = Sn, the dual to (2) is the maximum entropy problem max p H(p) : p ∈P, S = X x∈{0,1}n p(x)xxT , (3) where P is the set of distributions with support in {0, 1}n, and H is the entropy H(p) = − X x∈{0,1}n p(x) log p(x). (4) The constraints of problem (3) define a polytope in R2n called the marginal polytope. For general Q’s, computing the log-partition function is NP-hard. Hence, except for special choices of Q, the maximum-likelihood problem (2) is also NP-hard. It is thus desirable to find computationally tractable approximations to the log-partition function, such that the resulting maximum-likelihood problem is also tractable. In this regard, convex, upper bounds on the log-partition function are of particular interest, and our focus here: convexity usually brings about computational tractability, while using upper bounds yields a parameter Q that is suboptimal for the exact problem. Using an upper bound in lieu of Z(Q) in (2), leads to a problem we will generically refer to as the pseudo maximumlikelihood problem. This corresponds to a relaxation to the maximum-entropy problem, which is (3) when Q = Sn. Such relaxations may involve two ingredients: an upper bound on the entropy, and an outer approximation to the marginal polytope. 1.2 Prior work Due to the vast applicability of Ising models, the problem of approximating their log-partition function, and the related maximum-likelihood problem, has received considerable attention in the literature for decades, first in statistical physics, and more recently in machine learning. The so-called log-determinant bound has been recently introduced, for a large class of Markov random fields, by Wainwright and Jordan [2]. (Their paper provides an excellent overview of the prior work, in the general context of graphical models.) The log-determinant bound is based on an upper bound on the differential entropy of continuous random variable, that is attained for a Gaussian distribution. The log-determinant bound enjoys good tractability properties, both for the computation of the log-partition function, and in the context of the maximum-likelihood problem (2). A recent paper by Ravikumar and Lafferty [1] discusses using bounds on the log-partition function to estimate marginal probabilities for a large class of graphical models, which adds extra motivation for the present study. 1.3 Main results and outline The main purpose of this note is to introduce a new upper bound on the log-partition function that is computationally tractable. The new bound is convex in Q, and leads to a restriction to the maximum-likelihood problem that is also tractable. Our development crucially involves a specific class of Ising models, which we’ll refer to as standard Ising models, in which the model parameter Q has the form Q = µI + λ11T , where λ, µ are arbitrary scalars. Such models are indeed standard in statistical physics: the first term µI describes interaction with the external magnetic field, and the second (λ11T ) is a simple mean field approximation to ferro-magnetic coupling. For standard Ising models, it can be shown that the log-partition functions has a computationally tractable, closed-form expression. Due to space limitation, such proof is omitted in this paper. Our bound is constructed so as to be exact in the case of standard Ising models. In fact, the error between our bound and the true value of the log-partition function is bounded above by twice the l1-norm distance from the model parameters (Q) to the class of standard Ising models. The outline of the note reflects our main results: in section 2, we introduce our bound, and show that the approximation error is bounded above by the distance to the class of standard Ising models. We discuss in section 3 the use of our bound in the context of the maximum-likelihood problem (2) and its dual (3). In particular, we discuss how imposing a bound on the distance to the class of standard Ising models may be desirable, not only to obtain an accurate approximation to the log-partition function, but also to find a parsimonious model, having good interpretability properties. We then compare the new bound with the log-determinant bound of Wainwright and Jordan in section 4. We show that our new bound outperforms the log-determinant bound when the norm ∥Q∥1 is small enough (less than 0.08n), and provide numerical experiments supporting the claim that our comparison analysis is quite conservative: our bound appears to be better over a wide range of values of ∥Q∥1. Notation. Throughout the note, n is a fixed integer. For k ∈{0, . . . , n}, define ∆k := {x ∈{0, 1}n : Card(x) = k}. Let ck = |∆k| denote the cardinal of ∆k, and πk := 2−nck the probability of ∆k under the uniform distribution. For a distribution p, the notation Ep refers to the corresponding expectation operator, and Probp(S) to the probability of the event S under p. The set P is the set of distributions with support on {0, 1}n. For X ∈Rn×n, the notation ∥X∥1 denotes the sum of the absolute values of the elements of X, and ∥X∥∞the largest of these values. The set Sn is the set of symmetric matrices, Sn + the set of symmetric positive semidefinite matrices. We use the notation X ⪰0 for the statement X ∈Sn +. If x ∈Rn, D(x) is the diagonal matrix with x on its diagonal. If X ∈Rn×n, d(X) is the n-vector formed with the diagonal elements of X. Finally, X is the set {(X, x) ∈Sn × Rn : d(X) = x} and X+ = {(X, x) ∈Sn × Rn : X ⪰xxT , d(X) = x}. 2 The Cardinality Bound 2.1 The maximum bound To ease our derivation, we begin with a simple bound based on replacing each term in the log-partition function by its maximum over {0, 1}n. This leads to an upper bound on the log-partition function: Z(Q) ≤n log 2 + φmax(Q), where φmax(Q) := max x∈{0,1}n xT Qx. Computing the above quantity is in general NP-hard. Starting with the expression φmax(Q) = max (X,x)∈X+ TrQX : rank(X) = 1, and relaxing the rank constraint leads to the upper bound φmax(Q) ≤ψmax(Q), where ψmax(Q) is defined via a semidefinite program: ψmax(Q) = max (X,x)∈X+ TrQX, (5) where X+ = {(X, x) ∈Sn × Rn : X ⪰xxT , d(X) = x}. For later reference, we note the dual form: ψmax(Q) = min t,ν t : µ D(ν) −Q 1 2ν 1 2νT t ¶ ⪰0 (6) = min ν 1 4νT (D(ν) −Q)−1ν : D(ν) ≻Q. (7) The corresponding bound on the log-partition function, referred to as the maximum bound, is Z(Q) ≤Zmax(Q) := n log 2 + ψmax(Q). The complexity of this bound (using interior-point methods) is roughly O(n3). Let us make a few observations before proceeding. First, the maximum-bound is a convex function of Q, which is important in the context of the maximum-likelihood problem (2). Second, we have Zmax(Q) ≤n log 2 + ∥Q∥1, which follows from (5), together with the fact that any matrix X that is feasible for that problem satisfies ∥X∥∞≤1. Finally, we observe that the function Zmax is Lipschitz continuous, with constant 1 with respect to the l1-norm. It can be shown that the same property holds for the log-partition function Z itself. Due to space limitation such proof is omitted in this paper. Indeed, for every symmetric matrices Q, R we have the sub-gradient inequality Zmax(R) ≥Zmax(Q) + TrXopt(R −Q), where Xopt is any optimal variable for the dual problem (5). Since any feasible X satisfies ∥X∥∞≤1, we can bound the term TrXopt(Q−R) from below by −∥Q−R∥1, and after exchanging the roles of Q, R, obtain the desired result. 2.2 The cardinality bound For every k ∈{0, . . . , n}, consider the subset of variables with cardinality k, ∆k := {x ∈{0, 1}n : Card(x) = k}. This defines a partition of {0, 1}n, thus Z(Q) = log à n X k=0 X x∈∆k exp[xT Qx] ! . We can refine the maximum bound by replacing the terms in the log-partition by their maximum over ∆k, leading to Z(Q) ≤log à n X k=0 ck exp[φk(Q)] ! , where, for k ∈{0, . . . , n}, ck = |∆k|, and φk(Q) := max x∈∆k xT Qx. Computing φk(Q) for arbitrary k ∈{0, . . . , n} is NP-hard. Based on the identity φk(Q) = max (X,x)∈X+ TrQX : xT x = k, 1T X1 = k2, rankX = 1, (8) and using rank relaxation as before, we obtain the bound φk(Q) ≤ψk(Q), where ψk(Q) = max (X,x)∈X+ TrQX : xT x = k, 1T X1 = k2. (9) We define the cardinality bound, as Zcard(Q) := log à n X k=0 ck exp[ψk(Q)] ! . The complexity of computing ψk(Q) (using interior-point methods) is roughly O(n3). The upper bound Zcard(Q) is computed via n semidefinite programs of the form (9). Hence, its complexity is roughly O(n4). Problem (9) admits the dual form ψk(Q) := min t,µ,ν,λ t + kµ + λk2 : µ D(ν) + µI + λ11T −Q 1 2ν 1 2νT t ¶ ⪰0. (10) The fact that ψk(Q) ≤ψmax(Q) for every k is obtained upon setting λ = µ = 0 in the semi-definite programming problem (10). In fact, we have ψk(Q) = min µ,λ kµ + k2λ + ψmax(Q −µI −λ11T ). (11) The above expression can be directly obtained from the following, valid for every µ, λ: φk(Q) = kµ + k2λ + φk(Q −µI −λ11T ) ≤kµ + k2λ + φmax(Q −µI −λ11T ) ≤kµ + k2λ + ψmax(Q −µI −λ11T ). It can be shown (proof which we omit due to space limitation) that, in the case of standard Ising models, that is if Q has the form µI + λ11T for some scalars µ, λ, then the bound ψk(Q) is exact. Since the values of xT Qx when x ranges ∆k are constant, the cardinality bound is also exact. By construction, Zcard(Q) is guaranteed to be better (lower) than Zmax(Q), since the latter is obtained upon replacing ψk(Q) by its upper bound ψ(Q) for every k. The cardinality bound thus satisfies Z(Q) ≤Zcard(Q) ≤Zmax(Q) ≤n log 2 + ∥Q∥1. (12) Using the same technique as used in the context of the maximum bound, we can show that the function ψk is Lipschitzcontinuous, with constant 1 with respect to the l1-norm. Using the Lipschitz continuity of positively weighted log-sumexp functions (with constant 1 with respect to the l∞norm), we deduce that Zcard(Q) is also Lipschitz-continuous: for every symmetric matrices Q, R, |Zcard(Q) −Zcard(R)| ≤ ¯¯¯¯¯log à n X k=0 ck exp[ψk(Q)] ! −log à n X k=0 ck exp[ψk(R)] !¯¯¯¯¯ ≤max 0≤k≤n |ψk(Q) −ψk(R)| ≤∥Q −R∥1, as claimed. 2.3 Quality analysis We now seek to establish conditions on the model parameter Q, which guarantee that the approximation error Zcard(Q) −Z(Q) is small. The analysis relies on the fact that, for standard Ising models, the error is zero. We begin by establishing an upper bound on the difference between maximal and minimal values of xT Qx when x ∈∆k. We have the bound min x∈∆k xT Qx ≥ηk(Q) := min (X,x)∈X+ TrQX : xT x = k, 1T X1 = k2. In the same fashion as for the quantity ψk(Q), we can express ηk(Q) as ηk(Q) = max µ,λ kµ + k2λ + ψmin(Q −µI −λ11T ), where ψmin(Q) := min (X,x)∈X+ TrQX. Based on this expression , we have, for every k: 0 ≤ψk(Q) −ηk(Q) = min λ,µ, λ′,µ′ k(µ −µ′) + k2(λ −λ′) + ψmax(Q −µI −λ11T ) −ψmin(Q −µ′I −λ′11T ) ≤min λ,µ ψmax(Q −µI −λ11T ) −ψmin(Q −µI −λ11T ) ≤2 minλ,µ ∥Q −µI −λ11T ∥1, where we have used the fact that , for every symmetric matrix R, we have 0 ≤ψmax(R) −ψmin(R) = max (X,x),(Y,y)∈X+ TrR(X −Y ) ≤ max ∥X∥∞≤1, ∥Y ∥∞≤1 TrR(X −Y ) = 2∥R∥1. Using again the Lipschitz continuity properties of the weighted log-sum-exp function, we obtain that for every Q, the absolute error between Z(Q) and Zcard(Q) is bounded as follows: 0 ≤Zcard(Q) −Z(Q) ≤log à n X k=0 ck exp[ψk(Q)] ! −log à n X k=0 ck exp[ηk(Q)] ! ≤max 0≤k≤n (ψk(Q) −ηk(Q)) ≤2Dst(Q), Dst(Q) := min λ,µ ∥Q −µI −λ11T ∥1, (13) Thus, a measure of quality is Dst(Q), the distance, in l1-norm, between the model and the class of standard Ising models. Note that this measure is easily computed, in O(n2 log n) time, by first setting λ to be the median of the values Qij, 1 ≤i < j ≤n, and then setting µ to be the median of the values Qii −λ, i = 1, . . . , n. We summarize our findings so far with the following theorem: Theorem 1 (Cardinality bound) The cardinality bound is Zcard(Q) := log à n X k=0 ck exp[ψk(Q)] ! . where φk(Q), k = 0, . . . , n, is defined via the semidefinite program (9), which can be solved in O(n3). The approximation error is bounded above by twice the distance (in l1-norm) to the class of standard Ising models: 0 ≤Zcard(Q) −Z(Q) ≤2 min λ,µ ∥Q −µI −λ11T ∥1. 3 The Pseudo Maximum-Likelihood Problem 3.1 Tractable formulation Using the bound Zcard(Q) in lieu of Z(Q) in the maximum-likelihood problem (2) leads to a convex restriction of that problem, referred to as the pseudo-maximum likelihood problem. This problem can be cast as min t,µ,ν,Q log à n X k=0 ck exp[tk + kµk + k2λk] ! −TrQS s.t. Q ∈Q, µ D(νk) + µkI + λk11T −Q 1 2νk 1 2νT k tk ¶ ⪰0, k = 0, . . . , n. The complexity of this bound is XXX. For numerical reasons, and without loss of generality, it is advisable to scale the ck’s and replace them by πk := 2−nck ∈[0, 1]. 3.2 Dual and interpretation When Q = Sn, the dual to the above problem is max (Yk,yk,qk)n k=0 −D(q||π) : S = n X k=0 Yk, q ≥0, qT 1 = 1, µ Yk yk yT k qk ¶ ⪰0, d(Yk) = yk, 1T yk = kqk, 1T Yk1 = k2qk, k = 0 . . . , n. where π is the distribution on {0, . . . , n}, with πk = Probu∆k = 2−nck, and D(q||π) is the relative entropy (Kullback-Leibler divergence) between the distributions q, π: D(q||π) := n X k=0 qk log qk πk . To interpret this dual, we assume without loss of generality q > 0, and use the variables Xk := q−1 k Yk, xk := q−1 k yk. We obtain the equivalent (non-convex) formulation max (Xk,xk,qk)n k=0 −D(q||π) : S = n X k=0 qkXk, q ≥0, qT 1 = 1, (14) (Xk, xk) ∈X+, 1T xk = k, 1T Xk1 = k2, k = 0 . . . , n. The above problem can be obtained as a relaxation to the dual of the exact maximum-likelihood problem (2), which is the maximum entropy problem (3). The relaxation involves two steps: one is to form an outer approximation to the marginal polytope, the other is to find an upper bound on the entropy function (4). First observe that we can express any distribution on {0, 1}n as p(x) = n X k=0 qkpk(x), (15) where qk = Probp∆k = X x∈∆k p(x), pk(x) = ½ q−1 k p(x) if x ∈∆k, 0 otherwise. Note that the functions pk are valid distributions on {0, 1}n as well as ∆k. To obtain an outer approximation to the marginal polytope, we then write the moment-matching equality constraint in problem (3) as S = EpxxT = n X k=0 qkXk, where Xk’s are the second-order moment matrices with respect to pk: Xk = EpkxxT = q−1 k X x∈∆k p(x)xxT . To relax the constraints in the maximum-entropy problem (3), we simply use the valid constraints Xk ⪰xkxT k , d(Xk) = xk, 1T xk = k, 1T Xk1 = k2, where xk is the mean under pk: xk = Epkx = q−1 k X x∈∆k p(x)x. This process yields exactly the constraints of the relaxed problem (14). To finalize our relaxation, we now form an upper bound on the entropy function (4). To this end, we use the fact that, since each pk has support in ∆k, its entropy is bounded above by log |∆k|, as follows: −H(p) = X x∈{0,1}n p(x) log p(x) = n X k=0 X x∈∆k p(x) log p(x) = n X k=0 X x∈∆k qkpk(x) log(qkpk(x)) = n X k=0 qk(log qk −H(pk)) ≥ n X k=0 qk(log qk −log |∆k|) (|∆k| = 2nπk) ≥ n X k=0 qk log qk πk −n log 2, which is, up to a constant, the objective of problem (14). 3.3 Ensuring quality via bounds on Q We consider the (exact) maximum-likelihood problem (2), with Q = {Q = QT : ∥Q∥1 ≤ϵ}: min Q=QT Z(Q) −TrQS : ∥Q∥1 ≤ϵ, (16) and its convex relaxation: min Q=QT Zcard(Q) −TrQS : ∥Q∥1 ≤ϵ. (17) The feasible sets of problems (16) and (17) are the same, and on it the difference in the objective functions is uniformly bounded by 2ϵ. Thus, any ϵ-suboptimal solution of the relaxation (17) is guaranteed to by 3ϵ-suboptimal for the exact problem, (16). In practice, the l1-norm constraint in (17) encourages sparsity of Q, hence the interpretability of the model. It also has good properties in terms of the generalization error. As seen above, the constraint also implies a better approximation to the exact problem (16). All these benefits come at the expense of goodness-of-fit, as the constraint reduces the expressive power of the model. This is an illustration of the intimate connections between computational and statistical properties of the model. A more accurate bound on the approximation error can be obtained by imposing the following constraint on Q and two new variables λ, µ: ∥Q −µI −λ11T ∥1 ≤ϵ. We can draw similar conclusions as before. Here, the resulting model will not be sparse, in the sense of having many elements in Q equal to zero. However, it will still be quite interpretable, as the bound above will encourage the number of off-diagonal elements in Q that differ from their median, to be small. A yet more accurate control on the approximation error can be induced by the constraints ψk(Q) ≤ϵ + ηk(Q) for every k, each of which can be expressed as an LMI constraint. The corresponding constrained relaxation to the maximum-likelihood problem has the form min t,µ±,ν±,Q log à n X k=0 ck exp[t+ k + kµ+ k + k2λ+ k ] ! −TrQS s.t. µ diag(ν+ k ) + µ+ k I + λ+ k 11T −Q 1 2ν+ k 1 2ν+ k t+ k ¶ ⪰0, k = 0, . . . , n, µ Q −diag(ν− k ) −µ− k I −λ− k 11T 1 2ν− k 1 2ν− k t− k ¶ ⪰0, k = 0, . . . , n, t+ k −t− k ≤ϵ, k = 0, . . . , n. Using this model instead of ones we saw previously, we sacrifice less on the front of the approximation to the true likelihood, at the expense of increased computational effort. 4 Links with the Log-Determinant Bound 4.1 The log-determinant bounds The bound in Wainwright and Jordan [2] is based on an upper bound on the (differential) entropy of a continuous random variable, which is attained for a Gaussian distribution. It has the form Z(Q) ≤Zld(Q), with Zld(Q) := αn + max (X,x)∈X+ TrQX + 1 2 log det(X −xxT + 1 12I) (18) where α := (1/2) log(2πe) ≈1.42. Wainwright and Jordan suggest to further relax this bound to one which is easier to compute: Zld(Q) ≤Zrld(Q) := αn + max (X,x)∈X TrQX + 1 2 log det(X −xxT + 1 12I). (19) Like Z and the bounds examined previously, the bound Zld and Zrld are Lipschitz-continuous, with constant 1 with respect to the l1 norm. The proof starts with the representations above, and exploits the fact that ∥Q∥1 is an upper bound on TrQX when (X, x) ∈X+. The dual of the log-determinant bound has the form (see appendix (??)) Zld(Q) =n 2 log π −1 2 log 2+ min t,ν,F,g,h t + 1 12Tr(D(ν) −Q −F) −1 2 log det µ D(ν) −Q −F −1 2ν −g −1 2νT −gT t −h ¶ s.t. µ F g g h ¶ ⪰0. (20) The relaxed counterpart Zrld(Q) is obtained upon setting F, g, h to zero in the dual above: Zrld(Q) = n 2 log π −1 2 log 2 + min t,ν t + 1 12Tr(D(ν) −Q) −1 2 log det µ D(ν) −Q −1 2ν −1 2νT t ¶ . Using Schur complements to eliminate the variable t, we further obtain Zrld(Q) = n 2 log π + 1 2+ min ν 1 4νT (D(ν) −Q)−1ν + 1 12Tr(D(ν) −Q) −1 2 log det(D(ν) −Q). (21) 4.2 Comparison with the maximum bound We first note the similarity in structure between the dual problem (5) defining Zmax(Q) and that of the relaxed logdeterminant bound. Despite these connections, the log-determinant bound is neither better nor worse than the cardinality or maximum bounds. Actually, for some special choices of Q (e.g. when Q is diagonal), the cardinality bound is exact, while the log-determinant one is not. Conversely, one can choose Q so that Zcard(Q) > Zld(Q), so no bound dominates the other. The same can be said for Zmax(Q) (see section 4.4 for numerical examples). However, when we impose an extra condition on Q, namely a bound on its l1 norm, more can be said. The analysis is based on the case Q = 0, and exploits the Lipschitz continuity of the bounds with respect to the l1-norm. First notice (although not shown in this paper because of space limitation) that, for Q = 0, the relaxed log-determinant bound writes Zrld(0) = n 2 log 2πe 3 + 1 2 = Zmax(0) + n 2 log πe 6 + 1 2. Now invoke the Lipschitz continuity properties of the bounds Zrld(Q) and Zmax(Q), and obtain that Zrld(Q) −Zmax(Q) = (Zrld(Q) −Zrld(0)) + (Zrld(0) −Zmax(0)) + (Zmax(0) −Zmax(Q)) ≥−2∥Q∥1 + (Zrld(0) −Zmax(0)) = −2∥Q∥1 + +n 2 log πe 6 + 1 2. This proves that if ∥Q∥1 ≤n 4 log πe 6 + 1 4, then the relaxed log-determinant bound Zrld(Q) is worse (larger) than the maximum bound Zmax(Q). We can strengthen the above condition to ∥Q∥1 ≤0.08n. 4.3 Summary of comparison results To summarize our findings: Theorem 2 (Comparison) We have for every Q: Z(Q) ≤Zcard(Q) ≤Zmax(Q) ≤n log 2 + ∥Q∥1. In addition, we have Zmax(Q) ≤Zrld(Q) whenever ∥Q∥1 ≤0.08n. 4.4 A numerical experiment We now illustrate our findings on the comparison between the log-determinant bounds and the cardinality and maximum bounds. We set the size of our model to be n = 20, and for a range of values of a parameter ρ, generate N = 10 random instances of Q with ∥Q∥1 = ρ. Figure ?? shows the average values of the bounds, as well as the associated error bars. Clearly, the new bound outperforms the log-determinant bounds for a wide range of values of ρ. Our predicted threshold value of ∥Q∥1 for which the new bound becomes worse, namely ρ = 0.08n ≈1.6 is seen to be very conservative, with respect to the observed threshold of ρ ≈30. On the other hand, we observe that for large values of ∥Q∥1, the log-determinant bounds do behave better. Across the range of ρ, we note that the log-determinant bound is indistinguishable from its relaxed counterpart. 5 Conclusion and Remarks We have introduced a new upper bound (the cardinality bound) for the log-partition function corresponding to secondorder Ising models for binary distribution. We have shown that such a bound can be computed via convex optimization, and, when compared to the log-determinant bound introduced by Wainwright and Jordan (2006), the cardinality bound performs better when the l1-norm of the model parameter vector is small enough. Although not shown in the paper, the cardinality bound becomes exact in the case of standard Ising model, while the maximum bound (for example) is not exact for such model. As was shown in section 2, the cardinality bound was computed by defining a partition of {0, 1}. This idea can be generalized to form a class of bounds which we call partition bounds. It turns out that partitions bound are closely linked to the more general class bounds that are based on worst-case probability analysis. We acknowledge the importance of applying our bound to real-word data. We hope to include such results in subsequent versions of this paper. References [1] P. Ravikumar and J. Lafferty. Variational Chernoff bounds for graphical models. In Proc. Advances in Neural Information Processing Systems (NIPS), December 2007. [2] Martin J. Wainwright and Michael I. Jordan. Log-determinant relaxation for approximate inference in discrete Markov random fields. IEEE Trans. Signal Processing, 2006.
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Adaptive Forward-Backward Greedy Algorithm for Sparse Learning with Linear Models Tong Zhang Statistics Department Rutgers University, NJ tzhang@stat.rutgers.edu Abstract Consider linear prediction models where the target function is a sparse linear combination of a set of basis functions. We are interested in the problem of identifying those basis functions with non-zero coefficients and reconstructing the target function from noisy observations. Two heuristics that are widely used in practice are forward and backward greedy algorithms. First, we show that neither idea is adequate. Second, we propose a novel combination that is based on the forward greedy algorithm but takes backward steps adaptively whenever beneficial. We prove strong theoretical results showing that this procedure is effective in learning sparse representations. Experimental results support our theory. 1 Introduction Consider a set of input vectors x1, . . . , xn ∈Rd, with corresponding desired output variables y1, . . . , yn. The task of supervised learning is to estimate the functional relationship y ≈f(x) between the input x and the output variable y from the training examples {(x1, y1), . . . , (xn, yn)}. The quality of prediction is often measured through a loss function φ(f(x), y). In this paper, we consider linear prediction model f(x) = wT x. As in boosting or kernel methods, nonlinearity can be introduced by including nonlinear features in x. We are interested in the scenario that d ≫n. That is, there are many more features than the number of samples. In this case, an unconstrained empirical risk minimization is inadequate because the solution overfits the data. The standard remedy for this problem is to impose a constraint on w to obtain a regularized problem. An important target constraint is sparsity, which corresponds to the (non-convex) L0 regularization, where we define ∥w∥0 = |{j : wj ̸= 0}| = k. If we know the sparsity parameter k, a good learning method is L0 regularization: ˆw = arg min w∈Rd 1 n n X i=1 φ(wT xi, yi) subject to ∥w∥0 ≤k. (1) If k is not known, then one may regard k as a tuning parameter, which can be selected through crossvalidation. This method is often referred to as subset selection in the literature. Sparse learning is an essential topic in machine learning, which has attracted considerable interests recently. Generally speaking, one is interested in two closely related themes: feature selection, or identifying the basis functions with non-zero coefficients; estimation accuracy, or reconstructing the target function from noisy observations. It can be shown that the solution of the L0 regularization problem in (1) achieves good prediction accuracy if the target function can be approximated by a sparse ¯w. It can also solve the feature selection problem under extra identifiability assumptions. However, a fundamental difficulty with this method is the computational cost, because the number of subsets of {1, . . . , d} of cardinality k (corresponding to the nonzero components of w) is exponential in k. There are no efficient algorithms to solve the subset selection formulation (1). 1 Due to the computational difficult, in practice, there are three standard methods for learning sparse representations by solving approximations of (1). The first approach is L1-regularization (Lasso). The idea is to replace the L0 regularization in (1) by L1 regularization. It is the closest convex approximation to (1). It is known that L1 regularization often leads to sparse solutions. Its performance has been theoretically analyzed recently. For example, if the target is truly sparse, then it was shown in [10] that under some restrictive conditions referred to as irrepresentable conditions, L1 regularization solves the feature selection problem. The prediction performance of this method has been considered in [6, 2, 1, 9]. Despite its popularity, there are several problems with L1 regularization: first, the sparsity is not explicitly controlled, and good feature selection property requires strong assumptions; second, in order to obtain very sparse solution, one has to use a large regularization parameter that leads to suboptimal prediction accuracy because the L1 penalty not only shrinks irrelevant features to zero, but also shrinks relevant features to zero. A sub-optimal remedy is to threshold the resulting coefficients; however this requires additional tuning parameters, making the resulting procedures more complex and less robust. The second approach to approximately solve the subset selection problem is forward greedy algorithm, which we will describe in details in Section 2. The method has been widely used by practitioners. The third approach is backward greedy algorithm. Although this method is widely used by practitioners, there isn’t any theoretical analysis when n ≪d (which is the case we are interested in here). The reason will be discussed later. In this paper, we are particularly interested in greedy algorithms because they have been widely used but the effectiveness has not been well analyzed. As we shall explain later, neither the standard forward greedy idea nor th standard backward greedy idea is adequate for our purpose. However, the flaws of these methods can be fixed by a simple combination of the two ideas. This leads to a novel adaptive forward-backward greedy algorithm which we present in Section 3. The general idea works for all loss functions. For least squares loss, we obtain strong theoretical results showing that the method can solve the feature selection problem under moderate conditions. For clarity, this paper only considers the fixed design formulation. To simplify notations in our description, we will replace the optimization problem in (1) with a more general formulation. Instead of working with n input data vectors xi ∈Rd, we work with d feature vectors fj ∈Rn (j = 1, . . . , d), and y ∈Rn. Each fj corresponds to the j-th feature component of xi for i = 1, . . . , n. That is, fj,i = xi,j. Using this notation, we can generally rewrite (1) with in the form ˆw = arg minw∈Rd R(w) subject to ∥w∥0 ≤k, where weight w = [w1, . . . , wd] ∈Rd, and R(w) is a real-valued cost function which we are interested in optimization. For least squares regression, we have R(w) = n−1∥P j wjfj −y∥2 2. In the following, we also let ej ∈Rd be the vector of zeros, except for the j-component which is one. For convenience, we also introduce the following notations. Definition 1.1 Define supp(w) = {j : wj ̸= 0} as the set of nonzero coefficients of a vector w = [w1, . . . , wd] ∈Rd. For a weight vector w ∈Rd, we define mapping f : Rd →Rn as: f(w) = Pd j=1 wjfj. Given f ∈Rd and F ⊂{1, . . . , d}, let ˆw(F, f) = minw∈Rd ∥f(w) − f∥2 2 subject to supp(w) ⊂F, and let ˆw(F) = ˆw(F, y) be the solution of the least squares problem using features F. 2 Forward and Backward Greedy Algorithms Forward greedy algorithms have been widely used in applications. The basic algorithm is presented in Figure 1. Although a number of variations exist, they all share the basic form of greedily picking an additional feature at every step to aggressively reduce the cost function. The intention is to make most significant progress at each step in order to achieve sparsity. In this regard, the method can be considered as an approximation algorithm for solving (1). A major flaw of this method is that it can never correct mistakes made in earlier steps. As an illustration, we consider the situation plotted in Figure 2 with least squares regression. In the figure, y can be expressed as a linear combination of f1 and f2 but f3 is closer to y. Therefore using the forward greedy algorithm, we will find f3 first, then f1 and f2. At this point, we have already found all good features as y can be expressed by f1 and f2, but we are not able to remove f3 selected in the first step. The above argument implies that forward greedy method is inadequate for feature selection. The method only works when small subsets of the basis functions {fj} are near orthogonal 2 Input: f1, . . . , fd, y ∈Rn and ϵ > 0 Output: F (k) and w(k) let F (0) = ∅and w(0) = 0 for k = 1, 2, . . . let i(k) = arg mini minα R(w(k−1) + αei) let F (k) = {i(k)} ∪F (k−1) let w(k) = ˆw(F (k)) if (R(w(k−1)) −R(w(k)) ≤ϵ) break end Figure 1: Forward Greedy Algorithm f5 y f1 f2 f3 f4 Figure 2: Failure of Forward Greedy Algorithm [7]. In general, Figure 2 (which is the case we are interested in in this paper) shows that forward greedy algorithm will make errors that are not corrected later on. In order to remedy the problem, the so-called backward greedy algorithm has been widely used by practitioners. The idea is to train a full model with all the features, and greedily remove one feature (with the smallest increase of cost function) at a time. Although at the first sight, backward greedy method appears to be a reasonable idea that addresses the problem of forward greedy algorithm, it is computationally very costly because it starts with a full model with all features. Moreover, there are no theoretical results showing that this procedure is effective. In fact, under our setting, the method may only work when d ≪n (see, for example, [3]), which is not the case we are interested in. In the case d ≫n, during the first step, we start with a model with all features, which can immediately overfit the data with perfect prediction. In this case, the method has no ability to tell which feature is irrelevant and which feature is relevant because removing any feature still completely overfits the data. Therefore the method will completely fail when d ≫n, which explains why there is no theoretical result for this method. 3 Adaptive Forward-Backward Greedy Algorithm The main strength of forward greedy algorithm is that it always works with a sparse solution explicitly, and thus computationally efficient. Moreover, it does not significantly overfit the data due to the explicit sparsity. However, a major problem is its inability to correct any error made by the algorithm. On the other hand, backward greedy steps can potentially correct such an error, but need to start with a good model that does not completely overfit the data — it can only correct errors with a small amount of overfitting. Therefore a combination of the two can solve the fundamental flaws of both methods. However, a key design issue is how to implement a backward greedy strategy that is provably effective. Some heuristics exist in the literature, although without any effectiveness proof. For example, the standard heuristics, described in [5] and implemented in SAS, includes another threshold ϵ′ in addition to ϵ: a feature is deleted if the cost-function increase by performing the deletion is no more than ϵ′. Unfortunately we cannot provide an effectiveness proof for this heuristics: if the threshold ϵ′ is too small, then it cannot delete any spurious features introduced in the forward steps; if it is too large, then one cannot make progress because good features are also deleted. In practice it can be hard to pick a good ϵ′, and even the best choice may be ineffective. 3 This paper takes a more principled approach, where we specifically design a forward-backward greedy procedure with adaptive backward steps that are carried out automatically. The procedure has provably good performance and fixes the drawbacks of forward greedy algorithm illustrated in Figure 2. There are two main considerations in our approach: we want to take reasonably aggressive backward steps to remove any errors caused by earlier forward steps, and to avoid maintaining a large number of basis functions; we want to take backward step adaptively and make sure that any backward greedy step does not erase the gain made in the forward steps. Our algorithm, which we refer to as FoBa, is listed in Figure 3. It is designed to balance the above two aspects. Note that we only take a backward step when the increase of cost function is no more than half of the decrease of cost function in earlier forward steps. This implies that if we take ℓforward steps, then no matter how many backward steps are performed, the cost function is decreased by at least an amount of ℓϵ/2. It follows that if R(w) ≥0 for all w ∈Rd, then the algorithm terminates after no more than 2R(0)/ϵ steps. This means that the procedure is computationally efficient. Input: f1, . . . , fd, y ∈Rn and ϵ > 0 Output: F (k) and w(k) let F (0) = ∅and w(0) = 0 let k = 0 while true let k = k + 1 // forward step let i(k) = arg mini minα R(w(k−1) + αei) let F (k) = {i(k)} ∪F (k−1) let w(k) = ˆw(F (k)) let δ(k) = R(w(k−1)) −R(w(k)) if (δ(k) ≤ϵ) k = k −1 break endif // backward step (can be performed after each few forward steps) while true let j(k) = arg minj∈F (k) R(w(k) −w(k) j ej) let δ′ = R(w(k) −w(k) j(k)ej(k)) −R(w(k)) if (δ′ > 0.5δ(k)) break let k = k −1 let F (k) = F (k+1) −{j(k+1)} let w(k) = ˆw(F (k)) end end Figure 3: FoBa: Forward-Backward Greedy Algorithm Now, consider an application of FoBa to the example in Figure 2. Again, in the first three forward steps, we will be able to pick f3, followed by f1 and f2. After the third step, since we are able to express y using f1 and f2 only, by removing f3 in the backward step, we do not increase the cost. Therefore at this stage, we are able to successfully remove the incorrect basis f3 while keeping the good features f1 and f2. This simple illustration demonstrates the effectiveness of FoBa. In the following, we formally characterize this intuitive example, and prove the effectiveness of FoBa for feature selection as well as parameter estimation. Our analysis assumes the least squares loss. However, it is possible to handle more general loss functions with a more complicated derivation. We introduce the following definition, which characterizes how linearly independent small subsets of {fj} of size k are. For k ≪n, the number ρ(k) can be bounded away from zero even when d ≫n. For example, for random basis functions fj, we may take ln d = O(n/k) and still have ρ(k) to be bounded away from zero. This quantity is the smallest eigenvalue of the k × k diagonal blocks of the d × d design matrix [f T i fj]i,j=1,...,d, and has appeared in recent analysis of L1 regularization 4 methods such as in [2, 8], etc. We shall refer it to as the sparse eigenvalue condition. This condition is the least restrictive condition when compared to other conditions in the literature [1]. Definition 3.1 Define for all 1 ≤k ≤d: ρ(k) = inf 1 n∥f(w)∥2 2/∥w∥2 2 : ∥w∥0 ≤k . Assumption 3.1 Consider least squares loss R(w) = 1 n∥f(w(k)) −y∥2 2. Assume that the basis functions are normalized such that 1 n∥fj∥2 2 = 1 for all j = 1, . . . , d, and assume that {yi}i=1,...,n are independent (but not necessarily identically distributed) sub-Gaussians: there exists σ ≥0 such that ∀i and ∀t ∈R, Eyiet(yi−Eyi) ≤eσ2t2/2. Both Gaussian and bounded random variables are sub-Gaussian using the above definition. For example, we have the following Hoeffding’s inequality. If a random variable ξ ∈[a, b], then Eξet(ξ−Eξ) ≤e(b−a)2t2/8. If a random variable is Gaussian: ξ ∼N(0, σ2), then Eξetξ ≤eσ2t2/2. The following theorem is stated with an explicit ϵ for convenience. In applications, one can always run the algorithm with a smaller ϵ and use cross-validation to determine the optimal stopping point. Theorem 3.1 Consider the FoBa algorithm in Figure 3, where Assumption 3.1 holds. Assume also that the target is sparse: there exists ¯w ∈Rd such that ¯wT xi = Eyi for i = 1, . . . , n, and ¯F = supp( ¯w). Let ¯k = | ¯F|, and ϵ > 0 be the stopping criterion in Figure 3. Let s ≤d be an integer which either equals d or satisfies the condition 8¯k ≤sρ(s)2. If minj∈supp( ¯w) | ¯wj|2 ≥ 64 25ρ(s)−2ϵ, and for some η ∈(0, 1/3), ϵ ≥64ρ(s)−2σ2 ln(2d/η)/n, then with probability larger than 1 −3η, when the algorithm terminates, we have F (k) = ¯F and ∥w(k) −¯w∥2 ≤ σ p¯k/(nρ(¯k)) h 1 + p 20 ln(1/η) i . The result shows that one can identify the correct set of features ¯F as long as the weights ¯wj are not close to zero when j ∈¯F. This condition is necessary for all feature selection algorithms including previous analysis of Lasso. The theorem can be applied as long as eigenvalues of small s × s diagonal blocks of the design matrix [f T i fj]i,j=1,...,d are bounded away from zero (i.e., sparse eigenvalue condition). This is the situation under which the forward greedy step can make mistakes, but such mistakes can be corrected using FoBa. Because the conditions of the theorem do not prevent forward steps from making errors, the example described in Figure 2 indicates that it is not possible to prove a similar result for the forward greedy algorithm. The result we proved is also better than that of Lasso, which can successfully select features under irrepresentable conditions of [10]. It is known that the sparse eigenvalue condition considered here is generally weaker [8, 1]. Our result relies on the assumption that | ¯wj| (j ∈¯F) is larger than the noise level O(σ p ln d/n) in order to select features effectively. If any nonzero weight is below the noise level, then no algorithm can distinguish it from zero with large probability. That is, in this case, one cannot reliably perform feature selection due to the noise. Therefore FoBa is near optimal in term of its ability to perform reliable feature selection, except for the constant hiding in O(·). For target that is not truly sparse, similar results can be obtained. In this case, it is not possible to correctly identify all the features with large probability. However, we can show that FoBa can still select part of the features reliably, with good parameter estimation accuracy. Such results can be found in the full version of the paper, available from the author’s website. 4 Experiments We compare FoBa described in Section 3) to forward-greedy and L1-regularization on artificial and real data. They show that in practice, FoBa is closer to subset selection than the other two approaches, in the sense that FoBa achieves smaller training error given any sparsity level. In oder to compare with Lasso, we use the LARS [4] package in R, which generates a path of actions for adding and deleting features, along the L1 solution path. For example, a path of {1, 3, 5, −3, . . .} means that in the fist three steps, feature 1, 3, 5 are added; and the next step removes feature 3. Using such a solution path, we can compare Lasso to Forward-greedy and FoBa under the same framework. Similar to the Lasso path, FoBa also generates a path with both addition and deletion operations, while forward-greedy algorithm only adds features without deletion. 5 Our experiments compare the performance of the three algorithms using the corresponding feature addition/deletion paths. We are interested in features selected by the three algorithms at any sparsity level k, where k is the desired number of features presented in the final solution. Given a path, we can keep an active feature set by adding or deleting features along the path. For example, for path {1, 3, 5, −3}, we have two potential active feature sets of size k = 2: {1, 3} (after two steps) and {1, 5} (after four steps). We then define the k best features as the active feature set of size k with the smallest least squares error because this is the best approximation to subset selection (along the path generated by the algorithm). From the above discussion, we do not have to set ϵ explicitly in the FoBa procedure. Instead, we just generate a solution path which is five times as long as the maximum desired sparsity k, and then generate the best k features for any sparsity level using the above described procedure. 4.1 Simulation Data Since for real data, we do not know the true feature set ¯F, simulation is needed to compare feature selection performance. We generate n = 100 data points of dimension d = 500. The target vector ¯w is truly sparse with ¯k = 5 nonzero coefficients generated uniformly from 0 to 10. The noise level is σ2 = 0.1. The basis functions fj are randomly generated with moderate correlation: that is, some basis functions are correlated to the basis functions spanning the true target. Note that if there is no correlation (i.e., fj are independent random vectors), then both forward-greedy and L1-regularization work well because the basis functions are near orthogonal (this is the well-known case considered in the compressed sensing literature). Therefore in this experiment, we generate moderate correlation so that the performance of the three methods can be differentiated. Such moderate correlation does not violate the sparse eigenvalue condition in our analysis, but violates the more restrictive conditions for forward-greedy method and Lasso. FoBa Forward-greedy L1 least squares training error 0.093 ± 0.02 0.16 ± 0.089 0.25 ± 0.14 parameter estimation error 0.057 ± 0.2 0.52 ± 0.82 1.1 ± 1 feature selection error 0.76 ± 0.98 1.8 ± 1.1 3.2 ± 0.77 Table 1: Performance comparison on simulation data at sparsity level k = 5 Table 1 shows the performance of the three methods (including two versions of FoBa), where we repeat the experiments 50 times, and report the average ± standard-deviation. We use the three methods to select five best features, using the procedure described above. We report three metrics. Training error is the squared error of the least squares solution with the selected five features. Parameter estimation error is the 2-norm of the estimated parameter (with the five features) minus the true parameter. Feature selection error is the number of incorrectly selected features. It is clear from the table that for this data, FoBa achieves significantly smaller training error than the other two methods, which implies that it is closest to subset selection. Moreover, the parameter estimation performance and feature selection performance are also better. The two versions of FoBa perform very similarly for this data. 4.2 Real Data Instead of listing results for many datasets without gaining much insights, we present a more detailed study on a typical dataset, which reflect typical behaviors of the algorithms. Our study shows that FoBa does what it is designed to do well: that is, it gives a better approximation to subset selection than either forward-greedy or L1 regularization. Moreover, the difference between aggressive FoBa and conservative FoBa become more significant. In this study, we use the standard Boston Housing data, which is the housing data for 506 census tracts of Boston from the 1970 census, available from the UCI Machine Learning Database Repository: http://archive.ics.uci.edu/ml/. Each census tract is a data-point, with 13 features (we add a constant offset one as the 14th feature), and the desired output is the housing price. In the experiment, we randomly partition the data into 50 training plus 456 test points. We perform the experiments 50 times, and for each sparsity level from 1 to 10, we report the average training and test squared error. The results are plotted in Figure 4. From the results, we can see that FoBa achieves 6 better training error for any given sparsity, which is consistent with the theory and the design goal of FoBa. Moreover, it achieves better test accuracy with small sparsity level (corresponding to a more sparse solution). With large sparsity level (corresponding to a less sparse solution), the test error increase more quickly with FoBa. This is because it searches a larger space by more aggressively mimic subset selection, which makes it more prone to overfitting. However, at the best sparsity level of 2 or 3 (for aggressive and conservative FoBa, respectively), FoBa achieves significantly better test error. Moreover, we can observe with small sparsity level (a more sparse solution), L1 regularization performs poorly, due to the bias caused by using a large L1-penalty. G G G G G G G G G G 2 4 6 8 10 20 30 40 50 60 sparsity training error G FoBa forward−greedy L1 G G G G G G G G G G 2 4 6 8 10 35 40 45 50 55 60 65 70 sparsity test error G FoBa forward−greedy L1 Figure 4: Performance of the algorithms on Boston Housing data Left: average training squared error versus sparsity; Right: average test squared error versus sparsity For completeness, we also compare FoBa to the backward-greedy algorithm and the classical heuristic forward-backward greedy algorithm as implemented in SAS (see its description at the beginning of Section 3). We still use the Boston Housing data, but plot the results separately, in order to avoid cluttering. As we have pointed out, there is no theory for the SAS version of forward-backward greedy algorithm. It is difficult to select an appropriate backward threshold ϵ′: a too small value leads to few backward steps, and a too large value leads to overly aggressive deletion, and the procedure terminates very early. In this experiment, we pick a value of 10, because it is a reasonably large quantity that does not lead to an extremely quick termination of the procedure. The performance of the algorithms are reported in Figure 5. From the results, we can see that backward greedy algorithm performs reasonably well on this problem. Note that for this data, d ≪n, which is the scenario that backward does not start with a completely overfitted full model. Still, it is inferior to FoBa at small sparsity level, which means that some degree of overfitting still occurs. Note that backward-greedy algorithm cannot be applied in our simulation data experiment, because d ≫n which causes immediate overfitting. From the graph, we also see that FoBa is more effective than the SAS implementation of forward-backward greedy algorithm. The latter does not perform significant better than the forward-greedy algorithm with our choice of ϵ′. Unfortunately, using a larger backward threshold ϵ′ will lead to an undesirable early termination of the algorithm. This is why the provably effective adaptive backward strategies introduced in this paper are superior. 5 Discussion This paper investigates the problem of learning sparse representations using greedy algorithms. We showed that neither forward greedy nor backward greedy algorithms are adequate by themselves. However, through a novel combination of the two ideas, we showed that an adaptive forward-back greedy algorithm, referred to as FoBa, can effectively solve the problem under reasonable conditions. FoBa is designed to be a better approximation to subset selection. Under the sparse eigenvalue condition, we obtained strong performance bounds for FoBa for feature selection and parameter estimation. In fact, to the author’s knowledge, in terms of sparsity, the bounds developed for FoBa in this paper are superior to all earlier results in the literature for other methods. 7 G G G G G G G G G G 2 4 6 8 10 20 30 40 50 sparsity training error G G G G G G G G G G G G FoBa Forward−Backward (SAS) forward−greedy backward−greedy G G G G G G G G G G 2 4 6 8 10 40 50 60 70 sparsity test error G G G G G G G G G G G G FoBa Forward−Backward (SAS) forward−greedy backward−greedy Figure 5: Performance of greedy algorithms on Boston Housing data. Left: average training squared error versus sparsity; Right: average test squared error versus sparsity Our experiments also showed that FoBa achieves its design goal: that is, it gives smaller training error than either forward-greedy or L1 regularization for any given level of sparsity. Therefore the experiments are consistent with our theory. In real data, better sparsity helps on some data such as Boston Housing. However, we shall point out that while FoBa always achieves better training error for a given sparsity in our experiments on other datasets (thus it achieves our design goal), L1regularization some times achieves better test performance. This is not surprising because sparsity is not always the best complexity measure for all problems. In particular, the prior knowledge of using small weights, which is encoded in the L1 regularization formulation but not in greedy algorithms, can lead to better generalization performance on some data (when such a prior is appropriate). References [1] Peter Bickel, Yaacov Ritov, and Alexandre Tsybakov. Simultaneous analysis of Lasso and Dantzig selector. Annals of Statistics, 2008. to appear. [2] Florentina Bunea, Alexandre Tsybakov, and Marten H. Wegkamp. Sparsity oracle inequalities for the Lasso. Electronic Journal of Statistics, 1:169–194, 2007. [3] Christophe Couvreur and Yoram Bresler. On the optimality of the backward greedy algorithm for the subset selection problem. SIAM J. Matrix Anal. Appl., 21(3):797–808, 2000. [4] Bradley Efron, Trevor Hastie, Iain Johnstone, and Robert Tibshirani. Least angle regression. Annals of Statistics, 32(2):407–499, 2004. [5] T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning. Springer, 2001. [6] Vladimir Koltchinskii. Sparsity in penalized empirical risk minimization. Annales de l’Institut Henri Poincaré, 2008. [7] Joel A. Tropp. Greed is good: Algorithmic results for sparse approximation. IEEE Trans. Info. Theory, 50(10):2231–2242, 2004. [8] Cun-Hui Zhang and Jian Huang. Model-selection consistency of the Lasso in high-dimensional linear regression. Technical report, Rutgers University, 2006. [9] Tong Zhang. Some sharp performance bounds for least squares regression with L1 regularization. The Annals of Statistics, 2009. to appear. [10] Peng Zhao and Bin Yu. On model selection consistency of Lasso. 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Reconciling Real Scores with Binary Comparisons: A Unified Logistic Model for Ranking Nir Ailon Google Research NY 111 8th Ave, 4th FL New York NY 10011 nailon@gmail.com Abstract The problem of ranking arises ubiquitously in almost every aspect of life, and in particular in Machine Learning/Information Retrieval. A statistical model for ranking predicts how humans rank subsets V of some universe U. In this work we define a statistical model for ranking that satisfies certain desirable properties. The model automatically gives rise to a logistic regression based approach to learning how to rank, for which the score and comparison based approaches are dual views. This offers a new generative approach to ranking which can be used for IR. There are two main contexts for this work. The first is the theory of econometrics and study of statistical models explaining human choice of alternatives. In this context, we will compare our model with other well known models. The second context is the problem of ranking in machine learning, usually arising in the context of information retrieval. Here, much work has been done in the discriminative setting, where different heuristics are used to define ranking risk functions. Our model is built rigorously and axiomatically based on very simple desirable properties defined locally for comparisons, and automatically implies the existence of a global score function serving as a natural model parameter which can be efficiently fitted to pairwise comparison judgment data by solving a convex optimization problem. 1 Introduction Ranking is an important task in information sciences. The most notable application is information retrieval (IR), where it is crucial to return results in a sorted order for the querier. The subject of preference and ranking has been thoroughly studied in the context of statistics and econometric theory [8, 7, 29, 36, 34, 31], combinatorial optimization [26, 37, 20, 3, 4, 14] and machine learning [6, 9, 33, 21, 19, 35, 23, 22, 25, 16, 17, 1, 13, 15, 28, 18]. Recently Ailon and Mehryar [5] following Balcan et al [9] have made significant progress in reducing the task of learning ranking to the binary classification problem of learning preferences. This comparison based approach is in contrast with a score based approach which tries to regress to a score function on the elements we wish to rank, and sort the elements based on this score as a final step. The difference between the score based and comparison approaches is an example of ”local vs. global” views: A comparison is local (how do two elements compare with each other), and a score is global (how do we embed the universe on a scale). The score based approach seems reasonable in cases where the score can be defined naturally in terms of measurable utility. In some real world scenarios, either (i) an interpretable score is difficult to define (e.g. a relevance score in information retrieval) and (ii) an interpretable score is easy to define (e.g. how much a random person is willing to pay for product X in some population) but learning the score is difficult due to noisy or costly label acquisition for scores on individual points [7]. A well known phenomenon in the psychological study of human choice seems to potentially offer an elegant solution to the above difficulties: Human response to comparison questions is more stable in the sense that it is not easily affected by irrelevant alternatives. This phenomenon makes acquisition of comparison labels for learning tasks more appealing, but raises the question of how to go back and fit a latent score function that explains the comparisons. Moreover, the score parameter fitting must be computationally efficient. Much effort has been recently put in this subject from a machine learning perspective [6, 9, 33, 21, 19, 35, 23, 22, 25, 16, 17, 1, 13, 15, 28, 18]. 2 Ranking in Context The study of ranking alternatives has not been introduced by ML/IR, and has been studied throughly from the early years of the 20th century in the context of statistics and econometrics. We mention work in ML/IR by Lebanon and Lafferty [27] and Cao et al. [12] who also draw from the classic work for information retrieval purposes. ML/IR is usually interested in the question of how a machine should correctly rank alternatives based on experience from human feedback, whereas in statistics and econometrics the focus is on the question of how a human chooses from alternatives (for the purpose of e.g. effective marketing or policy making). Therefore, there are notable differences between the modern and classic foci. Notwithstanding these differences, the classic foci is relevant to modern applications, and vice versa. For example, any attempt to correctly choose from a set (predominantly asked in the classic context) can be converted into a ranking algorithm by repeatedly choosing and removing from the set. Definition 2.1 A ranking model for U is a function D mapping any finite subset V ⊆U to a distribution on rankings of V . In other words, D(V ) is a probability distribution on the |V |! possible orderings of V . A Thurstonian model for ranking (so named after L. Thurstone [36]) is one in which an independent random real valued variable Zv is associated with each v ∈V , and the ranking is obtained by sorting the elements if V in decreasing order (assuming the value represents utility). Often the distributions governing the Zv’s are members of a parametric family, with a location parameter representing an intrinsic ”value”. The source of variability in Zv is beyond the scope of this work. This model is related to the more general random utility model (RUM) approach studied in econometrics. A purely comparison based model is due to Babington and Smith: The parameter of the model is a matrix {puv}u,v∈U. Given items u, v, a subject would prefer u over v with probability puv = 1−pvu. Given a subset V , the subject flips a corresponding biased coin independently to decide on the preference of all pairs u, v ∈V , and repeats the process until the set of preferences is transitive. This model is unwieldy in full generality, and more succinct representations were proposed. Mallows [30] following Bradley and Terry [11] proposed to take puv as α(u)/(α(u) + α(v)), where the α(v)’s are constants attached to each element. Note that the marginal probability of u being preferred over v in the context of a set V ⊃{u, v} in the Babington-Smith model is in general not puv, even in Mallows’s special case. In distance based models it is assumed that there is a ”modal” ranking of the set V , and the probability of any ranking decreases with its distance from the mode. Several definitions of distances between permutations. Often the probability density itself is defined as an exponential model. We refer the reader to [31] for in depth analysis of such models. The Plackett-Luce model. The classic model most related to this work is Plackett and Luce’s [29, 34] multistage model for ranking. Each element v ∈U has an assigned ”value” parameter α(v). At each stage a choice is made. Given a set V , item u ∈V wins with probability α(u)/ P v∈V α(v).1 The winner is removed from V and the process is repeated for the remaining elements, until a ranking is obtained. Yellott [38] made the surprising observation that the Luce-Plackett model is exactly Thurstone’s model where the Zu’s are translated Gumbel (doubly-exponential) distributed 1This choice function is known as the multinomial logit (MNL) and is equivalent to the standard (dichotomous) logit when only two alternatives are available. variables. The underlying winner choice model satisfies Luce’s choice axiom [29] which, roughly speaking, stipulates that the probability of an element u winning in V is the same as the product of the probability of the winner contained in V ′ ⊆V and the probability of u winning in V ′. It turns out that this axiom (often used as criticism of the model) implies the underlying choice function of the Plackett-Luce model. An interesting property of Plackett-Luce for our purpose is that it is asymmetric in the sense that it is winner-centric and not loser-centric. The model cannot explain both ranking by successive loser choice and successive winner choice simultaneously unless it is trivial (this point was noticed by McCullagh [32]). It is clear however that breaking down the process of ranking by humans to an iterated choice of winners ignores the process of elimination (placing alternatives at the bottom of the list). In the following sections we propose a new symmetric model for ranking, in which the basic discrete task is a comparison of pairs of elements, and not choice of an element from arbitrarily large sets (as in Plackett-Luce). 3 An Axiomatic Approach for Defining a Pairwise-Stable Model for Ranking For a ranking π of some subset V ⊆U, we use the notation u ≺π v to denote that u precedes2 v according to π. We let π(v) ∈{1, . . . , n} denote the rank of v ∈V , where lower numbers designate precedence (hence u ≺π v if π(u) < π(v)). The inverse π−1(i) is the unique element v of V with π(v) = i. We overload notation and let π(u, v) denote the indicator variable taking the value of 1 if u ≺v and 0 otherwise. Definition 3.1 A ranking model D for U satisfies pairwise stability if for any u, v ∈U and for any V1, V2 ⊇{u, v}, Prπ∼D(V1)[u ≺π v] = Prπ∼D(V2)[u ≺v]. Pairwise stability means that the preference (or comparison) of u, v is statistically independent of the context (subset) they are ranked in. Note that Plackett-Luce is pairwise stable (this follows from the fact that the model is Thurstonian) but Babington-Smith/Mallows is not. If a ranking model D satisfies pairwise stability, then the probability PrD[u ≺v] is naturally defined and equals Prπ∼D(V )[u ≺π v] for any V ⊇{u, v}. Pairwise stability is a weak property which permits a very wide family of ranking distributions. In particular, if the universe U is a finite set then any distribution Π on rankings on the entire universe U gives rise to a model DΠ with DΠ(V ) defined as the restriction of Π to V . This model clearly satisfies pairwise stability but does not have a succint description and hence undesirable. We strengthen the conditions on our model by considering triplets of elements. Assume that a model D satisfies pairwise stability. Fix three elements u, v, w. Consider a process in which we randomly and independently decide how u and w should compare with v. What would be the induced distribution on the order of u and w, conditioned on them being placed on opposite sides of v? If we sample from the distributions D({u, v}) and D({v, w}) to independently decide how to compare u with v and w with v (respectively), then we get Pr[u ≺w |( u ≺v ≺w) ∨(w ≺v ≺u)] = PrD[u ≺v] PrD[v ≺w] PrD[u ≺v] PrD[v ≺w] + PrD[w ≺v] PrD[v ≺u] . What happens if we force this to equal PrD[u ≺w]? In words, this would mean that the comparison of u with w conditioned on the comparison being determined by pivoting around v is distributed like D({u, w}). We write this desired property as follows (the second line follows from the first): 2We choose in this work to use the convention that an element u precedes v if u is in a more favorable position. When a score function is introduced later, the convention will be that higher scores correspond to more favorable positions. We will use the symbol < (resp. >) to compare scores, which is semantically opposite to ≺(resp. ≻) by our convention. Pr D [u ≺w] = PrD[u ≺v] PrD[v ≺w] PrD[u ≺v] PrD[v ≺w] + PrD[w ≺v] PrD[v ≺u] Pr D [w ≺u] = PrD[w ≺v] PrD[v ≺u] PrD[w ≺v] PrD[v ≺u] + PrD[u ≺v] PrD[v ≺w] . (1) Definition 3.2 Assume D is a ranking model for U satisfying pairwise stability. For a pair u, w ∈U and another element v ∈U we say that u and w satisfy the pivot condition with respect to v if (1) holds. Dividing the two desired equalities in (1), we get (assuming the ratio exists): PrD[u ≺w] PrD[w ≺u] = PrD[u ≺v] PrD[v ≺w] PrD[w ≺v] PrD[v ≺u] . (2) If we denote by ∆D(a, b) the ”comparison logit3”: ∆D(a, b) = log(PrD[a ≺b]/ PrD[b ≺a]) , then (2) implies ∆D(u, v) + ∆D(v, w) + ∆D(w, u) = 0 . This in turn implies that there exist numbers s1, s2, s3 such that ∆(u, v) = s1 −s2, ∆(v, w) = s2 −s3 and ∆(w, u) = s1 −s3. These numbers, defined up to any additive constant, should be called (additive) scores. We will see in what follows that the score function can be extended to a larger set by patching scores on triplets. By the symmetry it is now clear that the pivoting condition of u and w with respect to v implies the pivoting condition of u and v with respect to w and of v and w with respect to u. In other words, the pivoting condition is a property of the triplet {u, v, w}. Definition 3.3 Assume a ranking model D for U satisfies pairwise stability, and let ∆D : U × U → R denote the comparison logit as defined above. A triplet {u, v, w} ⊆U is said to satisfy the pivot condition in D if ∆D(u, v)+∆D(v, w)+∆D(w, u) = 0 . We say that U satisfies the pivot condition in D if {u, v, w} satisfies the pivot condition for all {u, v, w} ⊆U. Lemma 3.1 If U satisfies the pivot condition in a pairwise stability model D for U, then there exists a real valued score function s : V →R such that for all a, b ∈V , ∆D(a, b) = s(a) −s(b) . Proof Fix some element v ∈U and set s(v) = 0. For every other element u ∈V \ {v} set s(v) = ∆D(v, u). It is now immediate to verify that for all a, b ⊆V one has ∆D(a, b) = s(a)−s(b). Indeed, by construction s(a) −s(b) = ∆D(a, u) −∆D(b, u) but by the pivot property this equals exactly ∆D(a, b), as required (remember that ∆D(a, b) = −∆D(a, b) by definition of ∆D). By starting with local assumptions (pairwise stability and the pivoting property), we obtained a natural global score function s on the universe of elements. The score function governs the probability of u preceding v via the difference s(u) −s(v) passed through the inverse logit. Note that we used the assumption that the comparison logit is finite on all u, v (equivalently, that 0 < PrD(u ≺v) < 1 for all u, v), but this assumption can be dropped if we allow the score function to obtain values in R + ωZ, where ω is the limit ordinal of R. The Plackett-Luce model satisfies both pairwise stability and the pivot condition with s(u) = log α(u). Hence our definitions are non empty. Inspired by recent work on the QuickSort algorithm [24] as a random process [4, 3, 5, 37], we define a new symmetric model based on a series of comparisons rather than choices from sets. 4 The New Ranking Model We define a model called QSs (short for QuickSort), parametrized by a score function s : U 7→R as follows. Given a finite subset V ⊂U: 1. Pick a ”pivot element” v uniformly at random from V . 3The ”logit of p” is standard shorthand for the log-odds, or log(p/(1 −p)). 2. For all u ∈V \ {v}, place u to the left of v with probability 1/(1 + es(v)−s(u)), and to the right with the remaining probability 1/(1 + es(u)−s(v)), independently of all other choices. 3. Recurse on the left and on the right sides, and output the ranking of V obtained by joining the results in an obvious way (left ≺pivot ≺right). (The function 1/(1 + e−x) is the inverse logit function.) We shall soon see that QuickSort gives us back all the desired statistical local properties of a ranking models. That the model QSs can be sampled efficiently is a simple consequence of the fact that QuickSort runs in expected time O(n log n) (some attention needs to be paid the fact that unlike in the textbook proofs for QuickSort the pivoting process is randomized, but this is not difficult [5]). Theorem 4.1 The ranking model QSs for U satisfies both pairwise stability and the pivoting condition. Additionally, for any subset V ⊆U the mode of QSs(V ) is any ranking π∗satisfying u ≺π∗v whenever s(u) > s(v). Proof (of Theorem 4.1): First we note that if QSs satisfies pairwise stability, then the pivot property will be implied as well. Indeed, by taking V = {u, v} we would get from the model that PrQSs(u ≺v) = 1/(1 + es(v)−s(u)), immediately implying the pivot property. To see that QSs satisfies pairwise stability, we show that for any u, v and V ⊇{u, v}, the probability of the event u ≺π v is exactly 1/(1 + es(v)−s(u)), where π ∼QSs(V ). Indeed, the order of u, v can be determined in one of two ways. (i) Directly: u or v are chosen as pivot when the other is present in the same recursive call. We call this event E{u,v}. Conditioned on this event, clearly the probability that u ≺π v is exactly the required probability 1/(1+es(v)−s(u)) by step 2 of QuickSort (note that it doesn’t matter which one of v or u is the pivot). (ii) Indirectly: A third element w ∈V is the pivot when both u and v are present in the recursive call, and w sends u and v to opposite recursion sides. We denote this event by E′ {u,v},w. Conditioned on this event, the probability that u ≺π v, is exactly as required (by using the same logit calculus we used in Section 3). To conclude the proof of pairwise stability, it remains to observe that the collection of events {E{u,v}} ∪ n E′ {u,v},w : w ∈V \ {u, v} o is a pairwise disjoint cover of the probability space. This implies that Prπ∼QSs(V )(u ≺π v) is the desired quantity 1/(1 + es(v)−s(u)), concluding the proof of pairwise stability. We need to work harder to prove the intuitive mode argument. Let τ, σ be two permutations on V such that a1 ≺τ a2 ≺τ · · · ≺τ ak ≺τ u ≺τ v ≺τ ak+1 ≺τ · · · ≺τ an−2 a1 ≺a2 ≺σ · · · ≺σ ak ≺σ v ≺σ u ≺σ ak+1 ≺σ · · · ≺σ an−2 , where V = {u, v}∪{a1, . . . , an−2}. In words, τ and σ differ on the order of exactly two consecutive elements u, v. Assume that s(u) > s(v) (so τ, placing u in a more favorable position than v, is intuitively more ”correct”). We will prove that the probability of getting τ is strictly higher than the probability of getting σ from QSs. Since π∗, the permutation sorting by s, can be obtained from any permutation by a sequence of swapping incorrectly ordered (according to s) adjacent pairs, this would prove the theorem by a standard inductive argument. Let qτ = Prπ∼QS[π = τ], and similarly define qσ. To prove that qτ > qσ we need extra notation. Our QuickSort generative model gives rise to a random integer node-labeled ordered binary tree4 implicitly constructed as an execution side effect. This tree records the final position of the pivots chosen in each step as follows: The label L of the root of the tree is the rank of the pivot in the final solution (which equals the size of the left recursion plus 1). The left subtree is the tree recursively constructed on the left, and the right subtree is the tree recursively constructed on the right with L added to the labels of all the vertices. Clearly the resulting tree has exactly n nodes with each label in {1 . . . n} appearing exactly once. Let pπ,T denote the probability that QuickSort outputs a permutation π and (implicitly) constructs a pivot selection tree T. Let T denote the collection of all ordered labeled binary trees with node labels in {1, . . . , n}. For T ∈T and a node x ∈T let ℓ(x) denote the integer label on x. Let Tx denote the subtree rooted by x and let ℓ(Tx) denote the 4By that we mean a tree in which each node has at most one left child node and at most one right child node, and the nodes are labeled with integers. collection of labels on those nodes. By construction, if QuickSort outputted a ranking π with an (implicitly constructed) tree T, then at some point the recursive call to QuickSort took π−1(ℓ(Tx)) as input and chose π−1(ℓ(x)) as pivot, for any node x of T. By a standard probability argument (summing over a disjoint cover of events): qτ = P T ∈T qτ,T and qσ = P T ∈T qσ,T . It suffices to show now that for any fixed T ∈T , qτ,T > qσ,T . To compute qπ,T for π = τ, σ we proceed as follows: At each node x of T we will attach a number Pπ(x) which is the likelihood of the decisions made at that level, namely, the choice of the pivot itself and the separation of the rest of the elements to its right and left. Pπ(x) = 1 |Tx| Y y∈TL(x) Pr QS[π−1(ℓ(y)) ≺π−1(ℓ(x))] × Y y∈TR(x) Pr QS[π−1(ℓ(x)) ≺π−1(ℓ(y))] , Where |Tx| is the number of nodes in Tx, TR(x) is the set of vertices in the left subtree of x and similarly for TL(x). The factor 1/|Tx| comes from the likelihood of uniformly at random having chosen the pivot π−1(ℓ(x)) from the set of nodes of Tx. The first product corresponds to the random comparison decisions made on the elements thrown to the left, and the second to right. By construction, pτ,T = Q x∈T Pτ(x) and similarly pσ,T = Q x∈T Pσ(x). Since u, v are adjacent in both τ and σ, it is clear that the two nodes x1, x2 ∈T labeled τ(u) and τ(v) respectively have an ancestor-descendent relation in T (otherwise their least common ancestor in T would have been placed between them, violating the consecutiveness of u and v in our construction and implying pτ,T = qτ,T = 0). Also recall that σ(u) = τ(v) and σ(v) = τ(u). By our assumption that τ and σ differ only on the order of the adjacent elements u, v, Pτ(x) and Pσ(x) could differ only on nodes x on the path between x1 and x2. Assume w.l.o.g. that x1 is an ancestor of x2, and that x2 is a node in the left subtree of x1. By our construction, x2 is the rightmost node5 in TL(x1). Let Y denote the set of nodes on the path from x1 to x2 (exclusive) in T. Let W denote the set of nodes in the left (and only) subtree of x2, and let Z denote the set of remaining nodes in TL(x1): Z = TL(x1) \ (W ∪Y ∪{x2}). Since τ −1(ℓ(z)) = σ−1(ℓ(z)) for all z ∈Z we can define elt(z) = τ −1(ℓ(z)) = σ−1(ℓ(z)) and similarly we can correspond each y ∈Y with a single element elt(y) and each w ∈W with a single elements elt(w) of V . As claimed above, we only need to compare between Pτ(x1) and Pσ(x1), between Pτ(x2) and Pσ(x2) and Pτ(y) and Pσ(y) for y ∈Y . Carefully unfolding these products node by node, we see that it suffices to notice that for all y ∈Y , the probability of throwing elt(y) to the left of u (pivoting on u) times the probability of throwing v to the right of elt(y) (pivoting on elt(y)) as appears inside the product Pσ(x1)Pσ(y) is exactly the probability of throwing elt(y) to the left of v (pivoting on v) times the probability of throwing u to the right of elt(y) (pivoting on elt(y)) as appears inside the product Pτ(x1)Pτ(y). Also for all w ∈W the probability of throwing elt(w) to the left of u (pivoting on u) times the probability of throwing elt(w) to the left of v (pivoting on v) appears exactly once in both Pτ(x1)Pτ(x2) and Pσ(x1)Pσ(x2) (though in reversed order). Following these observations one can be convinced by the desired result of the theorem by noting that in virtue of s(u) > s(v): (i) PrQS[v ≺u] > PrQS[u ≺v], and (ii) for all z ∈Z, PrQS[elt(z) ≺u] > PrQS[elt(z) ≺v]. 5 Comparison of Models The stochastic QuickSort model as just defined as well as Plackett-Luce share much in common, but they are not identical for strictly more than 2 elements. Both satisfy the intuitive property that the mode of the distribution corresponding to a set V is any ranking which sorts the elements of V in decreasing s(v) = log α(v) value. The stochastic QuickSort model, however, does not suffer from the asymmetry problem which is often stated as a criticism of Plackett-Luce. Indeed, the distributions QSs(V ) has the following property: If we draw from QSs(V ) and flip the resulting permutation, the resulting distribution is QS−s(V ). This property does not hold in general for Plackett-Luce, and hence serves as proof of their nonequivalence. Assume we want to fit s in the MLE sense by drawing random permutations from QSs(V ). This seems to be difficult due to the unknown choice of pivot. On the other hand, the log-likelihood function corresponding to Plackett-Luce is globally concave in the values of the function s on V , and hence a global maximum can be efficiently found. This also holds true in a generalized linear model, in which s(v) is given as the dot product of a feature vector φ(v) with an unknown weight 5The rightmost node of T is the root if it has no right descendent, or the rightmost node of its right subtree. vector which we estimate (as done in [10] in the context of predicting demand for electric cars). Hence, for the purpose of learning given full permutations of strictly more than two elements, the Plackett-Luce model is easier to work with. In practical IR settings, however, it is rare that training data is obtained as full permutations: such a task is tiresome. In most applications, the observables used for training are in the form of binary response vectors (either relevant or irrelevant for each alternative) or comparison of pairs of alternatives (either A better or B better given A,B). For the latter, Plackett-Luce is identical to QuickSort, and hence efficient fitting of parameters is easy (using logistic regression). As for the former, the process of generating a binary response vector can be viewed as the task performed at a single QuickSort recursive level. It turns out that by defining a nuisance parameter to represent the value s of an unknown pivot, MLE estimation can be performed efficiently and exactly [2]. References [1] Shivani Agarwal and Partha Niyogi. Stability and generalization of bipartite ranking algorithms. In COLT, pages 32–47, 2005. [2] N. Ailon. A simple linear ranking algorithm using query dependent intercept variables. arXiv:0810.2764v1. [3] Nir Ailon. Aggregation of partial rankings, p-ratings and top-m lists. 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Theory of matching pursuit Zakria Hussain and John Shawe-Taylor Department of Computer Science University College London, UK {z.hussain,j.shawe-taylor}@cs.ucl.ac.uk Abstract We analyse matching pursuit for kernel principal components analysis (KPCA) by proving that the sparse subspace it produces is a sample compression scheme. We show that this bound is tighter than the KPCA bound of Shawe-Taylor et al [7] and highly predictive of the size of the subspace needed to capture most of the variance in the data. We analyse a second matching pursuit algorithm called kernel matching pursuit (KMP) which does not correspond to a sample compression scheme. However, we give a novel bound that views the choice of subspace of the KMP algorithm as a compression scheme and hence provide a VC bound to upper bound its future loss. Finally we describe how the same bound can be applied to other matching pursuit related algorithms. 1 Introduction Matching pursuit refers to a family of algorithms that generate a set of bases for learning in a greedy fashion. A good example of this approach is the matching pursuit algorithm [4]. Viewed from this angle sparse kernel principal components analysis (PCA) looks for a small number of kernel basis vectors in order to maximise the Rayleigh quotient. The algorithm was proposed by [8]1 and motivated by matching pursuit [4], but to our knowledge sparse PCA has not been analysed theoretically. In this paper we show that sparse PCA (KPCA) is a sample compression scheme and can be bounded using the size of the compression set [3, 2] which is the set of training examples used in the construction of the KPCA subspace. We also derive a more general framework for this algorithm that uses the principle “maximise Rayleigh quotient and deflate”. A related algorithm called kernel matching pursuit (KMP) [10] is a sparse version of least squares regression but without the property of being a compression scheme. However, we use the number of basis vectors constructed by KMP to help upper bound the loss of the KMP algorithm using the VC dimension. This bound is novel in that it is applied in an empirically chosen low dimensional hypothesis space and applies independently of the actual dimension of the ambient feature space (including one constructed from the Gaussian kernel). In both cases we illustrate the use of the bounds on real and/or simulated data. Finally we also show that the KMP bound can be applied to a sparse kernel canonical correlation analysis that uses a similar matching pursuit technique. We do not describe the algorithm here due to space constraints and only concentrate on theoretical results. We begin with preliminary definitions. 2 Preliminary definitions Throughout the paper we consider learning from samples of data. For the regression section the data is a sample S = {(xi, yi)}m i=1 of input-output pairs drawn from a joint space X × Y where x ∈Rn and y ∈R. For the principal components analysis the data is a sample S = {xi}m i=1 of 1The algorithm was proposed as a low rank kernel approximation – however the algorithm turns out to be a sparse kernel PCA (to be shown). 1 multivariate examples drawn from a space X. For simplicity we always assume that the examples are already projected into the kernel defined feature space, so that the kernel matrix K has entries K[i, j] = ⟨xi, xj⟩. The notation K[i, :] and K[:, i] will denote the ith row and ith column of the matrix K, respectively. When using a set of indices i = {i1, . . . , ik} (say) then K[i, i] denotes the square matrix defined solely by the index set i. The transpose of a matrix X or vector x is denoted by X′ or x′ respectively. The input data matrix X will contain examples as row vectors. For analysis purposes we assume that the training examples are generated i.i.d. according to an unknown but fixed probability distribution that also governs the generation of the test data. Expectation over the training examples (empirical average) is denoted by ˆE[·], while expectation with respect to the underlying distribution is denoted E[·]. For the sample compression analysis the compression function Λ induced by a sample compression learning algorithm A on training set S is the map Λ : S 7−→Λ(S) such that the compression set Λ(S) ⊂S is returned by A. A reconstruction function Φ is a mapping from a compression set Λ(S) to a set F of functions Φ : Λ(S) 7−→F. Let A(S) be the function output by learning algorithm A on training set S. Therefore, a sample compression scheme is a reconstruction function Φ mapping a compression set Λ(S) to some set of functions F such that A(S) = Φ(Λ(S)). If F is the set of Boolean-valued or Real-valued functions then the sample compression scheme is said to be a classification or regression algorithm, respectively. 2.1 Sparse kernel principal components analysis Principal components analysis [6] can be expressed as the following maximisation problem: max w w′X′Xw w′w , (1) where w is the weight vector. In a sparse KPCA algorithm we would like to find a sparsely represented vector w = X[i, :]′ ˜α, that is a linear combination of a small number of training examples indexed by vector i. Therefore making this substitution into Equation (1) we have the following sparse dual PCA maximisation problem, max ˜α ˜α′X[i, :]X′XX[i, :]′ ˜α ˜α′X[i, :]X[i, :]′ ˜α , which is equivalent to sparse kernel PCA (SKPCA) with sparse kernel matrix K[:, i]′ = X[i, :]X′, max ˜α ˜α′K[:, i]′K[:, i]˜α ˜α′K[i, i]˜α , where ˜α is a sparse vector of length k = |i|. Clearly maximising the quantity above will lead to the maximisation of the generalised eigenvalues corresponding to ˜α – and hence a sparse subset of the original KPCA problem. We would like to find the optimal set of indices i. We proceed in a greedy manner (matching pursuit) in much the same way as [8]. The procedure involves choosing basis vectors that maximise the Rayleigh quotient without the set of eigenvectors. Choosing basis vectors iteratively until some pre-specified number of k vectors are chosen. An orthogonalisation of the kernel matrix at each step ensures future potential basis vectors will be orthogonal to those already chosen. The quotient to maximise is: max ρi = e′ iK2ei e′ iKei , (2) where ei is the ith unit vector. After this maximisation we need to orthogonalise (deflate) the kernel matrix to create a projection into the space orthogonal to the basis vectors chosen to ensure we find the maximum variance of the data in the projected space. The deflation step can be carried out as follows. Let τ = K[:, i] = XX′ei where ei is the ith unit vector. We know that primal PCA deflation can be carried out with respect to the features in the following way: ˆX′ = I −uu′ u′u X′, where u is the projection directions defined by the chosen eigenvector and ˆX is the deflated matrix. However, in sparse KPCA, u = X′ei because the projection directions are simply the examples in X. Therefore, for sparse KPCA we have: ˆX ˆX′ = X I −uu′ u′u I −uu′ u′u X′ = XX′ −XX′eie′ iXX′ e′ iXX′ei = K −K[:, i]K[:, i]′ K[i, i] . 2 Therefore, given a kernel matrix K the deflated kernel matrix ˆK can be computed as follows: ˆK = K − ττ ′ K[ik, ik] (3) where τ = K[:, ik] and ik denotes the latest element in the vector i. The algorithm is presented below in Algorithm 1 and we use the notation K.2 to denote component wise squaring. Also, division of vectors are assumed to be component wise. Algorithm 1: A matching pursuit algorithm for kernel principal components analysis (i.e., sparse KPCA) Input: Kernel K, sparsity parameter k > 0. 1: initialise i = [ ] 2: for j = 1 to k do 3: Set ij to index of max n (K.2)′1 diag{K} o 4: set τ = K[:, ij] to deflate kernel matrix like so: K = K − ττ ′ K[ij,ij] 5: end for 6: Compute ˜K using i and Equation (5) Output: Output sparse matrix approximation ˜K This algorithm is presented in Algorithm 1 and is equivalent to the algorithm proposed by [8]. However, their motivation comes from the stance of finding a low rank matrix approximation of the kernel matrix. They proceed by looking for an approximation ˜K = K[:, i]T for a set i such that the Frobenius norm between the trace residuals tr{K −K[:, i]T} = tr{K −˜K} is minimal. Their algorithm finds the set of indices i and the projection matrix T. However, the use of T in computing the low rank matrix approximation seems to imply the need for additional information from outside of the chosen basis vectors in order to construct this approximation. However, we show that a projection into the space defined solely by the chosen indices is enough to reconstruct the kernel matrix and does not require any extra information.2 The projection is the well known Nystr¨om method [11]. An orthogonal projection Pi(φ(xj)) of a feature vector φ(xj) into a subspace defined only by the set of indices i can be expressed as: Pi(xj) = ˜X′( ˜X ˜X′)−1 ˜Xφ(xj), where ˜X = X[i, :] are the i training examples from data matrix X. It follows that, Pi(xj)′Pi(xj) = φ(xj)′ ˜X′( ˜X ˜X′)−1 ˜X ˜X′( ˜X ˜X′)−1 ˜Xφ(xj) = K[i, j]K[i, i]−1K[j, i], (4) with K[i, j] denoting the kernel entries between the index set i and the feature vector φ(xj). Giving us the following projection into the space defined by i: ˜K = K[:, i]K[i, i]−1K[:, i]′. (5) Claim 1. The sparse kernel principal components analysis algorithm is a compression scheme. Proof. We can reconstruct the projection from the set of chosen indices i using Equation (4). Hence, i forms a compression set. We now prove that Smola and Sch¨olkopf’s low rank matrix approximation algorithm [8] (without sub-sampling)3 is equivalent to sparse kernel principal components analysis presented in this paper (Algorithm 1). Theorem 1. Without sub-sampling, Algorithm 1 is equivalent to Algorithm 2 of [8]. 2In their book, Smola and Sch¨olkopf redefine their kernel approximation in the same way as we have done [5], however they do not make the connection that it is a compression scheme (see Claim 1). 3We do not use the “59-trick” in our algorithm – although it’s inclusion would be trivial and would result in the same algorithm as in [8] 3 Proof. Let K be the kernel matrix and let K[:, i] be the ith column of the kernel matrix. Assume X is the input matrix containing rows of vectors that have already been mapped into a higher dimensional feature space using φ such that X = (φ(x1), . . . , φ(xm))′. Smola and Sch¨olkopf [8] state in section 4.2 of their paper that their algorithm 2 finds a low rank approximation of the kernel matrix such that it minimises the Frobenius norm ∥X−˜X∥2 Frob = tr{K−˜K} where ˜X is the low rank approximation of X. Therefore, we need to prove that Algorithm 1 also minimises this norm. We would like to show that the maximum reduction in the Frobenius norm between the kernel K and its projection ˜K is in actual fact the choice of basis vectors that maximise the Rayleigh quotient and deflate according to Equation (3). At each stage we deflate by, K = K − ττ ′ K[ik, ik]. The trace tr{K} = Pm i=1 K[i, i] is the sum of the diagonal elements of matrix K. Therefore, tr{K} = tr{K} −tr{ττ ′} K[ik, ik] = tr{K} −tr{τ ′τ} K[ik, ik] = tr{K} −K2[ik, ik] K[ik, ik] . The last term of the final equation corresponds exactly to the Rayleigh quotient of Equation (2). Therefore the maximisation of the Rayleigh quotient does indeed correspond to the maximum reduction in the Frobenius norm between the approximated matrix ˜X and X. 2.2 A generalisation error bound for sparse kernel principal components analysis We use the sample compression framework of [3] to bound the generalisation error of the sparse KPCA algorithm. Note that kernel PCA bounds [7] do not use sample compression in order to bound the true error. As pointed out above, we use the simple fact that this algorithm can be viewed as a compression scheme. No side information is needed in this setting and a simple application of [3] is all that is required. That said the usual application of compression bounds has been for classification algorithms, while here we are considering a subspace method. Theorem 2. Let Ak be any learning algorithm having a reconstruction function that maps compression sets to subspaces. Let m be the size of the training set S, let k be the size of the compression set, let ˆEm−k[ℓ(Ak(S))] be the residual loss between the m −k points outside of the compression set and their projections into a subspace, then with probability 1 −δ, the expected loss E[ℓ(Ak(S))] of algorithm Ak given any training set S can be bounded by, E[ℓ(Ak(S))] ≤ min 1≤t≤k " ˆEm−t[ℓ(At(S))] + s R 2(m −t) t ln em t + ln 2m δ # , where ℓ(·) ≥0 and R = sup ℓ(·). Proof. Consider the case where we have a compression set of size k. Then we have m k different ways of choosing the compression set. Given δ confidence we apply Hoeffding’s bound to the m−k points not in the compression set once for each choice by setting it equal to δ/ m k . Solving for ϵ gives us the theorem when we further apply a factor 1/m to δ to ensure one application for each possible choice of k. The minimisation over t chooses the best value making use of the fact that using more dimensions can only reduce the expected loss on test points. We now consider the application of the above bound to sparse KPCA. Let the corresponding loss function be defined as ℓ(At(S))(x) = ∥x −Pit(x)∥2, where x is a test point and Pit(x) its projection into the subspace determined by the set it of indices returned by At(S). Thus we can give a more specific loss bound in the case where we use a Gaussian kernel in the sparse kernel principal components analysis. Corollary 1 (Sample compression bound for sparse KPCA). Using a Gaussian kernel and all of the definitions from Theorem 2, we get the following bound: E[ℓ(A(S))] ≤ min 1≤t≤k " 1 m −t m−t X i=1 ∥xi −Pit(xi)∥2 + s 1 2(m −t) t ln em t + ln 2m δ # , 4 Note that R corresponds to the smallest radius of a ball that encloses all of the training points. Hence, for the Gaussian kernel R equals 1. We now compare the sample compression bound proposed above for sparse KPCA with the kernel PCA bound introduced by [7]. The left hand side of Figure 1 shows plots for the test error residuals (for the Boston housing data set) together with its upper bounds computed using the bound of [7] and the sample compression bound of Corollary 1. The sample compression bound is much tighter than the KPCA bound and also non-trivial (unlike the KPCA bound). The sample compression bound is at its lowest point after 43 basis vectors have been added. We speculate that at this point the “true” dimensions of the data have been found and that all other dimensions correspond to “noise”. This corresponds to the point at which the plot of residual becomes linear, suggesting dimensions with uniform noise. We carry out an extra toy experiment to help assess whether or not this is true and to show that the sample compression bound can help indicate when the principal components have captured most of the actual data. The right hand side plot of Figure 1 depicts the results of a toy experiment where we randomly sampled 1000 examples with 450 dimensions from a Gaussian distribution with zero mean and unit variance. We then ensured that 50 dimensions contained considerably larger eigenvalues than the remaining 400. From the right plot of Figure 1 we see that the test residual keeps dropping at a constant rate after 50 basis vectors have been added. The compression bound picks 46 dimensions with the largest eigenvalues, however, the KPCA bound of [7] is much more optimistic and is at its lowest point after 30 basis vectors, suggesting erroneously that SKPCA has captured most of the data in 30 dimensions. Therefore, as well as being tighter and non-trivial, the compression bound is much better at predicting the best choice for the number of dimensions to use with sparse KPCA. Note that we carried out this experiment without randomly permuting the projections into a subspace because SKPCA is rotation invariant and will always choose the principal components with the largest eigenvalues. 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Level of sparsity Residual Bound plots for sparse kernel PCA PCA bound sample compression bound test residual 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 0 0.5 1 1.5 2 2.5 X: 46 Y: 0.7984 X: 30 Y: 1.84 Level of sparsity Residual Bound plots for sparse kernel PCA PCA bound sample compression bound test residual Figure 1: Bound plots for sparse kernel PCA comparing the sample compression bound proposed in this paper and the already existing PCA bound. The plot on the left hand side is for the Boston Housing data set and the plot on the right is for a Toy experiment with 1000 training examples (and 450 dimensions) drawn randomly from a Gaussian distribution with zero mean and unit variance. 3 Kernel matching pursuit Unfortunately, the theory of the last section, where we gave a sample compression bound for SKPCA cannot be applied to KMP. This is because the algorithm needs information from outside of the compression set in order to construct its regressors and make predictions. However, we can use a VC argument together with a sample compression trick in order to derive a bound for KMP in terms of the level of sparsity achieved, by viewing the sparsity achieved in the feature space as a 5 compression scheme. Please note that we do not derive or reproduce the KMP algorithm here and advise the interested reader to read the manuscript of [10] for the algorithmic details. 3.1 A generalisation error bound for kernel matching pursuit VC bounds have commonly been used to bound learning algorithms whose hypothesis spaces are infinite. One problem with these results is that the VC-dimension can sometimes be infinite even in cases where learning is successful (e.g., the SVM). However, in this section we can avoid this issue by making use of the fact that the VC-dimension of the set of linear threshold functions is simply the dimensionality of the function class. In the kernel matching pursuit algorithm this translates directly into the number of basis vectors chosen and hence a standard VC argument. The natural loss function for KMP is regression – however in order to use standard VC bounds we map the regression loss into a classification loss in the following way. Definition 1. Let S ∼D be a regression training sample generated iid from a fixed but unknown probability distribution D. Given the error ℓ(f) = |f(x) −y| for a regression function f between training example x and regression output y we can define, for some fixed positive scalar α ∈R, the corresponding true classification loss (error) as ℓα(f) = Pr (x,y)∼D {|f(x) −y| > α} . Similarly, we can define the corresponding empirical classification loss as ˆℓα(f) = ℓS α(f) = Pr (x,y)∼S {|f(x) −y| > α} = E(x,y)∼S {I(|f(x) −y| > α)} , where I is the indicator function and S is suppressed when clear from context. Now that we have a loss function that is binary we can make a simple sample compression argument, that counts the number of possible subspaces, together with a traditional VC style bound to upper bound the expected loss of KMP. To help keep the notation consistent with earlier definitions we will denote the indices of the chosen basis vectors by i. The indices of i are chosen from the training sample S and we denote Si to be those samples indexed by the vector i. Given these definitions and the bound of Vapnik and Chervonenkis [9] we can upper bound the true loss of KMP as follows. Theorem 3. Fix α ∈R, α > 0. Let A be the regression algorithm of KMP, m the size of the training set S and k the size of the chosen basis vectors i. Let S be reordered so that the last m −k points are outside of the set i and let t = Pm i=m−k I(|f(xi) −yi| > α) be the number of errors for those points in S ∖Si. Then with probability 1 −δ over the generation of the training set S the expected loss E[ℓ(·)] of algorithm A can be bounded by, E[ℓ(A(S))] ≤ 2 m −k −t (k + 1) log 4e(m −k −t) k + 1 + k log em k +t log e(m −k) t + log 2m2 δ . Proof. First consider a fixed size k for the compression set and number of errors t. Let S1 = {xi1, . . . , xik} be the set of k training points chosen by the KMP regressor, S2 = {xik+1, . . . , xik+t} the set of points erred on in training and ¯S = S ∖(S1 ∪S2) the points outside of the compression set (S1) and training error set (S2). Suppose that the first k points form the compression set and the next t are the errors of the KMP regressor. Since the remaining m −k −t points ¯S are drawn independently we can apply the VC bound [9] to the ℓα loss to obtain the bound Pr n ¯S : ℓ ¯S α(f) = 0, ℓα(f) > ϵ o ≤2 4e(m −k −t) k + 1 k+1 2−ϵ(m−k−t)/2, where we have made use of a bound on the number of dichotomies that can be generated by parallel hyperplanes [1], which is Pk+d−1 i=0 md−1 i which is ≤ e(md−1) k+d−1 k+d−1 , where d is the number of parallel hyperplanes and equals 2 in our case. We now need to consider all of the ways that the 6 k basis vectors and t error points might have occurred and apply the union bound over all of these possibilities. This gives the bound Pr n S : ∃f ∈span{S1} s.t. ℓS2 α (f) = 1, ℓ ¯S α(f) = 0, ℓα(f) > ϵ o ≤ m k m −k t 2 4e(m −k −t) k + 1 k+1 2−ϵ(m−k−t)/2. (6) Finally we need to consider all possible choices of the values of k and t. The number of these possibilities is clearly upper bounded by m2. Setting m2 times the rhs of (6) equal to δ and solving for ϵ gives the result. This is the first upper bound on the generalisation error for KMP that we are aware of and as such we cannot compare the bound against any others. Figure 2 plots the KMP test error against the loss 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Level of sparsity Loss KMP error on Boston housing data set bound KMP test error Figure 2: Plot of KMP bound against its test error. We used 450 examples for training and the 56 for testing. Bound was scaled down by a factor of 5. bound given by Theorem 3. The bound value has been scaled by 5 in order to get the correct pictorial representation of the two plots. Figure 2 shows its minima directly coincides with the lowest test error (after 17 basis vectors). This motivates a training algorithm for KMP that would use the bound as the minimisation criteria and stop once the bound fails to become smaller. Hence, yielding a more automated training procedure. 4 Extensions The same approach that we have used for bounding the performance of kernel matching pursuit can be used to bound a matching pursuit version of kernel canonical correlation analysis (KCCA) [6]. By choosing the basis vectors greedily to optimise the quotient: max i ρi = e′ iKxKyei q e′ iK2xeie′ iK2yei , and proceeding in the same manner as Algorithm 1 by deflating after each pair of basis vectors are chosen, we create a sparsely defined subspace within which we can run the standard CCA algorithm. This again means that the overall algorithm fails to be a compression scheme as side information is required. However, we can use the same approach described for KMP to bound the expected fit of the projections from the two views. The resulting bound has the following form. Theorem 4. Fix α ∈R, α > 0. Let A be the SKCCA algorithm, m the size of the paired training sets SX×Y and k the cardinality of the set i of chosen basis vectors. Let SX×Y be reordered so that the last m−k paired data points are outside of the set i and define t = Pm i=m−k I(|fx(xi)−fy(yi)| > 7 α) to be the number of errors for those points in SX×Y ∖SX×Y i , where fx is the projection function of the X view and fy the projection function of the Y view. Then with probability 1 −δ over the generation of the paired training sets SX×Y the expected loss E[ℓ(·)] of algorithm A can be bounded by, E[ℓ(A(S))] ≤ 2 m −k −t (k + 1) log 4e(m −k −t) k + 1 + k log em k +t log e(m −k) t + log 2m2 δ . 5 Discussion Matching pursuit is a meta-scheme for creating learning algorithms for a variety of tasks. We have presented novel techniques that make it possible to analyse this style of algorithm using a combination of compression scheme ideas and more traditional learning theory. We have shown how sparse KPCA is in fact a compression scheme and demonstrated bounds that are able to accurately guide dimension selection in some cases. We have also used the techniques to bound the performance of the kernel matching pursuit (KMP) algorithm and to reinforce the generality of the approach indicated and how the approach can be extended to a matching pursuit version of KCCA. The results in this paper imply that the performance of any learning algorithm from the matching pursuit family can be analysed using a combination of sparse and traditional learning bounds. The bounds give a general theoretical justification of the framework and suggest potential applications of matching pursuit methods to other learning tasks such as novelty detection, ranking and so on. Acknowledgements The work was sponsored by the PASCAL network of excellence and the SMART project. References [1] M. Anthony. Partitioning points by parallel planes. Discrete Mathematics, 282:17–21, 2004. [2] S. Floyd and M. Warmuth. Sample compression, learnability, and the Vapnik-Chervonenkis dimension. Machine Learning, 21(3):269–304, 1995. [3] N. Littlestone and M. K. Warmuth. Relating data compression and learnability. Technical report, University of California Santa Cruz, Santa Cruz, CA, 1986. [4] S. Mallat and Z. Zhang. Matching pursuit with time-frequency dictionaries. IEEE Transactions on Signal Processing, 41(12):3397–3415, 1993. [5] B. Sch¨olkopf and A. Smola. Learning with Kernels. MIT Press, Cambridge, MA, 2002. [6] J. Shawe-Taylor and N. Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, Cambridge, U.K., 2004. [7] J. Shawe-Taylor, C. K. I. Williams, N. Cristianini, and J. Kandola. On the eigenspectrum of the Gram matrix and the generalization error of kernel-PCA. IEEE Transactions on Information Theory, 51(7):2510–2522, 2005. [8] A. J. Smola and B. Sch¨olkopf. Sparse greedy matrix approximation for machine learning. In Proceedings of 17th International Conference on Machine Learning, pages 911–918. Morgan Kaufmann, San Francisco, CA, 2000. [9] V. N. Vapnik and A. Y. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications, 16(2):264–280, 1971. [10] P. Vincent and Y. Bengio. Kernel matching pursuit. Machine Learning, 48:165–187, 2002. [11] C. K. I. Williams and M. Seeger. Using the Nystr¨om method to speed up kernel machines. In Advances in Neural Information Processing Systems, volume 13, pages 682–688. MIT Press, 2001. 8
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Policy Search for Motor Primitives in Robotics Jens Kober, Jan Peters Max Planck Institute for Biological Cybernetics Spemannstr. 38 72076 Tübingen, Germany {jens.kober,jan.peters}@tuebingen.mpg.de Abstract Many motor skills in humanoid robotics can be learned using parametrized motor primitives as done in imitation learning. However, most interesting motor learning problems are high-dimensional reinforcement learning problems often beyond the reach of current methods. In this paper, we extend previous work on policy learning from the immediate reward case to episodic reinforcement learning. We show that this results in a general, common framework also connected to policy gradient methods and yielding a novel algorithm for policy learning that is particularly well-suited for dynamic motor primitives. The resulting algorithm is an EM-inspired algorithm applicable to complex motor learning tasks. We compare this algorithm to several well-known parametrized policy search methods and show that it outperforms them. We apply it in the context of motor learning and show that it can learn a complex Ball-in-a-Cup task using a real Barrett WAMTM robot arm. 1 Introduction Policy search, also known as policy learning, has become an accepted alternative of value functionbased reinforcement learning [2]. In high-dimensional domains with continuous states and actions, such as robotics, this approach has previously proven successful as it allows the usage of domainappropriate pre-structured policies, the straightforward integration of a teacher’s presentation as well as fast online learning [2, 3, 10, 18, 5, 6, 4]. In this paper, we will extend the previous work in [17, 18] from the immediate reward case to episodic reinforcement learning and show how it relates to policy gradient methods [7, 8, 11, 10]. Despite that many real-world motor learning tasks are essentially episodic [14], episodic reinforcement learning [1] is a largely undersubscribed topic. The resulting framework allows us to derive a new algorithm called Policy Learning by Weighting Exploration with the Returns (PoWER) which is particularly well-suited for learning of trial-based tasks in motor control. We are especially interested in a particular kind of motor control policies also known as dynamic motor primitives [22, 23]. In this approach, dynamical systems are being used in order to encode a policy, i.e., we have a special kind of parametrized policy which is well-suited for robotics problems. We show that the presented algorithm works well when employed in the context of learning dynamic motor primitives in four different settings, i.e., the two benchmark problems from [10], the Underactuated Swing-Up [21] and the complex task of Ball-in-a-Cup [24, 20]. Both the Underactuated Swing-Up as well as the Ball-in-a-Cup are achieved on a real Barrett WAMTM robot arm. Please also refer to the video on the first author’s website. Looking at these tasks from a human motor learning perspective, we have a human acting as teacher presenting an example for imitation learning and, subsequently, the policy will be improved by reinforcement learning. Since such tasks are inherently single-stroke movements, we focus on the special class of episodic reinforcement learning. In our experiments, we show how a presented movement is recorded using kinesthetic teach-in and, subsequently, how a Barrett WAMTM robot arm is learning the behavior by a combination of imitation and reinforcement learning. 1 2 Policy Search for Parameterized Motor Primitives Our goal is to find reinforcement learning techniques that can be applied to a special kind of prestructured parametrized policies called motor primitives [22, 23], in the context of learning highdimensional motor control tasks. In order to do so, we first discuss our problem in the general context of reinforcement learning and introduce the required notation in Section 2.1. Using a generalization of the approach in [17, 18], we derive a new EM-inspired algorithm called Policy Learning by Weighting Exploration with the Returns (PoWER) in Section 2.3 and show how the general framework is related to policy gradients methods in 2.2. [12] extends the [17] algorithm to episodic reinforcement learning for discrete states; we use continuous states. Subsequently, we discuss how we can turn the parametrized motor primitives [22, 23] into explorative [19], stochastic policies. 2.1 Problem Statement & Notation In this paper, we treat motor primitive learning problems in the framework of reinforcement learning with a strong focus on the episodic case [1]. We assume that at time t there is an actor in a state st and chooses an appropriate action at according to a stochastic policy π(at|st, t). Such a policy is a probability distribution over actions given the current state. The stochastic formulation allows a natural incorporation of exploration and, in the case of hidden state variables, the optimal timeinvariant policy has been shown to be stochastic [8]. Upon the completion of the action, the actor transfers to a state st+1 and receives a reward rt. As we are interested in learning complex motor tasks consisting of a single stroke [23], we focus on finite horizons of length T with episodic restarts [1] and learn the optimal parametrized, stochastic policy for such reinforcement learning problems. We assume an explorative version of the dynamic motor primitives [22, 23] as parametrized policy π with parameters θ ∈Rn. However, in this section, we will keep most derivations sufficiently general that they would transfer to various other parametrized policies. The general goal in reinforcement learning is to optimize the expected return of the policy π with parameters θ defined by J(θ) = Tp(τ)R(τ)dτ, (1) where T is the set of all possible paths, rollout τ = [s1:T +1, a1:T ] (also called episode or trial) denotes a path of states s1:T +1 = [s1, s2, . . ., sT +1] and actions a1:T = [a1, a2, . . ., aT ]. The probability of rollout τ is denoted by p(τ) while R(τ) refers to its return. Using the standard assumptions of Markovness and additive accumulated rewards, we can write p(τ) = p(s1)QT t=1p(st+1|st, at)π(at|st, t), R(τ) = T −1PT t=1r(st, at, st+1, t), (2) where p(s1) denotes the initial state distribution, p(st+1|st, at) the next state distribution conditioned on last state and action, and r(st, at, st+1, t) denotes the immediate reward. While episodic Reinforcement Learning (RL) problems with finite horizons are common in motor control, few methods exist in the RL literature, e.g., Episodic REINFORCE [7], the Episodic Natural Actor Critic eNAC [10] and model-based methods using differential-dynamic programming [21]. Nevertheless, in the analytically tractable cases, it has been studied deeply in the optimal control community where it is well-known that for a finite horizon problem, the optimal solution is non-stationary [15] and, in general, cannot be represented by a time-independent policy. The motor primitives based on dynamical systems [22, 23] are a particular type of time-variant policy representation as they have an internal phase which corresponds to a clock with additional flexibility (e.g., for incorporating coupling effects, perceptual influences, etc.), thus, they can represent optimal solutions for finite horizons. We embed this internal clock or movement phase into our state and, thus, from optimal control perspective have ensured that the optimal solution can be represented. 2.2 Episodic Policy Learning In this section, we discuss episodic reinforcement learning in policy space which we will refer to as Episodic Policy Learning. For doing so, we first discuss the lower bound on the expected return suggested in [17] for guaranteeing that policy update steps are improvements. In [17, 18] only the immediate reward case is being discussed, we extend their framework to episodic reinforcement learning and, subsequently, derive a general update rule which yields the policy gradient theorem [8], a generalization of the reward-weighted regression [18] as well as the novel Policy learning by Weighting Exploration with the Returns (PoWER) algorithm. 2.2.1 Bounds on Policy Improvements Unlike in reinforcement learning, other machine learning branches have focused on optimizing lower bounds, e.g., resulting in expectation-maximization (EM) algorithms [16]. The reasons for this preference apply in policy learning: if the lower bound also becomes an equality for the sampling policy, 2 we can guarantee that the policy will be improved by optimizing the lower bound. Surprisingly, results from supervised learning can be transferred with ease. For doing so, we follow the scenario suggested in [17], i.e., generate rollouts τ using the current policy with parameters θ which we weight with the returns R (τ) and subsequently match it with a new policy parametrized by θ′. This matching of the success-weighted path distribution is equivalent to minimizing the KullbackLeibler divergence D (pθ′ (τ) ∥pθ (τ) R (τ)) between the new path distribution pθ′ (τ) and the reward-weighted previous one pθ (τ) R (τ). As shown in [17, 18], this results in a lower bound on the expected return using Jensen’s inequality and the concavity of the logarithm, i.e., log J(θ′) = log T pθ (τ) pθ (τ)pθ′ (τ) R (τ) dτ ≥ T pθ (τ) R (τ) log pθ′ (τ) pθ (τ) dτ + const, (3) ∝−D (pθ (τ) R (τ) ∥pθ′ (τ)) = Lθ(θ′), (4) where D (p (τ) ∥q (τ)) = p (τ) log(p (τ) /q (τ))dτ is the Kullback-Leibler divergence which is considered a natural distance measure between probability distributions, and the constant is needed for tightness of the bound. Note that pθ (τ) R (τ) is an improper probability distribution as pointed out in [17]. The policy improvement step is equivalent to maximizing the lower bound on the expected return Lθ(θ′) and we show how it relates to previous policy learning methods. 2.2.2 Resulting Policy Updates In the following part, we will discuss three different policy updates which directly result from Section 2.2.1. First, we show that policy gradients [7, 8, 11, 10] can be derived from the lower bound Lθ(θ′) (as was to be expected from supervised learning, see [13]). Subsequently, we show that natural policy gradients can be seen as an additional constraint regularizing the change in the path distribution resulting from a policy update when improving the policy incrementally. Finally, we will show how expectation-maximization (EM) algorithms for policy learning can be generated. Policy Gradients. When differentiating the function Lθ(θ′) that defines the lower bound on the expected return, we directly obtain ∂θ′Lθ(θ′) = Tpθ (τ) R (τ) ∂θ′ log pθ′ (τ) dτ, (5) where T is the set of all possible paths and ∂θ′ log pθ′ (τ) = PT t=1 ∂θ′ log π(at|st, t) denotes the log-derivative of the path distribution. As this log-derivative only depends on the policy, we can estimate a gradient from rollouts without having a model by simply replacing the expectation by a sum; when θ′ is close to θ, we have the policy gradient estimator which is widely known as Episodic REINFORCE [7], i.e., we have limθ′→θ ∂θ′Lθ(θ′) = ∂θJ(θ). Obviously, a reward which precedes an action in an rollout, can neither be caused by the action nor cause an action in the same rollout. Thus, when inserting Equations (2) into Equation (5), all cross-products between rt and ∂θ log π(at+δt|st+δt, t + δt) for δt > 0 become zero in expectation [10]. Therefore, we can omit these terms and rewrite the estimator as ∂θ′Lθ(θ′) = E nPT t=1∂θ′ log π(at|st, t)Qπ(s, a, t) o , (6) where Qπ(s, a, t) = E{PT ˜t=tr(s˜t, a˜t, s˜t+1, ˜t)|st = s, at = a} is called the state-action value function [1]. Equation (6) is equivalent to the policy gradient theorem [8] for θ′ →θ in the infinite horizon case where the dependence on time t can be dropped. The derivation results in the Natural Actor Critic as discussed in [9, 10] when adding an additional punishment to prevent large steps away from the observed path distribution. This can be achieved by restricting the amount of change in the path distribution and, subsequently, determining the steepest descent for a fixed step away from the observed trajectories. Change in probability distributions is naturally measured using the Kullback-Leibler divergence, thus, after adding the additional constraint of D(pθ′(τ)∥pθ(τ)) ≈0.5(θ′ −θ)TF(θ)(θ′ −θ) = δ using a second-order expansion as approximation where F(θ) denotes the Fisher information matrix [9, 10]. Policy Search via Expectation Maximization. One major drawback of gradient-based approaches is the learning rate, an open parameter which can be hard to tune in control problems but is essential for good performance. Expectation-Maximization algorithms are well-known to avoid this problem in supervised learning while even yielding faster convergence [16]. Previously, similar ideas have been explored in immediate reinforcement learning [17, 18]. In general, an EMalgorithm would choose the next policy parameters θn+1 such that θn+1 = argmaxθ′ Lθ(θ′). In the case where π(at|st, t) belongs to the exponential family, the next policy can be determined analytically by setting Equation (6) to zero, i.e., E nPT t=1∂θ′ log π(at|st, t)Qπ(s, a, t) o = 0, (7) 3 Algorithm 1 Policy learning by Weighting Exploration with the Returns for Motor Primitives Input: initial policy parameters θ0 repeat Sample: Perform rollout(s) using a = (θ + εt)Tφ(s, t) with [εt]ij ∼N(0, σ2 ij) as stochastic policy and collect all (t, st, at, st+1, εt, rt+1) for t = {1, 2, . . . , T + 1}. Estimate: Use unbiased estimate ˆQπ(s, a, t) = PT ˜t=t r(s˜t, a˜t, s˜t+1, ˜t). Reweight: Compute importance weights and reweight rollouts, discard low-importance rollouts. Update policy using θk+1 = θk + DPT t=1εtQπ(s, a, t) E w(τ) .DPT t=1Qπ(s, a, t) E w(τ). until Convergence θk+1 ≈θk and solving for θ′. Depending on the choice of a stochastic policy, we will obtain different solutions and different learning algorithms. It allows the extension of the reward-weighted regression to larger horizons as well as the introduction of the Policy learning by Weighting Exploration with the Returns (PoWER) algorithm. 2.3 Policy learning by Weighting Exploration with the Returns (PoWER) In most learning control problems, we attempt to have a deterministic mean policy ¯a = θTφ(s, t) with parameters θ and basis functions φ. In Section 3, we will introduce the basis functions of the motor primitives. When learning motor primitives, we turn this deterministic mean policy ¯a = θTφ(s, t) into a stochastic policy using additive exploration ε(s, t) in order to make modelfree reinforcement learning possible, i.e., we always intend to have a policy π(at|st, t) which can be brought into the form a = θTφ(s, t) + ϵ(φ(s, t)). Previous work in this context [7, 4, 10, 18], with the notable exception of [19], has focused on state-independent, white Gaussian exploration, i.e., ϵ(φ(s, t)) ∼N(0, Σ). It is straightforward to obtain the Reward-Weighted Regression for episodic RL by solving Equation (7) for θ′ which naturally yields a weighted regression method with the state-action values Qπ(s, a, t) as weights. This form of exploration has resulted into various applications in robotics such as T-Ball batting, Peg-In-Hole, humanoid robot locomotion, constrained reaching movements and operational space control, see [4, 10, 18] for both reviews and their own applications. However, such unstructured exploration at every step has a multitude of disadvantages: it causes a large variance which grows with the number of time-steps [19, 10], it perturbs actions too frequently ‘washing’ out their effects and can damage the system executing the trajectory. As a result, all methods relying on this state-independent exploration have proven too fragile for learning the Ballin-a-Cup task on a real robot system. Alternatively, as introduced by [19], one could generate a form of structured, state-dependent exploration ϵ(φ(s, t)) = εT t φ(s, t) with [εt]ij ∼N(0, σ2 ij), where σ2 ij are meta-parameters of the exploration that can also be optimized. This argument results into the policy a ∼π(at|st, t) = N(a|θTφ(s, t), ˆΣ(s, t)). Inserting the resulting policy into Equation (7), we obtain the optimality condition in the sense of Equation (7) and can derive the update rule θ′ = θ + E nPT t=1Qπ(s, a, t)W(s, t) o−1 E nPT t=1Qπ(s, a, t)W(s, t)εt o (8) with W(s, t) = φ(s, t)φ(s, t)T/(φ(s, t)Tφ(s, t)). Note that for our motor primitives W reduces to a diagonal, constant matrix and cancels out. Hence the simplified form in Algorithm 1. In order to reduce the number of rollouts in this on-policy scenario, we reuse the rollouts through importance sampling as described in the context of reinforcement learning in [1]. To avoid the fragility sometimes resulting from importance sampling in reinforcement learning, samples with very small importance weights are discarded. The expectations E{·} are replaced by the importance sampler denoted by ⟨·⟩w(τ). The resulting algorithm is shown in Algorithm 1. As we will see in Section 3, this PoWER method outperforms all other described methods significantly. 3 Application to Motor Primitive Learning for Robotics In this section, we demonstrate the effectiveness of the algorithm presented in Section 2.3 in the context of motor primitive learning for robotics. For doing so, we will first give a quick overview how the motor primitives work and how the algorithm can be used to adapt them. As first evaluation, we will show that the novel presented PoWER algorithm outperforms many previous well-known 4 methods, i.e., ‘Vanilla’ Policy Gradients, Finite Difference Gradients, the Episodic Natural Actor Critic and the generalized Reward-Weighted Regression on the two simulated benchmark problems suggested in [10] and a simulated Underactuated Swing-Up [21]. Real robot applications are done with our best benchmarked method, the PoWER method. Here, we first show PoWER can learn the Underactuated Swing-Up [21] even on a real robot. As a significantly more complex motor learning task, we show how the robot can learn a high-speed Ball-in-a-Cup [24] movement with motor primitives for all seven degrees of freedom of our Barrett WAMTM robot arm. 3.1 Using the Motor Primitives in Policy Search The motor primitive framework [22, 23] can be described as two coupled differential equations, i.e., we have a canonical system ˙y = f(y, z) with movement phase y and possible external coupling to z as well as a nonlinear system ¨x = g(x, ˙x, y, θ) which yields the current action for the system. Both dynamical systems are chosen to be stable and to have the right properties so that they are useful for the desired class of motor control problems. In this paper, we focus on single stroke movements as they frequently appear in human motor control [14, 23] and, thus, we will always choose the point attractor version of the motor primitives exactly as presented in [23] and not the older one in [22]. The biggest advantage of the motor primitive framework of [22, 23] is that the function g is linear in the policy parameters θ and, thus, well-suited for imitation learning as well as for our presented reinforcement learning algorithm. For example, if we would have to learn only a motor primitive for a single degree of freedom qi, then we could use a motor primitive in the form ¯¨qi = g(qi, ˙qi, y, θ) = φ(s)Tθ where s = [qi, ˙qi, y] is the state and where time is implicitly embedded in y. We use the output of ¯¨qi = φ(s)Tθ = ¯a as the policy mean. The perturbed accelerations ¨qi = a = ¯a+ε is given to the system. The details of φ are given in [23]. 10 2 10 3 −1000 −500 −250 number of rollouts average return (a) minimum motor command 10 2 10 3 −10 2 −10 1 number of rollouts average return (b) passing through a point FDG VPG eNAC RWR PoWER Figure 1: This figure shows the mean performance of all compared methods in two benchmark tasks averaged over twenty learning runs with the error bars indicating the standard deviation. Policy learning by Weighting Exploration with the Returns (PoWER) clearly outperforms Finite Difference Gradients (FDG), ‘Vanilla’ Policy Gradients (VPG), the Episodic Natural Actor Critic (eNAC) and the adapted Reward-Weighted Regression (RWR) for both tasks. In Sections 3.3 and 3.4, we use imitation learning for the initialization. For imitations, we follow [22]: first, extract the duration of the movement from initial and final zero velocity and use it to adjust the time constants. Second, use locally-weighted regression to solve for an imitation from a single example. 3.2 Benchmark Comparison As benchmark comparison, we intend to follow a previously studied scenario in order to evaluate which method is best-suited for our problem class. For doing so, we perform our evaluations on the exact same benchmark problems as [10] and use two tasks commonly studied in motor control literature for which the analytic solutions are known, i.e., a reaching task where a goal has to be reached at a certain time while the used motor commands have to be minimized and a reaching task of the same style with an additional via-point. In this comparison, we mainly want to show the suitability of our algorithm and show that it outperforms previous methods such as Finite Difference Gradient (FDG) methods [10], ‘Vanilla’ Policy Gradients (VPG) with optimal baselines [7, 8, 11, 10], the Episodic Natural Actor Critic (eNAC) [9, 10], and the episodic version of the Reward-Weighted Regression (RWR) algorithm [18]. For both tasks, we use the same rewards as in [10] but we use the newer form of the motor primitives from [23]. All open parameters were manually optimized for each algorithm in order to maximize the performance while not destabilizing the convergence of the learning process. When applied in the episodic scenario, Policy learning by Weighting Exploration with the Returns (PoWER) clearly outperformed the Episodic Natural Actor Critic (eNAC), ‘Vanilla’ Policy Gradient (VPG), Finite Difference Gradient (FDG) and the adapted Reward-Weighted Regression (RWR) for both tasks. The episodic Reward-Weighted Regression (RWR) is outperformed by all other algorithms suggesting that this algorithm does not generalize well from the immediate reward case. 5 Figure 2: This figure shows the time series of the Underactuated Swing-Up where only a single joint of the robot is moved with a torque limit ensured by limiting the maximal motor current of that joint. The resulting motion requires the robot to (i) first move away from the target to limit the maximal required torque during the swing-up in (ii-iv) and subsequent stabilization (v). The performance of the PoWER method on the real robot is shown in (vi). While FDG gets stuck on a plateau, both eNAC and VPG converge to the same, good final solution. PoWER finds the same (or even slightly better) solution while achieving it noticeably faster. The results are presented in Figure 1. Note that this plot has logarithmic scales on both axes, thus a unit difference corresponds to an order of magnitude. The omission of the first twenty rollouts was necessary to cope with the log-log presentation. 3.3 Underactuated Swing-Up 50 100 150 200 0.6 0.7 0.8 0.9 1 number of rollouts average return RWR PoWER FDG VPG eNAC Figure 3: This figure shows the performance of all compared methods for the swing-up in simulation and show the mean performance averaged over 20 learning runs with the error bars indicating the standard deviation. PoWER outperforms the other algorithms from 50 rollouts on and finds a significantly better policy. As additional simulated benchmark and for the realrobot evaluations, we employed the Underactuated Swing-Up [21]. Here, only a single degree of freedom is represented by the motor primitive as described in Section 3.1. The goal is to move a hanging heavy pendulum to an upright position and stabilize it there in minimum time and with minimal motor torques. By limiting the motor current for that degree of freedom, we can ensure that the torque limits described in [21] are maintained and directly moving the joint to the right position is not possible. Under these torque limits, the robot needs to (i) first move away from the target to limit the maximal required torque during the swing-up in (ii-iv) and subsequent stabilization (v) as illustrated in Figure 2 (i-v). This problem is similar to a mountain-car problem where the car would have to stop on top or experience a failure. The applied torque limits were the same as in [21] and so was the reward function was the except that the complete return of the trajectory was transformed by an exp(·) to ensure positivity. Again all open parameters were manually optimized. The motor primitive with nine shape parameters and one goal parameter was initialized by imitation learning from a kinesthetic teach-in. Subsequently, we compared the other algorithms as previously considered in Section 3.2 and could show that PoWER would again outperform them. The results are given in Figure 3. As it turned out to be the best performing method, we then used it successfully for learning optimal swing-ups on a real robot. See Figure 2 (vi) for the resulting real-robot performance. 3.4 Ball-in-a-Cup on a Barrett WAMTM The most challenging application in this paper is the children’s game Ball-in-a-Cup [24] where a small cup is attached at the robot’s end-effector and this cup has a small wooden ball hanging down from the cup on a 40cm string. Initially, the ball is hanging down vertically. The robot needs to move fast in order to induce a motion at the ball through the string, swing it up and catch it with the cup, a possible movement is illustrated in Figure 4 (top row). The state of the system is described in joint angles and velocities of the robot and the Cartesian coordinates of the ball. The actions are the joint space accelerations where each of the seven joints is represented by a motor primitive. All motor primitives are perturbed separately but employ the same joint final reward given by r(tc) = exp(−α(xc −xb)2 −α(yc −yb)2) while r(t) = 0 for all other t ̸= tc where tc is the moment where the ball passes the rim of the cup with a downward direction, the cup position denoted by [xc, yc, zc] ∈R3, the ball position [xb, yb, zb] ∈R3 and a scaling parameter α = 100. The task is quite complex as the reward is not modified solely by the movements of the cup but foremost by the movements of the ball and the movements of the ball are very sensitive to changes in the movement. A small perturbation of the initial condition or during the trajectory will drastically change the movement of the ball and hence the outcome of the rollout. 6 Figure 4: This figure shows schematic drawings of the Ball-in-a-Cup motion, the final learned robot motion as well as a kinesthetic teach-in. The green arrows show the directions of the current movements in that frame. The human cup motion was taught to the robot by imitation learning with 31 parameters per joint for an approximately 3 seconds long trajectory. The robot manages to reproduce the imitated motion quite accurately, but the ball misses the cup by several centimeters. After ca. 75 iterations of our Policy learning by Weighting Exploration with the Returns (PoWER) algorithm the robot has improved its motion so that the ball goes in the cup. Also see Figure 5. 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 number of rollouts average return Figure 5: This figure shows the expected return of the learned policy in the Ball-ina-Cup evaluation averaged over 20 runs. Due to the complexity of the task, Ball-in-a-Cup is even a hard motor learning task for children who usually only succeed at it by observing another person playing and a lot of improvement by trial-and-error. Mimicking how children learn to play Ball-in-a-Cup, we first initialize the motor primitives by imitation and, subsequently, improve them by reinforcement learning. We recorded the motions of a human player by kinesthetic teach-in in order to obtain an example for imitation as shown in Figure 4 (middle row). From the imitation, it can be determined by cross-validation that 31 parameters per motor primitive are needed. As expected, the robot fails to reproduce the the presented behavior and reinforcement learning is needed for self-improvement. Figure 5 shows the expected return over the number of rollouts where convergence to a maximum is clearly recognizable. The robot regularly succeeds at bringing the ball into the cup after approximately 75 iterations. 4 Conclusion In this paper, we have presented a new perspective on policy learning methods and an application to a highly complex motor learning task on a real Barrett WAMTM robot arm. We have generalized the previous work in [17, 18] from the immediate reward case to the episodic case. In the process, we could show that policy gradient methods are a special case of this more general framework. During initial experiments, we realized that the form of exploration highly influences the speed of the policy learning method. This empirical insight resulted in a novel policy learning algorithm, Policy learning by Weighting Exploration with the Returns (PoWER), an EM-inspired algorithm that outperforms several other policy search methods both on standard benchmarks as well as on a simulated Underactuated Swing-Up. We successfully applied this novel PoWER algorithm in the context of learning two tasks on a physical robot, i.e., the Underacted Swing-Up and Ball-in-a-Cup. Due to the curse of dimensionality, we cannot start with an arbitrary solution. Instead, we mimic the way children learn Ball-in-a-Cup and first present an example for imitation learning which is recorded using kinesthetic teach-in. Subsequently, our reinforcement learning algorithm takes over and learns how to move the ball into 7 the cup reliably. After only realistically few episodes, the task can be regularly fulfilled and the robot shows very good average performance. References [1] R. Sutton and A. Barto. Reinforcement Learning. MIT Press, 1998. [2] J. Bagnell, S. Kadade, A. Ng, and J. Schneider. Policy search by dynamic programming. In Advances in Neural Information Processing Systems (NIPS), 2003. [3] A. Ng and M. Jordan. PEGASUS: A policy search method for large MDPs and POMDPs. In International Conference on Uncertainty in Artificial Intelligence (UAI), 2000. [4] F. Guenter, M. Hersch, S. Calinon, and A. Billard. Reinforcement learning for imitating constrained reaching movements. RSJ Advanced Robotics, 21, 1521-1544, 2007. [5] M. Toussaint and C. Goerick. Probabilistic inference for structured planning in robotics. In International Conference on Intelligent Robots and Systems (IROS), 2007. [6] M. Hoffman, A. Doucet, N. de Freitas, and A. Jasra. Bayesian policy learning with transdimensional MCMC. In Advances in Neural Information Processing Systems (NIPS), 2007. [7] R. J. Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8:229–256, 1992. [8] R. S. Sutton, D. McAllester, S. Singh, and Y. Mansour. Policy gradient methods for reinforcement learning with function approximation. In Advances in Neural Information Processing Systems (NIPS), 2000. [9] J. Bagnell and J. Schneider. Covariant policy search. In International Joint Conference on Artificial Intelligence (IJCAI), 2003. [10] J. Peters and S. Schaal. Policy gradient methods for robotics. In International Conference on Intelligent Robots and Systems (IROS), 2006. [11] G. Lawrence, N. Cowan, and S. Russell. Efficient gradient estimation for motor control learning. In International Conference on Uncertainty in Artificial Intelligence (UAI), 2003. [12] H. Attias. Planning by probabilistic inference. In Ninth International Workshop on Artificial Intelligence and Statistics (AISTATS), 2003. [13] J. Binder, D. Koller, S. Russell, and K. Kanazawa. Adaptive probabilistic networks with hidden variables. Machine Learning, 29:213–244, 1997. [14] G. Wulf. Attention and motor skill learning. Human Kinetics, Champaign, IL, 2007. [15] D. E. Kirk. Optimal control theory. Prentice-Hall, Englewood Cliffs, New Jersey, 1970. [16] G. J. McLachan and T. Krishnan. The EM Algorithm and Extensions. Wiley Series in Probability and Statistics. John Wiley & Sons, 1997. [17] P. Dayan and G. E. Hinton. Using expectation-maximization for reinforcement learning. Neural Computation, 9(2):271–278, 1997. [18] J. Peters and S. Schaal. Reinforcement learning by reward-weighted regression for operational space control. In International Conference on Machine Learning (ICML), 2007. [19] T. Rückstieß, M. Felder, and J. Schmidhuber. State-dependent exploration for policy gradient methods. In European Conference on Machine Learning (ECML), 2008. [20] M. Kawato, F. Gandolfo, H. Gomi, and Y. Wada. Teaching by showing in kendama based on optimization principle. In International Conference on Artificial Neural Networks, 1994. [21] C. G. Atkeson. Using local trajectory optimizers to speed up global optimization in dynamic programming. In Advances in Neural Information Processing Systems (NIPS), 1994. [22] A. Ijspeert, J. Nakanishi, and S. Schaal. Learning attractor landscapes for learning motor primitives. In Advances in Neural Information Processing Systems (NIPS), 2003. [23] S. Schaal, P. Mohajerian, and A. Ijspeert. Dynamics systems vs. optimal control — a unifying view. Progress in Brain Research, 165(1):425–445, 2007. [24] Wikipedia, May 31, 2008. http://en.wikipedia.org/wiki/Ball_in_a_cup [25] J. 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Mortal Multi-Armed Bandits Deepayan Chakrabarti Yahoo! Research Sunnyvale, CA 94089 deepay@yahoo-inc.com Ravi Kumar Yahoo! Research Sunnyvale, CA 94089 ravikumar@yahoo-inc.com Filip Radlinski∗ Microsoft Research Cambridge, UK filiprad@microsoft.com Eli Upfal† Brown University Providence, RI 02912 eli@cs.brown.edu Abstract We formulate and study a new variant of the k-armed bandit problem, motivated by e-commerce applications. In our model, arms have (stochastic) lifetime after which they expire. In this setting an algorithm needs to continuously explore new arms, in contrast to the standard k-armed bandit model in which arms are available indefinitely and exploration is reduced once an optimal arm is identified with nearcertainty. The main motivation for our setting is online-advertising, where ads have limited lifetime due to, for example, the nature of their content and their campaign budgets. An algorithm needs to choose among a large collection of ads, more than can be fully explored within the typical ad lifetime. We present an optimal algorithm for the state-aware (deterministic reward function) case, and build on this technique to obtain an algorithm for the state-oblivious (stochastic reward function) case. Empirical studies on various reward distributions, including one derived from a real-world ad serving application, show that the proposed algorithms significantly outperform the standard multi-armed bandit approaches applied to these settings. 1 Introduction Online advertisements (ads) are a rapidly growing source of income for many Internet content providers. The content providers and the ad brokers who match ads to content are paid only when ads are clicked; this is commonly referred to as the pay-per-click model. In this setting, the goal of the ad brokers is to select ads to display from a large corpus, so as to generate the most ad clicks and revenue. The selection problem involves a natural exploration vs. exploitation tradeoff: balancing exploration for ads with better click rates against exploitation of the best ads found so far. Following [17, 16], we model the ad selection task as a multi-armed bandit problem [5]. A multiarmed bandit models a casino with k slot machines (one-armed bandits), where each machine (arm) has a different and unknown expected payoff. The goal is to sequentially select the optimal sequence of slot machines to play (i.e., slot machine arms to pull) to maximize the expected total reward. Considering each ad as a slot machine, that may or may not provide a reward when presented to users, allows any multi-armed bandit strategy to be used for the ad selection problem. ∗Most of this work was done while the author was at Yahoo! Research. †Part of this work was done while the author was visiting the Department of Information Engineering at the University of Padova, Italy, supported in part by the FP6 EC/IST Project 15964 AEOLUS. Work supported in part by NSF award DMI-0600384 and ONR Award N000140610607. A standard assumption in the multi-armed bandit setting, however, is that each arm exists perpetually. Although the payoff function of an arm is allowed to evolve over time, the evolution is assumed to be slow. Ads, on the other hand, are regularly created while others are removed from circulation. This occurs as advertisers’ budgets run out, when advertising campaigns change, when holiday shopping seasons end, and due to other factors beyond the control of the ad selection system. The advertising problem is even more challenging as the set of available ads is often huge (in the tens of millions), while standard multi-armed bandit strategies converge only slowly and require time linear in the number of available options. In this paper we initiate the study of a rapidly changing variant of the multi-armed bandit problem. We call it the mortal multi-armed bandit problem since ads (or equivalently, available bandit arms) are assumed to be born and die regularly. In particular, we will show that while the standard multiarmed bandit setting allows for algorithms that only deviate from the optimal total payoff by O(ln t) [21], in the mortal arm setting a regret of Ω(t) is possible. Our analysis of the mortal multi-arm bandit problem considers two settings. First, in the less realistic but simpler state-aware (deterministic reward) case, pulling arm i always provides a reward that equals the expected payoff of the arm. Second, in the more realistic state-oblivious (stochastic reward) case, the reward from arm i is a binomial random variable indicating the true payoff of the arm only in expectation. We provide an optimal algorithm for the state-aware case. This algorithm is based on characterizing the precise payoff threshold below which repeated arm pulls become suboptimal. This characterization also shows that there are cases when a linear regret is inevitable. We then extend the algorithm to the state-oblivious case, and show that it is near-optimal. Following this, we provide a general heuristic recipe for modifying standard multi-armed bandit algorithms to be more suitable in the mortal-arm setting. We validate the efficacy of our algorithms on various payoff distributions including one empirically derived from real ads. In all cases, we show that the algorithms presented significantly outperform standard multi-armed bandit approaches. 2 Modeling mortality Suppose we wish to select the ads to display on a webpage. Every time a user visits this webpage, we may choose one ad to display. Each ad has a different potential to provide revenue, and we wish to sequentially select the ads to maximize the total expected revenue. Formally, say that at time t, we have ads A(t) = {ad1t, . . . , adkt} from which we must pick one to show. Each adit has a payoff µit ∈[0, 1] that is drawn from some known cumulative distribution F(µ)1. Presenting adit at time t provides a (financial) reward R(µit); the reward function R(·) will be specified below. If the pool of available ads A(t) were static, or if the payoffs were only slowly changing with t, this problem could be solved using any standard multi-armed bandit approach. As described earlier, in reality the available ads are rapidly changing. We propose the following simple model for this change: at the end of each time step t, one or more ads may die and be replaced with new ads. The process then continues with time t + 1. Note that since change happens only through replacement of ads, the number of ads k = |A(t)| remains fixed. Also, as long as an ad is alive, we assume that its payoff is fixed. Death can be modeled in two ways, and we will address both in this work. An ad i may have a budget Li that is known a priori and revealed to the algorithm. The ad dies immediately after it has been selected Li times; we assume that Li values are drawn from a geometric distribution, with an expected budget of L. We refer to this case as budgeted death. Alternatively, each ad may die with a fixed probability p after every time step, whether it was selected or not. This is equivalent to each ad being allocated a lifetime budget Li, drawn from a geometric distribution with parameter p, that is fixed when the arm is born but is never revealed to the algorithm; in this case new arms have an expected lifetime of L = 1/p. We call this timed death. In both death settings, we assume in our theoretical analysis that at any time there is always at least one previously unexplored ad available. This reflects reality where the number of ads is practically unlimited. Finally, we model the reward function in two ways, the first being simpler to analyze and the latter more realistic. In the state-aware (deterministic reward) case, we assume R(µit) = µit. This 1We limit our analysis to the case where F(µ) is stationary and known, as we are particularly interested in the long-term steady-state setting. provides us with complete information about each ad immediately after it is chosen to be displayed. In the state-oblivious (stochastic reward) case, we take R(µit) to be a random variable that is 1 with probability µit and 0 otherwise. The mortal multi-armed bandit setting requires different performance measures than the ones used with static multi-armed bandits. In the static setting, very little exploration is needed once an optimal arm is identified with near-certainty; therefore the quality measure is the total regret over time. In our setting the algorithm needs to continuously explore newly available arms. We therefore study the long term, steady-state, mean regret per time step of various solutions. We define this regret as the expected payoff of the best currently alive arm minus the payoff actually obtained by the algorithm. 3 Related work Our work is most related to the study of dynamic versions of the multi-arm bandit (MAB) paradigm where either the set of arms or their expected reward may change over time. Motivated by task scheduling, Gittins [10] proposed a policy where only the state of the active arm (the arm currently being played) can change in a given step, and proved its optimality for the Bayesian formulation with time discounting. This seminal result gave rise to a rich line of work, a proper review of which is beyond the scope of this paper. In particular, Whittle [23] introduced an extension termed restless bandits [23, 6, 15], where the states of all arms can change in each step according to a known (but arbitrary) stochastic transition function. Restless bandits have been shown to be intractable: e.g., even with deterministic transitions the problem of computing an (approximately) optimal strategy is PSPACE-hard [18]. Sleeping bandits problem, where the set of strategies is fixed but only a subset of them available in each step, were studied in [9, 7] and recently, using a different evaluation criteria, in [13]. Strategies with expected rewards that change gradually over time were studied in [19]. The mixture-of-experts paradigm is related [11], but assumes that data tuples are provided to each expert, instead of the tuples being picked by the algorithm, as in the bandit setting. Auer et al. [3] adopted an adversarial approach: they defined the adversarial MAB problem where the reward distributions are allowed to change arbitrarily over time, and the goal is to approach the performance of the best time-invariant policy. This formulation has been further studied in several other papers. Auer et al. [3, 1] also considered a more general definition of regret, where the comparison is to the best policy that can change arms a limited number of times. Due to the overwhelming strength of the adversary, the guarantees obtained in this line of work are relatively weak when applied to the setting that we consider in this paper. Another aspect of our model is that unexplored arms are always available. Related work broadly comes in three flavors. First, new arms can become available over time; the optimality of Gittins’ index was shown to extend to this case [22]. The second case is that of infinite-armed bandits with discrete arms, first studied by [4] and recently extended to the case of unknown payoff distributions and an unknown time horizon [20]. Finally, the bandit arms may be indexed by numbers from the real line, implying uncountably infinite bandit arms, but where “nearby” arms (in terms of distance along the real line) have similar payoffs [12, 14]. However, none of these approaches allows for arms to appear then disappear, which as we show later critically affects any regret bounds. 4 Upper bound on mortal reward In this section we show that in the mortal multi-armed bandit setting, the regret per time step of any algorithm can never go to zero, unlike in the standard MAB setting. Specifically, we develop an upper bound on the mean reward per step of any such algorithm for the state-aware, budgeted death case. We then use reductions between the different models to show that this bound holds for the state-oblivious, timed death cases as well. We prove the bound assuming we always have new arms available. The expected reward of an arm is drawn from a cumulative distribution F(µ) with support in [0, 1]. For X ∼F(µ), let E[X] be the expectation of X over F(µ). We assume that the lifetime of an arm has an exponential distribution with parameter p, and denote its expectation by L = 1/p. The following function captures the tradeoff between exploration and exploitation in our setting and plays a major role in our analysis: Γ(µ) = E[X] + (1 −F(µ))(L −1)E[X|X ≥µ] 1 + (1 −F(µ))(L −1) . (1) Theorem 1. Let ¯µ(t) denote the maximum mean reward that any algorithm for the state-aware mortal multi-armed bandit problem can obtain in t steps in the budgeted death case. Then limt→∞¯µ(t) ≤maxµ Γ(µ). Proof sketch. We distinguish between fresh arm pulls, i.e., pulls of arms that were not pulled before, and repeat arm pulls. Assume that the optimal algorithm pulls τ(t) distinct (fresh) arms in t steps, and hence makes t −τ(t) repeat pulls. The expected number of repeat pulls to an arm before it expires is (1 −p)/p. Thus, using Wald’s equation [8], the expected number of different arms the algorithm must use for the repeat pulls is (t −τ(t)) · p/(1 −p). Let ℓ(t) ≤τ(t) be the number of distinct arms that get pulled more than once. Using Chernoff bounds, we can show that for any δ > 0, for sufficiently large t, with probability ≥1 −1/t2 the algorithm uses at least ℓ(t) = p(t −τ(t))/(1 −p) · (1 −δ) different arms for the repeat pulls. Call this event E1(δ). Next, we upper bound the expected reward of the best ℓ(t) arms found in τ(t) fresh probes. For any h > 0, let µ(h) = F −1(1 −(ℓ(t)/τ(t))(1 −h)). In other words, the probability of picking an arm with expected reward greater or equal to µ(h) is (ℓ(t)/τ(t))(1 −h). Applying the Chernoff bound, for any δ, h > 0 there exists t(δ, h) such that for all t ≥t(δ, h) the probability that the algorithm finds at least ℓ(t) arms with expected reward at least µ(δ, h) = µ(h)(1−δ) is bounded by 1/t2. Call this event E2(δ, h). Let E(δ, h) be the event E1(δ) ∧¬E2(δ, h). The expected reward of the algorithm in this event after t steps is then bounded by τ(t)E[X] + (t −τ(t))E[X | X ≥µ(δ, h)] Pr(E(δ, h)) + (t −τ(t))(1 − Pr(E(δ, h)). As δ, h →0, Pr(E(δ, h)) →1, and the expected reward per step when the algorithm pulls τ(t) fresh arms is given by lim sup t→∞ ¯µ(t) ≤1 t τ(t)E[X] + (t −τ(t))E[X | X ≥µ] , where µ = F −1(1 −ℓ(t)/τ(t)) and ℓ(t) = (t −τ(t))p/(1 −p). After some calculations, we get lim supt→∞¯µ(t) ≤maxµ Γ(µ). In Section 5.1 we present an algorithm that achieves this performance bound in the state-aware case. The following two simple reductions establish the lower bound for the timed death and the stateoblivious models. Lemma 2. Assuming that new arms are always available, any algorithm for the timed death model obtains at least the same reward per timestep in the budgeted death model. Although we omit the proof due to space constraints, the intuition behind this lemma is that an arm in the timed case can die no sooner than in the budgeted case (i.e., when it is always pulled). As a result, we get: Lemma 3. Let ¯µdet(t) and ¯µsto(t) denote the respective maximum mean expected rewards that any algorithm for the state-aware and state-oblivious mortal multi-armed bandit problems can obtain after running for t steps. Then ¯µsto(t) ≤¯µdet(t). We now present two applications of the upper bound. The first simply observes that if the time to find an optimal arm is greater than the lifetime of such an arm, the the mean reward per step of any algorithm must be smaller than the best value. This is in contrast to the standard MAB problem with the same reward distribution, where the mean regret per step tends to 0. Corollary 4. Assume that the expected reward of a bandit arms is 1 with probability p < 1/2 and 1 −δ otherwise, for some δ ∈(0, 1]. Let the lifetime of arms have geometric distribution with the same parameter p. The mean reward per step of any algorithm for this supply of arms is at most 1 −δ + δp, while the maximum expected reward is 1, yielding an expected regret per step of Ω(1). Corollary 5. Assume arm payoffs are drawn from a uniform distribution, F(x) = x, x ∈[0, 1]. Consider the timed death case with parameter p ∈(0, 1). Then the mean reward per step in bounded by 1−√p 1−p and expected regret per step of any algorithm is Ω(√p). 5 Bandit algorithms for mortal arms In this section we present and analyze a number of algorithms specifically designed for the mortal multi-armed bandit task. We develop the optimal algorithm for the state-aware case and then modify the algorithm to the state-oblivious case, yielding near-optimal regret. We also study a subset approach that can be used in tandem with any standard multi-armed bandit algorithm to substantially improve performance in the mortal multi-armed bandit setting. 5.1 The state-aware case We now show that the algorithm DETOPT is optimal for this deterministic reward setting. Algorithm DETOPT input: Distribution F(µ), expected lifetime L µ∗←argmaxµ Γ(µ) [Γ is defined in (1)] while we keep playing i ←random new arm Pull arm i; R ←R(µi) = µi if R > µ∗ [If arm is good, stay with it] Pull arm i every turn until it expires end if end while Assume the same setting as in the previous section, with a constant supply of new arms. The expected reward of an arm is drawn from cumulative distribution F(µ). Let X be a random variable with that distribution, and E[X] be its expectation over F(µ). Assume that the lifetime of an arm has an exponential distribution with parameter p, and denote its expectation by L = 1/p. Recall Γ(µ) from (1) and let µ∗= argmaxµ Γ(µ). Now, Theorem 6. Let DETOPT(t) denote the mean per turn reward obtained by DETOPT after running for t steps with µ∗= argmaxµ Γ(µ), then limt→∞DETOPT(t) = maxµ Γ(µ). Note that the analysis of the algorithm holds for both budgeted and timed death models. 5.2 The state-oblivious case We now present a modified version of DETOPT for the state-oblivious case. The intuition behind this modification, STOCHASTIC, is simple: instead of pulling an arm once to determine its payoff µi, the algorithm pulls each arm n times and abandons it unless it looks promising. A variant, called STOCHASTIC WITH EARLY STOPPING, abandons the arm earlier if its maximum possible future reward will still not justify its retention. For n = O log L/ϵ2 , STOCHASTIC gets an expected reward per step of Γ(µ∗−ϵ) and is thus near-optimal; the details are omitted due to space constraints. Algorithm STOCHASTIC input: Distribution F(µ), expected lifetime L µ∗←argmaxµ Γ(µ) [Γ is defined in (1)] while we keep playing [Play a random arm n times] i ←random new arm; r ←0 for d = 1, . . . , n Pull arm i; r ←r + R(µi) end for if r > nµ∗[If it is good, stay with it forever ] Pull arm i every turn until it dies end if end while Algorithm STOCH. WITH EARLY STOPPING input: Distribution F(µ), expected lifetime L µ∗←argmaxµ Γ(µ) [Γ is defined in (1)] while we keep playing [Play random arm as long as necessary] i ←random new arm; r ←0; d ←0 while d < n and n −d ≥nµ∗−r Pull arm i; r ←r + R(µi); d ←d + 1 end while if r > nµ∗ [If it is good, stay with it forever] Pull arm i every turn until it dies end if end while The subset heuristic. Why can’t we simply use a standard multi-armed bandit (MAB) algorithm for mortal bandits as well? Intuitively, MAB algorithms invest a lot of pulls on all arms (at least logarithmic in the total number of pulls) to guarantee convergence to the optimal arm. This is necessary in the traditional bandit settings, but in the limit as t →∞, the cost is recouped and leads to sublinear regret. However, such an investment is not justified for mortal bandits: the most gain we can get from an arm is L (if the arm has payoff 1), which reduces the importance of convergence to the best arm. In fact, as shown by Corollary 4, converging to a reasonably good arm suffices. However, standard MAB algorithms do identify better arms very well. This suggests the following epoch-based heuristic: (a) select a subset of k/c arms uniformly at random from the total k arms at the beginning of each epoch, (b) operate a standard bandit algorithm on these until the epoch ends, and repeat. Intuitively, step (a) reduces the load on the bandit algorithm, allowing it to explore less and converge faster, in return for finding an arm that is probably optimal only among the k/c subset. Picking the right c and the epoch length then depends on balancing the speed of convergence of the bandit algorithm, the arm lifetimes, and the difference between the k-th and the k/c-th order statistics of the arm payoff distribution; in our experiments, c is chosen empirically. Using the subset heuristic, we propose an extension of the UCB1 algorithm2 [2], called UCB1K/C, for the state-oblivious case. Note that this is just one example of the use of this heuristic; any standard bandit algorithm could have been used in place of UCB1 here. In the next section, UCB1K/C is shown to perform far better than UCB1 in the mortal arms setting. The ADAPTIVEGREEDY heuristic. Empirically, simple greedy MAB algorithms have previously been shown to perform well due to fast convergence. Hence for the purpose of evaluation, we also compare to an adaptive greedy heuristic for mortal bandits. Note that the ϵn-greedy algorithm [2] does not apply directly to mortal bandits since the probability ϵt of random exploration decays to zero for large t, which can leave the algorithm with no good choices should the best arm expire. Algorithm UCB1K/C input: k-armed bandit, c while we keep playing S ←k/c random arms dead ←0 AUCB1(S) ←Initialize UCB1 over arms S repeat i ←arm selected by AUCB1(S) Pull arm i, provide reward to AUCB1(S) x ←total arms that died this turn Check for newly dead arms in S, remove any dead ←dead + x until dead ≥k/2 or |S| = 0 end while Algorithm ADAPTIVEGREEDY input: k-armed bandit, c Initialization: ∀i ∈[1, k], ri, ni ←0 while we keep playing m ←argmaxi ri/ni [Find best arm so far] pm ←rm/nm With probability min(1, c · pm) j ←m Otherwise [Pull a random arm] j ←uniform(1, k) r ←R(j) rj ←rj + r [Update the observed rewards] nj ←nj + 1 end while 6 Empirical evaluation In this section we evaluate the performance of UCB1K/C, STOCHASTIC, STOCHASTIC WITH EARLY STOPPING, and ADAPTIVEGREEDY in the mortal arm state-oblivious setting. We also compare these to the UCB1 algorithm [2], that does not consider arm mortality in its policy but is among the faster converging standard multi-armed bandit algorithms. We present the results of simulation studies using three different distributions of arm payoffs F(·). Uniform distributed arm payoffs. Our performance analyses assume that the cumulative payoff distribution F(·) of new arms is known. A particularly simple one is the uniform distribution, µit ∼uniform(0, 1). Figure 1(a) shows the performance of these algorithms as a function of the expected lifetime of each arm, using a timed death and state-oblivious model. The evaluation was performed over k = 1000 arms, with each curve showing the mean regret per turn obtained by each algorithm when averaged over ten runs. Each run was simulated for ten times the expected lifetime of the arms, and all parameters were empirically optimized for each algorithm and each lifetime. Repeating the evaluation with k = 100, 000 arms produces qualitatively very similar performance. We first note the striking difference between UCB1 and UCB1K/C, with the latter performing far better. In particular, even with the longest lifetimes, each arm can be sampled in expectation at most 100 times. With such limited sampling, UCB1 spends almost all the time exploring and generates almost the same regret of 0.5 per turn as would an algorithm that pulls arms at random. In contrast, UCB1K/C is able to obtain a substantially lower regret by limiting the exploration to a subset of the arms. This demonstrates the usefulness of the K/C idea: by running the UCB1 algorithm on an appropriately sized subset of arms, the overall regret per turn is reduced drastically. In practice, 2UCB1 plays the arm j, previously pulled nj times, with highest mean historical payoff plus p (2 ln n)/nj. (a) 0 0.1 0.2 0.3 0.4 0.5 100 1000 10000 100000 Regret per time step Expected arm lifetime UCB1 UCB1-k/c Stochastic Stochastic with Early Stop. AdaptiveGreedy (b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 100 1000 10000 100000 Regret per time step Expected arm lifetime UCB1 UCB1-k/c Stochastic Stochastic with Early Stopping AdaptiveGreedy Figure 1: Comparison of the regret per turn obtained by five different algorithms assuming that new arm payoffs come from the (a) uniform distribution and (b) beta(1, 3) distribution. with k = 1000 arms, the best performance was obtained with K/C between 4 and 40, depending on the arm lifetime. Second, we see that STOCHASTIC outperformed UCB1K/C with optimally chosen parameters. Moreover, STOCHASTIC WITH EARLY STOPPING performs as well as ADAPTIVEGREEDY, which matches the best performance we were able to obtain by any algorithm. This demonstrates that (a) the state-oblivious versions of the optimal deterministic algorithm is effective in general, and (b) the early stopping criterion allows arms with poor payoff to be quickly weeded out. Beta distributed arm payoffs. While the strategies discussed perform well when arm payoffs are uniformly distributed, it is unlikely that in a real setting the payoffs would be so well distributed. In particular, if there are occasional arms with substantially higher payoffs, we could expect any algorithm that does not exhaustively search available arms may obtain very high regret per turn. Figure 1(b) shows the results when the arm payoff probabilities are drawn from the beta(1, 3) distribution. We chose this distribution as it has finite support yet tends to select small payoffs for most arms while selecting high payoffs occasionally. Once again, we see that STOCHASTIC WITH EARLY STOPPING and ADAPTIVEGREEDY perform best, with the relative ranking of all other algorithms the same as in the uniform case above. The absolute regret of the algorithms we have proposed is increased relative to that seen in Figure 1(a), but still substantially better than that of the UCB1. In fact, the regret of the UCB1 has increased more under this distribution than any other algorithm. Real-world arm payoffs. Considering the application that motivated this work, we now evaluate the performance of the four new algorithms when the arm payoffs come from the empirically observed distribution of clickthrough rates on real ads served by a large ad broker. Figure 2(a) shows a histogram of the payoff probabilities for a random sample of approximately 300 real ads belonging to a shopping-related category when presented on web pages classified as belonging to the same category. The probabilities have been linearly scaled such that all ads have payoff between 0 and 1. We see that the distribution is unimodal, and is fairly tightly concentrated. By sampling arm payoffs from a smoothed version of this empirical distribution, we evaluated the performance of the algorithms presented earlier. Figure 2(b) shows that the performance of all the algorithms is consistent with that seen for both the uniform and beta payoff distributions. In particular, while the mean regret per turn is somewhat higher than that seen for the uniform distribution, it is still lower than when payoffs are from the beta distribution. As before, STOCHASTIC WITH EARLY STOPPING and ADAPTIVEGREEDY perform best, indistinguishable from each other. 7 Conclusions We have introduced a new formulation of the multi-armed bandit problem motivated by the real world problem of selecting ads to display on webpages. In this setting the set of strategies available to a multi-armed bandit algorithm changes rapidly over time. We provided a lower bound of linear regret under certain payoff distributions. Further, we presented a number of algorithms that perform substantially better in this setting than previous multi-armed bandit algorithms, including one that is optimal under the state-aware setting, and one that is near-optimal under the state-oblivious setting. Finally, we provided an extension that allows any previous multi-armed bandit algorithm to be used (a) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 0.2 0.4 0.6 0.8 1 Fraction of arms Payoff probability (scaled) Ad Payoff Distribution (b) 0 0.1 0.2 0.3 0.4 0.5 100 1000 10000 100000 Regret per time step Expected arm lifetime Stochastic Stochastic with Early Stopping AdaptiveGreedy UCB1 UCB1-k/c Figure 2: (a) Distribution of real world ad payoffs, scaled linearly such that the maximum payoff is 1 and (b) Regret per turn under the real-world ad payoff distribution. in the case of mortal arms. Simulations on multiple payoff distributions, including one derived from real-world ad serving application, demonstrate the efficacy of our approach. Acknowledgments We would like to thank the anonymous reviewers for their helpful comments and suggestions. References [1] P. Auer. Using confidence bounds for exploitation-exploration trade-offs. J. Machine Learning Research, 3:397–422, 2002. [2] P. Auer, N. 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In SDM, pages 216–227, 2007. [17] S. Pandey, D. Chakrabarti, and D. Agarwal. Multi-armed bandit problems with dependent arms. In ICML, pages 721–728, 2007. [18] C. H. Papadimitriou and J. N. Tsitsiklis. The complexity of optimal queueing network control. In 9th CCC, pages 318–322, 1994. [19] A. Slivkins and E. Upfal. Adapting to a changing environment: The Brownian restless bandits. In 21st COLT, pages 343–354, 2008. [20] O. Teytaud, S. Gelly, and M. Sebag. Anytime many-armed bandits. In CAP, 2007. [21] T.Lai and H.Robbins. Asymptotically efficient adaptive allocation rules. Adv. Appl. Math., 6:4–22, 1985. [22] P. Whittle. Arm-acquiring bandits. The Annals of Probability, 9(2):284–292, 1981. [23] P. Whittle. Restless bandits: Activity allocation in a changing world. J. of Appl. Prob., 25A:287–298, 1988.
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Adapting to a Market Shock: Optimal Sequential Market-Making Sanmay Das Department of Computer Science Rensselaer Polytechnic Institute Troy, NY 12180 sanmay@cs.rpi.edu Malik Magdon-Ismail Department of Computer Science Rensselaer Polytechnic Institute Troy, NY 12180 magdon@cs.rpi.edu Abstract We study the profit-maximization problem of a monopolistic market-maker who sets two-sided prices in an asset market. The sequential decision problem is hard to solve because the state space is a function. We demonstrate that the belief state is well approximated by a Gaussian distribution. We prove a key monotonicity property of the Gaussian state update which makes the problem tractable, yielding the first optimal sequential market-making algorithm in an established model. The algorithm leads to a surprising insight: an optimal monopolist can provide more liquidity than perfectly competitive market-makers in periods of extreme uncertainty, because a monopolist is willing to absorb initial losses in order to learn a new valuation rapidly so she can extract higher profits later. 1 Introduction Designing markets to achieve certain goals is gaining renewed importance with the prevalence of many novel markets, ranging from prediction markets [13] to markets for e-services [11]. These markets tend to be thin (illiquid) when they first appear. Similarly, when a market shock occurs to the value of an instrument on a financial exchange, thousands of speculative traders suddenly possess new valuations on the basis of which they would like to trade. Periods of uncertainty, like those following a shock, are also periods of illiquidity, so trading may be sparse right after a shock. This is a chicken-and-egg problem. People do not want to trade in thin markets, and yet, having many people trading is what creates liquidity. These markets therefore need to be bootstrapped into a phase where they are sufficiently liquid to attract trading. This bootstrapping is often achieved through market-makers [12]. Market-makers are responsible for providing liquidity and maintaining order on the exchange. For example, the NYSE designates a single monopolist specialist (marketmaker) for each stock, while the NASDAQ allows multiple market-makers to compete. There has been much debate on whether one of these models is better than the other. This debate is again important today for those who are designing new markets. Should they employ a single monopolistic market-maker or multiple competitive market-makers? Alternatively, should the market-maker be based on some other criterion, and if so, what is the optimal design for this agent? Market makers want to maximize profit, which could run contrary to their “social responsibility” of providing liquidity. A monopolist market maker attempts to maximize expected discounted profits, while competitive (non-colluding) market makers may only expect zero profit, since any profits should be wiped out by competition. Therefore, one would expect markets with competitive marketmakers to be of better quality. However, this has not been observed in practice, especially in the well-studied case of the NASDAQ vs. the NYSE [1, 9]. Many explanations have been proposed in the empirical literature, and have explained parts of this phenomenon. One reason that has been speculated about anecdotally but never analyzed formally is the learning aspect of the problem. For 1 example, the NYSE’s promotional literature used to tout the benefits of a monopolist for “maintaining a fair and orderly market” in the face of market shocks [6]. The main challenge to formally analyzing this question is the complexity of the monopolistic market maker’s sequential decision problem. The market maker, when setting bid and ask prices, is plagued by a heavily path dependent exploitation-exploration dilema. There is a tradeoff between setting the prices to extract maximum profit from the next trade versus setting the prices to get as much information about the new value of the instrument so as to generate larger profits from future trades. There is no known solution to this sequential decision problem. We present the first such solution within an established model of market making. We show the surprising fact that a monopolist market maker leads to higher market liquidity in periods of extreme market shock than does a zero-profit competitive market maker. In various single period settings, it has been shown that monopolists can sometimes provide greater liquidity [6] by averaging expected profits across different trade sizes. We show for the first time that this can hold true with fixed trade sizes in a multi-period setting, because the market-maker is willing to take losses following a shock in order to learn the new valuation more quickly. 1.1 Market Microstructure Background Market microstructure has recently received much attention from a computational perspective [10, 4, 12]. The driving problem of this paper is price discovery. Suppose an instrument has just begun trading in a market where different people have different beliefs about its value. An example is shares in the “Barack Obama wins the presidential election” market. These shares should trade at prices that reflect the probability that the event will occur: if the outcome pays off $100, the shares should trade at about $55 if the aggregate public belief is 55% that the event will occur. Similarly, the price of a stock should reflect the aggregate public belief about future cash flows associated with a company. It is well-known that markets are good at aggregating information into prices, but different market structures possess different qualities in this regard. We are concerned with the properties of dealer markets, in which prices are set by one or more market-makers responsible for providing liquidity by taking one side of every trade. Market-making has been studied extensively in the theoretical market microstructure literature [8, 7, for example], but only recently has the dynamic multi-period problem gained attention [2, 3]. Since we are interested in the problem of how a market-maker learns a value for an asset, we follow the general model of Glosten and Milgrom which abstracts away from the problem of quantities by restricting attention to situations where the market-maker places bid and ask quotes for one unit of the asset at each time step. Das [3] has extended this model to consider the market-maker’s learning problem with competitive pricing, while Darley et al [2] have used similar modeling for simulations of the NASDAQ. The Glosten and Milgrom model has become a standard model in this area. Liquidity, which is not easy to quantify, is the prime social concern. In practice, it is a function of the depth of the limit order book. In our models, we measure liquidity using the bid-ask spread, or alternatively the probability that a trade will occur. This gives a good indication of the level of informational heterogeneity in the market, and of execution costs. The dynamic behavior of the spread gives insight into the price discovery process. 1.2 Our Contribution We consider the question of optimal sequential price-setting in the Glosten-Milgrom model. The market-maker sets bid and ask prices at each trading period1 and when a trader arrives she has the option of buying or selling at those prices, or of not executing a trade. There are many results relating to the properties of zero-profit (competitive) market-makers [7, 3]. The zero-profit problem is a single-period decision-making problem with online belief updates. Within this same framework, one can formulate the decision problem for a monopolist market-maker who maximizes her total discounted profit as a reinforcement learning problem. The market maker’s state is her belief about the instrument value, and her action is to set bid and ask prices. The market maker’s actions must trade off profit taking (exploitation) with price discovery (exploration). 1The MM is willing to buy at the bid price and sell at the ask price. 2 The complexity of the sequential problem arises from the complexity of the state space and the fact that the action space is continuous. The state of the market-maker must represent her belief about the true value of the asset being traded. As such, it is a probability density function. In a parametric setting, the state space is finite dimensional, but continuous. Even if we assume a Gaussian prior for the market-maker’s belief as well as for the beliefs of all the traders, the market-maker’s beliefs quickly become a complex product of error functions, and the exact dynamic programming problem becomes intractable. We solve the Bellman equation for the optimal sequential market maker within the framework of Gaussian state space evolution, a close approximation to the true state space evolution. We present simulation results which testify to how closely the Gaussian framework approximates the true evolution. The Gaussian approximation alone does not alleviate the difficulties associated with reinforcement learning in continuous action and state spaces.2 However within our setting, we prove a key monotonicity property for the state update. This property allows us to solve for the value function exactly using a single pass dynamic program. Thus, our first contribution is a complete solution to the optimal sequential market making problem within a Gaussian update framework. Our second contribution relates to the phenomenological implications for market behavior. We obtain the surprising result that in periods of extreme shock, when the market maker has large uncertainty relative to the traders, the monopolist provides greater liquidity than competitive zero-profit market-makers. The monopolist increases liquidity, possibly taking short term losses, in order to learn more quickly, and in doing so offers the better social outcome. Of course, once the monopolist has adapted to the shock, she equilibrates at a higher bid ask spread than the the corresponding zero-profit market maker with the same beliefs. 2 The Model and the Sequential Decision Problem 2.1 Market Model At time 0, a shock occurs causing an instrument to attain value V which will be held fixed through time (we consider one instrument in the market). This could represent a real market shock to a stock value (change in public beliefs), an IPO, or the introduction of a new contract in a prediction market. We use a model similar to Das’s [3] extension of the Glosten and Milgrom [7] model. We assume that trading is divided into a sequence of discrete trading time steps, each time step corresponding to the arrival of a trader. The value V is drawn from some distribution gV (v). The market-maker (MM), at each time step t ≥0, sets bid and ask prices bt ≤at at which she is willing to respectively buy and sell one unit. Traders arrive at time-steps t ≥0. Trader t arrives with a noisy estimate wt of V , where wt = V + ϵt. The {ϵt} are zero mean i.i.d. random variables with distribution function Fϵ. We will assume that Fϵ is symmetric, so that Fϵ(−x) = 1 −Fϵ(x). The trader decides whether to trade at either the bid or ask prices depending on the value of wt. The trader will buy at at if wt > at (she thinks the instrument is undervalued), sell at bt if wt < bt (she thinks the instrument is overvalued) and do nothing otherwise. MM receives a signal xt ∈{+1, 0, −1} indicating whether the trader bought, did nothing or sold. Note that information is conveyed only by the direction of the trade. Information can also be conveyed by the patterns and size of trades, but the present work abstracts away from those considerations. The market-maker’s objective is to maximize profit. In perfect competition, the MM is pushed to setting bid and ask prices that yield zero expected profit. In a monopolistic setting, she wants to optimize the profits she receives over time. As we will see below, this can be a difficult problem to solve. A commonly used alternative is to consider a greedy, or myopically optimal MM who only maximizes her expected profit from the next trade. This is a good approximation for agents with a high discount factor, since they are more concerned with immediate reward. We will consider all three types of market-makers, (1) Zero-profit, (2) Myopic, and (3) Optimal. 2Where one has to resort to unbounded value iteration methods whose convergence and uniqueness properties are little understood. 3 2.2 State Space The state space for the MM is determined by MM’s belief about the value V , described by a density function pt at time step t. The MM decides on actions (bid and ask prices) (bt, at) based on pt. The MM receives signal xt ∈{+1, 0, −1} as to whether the trader bought, sold, or did nothing. Let qt(V ; bt, at) be the probability of receiving signal xt given bid and ask (bt, at), conditioned on V . Assuming that Fϵ is continuous at bt −V and at −V , a straightforward calculation yields qt(V ; bt, at) = 1 −Fϵ(at −V ) xt = +1, Fϵ(at −V ) −Fϵ(bt −V ) xt = 0, Fϵ(bt −V ) xt = −1, or, qt(V ; bt, at) = Fϵ(z+ t −V ) −Fϵ(z− t −V ), where z+ t and z− t are respectively +∞, at, bt and at, bt, −∞when xt = +1, 0, −1. The Bayesian update to pt is then given by pt+1(v) = pt(v) qt(v;bt,at) At , where the normalization constant At = R ∞ −∞dv pt(v)qt(v; bt, at). Unfolding the recursion gives pt+1(v) = p0(v) Qt τ=1 qτ (v;bτ ,aτ ) Aτ 2.3 Solving for Market Maker Prices Let bt ≤at, and let rt be the expected profit at time t. The expected discounted return is then R = P∞ t=0 γtrt where 0 < γ < 1 is the discount factor. The optimal MM maximizes R. We can compute rt as rt = R ∞ −∞dv vFϵ(−v) (pt(v + bt) + pt(at −v)). rt decomposes into two terms which can be identified as the bid and ask side profits, rt = rbid t (bt) + rask t (at). In perfect competition, MM should not be expecting any profit on either the bid or ask side. This is because if the contrary were true, a competing MM could place bid or ask prices so as to obtain less profit, wiping out MM’s advantage. This should hold at every time step. Hence the MM will set bid and ask prices such that rbid t (bt) = 0 and rask t (at) = 0. Solving for bt, at, we find that bt and at must satisfy the following fixed point equations (these are also derived for the case of Gaussian noise by Das [3]), bt = R ∞ −∞dv vpt(v)Fϵ(bt −v) R ∞ −∞dv pt(v)Fϵ(bt −v) = Ept[V |xt = −1], at = R ∞ −∞dv vpt(v)Fϵ(v −at) R ∞ −∞dv pt(v)Fϵ(v −at) = Ept[V |xt = +1] (assuming the denominators, which are the conditional probabilities of hitting the bid or ask are non-zero). The myopic monopolist maximizes rt. For the typical case of well behaved distributions pt(v) and Fϵ, the bid and ask returns display a single maximum. In this case, we can obtain bmyp t and amyp t by setting the derivatives to zero (we assume the functions are well behaved so that the derivatives are defined). Letting fϵ(x) = F ′ ϵ(x) be the density function for the noise ϵt, bmyp t and amyp t satisfy the fixed point equations bt = R ∞ −∞dv pt(v)(vfϵ(bt −v) −Fϵ(bt −v)) R ∞ −∞dv pt(v)fϵ(bt −v) , at = R ∞ −∞dv pt(v)(vfϵ(at −v) + Fϵ(v −at)) R ∞ −∞dv pt(v)fϵ(at −v) The optimal strategy for MM is not as easy to obtain. When γ is large, the expected discounted return R could be significantly higher than the myopic return. The optimal MM might choose to sacrifice short term return for a substantially larger return over the long term. The only reason to do this is if choosing a sub-optimal short term strategy will lead to a significant decrease in the uncertainty in V (which translates to a narrowing of the probability distribution pt(v)). MM can then exploit this more certain information regarding V in the longer term. The optimal strategy for the MM is encapsulated in the Bellman equation for the value functional (where the state pt, is a function, (bt, at) is the action, and π is a policy): V (pt; π) = E[r0|pt, bπ t (pt), aπ t (pt)] + γE[V (pt+1; π)|pt, bπ t (pt), aπ t (pt)] This equation reflects the fact that the MM’s expected profit is a function of both her immediate expected return, and her future state, which is also affected by her bid and ask prices. The fact that V is a value functional leads to numerous technical problems when solving this Bellman equation. The problem is heavily path dependent with the number of paths being exponential in the number of trading periods. To make this tractable, we use a Gaussian approximation for the state space evolution. 4 1 2 3 4 5 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Exact vs. Approximate Belief Update Value Probability 2 steps 5 steps 20 steps Figure 1: Gaussian state update (dashed) versus true state update (solid) illustrating that the Gaussian approximation is valid. I(α, β) = Z ∞ −∞ dx N(x) Z α−βx −∞ dy N(y) = Φ α p 1 + β2 ! , J(α, β) = Z ∞ −∞ dx x · N(x) Z α−βx −∞ dy N(y) = − s β2 1 + β2 · N α p 1 + β2 ! , K(α, β) = Z ∞ −∞ dx x2 e−x2/2 √ 2π Z α−βx −∞ dy e−y2/2 √ 2π = I(α, β) − αβ2 (1 + β2)3/2 · N α p 1 + β2 ! L(α, β) = I(α, β) −K(α, β) A(z+, z−) = I z+ −µt σϵ , ρt ! −I z−−µt σϵ , ρt ! B(z+, z−) = J z+ −µt σϵ , ρt ! −J z−−µt σϵ , ρt ! C(z+, z−) = L z+ −µt σϵ , ρt ! −L z−−µt σϵ , ρt ! Figure 2: Gaussian integrals and normalization constants used in the derivation of the DP and the state updates. 2.4 The Gaussian Approximation From a Gaussian prior and performing Bayesian updates, one expects that the state distribution will be closely approximated by a Gaussian (see Figure 1). Thus, forcing the MM to maintain a Gaussian belief over the true value at each time t should give a good approximation to the true state space evolution, and the resulting optimal actions should closely match the true optimal actions. In making this reduction, we reduce the state space to a two parameter function class parameterized by the mean and variance, (µt, σ2 t ). The value function is independent of µt (hence dependent only on σt), and the optimal action is of the form bt = µt −δt, at = µt + δt. Thus, V (σt) = max δ {rt(σt, δ) + γE[V (σt+1)|δ]} (1) To compute the expectation on the RHS, we need the probabilistic dynamics in the (approximate) Gaussian state space, i.e., we need the evolution of µt, σt. Let N(·), Φ(·) denote the standard normal density and distribution. Let pt(v) = 1 σt N v−µt σt be Gaussian with mean µt and variance σ2 t . Assume that the noise is also Gaussian with variance σ2 ϵ , so Fϵ(x) = Φ( x σϵ ). At time t + 1, after the Bayesian update, we have pt+1 = 1 A · 1 σt N v−µt σt h Φ z+−v σϵ −Φ z−−v σϵ i . The normalization constant A(z+, z−) is given in Figure 2, and z+ t and z− t are respectively +∞, at, bt and at, bt, −∞when xt = +1, 0, −1. The updates µt+1 and σ2 t+1 are obtained from Ept+1[V ] = R dv vpt+1(v) and Ept+1[V 2] = R dv v2pt+1(v). After some tedious algebra (see supplementary information), we obtain µt+1 = µt + σt · B A, (2) σ2 t+1 = σ2 t 1 −AC + B2 A2 . (3) Figure 2 gives the expressions for A, B, C. Theorem 2.1 (Monotonic state update). σ2 t+1 ≤σ2 t (see supplementary information for proof). 5 Establishing that σt is decreasing in t allows us to solve the dynamic program efficiently (note that the property of decreasing variance is well-known for the case of an update to a Gaussian prior when the observation is also Gaussian – we are showing this for threshold observations). 2.5 Solving the Bellman Equation We now return to the Bellman equation (1). In light of Theorem 2.1, the RHS of this equation is dependent only on states σt+1 that are strictly smaller than the state σt on the LHS. We can thus solve this problem numerically by computing V (0) and then building up the solution for a fine grid on the real line. We use linear interpolation between previously computed points if the variance update leads to a point not on the grid. We need to explicitly construct the states on the RHS with respect to which the expectation is being taken. The expectation is with respect to the future state σt+1, which depends directly on the trade outcome xt ∈{−1, 0, +1}. We define ρt = σt/σϵ and q = δt/σϵ p 1 + ρ2 t, where at = µt + δt and bt = µt −δt. The following table sumarizes some of the useful quantities: xt Prob. µt+1 σt+1 +1 1 −Φ(qt) µt + κtσt αtσt 0 2Φ(qt) −1 µt βtσt −1 1 −Φ(qt) µt −κtσt αtσt where α2 t = 1 − ρ2 t N(qt)(N(qt) −qt[1 −Φ(qt)]) (1 + ρ2 t )(1 −Φ(qt))2 β2 t = 1 − 2ρ2 t qtN(qt) (1 + ρ2 t )(2Φ(qt) −1) κt = v u u t ρ2 t 1 + ρ2 t N(qt) 1 −Φ(qt) Note that qt > 0, αt, βt < 1 and κt > 0. We can now compute E[V (σt+1|δt)] as 2(1 −Φ(qt))V (αtσt) + (2Φ(qt) −1)V (βtσt). This allows us to complete the specification for the Bellman equation (with x = ρ2 t where ρt = σ σϵ is the MM’s information disadvantage) V (x; σϵ) = max q 2σ2 ϵ √ 1 + x „ q(1 −Φ(q)) − x 1 + xN(q) « + γ ˆ 2(1 −Φ(q))V (α2(x, q)x; σϵ) + (2Φ(q) −1)V (β2(x, q)x; σϵ) ˜ff where α2(x, q) and β2(x, q) are as defined above with ρ2 t = x and qt = q. We define the optimal action q∗(x) as the value of q that maximizes the RHS. When x = 0, the myopic and optimal MM coincide, and so we have that V (0) = 2q∗(1−Φ(q∗)) 1−γ , where q∗= q∗(0) ≈0.7518 satisfies q∗N(q∗) = 1 −Φ(q∗). Note that if we only maximize the first term in the value function, we obtain the myopic action qmyp(ρ), satisfying the fixed point equation: qmyp = (1 + ρ2 t) 1−Φ(qmyp) N(qmyp) . There is a similarly elegant solution for the zero-profit MM under the Gaussian assumption, obtained by setting rt = 0, yielding the fixed point equation: qzero = ρ2 t 1+ρ2 t N(qzero) 1−Φ(qzero). 10 standard fixed point iterations are sufficient to solve these equations accurately. 3 Experimental Results First, we validate the Gaussian approximation by simulating a market as follows. The initial value V is drawn from a Gaussian with mean 0 and standard deviation σ, and we set the discount rate γ = 0.9. Each simulation consists of 100 trading periods at which point discounted returns become negligible. At each trading step t, a new trader arrives with a valuation wt ∼N(V, 1) (Gaussian with mean V and variance 1). We report results averaged over more than 10,000 simulations, each with a randomly sampled value of V . In each simulation, the market-maker’s state updates are given by the Gaussian approximation (2), (3), according to which she sets bid and ask prices. The trader at time-step t trades by comparing wt to bt, at. We simulate the outcomes of the optimal, myopic, and zero-profit MMs. An alternative 6 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 MM Information Disadvantage ! Discounted Profit Profit Vs. MM Information Disadvantage Opt. Myopic Zero Profit 0 1 2 3 4 0 2 4 6 8 10 MM Information Disadvantage ! Spread (2"/#$) Spread Vs. MM Information Disadvantage Opt. Myopic Zero Profit 5 10 15 20 25 30 35 !1 !0.5 0 0.5 1 Time Step t Profit Profits Over Time Opt. Myopic Zero Profit (a) Realized vs theoretical value function in the Gaussian approximation (thin black line). The realized closely matches the theoretical, validating the Gaussian framework. (b) Bid-ask spreads as a function of the MM information disadvantage ρ indicating that once ρ exceeds about 1.5, the monopolist offers the greatest liquidity. (c) Realized average return as a function of time: the monopolist is willing to take significant short term loss to improve future profits as a result of better price discovery. Figure 3: MM Properties derived from the solution of the Bellman equation. is to maintain the exact state as a product of error functions, and extract the mean and variance for computing the optimal action. This is computationally prohibitive, and leads to no significant differences. If the real world conformed to the MM’s belief, a new value Vt would be drawn from N(µt, σt) at each trading period t, and then the trader would receive a sample wt ∼N(Vt, 1). All our computations are exact within this “Gaussian” world, however the point here is to test the degree to which the Gaussian and real worlds differ. The ideal test of our optimal MM is against the true optimal for the real world, which is intractable. However, if we find that the theoretical value function for the optimal MM in the Gaussian world matches the realized value function in the real world, then we have strong, though not necessarily conclusive, evidence for two conclusions: (1) The Gaussian world is a good approximation to the real world, otherwise the realized and theoretical value functions would not coincide; (2) Since the two worlds are nearly the same, the optimal MM in the Gaussian world should closely match the true optimal. Figure 3(a) presents results which show that the realized and theoretical value functions are essentially the same, presenting the desired evidence (note that with independent updates, the posterior should be asymptotically Gaussian). Figure 3(a) also demonstrates that the optimal significantly outperforms the myopic market-maker. Figure 3(b) shows how the bid-ask spread will behave as a function of the MM information disadvantage. Some phenomenological properties of the market are shown in Figure 4.3 For a starting MM information disadvantage of ρ = 3, the optimal MM initially has significantly lower spread, even compared with the zero profit market-maker. The reason for this outcome is illustrated in Figure 3(c) where we see that the optimal market maker is offering lower spreads and taking on significant initial loss to be compensated later by significant profits due to better price discovery. At equilibrium the optimal MM’s spread and the myopic spread are equal, as expected. 4 Discussion Our solution to the Bellman equation for the optimal monopolistic MM leads to the striking conclusion that the optimal MM is willing to take early losses by offering lower spreads in order to make significantly higher profits later (Figures 3(b,c) and 4). This is quantitative evidence that the optimal MM offers more liquidity than a zero-profit MM after a market shock, especially when the MM is at a large information disadvantage. In this regime, exploration is more important than exploitation. Competition may actually impede the price discovery process, since the market makers would have no incentive to take early losses for better price discovery – competitive pricing is not necessarily informationally efficient (there are quicker ways for the market to “learn” a new valuation). 3With both zero-profit and optimal MMs we reproduce one of the key findings of Das [3]: the market exhibits a two-regime behavior. Price jumps are immediately followed by a regime of high spreads (the pricediscovery regime), and then when the market-maker learns the new valuation, the market settles into an equilibrium regime of lower spreads (the efficient market regime). 7 0 10 20 30 40 0 2 4 6 8 10 Time Step t Bid!Ask Spread Expected Bid!Ask Spread Dynamics Opt. Myopic Zero Profit 5 10 15 20 25 30 35 0 0.2 0.4 0.6 0.8 1 Time Step t Trade Probability Trading Activity Opt. Myopic Zero Profit 0 1 2 3 4 5 0 10 20 30 40 50 60 70 80 MM Information Disadvantage ! Time To Stabilization Spread Stabilization Rate Opt. Myopic Zero Profit (a) Realized spread over time (σ = 3). The optimal MM starts with lowest spread, and converges quickest to equilibrium. (b) Liquidity over time (σ = 3), measured by probability of a trade. Initial liquidity is higher for the optimal MM. (c) Time to spread stabilization. When MM’s information disadvantage increases, the optimal MM is significantly better. Figure 4: Realized market properties based on simulating the three MMs. Our solution is based on reducing a functional state space to a finite-dimensional one in which the Bellman equation can be solved efficiently. When the state is a probability distribution, updated according to independent events, we expect the Gaussian approximation to closely match the real state evolution. Hence, our methods may be generally applicable to problems of this form. While this paper presents a stylized model, simple trading models have been shown to produce rich market behavior in many cases (for example, [5]). The results presented here are an example of the kinds of insights that can be be gained from studying market properties in these models while approaching agent decision problems from the perspective of machine learning. At the same time, this paper is not purely theoretical. The eventual algorithm we present is easy to implement, and we are in the process of evaluating this algorithm in test prediction markets. Another direction we are pursuing is to endow the traders with intelligence, so they may learn the true value too. We believe the Gaussian approximation admits a solution for a monopolistic market-maker and adaptive traders. References [1] W.G. Christie and P.H. Schulz. Why do NASDAQ market makers avoid odd-eighth quotes? J. Fin., 49(5), 1994. [2] V. Darley, A. Outkin, T. Plate, and F. Gao. Sixteenths or pennies? Observations from a simulation of the NASDAQ stock market. In IEEE/IAFE/INFORMS Conf. on Comp. Intel. for Fin. Engr., 2000. [3] S. Das. A learning market-maker in the Glosten-Milgrom model. Quant. Fin., 5(2):169–180, April 2005. [4] E. Even-Dar, S.M. Kakade, M. Kearns, and Y. Mansour. (In)stability properties of limit order dynamics. In Proc. ACM Conf. on Elect. Comm., 2006. [5] J.D. Farmer, P. Patelli, and I.I Zovko. The predictive power of zero intelligence in financial markets. PNAS, 102(11):2254–2259, 2005. [6] L.R. Glosten. Insider trading, liquidity, and the role of the monopolist specialist. J. Bus., 62(2), 1989. [7] L.R. Glosten and P.R. Milgrom. Bid, ask and transaction prices in a specialist market with heterogeneously informed traders. J. Fin. Econ., 14:71–100, 1985. [8] S.J. Grossman and M.H. Miller. Liquidity and market structure. J. Fin., 43:617–633, 1988. [9] Roger D. Huang and Hans R. Stoll. Dealer versus auction markets: A paired comparison of execution costs on NASDAQ and the NYSE. J. Fin. Econ., 41(3):313–357, 1996. [10] S.M. Kakade, M. Kearns, Y. Mansour, and L. Ortiz. Competitive algorithms for VWAP and limit-order trading. In Proc. ACM Conf. on Elect. Comm., pages 189–198, 2004. [11] Juong-Sik Lee and Boleslaw Szymanski. Auctions as a dynamic pricing mechanism for e-services. In Cheng Hsu, editor, Service Enterprise Integration, pages 131–156. Kluwer, New York, 2006. [12] D. Pennock and R. Sami. Computational aspects of prediction markets. In N. Nisan, T. Roughgarden, E. Tardos, and V.V. Vazirani, editors, Algorithmic Game Theory. Cambridge University Press, 2007. [13] Justin Wolfers and Eric Zitzewitz. Prediction markets. J. Econ. Persp., 18(2):107–126, 2004. 8
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Accelerating Bayesian Inference over Nonlinear Differential Equations with Gaussian Processes Ben Calderhead Dept. of Computing Sci. University of Glasgow bc@dcs.gla.ac.uk Mark Girolami Dept. of Computing Sci. University of Glasgow girolami@dcs.gla.ac.uk Neil D. Lawrence School of Computer Sci. University of Manchester neill@cs.man.ac.uk Abstract Identification and comparison of nonlinear dynamical system models using noisy and sparse experimental data is a vital task in many fields, however current methods are computationally expensive and prone to error due in part to the nonlinear nature of the likelihood surfaces induced. We present an accelerated sampling procedure which enables Bayesian inference of parameters in nonlinear ordinary and delay differential equations via the novel use of Gaussian processes (GP). Our method involves GP regression over time-series data, and the resulting derivative and time delay estimates make parameter inference possible without solving the dynamical system explicitly, resulting in dramatic savings of computational time. We demonstrate the speed and statistical accuracy of our approach using examples of both ordinary and delay differential equations, and provide a comprehensive comparison with current state of the art methods. 1 Introduction Mechanistic system modeling employing nonlinear ordinary or delay differential equations 1 (ODEs or DDEs) is oftentimes hampered by incomplete knowledge of the system structure or the specific parameter values defining the observed dynamics [16]. Bayesian, and indeed non-Bayesian, approaches for parameter estimation and model comparison [19] involve evaluating likelihood functions, which requires the explicit numerical solution of the differential equations describing the model. The computational cost of obtaining the required numerical solutions of the ODEs or DDEs can result in extremely slow running times. In this paper we present a method for performing Bayesian inference over mechanistic models by the novel use of Gaussian processes (GP) to predict the state variables of the model as well as their derivatives, thus avoiding the need to solve the system explicitly. This results in dramatically improved computational efficiency (up to four hundred times faster in the case of DDEs). We note that state space models offer an alternative approach for performing parameter inference over dynamical models particularly for on-line analysis of data, see [2]. Related to the work we present, we also note that in [6] the use of GPs has been proposed in obtaining the solution of fully parameterised linear operator equations such as ODEs. Likewise in [12] GPs are employed as emulators of the posterior response to parameter values as a means of improving the computational efficiency of a hybrid Monte Carlo sampler. Our approach is different and builds significantly upon previous work which has investigated the use of derivative estimates to directly approximate system parameters for models described by ODEs. A spline-based approach was first suggested in [18] for smoothing experimental data and obtaining derivative estimates, which could then be used to compute a measure of mismatch for derivative values obtained from the system of equations. More recent developments of this method are described in [11]. All of these approaches, however, are plagued by similar problems. The methods 1The methodology in this paper can also be straightforwardly extended to partial differential equations. are all critically dependent on additional regularisation parameters to determine the level of data smoothing. They all exhibit the problem of providing sub-optimal point estimates; even [11] may not converge to a reasonable solution depending on the initial values selected, as we demonstrate in Section 5.1. Furthermore, it is not at all obvious how these methods can be extended for partially observed systems, which are typical in, e.g. systems biology [10, 1, 8, 19]. Finally, these methods only provide point estimates of the “correct” parameters and are unable to cope with multiple solutions. (Although it should be noted that [11] does offer a local estimate of uncertainty based on second derivatives, at additional computational cost.) It is therefore unclear how objective model comparison could be implemented using these methods. In contrast we provide a Bayesian solution, which is capable of sampling from multimodal distributions. We demonstrate its speed and statistical accuracy and provide comparisons with the current best methods. It should also be noted that the papers mentioned above have focussed only on parameter estimation for fully observed systems of ODEs; we additionally show how parameter inference over both fully and partially observed ODE systems as well as DDEs may be performed efficiently using our state derivative approach. 2 Posterior Sampling by Explicit Integration of Differential Equations A dynamical system may be described by a collection of N ordinary differential equations and model parameters θ which define a functional relationship between the process state, x(t), and its time derivative such that ˙x(t) = f(x, θ, t). Likewise delay differential equations can be used to describe certain dynamic systems, where now an explicit time-delay τ is employed. A sequence of process observations, y(t), are usually contaminated with some measurement error which is modeled as y(t) = x(t) + ϵ(t) where ϵ(t) defines an appropriate multivariate noise process, e.g. a zero-mean Gaussian with variance σ2 n for each of the N states. If observations are made at T distinct time points the N ×T matrices summarise the overall observed system as Y = X+E. In order to obtain values for X the system of ODEs must be solved, so that in the case of an initial value problem X(θ, x0) denotes the solution of the system of equations at the specified time points for the parameters θ and initial conditions x0. Figure 1(a) illustrates graphically the conditional dependencies of the overall statistical model and from this the posterior density follows by employing appropriate priors such that p(θ, x0, σ|Y) ∝π(θ)π(x0)π(σ) Q n NYn,·(X(θ, x0)n,·, Iσ2 n). The desired marginal p(θ|Y) can be obtained from this joint posterior2. Various sampling schemes can be devised to sample from the joint posterior. However, regardless of the sampling method, each proposal requires the specific solution of the system of differential equations which, as will be demonstrated in the experimental sections, is the main computational bottleneck in running an MCMC scheme for models based on differential equations. The computational complexity of numerically solving such a system cannot be easily quantified since it depends on many factors such as the type of model and its stiffness, which in turn depends on the specific parameter values used. A method to alleviate this bottleneck is the main contribution of this paper. 3 Auxiliary Gaussian Processes on State Variables Let us assume independent3 Gaussian process priors on the state variables such that p(Xn,·|ϕn) = N(0, Cϕn), where Cϕn denotes the matrix of covariance function values with hyperparameters ϕn. With noise ϵn ∼N(0, σ2 nIT ), the state posterior, p(Xn,·|Yn,·, σn, ϕn) follows as N(µn, Σn) where µn = Cϕn(Cϕn + σ2 nI)−1Yn,· and Σn = σ2 nCϕn(Cϕn + σ2 nI)−1. Given priors π(σn) and π(ϕn) the corresponding posterior is p(ϕn, σn|Yn,·) ∝π(σn)π(ϕn)NYn,·(0, σ2 nI + Cϕn) and from this we can obtain the joint posterior, p(X, σn=1···N, ϕn=1···N|Y, ), over a non-parametric GP model of the state-variables. Note that a non-Gaussian noise model may alternatively be implemented using warped GPs [14]. The conditional distribution for the state-derivatives is 2This distribution is implicitly conditioned on the numerical solver and associated error tolerances. 3The dependencies between state variables can be modeled by defining the overall state vector as x = vec(X) and using a GP prior of the form x ∼N(0, Σ ⊗C) where ⊗denotes the Kronecker matrix product and Σ is an N × N positive semi-definite matrix specifying inter-state similarities with C, the T × T matrix defining intra-state similarities [13]. (a) (b) (c) Figure 1: (a) Graphical model representing explicit solution of an ODE system, (b) Graphical model representing approach developed in this paper with dashed lines showing how the two models are combined in product form, (c) Likelihood surface for a simple oscillator model p( ˙Xn,·|Xn,·, ϕn, σn) = N(mn, Kn), where the mean and covariance are given by mn = ′Cϕn(Cϕn + σ2 nI)−1Xn,· and Kn = C ′′ ϕn −′Cϕn(Cϕn + σ2 nI)−1C ′ ϕn where C ′′ ϕn denotes the auto-covariance for each state-derivative with C ′ ϕn and ′Cϕn denoting the cross-covariances between the state and its derivative [13, 15]. The main advantage of using the Gaussian process model now becomes apparent. The GP specifies a jointly Gaussian distribution over the function and its derivatives ([13], pg.191). This allows us to evaluate a posterior over parameters θ consistent with the differential equation based on the smoothed state and state derivative estimates, see Figure 1(b). Assuming Normal errors between the state-derivatives ˙Xn,· and the functional, fn(X, θ, t) evaluated at the GP generated state-values, X corresponding to time points t = t1 · · · tT then p( ˙Xn,·|X, θ, γn) = N(fn(X, θ, t), Iγn) with γn a state-specific error variance. Both statistical models p( ˙Xn,·|Xn,·, ϕn, σn) and p( ˙Xn,·|X, θ, γn) can be linked in the form of a Product of Experts [7] to define the overall density p( ˙Xn,·|X, θ, γn, ϕn, σn) ∝ N(mn, Kn)N(fn(X, θ, t), Iγn) [see e.g. 20]. Introducing priors π(θ) and π(γ) = Q n π(γn) p(θ, γ|X, ϕ, σ) = Z p( ˙X, θ, γ|X, ϕ, σ)d ˙X ∝ π(θ)π(γ) Y n Z N(mn, Kn)N(fn(X, θ, t), Iγn)d ˙Xn,· ∝ π(θ)π(γ) Q n Z(γn) exp ( −1 2 X n (fn −mn)T(Kn + Iγn)−1(fn −mn) ) where fn ≡fn(X, θ, t), and Z(γn) = |2π(Kn + Iγn)| 1 2 is a normalizing constant. Since the gradients appear only linearly and their conditional distribution given X is Gaussian they can be marginalized exactly. In other words, given observations Y, we can sample from the conditional distribution for X and marginalize the augmented derivative space. The differential equation need now never be explicitly solved, its implicit solution is integrated into the sampling scheme. 4 Sampling Schemes for Fully and Partially Observed Systems The introduction of the auxiliary model and its associated variables has enabled us to recast the differential equation as another component of the inference process. The relationship between the auxiliary model and the physical process that we are modeling is shown in Figure 1(b), where the dotted lines represent a transfer of information between the models. This information transfer takes place through sampling candidate solutions for the system in the GP model. Inference is performed by combining these approximate solutions with the system dynamics from the differential equations. It now remains to define an overall sampling scheme for the structural parameters. For brevity, we omit normalizing constants and assume that the system is defined in terms of ODEs. However, our scheme is easily extended for delay differential equations (DDEs) where now predictions at each time point t and the associated delay (t −τ) are required — we present results for a DDE system in Section 5.2. We can now consider the complete sampling scheme by also inferring the hyperparameters and corresponding predictions of the state variables and derivatives using the GP framework described in Section 3. We can obtain samples θ from the desired marginal posterior p(θ|Y)4 by sampling from the joint posterior p(θ, γ, X, ϕ, σ|Y) as follows ϕn, σn|Yn,· ∼ p(ϕn, σn|Yn,·) ∝π(σn)π(ϕn)NYn,·(0, σ2 nI + Cϕn) (1) Xn,·|Yn,·, σn, ϕn ∼ p(Xn,·|Yn,·, σn, ϕn) = NXn,·(µn, Σn) (2) θ, γ|X, ϕ, σ ∼ p(θ, γ|X, ϕ, σ) ∝π(θ)π(γ) exp ( −1 2 X n δT n(Kn + Iγn)−1δn ) (3) where δn ≡fn −mn. This requires two Metropolis sampling schemes; one for inferring the parameters of the GP, ϕ and σ, and another for the parameters of the structural system, θ and γ. However, as a consequence of the system induced dynamics the corresponding likelihood surface defined by p(Y|θ, x0, σ) can present formidable challenges to standard sampling methods. As an example Figure 1(c) illustrates the induced likelihood surface of a simple dynamic oscillator similar to that presented in the experimental section. Recent advances in MCMC methodology suggest solutions to this problem in the form of population-based MCMC methods [8], which we therefore implement to sample the structural parameters of our model. Population MCMC enables samples to be drawn from a target density p(θ) by defining a product of annealed densities indexed by a temperature parameter β, such that p(θ|β) = Q i p(θ|βi) and the desired target density p(θ) is defined for one value of βi. It is convenient to fix a geometric path between the prior and posterior, which we do in our implementation, although other sequences are possible [3]. A time homogeneous Markov transition kernel which has p(θ) as its stationary distribution can then be constructed from both local Metropolis proposal moves and global temperature switching moves between the tempered chains of the population [8], allowing freer movement within the parameter space. The computational scaling for each component of the sampler is now considered. Sampling of the GP covariance function parameters by a Metropolis step requires computation of a matrix determinant and its inverse, so for all N states in the system a dominant scaling of O(NT 3) will be obtained. This poses little problem for many applications in systems biology since T is often fairly small (T ≈10 to 100). For larger values of T, sparse approximations can offer much improved computational scaling of order O(NM 2T), where M is the number of time points selected [9]. Sampling from a multivariate Normal whose covariance matrix and corresponding decompositions have already been computed therefore incurs no dominating additional computational overhead. The final Metropolis step (Equation 3) requires each of the Kn matrices to be constructed and the associated determinants and inverses computed thus incurring a total O(NT 3) scaling per sample. An approximate scheme can be constructed by first obtaining the maximum a posteriori values for the GP hyperparameters and posterior mean state values, ˆϕ, ˆσ, ˆXn, and then employing these in Equation 3. This will provide samples from p(θ, γ| ˆX, ˆϕ, ˆσ, Y) which may be a useful surrogate for the full joint posterior incurring lower computational cost as all matrix operations will have been pre-computed, as will be demonstrated later in the paper. We can also construct a sampling scheme for the important special case where some states are unobserved. We partition X into Xo, and Xu. Let o index the observed states, then we may infer all the unknown variables as follows p(θ, γ, Xu|Xo, ϕ, σ) ∝ π(θ)π(γ)π(Xu) exp ( −1 2 X n∈o (δo,u n )T(Kn + Iγn)−1(δo,u n ) ) where δo,u n ≡fn(Xo, Xu, θ, t) −mn and π(Xu) is an appropriately chosen prior. The values of unobserved species are obtained by propagating their sampled initial values using the corresponding discrete versions of the differential equations and the smoothed estimates of observed species. The p53 transcriptional network example we include requires inference over unobserved protein species, see Section 5.3. 4Note that this is implicitly conditioned on the class of covariance function chosen. 5 Experimental Examples We now demonstrate our GP-based method using a standard squared exponential covariance function on a variety of examples involving both ordinary and delay differential equations, and compare the accuracy and speed with other state-of-the-art methods. 5.1 Example 1 - Nonlinear Ordinary Differential Equations We first consider the FitzHugh-Nagumo model [11] which was originally developed to model the behaviour of spike potentials in the giant axon of squid neurons and is defined as ˙V = c V −V 3/3 + R , ˙R = −(V −a + bR) /c. Although consisting of only 2 equations and 3 parameters, this dynamical system exhibits a highly nonlinear likelihood surface [11], which is induced by the sharp changes in the properties of the limit cycle as the values of the parameters vary. Such a feature is common to many nonlinear systems and so this model provides an excellent test for our GP-based parameter inference method. Data is generated from the model, with parameters a = 0.2, b = 0.2, c = 3, at {40, 80, 120} time points with additive Gaussian noise, N(0, v) for v = 0.1 × σn, where σn is the standard deviation for the nth species. The parameters were then inferred from these data sets using the full Bayesian sampling scheme and the approximate sampling scheme (Section 4), both employing population MCMC. Additionally, we inferred the parameters using 2 alternative methods, the profiled estimation method of Ramsay et al. [11] and a Population MCMC based sampling scheme, in which the ODEs were solved explicitly (Section 2), to complete the comparative study. All the algorithms were coded in Matlab, and the population MCMC algorithms were run with 30 temperatures, and used a suitably diffuse Γ(2, 1) prior distribution for all parameters, forming the base distribution for the sampler. Two of these population MCMC samplers were run in parallel and the ˆR statistic [5] was used to monitor convergence of all chains at all temperatures. The required numerical approximations to the ODE were calculated using the Sundials ODE solver, which has been demonstrated to be considerably (up to 100 times) faster than the standard ODE45/ODE15s solvers commonly used in Matlab. In our experiments the chains generally converged after around 5000 iterations, and 2000 samples were then drawn to form the posterior distributions. Ramsay’s method [11] was implemented using the Matlab code which accompanies their paper. The optimal algorithm settings were used, tuned for the FitzHugh-Nagumo model (see [11] for details) which they also investigated. Each experiment was repeated 100 times, and Table 1 shows summary statistics for each of the inferred parameters. All of the three sampling methods based on population MCMC produced low variance samples from posteriors positioned close to the true parameters values. Most noticeable from the results in Figure 2 is the dramatic speed advantage the GP based methods have over the more direct approach, whereby the differential equations are solved explicitly; the GP methods introduced in this paper offer up to a 10-fold increase in speed, even for this relatively simple system of ODEs. We found the performance of the profiled estimation method [11] to be very sensitive to the initial parameter values. In practice parameter values are unknown, indeed little may be known even about the range of possible values they may take. Thus it seems sensible to choose initial values from a wide prior distribution so as to explore as many regions of parameter space as possible. Employing FitzHugh-Nagumo ODE Model Samples Method a b c 40 GP MAP 0.1930 ± 0.0242 0.2070 ± 0.0453 2.9737 ± 0.0802 GP Fully Bayesian 0.1983 ± 0.0231 0.2097 ± 0.0481 3.0133 ± 0.0632 Explicit ODE 0.2015 ± 0.0107 0.2106 ± 0.0385 3.0153 ± 0.0247 80 GP MAP 0.1950 ± 0.0206 0.2114 ± 0.0386 2.9801 ± 0.0689 GP Fully Bayesian 0.2068 ± 0.0194 0.1947 ± 0.0413 3.0139 ± 0.0585 Explicit ODE 0.2029 ± 0.0121 0.1837 ± 0.0304 3.0099 ± 0.0158 120 GP MAP 0.1918 ± 0.0145 0.2088 ± 0.0317 3.0137 ± 0.0489 GP Fully Bayesian 0.1971 ± 0.0162 0.2081 ± 0.0330 3.0069 ± 0.0593 Explicit ODE 0.2071 ± 0.0112 0.2123 ± 0.0286 3.0112 ± 0.0139 Table 1: Summary statistics for each of the inferred parameters of the FitzHugh-Nagumo model. Each experiment was repeated 100 times and the mean parameter values are shown. We observe that all three populationbased MCMC methods converge close to the true parameter values, a = 0.2, b = 0.2 and c = 3. Figure 2: Summary statistics of the overall time taken for the algorithms to run to completion. Solid bars show mean time for 100 runs; superimposed boxplots display median results with upper and lower quartiles. profiled estimation using initial parameter values drawn from a wide gamma prior, however, yielded highly biased results, with the algorithm often converging to local maxima far from the true parameter values. The parameter estimates become more biased as the variance of the prior is increased, i.e. as the starting points move further from the true parameter values. E.g. consider parameter a; for 40 data points, for initial values a, b, c ∼N({0.2, 0.2, 3}, 0.2), the range of estimated values for ˆa was [Min, Median, Max] = [0.173, 0.203, 0.235]. For initial values a, b, c ∼Γ(1, 0.5), the ˆa had a range [Min, Median, Max] = [−0.329, 0.205, 9.3 × 109] and for a wider prior a, b, c ∼Γ(2, 1), then ˆa had range [Min, Median, Max] = [−1.4 × 1010, 0.195, 2.2 × 109]. Lack of robustness therefore seems to be a significant problem with this profiled estimation method. The speed of the profiled estimation method was also extremely variable, and this was observed to be very dependent on the initial parameter values e.g. for initial values a, b, c ∼N({0.2, 0.2, 3}, 0.2), the times recorded were [Min, Mean, Max] = [193, 308, 475]. Using a different prior for initial values such that a, b, c ∼Γ(1, 0.5), the times were [Min, Mean, Max] = [200, 913, 3265] and similarly for a wider prior a, b, c ∼Γ(2, 1), [Min, Mean, Max] = [132, 4171, 37411]. Experiments performed with noise v = {0.05, 0.2} × σn produced similar and consistent results, however they are omitted due to lack of space. 5.2 Example 2 - Nonlinear Delay Differential Equations This example model describes the oscillatory behaviour of the concentration of mRNA and its corresponding protein level in a genetic regulatory network, introduced by Monk [10]. The translocation of mRNA from the nucleus to the cytosol is explicitly described by a delay differential equation. dµ dt = 1 1 + (p(t −τ)/p0)n −µmµ dp dt = µ −µpp where µm and µp are decay rates, p0 is the repression threshold, n is a Hill coefficient and τ is the time delay. The application of our method to DDEs is of particular interest since numerical solutions to DDEs are generally much more computationally expensive to obtain than ODEs. Thus inference of such models using MCMC methods and explicitly solving the system at each iteration becomes less feasible as the complexity of the system of DDEs increases. We consider data generated from the above model, with parameters µm = 0.03, µp = 0.03, p0 = 100, τ = 25, at {40, 80, 120} time points with added random noise drawn from a Gaussian distribution, N(0, v) for v = 0.1 × σn, where σn is the standard deviation of the time-series data for the nth species. The parameters were then inferred from these data sets using our GP-based population MCMC methods. Figure 3 shows a time comparison for 10 iterations of the GP sampling algorithms and compares it to explicitly solving the DDEs using the Matlab solver DDE23 (which is generally faster than the Sundials solver for DDEs). The GP methods are around 400 times faster for 40 data points. Using the GP methods, samples from the full posterior can be obtained in less than an hour. Solving the DDEs explicitly, the population MCMC algorithm would take in excess of two weeks computation time, assuming the chains take a similar number of iterations to converge. Monk DDE Model Samples Method µm µp ×10−3 p0 ×10−3 τ 40 GP MAP 100.21 ± 2.08 29.7 ± 1.6 30.1 ± 0.3 25.65 ± 1.04 GP Full Bayes 99.75 ± 1.50 29.8 ± 1.2 30.1 ± 0.2 25.33 ± 0.85 80 GP MAP 99.48 ± 1.29 29.5 ± 0.9 30.1 ± 0.1 24.81 ± 0.59 GP Full Bayes 100.26 ± 1.03 30.1 ± 0.6 30.1 ± 0.1 24.87 ± 0.44 120 GP MAP 99.91 ± 1.02 30.0 ± 0.5 30.0 ± 0.1 24.97 ± 0.38 GP Full Bayes 100.23 ± 0.92 30.0 ± 0.4 30.0 ± 0.1 25.03 ± 0.25 Table 2: Summary statistics for each of the inferred parameters of the Monk model. Each experiment was repeated 100 times and we observe that both GP population-based MCMC methods converge close to the true parameter values, µm = 100, µp = 30 × 10−3 and p0 = 30 × 10−3. The time-delay parameter, τ = 25, is also successfully inferred. Figure 3: Summary statistics of the time taken for the algorithms to complete 10 iterations using DDE model. 5.3 Example 3 - The p53 Gene Regulatory Network with Unobserved Species Our third example considers a linear and a nonlinear model describing the regulation of 5 target genes by the tumour repressor transcription factor protein p53. We consider the following differential equations which relate the expression level xj(t) of the jth gene at time t to the concentration of the transcription factor protein f(t) which regulates it, ˙xj = Bj +Sjg(f(t))−Djxj(t), where Bj is the basal rate of gene j, Sj is the sensitivity of gene j to the transcription factor and Dj is the decay rate of the mRNA. Letting g(f(t)) = f(t) gives us the linear model originally investigated in [1], and letting g(f(t)) = exp(f(t)) gives us the nonlinear model investigated in [4]. The transcription factor f(t) is unobserved and must be inferred along with the other structural parameters Bj, Sj and Dj using the sampling scheme detailed in Section 4.1. In this experiment, priors on the unobserved species used were f(t) ∼Γ(2, 1) with a log-Normal proposal. We test our method using the (a) Linear Model (b) Nonlinear Model Figure 4: The predicted output of the p53 gene using data from Barenco et al. [1] and the accelerated GP inference method for (a) the linear model and (b) the nonlinear response model. Note that the asymmetric error bars in (b) are due to exp(y) being plotted, as opposed to just y in (a). Our results are compared to the results obtained by Barenco et al. [1] (shown as crosses) and are comparable to those obtained by Lawrence et al. [4]. leukemia data set studied in [1], which comprises 3 measurements at each of 7 time points for each of the 5 genes. Figure 4 shows the inferred missing species and the results are in good accordance with recent biological studies. For this example, our GP sampling algorithms ran to completion in under an hour on a 2.2GHz Centrino laptop, with no difference in speed between using the linear and nonlinear models; indeed the equations describing this biological system could be made more complex with little additional computational cost. 6 Conclusions Explicit solution of differential equations is a major bottleneck for the application of inferential methodology in a number of application areas, e.g. systems biology, nonlinear dynamic systems. We have addressed this problem and placed it within a Bayesian framework which tackles the main shortcomings of previous solutions to the problem of system identification for nonlinear differential equations. Our methodology allows the possibility of model comparison via the use of Bayes factors, which may be straightforwardly calculated from the samples obtained from the population MCMC algorithm. Possible extensions to this method include more efficient sampling exploiting control variable methods [17], embedding characteristics of a dynamical system in the design of covariance functions and application of our method to models involving partial differential equations. Acknowledgments Ben Calderhead is supported by Microsoft Research through its European PhD Scholarship Programme. Mark Girolami is supported by an EPSRC Advanced Research Fellowship EP/EO52029 and BBSRC Research Grant BB/G006997/1. References [1] Barenco, M., Tomescu, D., Brewer, D., Callard, D., Stark, J. and Hubank, M. (2006) Ranked prediction of p53 targets using hidden variable dynamic modeling, Genome Biology, 7 (3):R25. [2] Doucet, A., de Freitas, N. and Gordon, N., (2001) Sequential Monte Carlo Methods in Practice, Springer. [3] Friel, N. and Pettitt, A. N. (2008) Marginal Likelihood Estimation via Power Posteriors. Journal of the Royal Statistical Society: Series B, 70 (3), 589-607. [4] Gao, P., Honkela, A., Rattray, M. and Lawrence, N.D. (2008) Gaussian Process Modelling of Latent Chemical Species: Applications to Inferring Transcription Factor Activities, Bioinformatics, 24, i70-i75. [5] Gelman, A., Carlin, J.B., Stern, H.S. and Rubin, D.B. (2004) Bayesian Data Analysis, Chapman & Hall. [6] Graepel, T., (2003) Solving noisy linear operator equations by Gaussian processes: application to ordinary and partial differential equations, Proc. ICML 2003. [7] Mayraz, G. and Hinton, G. (2001) Recognizing Hand-Written Digits Using Hierarchical Products of Experts, Proc. NIPS 13. [8] Jasra, A., Stephens, D.A. and Holmes, C.C., (2007) On population-based simulation for static inference, Statistics and Computing, 17, 263-279. [9] Lawrence, N.D., Seeger, M. and Herbrich, R. (2003) Fast sparse Gaussian process methods: the informative vector machine, Proc. NIPS 15. [10] Monk, N. (2003) Oscillatory Expression of Hes1, p53, and NF-kB Driven by Transcriptional Time Delays. Current Biology, 13 (16), 1409-1413. [11] Ramsay, J., Hooker, G., Campbell, D. and Cao, J. (2007) Parameter Estimation for Differential Equations: A Generalized Smoothing Approach. Journal of the Royal Statistical Society: Series B, 69 (5), 741-796. [12] Rasmussen, C, E., (2003) Gaussian processes to speed up hybrid Monte Carlo for expensive Bayesian integrals, Bayesian Statistics, 7, 651-659. [13] Rasmussen, C.E. and Williams, C.K.I. (2006) Gaussian Processes for Machine Learning, The MIT Press. [14] Snelson, E., Rasmussen, C.E. and Ghahramani, Z. (2004), Warped Gaussian processes, Proc. NIPS 16. [15] Solak, E., Murray-Smith, R., Leithead, W.E., Leith, D.J. and Rasmussen, C.E. (2003) Derivative observations in Gaussian Process models of dynamic systems, Proc. NIPS 15. [16] Tarantola, A. (2005) Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM. [17] Titsias, M. and Lawrence, N. (2008) Efficient Sampling for Gaussian Process Inference using Control Variables, Proc. NIPS 22. [18] Varah, J.M. (1982) A spline least squares method for numerical parameter estimation in differential equations. SIAM J. Scient. Comput., 3, 28-46. [19] Vyshemirsky, V. and and Girolami, M., (2008), Bayesian ranking of biochemical system models Bioinformatics 24, 833-839. [20] Williams, C.K.I., Agakov, F.V., Felderof, S.N. (2002), Products of Gaussians, Proc. NIPS 14.
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DiscLDA: Discriminative Learning for Dimensionality Reduction and Classification Simon Lacoste-Julien Computer Science Division UC Berkeley Berkeley, CA 94720 Fei Sha Dept. of Computer Science University of Southern California Los Angeles, CA 90089 Michael I. Jordan Dept. of EECS and Statistics UC Berkeley Berkeley, CA 94720 Abstract Probabilistic topic models have become popular as methods for dimensionality reduction in collections of text documents or images. These models are usually treated as generative models and trained using maximum likelihood or Bayesian methods. In this paper, we discuss an alternative: a discriminative framework in which we assume that supervised side information is present, and in which we wish to take that side information into account in finding a reduced dimensionality representation. Specifically, we present DiscLDA, a discriminative variation on Latent Dirichlet Allocation (LDA) in which a class-dependent linear transformation is introduced on the topic mixture proportions. This parameter is estimated by maximizing the conditional likelihood. By using the transformed topic mixture proportions as a new representation of documents, we obtain a supervised dimensionality reduction algorithm that uncovers the latent structure in a document collection while preserving predictive power for the task of classification. We compare the predictive power of the latent structure of DiscLDA with unsupervised LDA on the 20 Newsgroups document classification task and show how our model can identify shared topics across classes as well as class-dependent topics. 1 Introduction Dimensionality reduction is a common and often necessary step in most machine learning applications and high-dimensional data analyses. There is a rich history and literature on the subject, ranging from classical linear methods such as principal component analysis (PCA) and Fisher discriminant analysis (FDA) to a variety of nonlinear procedures such as kernelized versions of PCA and FDA as well as manifold learning algorithms. A recent trend in dimensionality reduction is to focus on probabilistic models. These models, which include generative topological mapping, factor analysis, independent component analysis and probabilistic latent semantic analysis (pLSA), are generally specified in terms of an underlying independence assumption or low-rank assumption. The models are generally fit with maximum likelihood, although Bayesian methods are sometimes used. In particular, Latent Dirichlet Allocation (LDA) is a Bayesian model in the spirit of pLSA that models each data point (e.g., a document) as a collection of draws from a mixture model in which each mixture component is known as a topic [3]. The mixing proportions across topics are document-specific, and the posterior distribution across these mixing proportions provides a reduced representation of the document. This model has been used successfully in a number of applied domains, including information retrieval, vision and bioinformatics [8, 1]. The dimensionality reduction methods that we have discussed thus far are entirely unsupervised. Another branch of research, known as sufficient dimension reduction (SDR), aims at making use of supervisory data in dimension reduction [4, 7]. For example, we may have class labels or regression responses at our disposal. The goal of SDR is then to identify a subspace or other low-dimensional object that retains as much information as possible about the supervisory signal. Having reduced dimensionality in this way, one may wish to subsequently build a classifier or regressor in the reduced representation. But there are other goals for the dimension reduction as well, including visualization, domain understanding, and domain transfer (i.e., predicting a different set of labels or responses). In this paper, we aim to combine these two lines of research and consider a supervised form of LDA. In particular, we wish to incorporate side information such as class labels into LDA, while retaining its favorable unsupervised dimensionality reduction abilities. The goal is to develop parameter estimation procedures that yield LDA topics that characterize the corpus and maximally exploit the predictive power of the side information. As a parametric generative model, parameters in LDA are typically estimated with maximum likelihood estimation or Bayesian posterior inference. Such estimates are not necessarily optimal for yielding representations for prediction and regression. In this paper, we use a discriminative learning criterion—conditional likelihood—to train a variant of the LDA model. Moreover, we augment the LDA parameterization by introducing class-label-dependent auxiliary parameters that can be tuned by the discriminative criterion. By retaining the original LDA parameters and introducing these auxiliary parameters, we are able to retain the advantages of the likelihood-based training procedure and provide additional freedom for tracking the side information. The paper is organized as follows. In Section 2, we introduce the discriminatively trained LDA (DiscLDA) model and contrast it to other related variants of LDA models. In Section 3, we describe our approach to parameter estimation for the DiscLDA model. In Section 4, we report empirical results on applying DiscLDA to model text documents. Finally, in Section 5 we present our conclusions. 2 Model We start by reviewing the LDA model [3] for topic modeling. We then describe our extension to LDA that incorporates class-dependent auxiliary parameters. These parameters are to be estimated based on supervised information provided in the training data set. 2.1 LDA The LDA model is a generative process where each document in the text corpus is modeled as a set of draws from a mixture distribution over a set of hidden topics. A topic is modeled as a probability distribution over words. Let the vector wd be the bag-of-words representation of document d. The generative process for this vector is illustrated in Fig. 1 and has three steps: 1) the document is first associated with a K-dimensional topic mixing vector θd which is drawn from a Dirichlet distribution, θd ∼Dir(α); 2) each word wdn in the document is then assigned to a single topic zdn drawn from the multinomial variable, zdn ∼Multi(θd); 3) finally, the word wdn is drawn from a V -dimensional multinomial variable, wdn ∼Multi(φzdn), where V is the size of the vocabulary. Given a set of documents, {wd}D d=1, the principal task is to estimate the parameters {φk}K k=1. This can be done by maximum likelihood, Φ∗= arg maxΦ p({wd}; Φ), where Φ ∈ℜV ×K is a matrix parameter whose columns {φk}K k=1 are constrained to be members of a probability simplex. It is also possible to place a prior probability distribution on the word probability vectors {φk}K k=1—e.g., a Dirichlet prior, φk ∼Dir(β)—and treat the parameter Φ as well as the hyperparameters α and β via Bayesian methods. In both the maximum likelihood and Bayesian framework it is necessary to integrate over θd to obtain the marginal likelihood, and this is accomplished either using variational inference or Gibbs sampling [3, 8]. 2.2 DiscLDA In our setting, each document is additionally associated with a categorical variable or class label yd ∈{1, 2, . . ., C} (encoding, for example, whether a message was posted in the newsgroup alt.atheism vs. talk.religion.misc). To model this labeling information, we introduce a simple extension to the standard LDA model. Specifically, for each class label y, we introduce a linear transformation T y : ℜK →ℜL +, which transforms a K-dimensional Dirichlet variable θd to α θd zdn wdn N D Φ β Figure 1: LDA model. π α θd zdn wdn N D Φ β yd T Figure 2: DiscLDA. α π θd zdn wdn N D Φ β yd udn T Figure 3: DiscLDA with auxiliary variable u. a mixture of Dirichlet distributions: T yθd ∈ℜL. To generate a word wdn, we draw its topic zdn from T ydθd. Note that T y is constrained to have its columns sum to one to ensure the normalization of the transformed variable T yθd and is thus a stochastic matrix. Intuitively, every document in the text corpus is represented through θd as a point in the topic simplex {θ | P k θk = 1}, and we hope that the linear transformation {T y} will be able to reposition these points such that documents with the same class labels are represented by points nearby to each other. Note that these points can not be placed arbitrarily, as all documents—whether they have the same class labels or they do not— share the parameter Φ ∈ℜV ×L. The graphical model in Figure 2 shows the new generative process. Compared to standard LDA, we have added the nodes for the variable yd (and its prior distribution π), the transformation matrices T y and the corresponding edges. An alternative to DiscLDA would be a model in which there are class-dependent topic parameters φy k which determine the conditional distribution of the words: wdn | zdn, yd, Φ ∼Multi(φyd zdn). The problem with this approach is that the posterior p(y|w, Φ) is a highly non-convex function of Φ which makes its optimization very challenging given the high dimensionality of the parameter space in typical applications. Our approach circumvents this difficulty by learning a low-dimensional transformation of the φk’s in a discriminative manner instead. Indeed, transforming the topic mixture vector θ is actually equivalent to transforming the Φ matrix. To see this, note that by marginalizing out the hidden topic vector z, we get the following distribution for the word wdn given θ: wdn | yd, θd, T ∼Mult (ΦT yθd) . By the associativity of the matrix product, we see that we obtain an equivalent probabilistic model by applying the linear transformation to Φ instead, and, in effect, defining the class-dependent topic parameters as follows: φy k = X l φlT y lk. Another motivation for our approach is that it gives the model the ability to distinguish topics which are shared across different classes versus topics which are class-specific. For example, this separation can be accomplished by using the following transformations (for binary classification): T 1 = IK 0 0 0 0 IK ! , T 2 = 0 0 IK 0 0 IK ! (1) where IK stands for the identity matrix with K rows and columns. In this case, the last K topics are shared by both classes, whereas the two first groups of K topics are exclusive to one class or the other. We will explore this parametric structure later in our experiments. Note that we can give a generative interpretation to the transformation by augmenting the model with a hidden topic vector variable u, as shown in Fig. 3, where p(u = k|z = l, T, y) = T y kl. In this augmented model T can be interpreted as the probability transition matrix from z-topics to u-topics. By including a Dirichlet prior on the T parameters, the DiscLDA model can be related to the authortopic model [10], if we restrict to the special case in which there is only one author per document. In the author-topic model, the bag-of-words representation of a document is augmented by a list of the authors of the document. To generate a word in a document, one first picks at random the author associated with this document. Given the author (y in our notation), a topic is chosen according to corpus-wide author-specific topic-mixture proportions (which is a column vector T y in our notation). The word is then generated from the corresponding topic distribution as usual. According to this analogy, we see that our model not only enables us to predict the author of a document (assuming a small set of possible authors), but we also capture the content of documents (using θ) as well as the corpus-wide class properties (using T ). The focus of the author-topic model was to model the interests of authors, not the content of documents, explaining why there was no need to add document-specific topic-mixture proportions. Because we want to predict the class for a specific document, it is crucial that we also model the content of a document. Recently, there has been growing interest in topic modeling with supervised information. Blei and McAuliffe [2] proposed a supervised LDA model where the empirical topic vector z (sampled from θ) is used as a covariate for a regression on y (see also [6]). Mimno and McCallum [9] proposed a Dirichlet-multinomial regression which can handle various types of side information, including the case in which this side information is an indicator variable of the class (y)1. Our work differs from theirs, however, in that we train the transformation parameter by maximum conditional likelihood instead of a generative criterion. 3 Inference and learning Given a corpus of documents and their labels, we estimate the parameters {T y} by maximizing the conditional likelihood P d log p(yd | wd; {T y}, Φ) while holding Φ fixed. To estimate the parameters Φ, we hold the transformation matrices fixed and maximize the posterior of the model, in much the same way as in standard LDA models. Intuitively, the two different training objectives have two effects on the model: the optimization of the posterior with respect to Φ captures the topic structure that is shared in documents throughout a corpus, while the optimization of the conditional likelihood with respect to {T y} finds a transformation of the topics that discriminates between the different classes within the corpus. We use the Rao-Blackwellized version of Gibbs sampling presented in [8] to obtain samples of z and u with Φ and θ marginalized out. Those samples can be used to estimate the likelihood of p(w|y, T ), and thus the posterior p(y|w, T ) for prediction, by using the harmonic mean estimator [8]. Even though this estimator can be unstable in general model selection problems, we found that it gave reasonably stable estimates for our purposes. We maximize the conditional likelihood objective with respect to T by using gradient ascent, for a fixed Φ. The gradient can be estimated by Monte Carlo EM, with samples from the Gibbs sampler. More specifically, we use the matching property of gradients in EM to write the gradient as: ∂ ∂T log p(y|w, T , Φ) = Eqy t (z) ∂ ∂T log p(w, z|y, T , Φ) −Ert(z) ∂ ∂T log p(w, z|T , Φ) , (2) where qy t (z) = p(z|w, y, T t, Φ), rt(z) = p(z|w, T t, Φ) and the derivatives are evaluated at T = T t. We can approximate those expectations using the relevant Gibbs samples. After a few gradient updates, we refit Φ by its MAP estimate from Gibbs samples. 3.1 Dimensionality reduction We can obtain a supervised dimensionality reduction method by using the average transformed topic vector as the reduced representation of a test document. We estimate it using E [T yθ|Φ, w, T ] = P y p(y|Φ, w, T )E [T yθ|y, Φ, w, T ]. The first term on the right-hand side of this equation can 1In this case, their model is actually the same as Model 1 in [5] with an additional prior on the classdependent parameters for the Dirichlet distribution on the topics. Figure 4: t-SNE 2D embedding of the E [T yθ|Φ, w, T ] representation of Newsgroups documents, after fitting to the DiscLDA model (T was fixed).
Figure 5: t-SNE 2D embedding of the E [θ|Φ, w, T ] representation of Newsgroups documents, after fitting to the standard unsupervised LDA model. be estimated using the harmonic mean estimator and the second term can be approximated from MCMC samples of z. This new representation can be used as a feature vector for another classifier or for visualization purposes. 4 Experimental results We evaluated the DiscLDA model empirically on text modeling and classification tasks. Our experiments aimed to demonstrate the benefits of discriminative training of LDA for discovering a compact latent representation that contains both predictive and shared components across different types of data. We evaluated the performance of our model by contrasting it to standard LDA models that were not trained discriminatively. 4.1 Text modeling The 20 Newsgroups dataset contains postings to Usenet newsgroups. The postings are organized by content into 20 related categories and are therefore well suited for topic modeling. In this section, we investigate how DiscLDA can exploit the labeling information—the category—in discovering meaningful hidden structures that differ from those found using unsupervised techniques. We fit the dataset to both a standard 110-topic LDA model and a DiscLDA model with restricted forms of the transformation matrices {T y}y=20 y=1 . Specifically, the transformation matrix T y for class label c is fixed and given by the following blocked matrix T y = 0 0 ... ... IK0 0 ... ... 0 IK1 . (3) This matrix has (C + 1) rows and two columns of block matrices. All but two block matrices are zero matrices. At the first column and the row y, the block matrix is an identity matrix with dimensionality of K0 × K0. The last element of T y is another identity matrix with dimensionality K1. When applying the transformation to a topic vector θ ∈ℜK0+K1, we obtain a transformed topic vector θtr = T yθ whose nonzeros elements partition the components θtr into (C +1) disjoint sets: one set of K0 elements for each class label that does not overlap with the others, and a set of K1 components that is shared by all class labels. Intuitively, the shared components should use all class labels to model common latent structures, while nonoverlapping components should model specific characteristics of data from each class. Class Most popular words alt.atheism atheism, religion, bible, god, system, moral, atheists, keith, jesus, islam, comp.graphics files, color, images, file, image, format, software, graphics, jpeg, gif, comp.os.ms-windows.misc card, files, mouse, file, dos, drivers, win, ms, windows, driver, comp.sys.ibmpc.hardware drive, card, drives, bus, mb, os, disk, scsi, controller, ide, comp.sys.mac.hardware drive, apple, mac, speed, monitor, mb, quadra, mhz, lc, scsi, comp.windows.x server, entry, display, file, program, output, window, motif, widget, lib, misc.forsale price, mail, interested, offer, cover, condition, dos, sale, cd, shipping, rec.autos cars, price, drive, car, driving, speed, engine, oil, ford, dealer, rec.motorcycles ca, ride, riding, dog, bmw, helmet, dod, bike, motorcycle, bikes, rec.sport.baseball games, baseball, year, game, runs, team, hit, players, season, braves, rec.sport.hockey ca, period, play, games, game, team, win, players, season, hockey, sci.crypt government, key, public, security, chip, clipper, keys, db, privacy, encryption, sci.electronics current, power, ground, wire, output, circuit, audio, wiring, voltage, amp, sci.med gordon, food, disease, pitt, doctor, medical, pain, health, msg, patients, sci.space earth, space, moon, nasa, orbit, henry, launch, shuttle, satellite, lunar, soc.religion.christian christians, bible, church, truth, god, faith, christian, christ, jesus, rutgers, talk.politics.guns people, gun, guns, government, file, fire, fbi, weapons, militia, firearms, talk.politics.mideast people, turkish, government, jews, israel, israeli, turkey, armenian, armenians, armenia, talk.politics.misc american, men, war, mr, tax, government, president, health, cramer, stephanopoulos, talk.religion.misc religion, christians, bible, god, christian, christ, morality, objective, sandvik, jesus, Shared topics ca, people, post, wrote, group, system, world, work, ll, make, true, university, great, case, number, read, day, mail, information, send, back, article, writes, question, find, things, put, don, cs, didn, good, end, ve, long, point, years, doesn, part, time, state, fact, thing, made, problem, real, david, apr, give, lot, news Table 1: Most popular words from each group of class-dependent topics or a bucket of “shared” topics learned in the 20 Newsgroups experiment with fixed T matrix. In a first experiment, we examined whether the DiscLDA model can exploit the structure for T y given in (3). In this experiment, we first obtained an estimate of the Φ matrix by setting it to the MAP estimate from Gibbs samples as explained in Section 3. We then estimated a new representation for test documents by taking the conditional expectation of T yθ with y marginalized out as explained in Section 3.1. Finally, we then computed a 2D-embedding of this K1-dimensional representation of documents. To obtain an embedding, we first tried standard multidimensional scaling (MDS), using the symmetrical KL divergence between pairs of θtr topic vectors as a dissimilarity metric, but the results were hard to visualize. A more interpretable embedding was obtained using a modified version of the t-SNE stochastic neighborhood embedding presented by van der Maaten and Hinton [11]. Fig. 4 shows a scatter plot of the 2D–embedding of the topic representation of the 20 Newsgroups test documents, where the colors of the dots, each corresponding to a document, encode class labels. Clearly, the documents are well separated in this space. In contrast, the embedding computed from standard LDA, shown in Fig. 5, does not show a clear separation. In this experiment, we have set K0 = 5 and K1 = 10 for DiscLDA, yielding 110 possible topics; hence we set K = 110 for the standard LDA model for proper comparison. It is also instructive to examine in detail the topic structures of the fitted DiscLDA model. Given the specific setup of our transformation matrix T , each component of the topic vector u is either associated with a class label or shared across all class labels. For each component, we can compute the most popular words associated from the word-topic distribution Φ. In Table 1, we list these words and group them under each class labels and a special bucket “shared.” We see that the words are highly indicative of their associated class labels. Additionally, the words in the “shared” category are “neutral,” neither positively nor negatively suggesting proper class labels where they are likely LDA+SVM DiscLDA+SVM discLDA alone 20% 17% 17% Table 2: Binary classification error rates for two newsgroups to appear. In fact, these words confirm the intuition of the DiscLDA model: they reflect common English usage underlying different documents. We note that we had already taken out a standard list of stop words from the documents. 4.2 Document classification It is also of interest to consider the classification problem more directly and ask whether the features delivered by DiscLDA are more useful for classification than those delivered by LDA. Of course, we can also use DiscLDA as a classification method per se, by marginalizing over the latent variables and computing the probability of the label y given the words in a test document. Our focus in this section, however, is its featural representation. We thus use a different classification method (the SVM) to compare the features obtained by DiscLDA to those obtained from LDA. In a first experiment, we returned to the fixed T setting studied in Section 4.1 and considered the features obtained by DiscLDA for the 20 Newsgroups problem. Specifically, we constructed multiclass linear SVM classifiers using the expected topic proportion vectors from unsupervised LDA and DiscLDA models as features as described in Section 3.1. The results were as follows. Using the topic vectors from standard LDA the error rate of classification was 25%. When the topic vectors from the DiscLDA model were used we obtained an error rate of 20%. Clearly the DiscLDA features have retained information useful for classification. We also computed the MAP estimate of the class label y∗= arg max p(y|w) from DiscLDA and used this estimate directly as a classifier. The error rate was again 20%. In a second experiment, we considered the fully adaptive setting in which the transformation matrix T y is learned in a discriminative fashion as described in Section 3. We initialized the matrix T to a smoothed block diagonal matrix having a pattern similar to (1), with 20 shared topics and 20 class-dependent topics per class. We then sampled u and z for 300 Gibbs steps to obtain an initial estimate of the Φ vector. This was followed by the discriminative learning process in which we iteratively ran batch gradient (in the log domain, so that T remained normalized) using Monte Carlo EM with a constant step size for 10 epochs. We then re-estimated Φ by sampling u conditioned on (Φ, T ). This discriminative learning process was repeated until there was no improvement on a validation data set. The step size was chosen by grid search. In this experiment, we considered the binary classification problem of distinguishing postings of the newsgroup alt.atheism from postings of the newsgroup talk.religion.misc, a difficult task due to the similarity in content between these two groups. Table 2 summarizes the results of our experiment, where we have used topic vectors from unsupervised LDA and DiscLDA as input features to binary linear SVM classifiers. We also computed the prediction of the label of a document directly with DiscLDA. As shown in the table, the DiscLDA model clearly generates topic vectors with better predictive power than unsupervised LDA. In Table 3 we present the ten most probable words for a subset of topics learned using the discriminative DiscLDA approach. We found that the learned T had a block-diagonal structure similar to (3), though differing significantly in some ways. In particular, although we started with 20 shared topics the learned T had only 12 shared topics. We have grouped the topics in Table 3 according to whether they were class-specific or shared, uncovering an interesting latent structure which appears more discriminating than the topics presented in Table 1. 5 Discussion We have presented DiscLDA, a variation on LDA in which the LDA parametrization is augmented to include a transformation matrix and in which this matrix is learned via a conditional likelihood criterion. This approach allows DiscLDA to retain the ability of the LDA approach to find useful Topics for alt.atheism Topics for talk.religion.misc Shared topics god, atheism, religion, atheists, religious, atheist, belief, existence, strong evil, group, light, read, stop, religions, muslims, understand, excuse things, bobby, men, makes, bad, mozumder, bill, ultb, isc, rit argument, true, conclusion, fallacy, arguments, valid, form, false, logic, proof back, gay, convenient, christianity, homosexuality, long, nazis, love, homosexual, david system, don, moral, morality, murder, natural, isn, claim, order, animals peace, umd, mangoe, god, thing, language, cs, wingate, contradictory, problem bible, ra, jesus, true, christ, john, issue, church, lds, robert evidence, truth, statement, simply, accept, claims, explain, science, personal, left Table 3: Ten most popular words from a random selection of different types of topics learned in the discriminative learning experiment on the binary dataset. low-dimensional representations of documents, but to also make use of discriminative side information (labels) in forming these representations. Although we have focused on LDA, we view our strategy as more broadly useful. A virtue of the probabilistic modeling framework is that it can yield complex models that are modular and can be trained effectively with unsupervised methods. Given the high dimensionality of such models, it may be intractable to train all of the parameters via a discriminative criterion such as conditional likelihood. In this case it may be desirable to pursue a mixed strategy in which we retain the unsupervised criterion for the full parameter space but augment the model with a carefully chosen transformation so as to obtain an auxiliary low-dimensional optimization problem for which conditional likelihood may be more effective. Acknowledgements We thank the anonymous reviewers as well as Percy Liang, Iain Murray, Guillaume Obozinski and Erik Sudderth for helpful suggestions. Our work was supported by Grant 0509559 from the National Science Foundation and by a grant from Google. References [1] T. L. Berg, A. C. Berg, J. Edwards, M. Maire, R. White, Y. W. Teh, E. Learned-Miller, and D. A. Forsyth. Names and faces in the news. In Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, Washington, DC, 2004. [2] D. Blei and J. McAuliffe. Supervised topic models. In J. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 20, Cambridge, MA, 2008. MIT Press. [3] D. M. Blei, A. Y. Ng, and M. I. Jordan. Latent Dirichlet allocation. Journal of Machine Learning Research, 3:993–1022, 2003. [4] F. Chiaromonte and R. D. Cook. Sufficient dimension reduction and graphics in regression. Annals of the Institute of Statistical Mathematics, 54(4):768–795, 2002. [5] L. Fei-fei and P. Perona. A Bayesian hierarchical model for learning natural scene categories. In Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, San Diego, CA, 2005. [6] P. Flaherty, G. Giaever, J. Kumm, M. I. Jordan, and A. P. Arkin. A latent variable model for chemogenomic profiling. Bioinformatics, 21:3286–3293, 2005. [7] K. Fukumizu, F. R. Bach, and M. I. Jordan. Kernel dimension reduction in regression. Annals of Statistics, 2008. To appear. [8] T. Griffiths and M. Steyvers. Finding scientific topics. Proceedings of the National Academy of Sciences, 101:5228–5235, 2004. [9] D. Mimno and A. McCallum. Topic models conditioned on arbitrary features with Dirichletmultinomial regression. In Proceedings of the 24th Annual Conference on Uncertainty in Artificial Intelligence, Helsinki, Finland, 2008. [10] M. Rosen-Zvi, T. Griffiths T, M. Steyvers, and P. Smyth. The author-topic model for authors and documents. In Proceedings of the 20th Annual Conference on Uncertainty in Artificial Intelligence, Banff, Canada, 2004. [11] L. J. P. van der Maaten and G. E. Hinton. Visualizing data using t-SNE. Journal of Machine Learning Research, 9:2579–2605, 2008.
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Understanding Brain Connectivity Patterns during Motor Imagery for Brain-Computer Interfacing Moritz Grosse-Wentrup Max Planck Institute for Biological Cybernetics Spemannstr. 38 72076 T¨ubingen, Germany moritzgw@ieee.org Abstract EEG connectivity measures could provide a new type of feature space for inferring a subject’s intention in Brain-Computer Interfaces (BCIs). However, very little is known on EEG connectivity patterns for BCIs. In this study, EEG connectivity during motor imagery (MI) of the left and right is investigated in a broad frequency range across the whole scalp by combining Beamforming with Transfer Entropy and taking into account possible volume conduction effects. Observed connectivity patterns indicate that modulation intentionally induced by MI is strongest in the γ-band, i.e., above 35 Hz. Furthermore, modulation between MI and rest is found to be more pronounced than between MI of different hands. This is in contrast to results on MI obtained with bandpower features, and might provide an explanation for the so far only moderate success of connectivity features in BCIs. It is concluded that future studies on connectivity based BCIs should focus on high frequency bands and consider experimental paradigms that maximally vary cognitive demands between conditions. 1 Introduction Brain-Computer Interfaces (BCIs) are devices that enable a subject to communicate without utilizing the peripheral nervous system, i.e., without any overt movement requiring volitional motor control. The primary goal of research on BCIs is to enable basic communication for subjects unable to communicate by normal means due to neuro-degenerative diseases such as amyotrophic lateral sclerosis (ALS). In non-invasive BCIs, this is usually approached by measuring the electric field of the brain by EEG, and detecting changes intentionally induced by the subject (cf. [1] for a general introduction to BCIs). The most commonly used experimental paradigm in this context is motor imagery (MI) [2]. In MI subjects are asked to haptically imagine movements of certain limbs, e.g., the left or the right hand. MI is known to be accompanied by a decrease in bandpower (usually most prominent in the µ-band, i.e., roughly at 8-13 Hz) in that part of the motor cortex representing the specific limb [3]. These bandpower changes, termed event related (de-)synchronization (ERD/ERS), can be detected and subsequently used for inferring the subject’s intention. This approach to BCIs has been demonstrated to be very effective in healthy subjects, with only little subject training time required to achieve classification accuracies close to 100% in two-class paradigms [4–6]. Furthermore, satisfactory classification results have been reported with subjects in early to middle stages of ALS [7]. However, all subjects diagnosed with ALS and capable of operating a BCI still had residual motor control that enabled them to communicate without the use of a BCI. Until now, no communication has been established with a completely locked-in subject, i.e., a subject without any residual motor control. Establishing communication with a completely locked-in subject arguably constitutes the most important challenge in research on BCIs. 1 Unfortunately, reasons for the failure of establishing communication with completely locked-in subjects remain unknown. While cognitive deficits in completely locked-in patients can at present not be ruled out as the cause of this failure, another possible explanation is abnormal brain activity observed in patients in late stages of ALS [8]. Our own observations indicate that intentionally induced bandpower changes in the electric field of the brain might be reduced in subjects in late stages of ALS. To explore the plausibility of this explanation for the failure of current BCIs in completely locked-in subjects, it is necessary to devise feature extraction algorithms that do not rely on measures of bandpower. In this context, one promising approach is to employ connectivity measures between different brain regions. It is well known from fMRI-studies that brain activity during MI is not confined to primary motor areas, but rather includes a distributed network including pre-motor, parietal and frontal regions of the brain [9]. Furthermore, synchronization between different brain regions is known to be an essential feature of cognitive processing in general [10]. Subsequently, it can be expected that different cognitive tasks, such as MI of different limbs, are associated with different connectivity patterns between brain regions. These connectivity patterns should be detectable from EEG recordings, and thus offer a new type of feature space for inferring a subject’s intention. Since measures of connectivity are, at least in principle, independent of bandpower changes, this might offer a new approach to establishing communication with completely locked-in subjects. In recent years, several measures of connectivity have been developed for analyzing EEG recordings (cf. [11] for a good introduction and a comparison of several algorithms). However, very few studies exist that analyze connectivity patterns as revealed by EEG during MI [12,13]. Furthermore, these studies focus on differences in connectivity patterns between MI and motor execution, which is not of primary interest for research on BCIs. In the context of non-invasive BCIs, connectivity measures have been most notably explored in [14] and [15]. However, these studies only consider frequency bands and small subsets of electrodes known to be relevant for bandpower features, and do not take into account possible volume conduction effects. This might lead to misinterpreting bandpower changes as changes in connectivity. Consequently, a better understanding of connectivity patterns during MI of different limbs as measured by EEG is required to guide the design of new feature extraction algorithms for BCIs. Specifically, it is important to properly address possible volume conduction effects, not confine the analysis to a small subset of electrodes, and consider a broad range of frequency bands. In this work, these issues are addressed by combining connectivity analysis during MI of the left and right hand in four healthy subjects with Beamforming methods [6]. Since it is well known that MI includes primary motor cortex [3], this area is chosen as the starting point of the connectivity analysis. Spatial filters are designed that selectively extract those components of the EEG originating in the left and right motor cortex. Then, the concept of Transfer Entropy [16] is used to estimate class-conditional ’information flow’ from all 128 employed recording sites into the left and right motor cortex in frequency bands ranging from 5 - 55 Hz. In this way, spatial topographies are obtained for each frequency band that depict by how much each area of the brain is influencing the left/right motor cortex during MI of the left/right hand. Interestingly, the most pronounced changes in connectivity patterns are not observed in MI of the left vs. the right hand, but rather in rest vs. MI of either hand. Furthermore, these pattern changes are most pronounced in frequency bands not usually associated with MI. i.e., in the γ-band above 35 Hz. These results suggest that in order to fully exploit the capabilities of connectivity measures for BCIs, and establish communication with completely locked-in subjects, it might be advisable to consider γ-band oscillations and adapt experimental paradigms as to maximally vary cognitive demands between conditions. 2 Methods 2.1 Symmetric vs. Asymmetric Connectivity Analysis In analyzing interrelations between time-series data it is important to distinguish symmetric from asymmetric measures. Consider Fig. 1, depicting two graphs of three random processes s1 to s3, representing three EEG sources. The goal of symmetric connectivity analysis (Fig. 1.a) is to estimate some instantaneous measure of similarity between random processes, i.e., assigning weights to the undirected edges between the nodes of the graph in Fig. 1.a. Amplitude coupling and phase synchronization fall into this category, which are the measures employed in [14] and [15] for feature extraction in BCIs. However, interrelations between EEG sources originating in different regions of 2 a) s1[t] s2[t] s3[t] s1[t + 1] s2[t + 1] s3[t + 1] b) s1[t] s2[t] s3[t] s1[t + 1] s2[t + 1] s3[t + 1] Figure 1: Illustration of symmetric- vs. asymmetric connectivity analysis for three EEG sources within the brain. the brain can be expected to be asymmetric, with certain brain regions exerting stronger influence on other regions than vice versa. For this reason, asymmetric connectivity measures potentially provide more information on cognitive processes than symmetric measures. Considering asymmetric relations between random processes requires a definition of how the influence of one process on another process is to be measured, i.e., a quantitative definition of causal influence. The commonly adopted definition of causality in time-series analysis is that si causes sj if observing si helps in predicting future observations of sj, i.e., reduces the prediction error of sj. This implies that cause precedes effect, i.e., that the graph in Fig. 1.b may only contain directed arrows pointing forward in time. Note that there is some ambiguity in this definition of causality, since it does not specify a metric for reduction of the prediction error of sj due to observing si. In Granger causality (cf. [11]), reduction of the variance of the prediction error is chosen as a metric, essentially limiting Granger causality to linear systems. It should be noted, however, that any other metric is equally valid. Finally, note that for reasons of simplicity the graph in Fig. 1.b only contains directed edges from nodes at time t to nodes at time t + 1. In general, directed arrows from nodes at times t, . . . , t−k to nodes at time t+1 may be considered, with k the order of the random processes generating s[t + 1]. To assess Granger causality between bivariate time-series data a linear autoregressive model is fit to the data, which is then used to compute a 2x2 transfer matrix in the frequency domain (cf. [11]). The off-diagonal elements of the transfer matrix then provide a measure of the asymmetric interaction between the observed time-series. Extensions of Granger causality to multivariate time-series data, termed directed transfer function (DTF) and partial directed coherence (PDC), have been developed (cf. [11] and the references therein). However, in this work a related but different measure for asymmetric interrelations between time-series is utilized. The concept of Transfer Entropy (TE) [16] defines the causal influence of si on sj as the reduction in entropy of sj obtained by observing si. More precisely, let si and sj denote two random processes, and let sk i/j[t] := si/j[t], . . . , si/j[t −k] . TE is then defined as Tk (si[t] →sj[t + 1]) := H sj[t + 1]|sk j [t] −H sj[t + 1]|sk j [t], sk i [t] , (1) with k the order of the random processes and H(·) the Shannon entropy. TE can thus be understood as the reduction in uncertainty about the random process sj at time t + 1 due to observing the past k samples of the random process si. Both, Granger causality and TE, thus define causal influence as a reduction in the uncertainty of a process due to observing another process, but employ different metrics to measure reduction in uncertainty. While TE is a measure that applies to any type of random processes, it is difficult to compute in practice. Hence, in this study only Gaussian processes are considered, i.e., it is assumed that sj[t + 1], sk j [t], sk i [t] is jointly Gaussian distributed. TE can then be computed as T GP k (si[t] →sj[t + 1]) = 1 2 log det R(sk j [t],sk i [t]) det R(sj[t+1],sk j [t]) det R(sj[t+1],sk j [t],sk i [t]) det R(sk j [t]) , (2) with R(·) the (cross-)covariance matrices of the respective random processes [17]. In comparison to Granger causality and related measures, TE for Gaussian processes possesses several advantages. It is easy to compute from a numerical perspective, since it does not require fitting a multivariate autoregressive model including (implicit) inversion of large matrices. Furthermore, for continuous processes it is invariant under coordinate transformations [17]. Importantly, this entails invariance with regard to scaling of the random processes. Computing TE for Gaussian processes requires estimation of the (cross-)covariance matrices in (2). Consider a matrix S ∈R2×T ×N, corresponding to data recorded from two EEG 3 sources during an experimental paradigm with N trials of T samples each. In order to compute T GP k (s1[t] →s2[t + 1]) for t = k + 1, . . . , T −k −1, it is assumed that in each trial s1[t] and s2[t] are i.i.d. samples from the distribution p(s1[t], s2[t]), i.e., that the non-stationary Gaussian processes that give rise to the observation matrix S are identical for each of the N repetitions of the experimental paradigm. For each instant in time, TE can then be evaluated by computing the sample (cross-)covariance matrices required in (2) across trials. Note that evaluating (2) requires specification of k. In general, k should be chosen as large as possible in order to maximize information on the random processes contained in the (cross-)covariance matrices. However, choosing k too large leads to rank deficient matrices with a determinant of zero. Here, for each observation matrix S the highest possible k is chosen such that none of the matrices in (2) is rank deficient. 2.2 The Problem of Volume Conduction in EEG Connectivity Analysis The goal of connectivity analysis in EEG recordings is to estimate connectivity patterns between different regions of the brain. Unfortunately, EEG recordings do not offer direct access to EEG sources. Instead, each EEG electrode measures a linear and instantaneous superposition of EEG sources within the brain [18]. This poses a problem for symmetric connectivity measures, since these assess instantaneous coupling between electrodes [18]. Asymmetric connectivity measures such as TE, on the other hand, are not based on instantaneous coupling, but rather consider prediction errors. It is not obvious that instantaneous volume conduction also poses a problem for this type of measures. Unfortunately, the following example demonstrates that volume conduction also leads to incorrect connectivity estimates in asymmetric connectivity analysis based on TE. Example 1 (Volume Conduction Effects in Connectivity Analysis based on Transfer Entropy) Consider the EEG signals x1[t] and x2[t], recorded at two electrodes placed on the scalp, that consist of a linear superposition of three EEG sources s1[t] to s3[t] situated somewhere within the brain (Fig. 2.a). Let x[t] = (x1[t], x2[t])T and s[t] = (s1[t], s2[t], s3[t])T. Then x[t] = As[t], with A ∈R2×3 describing the projection strength of each source to each electrode. For sake of simplicity, assume that A = (1 0 1 ; 0 1 1 ), i.e., that the first source only projects to the first electrode with unit strength, the second source only projects to the second electrode with unit strength, and the third source projects to both electrodes with unit strength. Furthermore, assume that p(s[t + 1], s[t]) = N(0, Σ) with Σ = 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 γ 0 0 0 1 0 0 0 0 0 0 1 0 0 0 γ 0 0 1 , (3) i.e., that all sources have zero mean, unit variance, are mutually independent, and s1 and s2 are uncorrelated in time. Only s3[t] and s3[t + 1] are assumed to be correlated with covariance γ (Fig. 2.b). In this setting, it would be desirable to obtain zero TE between both electrodes, since there is no interaction between the sources giving rise to the EEG. However, some rather tedious algebraic manipulations reveal that in this case T GP 1 (x2[t] →x1[t + 1]) = 1 2 log 3 2 + 1 2 log 4 −γ2 6 −2γ2 . (4) Note that (4) is zero if and only if γ = 0, i.e., if s3 represents white noise. Otherwise, TE between the two electrodes is estimated to be greater than zero solely due to volume conduction effects from source s3. Further note that qualitatively this result holds independently of the strength of the projection of the third source to both electrodes. 2.3 Attenuation of Volume Conduction Effects via Beamforming One way to avoid volume conduction effects in EEG connectivity analysis is to perform source localization on the obtained EEG data, and apply connectivity measures on estimated current density time-series at certain locations within the brain [11]. This is feasible to test certain hypothesis, e.g., to evaluate whether there exists a causal link between two specific points within the brain. However, testing pairwise causal links between more than just a few points within the brain is computationally 4 s1[t] s2[t] s3[t] x1[t] x2[t] b) a) s1[t] s2[t] s3[t] s1[t + 1] s2[t + 1] s3[t + 1] Figure 2: Illustration of volume conduction effects in EEG connectivity analysis. intractable. Accordingly, attenuation of volume conduction effects via source localization is not feasible if a complete connectivity pattern considering the whole brain is desired. Here, a different approach is pursued. It is well known that primary motor cortex is central to MI as measured by EEG [3]. Accordingly, it is assumed that any brain region involved in MI displays some connectivity to the primary motor cortex. This (admittedly rather strong) assumption enables a complete analysis of the connectivity patterns during MI covering the whole brain in the following way. First, two spatial filters, commonly known as Beamformers, are designed that selectively extract EEG sources originating within the right and left motor cortex, respectively [6]. In brief, this can be accomplished by solving the optimization problem w∗= argmax w∈RM ( wTR˜xl/rw wTRxw ) , (5) with Rx ∈RM×M the covariance of the recorded EEG, and R˜xl/r ∈RM×M model-based spatial covariance matrices of EEG sources originating within the left/right motor cortex. In this way, spatial filters can be obtained that optimally attenuate the variance of all EEG sources not originating within the left/right motor cortex. The desired spatial filters are obtained as the eigenvectors with the largest eigenvalue of the generalized eigenvalue problem R˜xl/rw = λRxw (cf. [6] for a more detailed presentation). With EEG sources originating within the left and right motor cortex extracted, TE from all EEG electrodes into the left and right motor cortex can be computed. In this way, volume conduction effects from all sources within the brain into the left/right motor cortex can be optimally attenuated. However, volume conduction effects from the left/right motor cortex to any of the EEG electrodes still poses a problem. Accordingly, it has to be verified if any positive TE from an EEG electrode into the left/right motor cortex could be caused by bandpower changes within the left/right motor cortex. Positive TE from any electrode into the left/right motor cortex can only be considered as a genuine causal link if it is not accompanied by a bandpower change in the respective motor cortex. 3 Experimental Results To investigate connectivity patterns during MI the following experimental paradigm was employed. Subjects sat in a dimly lit and shielded room, approximately two meters in front of a silver screen. Each trial started with a centrally displayed fixation cross. After three seconds, the fixation cross was overlaid with a centrally placed arrow pointing to the left or right. This instructed subjects to begin MI of the left or right hand, respectively. Subjects were explicitly instructed to perform haptic MI, but the exact choice of the type of imaginary hand movement was left unspecified. After a further seven seconds the arrow was removed, indicating the end of the trial and start of the next trial. 150 trials per class were carried out by each subjects in randomized order. During the experiment, EEG was recorded at 128 electrodes placed according to the extended 10-20 system with electrode Cz as reference. EEG data was re-referenced to common average reference offline. Four healthy subjects participated in the experiment, all of which were male and right handed with an age of 27 ± 2.5 years. For each subject, electrode locations were recorded with an ultrasound tracking system. No artifact correction was employed and no trials were rejected. For each subject, model-based covariance matrices R˜xl/r for EEG sources within the left/right motor cortex were computed as described in [6]. The EEG covariance matrix Rx was computed for each subject using all available data, and the two desired Beamformers, extracting EEG sources from the left and right motor cortex, were computed by solving (5). The EEG sources extracted from the left/right motor cortex as well as the unfiltered data recorded at each electrode were then bandpass5 filtered with sixth-order Butterworth filters in five frequency bands ranging from 5 to 55 Hz in steps of 10 Hz. Then, TE was computed from all EEG electrodes into the left/right motor cortex at each sample point as described in Section 2.1. Furthermore, for each subject class-conditional bandpower changes (ERD/ERS) of sources extracted from the left/right motor cortex were computed in order to identify frequency bands with common modulations in bandpower and TE. Two subjects showed significant modulations of bandpower in all five frequency bands. These were excluded from further analysis, since any observed positive TEs could have been confounded by volume conduction. The resulting topographies of mean TE between conditions of the two remaining subjects are shown in Fig. 3. Here, the first two columns show mean TE from all electrodes into the left/right motor cortex during MI of either hand (3.5-10s) minus mean TE during baseline (0.5-3s) in each of the five frequency bands. The last two columns show mean differences in TE into the left/right motor cortex between MI of the left and right hand (both conditions also baseline corrected). Note that the topographies in Fig. 3 have been normalized to the maximum difference across conditions to emphasize differences between conditions. Interestingly, no distinct differences in TE are observed between MI of the left and right hand. Instead, strongest differences in TE are observed in rest vs. MI of either hand (left two columns). The amount of decrease in TE during MI relative to rest increases with higher frequencies, and is most pronounced in the γ-band from 45-55 Hz (last row, left two columns). Topographically, strongest differences are observed in frontal, pre-central, and post-central areas. Observed changes in TE are statistically significant with significance level α = 0.01 at all electrodes in Fig. 3 marked with red crosses (statistical significance was tested nonparametrically and individually for each subject, Beamformer, and condition by one thousand times randomly permuting the EEG data of each recorded trial in time and testing the null-hypothesis that changes in TE at least as large as those in Fig.3 are observed without any temporal structure being present in the data). Due to computational resources only a small subset of electrodes was tested for significance. The observed changes in TE display opposite modulations in comparison to mean bandpower changes observed in left/right motor cortex relative to baseline (Fig. 4, only significant (α = 0.01) bandpower changes relative to baseline (0-3s) plotted). Here, strongest modulation of bandpower is found in the µ- (∼10 Hz) and β-band (∼25 Hz). Frequencies above 35 Hz show very little modulation, indicating that the observed differences in TE at high frequencies in Fig. 3 are not due to volume conduction but genuine causal links. 4 Discussion In this study, Beamforming and TE were employed to investigate the topographies of ’information flow’ into the left and right motor cortex during MI as measured by EEG. To the best of the author’s knowledge, this is the first study investigating asymmetric connectivity patterns between brain regions during MI of different limbs considering a broad frequency range, a large number of recordings sites, and properly taking into account volume conduction effects. However, it should be pointed out that there are several issues that warrant further investigation. First, the presented results are obtained from only two subjects, since two subjects had to be excluded due to possible volume conduction effects. Future studies with more subjects are required to validate the obtained results. Also, no outflow from primary motor cortex and no TE between brain regions not including primary motor cortex have been considered. Finally, the methodology presented in this study can not be applied in a straight-forward manner to single-trial data, and is thus only of limited use for actual feature extraction in BCIs. Never the less, the obtained results indicate that bandpower changes in motor cortex and connectivity between motor cortex and other regions of the brain are processes that occupy distinct spectral bands and are modulated by different cognitive tasks. In conjunction with the observation of no distinct changes in connectivity patterns between MI of different limbs, this indicates that in [14] and [15] bandpower changes might have been misinterpreted as connectivity changes. This is further supported by the fact that these studies focused on frequency bands displaying significant modulation of bandpower (8-30 Hz) and did not control for volume conduction effects. In conclusion, the pronounced modulation of connectivity between MI of either hand vs. rest in the γ-band observed in this study underlines the importance of also considering high frequency bands in EEG connectivity analysis. Furthermore, since the γ-band is thought to be crucial for dynamic functional connectivity between brain regions [10], future studies on connectivity patterns in BCIs should consider experimental paradigms that maximally vary cognitive demands in order to activate different networks within the brain across conditions. 6 Left MC Left MC Right MC Right MC 5-15 Hz 15-25 Hz 25-35 Hz 35-45 Hz 45-55 Hz Motor Imagery - Rest Left - Right Motor Imagery 0 1 -1 C4 C4 C4 C4 C4 C3 C3 C3 C3 C3 Figure 3: Topographies of mean Transfer Entropy changes into left/right motor cortex (MC). C3/C4 mark electrodes over left/right motor cortex. Red crosses indicate statistically significant electrodes. Plotted with [19]. Left Motor Cortex Right Motor Cortex Left Hand Imagery Right Hand Imagery 0s 0s 3s 3s 10s 10s 10 Hz 10 Hz 20 Hz 20 Hz 30 Hz 30 Hz 40 Hz 40 Hz 50 Hz 50 Hz 0 dB 8 dB -8 dB Figure 4: Class-conditional mean ERD/ERS in left/right motor cortex relative to baseline (0-3s). Horizontal line marks start of motor imagery. Plotted with [19]. 7 References [1] J.R. Wolpaw, N. Birbaumer, D.J. McFarland, G. Pfurtscheller, and T.M. Vaughan. Braincomputer interfaces for communication and control. Clinical Neurophysiology, 113(6):767– 791, 2002. [2] S.G. Mason, A. Bashashati, M. Fatourechi, K.F. Navarro, and G.E. Birch. A comprehensive survey of brain interface technology designs. Annals of Biomedical Engineering, 35(2):137– 169, 2007. [3] G. Pfurtscheller and F.H. Lopes da Silva. Even-related EEG/MEG synchronization and desynchronization: basic principles. Clinical Neurophysiology, 110:1842–1857, 1999. [4] H. Ramoser, J. Mueller-Gerking, and G. Pfurtscheller. Optimal spatial filtering of single trial EEG during imagined hand movement. IEEE Transactions on Rehab. Eng., 8(4):441–446, 2000. [5] B. Blankertz, G. Dornhege, M. Krauledat, K.R. Mueller, and G. Curio. 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Brain Topography, 19:137–145, 2007. [14] E. Gysels and P. Celka. Phase synchronization for the recognition of mental tasks in a braincomputer interface. IEEE Transactions on Rehab. Eng., 12(4):406–415, 2004. [15] Q. Wei, Y. Wang, X. Gao, and S. Gao. Amplitude and phase coupling measures for feature extraction in an EEG-based brain-computer interface. Journal of Neural Engineering, 4:120– 129, 2007. [16] T. Schreiber. Measuring information transfer. Physical Review Letters, 85(2):461–464, 2000. [17] A. Kaiser and T. Schreiber. Information transfer in continuous processes. Physica D, 166:43– 62, 2002. [18] P.L. Nunez and R. Srinivasan. Electric Fields of the Brain: The Neurophysics of EEG. Oxford University Press, 2005. [19] A. Delorme and S. Makeig. EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics including independent component analysis. Journal of Neuroscience Methods, 134(1):9–21, 2004. 8
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Rademacher Complexity Bounds for Non-I.I.D. Processes Mehryar Mohri Courant Institute of Mathematical Sciences and Google Research 251 Mercer Street New York, NY 10012 mohri@cims.nyu.edu Afshin Rostamizadeh Department of Computer Science Courant Institute of Mathematical Sciences 251 Mercer Street New York, NY 10012 rostami@cs.nyu.edu Abstract This paper presents the first Rademacher complexity-based error bounds for noni.i.d. settings, a generalization of similar existing bounds derived for the i.i.d. case. Our bounds hold in the scenario of dependent samples generated by a stationary β-mixing process, which is commonly adopted in many previous studies of noni.i.d. settings. They benefit from the crucial advantages of Rademacher complexity over other measures of the complexity of hypothesis classes. In particular, they are data-dependent and measure the complexity of a class of hypotheses based on the training sample. The empirical Rademacher complexity can be estimated from such finite samples and lead to tighter generalization bounds. We also present the first margin bounds for kernel-based classification in this non-i.i.d. setting and briefly study their convergence. 1 Introduction Most learning theory models such as the standard PAC learning framework [13] are based on the assumption that sample points are independently and identically distributed (i.i.d.). The design of most learning algorithms also relies on this key assumption. In practice, however, the i.i.d. assumption often does not hold. Sample points have some temporal dependence that can affect the learning process. This dependence may appear more clearly in times series prediction or when the samples are drawn from a Markov chain, but various degrees of time-dependence can also affect other learning problems. A natural scenario for the analysis of non-i.i.d. processes in machine learning is that of observations drawn from a stationary mixing sequence, a standard assumption adopted in most previous studies, which implies a dependence between observations that diminishes with time [7,9,10,14,15]. The pioneering work of Yu [15] led to VC-dimension bounds for stationary β-mixing sequences. Similarly, Meir [9] gave bounds based on covering numbers for time series prediction [9]. Vidyasagar [14] studied the extension of PAC learning algorithms to these non-i.i.d. scenarios and proved that under some sub-additivity conditions, a PAC learning algorithm continues to be PAC for these settings. Lozano et al. studied the convergence and consistency of regularized boosting under the same assumptions [7]. Generalization bounds have also been derived for stable algorithms with weakly dependent observations [10]. The consistency of learning under the more general scenario of αmixing with non-stationary sequences has also been studied by Irle [3] and Steinwart et al. [12]. This paper gives data-dependent generalization bounds for stationary β-mixing sequences. Our bounds are based on the notion of Rademacher complexity. They extend to the non-i.i.d. case the Rademacher complexity bounds derived in the i.i.d. setting [2, 4, 5]. To the best of our knowledge, these are the first Rademacher complexity bounds derived for non-i.i.d. processes. Our proofs make 1 use of the so-called independent block technique due to Yu [15] and Bernstein and extend the applicability of the notion of Rademacher complexity to non-i.i.d. cases. Our generalization bounds benefit from all the advantageous properties of Rademacher complexity as in the i.i.d. case. In particular, since the Rademacher complexity can be bounded in terms of other complexity measures such as covering numbers and VC-dimension [1], it allows us to derive generalization bounds in terms of these other complexity measures, and in fact improve on existing bounds in terms of these other measures, e.g., VC-dimension. But, perhaps the most crucial advantage of bounds based on the empirical Rademacher complexity is that they are data-dependent: they measure the complexity of a class of hypotheses based on the training sample and thus better capture the properties of the distribution that has generated the data. The empirical Rademacher complexity can be estimated from finite samples and lead to tighter bounds. Furthermore, the Rademacher complexity of large hypothesis sets such as kernel-based hypotheses, decision trees, convex neural networks, can sometimes be bounded in some specific ways [2]. For example, the Rademacher complexity of kernel-based hypotheses can be bounded in terms of the trace of the kernel matrix. In Section 2, we present the essential notion of a mixing process for the discussion of learning in non-i.i.d. cases and define the learning scenario. Section 3 introduces the idea of independent blocks and proves a bound on the expected deviation of the error from its empirical estimate. In Section 4, we present our main Rademacher generalization bounds and discuss their properties. 2 Preliminaries This section introduces the concepts needed to define the non-i.i.d. scenario we will consider, which coincides with the assumptions made in previous studies [7,9,10,14,15]. 2.1 Non-I.I.D. Distributions The non-i.i.d. scenario we will consider is based on stationary β-mixing processes. Definition 1 (Stationarity). A sequence of random variables Z = {Zt}∞ t=−∞is said to be stationary if for any t and non-negative integers m and k, the random vectors (Zt, . . . , Zt+m) and (Zt+k, . . . , Zt+m+k) have the same distribution. Thus, the index t or time, does not affect the distribution of a variable Zt in a stationary sequence (note that this does not imply independence). Definition 2 (β-mixing). Let Z = {Zt}∞ t=−∞be a stationary sequence of random variables. For any i, j ∈Z ∪{−∞, +∞}, let σj i denote the σ-algebra generated by the random variables Zk, i ≤k ≤j. Then, for any positive integer k, the β-mixing coefficient of the stochastic process Z is defined as β(k) = sup n E B∈σn −∞ h sup A∈σ∞ n+k Pr[A | B] −Pr[A] i . (1) Z is said to be β-mixing if β(k) →0. It is said to be algebraically β-mixing if there exist real numbers β0 > 0 and r > 0 such that β(k) ≤β0/kr for all k, and exponentially mixing if there exist real numbers β0 and β1 such that β(k) ≤β0 exp(−β1kr) for all k. Thus, a sequence of random variables is mixing when the dependence of an event on those occurring k units of time in the past weakens as a function of k. 2.2 Rademacher Complexity Our generalization bounds will be based on the following measure of the complexity of a class of functions. Definition 3 (Rademacher Complexity). Given a sample S ∈Xm, the empirical Rademacher complexity of a set of real-valued functions H defined over a set X is defined as follows: bRS(H) = 2 m E σ sup h∈H m X i=1 σih(xi) S = (x1, . . . , xm) . (2) 2 The expectation is taken over σ = (σ1, . . . , σn) where σis are independent uniform random variables taking values in {−1, +1} called Rademacher random variables. The Rademacher complexity of a hypothesis set H is defined as the expectation of bRS(H) over all samples of size m: Rm(H) = E S bRS(H) |S| = m . (3) The definition of the Rademacher complexity depends on the distribution according to which samples S of size m are drawn, which in general is a dependent β-mixing distribution D. In the rare instances where a different distribution eD is considered, typically for an i.i.d. setting, we explicitly indicate that distribution as a superscript: R e D m(H). The Rademacher complexity measures the ability of a class of functions to fit noise. The empirical Rademacher complexity has the added advantage that it is data-dependent and can be measured from finite samples. This can lead to tighter bounds than those based on other measures of complexity such as the VC-dimension [2,4,5]. We will denote by bRS(h) the empirical average of a hypothesis h: X →R and by R(h) its expectation over a sample S drawn according to a stationary β-mixing distribution: bRS(h) = 1 m m X i=1 h(zi) R(h) = E S[ bRS(h)]. (4) The following proposition shows that this expectation is independent of the size of the sample S, as in the i.i.d. case. Proposition 1. For any sample S of size m drawn from a stationary distribution D, the following holds: ES∼Dm[ bRS(h)] = Ez∼D[h(z)]. Proof. Let S = (x1, . . . , xm). By stationarity, Ezi∼D[h(zi)] = Ezj∼D[h(zj)] for all 1 ≤i, j ≤m, thus, we can write: E S[ bRS(h)] = 1 m m X i=1 E S [h(zi)] = 1 m m X i=1 E zi[h(zi)] = E z [h(z)]. 3 Proof Components Our proof makes use of McDiarmid’s inequality [8] to show that the empirical average closely estimates its expectation. To derive a Rademacher generalization bound, we apply McDiarmid’s inequality to the following random variable, which is the quantity we wish to bound: Φ(S) = sup h∈H R(h) −bRS(h). (5) McDiarmid’s inequality bounds the deviation of Φ from its mean, thus, we must also bound the expectation E[Φ]. However, we immediately face two obstacles: both McDiarmid’s inequality and the standard bound on E[Φ] hold only for samples drawn in an i.i.d. fashion. The main idea behind our proof is to analyze the non-i.i.d. setting and transfer it to a close independent setting. The following sections will describe in detail our solution to these problems. 3.1 Independent Blocks We derive Rademacher generalization bounds for the case where training and test points are drawn from a stationary β-mixing sequence. As in previous non-i.i.d. analyses [7, 9, 10, 15], we use a technique transferring the original problem based on dependent points to one based on a sequence of independent blocks. The method consists of first splitting a sequence S into two subsequences S0 and S1, each made of µ blocks of a consecutive points. Given a sequence S = (z1, . . . , zm) with m = 2aµ, S0 and S1 are defined as follows: S0 = (Z1, Z2, . . . , Zµ), where Zi = (z(2i−1)+1, . . . , z(2i−1)+a), (6) S1 = (Z(1) 1 , Z(1) 2 , . . . , Z(1) µ ), where Z(1) i = (z2i+1, . . . , z2i+a). (7) 3 Instead of the original sequence of odd blocks S0, we will be working with a sequence eS0 of independent blocks of equal size a to which standard i.i.d. techniques can be applied: eS0 = ( eZ1, eZ2, . . . , eZµ) with mutually independent eZks, but, the points within each block eZk follow the same distribution as in Zk. As stated by the following result of Yu [15][Corollary 2.7], for a sufficiently large spacing a between blocks and a sufficiently fast mixing distribution, the expectation of a bounded measurable function h is essentially unchanged if we work with eS0 instead of S0. Corollary 1 ([15]). Let h be a measurable function bounded by M ≥0 defined over the blocks Zk, then the following holds: | E S0[h] −E e S0 [h]| ≤(µ −1)Mβ(a), (8) where ES0 denotes the expectation with respect to S0, EeS0 the expectation with respect to the eS0. We denote by eD the distribution corresponding to the independent blocks eZk. Also, to work with block sequences, we extend some of our definitions: we define the extension ha : Za →R of any hypothesis h∈H to a block-hypothesis by ha(B)= 1 a Pa i=1 h(Zi) for any block B =(z1, . . . , za)∈ Za, and define Ha as the set of all block-based hypotheses ha generated from h∈H. It will also be useful to define the subsequence Sµ, which consists of µ singleton points separated by a gap of 2a −1 points. This can be thought of as the sequence constructed from S0, or S1, by selecting only the jth point from each block, for any fixed j ∈{1, . . . , a}. 3.2 Concentration Inequality McDiarmid’s inequality requires the sample to be i.i.d. Thus, we first show that Pr[Φ(S)] can be bounded in terms of independent blocks and then apply McDiarmid’s inequality to the independent blocks. Lemma 1. Let H be a set of hypotheses bounded by M. Let S denote a sample, of size m, drawn according to a stationary β-mixing distribution and let eS0 denote a sequence of independent blocks. Then, for all a, µ, ǫ > 0 with 2µa = m and ǫ > EeS0[Φ(eS0)], the following bound holds: Pr S [Φ(S) > ǫ] ≤2 Pr eS0 [Φ(eS0) −E eS0 [Φ(eS0)] > ǫ′] + 2(µ −1)β(a), where ǫ′ = ǫ −E e S0[Φ(eS0)]. Proof. We first rewrite the left-hand side probability in terms of even and odd blocks and then apply Corollary 1 as follows: Pr S [Φ(S) > ǫ] = Pr S [sup h (R(h) −bRS(h)) > ǫ] = Pr S h sup h R(h)−b RS0(h) 2 + R(h)−b RS1(h) 2 > ǫ i (def. of bRS(h)) ≤Pr S h1 2 sup h (R(h) −bRS0(h)) + sup h (R(h) −bRS1(h)) > ǫ i (convexity of sup) = Pr S [Φ(S0) + Φ(S1) > 2ǫ] (def. of Φ) ≤Pr S0[Φ(S0) > ǫ] + Pr S1[Φ(S1) > ǫ] (union bound) = 2 Pr S0[Φ(S0) > ǫ] (stationarity) = 2 Pr S0[Φ(S0) −E e S0 [Φ(eS0)] > ǫ′]. (def. of ǫ′) The second inequality holds by the union bound and the fact that Φ(S0) or Φ(S1) must surpass ǫ for their sum to surpass 2ǫ. To complete the proof, we apply Corollary 1 to the expectation of the indicator variable of the event {Φ(S0) −EeS0[Φ(eS0)] > ǫ′}, which yields 2 Pr S0[Φ(S0) −E e S0 [Φ(eS0)] > ǫ′] ≤2 Pr eS0 [Φ(eS0) −E e S0 [Φ(eS0)] > ǫ′] + 2(µ −1)β(a). We can now apply McDiarmid’s inequality to the independent blocks of Lemma 1. 4 Proposition 2. For the same assumptions as in Lemma 1, the following bound holds for all ǫ > EeS0[Φ(eS0)]: Pr S [Φ(S) > ǫ] ≤2 exp −2µǫ′2 M 2 + 2(µ −1)β(a), where ǫ′ = ǫ −E e S0[Φ(eS0)]. Proof. To apply McDiarmid’s inequality, we view each block as an i.i.d. point with respect to ha. Φ(eS0) can be written in terms of ha as: Φ(eS0) = R(ha) −bR e S0(ha) = R(ha) −1 µ Pµ k=1 ha( eZk). Thus, changing a block eZk of the sample eS0 can change Φ(eS0) by at most 1 µ|h( eZk)| ≤M/µ. By McDiarmid’s inequality, the following holds for any ǫ > 2(µ −1)Mβ(a): Pr eS0 [Φ(eS0) −E eS0 [Φ(eS0)] > ǫ′] ≤exp −2ǫ′2 Pµ i=1(M/µ)2 = exp −2µǫ′2 M 2 . Plugging in the right-hand side in the statement of Lemma 1 proves the proposition. 3.3 Bound on the Expectation Here, we give a bound on Ee S0[Φ(S0)] based on the Rademacher complexity, as in the i.i.d. case [2]. But, unlike the standard case, the proof requires an analysis in terms of independent blocks. Lemma 2. The following inequality holds for the expectation EeS0[Φ(eS0)] defined in terms of an independent block sequence:Ee S0[Φ(eS0)] ≤R e D µ (H). Proof. By the convexity of the supremum function and Jensen’s inequality, Ee S0[Φ(eS0)] can be bounded in terms of empirical averages over two samples: E e S0 [Φ(eS0)] = E eS0 [sup h∈H E e S′ 0 [ bR e S′ 0(h)] −bR e S0(h)] ≤ E e S0, e S′ 0 [sup h∈H bR e S′ 0(h) −bR e S0(h)]. We now proceed with a standard symmetrization argument with the independent blocks thought of as i.i.d. points: E e S0 [Φ(eS0)] ≤E e S0, e S′ 0 [sup h∈H bR e S′ 0(h) −bR e S0(h)] = E e S0, e S′ 0 sup ha∈Ha 1 µ µ X i=1 ha(Zi) −ha(Z′ i) (def. of bR) = E e S0, e S′ 0,σ sup ha∈Ha 1 µ µ X i=1 σi(ha(Zi) −ha(Z′ i)) (Rad. var.’s) ≤ E e S0, e S′ 0,σ sup ha∈Ha 1 µ µ X i=1 σiha(Zi) + E eS0, e S′ 0,σ sup ha∈Ha 1 µ µ X i=1 σiha(Z′ i) (sub-add. of sup) = 2 E eS0,σ sup ha∈Ha 1 µ µ X i=1 σiha(Zi) . In the second equality, we introduced the Rademacher random variables σis. With probability 1/2, σi = 1 and the difference ha(Zi) −ha(Z′ i) is left unchanged; and, with probability 1/2, σi = −1 and Zi and Z′ i are permuted. Since the blocks Zi, or Z′ i are independent, taking the expectation over σ leaves the expectation unchanged. The inequality follows from the sub-additivity of the supremum function and the linearity of expectation. The final equality holds because eS0 and eS′ 0 are identically distributed due to the assumption of stationarity. We now relate the Rademacher block sequence to a sequence over independent points. The righthand side of the inequality just presented can be rewritten as 2 E eS0,σ sup ha∈Ha 1 µ µ X i=1 σiha(Zi) = E e S0,σ sup h∈H 2 µ µ X i=1 σi 1 a a X j=1 h(z(i) j ) , 5 where z(i) j denotes the jth point of the ith block. For j ∈[1, a], let eSj 0 denote the i.i.d. sample constructed from the jth point of each independent block Zi, i ∈[1, µ]. By reversing the order of summations and using the convexity of the supremum function, we obtain the following: E e S0 [Φ(eS0)] ≤E eS0,σ sup h∈H 1 a a X j=1 2 µ µ X i=1 σih(z(i) j ) (reversing order of sums) ≤1 a a X j=1 E e S0,σ sup h∈H 2 µ µ X i=1 σih(z(i) j ) (convexity of sup) =1 a a X j=1 E e Sj 0,σ sup h∈H 2 µ µ X i=1 σih(z(i) j ) (marginalization) = E eSµ,σ sup h∈H 2 µ µ X i=1 zi∈e Sµ σih(zi) ≤R e D µ (H). The first equality in this derivation is obtained by marginalizing over the variables that do not appear within the inner sum. Then, the second equality holds since, by stationarity, the choice of j does not change the value of the expectation. The remaining quantity, modulo absolute values, is the Rademacher complexity over µ independent points. 4 Non-i.i.d. Rademacher Generalization Bounds 4.1 General Bounds This section presents and analyzes our main Rademacher complexity generalization bounds for stationary β-mixing sequences. Theorem 1 (Rademacher complexity bound). Let H be a set of hypotheses bounded by M ≥0. Then, for any sample S of size m drawn from a stationary β-mixing distribution, and for any µ, a > 0 with 2µa = m and δ > 2(µ −1)β(a), with probability at least 1 −δ, the following inequality holds for all hypotheses h ∈H: R(h) ≤bRS(h) + R e D µ (H) + M s log 2 δ′ 2µ , where δ′ = δ −2(µ −1)β(a). Proof. Setting the right-hand side of Proposition 2 to δ and using Lemma 2 to bound EeS0[Φ(eS0)] with the Rademacher complexity R e D µ (H) shows the result. As pointed out earlier, a key advantage of the Rademacher complexity is that it can be measured from data, assuming that the computation of the minimal empirical error can be done effectively and efficiently. In particular we can closely estimate bRSµ(H), where Sµ is a subsample of the sample S drawn from a β-mixing distribution, by considering random samples of σ. The following theorem gives a bound precisely with respect to the empirical Rademacher complexity bRSµ. Theorem 2 (Empirical Rademacher complexity bound). Under the same assumptions as in Theorem 1, for any µ, a > 0 with 2µa = m and δ > 4(µ −1)β(a), with probability at least 1 −δ, the following inequality holds for all hypotheses h ∈H: R(h) ≤bRS(h) + bRSµ(H) + 3M s log 4 δ′ 2µ , where δ′ = δ −4(µ −1)β(a). 6 Proof. To derive this result from Theorem 1, it suffices to bound R e D µ (H) in terms of bRSµ(H). The application of Corollary 1 to the indicator variable of the event {R e D µ (H) −bRSµ(H) > ǫ} yields Pr R e D µ (H) −bRSµ(H) > ǫ ≤Pr R e D µ (H) −bR e Sµ(H) > ǫ + (µ −1)β(2a −1). (9) Now, we can apply McDiarmid’s inequality to R e D µ (H) −bR e Sµ(H) which is defined over points drawn in an i.i.d. fashion. Changing a point of Sµ can affect bR e Sµ by at most (2M/µ), thus, McDiarmid’s inequality gives Pr R e D µ (H) −bRSµ(H) > ǫ ≤exp −µǫ2 2M 2 + (µ −1)β(2a −1). (10) Note β is a decreasing function, which implies β(2a −1) ≤β(a). Thus, with probability at least 1 −δ/2, Rµ(H) ≤bRSµ(H) + M q 2 log 1 δ′ µ , with δ′ = δ/2 −(µ −1)β(a), a fortiori with δ′ = δ/4 −(µ −1)β(a). The result follows this inequality combined with the statement of Theorem 1 for a confidence parameter δ/2. This theorem can be used to derive generalization bounds for a variety of hypothesis sets and learning settings. In the next section, we present margin bounds for kernel-based classification. 4.2 Classification Let X denote the input space, Y ={−1, +1} the target values in classification, and Z =X × Y . For any hypothesis h and margin ρ>0, let bRρ S(h) denote the average amount by which yh(x) deviates from ρ over a sample S: bRρ S(h) = 1 m Pm i=1(ρ −yih(xi))+. Given a positive definite symmetric kernel K : X ×X →R, let K denote its Gram matrix for the sample S and HK the kernel-based hypothesis set {x 7→Pm i=1 αiK(xi, x): αKαT ≤1}, where α ∈Rm×1 denotes the column-vector with components αi, i = 1, . . . , m. Theorem 3 (Margin bound). Let ρ>0 and K be a positive definite symmetric kernel. Then, for any µ, a>0 with 2µa = m and δ>4(µ −1)β(a), with probability at least 1 −δ over samples S of size m drawn from a stationary β-mixing distribution, the following inequality holds for all hypotheses h∈HK: Pr[yh(x) ≤0] ≤1 ρ bRρ S(h) + 4 µρ p Tr[K] + 3 s log 4 δ′ 2µ , where δ′ = δ −4(µ −1)β(a). Proof. For any h∈H, let h denote the corresponding hypothesis defined over Z by: ∀z ∈Z, h(z)= −yh(x); and HK the hypothesis set {z ∈Z 7→h(z): h ∈HK}. Let L denote the loss function associated to the margin loss bRρ S(h). Then, Pr[yh(x) ≤0] ≤Pr[(L ◦h)(z) ≤0] = R(L ◦h). Since L −1 is 1/ρ-Lipschitz and (L −1)(0)=0, by Talagrand’s lemma [6], bRS((L −1) ◦HK)≤ 2bRS(HK)/ρ. The result is then obtained by applying Theorem 2 to R((L −1) ◦h) = R(L ◦h) −1 with bR((L −1) ◦h) = bR(L ◦h) −1, and using the known bound for the empirical Rademacher complexity of kernel-based classifiers [2,11]: bRS(HK)≤2 |S| p Tr[K]. In order to show that this bound converges, we must appropriately choose the parameter µ, or equivalently a, which will depend on the mixing parameter β. In the case of algebraic mixing and using the straightforward bound Tr[K] ≤mR2 for the kernel trace, where R is the radius of the ball that contains the data, the following corollary holds. Corollary 2. With the same assumptions as in Theorem 3, if β is further algebraically β-mixing, β(a) = β0a−r, then, with probability at least 1 −δ, the following bound holds for all hypotheses h∈HK: Pr[yh(x) ≤0] ≤1 ρ bRρ S(h) + 8Rmγ1 ρ + 3mγ2 r log 4 δ′ , where γ1 = 1 2 3 r+2 −1 , γ2 = 1 2 3 2r+4 −1 and δ′ = δ −2β0mγ1. 7 This bound is obtained by choosing µ = 1 2m 2r+1 2r+4 , which, modulo a multiplicative constant, is the minimizer of (√m/µ + µβ(a)). Note that for r > 1 we have γ1, γ2 < 0 and thus, it is clear that the bound converges, while the actual rate will depend on the distribution parameter r. A tighter estimate of the trace of the kernel matrix, possibly derived from data, would provide a better bound, as would stronger mixing assumptions, e.g., exponential mixing. Finally, we note that as r →∞ and β0 →0, that is as the dependence between points vanishes, the right-hand side of the bound approaches O( bRρ S +1/√m), which coincides with the asymptotic behavior in the i.i.d. case [2,4,5]. 5 Conclusion We presented the first Rademacher complexity error bounds for dependent samples generated by a stationary β-mixing process, a generalization of similar existing bounds derived for the i.i.d. case. We also gave the first margin bounds for kernel-based classification in this non-i.i.d. setting, including explicit bounds for algebraic β-mixing processes. Similar margin bounds can be obtained for the regression setting by using Theorem 2 and the properties of the empirical Rademacher complexity, as in the i.i.d. case. Many non-i.i.d. bounds based on other complexity measures such as the VC-dimension or covering numbers can be retrieved from our framework. Our framework and the bounds presented could serve as the basis for the design of regularization-based algorithms for dependent samples generated by a stationary β-mixing process. Acknowledgements This work was partially funded by the New York State Office of Science Technology and Academic Research (NYSTAR). References [1] M. Anthony and P. Bartlett. Neural Network Learning: Theoretical Foundations. Cambridge University Press, Cambridge, UK, 1999. [2] P. L. Bartlett and S. Mendelson. Rademacher and Gaussian complexities: Risk bounds and structural results. Journal of Machine Learning Research, 3:2002, 2002. [3] A. Irle. On the consistency in nonparametric estimation under mixing assumptions. Journal of Multivariate Analysis, 60:123–147, 1997. [4] V. Koltchinskii and D. Panchenko. Rademacher processes and bounding the risk of function learning. In High Dimensional Probability II, pages 443–459. preprint, 2000. [5] V. Koltchinskii and D. Panchenko. Empirical margin distributions and bounding the generalization error of combined classifiers. Annals of Statistics, 30, 2002. [6] M. Ledoux and M. Talagrand. Probability in Banach Spaces: Isoperimetry and Processes. Springer, 1991. [7] A. Lozano, S. Kulkarni, and R. Schapire. Convergence and consistency of regularized boosting algorithms with stationary β-mixing observations. Advances in Neural Information Processing Systems, 18, 2006. [8] C. McDiarmid. On the method of bounded differences. In Surveys in Combinatorics, pages 148–188. Cambridge University Press, 1989. [9] R. Meir. Nonparametric time series prediction through adaptive model selection. Machine Learning, 39(1):5–34, 2000. [10] M. Mohri and A. Rostamizadeh. Stability bounds for non-iid processes. Advances in Neural Information Processing Systems, 2007. [11] J. Shawe-Taylor and N. Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, 2004. [12] I. Steinwart, D. Hush, and C. Scovel. Learning from dependent observations. Technical Report LA-UR06-3507, Los Alamos National Laboratory, 2007. [13] L. G. Valiant. A theory of the learnable. ACM Press New York, NY, USA, 1984. [14] M. Vidyasagar. Learning and Generalization: with Applications to Neural Networks. Springer, 2003. [15] B. Yu. Rates of convergence for empirical processes of stationary mixing sequences. Annals Probability, 22(1):94–116, 1994. 8
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A rational model of preference learning and choice prediction by children Christopher G. Lucas Department of Psychology University of California, Berkeley Berkeley, CA 94720, USA clucas@berkeley.edu Thomas L. Griffiths Department of Psychology University of California, Berkeley Berkeley, CA 94720, USA tom griffiths@berkeley.edu Fei Xu Department of Psychology University of British Columbia Vancouver, B.C., Canada V6T 1Z4 fei@psych.ubc.ca Christine Fawcett Max-Planck-Institute for Psycholinguistics Wundtlaan 1, Postbus 310 6500AH, Nijmegen, The Netherlands christine.fawcett@mpi.nl Abstract Young children demonstrate the ability to make inferences about the preferences of other agents based on their choices. However, there exists no overarching account of what children are doing when they learn about preferences or how they use that knowledge. We use a rational model of preference learning, drawing on ideas from economics and computer science, to explain the behavior of children in several recent experiments. Specifically, we show how a simple econometric model can be extended to capture two- to four-year-olds’ use of statistical information in inferring preferences, and their generalization of these preferences. 1 Introduction Economists and computer scientists are often concerned with inferring people’s preferences from their choices, developing econometric methods (e.g., [1, 2]) and collaborative filtering algorithms (e.g., [3, 4, 5]) that will allow them to assess the subjective value of an item or determine which other items a person might like. However, identifying the preferences of others is also a key part of social cognitive development, allowing children to understand how people act and what they want. Young children are thus often in the position of economists or computer scientists, trying to infer the nebulous preferences of the people around them from the choices they make. In this paper, we explore whether the inferences that children draw about preferences can be explained within the same kind of formalism as that used in economics and computer science, testing the hypothesis that children are making rational inferences from the limited data available to them. Before about 18 months of age, children seem to assume that everyone likes the same things as themselves, having difficulty understanding the subjective nature of preferences (e.g., [6]). However, shortly after coming to recognize that different agents can maintain different preferences, children demonstrate a remarkably sophisticated ability to draw conclusions about the preferences of others from their behavior. For example, two-year-olds seem to be capable of using shared preferences between an agent and themselves as the basis for generalization of other preferences [7], while threeand four-year-olds can use statistical information to reason about preferences, inferring a preference for an object when an agent chooses the object more often than expected by chance [8]. This literature in developmental psychology is paralleled by work in econometrics on statistical models for inferring preferences from choices. In this paper, we focus on an approach that grew out 1 of the Nobel prize-winning work of McFadden (see [1] for a review), exploring a class of models known as mixed multinomial logit models [2]. These models assume that agents assign some utility to every option in a choice, and choose in a way that is stochastically related to these utilities. By observing the choices people make, we can recover their utilities by applying statistical inference, providing a simple rational standard against which the inferences of children can be compared. Research on preferences in computer science has tended to go beyond modeling individual choice, focusing on predicting which options people will like based not just on their own previous choice patterns but also drawing on the choices of other people – a problem known as collaborative filtering [3]. This work has led to the development of the now-ubiquitous recommendation systems that suggest which items one might like to purchase based on previous purchases, and has reached notoriety through the recent Netflix challenge. Economists have also explored models for the choices of multiple agents, using hierarchical Bayesian statistics [9]. These models combine information across agents to make inferences about the properties or value of different options. Our contribution in this paper is to bring together these different threads of research to develop rational models of children’s inferences about preferences. Section 2 summarizes developmental work on children’s inferences about preferences. Section 3 outlines the basic idea behind rational choice models, drawing on previous work in economics and computer science. We then consider how these models can be used to explain developmental data, with Section 4 concerned with inferences about preferences from choices, and Section 5 focusing on inferences about the properties of objects from preferences. Section 6 discusses the implications of our results and concludes the paper. 2 Children’s inferences about preferences The basic evidence that children do not differentiate the preferences of others before about 18 months of age comes from Repacholi and Gopnik [6]. Subsequent work has built on these results to explore the kinds of cues that children can use in inferring preferences, and how children generalize consistent patterns of preferences. 2.1 Learning preferences from statistical evidence While 18-month-olds are able to infer preferences from affective responses, we often need to make inferences from more impoverished data, such as the patterns of choices that people make when faced with various options. Recent work by Kushnir and colleagues [8] provided the first evidence that 3- and 4-year-old children can use statistical sampling information as the basis for inferring an agent’s preference for toys. Three groups of children were tested in a simple task. Each child was shown a big box of toys. For the first group, the box was filled with just one type of toy (e.g., 100% red discs). For the second group, the box was filled with two types of toys (e.g., 50% red discs and 50% blue plastic flowers). For the third group, the box was also filled with two types of toys, but in different proportions (e.g., 18% red discs and 82% blue plastic flowers). A puppet named Squirrel came in to play a game with the child. Squirrel looked into the box and picked out five toys. The sample always consisted of five red discs for all three conditions. Then the child was given three toys – a red disc (the target), a blue plastic flower (the alternative), and a yellow cylinder (the distractor) – and was asked to give Squirrel the one he liked. Each child received two trials with different objects. The results of the experiment showed that the children chose the target (the red disc) 0.96, 1.29, and 1.67 times (out of 2) in the 100%, 50%, and 18% conditions, respectively, suggesting that children used the non-random sampling behavior of Squirrel as the basis for inferring his preferences. 2.2 Generalizing from shared preferences Recognizing that preferences can vary from one agent to another also establishes an opportunity to discover that those preferences can differ in the degree to which they are related to one’s own. Fawcett and Markson [7] asked under what conditions children would use shared preferences between themselves and another agent as the basis for generalization, using a task similar to the “collaborative filtering” problem explored in computer science. Their experiments began with four blocks of training involving two actors. In each block the actors introduced two objects from a common category, including toys, television shows and foods. Each actor expressed liking the object she introduced and dislike for the other’s object. One actor had preferences that were matched to the 2 child’s in all blocks, in that her objects had features chosen to be more interesting to the child. After each actor reacted to the objects, the child was given an opportunity to play with the objects, and his or her preference for one object over the other was judged by independent coders, based on relative interest in and play with each object. After the training blocks, the first test block began. Each actor brought out a new object that was described as being in the same category as the training objects, but was hidden from the child by an opaque container. Each actor then reacted to her novel object in a manner that varied by condition. In the like condition, the actor’s reaction was to examine the object and describe it as her favorite object of the category. In the dislike condition, she examined the object and expressed dislike. In the indifferent condition the actor did not examine the toy, and professed ignorance about it. The child was then given an opportunity to choose one hidden object to play with. Finally, a second test block began, identical to the first except that the hidden objects were members of a different category from those seen in training. In Experiment 1, members of the new category could be taken to share features with members of the training category, e.g., toys versus books, while in Experiment 2 the new category was chosen to minimize such overlap, e.g., food versus television shows. Children consistently chose the test items that were favored by the agent who shared their own preferences during training, for both toys and the similar category, books. In contrast, when a highly distant category was used during test, children did not show any systematic generalization behaviors. These results suggest that children use shared preferences as the basis for generalization, but they also take into account whether the categories are related or not. 2.3 Summary and prospectus Recent results in developmental psychology indicate that young children are capable of making remarkably sophisticated inferences about the preferences of others. This raises the question of how they make these inferences, and whether the kinds of conclusions that children draw from the behavior of others are justified. We explore this question in the remainder of the paper. In the tradition of rational analysis [10], we consider the problem of how one might optimally infer people’s preferences from their choices, and compare the predictions of such a model with the developmental data. The results of this analysis will help us understand how children might conceive of the relationship between the choices that people make and the preferences they have. 3 A rational model connecting choice and preference In developing a rational account of how an agent might learn others’ preferences from choice information, we must first posit a specific ecological relationship between people’s preferences and their choices, and then determine how an agent would make optimal inferences from others’ behavior given knowledge of this relationship. Fortunately, the relationship between preferences and choices has been the subject of extensive research in economics and psychology. One of the most basic models of choice behavior is the Luce-Shepard choice rule [11, 12], which asserts that when presented with a set of J options with utilities u = (u1, . . . , uJ), people will choose option i with probability P(c = i|u) = exp(ui) P j exp(uj) (1) where j ranges over the options considered in the choice. Given this choice rule, learning about an agent’s preferences is a simple matter of Bayesian inference. Specifically, having observed a sequence of choices c = (c1, . . . , cN), we can compute a posterior distribution over the utilities of the options involved by applying Bayes’ rule p(u|c) = P(c|u)p(u) R P(c|u)p(u) du (2) where p(u) is a probability density expressing the prior probability of a vector of utilities u, and the likelihood P(c|u) is obtained by assuming that the choices are independent given u, being the product of the probabilities of the individual choices as in Equation 1. While this simple model is sufficient to capture preferences among a constrained set of objects, most models used in econometrics aim to predict the choices that agents will make about novel objects. 3 This can be done by assuming that options have features that determine their utility, with the utility of option i being a function of the utility of its features. If we let xi be a binary vector indicating whether an option possesses each of a finite set of features, and βa be the utility that agent a assigns to those features1, we can express the utility of option i for agent a as the inner product of these two vectors. The probability of agent a choosing option i is then P(c = j|X, βa) = exp(βT a xi) P j exp(βT a xj) (3) where X collects the features of all of the options. We can also integrate out βa to obtain the choice probabilities given just the features of the options, with P(c = j|X) = Z exp(βT a xi) P j exp(βT a xj) p(βa) dβa (4) This corresponds to the mixed multinomial logit (MML; [2]) model, which has been used for several decades in econometrics to model discrete-choice preferences in populations of consumers. The MML model and the Luce-Shepard rule on which it is built are theoretically appealing for several reasons. First, the Luce-Shepard rule reflects the choice probabilities that result when agents seek to maximize their utility in the presence of noise on utilities that follows a Weibull distribution [13], and is thus compatible with the standard assumptions of statistical decision theory. Second, the MML can approximate the distribution of choices for essentially any heterogeneous population of utility-maximizing agents given appropriate choice of p(βa) [1]. Finally, this approach has been widely used and generally successful in applications in marketing and econometrics (e.g., [9, 14]). While a wide range of random utility models can be represented with an appropriate choice of prior over βa, one common variation supposes that βa follows a Gaussian distribution around a population mean which in turn has a Gaussian prior. Given that the individuals in the experiments we examine are only dealing with two actors, we will assume a single-parameter prior in which different agents’ preferences are independent and preferences for individual features are uncorrelated with a Gaussian distribution with mean zero and variance σ2: βa ∼N(0, σ2I). The model outlined in this section provides a way to optimally answer the question of how to infer the preferences of an agent from their choices. In the remainder of the paper, we explore how well this simple rational model accounts for the inferences that children make about preferences, applying the model to the key developmental phenomena introduced in the previous section. 4 Using statistical information to infer preferences The experiment conducted by Kushnir and colleagues [8], discussed in Section 2.1, provides evidence that children are sensitive to statistical information when inferring the preferences of agents. In this section, we examine whether this inference is consistent with the predictions of the rational model outlined above. We first consider how to apply the MML model in this context, then discuss the model predictions and alternative explanations. 4.1 Applying the MML model The child’s goal is to learn what Squirrel’s preferences are, so as to offer an appropriate toy. Let βa be Squirrel’s preferences, c = (c1 . . . cN) the sequence of N choices Squirrel makes, and Xn = [xn1 . . . xnJn]T the observed features of Squirrel’s Jn options at choice event n. The set {X1, . . . , XN} will be denoted with X. Estimating βa entails computing p(βa|c, X) ∝ P(c|βa, X)p(βa), analogous to the inference of u in Equation 2. The probability of Squirrel’s choices is P(c|X, βa) = QN n=1 P(cn|Xn, βa), where P(cn = j|Xn, βa) is given by Equation 3. We chose to represent the objects as having minimal and orthogonal feature vectors, so that red discs (Squirrel’s target toy) had features [1 0 0]T , blue flowers (the alternative option in his choices) had features [0 1 0]T , and yellow cylinders (the distractor) had features [0 0 1]T , respectively. The 1We will refer to the utilities of features as “preferences” to distinguish them from the utilities of options. 4 (a) target alternate distractor 0 0.5 1 Predictions vs choice probabilities for σ2=2 100 percent target target alternate distractor 0 0.5 1 P(choice) 50 percent target target alternate distractor 0 0.5 1 object 18 percent target model children (b) 0 1 2 3 4 5 6 7 8 9 10 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 variance Sum squared error (SSE) Variance (σ ) parameter versus sum squared error 2 Figure 1: Model predictions for data in [8]. (a) Predicted probability that objects will be selected, plotted against observed proportions. (b) Sensitivity of model to setting of variance parameter. number of options in each choice Squirrel made from the box was the total number of objects in the box (38), with each type of object being represented with the appropriate frequency. The N = 5 choices made by Squirrel thus provide the data c from which its preferences can be inferred. We constructed an approximation to the posterior distribution over βa given c using importance sampling (see, e.g., [15] for details), drawing a sample of values of βa from the prior distribution p(βa) and giving each value weight proportional to the corresponding likelihood P(c|X, βa). The child must now select an object to give Squirrel, with one target, one alternative, and one distractor as options. If we suppose each child is matching Squirrel’s choice distribution, we can use the Luce-Shepard choice rule (Equation 3) to predict the rates at which children should choose the different objects for a particular value of βa, P(cchild = j|X, βa). The probability of a particular choice is then obtained by averaging over βa, with P(cchild = j|X, c) = Z P(cchild = j|X, βa)p(βa|X, c) dβa (5) which we compute using the approximate posterior distribution yielded by importance sampling. All simulations presented here use 106 samples, and were performed for a range of values of σ2, the parameter that determines the variance of the prior on βa. 4.2 Results Figure 1 (a) compares the predictions of the model with σ2 = 2 to the participants’ choice probabilities. The sum squared error (SSE) of the predictions was .0758 when compared with the observed probabilities of selecting the target object, with the correlation of the model predictions and observed data being r = 0.93. Figure 1 (b) shows that the goodness of fit is generally insensitive to the variance of the prior σ2, provided σ2 > 1. This is essentially the only free parameter of the model, as it sets the scale for the features X, indicating that there is a close correspondence between the predictions of the rational model and the inferences of the children under a variety of reasonable assumptions about the distribution of preferences. The only conspicuous difference between the model’s predictions and the children’s choices was the tendency of children to choose the target object more frequently than alternatives even when it was the only object in the box. This can be explained by observing that under the cover story used in the experiment, Squirrel was freely choosing to select objects from the box, implicitly indicating that it was choosing these objects over other unobserved options. As a simple test of this explanation, we generated new predictions under the assumption that each choice included one other unobserved option, with features orthogonal to the choices in the box. This improved the fit of the model, resulting in an SSE of .05 and a correlation of r = .95. 5 4.3 Alternative explanations Kushnir and colleagues [8] suggest that the children in the experiment may be learning preferences by using statistical information to identify situations where the agent’s behavior is not consistent with random sampling. We do not dispute that this may be correct; our analysis does not entail a commitment to a procedure by which children make inferences about preferences, but to the idea that whatever the process is, it should provides a good solution to the problem with which children are faced given the constraints under which they operate. We will add two observations. The first is that we need to consider how such an explanation might be generalized to explain behavior in other preference-learning situations, and if not, what additional processes might be at work. The second is that it is not difficult to test salient variations on the sampling-versus-preference view that are inconsistent with our own, as they predict that children make a dichotomous judgment – distinguishing random from biased sampling – rather than one that reveals the extent of a preference in addition to its presence and valence. The former predicts that over a wide range of sets of evidence that indicate an agent strongly prefers objects of type X to those of type Y, one can generate evidence consistent with a weaker preference for type W over Y (over more data points) that will lead children to offer the agent objects of type W over X. 5 Generalizing preferences to novel objects The study of Fawcett and Markson [7] introduced in Section 2.2 provides a way to go beyond simple estimation of preferences from choices, exploring how children solve the “collaborative filtering” problem of generalizing preferences to novel objects. We will outline how this can be captured using the MML model, present simulation results, and then consider alternative explanations. 5.1 Applying the MML model Forming an appropriate generalization in this task requires two kinds of inferences. The first inference the child must make – learning the actors’ preferences by computing p(βa|X, c) for a ∈(1, 2) – is the same as that necessary for the first set of experiments discussed above. The second inference is estimating the two hidden objects’ features via those preferences. In order to solve this problem, we need to modify the model slightly to allow us to predict actions other than choices. Specifically, we need to define how preferences are related to affective responses, since the actors simply indicated their affective response to the novel object. We will refer to the actor whose preferences matched those of the child as Actor 1, and the other actor as Actor 2. Let the features of Actor a’s preferred object in round n be xna, the features of the same-category novel object be xsa, and the features of the different-category novel object be xda. When the category is irrelevant, we will use x∗a ∈{xsa, xda} to indicate the features of the novel object. The goal of the child is to infer x∗a (and thus whether they themselves will like the novel object) from the observed affective response of agent a, the features of the objects from the previous rounds X, and the choices of the agent on the previous rounds c. This can be done by evaluating P(x∗a|X, c, ra) = Z P(x∗a|βa, ra)p(βa|X, c) dβa (6) where P(x∗a|βa, ra) is the posterior distribution over the features of the novel object given the preferences and affective response of the agent. Computing this distribution requires defining a likelihood P(ra|x∗a, βa) and a prior on features P(x∗a). We deal with these problems in turn. The likelihood P(ra|x∗a, βa) reflects the probability of the agent producing a particular affective response given the properties of the object and the agent’s preferences. In the experiment, the affective responses produced by the actors were of two types. In the like condition, the actor declared this to be her favorite object. If one takes the actor’s statement at face value and supposes that the actor has encountered an arbitrarily large number of such objects, then P(x∗a|βa, ra = “like”) = 1 for x∗a = arg max x∗a βT a x∗a, else 0 (7) In the dislike condition, the action – saying “there’s a toy in here, but I don’t like it”– communicates negative utility, or at least utility below some threshold which we will take to be zero, hence: P(x∗a|βa, ra = “dislike”) = 1 for βT a x∗a < 0, else 0 (8) 6 C4LS C4LD C4DS C4DD C3LS C3LD C3DS C3DD 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Condition P(choice=1) Model predictions versus child choices, 4 features, σ2=2 model children Figure 2: Model predictions for data in Experiment 1 of [7], excluding cases where children had fewer than 4 chances to play with training objects. The prefixes C4 and C3 denote cases where children chose to play with the objects presented by the matched actor (i.e., Actor 1) in training 4 and 3 times out of 4, respectively. The third character denotes the like (L) or dislike (D) condition, and the fourth character denotes whether the object is in the same category (S) or a different category (D) from those seen in training. P(choice = 1) is the probability of selecting Actor 1’s novel object. In defining the prior distributions from which the features of both the observed objects and the novel objects are sampled, it is important to represent differences between categories. Our feature vectors were concatenations of category-1 features, category-2 features, and multiple-category features, where each feature was present with probability 0.5 if its category could possess the feature, otherwise zero. We arbitrarily chose four features per category, for a total of twelve. Having computed a posterior distribution over x∗a using this prior and likelihood, the child must combine this information with his or her own preferences to select an object. Unfortunately, we do not have direct access to the utilities of the children in this experiment, so we must estimate them from the children’s choice data. We did this using the same procedure as for the adults’ utilities. That done, we apply the choice rule a final time and obtain choice probabilities for the objects selected by the children on the final trial. 5.2 Results Figure 2 shows the rates at which children chose actor 1’s object and the model’s predictions for the same. With the variance parameter set to σ2 = 2 and twelve features, the correlation between the predictions and the data was r = 0.88. When examining only the 4-year-old participants (N = 68), the correlation rose to r = 0.94. The number of features has little influence on predictive accuracy: with 30 features, the correlations were r = 0.85 and r = 0.93 for all participants and 4-year-old participants, respectively. The model predicts less-extreme probabilities than were observed in the choices of the children, in particular in the cases where children chose to play with one of actor 2’s objects. This may be attributed to actor 1’s objects having features one would expect people to like a priori, making the zero-mean preference prior inappropriate. 5.3 Alternative explanations The model described above provides good predictions of children’s inferences in this experiment, but also attributes relatively complex beliefs to the child. A natural question is whether a simpler mechanism might explain behavior in this case. Fawcett and Markson [7] discussed several alternative explanations of their findings, so this section will briefly recapitulate their discussion with a specific view towards alternative models. The simplest model might suggest that children were simply learning associations between specific behaviors by the matching actor and the presence of a desirable object. The most basic model of this kind, assuming that the matching actor is generally associated with desirable objects, is falsified by the children showing no bias towards the matching actor’s objects in the dislike condition. A more elaborate version, in which children associate positive affect by the matching actor with desirable objects, is inconsistent with children’s stronger preferences when the novel objects were similar to the training objects. This evidence also runs against explanations that the children are selecting objects based on liking the matched actor or believing that the matched actor is a more reliable judge of quality. Any of these alternatives might be made to fit the data via ad hoc assumptions 7 about subjective feature correlations or category similarity, but we see no reason to adopt a more complex model without better explanatory power. 6 Conclusion We have outlined a simple rational model for inferences about preferences from choices, drawing on ideas from economics and computer science, and shown that this model produces predictions that closely parallel the behavior of children reasoning about the preferences of others. These results shed light on how children may think about choice, desire, and other minds, and highlight new questions and possible extensions. In future work, we intend to explore whether the developmental shift discovered by Repacholi and Gopnik [6] can be explained in terms of rational model selection under an MML view. We believe that a hierarchical MML and the “Bayesian Ockham’s razor” provide a simple account, resembling a recent Bayesian treatment of false-belief learning [16]. Our approach also provides a framework for predicting how children might make inferences to preferences at the population level and exploring the information provided by correlated features. Acknowledgments. This work was supported by AFOSR grant number FA9550-07-1-0351, NSERC and SSHRC Canada, and the McDonnell Causal Learning Collaborative. References [1] D. McFadden and K. E. Train. Mixed MNL models of discrete response. Journal of Applied Econometrics, 15:447–470, 2000. [2] J. Hayden Boyd and R. E. Mellman. Effect of fuel economy standards on the u.s. automotive market: An hedonic demand analysis. Transportation Research A, 14:367–378, 1980. [3] D. Goldberg, David N., B. M. Oki, and T. Douglas. Using collaborative filtering to weave an information tapestry. Communications of the ACM, 35(12):61–70, 1992. [4] J. A. Konstan, B. N. Miller, D. Maltz, J. L. Herlocker, L. R. Gordon, and J. Riedl. Grouplens: applying collaborative filtering to usenet news. Communications of the ACM, 40:77–87, 1997. [5] C. Kadie J. Breese, D. Heckerman. Empirical analysis of predictive algorithms for collaborative filtering. In Proceedings of the Fourteenth Annual Conference on Uncertainty in Artificial Intelligence (UAI 98), San Francisco, CA, 1998. Morgan Kaufmann. [6] B. M. Repacholi and A. Gopnik. Early reasoning about desires: Evidence from 14- and 18-month-olds. Developmental Psychology, 33(1):12–21, 1997. [7] C. A. Fawcett and L. Markson. Children reason about shared preferences. revised manuscript submitted for publication. Developmental Psychology, under review. [8] T. Kushnir, F. Xu, and H. Wellman. Preschoolers use sampling information to infer the preferences of others. In 28th Annual Conference of the Cognitive Science Society, 2008. [9] K. E. Train, D. McFadden, and M. Ben-Akiva. The demand for local telephone service: A fully discrete model of residential calling patterns and service choices. The RAND Journal of Economics, 18(1):109– 123, 1987. [10] J. R. Anderson. The adaptive character of thought. Erlbaum, Hillsdale, NJ, 1990. [11] R. D. Luce. Individual choice behavior. John Wiley, New York, 1959. [12] R. N. Shepard. Stimulus and response generalization: A stochastic model relating generalization to distance in psychological space. Psychometrika, 22:325–345, 1957. [13] D. McFadden. Conditional logit analysis of qualitative choice behavior. In P. Zarembka, editor, Frontiers in Econometrics. Academic Press, New York, 1973. [14] D. Revelt and K. E. Train. Mixed logit with repeated choices: Households’ choices of appliance efficiency level. The Review of Economics and Statistics, 80(4):647–657, 1998. [15] R. M. Neal. Probabilistic inference using Markov chain Monte Carlo methods. Technical Report CRGTR-93-1, University of Toronto, 1993. [16] N. D. Goodman, C. L. Baker, E. Baraff Bonawitz, V. K. Mansinghka, A. Gopnik, H. Wellman, L. Schulz, and J. B. Tenenbaum. Intuitive theories of mind: A rational approach to false belief. In 28th Annual Conference of the Cognitive Science Society, 2006. 8
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An ideal observer model of infant object perception Charles Kemp Department of Psychology Carnegie Mellon University ckemp@cmu.edu Fei Xu Department of Psychology University of British Columbia fei@psych.ubc.ca Abstract Before the age of 4 months, infants make inductive inferences about the motions of physical objects. Developmental psychologists have provided verbal accounts of the knowledge that supports these inferences, but often these accounts focus on categorical rather than probabilistic principles. We propose that infant object perception is guided in part by probabilistic principles like persistence: things tend to remain the same, and when they change they do so gradually. To illustrate this idea we develop an ideal observer model that incorporates probabilistic principles of rigidity and inertia. Like previous researchers, we suggest that rigid motions are expected from an early age, but we challenge the previous claim that the inertia principle is relatively slow to develop [1]. We support these arguments by modeling several experiments from the developmental literature. Over the past few decades, ingenious experiments [1, 2] have suggested that infants rely on systematic expectations about physical objects when interpreting visual scenes. Looking time studies suggest, for example, that infants expect objects to follow continuous trajectories through time and space, and understand that two objects cannot simultaneously occupy the same location. Many of these studies have been replicated several times, but there is still no consensus about the best way to characterize the knowledge that gives rise to these findings. Two main approaches can be found in the literature. The verbal approach uses natural language to characterize principles of object perception [1, 3]: for example, Spelke [4] proposes that object perception is consistent with principles including continuity (“a moving object traces exactly one connected path over space and time”) and cohesion (“a moving object maintains its connectedness and boundaries”). The mechanistic approach proposes that physical knowledge is better characterized by describing the mechanisms that give rise to behavior, and researchers working in this tradition often develop computational models that support their theoretical proposals [5]. We pursue a third approach—the ideal observer approach [6, 7, 8]—that combines aspects of both previous traditions. Like the verbal approach, our primary goal is to characterize principles that account for infant behavior, and we will not attempt to characterize the mechanisms that produce this behavior. Like the mechanistic approach, we emphasize the importance of formal models, and suggest that these models can capture forms of knowledge that are difficult for verbal accounts to handle. Ideal observer models [6, 9] specify the conclusions that normatively follow given a certain source of information and a body of background knowledge. These models can therefore address questions about the information and the knowledge that support perception. Approaches to the information question characterize the kinds of perceptual information that human observers use. For example, Geisler [9] discusses which components of the information available at the retina contribute to visual perception, and Banks and Shannon [10] use ideal observer models to study the perceptual consequences of immaturities in the retina. Approaches to the knowledge question characterize the background assumptions that are combined with the available input in order to make inductive inferences. For example, Weiss and Adelson [7] describe several empirical phenomena that are consistent with the a priori assumption that motions tend to be slow and smooth. There are few previous attempts to develop ideal observer models of infant perception, and most of them focus only on the information question [10]. This paper addresses the knowledge question, and proposes that the ideal observer approach can help to identify the minimal set of principles needed to account for the visual competence of young infants. Most verbal theories of object perception focus on categorical principles [4], or principles that make a single distinction between possible and impossible scenes. We propose that physical knowledge in infancy is also characterized by probabilistic principles, or expectations that make some possible scenes more surprising than others. We demonstrate the importance of probabilistic principles by focusing on two examples: the rigidity principle states that objects usually maintain their shape and size when they move, and the inertia principle states that objects tend to maintain the same pattern of motion over time. Both principles capture important regularities, but exceptions to these regularities are relatively common. Focusing on rigidity and inertia allows us to demonstrate two contributions that probabilistic approaches can make. First, probabilistic approaches can reinforce current proposals about infant perception. Spelke [3] suggests that rigidity is a core principle that guides object perception from a very early age, and we demonstrate how this idea can be captured by a model that also tolerates exceptions, such as non-rigid biological motion. Second, probabilistic approaches can identify places where existing proposals may need to be revised. Spelke [3] argues that the principle of inertia is slow to develop, but we suggest that a probabilistic version of this principle can help to account for inferences made early in development. 1 An ideal observer approach An ideal observer approach to object perception can be formulated in terms of a generative model for scenes. Scenes can be generated in three steps. First we choose the number n of objects that will appear in the scene, and generate the shape, visual appearance, and initial location of each object. We then choose a velocity field for each object which specifies how the object moves and changes shape over time. Finally, we create a visual scene by taking a two-dimensional projection of the moving objects generated in the two previous steps. An ideal observer approach explores the idea that the inferences made by infants approximate the optimal inferences with respect to this generative model. We work within this general framework but make two simplifications. We will not discuss how the shapes and visual appearances of objects are generated, and we make the projection step simple by working with a two-dimensional world. These simplifications allow us to focus on the expectations about velocity fields that guide motion perception in infants. The next two sections present two prior distributions that can be used to generate velocity fields. The first is a baseline prior that does not incorporate probabilistic principles, and the second incorporates probabilistic versions of rigidity and inertia. The two priors capture different kinds of knowledge, and we argue that the second provides the more accurate characterization of the knowledge that infants bring to object perception. 1.1 A baseline prior on velocity fields Our baseline prior is founded on five categorical principles that are closely related to principles discussed by Spelke [3, 4]. The principles we consider rely on three basic notions: space, time, and matter. We also refer to particles, which are small pieces of matter that occupy space-time points. Particles satisfy several principles: C1. Temporal continuity. Particles are not created or destroyed. In other words, every particle that exists at time t1 must also exist at time t2. C2. Spatial continuity. Each particle traces a continuous trajectory through space. C3. Exclusion. No two particles may occupy the same space-time point. An object is a collection of particles, and these collections satisfy two principles: C4. Discreteness. Each particle belongs to exactly one object. C5. Cohesion. At each point in time, the particles belonging to an object occupy a single connected region of space. Suppose that we are interested in a space-time window specified by a bounded region of space and a bounded interval of time. For simplicity, we will assume that space is two-dimensional, and that the space-time window corresponds to the unit cube. Suppose that a velocity field ⃗v assigns a velocity (vx, vy) to each particle in the space-time window, and let ⃗vi be the field created by considering only particles that belong to object i. We develop a theory of object perception by defining a prior distribution p(⃗v) on velocity fields. Consider first the distribution p(⃗v1) on fields for a single object. Any field that violates one or more of principles C1–C5 is assigned zero probability. For instance, fields where part of an object winks out of existence violate the principle of temporal continuity, and fields where an object splits into two distinct pieces violate the principle of cohesion. Many fields, however, remain, including fields that specify non-rigid motions and jagged trajectories. For now, assume that we are working with a space of fields that is bounded but very large, and that the prior distribution over this space is uniform for all fields consistent with principles C1–C5: p(⃗v1) ∝f(⃗v1) = 0 if ⃗v1 violates C1–C5 1 otherwise. (1) Consider now the distribution p(⃗v1, ⃗v2) on fields for pairs of objects. Principles C1 through C5 rule out some of these fields, but again we must specify a prior distribution on those that remain. Our prior is induced by the following principle: C6. Independence. Velocity fields for multiple objects are independently generated subject to principles C1 through C5. More formally, the independence principle specifies how the prior for the multiple object case is related to the prior p(⃗v1) on velocity fields for a single object (Equation 1): p(⃗v1, . . . , ⃗vn) ∝f(⃗v1, . . . , ⃗vn) = 0 if {⃗vi} collectively violate C1–C5 f(⃗v1) . . . f( ⃗vn) otherwise. (2) 1.2 A smoothness prior on velocity fields We now develop a prior p(⃗v) that incorporates probabilistic expectations about the motion of physical objects. Consider again the prior p(⃗v1) on the velocity field ⃗v1 of a single object. Principles C1–C5 make a single cut that distinguishes possible from impossible fields, but we need to consider whether infants have additional knowledge that makes some of the possible fields less surprising than others. One informal idea that seems relevant is the notion of persistence[11]: things tend to remain the same, and when they change they do so gradually. We focus on two versions of this idea that may guide expectations about velocity fields: S1. Spatial smoothness. Velocity fields tend to be smooth in space. S2. Temporal smoothness. Velocity fields tend to be smooth in time. A field is “smooth in space” if neighboring particles tend to have similar velocities at any instant of time. The smoothest possible field will be one where all particles have the same velocity at any instant—in other words, where an object moves rigidly. The principle of spatial smoothness therefore captures the idea that objects tend to maintain the same shape and size. A field is “smooth in time” if any particle tends to have similar velocities at nearby instants of time. The smoothest possible field will be one where each particle maintains the same velocity throughout the entire interval of interest. The principle of temporal smoothness therefore captures the idea that objects tend to maintain their initial pattern of motion. For instance, stationary objects tend to remain stationary, moving objects tend to keep moving, and a moving object following a given trajectory tends to continue along that trajectory. Principles S1 and S2 are related to two principles— rigidity and inertia—that have been discussed in the developmental literature. The rigidity principle states that objects “tend to maintain their size and shape over motion”[3], and the inertia principle states that objects move smoothly in the absence of obstacles [4]. Some authors treat these principles rather differently: for instance, Spelke suggests that rigidity is one of the core principles that guides object perception from a very early age [3], but that the principle of inertia is slow to develop and is weak or fragile once acquired. Since principles S1 and S2 seem closely related, the suggestion that one develops much later than the other seems counterintuitive. The rest of this paper explores the idea that both of these principles are needed to characterize infant perception. Our arguments will be supported by formal analyses, and we therefore need formal versions of S1 and S2. There may be different ways to formalize these principles, but we present a simple baseline smoothness −200 0 200 L1 L2 U log “ p(H1|⃗v) p(H2|⃗v) ” b) U L1 L2 a) Figure 1: (a) Three scenes inspired by the experiments of Spelke and colleagues [12, 13]. Each scene can be interpreted as a single object, or as a small object on top of a larger object. (b) Relative preferences for the one-object and two-object interpretations according to two models. The baseline model prefers the one-object interpretation in all three cases, but the smoothness model prefers the one-object interpretation only for scenes L1 and L2. approach that builds on existing models of motion perception in adults [7, 8]. We define measures of instantaneous roughness that capture how rapidly a velocity field ⃗v varies in space and time: Rspace(⃗v, t) = 1 vol(O(t)) Z O(t) ∂⃗v(x, y, t) ∂x 2 + ∂⃗v(x, y, t) ∂y 2 dxdy (3) Rtime(⃗v, t) = 1 vol(O(t)) Z O(t) ∂⃗v(x, y, t) ∂t 2 dxdy (4) where O(t) is the set of all points that are occupied by the object at time t, and vol(O(t)) is the volume of the object at time t. Rspace(⃗v, t) will be large if neighboring particles at time t tend to have different velocities, and Rtime(⃗v, t) will be large if many particles are accelerating at time t. We combine our two roughness measures to create a single smoothness function S(·) that measures the smoothness of a velocity field: S(⃗v) = −λspace Z Rspace(⃗v, t)dt −λtime Z Rtime(⃗v, t)dt (5) where λspace and λtime are positive weights that capture the importance of spatial smoothness and temporal smoothness. For all analyses in this paper we set λspace = 10000 and λtime = 250, which implies that violations of spatial smoothness are penalized more harshly than violations of temporal smoothness. We now replace Equation 1 with a prior on velocity fields that takes smoothness into account: p(⃗v1) ∝f(⃗v1) = 0 if ⃗v1 violates C1–C5 exp (S(⃗v1)) otherwise. (6) Combining Equation 6 with Equation 2 specifies a model of object perception that incorporates probabilistic principles of rigidity and inertia. 2 Empirical findings: spatial smoothness There are many experiments where infants aged 4 months and younger appear to make inferences that are consistent with the principle of rigidity. This section suggests that the principle of spatial smoothness can account for these results. We therefore propose that a probabilistic principle (spatial smoothness) can explain all of the findings previously presented in support of a categorical principle (rigidity), and can help in addition to explain how infants perceive non-rigid motion. One set of studies explores inferences about the number of objects in a scene. When a smaller block is resting on top of a larger block (L1 in Figure 1a), 3-month-olds infer that the scene includes a single object [12]. The same result holds when the small and large blocks are both moving in the same direction (L2 in Figure 1a) [13]. When these blocks are moving in opposite directions (U in Figure 1a), however, infants appear to infer that the scene contains two objects [13]. Results like these suggest that infants may have a default expectation that objects tend to move rigidly. We compared the predictions made by two models about the scenes in Figure 1a. The smoothness model uses a prior p(⃗v1) that incorporates principles S1 and S2 (Equation 6), and the baseline model is identical except that it sets λspace = λtime = 0. Both models therefore incorporate principles C1– C6, but only the smoothness model captures the principle of spatial smoothness. Given any of the scenes in Figure 1a, an infant must solve two problems: she must compute the velocity field ⃗v for the scene and must decide whether this field specifies the motion of one or two objects. Here we focus on the second problem, and assume that the infant’s perceptual system has already computed a veridical velocity field for each scene that we consider. In principle, however, the smoothness prior in Equation 6 can address both problems. Previous authors have shown how smoothness priors can be used to compute velocity fields given raw image data [7, 8]. Let H1 be the hypothesis that a given velocity field corresponds to a single object, and let H2 be the hypothesis that the field specifies the motions of two objects. We assume that the prior probabilities of these hypotheses are equal, and that P(H1) = P(H2) = 0.5. An ideal observer can use the posterior odds ratio to choose between these hypotheses: P(H1|⃗v) P(H2|⃗v) = P(⃗v|H1) P(⃗v|H2) P(H1) P(H2) ≈ f(⃗v) R f(⃗v1)d⃗v1 R f(⃗v1, ⃗v2)d⃗v1d⃗v2 f( ⃗vA, ⃗vB) (7) Equation 7 follows from Equations 2 and 6, and from approximating P(⃗v|H2) by considering only the two object interpretation ( ⃗vA, ⃗vB) with maximum posterior probability. For each scene in Figure 1a, the best two object interpretation will specify a field ⃗vA for the small upper block, and a field ⃗vB for the large lower block. To approximate the posterior odds ratio in Equation 7 we compute rough approximations of R f(⃗v1)d⃗v1 and R f(⃗v1, ⃗v2)d⃗v1d⃗v2 by summing over a finite space of velocity fields. As described in the supporting material, we consider all fields that can be built from objects with 5 possible shapes, 900 possible starting locations, and 10 possible trajectories. For computational tractability, we convert each continuous velocity field to a discrete field defined over a space-time grid with 45 cells along each spatial dimension and 21 cells along the temporal dimension. Our results show that both models prefer the one-object hypothesis H1 when presented with scenes L1 and L2 (Figure 1b). Since there are many more two-object scenes than one-object scenes, any typical two-object interpretation is assigned lower prior probability than a typical one-object interpretation. This preference for simpler interpretations is a consequence of the Bayesian Occam’s razor. The baseline model makes the same kind of inference about scene U, and again prefers the one-object interpretation. Like infants, however, the smoothness model prefers the two-object interpretation of scene U. This model assigns low probability to a one-object interpretation where adjacent points on the object have very different velocities, and this preference for smooth motion is strong enough to overcome the simplicity preference that makes the difference when interpreting the other two scenes. Other experiments from the developmental literature have produced results consistent with the principle of spatial smoothness. For example, 3.5-month olds are surprised when a tall object is fully hidden behind a short screen, 4 month olds are surprised when a large object appears to pass through a small slot, and 4.5-month olds expect a swinging screen to be interrupted when an object is placed in its path [1, 2]. All three inferences appear to rely on the expectation that objects tend not to shrink or to compress like foam rubber. Many of these experiments are consistent with an account that simply rules out non-rigid motion instead of introducing a graded preference for spatial smoothness. Biological motions, however, are typically non-rigid, and experiments suggest that infants can track and make inferences about objects that follow non-rigid trajectories [14]. Findings like these call for a theory like ours that incorporates a preference for rigid motion, but recognizes that non-rigid motions are possible. 3 Empirical findings: temporal smoothness We now turn to the principle of temporal smoothness (S2) and discuss some of the experimental evidence that bears on this principle. Some researchers suggest that a closely related principle (inertia) is slow to develop, but we argue that expectations about temporal smoothness are needed to capture inferences made before the age of 4 months. Baillargeon and DeVos [15] describe one relevant experiment that explores inferences about moving objects and obstacles. During habituation, 3.5-month-old infants saw a car pass behind an occluder and emerge from the other side (habituation stimulus H in Figure 2a). An obstacle was then placed in the direct path of the car (unlikely scenes U1 and U2) or beside this direct path (likely scene L), and the infants again saw the car pass behind the occluder and emerge from the other side. Looking H U1 U2 L baseline smoothness 0 200 400 600 X X X log “ pH(L) pH(U2) ” log “ p(L) p(U2) ” log “ p(L) p(U1) ” log “ pH(L) pH(U1) ” a) b) Figure 2: (a) Stimuli inspired by the experiments of [15]. The habituation stimulus H shows a block passing behind a barrier and emerging on the other side. After habituation, a new block is added either out of the direct path of the first block (L) or directly in the path of the first block (U1 and U2). In U1, the first block leaps over the second block, and in U2 the second block hops so that the first block can pass underneath. (b) Relative probabilities of scenes L, U1 and U2 according to two models. The baseline model finds all three scenes equally likely a priori, and considers L and U2 equally likely after habituation. The smoothness model considers L more likely than the other scenes both before and after habituation. b) H1 L U H2 baseline smoothness −100 0 100 200 300 X X log “ pH2(L) pH2(U) ” log “ pH1(L) pH1(U) ” log “ p(L) p(U) ” a) c) Figure 3: (a) Stimuli inspired by the experiments of Spelke et al. [16]. (b) Model predictions. After habituation to H1, the smoothness model assigns roughly equal probabilities to L and U. After habituation to H2, the model considers L more likely. (c) A stronger test of the inertia principle. Now the best interpretation of stimulus U involves multiple changes of direction. time measurements suggested that the infants were more surprised to see the car emerge when the obstacle lay within the direct path of the car. This result is consistent with the principle of temporal smoothness, which suggests that infants expected the car to maintain a straight-line trajectory, and the obstacle to remain stationary. We compared the smoothness model and the baseline model on a schematic version of this task. To model this experiment, we again assume that the infant’s perceptual system has recovered a veridical velocity field, but now we must allow for occlusion. An ideal observer approach that treats a two dimensional scene as a projection of a three dimensional world can represent the occluder as an object in its own right. Here, however, we continue to work with a two dimensional world, and treat the occluded parts of the scene as missing data. An ideal observer approach should integrate over all possible values of the missing data, but for computational simplicity we approximate this approach by considering only one or two high-probability interpretations of each occluded scene. We also need to account for habituation, and for cases where the habituation stimulus includes occlusion. We assume that an ideal observer computes a habituation field ⃗vH, or the velocity field with maximum posterior probability given the habituation stimulus. In Figure 2a, the inferred habituation field ⃗vH specifies a trajectory where the block moves smoothly from the left to the right of the scene. We now assume that the observer expects subsequent velocity fields to be similar to ⃗vH. Formally, we use a product-of-experts approach to define a post-habituation distribution on velocity fields: pH(⃗v) ∝p(⃗v)p(⃗v| ⃗vH) (8) The first expert p(⃗v) uses the prior distribution in Equation 6, and the second expert p(⃗v| ⃗vH) assumes that field ⃗v is drawn from a Gaussian distribution centered on ⃗vH. Intuitively, after habituation to ⃗vH the second expert expects that subsequent velocity fields will be similar to ⃗vH. More information about this model of habituation is provided in the supporting material. Given these assumptions, the black and dark gray bars in Figure 2 indicate relative a priori probabilities for scenes L, U1 and U2. The baseline model considers all three scenes equally probable, but the smoothness model prefers L. After habituation, the baseline model is still unable to account for the behavioral data, since it considers scenes L and U2 to be equally probable. The smoothness model, however, continues to prefer L. We previously mentioned three consequences of the principle of temporal smoothness: stationary objects tend to remain stationary, moving objects tend to keep moving, and moving objects tend to maintain a steady trajectory. The “car and obstacle” task addresses the first and third of these proposals, but other tasks provide support for the second. Many authors have studied settings where one moving object comes to a stop, and a second object starts to move [17]. Compared to the case where the first object collides with the second, infants appear to be surprised by the “no-contact” case where the two objects never touch. This finding is consistent with the temporal smoothness principle, which predicts that infants expect the first object to continue moving until forced to stop, and expect the second object to remain stationary until forced to start. Other experiments [18] provide support for the principle of temporal smoothness, but there are also studies that appear inconsistent with this principle. In one of these studies [16], infants are initially habituated to a block that moves from one corner of an enclosure to another (H1 in Figure 3a). After habituation, infants see a block that begins from a different corner, and now the occluder is removed to reveal the block in a location consistent with a straight-line trajectory (L) or in a location that matches the final resting place during the habituation phase (U). Looking times suggest that infants aged 4-12 months are no more surprised by the inertia-violating outcome (U) than the inertia-consistent outcome (L). The smoothness model, however, can account for this finding. The outcome in U is contrary to temporal smoothness but consistent with habituation, and the tradeoff between these factors leads the model to assign roughly the same probability to scenes L and U (Figure 3b). Only one of the inertia experiments described by Spelke et al. [16] and Spelke et al. [1] avoids this tradeoff between habituation and smoothness. This experiment considers a case where the habituation stimulus (H2 in Figure 3a) is equally similar to the two test stimuli. The results suggest that 8 month olds are now surprised by the inertia-violating outcome, and the predictions of our model are consistent with this finding (Figure 3b). 4 and 6 month olds, however, continue to look equally at the two outcomes. Note, however, that the trajectories in Figure 3 include at most one inflection point. Experiments that consider trajectories with many inflection points can provide a more powerful way of exploring whether 4 month olds have expectations about temporal smoothness. One possible experiment is sketched in Figure 3c. The task is very similar to the task in Figure 3a, except that a barrier is added after habituation. In order for the block to end up in the same location as before, it must now follow a tortuous path around the barrier (U). Based on the principle of temporal smoothness, we predict that 4-month-olds will be more surprised to see the outcome in stimulus U than the outcome in stimulus L. This experimental design is appealing in part because previous work shows that infants are surprised by a case similar to U where the barrier extends all the way from one wall to the other [16], and our proposed experiment is a minor variant of this task. Although there is room for debate about the status of temporal smoothness, we presented two reasons to revisit the conclusion that this principle develops relatively late. First, some version of this principle seems necessary to account for experiments like the car and obstacle experiment in Figure 2. Second, most of the inertia experiments that produced null results use a habituation stimulus which may have prevented infants from revealing their default expectations, and the one experiment that escapes this objection considers a relatively minor violation of temporal smoothness. Additional experiments are needed to explore this principle, but we predict that the inertia principle will turn out to be yet another example of knowledge that is available earlier than researchers once thought. 4 Discussion and Conclusion We argued that characterizations of infant knowledge should include room for probabilistic expectations, and that probabilistic expectations about spatial and temporal smoothness appear to play a role in infant object perception. To support these claims we described an ideal observer model that includes both categorical (C1 through C5) and probabilistic principles (S1 and S2), and demonstrated that the categorical principles alone are insufficient to account for several experimental findings. Our two probabilistic principles are related to principles (rigidity and inertia) that have previously been described as categorical principles. Although rigidity and inertia appear to play a role in some early inferences, formulating these principles as probabilistic expectations helps to explain how infants deal with non-rigid motion and violations of inertia. Our analysis focused on some of the many existing experiments in the developmental literature, but new experiments will be needed to explore our probabilistic approach in depth. Categorical versions of a given principle (e.g. rigidity) allow room for only two kinds of behavior depending on whether the principle is violated or not. Probabilistic principles can be violated to a greater or lesser extent, and our approach predicts that violations of different magnitude may lead to different behaviors. Future studies of rigidity and inertia can consider violations of these principles that range from mild (Figure 3a) to severe (Figure 3c), and can explore whether infants respond to these violations differently. Future work should also consider whether the categorical principles we described (C1 through C5) are better characterized as probabilistic expectations. In particular, future studies can explore whether young infants consider large violations of cohesion (C5) or spatial continuity (C2) more surprising than smaller violations of these principles. Although we did not focus on learning, our approach allows us to begin thinking formally about how principles of object perception might be acquired. First, we can explore how parameters like the smoothness parameters in our model (λspace and λtime) might be tuned by experience. Second, we can use statistical model selection to explore transitions between different sets of principles. For instance, if a learner begins with the baseline model we considered (principles C1–C6), we can explore which subsequent observations provide the strongest statistical evidence for smoothness principles S1 and S2, and how much of this evidence is required before an ideal learner would prefer our smoothness model over the baseline model. It is not yet clear which principles of object perception could be learned, but the ideal observer approach can help to resolve this question. References [1] E. S. Spelke, K. Breinlinger, J. Macomber, and K. Jacobson. Origins of knowledge. Psychological Review, 99:605–632, 1992. [2] R. Baillargeon, L. Kotovsky, and A. Needham. The acquisition of physical knowledge in infancy. In D. Sperber, D. Premack, and A. J. Premack, editors, Causal Cognition: A multidisciplinary debate, pages 79–116. Clarendon Press, Oxford, 1995. [3] E. S. Spelke. Principles of object perception. Cognitive Science, 14:29–56, 1990. [4] E. Spelke. Initial knowledge: six suggestions. Cognition, 50:431–445, 1994. [5] D. Mareschal and S. P. Johnson. Learning to perceive object unity: a connectionist account. Developmental Science, 5:151–172, 2002. [6] D. Kersten and A. Yuille. Bayesian models of object perception. Current opinion in Neurobiology, 13: 150–158, 2003. [7] Y. Weiss and E. H. Adelson. Slow and smooth: a Bayesian theory for the combination of local motion signals in human vision. Technical Report A.I Memo No. 1624, MIT, 1998. [8] A. L. Yuille and N. M. Grzywacz. A mathematical analysis of the motion coherence theory. International Journal of Computer Vision, 3:155–175, 1989. [9] W. S. Geisler. Physical limits of acuity and hyperacuity. Journal of the Optical Society of America, 1(7): 775–782, 1984. [10] M. S. Banks and E. Shannon. Spatial and chromatic visual efficiency in human neonates. In Visual perception and cognition in infancy, pages 1–46. Lawrence Erlbaum Associates, Hillsdale, NJ, 1993. [11] R. Baillargeon. Innate ideas revisited: for a principle of persistence in infants’ physical reasoning. Perspectives on Psychological Science, 3(3):2–13, 2008. [12] R. Kestenbaum, N. Termine, and E. S. Spelke. Perception of objects and object boundaries by threemonth-old infants. British Journal of Developmental Psychology, 5:367–383, 1987. [13] E. S. Spelke, C. von Hofsten, and R. Kestenbaum. Object perception and object-directed reaching in infancy: interaction of spatial and kinetic information for object boundaries. Developmental Psychology, 25:185–196, 1989. [14] G. Huntley-Fenner, S. Carey, and A. Solimando. Objects are individuals but stuff doesn’t count: perceived rigidity and cohesiveness influence infants’ representations of small groups of discrete entities. Cognition, 85:203–221, 2002. [15] R. Baillargeon and J. DeVos. Object permanence in young infants: further evidence. Child Development, 61(6):1227–1246, 1991. [16] E. S. Spelke, G. Katz, S. E. Purcell, S. M. Ehrlich, and K. Breinlinger. Early knowledge of object motion: continuity and inertia. Cognition, 51:131–176, 1994. [17] L. Kotovsky and R. Baillargeon. Reasoning about collisions involving inert objects in 7.5-month-old infants. Developmental Science, 3(3):344–359, 2000. [18] T. Wilcox and A. Schweinle. Infants’ use of speed information to individuate objects in occlusion events. Infant Behavior and Development, 26:253–282, 2003.
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Empirical performance maximization for linear rank statistics St´ephan Cl´emenc¸on Telecom Paristech (TSI) - LTCI UMR Institut Telecom/CNRS 5141 stephan.clemencon@telecom-paristech.fr Nicolas Vayatis ENS Cachan & UniverSud - CMLA UMR CNRS 8536 vayatis@cmla.ens-cachan.fr Abstract The ROC curve is known to be the golden standard for measuring performance of a test/scoring statistic regarding its capacity of discrimination between two populations in a wide variety of applications, ranging from anomaly detection in signal processing to information retrieval, through medical diagnosis. Most practical performance measures used in scoring applications such as the AUC, the local AUC, the p-norm push, the DCG and others, can be seen as summaries of the ROC curve. This paper highlights the fact that many of these empirical criteria can be expressed as (conditional) linear rank statistics. We investigate the properties of empirical maximizers of such performance criteria and provide preliminary results for the concentration properties of a novel class of random variables that we will call a linear rank process. 1 Introduction In the context of ranking, several performance measures may be considered. Even in the simplest framework of bipartite ranking, where a binary label is available, there is not one and only natural criterion, but many possible options. The ROC curve provides a complete description of performance but its functional nature renders direct optimization strategies rather complex. Empirical risk minimization strategies are thus based on summaries of the ROC curve, which take the form of empirical risk functionals where the averages involved are no longer taken over i.i.d. sequences. The most popular choice is the so-called AUC criterion (see [AGH+05] or [CLV08] for instance), but when top-ranked instances are more important, various choices can be considered: the Discounted Cumulative Gain or DCG [CZ06], the p-norm push (see [Rud06]), or the local AUC (refer to [CV07]). The present paper starts from the simple observation that all these summary criteria have a common feature: conditioned upon the labels, they all belong to the class of linear rank statistics. Such statistics have been extensively studied in the mathematical statistics literature because of their optimality properties in hypothesis testing, see [HS67]. Now, in the statistical learning view, with the importance of excess risk bounds, the theory of rank tests needs to be revisited and new problems come up. The arguments required to deal with risk functionals based on linear rank statistics have been sketched in [CV07] in a special case. The empirical AUC, known as the Wilcoxon-Mann-Whitney statistic, is also a U-statistic and this particular dependence structure was extensively exploited in [CLV08]. In the present paper, we describe the generic structure of linear rank statistics as an orthogonal decomposition after projection onto the space of sums of i.i.d. random variables (Section 2). This projection method is the key to all statistical results related to maximizers of such criteria: consistency, (fast) rates of convergence or model selection. We relate linear rank statistics to performance measures relevant for the ranking problem by showing that the target of ranking algorithms 1 correspond to optimal ordering rules in that sense (Section 3). Eventually, we provide some preliminary results in Section 4 for empirical maximizers of performance criteria based on linear rank statistics with smooth score-generating functions. 2 Criteria based on linear rank statistics Along the paper, we shall consider the standard binary classification model. Take a random pair (X, Y ) ∈X ×{−1, +1}, where X is an observation vector in a high dimensional space X ⊂Rd and Y is a binary label, and denote by P the distribution of (X, Y ). The dependence structure between X and Y can be described by conditional distributions. We can consider two descriptions: either P = (µ, η) where µ is the marginal distribution of X and η is the posterior distribution defined by η(x) = P{Y = 1 | X = x} for all x ∈Rd, or else P = (p, G, H) with p = P{Y = 1} being the proportion of positive instances, G = L(X | Y = +1) the conditional distribution of positive instances and H = L(X | Y = −1) the conditional distribution of negative instances. A sample of size n of i.i.d. realizations of this statistical model can be represented as a set of pairs {(Xi, Yi)}1≤i≤n, where (Xi, Yi) is a copy of (X, Y ), but also as a set {X+ 1 , . . . , X+ k , X− 1 , . . . , X− m, } where L(X+ i ) = G, L(X− i ) = H, and k + m = n. In this setup, the integers k and m are random, drawn as binomials of size n and respective parameters p and 1 −p. 2.1 Motivation Most of the statistical learning theory has been developed for empirical risk minimizers (ERM) of sums of i.i.d. random variables. Mathematical results were elaborated with the use of empirical processes techniques and particularly concentration inequalities for such processes (see [BBL05] for an overview). This was made possible by the standard assumption that, in a batch setup, for the usual prediction problems (classification, regression or density estimation), the sample data {(Xi, Yi)}i=1,...,n are i.i.d. random variables. Another reason is that the error probability in these problems involves only ”first-order” events, depending only on (X1, Y1). In classification, for instance, most theoretical developments were focused on the error probability P{Y1 ̸= g(X1)} of a classifier g : X →{−1, +1}, which is hardly considered in practice because the two populations are rarely symmetric in terms of proportions or costs. For prediction tasks such as ranking or scoring, more involved statistics need to be considered, such as the Area Under the ROC curve (AUC), the local AUC, the Discounted Cumulative Gain (DCG), the p-norm push, etc. For instance, the AUC, a very popular performance measure in various scoring applications, such as medical diagnosis or credit-risk screening, can be seen as a probability of an ”event of order two”, i.e. depending on (X1, Y1), (X2, Y2). In information retrieval, the DCG is the reference measure and it seems to have a rather complicated statistical structure. The first theoretical studies either attempt to get back to sums of i.i.d. random variables by artificially reducing the information available (see [AGH+05], [Rud06]) or adopt a plug-in strategy ([CZ06]). Our approach is to i) avoid plug-in in order to understand the intimate nature of the learning problem, ii) keep all the information available and provide the analysis of the full statistic. We shall see that this approach requires the development of new tools for handling the concentration properties of rank processes, namely collections of rank statistics indexed by classes of functions, which have never been studied before. 2.2 Empirical performance of scoring rules The learning task on which we focus here is known as the bipartite ranking problem. The goal of ranking is to order the instances Xi by means of a real-valued scoring function s : X →R , given the binary labels Yi. We denote by S the set of all scoring functions. It is natural to assume that a good scoring rule s would assign higher ranks to the positive instances (those for which Yi = +1) than to the negative ones. The rank of the observation Xi induced by the scoring function s is expressed as Rank(s(Xi)) = Pn j=1 I{s(Xj)≤s(Xi)} and ranges from 1 to n. In the present paper, we consider a particular class of simple (conditional) linear rank statistics inspired from the Wilcoxon statistic. 2 Definition. 1 Let φ : [0, 1] →[0, 1] be a nondecreasing function. We define the ”empirical Wranking performance measure” as the empirical risk functional c Wn(s) = n X i=1 I{Yi=+1}φ Rank(s(Xi)) n + 1 , ∀s ∈S. The function φ is called the ”score-generating function” of the ”rank process” {c Wn(s)}s∈S. We refer to the book by Serfling [Ser80] for properties and asymptotic theory of rank statistics. We point out that our definition does not match exactly with the standard definition of linear rank statistics. Indeed, in our case, coefficients of the ranks in the sum are random because they involve the variables Yi. We will call statistics c Wn(s) conditional linear rank statistics. It is a very natural idea to consider ranking criteria based on ranks. Observe indeed that the performance of a given scoring function s is invariant by increasing transforms of the latter, when evaluated through the empirical W-ranking performance measure. For specific choices of the score-generating function φ, we recover the main examples mentioned in the introduction and many relevant criteria can be accurately approximated by statistics of this form: • φ(u) = u - this choice leads to the celebrated Wilcoxon-Mann-Whitney statistic which is related to the empirical version of the AUC (see [CLV08]). • φ(u) = u · I{u≥u0}, for some u0 ∈(0, 1) - such a score-generating function corresponds to the local AUC criterion, introduced recently in [CV07]. Such a criterion is of interest when one wants to focus on the highest ranks. • φ(u) = up - this is another choice which puts emphasis on high ranks but in a smoother way than the previous one. This is related to the p-norm push approach taken in [Rud06]. However, we point out that the criterion studied in the latter work relies on a different definition of the rank of an observation. Namely, the rank of positive instances among negative instances (and not in the pooled sample) is used. This choice permits to use independence which makes the technical part much simpler, at the price of increasing the variance of the criterion. • φ(u) = φn(u) = c ((n + 1) u)·I{u≥k/(n+1)} - this corresponds to the DCG criterion in the bipartite setup, one of the ”gold standard quality measure” in information retrieval, when grades are binary (namely I{Yi=+1}). The c(i)’s denote the discount factors, c(i) measuring the importance of rank i. The integer k denotes the number of top-ranked instances to take into account. Notice that, with our indexation, top positions correspond to the largest ranks and the sequence {ci} should be chosen to be increasing. 2.3 Uniform approximation of linear rank statistics This subsection describes the main result of the present analysis, which shall serve as the essential tool for deriving statistical properties of maximizers of empirical W-ranking performance measures. For a given scoring function s, we denote by Gs, respectively Hs, the conditional cumulative distribution function of s(X) given Y = +1, respectively Y = −1. With these notations, the unconditional cdf of s(X) is then Fs = pGs +(1−p)Hs. For averages of non-i.i.d. random variables, the underlying statistical structure can be revealed by orthogonal projections onto the space of sums of i.i.d. random variables in many situations. This projection argument was the key for the study of empirical AUC maximization, which involved U-processes, see [CLV08]. In the case of U-statistics, this orthogonal decomposition is known as the Hoeffding decomposition and the remainder may be expressed as a degenerate U-statistic, see [Hoe48]. For rank statistics, a similar though less accurate decomposition can be considered. We refer to [Haj68] for a systematic use of the projection method for investigating the asymptotic properties of general statistics. Lemma. 2 ([Haj68]) Let Z1, . . . , Zn be independent r.v.’s and T = T(Z1, . . . , Zn) be a square integrable statistic. The r.v. bT = Pn i=1 E[T | Zi] −(n −1)E[Z] is called the H´ajek projection of T. It satisfies E[ bT] = E[T] and E[( bT −T)2] = E[(T −E[T])2] −E[( bT −E[ bT])2]. 3 From the perspective of ERM in statistical learning theory, through the projection method, wellknown concentration results for standard empirical processes may carry over to more complex collections of r.v. such as rank processes, as shown by the next approximation result. Proposition. 3 Consider a score-generating function φ which is twice continuously differentiable on [0, 1]. We set Φs(x) = φ(Fs(s(x))) + p R +∞ s(x) φ′(Fs(u))dGs(u) for all x ∈X. Let S0 ⊂S be a VC major class of functions. Then, we have: ∀s ∈S0, c Wn(s) = bVn(s) + bRn(s), where bVn(s) = Pn i=1 I{Yi=+1}Φs(Xi) and bRn(s) = OP(1) as n →∞uniformly over s ∈S . The notation OP(1) means bounded in probability and the integrals are represented in the sense of the Lebesgue-Stieltjes integral. Details of the proof can be found in the Appendix. Remark 1 (ON THE COMPLEXITY ASSUMPTION.) On the terminology of major sets and major classes, we refer to [Dud99]. In the Proposition 3’s proof, we need to control the complexity of subsets of the form {x ∈X : s(x) ≤t}. The stipulated complexity assumption garantees that this collection of sets indexed by (s, t) ∈S0 × R forms a VC class. Remark 2 (ON THE SMOOTHNESS ASSUMPTION.) We point out that it is also possible to deal with discontinuous score-generating functions as seen in [CV07]. In this case, the lack of smoothness of φ has to be compensated by smoothness assumptions on the underlying conditional distributions. Another approach would consist of approximating c Wn(s) by the empirical W-ranking criterion where the score-generating function ψ would be a smooth approximation of φ. Owing to space limitations, here we only handle the smooth case. An essential hint to the study of the asymptotic behavior of a linear rank statistic consists in rewriting it as a function of the sampling cdf. Denoting by bFs(x) = n−1 Pn i=1 I{s(Xi)≤x} the empirical counterpart of Fs(x), we have: c Wn(s) = k X i=1 φ n n + 1 bFs s(X+ i ) . which may easily shown to converge to E[φ(Fs(s(X)) | Y = +1] as n →∞, see [CS58]. Definition. 4 For a given score-generating function φ, we will call the functional Wφ(s) = E[φ(Fs(s(X)) | Y = +1] , a ”W-ranking performance measure”. The following result is a consequence of Proposition 3 and its proof can be found in the Appendix. Proposition. 5 Let S0 ⊂S be a VC major class of functions with VC dimension V and φ be a score-generating function of class C1. Then, as n →∞, we have with probability one: 1 n sup s∈S0 |c Wn(s) −kWφ(s)| →0. 3 Optimality We introduce the class S∗of scoring functions obtained as strictly increasing transformations of the regression function η: S∗= { s∗= T ◦η | T : [0, 1] →R strictly increasing }. The class S∗contains the optimal scoring rules for the bipartite ranking problem. The next paragraphs motivate the use of W-ranking performance measures as optimization criteria for this problem. 4 3.1 ROC curves A classical tool for measuring the performance of a scoring rule s is the so-called ROC curve ROC(s, .) : α ∈[0, 1] 7→1 −Gs ◦H−1 s (1 −α), where H−1 s (x) = inf{t ∈R | Hs(t) ≥x}. In the case where s = η, we will denote ROC(η, α) = ROC∗(α), for any α ∈[0, 1]. The set of points (α, β) ∈[0, 1]2 which can be achieved as (α, ROC(s, α)) for some scoring function s is called the ROC space. It is a well-known fact that the regression function provides an optimal scoring function for the ROC curve. This fact relies on a simple application of Neyman-Pearson’s lemma. We refer to [CLV08] for the details. Using the fact that, for a given scoring function, the ROC curve is invariant by increasing transformations of the scoring function, we get the following result: Lemma. 6 For any scoring function s and any α ∈[0, 1], we have: ∀s∗∈S∗, ROC(s, α) ≤ROC(s∗, α) .= ROC∗(α) . The next result states that the set of optimal scoring functions coincides with the set of maximizers of the Wφ-ranking performance, provided that the score-generating function φ is strictly increasing. Proposition. 7 Assume that the score-generating function φ is strictly increasing. Then, we have: ∀s ∈S , Wφ(s) ≤Wφ(η) . Moreover W ∗ φ .= Wφ(η) = Wφ(s∗) for any s∗∈S∗. Remark 3 (ON PLUG-IN RANKING RULES) Theoretically, a possible approach to ranking is the plug-in method ([DGL96]), which consists of using an estimate ˆη of the regression function as a scoring function. As shown by the subsequent bound, when φ is differentiable with a bounded derivative, when ˆη is close to η in the L1-sense, it leads to a nearly optimal ordering in terms of W-ranking criterion: W ∗ φ −Wφ (bη) ≤(1 −p)||φ′||∞E[|bη(X) −η(X)|]. However, one faces difficulties with the plug-in approach when dealing with high-dimensional data, see [GKKW02]), which provides the motivation for exploring algorithms based on W-ranking performance maximization. 3.2 Connection to hypothesis testing From the angle embraced in this paper, the ranking problem is tightly related to hypothesis testing. Denote by X+ and X−two r.v. distributed as G and H respectively. As a first go, we can reformulate the ranking problem as the one of finding a scoring function s such that s(X−) is stochastically smaller than s(X+), which means, for example, that: ∀t ∈R, P{s(X−) ≥t} ≤P{s(X+) ≥t}. It is easy to see that the latter statement means that the ROC curve of s dominates the first diagonal of the ROC space. We point out the fact that the first diagonal corresponds to nondiscriminating scoring functions s0 such that Hs0 = Gs0. However, searching for a scoring function s fulfilling this property is generally not sufficient in practice. Heuristically, one would like to pick an s in order to be as far as possible from the case where ”Gs = Hs”. This requires to specify a certain measure of dissimilarity between distributions. In this respect, various criteria may be considered such as the L1-Mallows metric (see the next remark). Indeed, assuming temporarily that s is fixed and considering the problem of testing similarity vs. dissimilarity between two distributions Hs and Gs based on two independent samples s(X+ 1 ), . . . , s(X+ k ) and s(X− 1 ), . . . , s(X− m), it is well-known that nonparametric tests based on linear rank statistics have optimality properties. We refer to Chapter 9 in [Ser80] for an overview of rank procedures for testing homogeneity, which may yield relevant criteria in the ranking context. Remark 4 (CONNECTION BETWEEN AUC AND THE L1-MALLOWS METRIC ) Consider the AUC criterion: AUC(s) = R 1 α=0 ROC(s, α)dα. It is well-known that this criterion may be interpreted as 5 the ”rate of concording pairs”: AUC(s) = P{s(X) < s(X′) | Y = −1, Y ′ = +1} where (X, Y ) and (X′, Y ′) denote independent copies. Furthermore, it may be easily shown that AUC(s) = 1 2 + Z ∞ −∞ {Hs(t) −Gs(t)}dF(t), where the cdf F may be taken as any linear convex combination of Hs and Gs. Provided that Hs is stochastically smaller than Gs and that F(dt) is the uniform distribution over (0, 1) (this is always possible, even if it means replacing s by F ◦s, which leaves the ordering untouched), the second term may be identified as the L1-Mallows distance between Hs and Gs, a well-known probability metric widely considered in the statistical literature (also known as the L1-Wasserstein metric). 4 A generalization error bound We now provide a bound on the generalization ability of scoring rules based on empirical maximization of W-ranking performance criteria. Theorem. 8 Set the empirical W-ranking performance maximizer ˆsn = arg maxs∈S c Wn(s). Under the same assumptions as in Proposition 3 and assuming in addition that the class of functions Φs induced by S0 is also a VC major class of functions, we have, for any δ > 0, and with probability 1 −δ: W ∗ φ −Wφ(ˆsn) ≤c1 r V n + c2 r log(1/δ) n , for some positive constants c1, c2. The proof is a straightforward consequence from Proposition 3 and it can be found in the Appendix. 5 Conclusion In this paper, we considered a general class of performance measures for ranking/scoring which can be described as conditional linear rank statistics. Our overall setup encompasses in particular known criteria used in medical diagnosis and information retrieval. We have described the statistical nature of such statistics, proved that they ar compatible with optimal scoring functions in the bipartite setup, and provided a preliminary generalization bound with a √n-rate of convergence. By doing so, we provided the very results on a class of linear rank processes. Further work is needed to identify a variance control assumption in order to derive fast rates of convergence and to obtain consistency under weaker complexity assumptions. Moreover, it is not clear how to formulate convex surrogates for such functionals yet. Appendix - Proofs Proof of Proposition 5 By virtue of the finite increment theorem, we have: sup s∈S0 |c Wn(s) −kWφ(s)| ≤k||φ′||∞ 1 n + 1 + sup (s,t)∈S0×R | bFs(t) −Fs(t)| ! and the desired result immediately follows from the application of the VC inequality, see Remark 1. Proof of Proposition 3 Since φ is of class C2, a Taylor expansion at the second order immediately yields: c Wn(s) = k X i=1 φ(Fs(s(X+ i ))) + bBn(s) + bRn(s), 6 with bBn(s) = k X i=1 Rank(s(X+ i )) n + 1 −Fs(s(X+ i )) φ′(Fs(s(X+ i ))) | bRn(s)| ≤ k X i=1 Rank(s(X+ i )) n + 1 −Fs(s(X+ i )) 2 ||φ′′||∞. Following in the footsteps of [Haj68], we first compute the projection of bBn(s) onto the space Σ of r.v.’s of the form P i≤n fi(Xi, Yi) such that E[f 2 i (Xi, Yi)] < ∞for all i ∈{1, . . . , n}: PΣ( bBn(s)) = k X i=1 n X j=1 E Rank(s(X+ i )) n + 1 −Fs(s(X+ i )) φ′(Fs(s(Xi))) | Xj, Yj , This projection may be splitted into two terms: (I) = n X i=1 I{Yi=+1} 1 n + 1E[Rank(s(Xi)) | s(Xi)] −Fs(s(Xi)) φ′(Fs(s(Xi))), (II) = n X i=1 I{Yi=+1} X j̸=i E Rank(s(X+ i )) n + 1 −Fs(s(X+ i )) φ′(Fs(s(Xi))) | s(Xj), Yj . The first term is easily handled and may be seen as negligible (it is of order OP(n−1/2)), since we have E[Rank(s(Xi)) | s(Xi)] = n bFs(s(Xi)) and, by assumption, sup(s,t)∈S×R | bFs(t) −Fs(t)| = OP(n−1/2) (see Remark 1). Up to an additive term of order OP(1) uniformly over s ∈S, the second term may be rewritten as (II) = 1 n + 1 n X i=1 I{Yi=+1} X j̸=i E I{s(Xj)≤s(Xi)}φ′(Fs(s(Xi))) | s(Xj), Yj = k n + 1 n X j=1 Z ∞ s(Xj) φ′(Fs(u))dGs(u) − 1 n + 1 n X i=1 I{Yi=+1} Z ∞ s(Xi) φ′(Fs(u))dGs(u). As Pn i=1 I{Yi=+1} R ∞ s(Xi) φ′(Fs(u))dGs(u)/(n + 1) ≤supu∈[0,1] φ′(t) and k/(n + 1) ∼p, we get that, uniformly over s ∈S0: k X i=1 φ(Fs(s(X+ i ))) + PΣ( bBn(s))) = bVn(s) + OP(1) as n →∞. The term bRn(s) is negligible, since, up to the multiplicative constant ||φ′′||∞, it is bounded by 1 (n + 1)2 n X i=1 E 2Fs(s(Xi)) + X k̸=i {I{s(Xk)≤s(Xi)} −Fs(s(Xi))} 2 . As Fs is bounded by 1, it suffices to observe that for all i: E X k̸=i {I{s(Xk)≤s(Xi)} −Fs(s(Xi))} 2 | s(Xi) = X k̸=i E (I{s(Xk)≤s(Xi)} −Fs(s(Xi))})2 | s(Xi) . Bounding the variance of the binomial r.v. E[I{s(Xk) ≤s(Xi)} −Fs(s(Xi))})2 | s(Xi)] by 1/4, one finally gets that ˆRn(s) is of order OP(1) uniformly over s ∈S0. Eventually, one needs to evaluate the accuracy of the approximation yield by the projection bBn(s)− {PΣ( bBn(s)) −(n −1)E[ bBn(s)]}. Write, for all s ∈S0, bBn(s) = nbUn(s) + n X i=1 I{Yi=+1} 1 n + 1 −Fs(s(Xi)) φ′(Fs(s(Xi))), 7 where Un(s) = P i̸=j qs((Xi, Yi), (Xj, Yj))/(n(n + 1)) is a U-statistic with kernel: qs((x, y), (x′, y′)) = I{y=+1} · I{s(x′)≤s(x)} · φ′(Fs(s(x))). Hence, we have n−1 bBn(s) −{PΣ( bBn(s)) −(n −1)E[ bBn(s)]} = bUn(s) −{PΣ(bUn(s)) −(n −1)E[bUn(s)]} which actually corresponds to the degenerate part of the Hoeffding decomposition of the U-statistic bUn(s). Now, given that sups∈S0 ||qs||∞< ∞, it follows from Theorem 11 in [CLV08] for instance, combined with the basic symmetrization device of the kernel qs, that sup s∈S0 |bUn(s) −{PΣ(bUn(s)) −(n −1)E[bUn(s)]}| = OP(n−1) as n →∞, which concludes the proof. Proof of Proposition 7 Using the decomposition Fs = pGs + (1 −p)Hs, we are led to the following expression: pWφ(s) = Z 1 0 φ(u) du −(1 −p)E[φ(Fs(s(X))) | Y = −1]. Then, using a change of variable: E[φ(Fs(s(X))) | Y = −1] = Z 1 0 φ(p(1 −ROC(s, α)) + (1 −p)(1 −α)) dα . It is now easy to conclude since φ is increasing (by assumption) and because of the optimality of elements of S∗in the sense of Lemma 6. Proof of Theorem 8 Observe that, by virtue of Proposition 3, W ∗ φ −Wφ(ˆsn) ≤2 sup s∈S0 |c Wn(s)/k −Wφ(s)| ≤2 k sup s∈S0 |bVn(s) −kWφ(s)| + OP(n−1), and the desired bound derives from the VC inequality applied to the sup term, noticing that it follows from our assumptions that {(x, y) 7→I{y=+1}Φs(x)}s∈S0 is a VC class of functions. References [AGH+05] S. Agarwal, T. Graepel, R. Herbrich, S. Har-Peled, and D. Roth. Generalization bounds for the area under the ROC curve. Journal of Machine Learning Research, 6:393–425, 2005. [BBL05] S. Boucheron, O. Bousquet, and G. Lugosi. Theory of Classification: A Survey of Some Recent Advances. ESAIM: Probability and Statistics, 9:323–375, 2005. [CLV08] S. Cl´emenc¸on, G. Lugosi, and N. Vayatis. Ranking and empirical risk minimization of U-statistics. The Annals of Statistics, 36(2):844–874, 2008. [CS58] J. Chernoff and Savage. Asymptotic normality and efficiency of certain non parametric test statistics. Ann. Math. Stat., 29:972–994, 1958. [CV07] S. Cl´emenc¸on and N. Vayatis. Ranking the best instances. Journal of Machine Learning Research, 8:2671–2699, 2007. [CZ06] D. Cossock and T. Zhang. Subset ranking using regression. In H.U. Simon and G. Lugosi, editors, Proceedings of COLT 2006, volume 4005 of Lecture Notes in Computer Science, pages 605–619, 2006. [DGL96] L. Devroye, L. Gy¨orfi, and G. Lugosi. A Probabilistic Theory of Pattern Recognition. Springer, 1996. 8 [Dud99] R.M. Dudley. Uniform Central Limit Theorems. Cambridge University Press, 1999. [GKKW02] L. Gy¨orfi, M. K¨ohler, A. Krzyzak, and H. Walk. A Distribution-Free Theory of Nonparametric Regression. Springer, 2002. [Haj68] J. Hajek. Asymptotic normality of simple linear rank statistics under alternatives. Ann. Math. Stat., 39:325–346, 1968. [Hoe48] W. Hoeffding. A class of statistics with asymptotically normal distribution. Ann. Math. Stat., 19:293–325, 1948. [HS67] J. H´ajek and Z. Sid´ak. Theory of Rank Tests. Academic Press, 1967. [Rud06] C. Rudin. Ranking with a P-Norm Push. In H.U. Simon and G. Lugosi, editors, Proceedings of COLT 2006, volume 4005 of Lecture Notes in Computer Science, pages 589–604, 2006. [Ser80] R.J. Serfling. Approximation theorems of mathematical statistics. John Wiley & Sons, 1980. 9
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Resolution Limits of Sparse Coding in High Dimensions∗ Alyson K. Fletcher,† Sundeep Rangan,‡ and Vivek K Goyal§ Abstract This paper addresses the problem of sparsity pattern detection for unknown ksparse n-dimensional signals observed through m noisy, random linear measurements. Sparsity pattern recovery arises in a number of settings including statistical model selection, pattern detection, and image acquisition. The main results in this paper are necessary and sufficient conditions for asymptotically-reliable sparsity pattern recovery in terms of the dimensions m, n and k as well as the signal-tonoise ratio (SNR) and the minimum-to-average ratio (MAR) of the nonzero entries of the signal. We show that m > 2k log(n −k)/(SNR · MAR) is necessary for any algorithm to succeed, regardless of complexity; this matches a previous sufficient condition for maximum likelihood estimation within a constant factor under certain scalings of k, SNR and MAR with n. We also show a sufficient condition for a computationally-trivial thresholding algorithm that is larger than the previous expression by only a factor of 4(1 + SNR) and larger than the requirement for lasso by only a factor of 4/MAR. This provides insight on the precise value and limitations of convex programming-based algorithms. 1 Introduction Sparse signal models have been used successfully in a variety of applications including waveletbased image processing and pattern recognition. Recent research has shown that certain naturallyoccurring neurological processes may exploit sparsity as well [1–3]. For example, there is now evidence that the V1 visual cortex naturally generates a sparse representation of the visual data relative to a certain Gabor-like basis. Due to the nonlinear nature of sparse signal models, developing and analyzing algorithms for sparse signal processing has been a major research challenge. This paper considers the problem of estimating sparse signals in the presence of noise. We are specifically concerned with understanding the theoretical estimation limits and how far practical algorithms are from those limits. In the context of visual cortex modeling, this analysis may help us understand what visual features are resolvable from visual data. To keep the analysis general, we consider the following abstract estimation problem: An unknown sparse signal x is modeled as an n-dimensional real vector with k nonzero components. The locations of the nonzero components is called the sparsity pattern. We consider the problem of detecting the sparsity pattern of x from an m-dimensional measurement vector y = Ax + d, where A ∈Rm×n is a known measurement matrix and d ∈Rm is an additive noise vector with a known distribution. We are interested in ∗This work was supported in part by a University of California President’s Postdoctoral Fellowship, NSF CAREER Grant CCF-643836, and the Centre Bernoulli at ´Ecole Polytechnique F´ed´erale de Lausanne. †A. K. Fletcher (email: alyson@eecs.berkeley.edu) is with the Department of Electrical Engineering and Computer Sciences, University of California, Berkeley. ‡S. Rangan (email: srangan@qualcomm.com) is with Qualcomm Technologies, Bedminster, NJ. §V. K. Goyal (email: vgoyal@mit.edu) is with the Department of Electrical Engineering and Computer Science and the Research Laboratory of Electronics, Massachusetts Institute of Technology. finite SNR SNR →∞ Any algorithm must fail m < 2 MAR·SNRk log(n −k) + k −1 m ≤k Theorem 1 (elementary) Necessary and unknown (expressions above m ≍2k log(n −k) + k + 1 sufficient for lasso and right are necessary) Wainwright [14] Sufficient for m > 8(1+SNR) MAR·SNR k log(n −k) m > 8 MARk log(n −k) thresholding estimator (11) Theorem 2 from Theorem 2 Table 1: Summary of Results on Measurement Scaling for Reliable Sparsity Recovery (see body for definitions and technical limitations) determining necessary and sufficient conditions on the ability to reliably detect the sparsity pattern based on problem dimensions m, n and k, and signal and noise statistics. Previous work. While optimal sparsity pattern detection is NP-hard [4], greedy heuristics (matching pursuit [5] and its variants) and convex relaxations (basis pursuit [6], lasso [7], and others) have been widely-used since at least the mid 1990s. While these algorithms worked well in practice, until recently, little could be shown analytically about their performance. Some remarkable recent results are sets of conditions that can guarantee exact sparsity recovery based on certain simple “incoherence” conditions on the measurement matrix A [8–10]. These conditions and others have been exploited in developing the area of “compressed sensing,” which considers large random matrices A with i.i.d. components [11–13]. The main theoretical result are conditions that guarantee sparse detection with convex programming methods. The best of these results is due to Wainwright [14], who shows that the scaling m ≍2k log(n −k) + k + 1. (1) is necessary and sufficient for lasso to detect the sparsity pattern when A has Gaussian entries, provided the SNR scales to infinity. Preview. This paper presents new necessary and sufficient conditions, summarized in Table 1 along with Wainwright’s lasso scaling (1). The parameters MAR and SNR represent the minimumto-average and signal-to-noise ratio, respectively. The exact definitions and measurement model are given below. The necessary condition applies to all algorithms, regardless of complexity. Previous necessary conditions had been based on information-theoretic analyses such as [15–17]. More recent publications with necessary conditions include [18–21]. As described in Section 3, our new necessary condition is stronger than previous bounds in certain important regimes. The sufficient condition is derived for a computationally-trivial thresholding estimator. By comparing with the lasso scaling, we argue that main benefits of more sophisticated methods, such as lasso, is not generally in the scaling with respect to k and n but rather in the dependence on the minimum-to-average ratio. 2 Problem Statement Consider estimating a k-sparse vector x ∈Rn through a vector of observations, y = Ax + d, (2) where A ∈Rm×n is a random matrix with i.i.d. N(0, 1/m) entries and d ∈Rm is i.i.d. unitvariance Gaussian noise. Denote the sparsity pattern of x (positions of nonzero entries) by the set Itrue, which is a k-element subset of the set of indices {1, 2, . . . , n}. Estimates of the sparsity pattern will be denoted by ˆI with subscripts indicating the type of estimator. We seek conditions under which there exists an estimator such that ˆI = Itrue with high probability. In addition to the signal dimensions, m, n and k, we will show that there are two variables that dictate the ability to detect the sparsity pattern reliably: the signal-to-noise ratio (SNR), and what we will call the minimum-to-average ratio (MAR). The SNR is defined by SNR = E[∥Ax∥2] E[∥d∥2] = E[∥Ax∥2] m . (3) Since we are considering x as an unknown deterministic vector, the SNR can be further simplified as follows: The entries of A are i.i.d. N(0, 1/m), so columns ai ∈Rm and aj ∈Rm of A satisfy E[a′ iaj] = δij. Therefore, the signal energy is given by E ∥Ax∥2 = X X i,j∈Itrue E [a′ iajxixj] = X X i,j∈Itrue xixjδij = ∥x∥2. Substituting into the definition (3), the SNR is given by SNR = 1 m∥x∥2. (4) The minimum-to-average ratio of x is defined as MAR = minj∈Itrue |xj|2 ∥x∥2/k . (5) Since ∥x∥2/k is the average of {|xj|2 | j ∈Itrue}, MAR ∈(0, 1] with the upper limit occurring when all the nonzero entries of x have the same magnitude. One final value that will be important is the minimum component SNR, defined as SNRmin = 1 E∥d∥2 min j∈Itrue E∥ajxj∥2 = 1 m min j∈Itrue |xj|2. (6) The quantity SNRmin has a natural interpretation: The numerator, min E∥ajxj∥2, is the signal power due to the smallest nonzero component of x, while the denominator, E∥d∥2, is the total noise power. The ratio SNRmin thus represents the contribution to the SNR from the smallest nonzero component of the unknown vector x. Observe that (3) and (5) show SNRmin = 1 k SNR · MAR. (7) Normalizations. Other works use a variety of normalizations, e.g.: the entries of A have variance 1/n in [13,19]; the entries of A have unit variance and the variance of d is a variable σ2 in [14,17, 20,21]; and our scaling of A and a noise variance of σ2 are used in [22]. This necessitates great care in comparing results. To facilitate the comparison we have expressed all our results in terms of SNR, MAR and SNRmin as defined above. All of these quantities are dimensionless, in that if either A and d or x and d are scaled together, these ratios will not change. Thus, the results can be applied to any scaling of A, d and x, provided that the quantities SNR, MAR and SNRmin are computed appropriately. 3 Necessary Condition for Sparsity Recovery We first consider sparsity recovery without being concerned with computational complexity of the estimation algorithm. Since the vector x ∈Rn is k-sparse, the vector Ax belongs to one of L = n k subspaces spanned by k of the n columns of A. Estimation of the sparsity pattern is the selection of one of these subspaces, and since the noise d is Gaussian, the probability of error is minimized by choosing the subspace closest to the observed vector y. This results in the maximum likelihood (ML) estimate. Mathematically, the ML estimator can be described as follows. Given a subset J ⊆{1, 2, . . . , n}, let PJy denote the orthogonal projection of the vector y onto the subspace spanned by the vectors {aj | j ∈J}. The ML estimate of the sparsity pattern is ˆIML = arg max J : |J|=k ∥PJy∥2, where |J| denotes the cardinality of J. That is, the ML estimate is the set of k indices such that the subspace spanned by the corresponding columns of A contain the maximum signal energy of y. Since the number of subspaces L grows exponentially in n and k, an exhaustive search is, in general, computationally infeasible. However, the performance of ML estimation provides a lower bound on the number of measurements needed by any algorithm that cannot exploit a priori information on x other than it being k-sparse. ML estimation for sparsity recovery was first examined in [17]. There, it was shown that there exists a constant C > 0 such that the condition m > C max log(n −k) SNRmin , k log n k = C max k log(n −k) SNR · MAR , k log n k (8) is sufficient for ML to asymptotically reliably recover the sparsity pattern. Note that the equality between the two expressions in (8) is a consequence of (7). Our first theorem provides a corresponding necessary condition. Theorem 1 Let k = k(n), m = m(n), SNR = SNR(n) and MAR = MAR(n) be deterministic sequences in n such that limn→∞k(n) = ∞and m(n) < 2 −δ SNRmin log(n −k) + k −1 = 2 −δ MAR · SNR k log(n −k) + k −1 (9) for some δ > 0. Then even the ML estimator asymptotically cannot detect the sparsity pattern, i.e., lim n→∞Pr ˆIML = Itrue = 0. Proof sketch: The basic idea in the proof is to consider an “incorrect” subspace formed by removing one of the k correct vectors with the least energy, and adding one of the n −k incorrect vectors with largest energy. The change in energy can be estimated using tail distributions of chi-squared random variables. A complete proof appears in [23]. The theorem provides a simple lower bound on the minimum number of measurements required to recover the sparsity pattern in terms of k, n and the minimum component SNR, SNRmin. Note that the equivalence between the two expressions in (9) is due to (7). Remarks. 1. The theorem strengthens an earlier necessary condition in [18] which showed that there exists a C > 0 such that m = C SNRk log(n −k) is necessary for asymptotic reliable recovery. Theorem 1 strengthens the result to reflect the dependence on MAR and make the constant explicit. 2. The theorem applies for any k(n) such that limn→∞k(n) = ∞, including both cases with k = o(n) and k = Θ(n). In particular, under linear sparsity (k = αn for some constant α), the theorem shows that m ≍ 2α MAR · SNRn log n measurements are necessary for sparsity recovery. Similarly, if m/n is bounded above by a constant, then sparsity recovery will certainly fail unless k = O (SNR · MAR · n/ log n) . In particular, when SNR · MAR is bounded, the sparsity ratio k/n must approach zero. 3. In the case where SNR · MAR and the sparsity ratio k/n are both constant, the sufficient condition (8) reduces to m = (C/(SNR · MAR))k log(n −k), which matches the necessary condition in (9) within a constant factor. 4. In the case of MAR· SNR < 1, the bound (9) improves upon the necessary condition of [14] for the asymptotic success of lasso by the factor (MAR · SNR)−1. (1, 1) 2 4 5 10 15 20 25 30 35 40 (2, 1) 2 4 5 10 15 20 25 30 35 40 (5, 1) 2 4 5 10 15 20 25 30 35 40 (10, 1) 2 4 5 10 15 20 25 30 35 40 (10, 0.5) 2 4 5 10 15 20 25 30 35 40 (10, 0.2) 2 4 5 10 15 20 25 30 35 40 (10, 0.1) 2 4 5 10 15 20 25 30 35 40 0 0.2 0.4 0.6 0.8 1 k k k k k k k m Figure 1: Simulated success probability of ML detection for n = 20 and many values of k, m, SNR, and MAR. Each subfigure gives simulation results for k ∈{1, 2, . . ., 5} and m ∈{1, 2, . . ., 40} for one (SNR, MAR) pair. Each subfigure heading gives (SNR, MAR). Each point represents at least 500 independent trials. Overlaid on the color-intensity plots is a black curve representing (9). 5. The bound (9) can be compared against information-theoretic bounds such as those in [15–17, 20,21]. For example, a simple capacity argument in [15] shows that m ≥ 2 log2 n k log2(1 + SNR) (10) is necessary. When the sparsity ratio k/n and SNR are both fixed, m can satisfy (10) while growing only linearly with k. In contrast, Theorem 1 shows that with sparsity ratio and SNR · MAR fixed, m = Ω(k log(n−k)) is necessary for reliable sparsity recovery. That is, the number of measurements must grow superlinearly in k in the linear sparsity regime with bounded SNR. In the sublinear regime where k = o(n), the capacity-based bound (10) may be stronger than (9) depending on the scaling of SNR, MAR and other terms. 6. Results more similar to Theorem 1—based on direct analyses of error events rather than information-theoretic arguments—appeared in [18,19]. The previous results showed that with fixed SNR as defined here, sparsity recovery with m = Θ(k) must fail. The more refined analysis in this paper gives the additional log(n −k) factor and the precise dependence on MAR · SNR. 7. Theorem 1 is not contradicted by the relevant sufficient condition of [20, 21]. That sufficient condition holds for scaling that gives linear sparsity and MAR · SNR = Ω(√n log n). For MAR · SNR = √n log n, Theorem 1 shows that fewer than m ≍2√k log k measurements will cause ML decoding to fail, while [21, Thm. 3.1] shows that a typicality-based decoder will succeed with m = Θ(k) measurements. 8. The necessary condition (9) shows a dependence on the minimum-to-averageratio MAR instead of just the average power through SNR. Thus, the bound shows the negative effects of relatively small components. Note that [17, Thm. 2] appears to have dependence on the minimum power as well, but is actually only proven for the case MAR = 1. Numerical validation. Computational confirmation of Theorem 1 is technically impossible, and even qualitative support is hard to obtain because of the high complexity of ML detection. Nevertheless, we may obtain some evidence through Monte Carlo simulation. Fig. 1 shows the probability of success of ML detection for n = 20 as k, m, SNR, and MAR are varied. Signals with MAR < 1 are created by having one small nonzero component and k −1 equal, larger nonzero components. Taking any one column of one subpanel from bottom to top shows that as m is increased, there is a transition from ML failing to ML succeeding. One can see that (9) follows the failure-success transition qualitatively. In particular, the empirical dependence on SNR and MAR approximately follows (9). Empirically, for the (small) value of n = 20, it seems that with MAR · SNR held fixed, sparsity recovery becomes easier as SNR increases (and MAR decreases). 4 Sufficient Condition for Thresholding Consider the following simple estimator. As before, let aj be the jth column of the random matrix A. Define the thresholding estimate as ˆIthresh = j : |a′ jy|2 > µ , (11) where µ > 0 represents a threshold level. This algorithm simply correlates the observed signal y with all the frame vectors aj and selects the indices j where the correlation energy exceeds a certain level µ. It is significantly simpler than both lasso and matching pursuit and is not meant to be proposed as a competitive alternative. Rather, we consider thresholding simply to illustrate what precise benefits lasso and more sophisticated methods bring. Sparsity pattern recovery by thresholding was studied in [24], which proves a sufficient condition when there is no noise. The following theorem improves and generalizes the result to the noisy case. Theorem 2 Let k = k(n), m = m(n), SNR = SNR(n) and MAR = MAR(n) be deterministic sequences in n such that limn→∞k = ∞and m > 8(1 + δ)(1 + SNR) MAR · SNR k log(n −k) (12) for some δ > 0. Then, there exists a sequence of threshold levels µ = µ(n), such that thresholding asymptotically detects the sparsity pattern, i.e., lim n→∞Pr ˆIthresh = Itrue = 1. Proof sketch: Using tail distributions of chi-squared random variables, it is shown that the minimum value for the correlation |a′ jy|2 when j ∈Itrue is greater than the maximum correlation when j ̸∈Itrue. A complete proof appears in [23]. Remarks. 1. Comparing (9) and (12), we see that thresholding requires a factor of at most 4(1 + SNR) more measurements than ML estimation. Thus, for a fixed SNR, the optimal scaling not only does not require ML estimation, it does not even require lasso or matching pursuit—it can be achieved with a remarkably simply method. 2. Nevertheless, the gap between thresholding and ML of 4(1+SNR) measurements can be large. This is most apparent in the regime where the SNR →∞. For ML estimation, the lower bound on the number of measurements required by ML decreases to k −1 as SNR →∞.1 In contrast, with thresholding, increasing the SNR has diminishing returns: as SNR →∞, the bound on the number of measurements in (12) approaches m > 8 MARk log(n −k). (13) Thus, even with SNR →∞, the minimum number of measurements is not improved from m = Ω(k log(n −k)). This diminishing returns for improved SNR exhibited by thresholding is also a problem for more sophisticated methods such as lasso. For example, as discussed earlier, the analysis of [14] shows that when SNR · MAR →∞, lasso requires m > 2k log(n −k) + k + 1 (14) for reliable recovery. Therefore, like thresholding, lasso does not achieve a scaling better than m = O(k log(n −k)), even at infinite SNR. 3. There is also a gap between thresholding and lasso. Comparing (13) and (14), we see that, at high SNR, thresholding requires a factor of up to 4/MAR more measurements than lasso. This factor is largest when MAR is small, which occurs when there are relatively small nonzero components. Thus, in the high SNR regime, the main benefit of lasso is its ability to detect small coefficients, even when they are much below the average power. However, if the range of component magnitudes is not large, so MAR is close to one, lasso and thresholding have equal performance within a constant factor. 1Of course, at least k + 1 measurements are necessary. 4. The high SNR limit (13) matches the sufficient condition in [24] for the noise free case, except that the constant in (13) is tighter. Numerical validation. Thresholding is extremely simple and can thus be simulated easily for large problem sizes. The results of a large number of Monte Carlo simulations are presented in [23], which also reports additional simulations of maximum likelihood estimation. With n = 100, the sufficient condition predicted by (12) matches well to the parameter combinations where the probability of success drops below about 0.995. 5 Conclusions We have considered the problem of detecting the sparsity pattern of a sparse vector from noisy random linear measurements. Necessary and sufficient scaling laws for the number of measurements to recover the sparsity pattern for different detection algorithms were derived. The analysis reveals the effect of two key factors: the total signal-to-noise ratio (SNR), as well as the minimum-toaverage ratio (MAR), which is a measure of the spread of component magnitudes. The product of these factors is k times the SNR contribution from the smallest nonzero component; this product often appears. Our main conclusions are: • Tight scaling laws for constant SNR and MAR. In the regime where SNR = Θ(1) and MAR = Θ(1), our results show that the scaling of the number of measurements m = O(k log(n −k)) is both necessary and sufficient for asymptotically reliable sparsity pattern detection. Moreover, the scaling can be achieved with a thresholding algorithm, which is computationally simpler than even lasso or basis pursuit. Under the additional assumption of linear sparsity (k/n fixed), this scaling is a larger number of measurements than predicted by previous informationtheoretic bounds. • Dependence on SNR. While the number of measurements required for exhaustive ML estimation and simple thresholding have the same dependence on n and k with the SNR fixed, the dependence on SNR differs significantly. Specifically, thresholding requires a factor of up to 4(1 + SNR) more measurements than ML. Moreover, as SNR →∞, the number of measurements required by ML may be as low as m = k + 1. In contrast, even letting SNR →∞, thresholding and lasso still require m = O(k log(n −k)) measurements. • Lasso and dependence on MAR. Thresholding can also be compared to lasso, at least in the high SNR regime. There is a potential gap between thresholding and lasso, but the gap is smaller than the gap to ML. Specifically, in the high SNR regime, thresholding requires at most 4/MAR more measurements than lasso. Thus, the benefit of lasso over simple thresholding is its ability to detect the sparsity pattern even in the presence of relatively small nonzero coefficients (i.e. low MAR). However, when the components of the unknown vector have similar magnitudes (MAR close to one), the gap between lasso and simple thresholding is reduced. While our results provide both necessary and sufficient scaling laws, there is clearly a gap in terms of the scaling with the SNR. We have seen that full ML estimation could potentially have a scaling in SNR as small as m = O(1/SNR) + k −1. An open question is whether there is any practical algorithm that can achieve a similar scaling. A second open issue is to determine conditions for partial sparsity recovery. The above results define conditions for recovering all the positions in the sparsity pattern. However, in many practical applications, obtaining some large fraction of these positions would be sufficient. Neither the limits of partial sparsity recovery nor the performance of practical algorithms are completely understood, though some results have been reported in [19–21,25]. References [1] M. Lewicki. 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Measurements vs. bits: Compressed sensing meets information theory. In Proc. 44th Ann. Allerton Conf. on Commun., Control and Comp., Monticello, IL, Sept. 2006. [16] A. K. Fletcher, S. Rangan, and V. K. Goyal. Rate-distortion bounds for sparse approximation. In IEEE Statist. Sig. Process. Workshop, pages 254–258, Madison, WI, Aug. 2007. [17] M. J. Wainwright. Information-theoretic limits on sparsity recovery in the high-dimensional and noisy setting. Tech. Report 725, Univ. of California, Berkeley, Dept. of Stat., Jan. 2007. [18] V. K. Goyal, A. K. Fletcher, and S. Rangan. Compressive sampling and lossy compression. IEEE Sig. Process. Mag., 25(2):48–56, March 2008. [19] G. Reeves. Sparse signal sampling using noisy linear projections. Tech. Report UCB/EECS2008-3, Univ. of California, Berkeley, Dept. of Elec. Eng. and Comp. Sci., Jan. 2008. [20] M. Akc¸akaya and V. Tarokh. Shannon theoretic limits on noisy compressive sampling. arXiv:0711.0366v1 [cs.IT]., Nov. 2007. [21] M. Akc¸akaya and V. Tarokh. Noisy compressive sampling limits in linear and sublinear regimes. In Proc. Conf. on Inform. Sci. & Sys., Princeton, NJ, March 2008. [22] J. Haupt and R. Nowak. Signal reconstruction from noisy random projections. IEEE Trans. Inform. Theory, 52(9):4036–4048, Sept. 2006. [23] A. K. Fletcher, S. Rangan, and V. K. Goyal. Necessary and sufficient conditions on sparsity pattern recovery. arXiv:0804.1839v1 [cs.IT]., April 2008. [24] H. Rauhut, K. Schnass, and P. Vandergheynst. Compressed sensing and redundant dictionaries. IEEE Trans. Inform. Theory, 54(5):2210–2219, May 2008. [25] S. Aeron, M. Zhao, and V. Saligrama. On sensing capacity of sensor networks for the class of linear observation, fixed SNR models. arXiv:0704.3434v3 [cs.IT]., June 2007.
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Privacy-preserving logistic regression Kamalika Chaudhuri Information Theory and Applications University of California, San Diego kamalika@soe.ucsd.edu Claire Monteleoni∗ Center for Computational Learning Systems Columbia University cmontel@ccls.columbia.edu Abstract This paper addresses the important tradeoff between privacy and learnability, when designing algorithms for learning from private databases. We focus on privacy-preserving logistic regression. First we apply an idea of Dwork et al. [6] to design a privacy-preserving logistic regression algorithm. This involves bounding the sensitivity of regularized logistic regression, and perturbing the learned classifier with noise proportional to the sensitivity. We then provide a privacy-preserving regularized logistic regression algorithm based on a new privacy-preserving technique: solving a perturbed optimization problem. We prove that our algorithm preserves privacy in the model due to [6]. We provide learning guarantees for both algorithms, which are tighter for our new algorithm, in cases in which one would typically apply logistic regression. Experiments demonstrate improved learning performance of our method, versus the sensitivity method. Our privacy-preserving technique does not depend on the sensitivity of the function, and extends easily to a class of convex loss functions. Our work also reveals an interesting connection between regularization and privacy. 1 Introduction Privacy-preserving machine learning is an emerging problem, due in part to the increased reliance on the internet for day-to-day tasks such as banking, shopping, and social networking. Moreover, private data such as medical and financial records are increasingly being digitized, stored, and managed by independent companies. In the literature on cryptography and information security, data privacy definitions have been proposed, however designing machine learning algorithms that adhere to them has not been well-explored. On the other hand, data-mining algorithms have been introduced that aim to respect other notions of privacy that may be less formally justified. Our goal is to bridge the gap between approaches in the cryptography and information security community, and those in the data-mining community. This is necessary, as there is a tradeoff between the privacy of a protocol, and the learnability of functions that respect the protocol. In addition to the specific contributions of our paper, we hope to encourage the machine learning community to embrace the goals of privacy-preserving machine learning, as it is still a fledgling endeavor. In this work, we provide algorithms for learning in a privacy model introduced by Dwork et al. [6]. The ϵ-differential privacy model limits how much information an adversary can gain about a particular private value, by observing a function learned from a database containing that value, even if she knows every other value in the database. An initial positive result [6] in this setting depends on the sensitivity of the function to be learned, which is the maximum amount the function value can change due to an arbitrary change in one input. Using this method requires bounding the sensitivity of the function class to be learned, and then adding noise proportional to the sensitivity. This might be difficult for some functions that are important for machine learning. ∗The majority of this work was done while at UC San Diego. 1 The contributions of this paper are as follows. First we apply the sensitivity-based method of designing privacy-preserving algorithms [6] to a specific machine learning algorithm, logistic regression. Then we present a second privacy-preserving logistic regression algorithm. The second algorithm is based on solving a perturbed objective function, and does not depend on the sensitivity. We prove that the new method is private in the ϵ-differential privacy model. We provide learning performance guarantees for both algorithms, which are tighter for our new algorithm, in cases in which one would typically apply logistic regression. Finally, we provide experiments demonstrating superior learning performance of our new method, with respect to the algorithm based on [6]. Our technique may have broader applications, and we show that it can be applied to certain classes of optimization problems. 1.1 Overview and related work At the first glance, it may seem that anonymizing a data-set – namely, stripping it of identifying information about individuals, such as names, addresses, etc – is sufficient to preserve privacy. However, this is problematic, because an adversary may have some auxiliary information, which may even be publicly available, and which can be used to breach privacy. For more details on such attacks, see [12]. To formally address this issue, we need a definition of privacy which works in the presence of auxiliary knowledge by the adversary. The definition we use is due to Dwork et al. [6], and has been used in several applications [4, 11, 2]. We explain this definition and privacy model in more detail in Section 2. Privacy and learning. The work most related to ours is [8] and [3]. [8] shows how to find classifiers that preserve ϵ-differential privacy; however, their algorithm takes time exponential in d for inputs in Rd. [3] provides a general method for publishing data-sets while preserving ϵ-differential privacy such that the answers to all queries of a certain type with low VC-dimension are approximately correct. However, their algorithm can also be computationally inefficient. Additional related work. There has been a substantial amount of work on privacy in the literature, spanning several communities. Much work on privacy has been done in the data-mining community [1, 7], [14, 10], however the privacy definitions used in these papers are different, and some are susceptible to attacks when the adversary has some prior information. In contrast, the privacy definition we use avoids these attacks, and is very strong. 2 Sensitivity and the ϵ-differential privacy model Before we define the privacy model that we study, we will note a few preliminary points. Both in that model, and for our algorithm and analyses, we assume that each value in the database is a real vector with norm at most one. That is, a database contains values x1, . . . , xn, where xi ∈Rd, and ∥xi∥≤1 for all i ∈{1, . . . , n}. This assumption is used in the privacy model. In addition, we assume that when learning linear separators, the best separator passes through the origin. Note that this is not an assumption that the data is separable, but instead an assumption that a vector’s classification is based on its angle, regardless of its norm. In both privacy-preserving logistic regression algorithms that we state, the output is a parameter vector, w, which makes prediction SGN(w · x), on a point x. For a vector x, we use ||x|| to denote its Euclidean norm. For a function G(x) defined on Rd, we use ∇G to denote its gradient and ∇2G to denote its Hessian. Privacy Definition. The privacy definition we use is due to Dwork et al. [6, 5]. In this model, users have access to private data about individuals through a sanitization mechanism, usually denoted by M. The interaction between the sanitization mechanism and an adversary is modelled as a sequence of queries, made by the adversary, and responses, made by the sanitizer. The sanitizer, which is typically a randomized algorithm, is said to preserve ϵ-differential privacy, if the private value of any one individual in the data set does not affect the likelihood of a specific answer by the sanitizer by more than ϵ. More formally, ϵ-differential privacy can be defined as follows. 2 Definition 1 A randomized mechanism M provides ϵ-differential privacy, if, for all databases D1 and D2 which differ by at most one element, and for any t, Pr[M(D1) = t] Pr[M(D2) = t] ≤eϵ It was shown in [6] that if a mechanism satisfies ϵ-differential privacy, then an adversary who knows the private value of all the individuals in the data-set, except for one single individual, cannot figure out the private value of the unknown individual, with sufficient confidence, from the responses of the sanitizer. ϵ-differential privacy is therefore a very strong notion of privacy. [6] also provides a general method for computing an approximation to any function f while preserving ϵ-differential privacy. Before we can describe their method, we need a definition. Definition 2 For any function f with n inputs, we define the sensitivity S(f) as the maximum, over all inputs, of the difference in the value of f when one input of f is changed. That is, S(f) = max (a,a′) |f(x1, . . . , xn−1, xn = a) −f(x1, . . . , xn−1, xn = a′)| [6] shows that for any input x1, . . . , xn, releasing f(x1, . . . , xn) + η, where η is a random variable drawn from a Laplace distribution with mean 0 and standard deviation S(f) ϵ preserves ϵ-differential privacy. In [13], Nissim et al. showed that given any input x to a function, and a function f, it is sufficient to draw η from a Laplace distribution with standard deviation SS(f) ϵ , where SS(f) is the smoothedsensitivity of f around x. Although this method sometimes requires adding a smaller amount of noise to preserve privacy, in general, smoothed sensitivity of a function can be hard to compute. 3 A Simple Algorithm Based on [6], one can come up with a simple algorithm for privacy-preserving logistic regression, which adds noise to the classifier obtained by logistic regression, proportional to its sensitivity. From Corollary 2, the sensitivity of logistic regression is at most 2 nλ. This leads to Algorithm 1, which obeys the privacy guarantees in Theorem 1. Algorithm 1: 1. Compute w∗, the classifier obtained by regularized logistic regression on the labelled examples (x1, y1), . . . , (xn, yn). 2. Pick a noise vector η according to the following density function: h(η) ∝e−nϵλ 2 ||η||. To pick such a vector, we choose the norm of η from the Γ(d, 2 nϵλ) distribution, and the direction of η uniformly at random. 3. Output w∗+ η. Theorem 1 Let (x1, y1), . . . , (xn, yn) be a set of labelled points over Rd such that ||xi|| ≤1 for all i. Then, Algorithm 1 preserves ϵ-differential privacy. PROOF: The proof follows by a combination of [6], and Corollary 2, which states that the sensitivity of logistic regression is at most 2 nλ. □ Lemma 1 Let G(w) and g(w) be two convex functions, which are continuous and differentiable at all points. If w1 = argminwG(w) and w2 = argminwG(w) + g(w), then, ||w1 −w2|| ≤g1 G2 . Here, g1 = maxw ||∇g(w)|| and G2 = minv minw vT ∇2G(w)v, for any unit vector v. The main idea of the proof is to examine the gradient and the Hessian of the functions G and g around w1 and w2. Due to lack of space, the full proof appears in the full version of our paper. Corollary 2 Given a set of n examples x1, . . . , xn in Rd, with labels y1, . . . , yn, such that for all i, ||xi|| ≤1, the sensitivity of logistic regression with regularization parameter λ is at most 2 nλ. 3 PROOF: We use a triangle inequality and the fact that G2 ≥λ and g1 ≤1 n. □ Learning Performance. In order to assess the performance of Algorithm 1, we first try to bound the performance of Algorithm 1 on the training data. To do this, we need to define some notation. For a classifier w, we use L(w) to denote the expected loss of w over the data distribution, and ˆL(w) to denote the empirical average loss of w over the training data. In other words, ˆL(w) = 1 n Pn i=1 log(1 + e−yiwT xi), where, (xi, yi), i = 1, . . . , n are the training examples. Further, for a classifier w, we use the notation fλ(w) to denote the quantity 1 2λ||w||2 + L(w) and ˆfλ(w) to denote the quantity 1 2λ||w||2 + ˆL(w). Our guarantees on this algorithm can be summarized by the following lemma. Lemma 3 Given a logistic regression problem with regularization parameter λ, let w1 be the classifier that minimizes ˆfλ, and w2 be the classifier output by Algorithm 1 respectively. Then, with probability 1−δ over the randomness in the privacy mechanism, ˆfλ(w2) ≤ˆfλ(w1)+ 2d2(1+λ) log2(d/δ) λ2n2ϵ2 . Due to lack of space, the proof is deferred to the full version. From Lemma 3, we see that performance of Algorithm 1 degrades with decreasing λ, and is poor in particular when λ is very small. One question is, can we get a privacy-preserving approximation to logistic regression, which has better performance bounds for small λ? To explore this, in the next section, we look at a different algorithm. 4 A New Algorithm In this section, we provide a new privacy-preserving algorithm for logistic regression. The input to our algorithm is a set of examples x1, . . . , xn over Rd such that ||xi|| ≤1 for all i, a set of labels y1, . . . , yn for the examples, a regularization constant λ and a privacy parameter ϵ, and the output is a vector w∗in Rd. Our algorithm works as follows. Algorithm 2: 1. We pick a random vector b from the density function h(b) ∝e−ϵ 2 ||b||. To implement this, we pick the norm of b from the Γ(d, 2 ϵ ) distribution, and the direction of b uniformly at random. 2. Given examples x1, . . . , xn, with labels y1, . . . , yn and a regularization constant λ, we compute w∗= argminw 1 2λwT w + bT w n + 1 n Pn i=1 log(1 + e−yiwT xi). Output w∗. We observe that our method solves a convex programming problem very similar to the logistic regression convex program, and therefore it has running time similar to that of logistic regression. In the sequel, we show that the output of Algorithm 2 is privacy preserving. Theorem 2 Given a set of n examples x1, . . . , xn over Rd, with labels y1, . . . , yn, where for each i, ||xi|| ≤1, the output of Algorithm 2 preserves ϵ-differential privacy. PROOF: Let a and a′ be any two vectors over Rd with norm at most 1, and y, y′ ∈ {−1, 1}. For any such (a, y), (a′, y′), consider the inputs (x1, y1), . . . , (xn−1, yn−1), (a, y) and (x1, y1) . . . , (xn−1, yn−1), (a′, y′). Then, for any w∗output by our algorithm, there is a unique value of b that maps the input to the output. This uniqueness holds, because both the regularization function and the loss functions are differentiable everywhere. Let the values of b for the first and second cases respectively, be b1 and b2. Since w∗is the value that minimizes both the optimization problems, the derivative of both optimization functions at w∗is 0. This implies that for every b1 in the first case, there exists a b2 in the second case such that: b1 − ya 1+eyw∗T a = b2 − y′a′ 1+ey′w∗T a′ . Since ||a|| ≤1, ||a′|| ≤1, and 1 1+eyw∗T a ≤1, and 1 1+ey′w∗T a′ ≤1 4 for any w∗, ||b1 −b2|| ≤2. Using the triangle inequality, ||b1|| −2 ≤||b2|| ≤||b1|| + 2. Therefore, for any pair (a, y), (a′, y′), Pr[w∗|x1, . . . , xn−1, y1, . . . , yn−1, xn = a, yn = y] Pr[w∗|x1, . . . , xn−1, y1, . . . , yn−1, xn = a′, yn = y′] = h(b1) h(b2) = e−ϵ 2 (||b1||−||b2||) where h(bi) for i = 1, 2 is the density of bi. Since −2 ≤||b1|| −||b2|| ≤2, this ratio is at most eϵ. theorem follows. □ We notice that the privacy guarantee for our algorithm does not depend on λ; in other words, for any value of λ, our algorithm is private. On the other hand, as we show in Section 5, the performance of our algorithm does degrade with decreasing λ in the worst case, although the degradation is better than that of Algorithm 1 for λ < 1. Other Applications. Our algorithm for privacy-preserving logistic regression can be generalized to provide privacy-preserving outputs for more general convex optimization problems, so long as the problems satisfy certain conditions. These conditions can be formalized in the theorem below. Theorem 3 Let X = {x1, . . . , xn} be a database containing private data of individuals. Suppose we would like to compute a vector w that minimizes the function F(w) = G(w) + Pn i=1 l(w, xi), over w ∈Rd for some d, such that all of the following hold: 1. G(w) and l(w, xi) are differentiable everywhere, and have continuous derivatives 2. G(w) is strongly convex and l(w, xi) are convex for all i 3. ||∇wl(w, x)|| ≤κ, for any x. Let b = B · ˆb, where B is drawn from Γ(d, 2κ ϵ ), and ˆb is drawn uniformly from the surface of a ddimensional unit sphere. Then, computing w∗, where w∗= argminwG(w) + Pn i=1 l(w, xi) + bT w, provides ϵ-differential privacy. 5 Learning Guarantees In this section, we show theoretical bounds on the number of samples required by the algorithms to learn a linear classifier. For the rest of the section, we use the same notation used in Section 3. First we show that, for Algorithm 2, the values of ˆfλ(w2) and ˆfλ(w1) are close. Lemma 4 Given a logistic regression problem with regularization parameter λ, let w1 be the classifier that minimizes ˆfλ, and w2 be the classifier output by Algorithm 2 respectively. Then, with probability 1 −δ over the randomness in the privacy mechanism, ˆfλ(w2) ≤ˆfλ(w1) + 8d2 log2(d/δ) λn2ϵ2 . The proof is in the full version of our paper. As desired, for λ < 1, we have attained a tighter bound using Algorithm 2, than Lemma 3 for Algorithm 1. Now we give a performance guarantee for Algorithm 2. Theorem 4 Let w0 be a classifier with expected loss L over the data distribution. If the training examples are drawn independently from the data distribution, and if n > C max( ||w0||2 ϵ2 g , d log( d δ )||w0|| ϵgϵ ), for some constant C, then, with probability 1 −δ, the classifier output by Algorithm 2 has loss at most L + ϵg over the data distribution. PROOF: Let w∗be the classifier that minimizes fλ(w) over the data distribution, and let w1 and w2 be the classifiers that minimize ˆfλ(w) and ˆfλ(w) + bT w n over the data distribution respectively. We can use an analysis as in [15] to write that: L(w2) = L(w0) + (fλ(w2) −fλ(w∗)) + (fλ(w∗) −fλ(w0)) + λ 2 ||w0||2 −λ 2 ||w2||2 (1) 5 Notice that from Lemma 4, ˆfλ(w2) −ˆfλ(w1) ≤8d2 log2(d/δ) λn2ϵ2 . Using this and [16], we can bound the second quantity in equation 1 as fλ(w2) −fλ(w∗) ≤16d2 log2(d/δ) λn2ϵ2 + O( 1 λn). By definition of w∗, the third quantity in Equation 1 is non-positive. If λ is set to be ϵg ||w0||2 , then, the fourth quantity in Equation 1 is at most ϵg 2 . Now, if n > C · ||w0||2 ϵ2 g for a suitable constant C, 1 λn ≤ϵg 4 . In addition, if n > C · ||w0||d log( d δ ) ϵϵg , then, 16d2 log2( d δ ) λn2ϵ2 ≤ϵg 4 . In either case, the total loss of the classifier w2 output by Algorithm 2 is at most L(w0) + ϵg. □ The same technique can be used to analyze the sensitivity-based algorithm, using Lemma 3, which yields the following. Theorem 5 Let w0 be a classifier with expected loss L over the data distribution. If the training examples are drawn independently from the data distribution, and if n > C max( ||w0||2 ϵ2 g , d log( d δ )||w0|| ϵgϵ , d log( d δ )||w0||2 ϵ3/2 g ϵ ), for some constant C, then, with probability 1 −δ, the classifier output by Algorithm 2 has loss at most L + ϵg over the data distribution. It is clear that this bound is never lower than the bound for Algorithm 2. Note that for problems in which (non-private) logistic regression performs well, ∥w0∥≥1 if w0 has low loss, since otherwise one can show that the loss of w0 would be lower bounded by log(1 + 1 e). Thus the performance guarantee for Algorithm 2 is significantly stronger than for Algorithm 1, for problems in which one would typically apply logistic regression. 6 Results in Simulation Uniform, margin=0.03 Unseparable (uniform with noise 0.2 in margin 0.1) Sensitivity method 0.2962±0.0617 0.3257±0.0536 New method 0.1426±0.1284 0.1903±0.1105 Standard LR 0±0.0016 0.0530±0.1105 Figure 1: Test error: mean ± standard deviation over five folds. N=17,500. We include some simulations that compare the two privacy-preserving methods, and demonstrate that using our privacy-preserving approach to logistic regression does not degrade learning performance terribly, from that of standard logistic regression. Performance degradation is inevitable however, as in both cases, in order to address privacy concerns, we are adding noise, either to the learned classifier, or to the objective. In order to obtain a clean comparison between the various logistic regression variants, we first experimented with artificial data that is separable through the origin. Because the classification of a vector by a linear separator through the origin depends only its angle, not its norm, we sampled the data from the unit hypersphere. We used a uniform distribution on the hypersphere in 10 dimensions with zero mass within a small margin (0.03) from the generating linear separator. Then we experimented on uniform data that is not linearly separable. We sampled data from the surface of the unit ball in 10 dimensions, and labeled it with a classifier through the origin. In the band of margin ≤0.1 with respect to the perfect classifier, we performed random label flipping with probability 0.2. For our experiments, we used convex optimization software provided by [9]. Figure 1 gives mean and standard deviation of test error over five-fold cross-validation, on 17,500 points. In both simulations, our new method is superior to the sensitivity method, although incurs more error than standard (non-private) logistic regression. For both problems, we tuned the logistic regression parameter, λ, to minimize the test error of standard logistic regression, using five-fold cross-validation on a holdout set of 10,000 points (the tuned values are: λ = 0.01 in both cases). For each test error computation, the performance of each of the privacy-preserving algorithms was evaluated by averaging over 200 random restarts, since they are both randomized algorithms. In Figure 2a)-b) we provide learning curves. We graph the test error after each increment of 1000 points, averaged over five-fold cross validation. The learning curves reveal that, not only does the 6 2 4 6 8 10 12 14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 N/1000. Learning curve for uniform data. d=10, epsilon=0.1, margin=0.03, lambda=0.01 Avg test error over 5−fold cross−valid. 200 random restarts. Our method Standard LR Sensitivity method 2 4 6 8 10 12 14 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 N/1000. Learning curve for unseparable data. d=10, epsilon=0.1, lambda=0.01 Avg test error over 5−fold cross−valid. 200 random restarts. Our method Standard LR Sensitivity method 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Epsilon. Uniform data, d=10, n=10,000, margin=0.03, lambda=0.01 Avg test error over 5!fold cross!valid. 200 random restarts. Our method Sensitivity method 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Epsilon. Unseparable data, d=10, n=10,000, lambda=0.01 Avg test error over 5−fold cross−valid. 200 random restarts. Our method Sensitivity method Figure 2: Learning curves: a) Uniform distribution, margin=0.03, b) Unseparable data. Epsilon curves: c) Uniform distribution, margin=0.03, d) Unseparable data. new method reach a lower final error than the sensitivity method, but it also has better performance at most smaller training set sizes. In order to observe the effect of the level of privacy on the learning performance of the privacypreserving learning algorithms, in Figure 2c)-d) we vary ϵ, the privacy parameter to the two algorithms, on both the uniform, low margin data, and the unseparable data. As per the definition of ϵ-differential privacy in Section 2, strengthening the privacy guarantee corresponds to reducing ϵ. Both algorithms’ learning performance degrades in this direction. For the majority of values of ϵ that we tested, the new method is superior in managing the tradeoff between privacy and learning performance. For very small ϵ, corresponding to extremely stringent privacy requirements, the sensitivity method performs better but also has a predication accuracy close to chance, which is not useful for machine learning purposes. 7 Conclusion In conclusion, we show two ways to construct a privacy-preserving linear classifier through logistic regression. The first one uses the methods of [6], and the second one is a new algorithm. Using the ϵ-differential privacy definition of Dwork et al. [6], we prove that our new algorithm is privacy-preserving. We provide learning performance guarantees for the two algorithms, which are tighter for our new algorithm, in cases in which one would typically apply logistic regression. In simulations, our new algorithm outperforms the method based on [6]. Our work reveals an interesting connection between regularization and privacy: the larger the regularization constant, the less sensitive the logistic regression function is to any one individual example, and as a result, the less noise one needs to add to make it privacy-preserving. Therefore, regularization not only prevents overfitting, but also helps with privacy, by making the classifier less 7 sensitive. An interesting future direction would be to explore whether other methods that prevent overfitting also have such properties. Other future directions would be to apply our techniques to other commonly used machine-learning algorithms, and to explore whether our techniques can be applied to more general optimization problems. Theorem 3 shows that our method can be applied to a class of optimization problems with certain restrictions. An open question would be to remove some of these restrictions. Acknowledgements. We thank Sanjoy Dasgupta and Daniel Hsu for several pointers. References [1] R. Agrawal and R. Srikant. Privacy-preserving data mining. SIGMOD Rec., 29(2):439–450, 2000. [2] B. Barak, K. Chaudhuri, C. Dwork, S. Kale, F. McSherry, and K. Talwar. Privacy, accuracy, and consistency too: a holistic solution to contingency table release. In PODS, pages 273–282, 2007. [3] A. Blum, K. Ligett, and A. Roth. A learning theory approach to non-interactive database privacy. In R. E. Ladner and C. Dwork, editors, STOC, pages 609–618. ACM, 2008. [4] K. Chaudhuri and N. Mishra. When random sampling preserves privacy. In C. Dwork, editor, CRYPTO, volume 4117 of Lecture Notes in Computer Science, pages 198–213. Springer, 2006. [5] C. Dwork. Differential privacy. In M. Bugliesi, B. Preneel, V. Sassone, and I. Wegener, editors, ICALP (2), volume 4052 of Lecture Notes in Computer Science, pages 1–12. Springer, 2006. [6] C. Dwork, F. McSherry, K. Nissim, and A. Smith. Calibrating noise to sensitivity in private data analysis. In Theory of Cryptography Conference, pages 265–284, 2006. [7] A. Evfimievski, J. Gehrke, and R. Srikant. Limiting privacy breaches in privacy preserving data mining. In PODS, pages 211–222, 2003. [8] S. P. Kasiviswanathan, H. K. Lee, K. Nissim, S. Raskhodnikova, and A. Smith. What can we learn privately? In Proc. of Foundations of Computer Science, 2008. [9] C. T. Kelley. Iterative Methods for Optimization. SIAM, 1999. [10] A. Machanavajjhala, J. Gehrke, D. Kifer, and M. Venkitasubramaniam. l-diversity: Privacy beyond kanonymity. In ICDE, page 24, 2006. [11] F. McSherry and K. Talwar. Mechanism design via differential privacy. In FOCS, pages 94–103, 2007. [12] A. Narayanan and V. Shmatikov. Robust de-anonymization of large sparse datasets. In IEEE Symposium on Security and Privacy, pages 111–125. IEEE Computer Society, 2008. [13] K. Nissim, S. Raskhodnikova, and A. Smith. Smooth sensitivity and sampling in private data analysis. In D. S. Johnson and U. Feige, editors, STOC, pages 75–84. ACM, 2007. [14] P. Samarati and L. Sweeney. Protecting privacy when disclosing information: k-anonymity and its enforcement through generalization and suppression. In Proc. of the IEEE Symposium on Research in Security and Privacy, 1998. [15] S. Shalev-Shwartz and N. Srebro. Svm optimization: Inverse dependence on training set size. In International Conference on Machine Learning(ICML), 2008. [16] K. Sridharan, N. Srebro, and S. Shalev-Schwartz. Fast rates for regularized objectives. In Neural Information Processing Systems, 2008. 8
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Efficient Exact Inference in Planar Ising Models Nicol N. Schraudolph Dmitry Kamenetsky nips@schraudolph.org dkamen@cecs.anu.edu.au National ICT Australia, Locked Bag 8001, Canberra ACT 2601, Australia & RSISE, Australian National University, Canberra ACT 0200, Australia Abstract We give polynomial-time algorithms for the exact computation of lowest-energy states, worst margin violators, partition functions, and marginals in certain binary undirected graphical models. Our approach provides an interesting alternative to the well-known graph cut paradigm in that it does not impose any submodularity constraints; instead we require planarity to establish a correspondence with perfect matchings in an expanded dual graph. Maximum-margin parameter estimation for a boundary detection task shows our approach to be efficient and effective. A C++ implementation is available from http://nic.schraudolph.org/isinf/. 1 Introduction Undirected graphical models are a popular tool in machine learning; they represent real-valued energy functions of the form E′(y) := X i∈V E′ i(yi) + X (i,j)∈E E′ ij(yi, yj) , (1) where the terms in the first sum range over the nodes V = {1, 2, . . . n}, and those in the second sum over the edges E ⊆V × V of an undirected graph G(V, E). The junction tree decomposition provides an efficient framework for exact statistical inference in graphs that are (or can be turned into) trees of small cliques. The resulting algorithms, however, are exponential in the clique size, i.e., the treewidth of the original graph. This is prohibitively large for many graphs of practical interest — for instance, it grows as O(n) for an n × n square lattice. Many approximate inference techniques have been developed so as to deal with such graphs, such as pseudo-likelihood, mean field approximation, loopy belief propagation, and tree reweighting. 1.1 The Ising Model Efficient exact inference is possible in certain graphical models with binary node labels. Here we focus on Ising models, whose energy functions have the form E : {0, 1}n →R with E(y) := X (i,j)∈E [yi ̸= yj] Eij, (2) where [·] denotes the indicator function, i.e., the cost Eij is incurred only in those states y where yi and yj disagree. Compared to the general model (1) for binary nodes, (2) imposes two additional restrictions: zero node energies, and edge energies in the form of disagreement costs. At first glance these constraints look severe; for instance, such systems must obey the symmetry E(y) = E(¬ y), where ¬ denotes Boolean negation (ones’ complement). It is well known, however, that adding a single node makes the Ising model (2) as expressive as the general model (1) for binary variables: Theorem 1 Every energy function of the form (1) over n binary variables is equivalent to an Ising energy function of the form (2) over n + 1 variables, with the additional variable held constant. (a) k i k 0 E′ i(1)−E′ i(0) (b) k i k 0 k j @ @ @ @ @ E′ ij(1,0)−E′ ij(0,0) −Eij E′ ij(0,1)−E′ ij(0,0) −Eij Eij := 1 2 [(E′ ij(0,1) + E′ ij(1,0)) −(E′ ij(0,0) + E′ ij(1,1))] (c) k i k 0 k j k 1 E0i Eij −E0j (d) k i k 0 k j k 1 Eij +E0i 2Eij Eij −E0j Figure 1: Equivalent Ising model (with disagreement costs) for a given (a) node energy E′ i, (b) edge energy E′ ij in a binary graphical model; (c) equivalent submodular model if Eij > 0 and E0i > 0 but E0j < 0; (d) equivalent directed model of Kolmogorov and Zabih [1], Fig. 2d. Proof by construction: Two energy functions are equivalent if they differ only by a constant. Without loss of generality, denote the additional variable y0 and hold it constant at y0 := 0. Given an energy function of the form (1), construct an Ising model with disagreement costs as follows: 1. For each node energy function E′ i(yi), add a disagreement cost term E0i := E′ i(1)−E′ i(0); 2. For each edge energy function E′ ij(yi, yj), add the three disagreement cost terms Eij := 1 2[(E′ ij(0, 1) + E′ ij(1, 0)) −(E′ ij(0, 0) + E′ ij(1, 1))], E0i := E′ ij(1, 0) −E′ ij(0, 0) −Eij, and (3) E0j := E′ ij(0, 1) −E′ ij(0, 0) −Eij. Summing the above terms, the total bias of node i (i.e., its disagreement cost with the bias node) is E0i = E′ i(1) −E′ i(0) + X j:(i,j)∈E [E′ ij(1, 0) −E′ ij(0, 0) −Eij] . (4) This construction defines an Ising model whose energy in every configuration y is shifted, relative to that of the general model we started with, by the same constant amount, namely E′(0): ∀y ∈{0, 1}n : E h 0y i = E′(y) − X i∈V E′ i(0) − X (i,j)∈E E′ ij(0, 0) = E′(y) −E′(0). (5) The two models’ energy functions are therefore equivalent. Note how in the above construction the label symmetry E(y) = E(¬ y) of the plain Ising model (2) is conveniently broken by the introduction of a bias node, through the convention that y0 := 0. 1.2 Energy Minimization via Graph Cuts Definition 2 The cut C of a binary graphical model G(V, E) induced by state y ∈{0, 1}n is the set C(y) := {(i, j) ∈E : yi ̸= yj}; its weight |C(y)| is the sum of the weights of its edges. Any given state y partitions the nodes of a binary graphical model into two sets: those labeled ‘0’, and those labeled ‘1’. The corresponding graph cut is the set of edges crossing the partition; since only they contribute disagreement costs to the Ising model (2), we have ∀y : |C(y)| = E(y). The lowest-energy state of an Ising model therefore induces its minimum-weight cut. Minimum-weight cuts can be computed in polynomial time in graphs whose edge weights are all non-negative. Introducing one more node, with the constraint yn+1 := 1, allows us to construct an equivalent energy function by replacing each negatively weighted bias edge E0i < 0 by an edge to the new node n + 1 with the positive weight Ei,n+1 := −E0i > 0 (Figure 1c). This still leaves us with the requirement that all non-bias edges be non-negative. This submodularity constraint implies that agreement between nodes must be locally preferable to disagreement — a severe limitation. Graph cuts have been widely used in machine learning to find lowest-energy configurations, in particular in image processing. Our construction (Figure 1c) differs from that of Kolmogorov and Zabih [1] (Figure 1d) in that we do not employ the notion of directed edges. (In directed graphs, the weight of a cut is the sum of the weights of only those edges crossing the cut in a given direction.) (a) 1 2 3 4 5 (b) 2 1 3 4 5 (c) 1 2 3 4 5 (d) 1 2 3 4 5 Figure 2: (a) a non-plane drawing of a planar graph; (b) a plane drawing of the same graph; (c) a different plane drawing of same graph, with the same planar embedding as (b); (d) a plane drawing of the same graph with a different planar embedding. 2 Planarity Unlike graph cut methods, the inference algorithms we describe below do not depend on submodularity; instead they require that the model graph be planar, and that a planar embedding be provided. 2.1 Embedding Planar Graphs Definition 3 Let G(V, E) be an undirected, connected graph. For each vertex i ∈V, let Ei denote the set of edges in E incident upon i, considered as being oriented away from i, and let πi be a cyclic permutation of Ei. A rotation system for G is a set of permutations Π = {πi : i ∈V}. Rotation systems [2] directly correspond to topological graph embeddings in orientable surfaces: Theorem 4 (White and Beineke [2], p. 22f) Each rotation system determines an embedding of G in some orientable surface S such that ∀i ∈V, any edge (i, j) ∈Ei is followed by πi(i, j) in (say) clockwise orientation, and such that the faces F of the embedding, given by the orbits of the mapping (i, j) →πj(j, i), are 2-cells (topological disks). Note that while in graph visualisation “embedding” is often used as a synonym for “drawing”, in modern topological graph theory it stands for “rotation system”. We adopt the latter usage, which views embeddings as equivalence classes of graph drawings characterized by identical cyclic ordering of the edges incident upon each vertex. For instance, π4(4, 5) = (4, 3) in Figures 2b and 2c (same embedding) but π4(4, 5) = (4, 1) in Figure 2d (different embedding). A sample face in Figures 2b–2d is given by the orbit (4, 1) →π1(1, 4)=(1, 2) →π2(2, 1)=(2, 4) →π4(4, 2)=(4, 1). The genus g of the embedding surface S can be determined from the Euler characteristic |V| −|E| + |F| = 2 −2g, (6) where |F| is found by counting the orbits of the rotation system, as described in Theorem 4. Since planar graphs are exactly those that can be embedded on a surface of genus g = 0 (a topological sphere), we arrive at a purely combinatorial definition of planarity: Definition 5 A graph G(V, E) is planar iff it has a rotation system Π producing exactly 2+|E|−|V| orbits. Such a system is called a planar embedding of G, and G(V, E, Π) is called a plane graph. Our inference algorithms require a plane graph as input. In certain domains (e.g., when working with geographic information) a plane drawing of the graph (from which the corresponding embedding is readily determined) may be available. Where it is not, we employ the algorithm of Boyer and Myrvold [3] which, given any connected graph G as input, produces in linear time either a planar embedding for G or a proof that G is non-planar. Source code for this step is freely available [3, 4]. 2.2 The Planarity Constraint In Section 1.1 we have mapped a general binary graphical model to an Ising model with an additional bias node; now we require that that Ising model be planar. What does that imply for the original, general model? If all nodes of the graph are to be connected to the bias node without violating planarity, the graph has to be outerplanar, i.e., have a planar embedding in which all its nodes lie on the external face — a very severe restriction. Figure 3: Possible cuts (bold blue dashes) of a square face of the model graph (dashed) and complementary perfect matchings (bold red lines) of its expanded dual (solid lines). The situation improves, however, if only a subset B ⊂V of nodes have non-zero bias (4): Then the graph only has to be B-outerplanar, i.e., have a planar embedding in which all nodes in B lie on the same face. In image processing, for instance, where it is common to operate on a square grid of pixels, we can permit bias for all nodes on the perimeter of the grid. In general, a planar embedding which maximizes a weighted sum over the nodes bordering a given face can be found in linear time [5]; by setting node weights to some measure of bias (such as E 2 0i ) we can efficiently obtain the planar Ising model closest (in that measure) to any given planar binary graphical model. In contrast to submodularity, B-outerplanarity is a structural constraint. This has the advantage that once a model obeying the constraint is selected, inference (e.g., parameter estimation) can proceed via unconstrained methods (e.g., optimization). Finally, we note that all our algorithms can be extended to work for non-planar graphs as well. They then take time exponential in the genus of the embedding though still polynomial in the size of the graph; for graphs of low genus this may well be preferable to current approximative methods. 3 Computing Optimal States via Maximum-Weight Perfect Matching The relationship between the states of a planar Ising model and perfect matchings (“dimer coverings” to physicists) was first discovered by Kasteleyn [6] and Fisher [7]. Globerson and Jaakkola [8] presented a more direct construction for triangulated graphs, which we generalize here. 3.1 The Expanded Dual Graph Definition 6 The dual G∗(F, E) of an embedded graph G(V, E, Π) has a vertex for each face of G, with edges connecting vertices corresponding to faces that are adjacent (i.e., share an edge) in G. Each edge of the dual crosses exactly one edge of the original graph; due to this one-to-one relationship we will consider the dual to have the same set of edges E (with the same energies) as the original. We now expand the dual graph by replacing each node with a q-clique, where q is the degree of the node, as shown in Figure 3 for q = 4. The additional edges internal to each q-clique are given zero energy so as to leave the model unaffected. For large q the introduction of these O(q2) internal edges slows down subsequent computations (solid line in Figure 4, left); this can be avoided by subdividing the offending q-gonal face with chords (which are also given zero energy) before constructing the dual. Our implementation performs best when “octangulating” the graph, i.e., splitting octagons off all faces with q > 13; this is more efficient than a full triangulation (Figure 4, left). 3.2 Complementary Perfect Matchings Definition 7 A perfect matching of a graph G(V, E) is a subset M ⊆E of edges wherein exactly one edge is incident upon each vertex in V; its weight |M| is the sum of the weights of its edges. Theorem 8 For every cut C of an embedded graph G(V, E, Π) there exists at least one (if G is triangulated: exactly one) perfect matching M of its expanded dual complementary to C, i.e., E\M = C. Proof sketch Consider the complement E\C of the cut as a partial matching of the expanded dual. By definition, C intersects any cycle of G, and therefore also the perimeters of G’s faces F, in an even number of edges. In each clique of the expanded dual, C’s complement thus leaves an even number of nodes unmatched; M can therefore be completed using only edges interior to the cliques. In a 3-clique, there is only one way to do this, so M is unique if G is triangulated. In other words, there exists a surjection from perfect matchings in the expanded dual of G to cuts in G. Furthermore, since we have given edges interior to the cliques of the expanded dual zero energy, every perfect matching M complementary to a cut C of our Ising model (2) obeys the relation |M| + |C| = X (i,j)∈E Eij = const. (7) This means that instead of a minimum-weight cut in a graph we can look for a maximum-weight perfect matching in its expanded dual. But will that matching always be complementary to a cut? Theorem 9 Every perfect matching M of the expanded dual of a plane graph G(V, E, Π) is complementary to a cut C of G, i.e., E\M = C. Proof sketch In each clique of the expanded dual, an even number of nodes is matched by edges interior to the clique. The complement E\M of the matching in G thus contains an even number of edges around the perimeter of each face of the embedding. By induction over faces, this holds for every contractible (on the embedding surface) cycle of G. Because a plane is simply connected, all cycles in a plane graph are contractible; thus E\M is a cut. This is where planarity matters: Surfaces of non-zero genus are not simply connected, and thus non-plane graphs may contain non-contractible cycles; our construction does not guarantee that the complement E\M of a perfect matching of the expanded dual contains an even number of edges along such cycles. For planar graphs, however, the above theorems allow us to leverage known polynomial-time algorithms for perfect matchings into inference methods for Ising models. 3.3 The Lowest-Energy (MAP or Ground) State The blossom-shrinking algorithm [9, 10] is a sophisticated method to efficiently compute the maximum-weight perfect matching of a graph. It can be implemented to run in as little as O(|E| |V| log |V|) time. Although the Blossom IV code we are using [11] is asymptotically less efficient — O(|E| |V|2) — we have found it to be very fast in practice (Figure 4, left). We can now efficiently compute the lowest-energy state of a planar Ising model as follows: Find a planar embedding of the model graph (Section 2.1), construct its expanded dual (Section 3.1), and run the blossom-shrinking algorithm on that to compute its maximum-weight perfect matching. Its complement in the original model is the minimum-weight graph cut (Section 3.2). We can identify the state which induces this cut via a depth-first graph traversal that labels nodes as it encounters them, starting by labeling the bias node y0 := 0; this is shown below as Algorithm 1. Algorithm 1 Find State from Corresponding Graph Cut Input: Ising model graph G(V, E) procedure dfs state(i ∈{0, 1, 2, . . . n}, s ∈{0, 1}) graph cut C(y) ⊆E if yi = unknown then 1. ∀i ∈{0, 1, 2, . . . n} : yi := s; ∀(i, j) ∈Ei : yi := unknown; if (i, j) ∈C then dfs state(j, ¬s); 2. dfs state(0, 0); else dfs state(j, s); Output: state vector y else assert yi = s; 3.4 The Worst Margin Violator Maximum-margin parameter estimation in graphical models involves determining the worst margin violator — the state that minimizes, relative to a given target state y∗, the margin energy M(y|y∗) := E(y) −d(y|y∗), (8) where d(·|·) is a measure of divergence in state space. If d(·|·) is the weighted Hamming distance d(y|y∗) := X (i,j)∈E [[yi ̸= yj] ̸= [y∗ i ̸= y∗ j ]] vij, (9) 10 100 1e3 1e4 1e5 1 0.1 0.01 1e-3 1e-4 CPU time (seconds) original triang. octang. 10 100 1000 0.001 0.01 0.1 1 10 CPU time (seconds) no prefact. prefactored 10 100 1000 1e3 1e6 1e9 memory (bytes) full-size K prefact. H Figure 4: Cost of inference on a ring graph, plotted against ring size. Left & center: CPU time on Apple MacBook with 2.2 GHz Intel Core2 Duo processor; right: storage size. Left: MAP state via Blossom IV [11] on original, triangulated, and octangulated ring; center & right: marginal probabilities, full matrix K (double precision, no prefactoring) vs. prefactored half-Kasteleyn bitmatrix H. where the vij ≥0 are constant weighting factors (in the simplest case: all ones) on the edges of our Ising model, then it is easily verified that the margin energy (8) is implemented (up to a shift that depends only on y∗) by an isomorphic Ising model with disagreement costs Eij + (2 [y∗ i ̸= y∗ j ] −1) vij. (10) We can thus use our algorithm of Section 3.3 to efficiently find the worst margin violator, argminy M(y|y∗), for maximum-margin parameter estimation. 4 Computing the Partition Function and Marginal Probabilities1 A Markov random field (MRF) over our Ising model (2) models the distribution P(y) = 1 Z e−E(y), where Z := X y e−E(y) (11) is the MRF’s partition function. As it involves a summation over exponentially many states y, calculating the partition function is generally intractable. For planar graphs, however, the generating function for perfect matchings can be calculated in polynomial time via the determinant of a skewsymmetric matrix [6, 7]. Due to the close relationship with graph cuts (Section 3.2) we can calculate Z in (11) likewise. Elaborating on work of Globerson and Jaakkola [8], we first convert the Ising model graph into a Boolean “half-Kasteleyn” matrix H: 1. plane triangulate the embedded graph so as to make the relationship between cuts and complementary perfect matchings a bijection (cf. Section 3.2); 2. orient the edges of the graph such that the in-degree of every node is odd; 3. construct the Boolean half-Kasteleyn matrix H from the oriented graph; 4. prefactor the triangulation edges (added in Step 1) out of H. Our Step 2 simplifies equivalent operations in previous constructions [6–8], Step 3 differs in that it only sets unit (i.e., +1) entries in a Boolean matrix, and Step 4 can dramatically reduce the size of H for compact storage (as a bit matrix) and faster subsequent computations (Figure 4, center & right). For a given set of disagreement edge costs Ek, k = {1, 2, , . . . |E|} on that graph, we then build from H and the Ek the skew-symmetric, real-valued Kasteleyn matrix K: 1. K := H; 2. ∀k ∈{1, 2, , . . . |E|} : K2k−1,2k := K2k−1,2k + eEk; 3. K := K −K⊤. The partition function for perfect matchings is p |K| [6–8], so we factor K and use (7) to compute the log partition function for (11) as ln Z = 1 2 ln |K| −P k∈E Ek. Its derivative yields the marginal probability of disagreement on the kth edge, and is computed via the inverse of K: P(k ∈C) := −∂ln Z ∂Ek = 1 − 1 2|K| ∂|K| ∂Ek = 1 −1 2 tr K−1 ∂K ∂Ek = 1 + K−1 2k−1,2k K2k−1,2k. (12) 1 We only have space for a high-level overview here; see [12] for full details. Figure 5: Boundary detection by maximum-margin training of planar Ising grids; from left to right: Ising model (100 × 100 grid), original image, noisy mask, and MAP segmentation of the Ising grid. 5 Maximum Likelihood vs. Maximum Margin CRF Parameter Estimation Our algorithms can be applied to regularized parameter estimation in conditional random fields (CRFs). In a linear planar Ising CRF, the disagreement costs Eij in (2) are computed as inner products between features (sufficient statistics) x of the modeled data and corresponding parameters θ of the model, and (11) is used to model the conditional distribution P(y|x, θ). Maximum likelihood (ML) parameter estimation then seeks to minimize wrt. θ the L2-regularized negative log likelihood LML(θ) := 1 2 λ∥θ∥2 −ln P(y∗|x, θ) = 1 2 λ∥θ∥2 + E(y∗|x, θ) + ln Z(θ|x) (13) of a given target labeling y∗,2 with regularization parameter λ. This is a smooth, convex objective that can be optimized via batch or online implementations of gradient methods such as LBFGS [13]; the gradient of the log partition function in (13) is obtained by computing the marginals (12). For maximum margin (MM) parameter estimation [14] we instead minimize LMM(θ) := 1 2 λ∥θ∥2 + E(y∗|x, θ) −min y M(y|y∗, x, θ) (14) = 1 2 λ∥θ∥2 + E(y∗|x, θ) −E(ˆy|x, θ) + d(ˆy|y∗), where ˆy := argminy M(y|y∗, x, θ) is the worst margin violator, i.e., the state that minimizes the margin energy (8). LMM(θ) is convex but non-smooth; we can minimize it via bundle methods such as the BT bundle trust algorithm [15], making use of the convenient lower bound ∀θ: LMM(θ) ≥0. To demonstrate the scalability of planar Ising models, we designed a simple boundary detection task based on images from the GrabCut Ground Truth image segmentation database [16]. We took 100 × 100 pixel subregions of images that depicted a segmentation boundary, and corrupted the segmentation mask with pink noise, produced by convolving a white noise image (all pixels i.i.d. uniformly random) with a Gaussian density with one pixel standard deviation. We then employed a planar Ising model to recover the original boundary — namely, a 100 × 100 square grid with one additional edge pegged to a high energy, encoding prior knowledge that two opposing corners of the grid depict different regions (Figure 5, left). The energy of the other edges was Eij := ⟨[1, |xi −xj|], θ⟩, where xi is the pixel intensity at node i. We did not employ a bias node for this task, and simply set λ = 1. Note that this is a huge model: 10 000 nodes and 19 801 edges. Computing the partition function or marginals would require inverting a Kasteleyn matrix with over 1.5 · 109 entries; minimizing (13) is therefore computationally infeasible for us. Computing a ground state via the algorithm described in Section 3, by contrast, takes only 0.3 seconds on an Apple MacBook with 2.2 GHz Intel Core2 Duo processor. We can therefore efficiently minimize (14) to obtain the MM parameter vector θ∗, then compute the CRF’s MAP (i.e., ground) state for rapid prediction. Figure 5 (right) shows how even for a signal-to-noise (S/N) ratio of 1:8, our approach is capable of recovering the original segmentation boundary quite well, with only 0.67% of nodes mislabeled here. For S/N ratios of 1:9 and lower the system was unable to locate the boundary; for S/N ratios of 1:7 and higher we obtained perfect reconstruction. Further experiments are reported in [12]. On smaller grids, ML parameter estimation and marginals for prediction become computationally feasible, if slower than the MM/MAP approach. This will allow direct comparison of ML vs. MM for parameter estimation, and MAP vs. marginals for prediction, to our knowledge for the first time on graphs intractable for the junction tree appproach, such as the grids often used in image processing. 2 For notational clarity we suppress here the fact that we are usually modeling a collection of data items. 6 Discussion We have proposed an alternative algorithmic framework for efficient exact inference in binary graphical models, which replaces the submodularity constraint of graph cut methods with a planarity constraint. Besides proving efficient and effective in first experiments, our approach opens up a number of interesting research directions to be explored: Our algorithms can all be extended to nonplanar graphs, at a cost exponential in the genus of the embedding. We are currently developing these extensions, which may prove of great practical value for graphs that are “almost” planar; examples include road networks (where edge crossings arise from overpasses without on-ramps) and graphs describing the tertiary structure of proteins [17]. These algorithms also provide a foundation for the future development of efficient approximate inference methods for nonplanar Ising models. Our method for calculating the ground state (Section 3) actually works for nonplanar graphs whose ground state does not contain frustrated non-contractible cycles. The QPBO graph cut method [18] finds ground states that do not contain any frustrated cycles, and otherwise yields a partial labeling. Can we likewise obtain a partial labeling of ground states with frustrated non-contractible cycles? The existence of two distinct tractable frameworks for inference in binary graphical models implies a yet more powerful hybrid: Consider a graph each of whose biconnected components is either planar or submodular. As a whole, this graph may be neither planar nor submodular, yet efficient exact inference in it is clearly possible by applying the appropriate framework to each component. Can this hybrid approach be extended to cover less obvious situations? References [1] V. Kolmogorov and R. Zabih. What energy functions can be minimized via graph cuts? IEEE Trans. Pattern Analysis and Machine Intelligence, 26(2):147–159, 2004. [2] A. T. White and L. W. Beineke. Topological graph theory. In L. W. Beineke and R. J. Wilson, editors, Selected Topics in Graph Theory, chapter 2, pages 15–49. Academic Press, 1978. [3] J. M. Boyer and W. J. Myrvold. On the cutting edge: Simplified O(n) planarity by edge addition. Journal of Graph Algorithms and Applications, 8(3):241–273, 2004. Reference implementation (C source code): http://jgaa.info/accepted/2004/BoyerMyrvold2004.8.3/planarity.zip [4] A. Windsor. Planar graph functions for the boost graph library. C++ source code, boost file vault: http: //boost-consulting.com/vault/index.php?directory=Algorithms/graph, 2007. [5] C. Gutwenger and P. Mutzel. Graph embedding with minimum depth and maximum external face. In G. Liotta, editor, Graph Drawing 2003, volume 2912 of LNCS, pages 259–272. Springer Verlag, 2004. [6] P. W. Kasteleyn. The statistics of dimers on a lattice: I. the number of dimer arrangements on a quadratic lattice. Physica, 27(12):1209–1225, 1961. [7] M. E. Fisher. Statistical mechanics of dimers on a plane lattice. Phys Rev, 124(6):1664–1672, 1961. [8] A. Globerson and T. Jaakkola. Approximate inference using planar graph decomposition. In B. Sch¨olkopf, J. Platt, and T. Hofmann (eds), Advances in Neural Information Processing Systems 19, 2007. MIT Press. [9] J. Edmonds. Maximum matching and a polyhedron with 0,1-vertices. Journal of Research of the National Bureau of Standards, 69B:125–130, 1965. [10] J. Edmonds. Paths, trees, and flowers. Canadian Journal of Mathematics, 17:449–467, 1965. [11] W. Cook and A. Rohe. Computing minimum-weight perfect matchings. INFORMS Journal on Computing, 11(2):138–148, 1999. C source code: http://www.isye.gatech.edu/˜wcook/blossom4 [12] N. N. Schraudolph and D. Kamenetsky. Efficient exact inference in planar Ising models. Technical Report 0810.4401, arXiv, 2008. http://aps.arxiv.org/abs/0810.4401 [13] S. V. N. Vishwanathan, N. N. Schraudolph, M. Schmidt, and K. Murphy. Accelerated training conditional random fields with stochastic gradient methods. In Proc. Intl. Conf. Machine Learning, pages 969–976, New York, NY, USA, 2006. ACM Press. [14] B. Taskar, C. Guestrin, and D. Koller. Max-margin Markov networks. In S. Thrun, L. Saul, and B. Sch¨olkopf (eds), Advances in Neural Information Processing Systems 16, pages 25–32, 2004. MIT Press. [15] H. Schramm and J. Zowe. A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results. SIAM J. Optimization, 2:121–152, 1992. [16] C. Rother, V. Kolmogorov, A. Blake, and M. Brown. GrabCut ground truth database, 2007. http:// research.microsoft.com/vision/cambridge/i3l/segmentation/GrabCut.htm [17] S. V. N. Vishwanathan, K. Borgwardt, and N. N. Schraudolph. Fast computation of graph kernels. In B. Sch¨olkopf, J. Platt, and T. Hofmann (eds), Advances in Neural Information Processing Systems 19, 2007. [18] V. Kolmogorov and C. Rother. Minimizing nonsubmodular functions with graph cuts – a review. IEEE Trans. Pattern Analysis and Machine Intelligence, 29(7):1274–1279, 2007.
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Deflation Methods for Sparse PCA Lester Mackey Computer Science Division University of California, Berkeley Berkeley, CA 94703 Abstract In analogy to the PCA setting, the sparse PCA problem is often solved by iteratively alternating between two subtasks: cardinality-constrained rank-one variance maximization and matrix deflation. While the former has received a great deal of attention in the literature, the latter is seldom analyzed and is typically borrowed without justification from the PCA context. In this work, we demonstrate that the standard PCA deflation procedure is seldom appropriate for the sparse PCA setting. To rectify the situation, we first develop several deflation alternatives better suited to the cardinality-constrained context. We then reformulate the sparse PCA optimization problem to explicitly reflect the maximum additional variance objective on each round. The result is a generalized deflation procedure that typically outperforms more standard techniques on real-world datasets. 1 Introduction Principal component analysis (PCA) is a popular change of variables technique used in data compression, predictive modeling, and visualization. The goal of PCA is to extract several principal components, linear combinations of input variables that together best account for the variance in a data set. Often, PCA is formulated as an eigenvalue decomposition problem: each eigenvector of the sample covariance matrix of a data set corresponds to the loadings or coefficients of a principal component. A common approach to solving this partial eigenvalue decomposition is to iteratively alternate between two subproblems: rank-one variance maximization and matrix deflation. The first subproblem involves finding the maximum-variance loadings vector for a given sample covariance matrix or, equivalently, finding the leading eigenvector of the matrix. The second involves modifying the covariance matrix to eliminate the influence of that eigenvector. A primary drawback of PCA is its lack of sparsity. Each principal component is a linear combination of all variables, and the loadings are typically non-zero. Sparsity is desirable as it often leads to more interpretable results, reduced computation time, and improved generalization. Sparse PCA [8, 3, 16, 17, 6, 18, 1, 2, 9, 10, 12] injects sparsity into the PCA process by searching for “pseudoeigenvectors”, sparse loadings that explain a maximal amount variance in the data. In analogy to the PCA setting, many authors attempt to solve the sparse PCA problem by iteratively alternating between two subtasks: cardinality-constrained rank-one variance maximization and matrix deflation. The former is an NP-hard problem, and a variety of relaxations and approximate solutions have been developed in the literature [1, 2, 9, 10, 12, 16, 17]. The latter subtask has received relatively little attention and is typically borrowed without justification from the PCA context. In this work, we demonstrate that the standard PCA deflation procedure is seldom appropriate for the sparse PCA setting. To rectify the situation, we first develop several heuristic deflation alternatives with more desirable properties. We then reformulate the sparse PCA optimization problem to explicitly reflect the maximum additional variance objective on each round. The result is a generalized deflation procedure that typically outperforms more standard techniques on real-world datasets. 1 The remainder of the paper is organized as follows. In Section 2 we discuss matrix deflation as it relates to PCA and sparse PCA. We examine the failings of typical PCA deflation in the sparse setting and develop several alternative deflation procedures. In Section 3, we present a reformulation of the standard iterative sparse PCA optimization problem and derive a generalized deflation procedure to solve the reformulation. Finally, in Section 4, we demonstrate the utility of our newly derived deflation techniques on real-world datasets. Notation I is the identity matrix. Sp + is the set of all symmetric, positive semidefinite matrices in Rp×p. Card(x) represents the cardinality of or number of non-zero entries in the vector x. 2 Deflation methods A matrix deflation modifies a matrix to eliminate the influence of a given eigenvector, typically by setting the associated eigenvalue to zero (see [14] for a more detailed discussion). We will first discuss deflation in the context of PCA and then consider its extension to sparse PCA. 2.1 Hotelling’s deflation and PCA In the PCA setting, the goal is to extract the r leading eigenvectors of the sample covariance matrix, A0 ∈Sp +, as its eigenvectors are equivalent to the loadings of the first r principal components. Hotelling’s deflation method [11] is a simple and popular technique for sequentially extracting these eigenvectors. On the t-th iteration of the deflation method, we first extract the leading eigenvector of At−1, xt = argmax x:xT x=1 xT At−1x (1) and we then use Hotelling’s deflation to annihilate xt: At = At−1 −xtxT t At−1xtxT t . (2) The deflation step ensures that the t + 1-st leading eigenvector of A0 is the leading eigenvector of At. The following proposition explains why. Proposition 2.1. If λ1 ≥. . . ≥λp are the eigenvalues of A ∈Sp +, x1, . . . , xp are the corresponding eigenvectors, and ˆA = A −xjxT j AxjxT j for some j ∈1, . . . , p, then ˆA has eigenvectors x1, . . . , xp with corresponding eigenvalues λ1, . . . , λj−1, 0, λj+1, . . . , λp. PROOF. ˆAxj = Axj −xjxT j AxjxT j xj = Axj −xjxT j Axj = λjxj −λjxj = 0xj. ˆAxi = Axi −xjxT j AxjxT j xi = Axi −0 = λixi, ∀i ̸= j. Thus, Hotelling’s deflation preserves all eigenvectors of a matrix and annihilates a selected eigenvalue while maintaining all others. Notably, this implies that Hotelling’s deflation preserves positivesemidefiniteness. In the case of our iterative deflation method, annihilating the t-th leading eigenvector of A0 renders the t + 1-st leading eigenvector dominant in the next round. 2.2 Hotelling’s deflation and sparse PCA In the sparse PCA setting, we seek r sparse loadings which together capture the maximum amount of variance in the data. Most authors [1, 9, 16, 12] adopt the additional constraint that the loadings be produced in a sequential fashion. To find the first such ”pseudo-eigenvector”, we can consider a cardinality-constrained version of Eq. (1): x1 = argmax x:xT x=1,Card(x)≤k1 xT A0x. (3) 2 That leaves us with the question of how to best extract subsequent pseudo-eigenvectors. A common approach in the literature [1, 9, 16, 12] is to borrow the iterative deflation method of the PCA setting. Typically, Hotelling’s deflation is utilized by substituting an extracted pseudo-eigenvector for a true eigenvector in the deflation step of Eq. (2). This substitution, however, is seldom justified, for the properties of Hotelling’s deflation, discussed in Section 2.1, depend crucially on the use of a true eigenvector. To see what can go wrong when Hotelling’s deflation is applied to a non-eigenvector, consider the following example. Example. Let C = 2 1 1 1 , a 2 × 2 matrix. The eigenvalues of C are λ1 = 2.6180 and λ2 = .3820. Let x = (1, 0)T , a sparse pseudo-eigenvector, and ˆC = C −xxT CxxT , the corresponding deflated matrix. Then ˆC = 0 1 1 1 with eigenvalues ˆλ1 = 1.6180 and ˆλ2 = −.6180. Thus, Hotelling’s deflation does not in general preserve positive-semidefiniteness when applied to a noneigenvector. That Sp + is not closed under pseudo-eigenvector Hotelling’s deflation is a serious failing, for most iterative sparse PCA methods assume a positive-semidefinite matrix on each iteration. A second, related shortcoming of pseudo-eigenvector Hotelling’s deflation is its failure to render a pseudoeigenvector orthogonal to a deflated matrix. If A is our matrix of interest, x is our pseudo-eigenvector with variance λ = xT Ax, and ˆA = A −xxT AxxT is our deflated matrix, then ˆAx = Ax − xxT AxxT x = Ax −λx is zero iff x is a true eigenvector. Thus, even though the “variance” of x w.r.t. ˆA is zero (xT ˆAx = xT Ax −xT xxT AxxT x = λ −λ = 0), “covariances” of the form yT ˆAx for y ̸= x may still be non-zero. This violation of the Cauchy-Schwarz inequality betrays a lack of positive-semidefiniteness and may encourage the reappearance of x as a component of future pseudo-eigenvectors. 2.3 Alternative deflation techniques In this section, we will attempt to rectify the failings of pseudo-eigenvector Hotelling’s deflation by considering several alternative deflation techniques better suited to the sparse PCA setting. Note that any deflation-based sparse PCA method (e.g. [1, 9, 16, 12]) can utilize any of the deflation techniques discussed below. 2.3.1 Projection deflation Given a data matrix Y ∈Rn×p and an arbitrary unit vector in x ∈Rp, an intuitive way to remove the contribution of x from Y is to project Y onto the orthocomplement of the space spanned by x: ˆY = Y (I −xxT ). If A is the sample covariance matrix of Y , then the sample covariance of ˆY is given by ˆA = (I −xxT )A(I −xxT ), which leads to our formulation for projection deflation: Projection deflation At = At−1 −xtxT t At−1 −At−1xtxT t + xtxT t At−1xtxT t = (I −xtxT t )At−1(I −xtxT t ) (4) Note that when xt is a true eigenvector of At−1 with eigenvalue λt, projection deflation reduces to Hotelling’s deflation: At = At−1 −xtxT t At−1 −At−1xtxT t + xtxT t At−1xtxT t = At−1 −λtxtxT t −λtxtxT t + λtxtxT t = At−1 −xtxT t At−1xtxT t . However, in the general case, when xt is not a true eigenvector, projection deflation maintains the desirable properties that were lost to Hotelling’s deflation. For example, positive-semidefiniteness is preserved: ∀y, yT Aty = yT (I −xtxT t )At−1(I −xtxT t )y = zT At−1z where z = (I −xtxT t )y. Thus, if At−1 ∈Sp +, so is At. Moreover, At is rendered left and right orthogonal to xt, as (I −xtxT t )xt = xt−xt = 0 and At is symmetric. Projection deflation therefore annihilates all covariances with xt: ∀v, vT Atxt = xT t Atv = 0. 3 2.3.2 Schur complement deflation Since our goal in matrix deflation is to eliminate the influence, as measured through variance and covariances, of a newly discovered pseudo-eigenvector, it is reasonable to consider the conditional variance of our data variables given a pseudo-principal component. While this conditional variance is non-trivial to compute in general, it takes on a simple closed form when the variables are normally distributed. Let x ∈Rp be a unit vector and W ∈Rp be a Gaussian random vector, representing the joint distribution of the data variables. If W has covariance matrix Σ, then (W, Wx) has covariance matrix V = Σ Σx xT Σ xT Σx , and V ar(W|Wx) = Σ −ΣxxT Σ xT Σx whenever xT Σx ̸= 0 [15]. That is, the conditional variance is the Schur complement of the vector variance xT Σx in the full covariance matrix V . By substituting sample covariance matrices for their population counterparts, we arrive at a new deflation technique: Schur complement deflation At = At−1 −At−1xtxT t At−1 xT t At−1xt (5) Schur complement deflation, like projection deflation, preserves positive-semidefiniteness. To see this, suppose At−1 ∈Sp +. Then, ∀v, vT Atv = vT At−1v −vT At−1xtxT t At−1v xT t At−1xt ≥0 as vT At−1vxT t At−1xt −(vT At−1xt)2 ≥0 by the Cauchy-Schwarz inequality and xT t At−1xt ≥0 as At−1 ∈Sp +. Furthermore, Schur complement deflation renders xt left and right orthogonal to At, since At is symmetric and Atxt = At−1xt −At−1xtxT t At−1xt xT t At−1xt = At−1xt −At−1xt = 0. Additionally, Schur complement deflation reduces to Hotelling’s deflation when xt is an eigenvector of At−1 with eigenvalue λt ̸= 0: At = At−1 −At−1xtxT t At−1 xT t At−1xt = At−1 −λtxtxT t λt λt = At−1 −xtxT t At−1xtxT t . While we motivated Schur complement deflation with a Gaussianity assumption, the technique admits a more general interpretation as a column projection of a data matrix. Suppose Y ∈Rn×p is a mean-centered data matrix, x ∈Rp has unit norm, and ˆY = (I −Y xxT Y T ||Y x||2 )Y , the projection of the columns of Y onto the orthocomplement of the space spanned by the pseudo-principal component, Y x. If Y has sample covariance matrix A, then the sample covariance of ˆY is given by ˆA = 1 nY T (I −Y xxT Y T ||Y x||2 )T (I −Y xxT Y T ||Y x||2 )Y = 1 nY T (I −Y xxT Y T ||Y x||2 )Y = A −AxxT A xT Ax . 2.3.3 Orthogonalized deflation While projection deflation and Schur complement deflation address the concerns raised by performing a single deflation in the non-eigenvector setting, new difficulties arise when we attempt to sequentially deflate a matrix with respect to a series of non-orthogonal pseudo-eigenvectors. Whenever we deal with a sequence of non-orthogonal vectors, we must take care to distinguish between the variance explained by a vector and the additional variance explained, given all previous vectors. These concepts are equivalent in the PCA setting, as true eigenvectors of a matrix are orthogonal, but, in general, the vectors extracted by sparse PCA will not be orthogonal. The additional variance explained by the t-th pseudo-eigenvector, xt, is equivalent to the variance explained by the component of xt orthogonal to the space spanned by all previous pseudo-eigenvectors, qt = xt −Pt−1xt, where Pt−1 is the orthogonal projection onto the space spanned by x1, . . . , xt−1. On each deflation step, therefore, we only want to eliminate the variance associated with qt. Annihilating the full vector xt will often lead to “double counting” and could re-introduce components parallel to previously annihilated vectors. Consider the following example: 4 Example. Let C0 = I. If we apply projection deflation w.r.t. x1 = ( √ 2 2 , √ 2 2 )T , the result is C1 = 1 2 −1 2 −1 2 1 2 , and x1 is orthogonal to C1. If we next apply projection deflation to C1 w.r.t. x2 = (1, 0)T , the result, C2 = 0 0 0 1 2 , is no longer orthogonal to x1. The authors of [12] consider this issue of non-orthogonality in the context of Hotelling’s deflation. Their modified deflation procedure is equivalent to Hotelling’s deflation (Eq. (2)) for t = 1 and can be easily expressed in terms of a running Gram-Schmidt decomposition for t > 1: Orthogonalized Hotelling’s deflation (OHD) qt = (I −Qt−1QT t−1)xt (I −Qt−1QT t−1)xt (6) At = At−1 −qtqT t At−1qtqT t where q1 = x1, and q1, . . . , qt−1 form the columns of Qt−1. Since q1, . . . , qt−1 form an orthonormal basis for the space spanned by x1, . . . , xt−1, we have that Qt−1QT t−1 = Pt−1, the aforementioned orthogonal projection. Since the first round of OHD is equivalent to a standard application of Hotelling’s deflation, OHD inherits all of the weaknesses discussed in Section 2.2. However, the same principles may be applied to projection deflation to generate an orthogonalized variant that inherits its desirable properties. Schur complement deflation is unique in that it preserves orthogonality in all subsequent rounds. That is, if a vector v is orthogonal to At−1 for any t, then Atv = At−1v −At−1xtxT t At−1v xT t At−1xt = 0 as At−1v = 0. This further implies the following proposition. Proposition 2.2. Orthogonalized Schur complement deflation is equivalent to Schur complement deflation. Proof. Consider the t-th round of Schur complement deflation. We may write xt = ot + pt, where pt is in the subspace spanned by all previously extracted pseudo-eigenvectors and ot is orthogonal to this subspace. Then we know that At−1pt = 0, as pt is a linear combination of x1, . . . , xt−1, and At−1xi = 0, ∀i < t. Thus, xT t Atxt = pT t Atpt + oT t Atpt + pT t Atot + oT t Atot = oT t Atot. Further, At−1xtxT t At−1 = At−1ptpT t At−1+At−1ptoT t At−1+At−1otpT t At−1+At−1otoT t At−1 = At−1otoT t At−1. Hence, At = At−1 −At−1otoT t At−1 oT t At−1ot = At−1 −At−1qtqT t At−1 qT t At−1qt as qt = ot ||ot||. Table 1 compares the properties of the various deflation techniques studied in this section. Method xT t Atxt = 0 Atxt = 0 At ∈Sp + Asxt = 0, ∀s > t Hotelling’s ✓ × × × Projection ✓ ✓ ✓ × Schur complement ✓ ✓ ✓ ✓ Orth. Hotelling’s ✓ × × × Orth. Projection ✓ ✓ ✓ ✓ Table 1: Summary of sparse PCA deflation method properties 3 Reformulating sparse PCA In the previous section, we focused on heuristic deflation techniques that allowed us to reuse the cardinality-constrained optimization problem of Eq. (3). In this section, we explore a more principled alternative: reformulating the sparse PCA optimization problem to explicitly reflect our maximization objective on each round. Recall that the goal of sparse PCA is to find r cardinality-constrained pseudo-eigenvectors which together explain the most variance in the data. If we additionally constrain the sparse loadings to 5 be generated sequentially, as in the PCA setting and the previous section, then a greedy approach of maximizing the additional variance of each new vector naturally suggests itself. On round t, the additional variance of a vector x is given by qT A0q qT q where A0 is the data covariance matrix, q = (I −Pt−1)x, and Pt−1 is the projection onto the space spanned by previous pseudo-eigenvectors x1, . . . , xt−1. As qT q = xT (I −Pt−1)(I −Pt−1)x = xT (I −Pt−1)x, maximizing additional variance is equivalent to solving a cardinality-constrained maximum generalized eigenvalue problem, max x xT (I −Pt−1)A0(I −Pt−1)x subject to xT (I −Pt−1)x = 1 Card(x) ≤kt. (7) If we let qs = (I −Ps−1)xs, ∀s ≤t −1, then q1, . . . , qt−1 form an orthonormal basis for the space spanned by x1, . . . , xt−1. Writing I −Pt−1 = I −Pt−1 s=1 qsqT s = Qt−1 s=1 (I −qsqT s ) suggests a generalized deflation technique that leads to the solution of Eq. (7) on each round. We imbed the technique into the following algorithm for sparse PCA: Algorithm 1 Generalized Deflation Method for Sparse PCA Given: A0 ∈Sp +, r ∈N, {k1, . . . , kr} ⊂N Execute: 1. B0 ←I 2. For t := 1, . . . , r • xt ← argmax x:xT Bt−1x=1,Card(x)≤kt xT At−1x • qt ←Bt−1xt • At ←(I −qtqT t )At−1(I −qtqT t ) • Bt ←Bt−1(I −qtqT t ) • xt ←xt/ ||xt|| Return: {x1, . . . , xr} Adding a cardinality constraint to a maximum eigenvalue problem renders the optimization problem NP-hard [10], but any of several leading sparse eigenvalue methods, including GSLDA of [10], DCPCA of [12], and DSPCA of [1] (with a modified trace constraint), can be adapted to solve this cardinality-constrained generalized eigenvalue problem. 4 Experiments In this section, we present several experiments on real world datasets to demonstrate the value added by our newly derived deflation techniques. We run our experiments with Matlab implementations of DCPCA [12] (with the continuity correction of [9]) and GSLDA [10], fitted with each of the following deflation techniques: Hotelling’s (HD), projection (PD), Schur complement (SCD), orthogonalized Hotelling’s (OHD), orthogonalized projection (OPD), and generalized (GD). 4.1 Pit props dataset The pit props dataset [5] with 13 variables and 180 observations has become a de facto standard for benchmarking sparse PCA methods. To demonstrate the disparate behavior of differing deflation methods, we utilize each sparse PCA algorithm and deflation technique to successively extract six sparse loadings, each constrained to have cardinality less than or equal to kt = 4. We report the additional variances explained by each sparse vector in Table 2 and the cumulative percentage variance explained on each iteration in Table 3. For reference, the first 6 true principal components of the pit props dataset capture 87% of the variance. 6 DCPCA GSLDA HD PD SCD OHD OPD GD HD PD SCD OHD OPD GD 2.938 2.938 2.938 2.938 2.938 2.938 2.938 2.938 2.938 2.938 2.938 2.938 2.209 2.209 2.076 2.209 2.209 2.209 2.107 2.280 2.065 2.107 2.280 2.280 0.935 1.464 1.926 0.935 1.464 1.477 1.988 2.067 2.243 1.985 2.067 2.072 1.301 1.464 1.164 0.799 1.464 1.464 1.352 1.304 1.120 1.335 1.305 1.360 1.206 1.057 1.477 0.901 1.058 1.178 1.067 1.120 1.164 0.497 1.125 1.127 0.959 0.980 0.725 0.431 0.904 0.988 0.557 0.853 0.841 0.489 0.852 0.908 Table 2: Additional variance explained by each of the first 6 sparse loadings extracted from the Pit Props dataset. On the DCPCA run, Hotelling’s deflation explains 73.4% of the variance, while the best performing methods, Schur complement deflation and generalized deflation, explain approximately 79% of the variance each. Projection deflation and its orthogonalized variant also outperform Hotelling’s deflation, while orthogonalized Hotelling’s shows the worst performance with only 63.2% of the variance explained. Similar results are obtained when the discrete method of GSLDA is used. Generalized deflation and the two projection deflations dominate, with GD achieving the maximum cumulative variance explained on each round. In contrast, the more standard Hotelling’s and orthogonalized Hotelling’s underperform the remaining techniques. DCPCA GSLDA HD PD SCD OHD OPD GD HD PD SCD OHD OPD GD 22.6% 22.6% 22.6% 22.6% 22.6% 22.6% 22.6% 22.6% 22.6% 22.6% 22.6% 22.6% 39.6% 39.6% 38.6% 39.6% 39.6% 39.6% 38.8% 40.1% 38.5% 38.8% 40.1% 40.1% 46.8% 50.9% 53.4% 46.8% 50.9% 51.0% 54.1% 56.0% 55.7% 54.1% 56.0% 56.1% 56.8% 62.1% 62.3% 52.9% 62.1% 62.2% 64.5% 66.1% 64.4% 64.3% 66.1% 66.5% 66.1% 70.2% 73.7% 59.9% 70.2% 71.3% 72.7% 74.7% 73.3% 68.2% 74.7% 75.2% 73.4% 77.8% 79.3% 63.2% 77.2% 78.9% 77.0% 81.2% 79.8% 71.9% 81.3% 82.2% Table 3: Cumulative percentage variance explained by the first 6 sparse loadings extracted from the Pit Props dataset. 4.2 Gene expression data The Berkeley Drosophila Transcription Network Project (BDTNP) 3D gene expression data [4] contains gene expression levels measured in each nucleus of developing Drosophila embryos and averaged across many embryos and developmental stages. Here, we analyze 03 1160524183713 s10436-29ap05-02.vpc, an aggregate VirtualEmbryo containing 21 genes and 5759 example nuclei. We run GSLDA for eight iterations with cardinality pattern 9,7,6,5,3,2,2,2 and report the results in Table 4. GSLDA additional variance explained GSLDA cumulative percentage variance HD PD SCD OHD OPD GD HD PD SCD OHD OPD GD PC 1 1.784 1.784 1.784 1.784 1.784 1.784 21.0% 21.0% 21.0% 21.0% 21.0% 21.0% PC 2 1.464 1.453 1.453 1.464 1.453 1.466 38.2% 38.1% 38.1% 38.2% 38.1% 38.2% PC 3 1.178 1.178 1.179 1.176 1.178 1.187 52.1% 51.9% 52.0% 52.0% 51.9% 52.2% PC 4 0.716 0.736 0.716 0.713 0.721 0.743 60.5% 60.6% 60.4% 60.4% 60.4% 61.0% PC 5 0.444 0.574 0.571 0.460 0.571 0.616 65.7% 67.4% 67.1% 65.9% 67.1% 68.2% PC 6 0.303 0.306 0.278 0.354 0.244 0.332 69.3% 71.0% 70.4% 70.0% 70.0% 72.1% PC 7 0.271 0.256 0.262 0.239 0.313 0.304 72.5% 74.0% 73.4% 72.8% 73.7% 75.7% PC 8 0.223 0.239 0.299 0.257 0.245 0.329 75.1% 76.8% 77.0% 75.9% 76.6% 79.6% Table 4: Additional variance and cumulative percentage variance explained by the first 8 sparse loadings of GSLDA on the BDTNP VirtualEmbryo. The results of the gene expression experiment show a clear hierarchy among the deflation methods. The generalized deflation technique performs best, achieving the largest additional variance on every round and a final cumulative variance of 79.6%. Schur complement deflation, projection deflation, and orthogonalized projection deflation all perform comparably, explaining roughly 77% of the total variance after 8 rounds. In last place are the standard Hotelling’s and orthogonalized Hotelling’s deflations, both of which explain less than 76% of variance after 8 rounds. 7 5 Conclusion In this work, we have exposed the theoretical and empirical shortcomings of Hotelling’s deflation in the sparse PCA setting and developed several alternative methods more suitable for non-eigenvector deflation. Notably, the utility of these procedures is not limited to the sparse PCA setting. Indeed, the methods presented can be applied to any of a number of constrained eigendecomposition-based problems, including sparse canonical correlation analysis [13] and linear discriminant analysis [10]. Acknowledgments This work was supported by AT&T through the AT&T Labs Fellowship Program. References [1] A. d’Aspremont, L. El Ghaoui, M. I. Jordan, and G. R. G. Lanckriet. A Direct Formulation for Sparse PCA using Semidefinite Programming. In Advances in Neural Information Processing Systems (NIPS). Vancouver, BC, December 2004. [2] A. d’Aspremont, F. R. Bach, and L. E. Ghaoui. Full regularization path for sparse principal component analysis. In Proceedings of the 24th international Conference on Machine Learning. Z. Ghahramani, Ed. ICML ’07, vol. 227. ACM, New York, NY, 177-184, 2007. [3] J. Cadima and I. Jolliffe. Loadings and correlations in the interpretation of principal components. Applied Statistics, 22:203.214, 1995. [4] C.C. Fowlkes, C.L. Luengo Hendriks, S.V. Kernen, G.H. Weber, O. Rbel, M.-Y. Huang, S. Chatoor, A.H. DePace, L. Simirenko and C. Henriquez et al. Cell 133, pp. 364-374, 2008. [5] J. Jeffers. Two case studies in the application of principal components. Applied Statistics, 16, 225-236, 1967. [6] I.T. Jolliffe and M. Uddin. A Modified Principal Component Technique based on the Lasso. Journal of Computational and Graphical Statistics, 12:531.547, 2003. [7] I.T. Jolliffe, Principal component analysis, Springer Verlag, New York, 1986. [8] I.T. Jolliffe. Rotation of principal components: choice of normalization constraints. Journal of Applied Statistics, 22:29-35, 1995. [9] B. Moghaddam, Y. Weiss, and S. Avidan. Spectral bounds for sparse PCA: Exact and greedy algorithms. Advances in Neural Information Processing Systems, 18, 2006. [10] B. Moghaddam, Y. Weiss, and S. Avidan. Generalized spectral bounds for sparse LDA. In Proc. ICML, 2006. [11] Y. Saad, Projection and deflation methods for partial pole assignment in linear state feedback, IEEE Trans. Automat. Contr., vol. 33, pp. 290-297, Mar. 1998. [12] B.K. Sriperumbudur, D.A. Torres, and G.R.G. Lanckriet. Sparse eigen methods by DC programming. Proceedings of the 24th International Conference on Machine learning, pp. 831838, 2007. [13] D. Torres, B.K. Sriperumbudur, and G. Lanckriet. Finding Musically Meaningful Words by Sparse CCA. Neural Information Processing Systems (NIPS) Workshop on Music, the Brain and Cognition, 2007. [14] P. White. The Computation of Eigenvalues and Eigenvectors of a Matrix. Journal of the Society for Industrial and Applied Mathematics, Vol. 6, No. 4, pp. 393-437, Dec., 1958. [15] F. Zhang (Ed.). The Schur Complement and Its Applications. Kluwer, Dordrecht, Springer, 2005. [16] Z. Zhang, H. Zha, and H. Simon, Low-rank approximations with sparse factors I: Basic algorithms and error analysis. SIAM J. Matrix Anal. Appl., 23 (2002), pp. 706-727. [17] Z. Zhang, H. Zha, and H. Simon, Low-rank approximations with sparse factors II: Penalized methods with discrete Newton-like iterations. SIAM J. Matrix Anal. Appl., 25 (2004), pp. 901-920. [18] H. Zou, T. Hastie, and R. Tibshirani. Sparse Principal Component Analysis. Technical Report, Statistics Department, Stanford University, 2004. 8
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Linear Classification and Selective Sampling Under Low Noise Conditions Giovanni Cavallanti DSI, Universit`a degli Studi di Milano, Italy cavallanti@dsi.unimi.it Nicol`o Cesa-Bianchi DSI, Universit`a degli Studi di Milano, Italy cesa-bianchi@dsi.unimi.it Claudio Gentile DICOM, Universit`a dell’Insubria, Italy claudio.gentile@uninsubria.it Abstract We provide a new analysis of an efficient margin-based algorithm for selective sampling in classification problems. Using the so-called Tsybakov low noise condition to parametrize the instance distribution, we show bounds on the convergence rate to the Bayes risk of both the fully supervised and the selective sampling versions of the basic algorithm. Our analysis reveals that, excluding logarithmic factors, the average risk of the selective sampler converges to the Bayes risk at rate N −(1+α)(2+α)/2(3+α) where N denotes the number of queried labels, and α > 0 is the exponent in the low noise condition. For all α > √ 3 −1 ≈0.73 this convergence rate is asymptotically faster than the rate N −(1+α)/(2+α) achieved by the fully supervised version of the same classifier, which queries all labels, and for α →∞the two rates exhibit an exponential gap. Experiments on textual data reveal that simple variants of the proposed selective sampler perform much better than popular and similarly efficient competitors. 1 Introduction In the standard online learning protocol for binary classification the learner receives a sequence of instances generated by an unknown source. Each time a new instance is received the learner predicts its binary label, and is then given the true label of the current instance before the next instance is observed. This protocol is natural in many applications, for instance weather forecasting or stock market prediction, because Nature (or the market) is spontaneously disclosing the true label after each learner’s guess. On the other hand, in many other applications obtaining labels may be an expensive process. In order to address this problem, a variant of online learning that has been proposed is selective sampling. In this modified protocol the true label of the current instance is never revealed unless the learner decides to issue an explicit query. The learner’s performance is then measured with respect to both the number of mistakes (made on the entire sequence of instances) and the number of queries. A natural sampling strategy is one that tries to identify labels which are likely to be useful to the algorithm, and then queries those ones only. This strategy somehow needs to combine a measure of utility of examples with a measure of confidence. In the case of learning with linear functions, a statistic that has often been used to quantify both utility and confidence is the margin. In [10] this approach was employed to define a selective sampling rule that queries a new label whenever the margin of the current instance, with respect to the current linear hypothesis, is smaller (in magnitude) than an adaptively adjusted threshold. Margins were computed using a linear learning algorithm based on an incremental version of Regularized linear Least-Squares (RLS) for classification. Although this selective sampling algorithm is efficient, and has simple variants working quite well in practice, the rate of convergence to the Bayes risk was never assessed in terms of natural distributional parameters, thus preventing a full understanding of the properties of this algorithm. We improve on those results in several ways making three main contributions: (i) By coupling the Tsybakov low noise condition, used to parametrize the instance distribution, with the linear model of [10], defining the conditional distribution of labels, we prove that the fully supervised RLS (all labels are queried) converges to the Bayes risk at rate e O n−(1+α)/(2+α) where α ≥0 is the noise exponent in the low noise condition. (ii) Under the same low noise condition, we prove that the RLS-based selective sampling rule of [10] converges to the Bayes risk at rate e O n−(1+α)/(3+α) , with labels being queried at rate e O n−α/(2+α) . Moreover, we show that similar results can be established for a mistake-driven (i.e., space and time efficient) variant. (iii) We perform experiments on a real-world medium-size dataset showing that variants of our mistake-driven sampler compare favorably with other selective samplers proposed in the literature, like the ones in [11, 16, 20]. Related work. Selective sampling, originally introduced by Cohn, Atlas and Ladner in [13, 14], differs from the active learning framework as in the latter the learner has more freedom in selecting which instances to query. For example, in Angluin’s adversarial learning with queries (see [1] for a survey), the goal is to identify an unknown boolean function f from a given class, and the learner can query the labels (i.e., values of f) of arbitrary boolean instances. Castro and Nowak [9] study a framework in which the learner also queries arbitrary domain points. However, in their case labels are stochastically related to instances (which are real vectors). They prove risk bounds in terms of nonparametric characterizations of both the regularity of the Bayes decision boundary and the behavior of the noise rate in its proximity. In fact, a large statistical literature on adaptive sampling and sequential hypothesis testing exists (see for instance the detailed description in [9]) which is concerned with problems that share similarities with active learning. The idea of querying small margin instances when learning linear classifiers has been explored several times in different active learning contexts. Campbell, Cristianini and Smola [8], and also Tong and Koller [23], study a poolbased model of active learning, where the algorithm is allowed to interactively choose which labels to obtain from an i.i.d. pool of unlabeled instances. A landmark result in the selective sampling protocol is the query-by-committee algorithm of Freund, Seung, Shamir and Tishby [17]. In the realizable (noise-free) case, and under strong distributional assumptions, this algorithm is shown to require exponentially fewer labels than instances when learning linear classifiers (see also [18] for a more practical implementation). An exponential advantage in the realizable case is also obtained with a simple variant of the Perceptron algorithm by Dasgupta, Kalai and Monteleoni [16], under the sole assumption that instances are drawn from the uniform distribution over the unit ball in Rd. In the general statistical learning case, under no assumptions on the joint distribution of label and instances, selective sampling bears no such exponential advantage. For instance, K¨a¨ari¨ainen shows that, in order to approach the risk of the best linear classifier f ∗within error ε, at least Ω((η/ε)2) labels are needed, where η is the risk of f ∗. A much more general nonparametric lower bound for active learning is obtained by Castro and Nowak [9]. General selective sampling strategies for the nonrealizable case have been proposed in [3, 4, 15]. However, none of these learning algorithms seems to be computationally efficient when learning linear classifiers in the general agnostic case. 2 Learning protocol and data model We consider the following online selective sampling protocol. At each step t = 1, 2, . . . the sampling algorithm (or selective sampler) receives an instance xt ∈Rd and outputs a binary prediction for the associated label yt ∈{−1, +1}. After each prediction, the algorithm has the option of “sampling” (issuing a query) in order to receive the label yt. We call the pair (xt, yt) an example. After seeing the label yt, the algorithm can choose whether or not to update its internal state using the new information encoded by (xt, yt). We assume instances xt are realizations of i.i.d. random variables Xt drawn from an unknown distribution on the surface of the unit Euclidean sphere in Rd, so that ∥Xt∥= 1 for all t ≥1. Following [10], we assume that labels yt are generated according to the following simple linear noise model: there exists a fixed and unknown vector u ∈Rd, with Euclidean norm ∥u∥= 1, such that E Yt Xt = xt = u⊤xt for all t ≥1. Hence Xt = xt has label 1 with probability (1 + u⊤xt)/2 ∈[0, 1]. Note that SGN(f ∗), for f ∗(x) = u⊤x, is the Bayes optimal classifier for this noise model. In the following, all probabilities P and expectations E are understood with respect to the joint distribution of the i.i.d. data process {(X1, Y1), (X2, Y2), . . . }. We use Pt to denote conditioning on (X1, Y1), . . . , (Xt, Yt). Let f : Rd →R be an arbitrary measurable function. The instantaneous regret R(f) is the excess risk of SGN(f) w.r.t. the Bayes risk, i.e., R(f) = P(Y1 f(X1) < 0) −P(Y1 f ∗(X1) < 0). Let f1, f2, . . . be a sequence of real functions where each ft is measurable w.r.t. the σ-algebra generated by (X1, Y1), . . . , (Xt−1, Yt−1), Xt. When (X1, Y1), . . . , (Xt−1, Yt−1) is understood from the context, we write ft as a function of Xt only. Let Rt−1(ft) be the instantaneous conditional regret Rt−1(ft) = Pt−1(Yt ft(Xt) < 0) −Pt−1(Yt f ∗(Xt) < 0). Our goal is to bound the expected cumulative regret E R0(f1) + R1(f2) + · · · + Rn−1(fn) , as a function of n, and other relevant quantities. Observe that, although the learner’s predictions can only depend on the queried examples, the regret is computed over all time steps, including the ones when the selective sampler did not issue a query. In order to model the distribution of the instances around the hyperplane u⊤x = 0, we use Mammen-Tsybakov low noise condition [24]: There exist c > 0 and α ≥0 such that P |f ∗(X1)| < ε ≤c εα for all ε > 0. (1) When the noise exponent α is 0 the low noise condition becomes vacuous. In order to study the case α →∞, one can use the following equivalent formulation of (1) —see, e.g., [5], P f ∗(X1)f(X1) < 0 ≤c R(f)α/(1+α) for all measurable f : Rd →R. With this formulation, one can show that α →∞implies the hard margin condition |f ∗(X1)| ≥1/(2c) w.p. 1. 3 Algorithms and theoretical analysis We consider linear classifiers predicting the value of Yt through SGN(w⊤ t Xt), where wt ∈Rd is a dynamically updated weight vector which might be intended as the current estimate for u. Our wt is an RLS estimator defined over the set of previously queried examples. More precisely, let Nt be the number of queried examples during the first t time steps, St−1 = x′ 1, . . . , x′ Nt−1 be the matrix of the queried instances up to time t −1, and yt−1 = y′ 1, . . . , y′ Nt−1 ⊤be the vector of the corresponding labels. Then the RLS estimator is defined by wt = I + St−1 S⊤ t−1 + xtx⊤ t −1 St−1 yt−1 , (2) where I is the d × d identity matrix. Note that wt depends on the current instance xt. The RLS estimator in this particular form has been first considered by Vovk [25] and by Azoury and Warmuth [2]. Compared to standard RLS, here xt acts by futher reducing the variance of wt. We use b∆t to denote the margin w⊤ t Xt whenever wt is understood from the context. Thus b∆t is the current approximation to ∆t. Note that b∆t is measurable w.r.t. the σ-algebra generated by (X1, Y1), . . . , (Xt−1, Yt−1), Xt. We also use ∆t to denote the Bayes margin f ∗(Xt) = u⊤Xt. The RLS estimator (2) can be stored in space Θ(d2), which we need for the inverse of I + St−1 S⊤ t−1 + xtx⊤ t . Moreover, using a standard formula for small-rank adjustments of inverse matrices, we can compute updates and predictions in time Θ(d2). The algorithm in (2) can also be expressed in dual variable form. This is needed, for instance, when we want to use the feature expansion facility provided by kernel functions. In this case, at time t the RLS estimator (2) can be represented in O(N 2 t−1) space. The update time is also quadratic in Nt−1. Our first result establishes a regret bound for the fully supervised algorithm, i.e., the algorithm that predicts using RLS as in (2), queries the label of every instance, and stores all examples. This result is the baseline against which we measure the performance of our selective sampling algorithm. The regret bound is expressed i.t.o. the whole spectrum of the process covariance matrix E[X1X⊤ 1 ]. Theorem 1 Assume the low noise condition (1) holds with exponent α ≥0 and constant c > 0. Then the expected cumulative regret after n steps of the fully supervised algorithm based on (2) is bounded by E 4c(1 + ln |I + SnS⊤ n |) 1+α 2+α n 1 2+α . This, in turn, is bounded from above by 4c 1 + Pd i=1 ln(1 + nλi) 1+α 2+α n 1 2+α = O d ln n 1+α 2+α n 1 2+α . Here | · | denotes the determinant of a matrix, Sn = X1, X2, . . . , Xn , and λi is the i-th eigenvalue of E[X1X⊤ 1 ]. When α = 0 (corresponding to a vacuous noise condition) the bound of Theorem 1 reduces to O √ d n ln n . When α →∞(corresponding to a hard margin condition) the bound gives the logarithmic behavior O d ln n . Notice that Pd i=1 ln(1 + nλi) is substantially smaller than d ln n whenever the spectrum of E[X1X⊤ 1 ] is rapidly decreasing. In fact, the second bound is clearly meaningful even when d = ∞, while the third one only applies to the finite dimensional case. Parameters: λ > 0, ρt > 0 for each t ≥1. Initialization: weight vector w = (0, . . . , 0)⊤; storage counter N = 0. At each time t = 1, 2, . . . do the following: 1. Observe instance xt ∈Rd : ||xt|| = 1; 2. Predict the label yt ∈{−1, 1} with SGN(w⊤ t xt), where wt is as in (2). 3. If N ≤ρt then query label yt and store (xt, yt); 4. Else if b∆2 t ≤128 ln t λ N then schedule the query of yt+1; 5. If (xt, yt) is scheduled to be stored, then increment N and update wt using (xt+1, yt+1). Figure 1: The selective sampling algorithm. Fast rates of convergence have typically been proven for batch-style algorithms, such as empirical risk minimizers and SVM (see, e.g., [24, 22]), rather than for online algorithms. A reference closer to our paper is Ying and Zhou [26], where the authors prove bounds for online linear classification using the low noise condition (1), though under different distributional assumptions. Our second result establishes a new regret bound, under low noise conditions, for the selective sampler introduced in [10]. This variant, described in Figure 1, queries all labels (and stores all examples) during an initial stage of length at least (16d)/λ2, where λ denotes the smallest nonzero eigenvalue of the process covariance matrix E[X1X⊤ 1 ]. When this transient regime is over, the sampler issues a query at time t based on both the query counter Nt−1 and the margin b∆t. Specifically, if evidence is collected that the number Nt−1 of stored examples is smaller than our current estimate of 1/∆2 t, that is if b∆2 t ≤(128 ln t)/(λNt−1), then we query (and store) the label of the next instance xt+1. Note that the margin threshold explicitly depends, through λ, on additional information about the data-generating process. This additional information is needed because, unlike the fully supervised classifier of Theorem 1, the selective sampler queries labels at random steps. This prevents us from bounding the sum of conditional variances of the involved RLS estimator through ln I + Sn S⊤ n , as we can do when proving Theorem 1 (see below). Instead, we have to individually bound each conditional variance term via the smallest empirical eigenvalue of the correlation matrix. The transient regime in Figure 1 is exactly needed to ensure that this smallest empirical eigenvalue gets close enough to λ. Compared to the analysis contained in [10], we are able to better capture the two main aspects of the selective sampling protocol: First, we control the probability of making a mistake when we do not query labels; second, the algorithm is able to adaptively optimize the sampling rate by exploiting the additional information provided by the examples having small margin. The appropriate sampling rate clearly depends on the (unknown) amount of noise α which the algorithm implicitly learns on the fly. In this respect, our algorithm is more properly an adaptive sampler, rather than a selective sampler. Finally, we stress that it is fairly straightforward to add to the algorithm in Figure 1 a mistake-driven rule for storing examples. Such a rule provides that, when a small margin is detected, a query be issued (and the next example be stored) only if SGN(b∆t) ̸= yt (i.e., only if the current prediction is mistaken). This turns out to be highly advantageous from a computational standpoint, because of the sparsity of the computed solution. It is easy to adapt our analysis to obtain even for this algorithm the same regret bound as the one established in Theorem 2. However, in this case we can only give guarantees on the expected number of stored examples (which can indeed be much smaller than the actual number of queried labels). Theorem 2 Assume the low noise condition (1) holds with unknown exponent α ≥0 and assume the selective sampler of Figure 1 is run with ρt = 16 λ2 max{d, ln t}. Then, after n steps, the expected cumulative regret is bounded by O d + ln n λ2 + ln n λ 1+α 3+α n 2 3+α whereas the expected number of queried labels (including the stored ones) is bounded by O d + ln n λ2 + ln n λ α 2+α n 2 2+α . The proof, sketched below, hinges on showing that b∆t is an almost unbiased estimate of the true margin ∆t, and relies on known concentration properties of i.i.d. processes. In particular, we show that our selective sampler is able to adaptively estimate the number of queries needed to ensure a 1/t increase of the regret when a query is not issued at time t. As expected, when we compare our semi-supervised selective sampler (Theorem 2) to the fully supervised “yardstick” (Theorem 1), we see that the per-step regret of the former vanishes at a significantly slower rate than the latter, i.e., n−1+α 3+α vs. n−1+α 2+α . Note, however, that the per-step regret of the semi-supervised algorithm vanishes faster than its fully-supervised counterpart when both regrets are expressed in terms of the number N of issued queries. To see this consider first the case α →∞(the hard margin case, essentially analyzed in [10]). Then both algorithms have a per-step regret of order (ln n)/n. However, since the semi-supervised algorithm makes only N = O(ln n) queries, we have that, as a function of N, the per-step regret of the semi-supervised algorithm is of order N/eN where the fully supervised has only (ln N)/N. We have thus recovered the exponential advantage observed in previous works [16, 17]. When α = 0 (vacuous noise conditions), the per-step regret rates in terms of N become (excluding logarithmic factors) of order N −1/3 in the semi-supervised case and of order N −1/2 in the fully supervised case. Hence, there is a critical value of α where the semi-supervised bound becomes better. In order to find this critical value we write the rates of the per-step regret for 0 ≤α < ∞obtaining N −(1+α)(2+α) 2(3+α) (semi-supervised algorithm) and N −1+α 2+α (fully supervised algorithm). By comparing the two exponents we find that, asymptotically, the semi-supervised rate is better than the fully supervised one for all values of α > √ 3 −1. This indicates that selective sampling is advantageous when the noise level (as modeled by the MammenTsybakov condition) is not too high. Finally, observe that the way it is stated now, the bound of Theorem 2 only applies to the finite-dimensional (d < ∞) case. It turns out this is a fixable artifact of our analysis, rather than an intrinsic limitation of the selective sampling scheme in Figure 1. See Remark 3 below. Proof of Theorem 1. The proof proceeds by relating the classification regret to the square loss regret via a comparison theorem. The square loss regret is then controlled by applying a known pointwise bound. For all measurable f : Rd →R, let Rφ(f) = E[ 1−Y1 f(X1) 2 − 1−Y1 f ∗(X1)2 ] be the square loss regret, and Rt−1,φ its conditional version. We apply the comparison theorem from [5] with the ψ-transform function ψ(z) = z2 associated with the square loss. Under the low noise condition (1) this yields R(f) ≤ 4c Rφ(f) 1+α 2+α for all measurable f. We thus have E Pn t=1 Rt−1(ft) ≤E hPn t=1 4c Rφ,t−1(ft) 1+α 2+α i ≤E h n 4c n Pn t=1 Rφ,t−1(ft) 1+α 2+α i , the last term following from Jensen’s inequality. Further, we observe that in our probabilistic model f ∗(x) = u⊤x is Bayes optimal for the square loss. In fact, for any unit norm x ∈Rd, we have f ∗(x) = arginfz∈R (1 −z)2 1+u⊤x 2 + (1 + z)2 1−u⊤x 2 = u⊤x . Hence Pn t=1 Rφ,t−1(ft) = Pn t=1 (Yt −w⊤ t Xt)2 −(Yt −u⊤Xt)2 which, in turn, can be bounded pointwise (see, e.g., [12, Theorem 11.8]) by 1 + ln I + Sn S⊤ n . Putting together gives the first bound. Next, we take the bound just obtained and apply Jensen’s inequality twice, first to the concave function (·) 1+α 2+α of a real argument, and then to the concave function ln |·| of a (positive definite) matrix argument. Observing that ESnS⊤ n = E[Pn t=1 XtX⊤ t ] = n EX1X⊤ 1 yields the second bound. The third bound derives from the second one just by using λi ≤1. □ Proof sketch of Theorem 2. We aim at bounding from above the cumulative regret Pn t=1 P(Yt b∆t < 0) −P(Yt ∆t < 0) which, according to our probabilistic model, can be shown to be at most c n ε1+α + Pn t=1 P(∆t b∆t ≤0, |∆t| ≥ε) . The last sum is upper bounded by n X t=1 P (Nt−1 ≤ρt) | {z } (I) + n X t=1 P b∆2 t ≤128 ln t λNt−1 , Nt−1 > ρt, |∆t| ≥ε | {z } (II) + n X t=1 P ∆t b∆t ≤0, b∆2 t > 128 ln t λNt−1 , Nt−1 > ρt | {z } (III) . where: (I) are the initial time steps; (II) are the time steps on which we trigger the query of the next label (because b∆2 t is smaller than the threshold at time t); (III) are the steps that do not trigger any queries at all. Note that (III) bounds the regret over non-sampled examples. In what follows, we sketch the way we bound each of the three terms separately. A bound on (I) is easily obtained as (I) ≤ρn = O( d+ln n λ2 ) just because ρn ≥ρt for all t ≤n. To bound (II) and (III) we need to exploit the fact that the subsequence of stored instances and labels is a sequence of i.i.d. random variables distributed as (X1, Y1), see [10]. This allows us to carry out a (somewhat involved) bias-variance analysis showing that for any fixed number Nt−1 = s of stored examples, b∆t is an almost unbiased estimator of ∆t, whose bias and variance tend to vanish as 1/s when s is sufficiently large. In particular, if |∆t| ≥ε then b∆t ≈∆t as long as Nt−1 is of the order of ln n λ ε2 . The variance of b∆t is controlled by known results (the one we used is [21, Theorem 4.2]) on the concentration of eigenvalues of an empirical correlation matrix 1 s P i XiX⊤ i to the eigenvalues of the process covariance matrix E[X1X⊤ 1 ]. For such a result to apply, we have to impose that Nt−1 ≥ρt. By suitably combining these concentration results we can bound term (II) by O( d+ln n λ2 + ln n λε2 ) and term (III) by O(ln n). Putting together and choosing ε of the order of ln n λ n 1+α 3+α gives the desired regret bound. The bound on the number of queried labels is obtained in a similar way. □ Remark 3 The linear dependence on d in Theorem 2 derives from a direct application of the concentration results in [21]. In fact, it is possible to take into account in a fairly precise manner the way the process spectrum decreases (e.g., [6, 7]), thereby extending the above analysis to the infinite-dimensional case. In this paper, however, we decided to stick to the simpler analysis leading to Theorem 2, since the resulting bounds would be harder to read, and would somehow obscure understanding of regret and sampling rate behavior as a function of n. 4 Experimental analysis In evaluating the empirical performance of our selective sampling algorithm, we consider two additional variants obtained by slightly modifying Step 4 in Figure 1. The first variant (which we just call SS, Selective Sampler) queries the current label instead of the next one. The rationale here is that we want to leverage the more informative content of small margin instances. The second variant is a mistake-driven version (referred to as SSMD, Selective Sampling Mistake Driven) that queries the current label (and stores the corresponding example) only if the label gets mispredicted. For clarity, the algorithm in Figure 1 will then be called SSNL (Selective Sampling Next Label) since it queries the next label whenever a small margin is observed. For all three algorithms we dropped the intial transient regime (Step 3 in Figure 1). We run our experiments on the first, in chronological order, 40,000 newswire stories from the Reuters Corpus Volume 1 dataset (RCV1). Every example in this dataset is encoded as a vector of real attributes computed through a standard TF-IDF bag-of-words processing of the original news stories, and is tagged with zero or more labels from a set of 102 classes. The online categorization of excerpts from a newswire feed is a realistic learning problem for selective sampling algorithms since a newswire feed consists of a large amount of uncategorized data with a high labeling cost. The classification performance is measured using a macroaveraged F-measure 2RP/(R + P), where P is the precision (fraction of correctly classified documents among all documents that were classified positive for the given topic) and R is the recall (fraction of correctly classified documents among all documents that are labelled with the given topic). All algorithms presented here are evaluated using dual variable implementations and linear kernels. The results are summarized in Figures 2 and 3. The former only refers to (an average over) the 50 most frequent categories, while the latter includes them all. In Figure 2 (left) we show how SSMD compares to SSNL, and to its most immediate counterpart, SS. In Figure 2 (right) we compare SSMD to other algorithms that are known to have good empirical performance, including the second-order version of the label efficient classifier (SOLE), as described in [11], and the DKMPERC variant of the DKM algorithm (see, e.g., [16, 20]). DKMPERC differs from DKM since it adopts a standard perceptron update rule. The perceptron algorithm (PERC) and its second-order counterpart (SOP) are reported here as a reference, since they are designed to query all labels. In particular, SOP is a mistake-driven variant of the algorithm analyzed in Theorem 1. It is reasonable to assume that in a selective sampling setup we are interested in the performance achieved when the fraction of queried labels stays below some threshold, say 10%. In this range of sampling rate, SSMD has the steepest increase in the achieved F-measure, and surpasses any other algorithm. Unsurprisingly, as the number of queried labels gets larger, SSMD, SOLE and SOP exhibit similar behaviors. Moreover, the less than ideal plot of SSNL seems to confirm the intuition that querying small margin instances 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Fraction of queried labels SSMD SSNL SS 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 F-measure Fraction of queried labels SSMD DKMperc SOLE SOP PERC Figure 2: Average F-measure obtained by different algorithms after 40,000 examples, as a function of the number of queried labels. The average only refers to the 50 most frequent categories. Points are obtained by repeatedly running each algorithm with different values of parameters (in Figure 1, the relevant parameter is λ). Trend lines are computed as approximate cubic splines connecting consecutive points. 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Topics Number of stored examples (normalized) Norm of the SVM weight vector (normalized) 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Topics F-measure Fraction of positive examples Fraction of queried labels Figure 3: Left: Correlation between the fraction of stored examples and the difficulty of each binary task, as measured by the separation margin. Right: F-measure achieved on the different binary classification tasks compared to the number of positive examples in each topic, and to the fraction of queried labels (including the stored ones). In both plots, topics are sorted by decreasing frequency of positive examples. The two plots are produced by SSMD with a specific value of the λ parameter. Varying λ does not significantly alter the reported trend. provides a significant advantage. Under our test conditions DKMPERC proved ineffective, probably because most tasks in the RCV1 dataset are not linearly separable. A similar behavior was observed in [20]. It is fair to remark that DKMPERC is a perceptron-like linear-threshold classifier while the other algorithms considered here are based on the more computationally intensive ridge regressionlike procedure. In our selective sampling framework it is important to investigate how harder problems influence the sampling rate of an algorithm and, for each binary problem, to assess the impact of the number of positive examples on F-measure performance. Coarsely speaking, we would expect that the hard topics are the infrequent ones. Here we focus on SSMD since it is reasonably the best candidate, among our selective samplers, as applied to real-world problems. In Figure 3 (left) we report the fraction of examples stored by SSMD on each of the 102 binary learning tasks (i.e., on each individual topic, including the infrequent ones), and the corresponding levels of F-measure and queried labels (right). Note that in both plots topics are sorted by frequency with the most frequent categories appearing on the left. We represent the difficulty of a learning task by the norm of the weight vector obtained by running the C-SVM algorithm on that task1. Figure 3 (left) clearly shows that SSMD rises the storage rate on difficult problems. In particular, even if two different tasks have largely different numbers of positive examples, the storage rate achieved by SSMD on those tasks may be 1The actual values were computed using SVM-LIGHT [19] with default parameters. Since the examples in the Reuters Corpus Volume 1 are cosine normalized, the choice of default parameters amounts to indirectly setting the parameter C to approximately 1.0. similar when the norm of the weight vectors computed by C-SVM is nearly the same. On the other hand, the right plot shows (to our surprise) that the achieved F-measure is fairly independent of the number of positive examples, but this independence is obtained at the cost of querying more and more labels. In other words, SSMD seems to realize the difficulty of learning infrequent topics and, in order to achieve a good F-measure performance, it compensates by querying many more labels. References [1] D. Angluin. Queries revisited. In 12th ALT, pages 12–31. Springer, 2001. [2] K.S. Azoury and M.K. Warmuth. Relative loss bounds for on-line density estimation with the exponential family of distributions. Machine Learning, 43(3):211–246, 2001. [3] M.F. Balcan, A. Beygelzimer, and J. Langford. Agnostic active learning. In 23rd ICML, pages 65–72. ACM Press, 2006. [4] M.F. Balcan, A. Broder, and T. Zhang. Margin-based active learning. In 20th COLT, pages 35–50. Springer, 2007. [5] P.L. Bartlett, M.I. Jordan, and J.D. McAuliffe. Convexity, classification, and risk bounds. JASA, 101(473):138–156, 2006. [6] G. Blanchard, O. Bousquet, and L. Zwald. Statistical properties of kernel principal component analysis. Machine Learning, 66:259–294, 2007. [7] M.L. Braun. Accurate error bounds for the eigenvalues of the kernel matrix. JMLR, 7:2303–2328, 2006. [8] C. Campbell, N. Cristianini, and A. Smola. Query learning with large margin classifiers. In 17th ICML, pages 111–118. Morgan Kaufmann, 2000. [9] R. Castro and R.D. Nowak. Minimax bounds for active learning. IEEE Trans. IT, 2008. To appear. [10] N. Cesa-Bianchi, A. Conconi, and C. Gentile. Learning probabilistic linear-threshold classifiers via selective sampling. In 16th COLT, pages 373–387. Springer, 2003. [11] N. Cesa-Bianchi, C. Gentile, and L. Zaniboni. Worst-case analysis of selective sampling for linear classification. JMLR, 7:1205–1230, 2006. [12] N. Cesa-Bianchi and G. Lugosi. Prediction, Learning, and Games. Cambridge University Press, 2006. [13] D. Cohn, L. Atlas, and R. Ladner. Improving generalization with active learning. Machine Learning, 15(2):201–221, 1994. [14] R. Cohn, L. Atlas, and R. Ladner. Training connectionist networks with queries and selective sampling. In NIPS 2. MIT Press, 1990. [15] S. Dasgupta, D. Hsu, and C. Monteleoni. A general agnostic active learning algorithm. In NIPS 20, pages 353–360. MIT Press, 2008. [16] S. Dasgupta, A. T. Kalai, and C. Monteleoni. Analysis of Perceptron-based active learning. In 18th COLT, pages 249–263. Springer, 2005. [17] Y. Freund, S. Seung, E. Shamir, and N. Tishby. Selective sampling using the query by committee algorithm. Machine Learning, 28(2/3):133–168, 1997. [18] R. Gilad-Bachrach, A. Navot, and N. Tishby. Query by committee made real. NIPS, 18, 2005. [19] T. Joachims. Making large-scale SVM learning practical. In B. Sch¨olkopf, C. Burges, and A. Smola, editors, Advances in Kernel Methods: Support Vector Learning. MIT Press, 1999. [20] C. Monteleoni and M. K¨a¨ari¨ainen. Practical online active learning for classification. In 24th IEEE CVPR, pages 249–263. IEEE Computer Society Press, 2007. [21] J. Shawe-Taylor, C.K.I. Williams, N. Cristianini, and J. Kandola. On the eigenspectrum of the Gram matrix and the generalization error of kernel-PCA. IEEE Trans. IT, 51(7):2510–2522, 2005. [22] I. Steinwart and C. Scovel Fast Rates for Support Vector Machines using Gaussian Kernels Annals of Statistics, 35: 575-607, 2007. [23] S. Tong and D. Koller. Support vector machine active learning with applications to text classification. In 17th ICML, pages 999–1006. Morgan Kaufmann, 2000. [24] A. Tsybakov. Optimal aggregation of classifiers in statistical learning. The Annals of Statistics, 32(1):135– 166, 2004. [25] V. Vovk. Competitive on-line statistics. 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Mixed Membership Stochastic Blockmodels Edoardo M. Airoldi 1,2, David M. Blei 1, Stephen E. Fienberg 3,4 & Eric P. Xing 4∗ 1 Department of Computer Science, 2 Lewis-Sigler Institute, Princeton University 3 Department of Statistics, 4 School of Computer Science, Carnegie Mellon University eairoldi@Princeton.EDU Abstract In many settings, such as protein interactions and gene regulatory networks, collections of author-recipient email, and social networks, the data consist of pairwise measurements, e.g., presence or absence of links between pairs of objects. Analyzing such data with probabilistic models requires non-standard assumptions, since the usual independence or exchangeability assumptions no longer hold. In this paper, we introduce a class of latent variable models for pairwise measurements: mixed membership stochastic blockmodels. Models in this class combine a global model of dense patches of connectivity (blockmodel) with a local model to instantiate node-specific variability in the connections (mixed membership). We develop a general variational inference algorithm for fast approximate posterior inference. We demonstrate the advantages of mixed membership stochastic blockmodel with applications to social networks and protein interaction networks. 1 Introduction The problem of modeling relational information among objects, such as pairwise relations represented as graphs, arises in a number of settings in machine learning. For example, scientific literature connects papers by citation, the Web connects pages by links, and protein-protein interaction data connect proteins by physical interaction records. In these settings, we often wish to infer hidden attributes of the objects from the pairwise observations. For example, we might want to compute a clustering of the web-pages, predict the functions of a protein, or assess the degree of relevance of a scientific abstract to a scholar’s query. Unlike traditional attribute data measured over individual objects, relational data violate the classical independence or exchangeability assumptions made in machine learning and statistics. The objects are dependent by their very nature, and this interdependence suggests that a different set of assumptions is more appropriate. Recently proposed models aim at resolving relational information into a collection of connectivity motifs. Such models are based on assumptions that often ignore useful technical necessities, or important empirical regularities. For instance, exponential random graph models [11] summarize the variability in a collection of paired measurements with a set of relational motifs, but do not provide a representation useful for making unit-specific predictions. Latent space models [4] project individual units of analysis into a low-dimensional latent space, but do not provide a group structure into such space useful for clustering. Stochastic blockmodels [8, 6] resolve paired measurements into groups and connectivity between pairs of groups, but constrain each unit to instantiate the connectivity patterns of a single group as observed in most applications. Mixed membership models, such as latent Dirichlet allocation [1], have emerged in recent years as a flexible modeling tool for data where the single group assumption is violated by the heterogeneity within a unit of analysis—e.g., a document, or a node in a graph. They have been successfully applied in many domains, such as document analysis [1], image processing [7], and population genetics [9]. Mixed membership models associate each unit of analysis with multiple groups rather than a single groups, via a membership ∗A longer version of this work is available online, at http://jmlr.csail.mit.edu/papers/v9/airoldi08a.html 1 probability-like vector. The concurrent membership of a data in different groups can capture its different aspects, such as different underlying topics for words constituting each document. The mixed membership formalism is a particularly natural idea for relational data, where the objects can bear multiple latent roles or cluster-memberships that influence their relationships to others. Existing mixed membership models, however, are not appropriate for relational data because they assume that the data are conditionally independent given their latent membership vectors. Conditional independence assumptions that technically instantiate mixed membership in recent work, however, are inappropriate for the relational data settings. In such settings, an objects is described by its relationships to others. Thus assuming that the ensemble of mixed membership vectors help govern the relationships of each object would be more appropriate. Here we develop mixed membership models for relational data and we describe a fast variational inference algorithm for inference and estimation. Our model captures the multiple roles that objects exhibit in interaction with others, and the relationships between those roles in determining the observed interaction matrix. We apply our model to protein interaction and social networks. 2 The Basic Mixed Membership Blockmodel Observations consist of pairwise measurements, represented as a graph G = (N, Y ), where Y (p, q) denotes the measurement taken on the pair of nodes (p, q). In this section we consider observations consisting of a single binary matrix, where Y (p, q) ∈{0, 1}, i.e., the data can be represented with a directed graph. The model generalizes to two important settings, however, as we discuss below—a collection of matrices and/or other types of measurements. We summarize a collection of pairwise measurements with a mapping from nodes to sets of nodes, called blocks, and pairwise relations among the blocks themselves. Intuitively, the inference process aims at identifying nodes that are similar to one another in terms of their connectivity to blocks of nodes. Similar nodes are mapped to the same block. Individual nodes are allowed to instantiate connectivity patterns of multiple blocks. Thus, the goal of the analysis with a Mixed Membership Blockmodel (MMB) is to identify (i) the mixed membership mapping of nodes, i.e., the units of analysis, to a fixed number of blocks, K, and (ii) the pairwise relations among the blocks. Pairwise measurements among N nodes are then generated according to latent distributions of block-membership for each node and a matrix of block-to-block interaction strength. Latent per-node distributions are specified by simplicial vectors. Each node is associated with a randomly drawn vector, say ⃗πi for node i, where πi,g denotes the probability of node i belonging to group g. In this fractional sense, each node can belong to multiple groups with different degrees of membership. The probabilities of interactions between different groups are defined by a matrix of Bernoulli rates B(K×K), where B(g, h) represents the probability of having a connection from a node in group g to a node in group h. The indicator vector ⃗zp→q denotes the specific block membership of node p when it connects to node q, while ⃗zp←q denotes the specific block membership of node q when it is connected from node p. The complete generative process for a graph G = (N, Y ) is as follows: • For each node p ∈N: – Draw a K dimensional mixed membership vector ⃗πp ∼Dirichlet ⃗α . • For each pair of nodes (p, q) ∈N × N: – Draw membership indicator for the initiator, ⃗zp→q ∼Multinomial ⃗πp . – Draw membership indicator for the receiver, ⃗zq→p ∼Multinomial ⃗πq . – Sample the value of their interaction, Y (p, q) ∼Bernoulli ⃗z ⊤ p→qB ⃗zp←q . Note that the group membership of each node is context dependent, i.e., each node may assume different membership when interacting with different peers. Statistically, each node is an admixture of group-specific interactions. The two sets of latent group indicators are denoted by {⃗zp→q : p, q ∈ N} =: Z→and {⃗zp←q : p, q ∈N} =: Z←. Further, the pairs of group memberships that underlie interactions, e.g., (⃗zp→q,⃗zp←q) for Y (p, q), need not be equal; this fact is useful for characterizing asymmetric interaction networks. Equality may be enforced when modeling symmetric interactions. The joint probability of the data Y and the latent variables {⃗π1:N, Z→, Z←} sampled according to the MMB is: p(Y,⃗π1:N, Z→, Z←|⃗α, B) = Y p,q P(Y (p, q)|⃗zp→q,⃗zp←q, B)P(⃗zp→q|⃗πp)P(⃗zp←q|⃗πq) Y p P(⃗πp|⃗α). 2 z 1→1 y11 z 1←1 z 1→2 y12 z 1←2 z 1→3 y13 z 1←3 z 1→N y1N z 1←N . . . z 2→1 y21 z 2←1 z 2→2 y22 z 2←2 z 2→3 y23 z 2←3 z 2→N y2N z 2←N . . . z 3→1 y31 z 3←1 z 3→2 y32 z 3←2 z 3→3 y33 z 3←3 z 3→N y3N z 3←N . . . z N→1 yN1 z N←1 z N→2 yN2 z N←2 z 1→1 yN3 z N←3 z N→N yNN z N←N . . . . . . . . . . . . . . . . . . α . . . π 1 2 π 3 π n π B Figure 1: The graphical model of the mixed membership blockmodel (MMB). We did not draw all the arrows out of the block model B for clarity. All the pairwise measurements, Y (p, q), depend on it. Introducing Sparsity. Adjacency matrices encoding binary pairwise measurements often contain a large amount of zeros, or non-interactions; they are sparse. It is useful to distinguish two sources of non-interaction: they may be the result of the rarity of interactions in general, or they may be an indication that the pair of relevant blocks rarely interact. In applications to social sciences, for instance, nodes may represent people and blocks may represent social communities. In this setting, it is reasonable to expect that a large portion of the non-interactions is due to limited opportunities of contact between people in a large population, or by design of the questionnaire, rather than due to deliberate choices, the structure of which the blockmodel is trying to estimate. It is useful to account for these two sources of sparsity at the model level. A good estimate of the portion of zeros that should not be explained by the blockmodel B reduces the bias of the estimates of B’s elements. We introduce a sparsity parameter ρ ∈[0, 1] in the model above to characterize the source of noninteraction. Instead of sampling a relation Y (p.q) directly the Bernoulli with parameter specified as above, we down-weight the probability of successful interaction to (1 −ρ) · ⃗z ⊤ p→qB ⃗zp←q. This is the result of assuming that the probability of a non-interaction comes from a mixture, 1 −σpq = (1 −ρ) · ⃗z ⊤ p→q(1 −B) ⃗zp←q + ρ, where the weight ρ capture the portion zeros that should not be explained by the blockmodel B. A large value of ρ will cause the interactions in the matrix to be weighted more than non-interactions, in determining plausible values for {⃗α, B,⃗π1:N}. Recall that {⃗α, B} are constant quantities to be estimated, while {⃗π1:N, Z→, Z←} are unknown variable quantities whose posterior distribution needs to be determined. Below, we detail the variational expectation-maximization (EM) procedure to carry out approximate estimation and inference. 2.1 Variational E-Step During the E-step, we update the posterior distribution over the unknown variable quantities {⃗π1:N, Z→, Z←}. The normalizing constant of the posterior is the marginal probability of the data, which requires an intractable integral over the simplicial vectors ⃗πp, p(Y | ⃗α, B) = Z ⃗π1:N X zp←q,zp→q p(Y,⃗π1:N, Z→, Z←|⃗α, B). (1) We appeal to mean-field variational methods [5] to approximate the posterior of interest. The main idea behind variational methods is to posit a simple distribution of the latent variables with free parameters, which are fit to make the approximation close in Kullback-Leibler divergence to the true posterior of interest. The log of the marginal probability in Equation 1 can be bound as follows, log p(Y | α, B) ≥Eq log p(Y,⃗π1:N, Z→, Z←|α, B) −Eq log q(⃗π1:N, Z→, Z←) , (2) by introducing a distribution of the latent variables q that depends on a set of free parameters. We specify q as the mean-field fully-factorized family, q(⃗π1:N, Z→, Z←| ⃗γ1:N, Φ→, Φ←), where {⃗γ1:N, Φ→, Φ←} is the set of free variational parameters that must be set to tighten the bound. We 3 tighten the bound with respect to the variational parameters, to minimize the KL divergence between q and the true posterior. The update for the variational multinomial parameters is ˆφp→q,g ∝ e Eq log πp,g · Y h B(g, h)Y (p,q)· 1 −B(g, h) 1−Y (p,q) φp←q,h (3) ˆφp←q,h ∝ e Eq log πq,h · Y g B(g, h)Y (p,q)· 1 −B(g, h) 1−Y (p,q) φp→q,g , (4) for g, h = 1, . . . , K. The update for the variational Dirichlet parameters γp,k is ˆγp,k = αk + X q φp→q,k + X q φp←q,k, (5) for all nodes p = 1, . . . , N and k = 1, . . . , K. Nested Variational Inference. To improve convergence, we developed a nested variational inference scheme based on an alternative schedule of updates to the traditional ordering [5]. In a na¨ıve iteration scheme for variational inference, one initializes the variational Dirichlet parameters ⃗γ1:N and the variational multinomial parameters (⃗φp→q, ⃗φp←q) to non-informative values, and then iterates until convergence the following two steps: (i) update ⃗φp→q and φp←q for all edges (p, q), and (ii) update ⃗γp for all nodes p ∈N. At each variational inference cycle one needs to allocate NK + 2N 2K scalars. In our experiments, the na¨ıve variational algorithm often failed to converge, or converged only after many iterations. We attribute this behavior to dependence between ⃗γ1:N and B in the model, which is not accounted for by the na¨ıve algorithm. The nested variational inference algorithm retains portion of this dependence across iterations by following a particular path to convergence. We keep the block of free parameters (⃗φp→q, ⃗φp←q) at their optimal values conditionally on the other variational parameters. These parameters are involved in the updates of parameters in ⃗γ1:N and in B, thus effectively providing a channel to maintain some dependence among them. From a computational perspective, the nested algorithm trades time for space thus allowing us to deal with large graphs. At each variational cycle we allocate NK + 2K scalars only. The algorithm can be parallelized, and, empirically, leads to a better likelihood bound per unit of running time. 2.2 M-Step During the M-step, we maximize the lower bound in Equation 2, used as a surrogate for the likelihood, with respect to the unknown constants {⃗α, B}. In other words, we compute the empirical Bayes estimates of the hyper-parameters. The M-step is equivalent to finding the MLE using expected sufficient statistics under the variational distribution. We consider the maximization step for each parameter in turn. A closed form solution for the approximate maximum likelihood estimate of ⃗α does not exist. We used linear-time Newton-Raphson, with gradient and Hessian ∂L⃗α ∂αk = N ψ X k αk −ψ(αk) + X p ψ(γp,k) −ψ X k γp,k , and ∂L⃗α ∂αk1αk2 = N I(k1=k2) · ψ′(αk1) −ψ′ X k αk , to find optimal values for ⃗α, numerically. The approximate MLE of B is ˆB(g, h) = P p,q Y (p, q) · φp→qg φp←qh P p,q φp→qg φp←qh , (6) for every pair (g, h) ∈[1, K]2. Finally, the approximate MLE of the sparsity parameter ρ is ˆρ = P p,q 1 −Y (p, q) · P g,h φp→qg φp←qh P p,q P g,h φp→qg φp←qh . (7) Alternatively, we can fix ρ prior to the analysis; the density of the interaction matrix is estimated with ˆd = P p,q Y (p, q)/N 2, and the sparsity parameter is set to ˜ρ = (1 −ˆd). This latter estimator 4 attributes all the information in the non-interactions to the point mass, i.e., to latent sources other than the block model B or the mixed membership vectors ⃗π1:N. It can be used, however, as a quick recipe to reduce the computational burden during exploratory analyses. Several model selection strategies exist for hierarchical models. In our setting, model selection translates into the choice of the number of blocks, K. Below, we chose K with held-out likelihood in a cross-validation experiment, on large networks, and with approximate BIC, on small networks. 2.3 Summarizing and De-Noising Pairwise Measurements It is useful to consider two data analysis perspectives the MMB can offer: (i) it summarizes the data, Y , in terms of the global blockmodel, B, and the node-specific mixed memberships, Πs, (ii) it de-noises the data, Y , in terms of the global blockmodel, B, and interaction-specific single memberships, Zs. In both cases the model depends on a small set of unknown constants to be estimated: α, and B. The likelihood is the same in both cases, although, the reasons for including the set of latent variables Zs differ. When summarizing data, we could integrate out the Zs analytically; this leads to numerical optimization of a smaller set of variational parameters, Γs. We choose to keep the Zs to simplify inference. When de-noising, the Zs are instrumental in estimating posterior expectations of each interactions individually—a network analog to the Kalman Filter. The posterior expectations of an interaction is computed as ⃗πp ′ B ⃗πq, and ⃗φp→q ′ B ⃗φp←q, in the two cases. 3 Empirical Results We evaluated the MMB on simulated data and on three collections of pairwise measurements. Results on simulated data sampled accordingly to the model show that variational EM accurately recovers the mixed membership map, ⃗π1:N, and the blockmodel, B. Cross-validation suggests an accurate estimate for K. Nested variational scheduling of parameter updates makes inference parallelizable and a typically reaches a better solution than the na¨ıve scheduling. First we consider, whom-do-like relations among 18 novices in a New England monastery. The unsupervised analysis demonstrates the type of patterns that MMB recovers from data, and allows us to contrast the summaries of the original measurements achieved through prediction and de-noising. The data was collected by Sampson during his stay at the monastery, while novices were preparing to join the monastic order [10]. Sampson’s original analysis is rooted in direct anthropological observations. He made a strong case for the existence of tight factions among the novices: the loyal opposition (whose members joined the monastery first), the young turks (who joined later on), the outcasts (who were not accepted in the two main factions), and the waverers (who did not take sides). The events that took place during Sampson’s stay at the monastery supported his observations— members of the young turks resigned or were expelled over religious differences (John and Gregory). Scholars in the social sciences typically regard the faction labels assigned by Sampson to the novices (and his conclusions, more in general) as ground truth to the extent of assessing the quality of results of quantitative analyses; we shall do the same here. Using the nested variational EM algorithm above, we fit an array of mixed membership blockmodels with different values of K, and collected model estimates {ˆα, ˆB} and posterior mixed membership vectors ⃗π1:18 for the novices. We used an approximation of BIC to choose the value of K supported by the data. This criterion selects ˆK = 3, the same number of proper groups that Sampson identified based on anthropological observations— the waverers are interstitial members, rather than a group. Figure 2 shows the patterns that the mixed membership blockmodel with ˆK = 3 recovers from data. In particular, the top-left panel shows a graphical representation of the blockmodel ˆB. The block that we can identify a-posteriori with the loyal opposition is portrayed as central to the monastery, while the block identified with the outcasts shows the lowest internal coherence, in accordance with Sampson’s observations. The top-right panel illustrates the posterior means of the mixed membership scores, E[⃗π|Y ], for the 18 monks in the monastery. The model (softly) partitions the monks according to Sampson’s classification, with Young Turks, Loyal Opposition, and Outcasts dominating each corner respectively. Notably, we can quantify the central role played by John Bosco and Gregory, who exhibit relations in all three groups, as well as the uncertain affiliations of Ramuald and Victor; Amand’s uncertain affiliation, however, is not captured. The bottom panels contrast the different resolution of the original adjacency matrix of whom-do-like sociometric relations (left panel) obtained with the two analyses MMB enables. 5 Figure 2: Top-Left: Estimated blockmodel, ˆB. Top-Right: Posterior mixed membership vectors, ⃗π1:18, projected in the simplex. The estimates correspond to a model with ˆB top-left, and ˆα = 0.058. Numbered points can be mapped to monks’ names using the legend on the right. The colors identify the four factions defined by Sampson’s anthropological observations. Bottom: Original adjacency matrix of whom-do-like sociometric relations (left), relations predicted using approximate MLEs for ⃗π1:N and B (center), and relations de-noised using the model including Zs indicators (right). If the goal of the analysis if to find a parsimonious summary of the data, the amount of relational information that is captured by in ˆα, ˆB, and E[⃗π|Y ] leads to a coarse reconstruction of the original sociomatrix (central panel). If the goal of the analysis if to de-noising a collection of pairwise measurements, the amount of relational information that is revealed by ˆα, ˆB and E[Z→, Z←|Y ] leads to a finer reconstruction of the original sociomatrix, Y —relations in Y are re-weighted according to how much they make sense to the model (right panel). Substantively, the unsupervised analysis of the sociometric relations with MMB offers quantitative support to several of Sampson’s observations. Second, we consider a friendship network among a group of 69 students in grades 7–12. The analysis here directly compares clustering results obtained with MMB to published results obtained with competing models, in a setting where a fair amount of social segregation is expected [2, 3]. The data is a collection of friendship relations among 69 students in a school surveyed in the National Study of Adolescent Health. The original population in the school of interest consisted of 71 students. Two students expressed no friendship preferences and were excluded from the analysis. We used variational EM algorithm to fit an array of mixed membership blockmodels with different values of K, collected model estimates, and used an approximation to BIC to select K. This procedure identified ˆK = 6 as the model-size that best explains the data; note that six is the number of grade-groups in the student population. The blocks are clearly interpretable a-posteriori in terms of grades, thus providing a mapping between grades and blocks. Conditionally on such a mapping, we assign students to the grade they are most associated with, according to their posterior-mean mixed membership vectors, E[⃗πn|Y ]. To be fair in the comparison with competing models, we assign students with a unique grade—despite MMB allows for mixed membership. Table 1 computes the correspondence of grades to blocks by quoting the number of students in each grade-block pair, for MMB versus the mixture blockmodel (MB) in [2], and the latent space cluster model (LSCM) in [3]. The higher the sum of counts on diagonal elements is the better is the correspondence, while the higher the sum of counts off diagonal elements is the worse is the correspondence. MMB performs best by allocating 63 students to their grades, versus 57 of MB, and 37 of LSCM. Correspondence only partially partially captures goodness of fit, however, it is a good metric in the setting we consider, where a fair amount of clustering is present. The extra-flexibility MMB offers over MB and LSCM reduces bias in the prediction of the membership of students to blocks, in this problem. In other words, mixed membership does not absorb noise in this example, rather it accommodates variability in the friendship relation that is instrumental in producing better predictions. 6 MMB Clusters MB Clusters LSCM Clusters Grade 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 7 13 1 0 0 0 0 13 1 0 0 0 0 13 1 0 0 0 0 8 0 9 2 0 0 1 0 10 2 0 0 0 0 11 1 0 0 0 9 0 0 16 0 0 0 0 0 10 0 0 6 0 0 7 6 3 0 10 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 0 3 7 11 0 0 1 0 11 1 0 0 1 0 11 1 0 0 0 0 3 10 12 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4 Table 1: Grade levels versus (highest) expected posterior membership for the 69 students, according to three alternative models. MMSB is the proposed mixed membership stochastic blockmodel, MSB is the mixture blockmodel in [2], and LSCM is the latent space cluster model in [3]. Third, we consider physical interactions among 871 proteins in yeast. The analysis allows us to evaluate the utility of MMB in summarizing and de-noising complex connectivity patterns quantitatively, using an independent set of functional annotations—consider two models that suggest different sets of interactions as reliable; we prefer the model that reveals functionally relevant interactions. The pairwise measurements consist of a hand-curated collection of physical protein interactions made available by the Munich Institute of Protein Sequencing (MIPS). The yeast genome database provides independent functional annotations for each protein, which we use for evaluating the functional content of the protein networks estimated with the MMB from the MIPS data, as detailed below. We explored a large model space, K = 2 . . . 225, and used five-fold cross-validation to identify a blockmodel B that reduces the dimensionality of the physical interactions among proteins in the training set, while revealing robust aspects of connectivity that can be leveraged to predict physical interactions among proteins in the test set. We determined that a fairly parsimonious model, K = 50, provides a good description of the observed physical interaction network. This finding supports the hypothesis that proteins derived from the MIPS data are interpretable in terms functional biological contexts. Alternatively, the blocks might encode signal at a finer resolution, such as that of protein complexes. If that was the case, however, we would expect the optimal number of blocks to be significantly higher; 871/5 ≈175, given an average size of five proteins in a protein complex. We then evaluated the functional content of the posterior induced by MMB. The goal is to assess to what extent MMB reveals substantive information about the functionality of proteins that can be used to inform subsequent analyses. To do this, first, we fit a model on the whole data set to estimate the blockmodel, B(50×50), and the mixed membership vectors between proteins and blocks, ⃗π1:871, and second, we either impute physical interactions by thresholding the posterior expectations computed using blockmodel and node-specific memberships (summarization task), or we de-noise the observed interactions using blockmodel and pair-specific memberships (de-noising task). Posterior expectations of each interaction are in [0, 1]. Thresholding such expectations at q, for instance, leads to a collection of binary physical interactions that are at reliable with probability p ≥q. We used an independent set of functional annotations from the yeast database (SGD at www.yeastgenome.org) to decide which interactions are functionally meaningful; namely those between pairs of proteins that share at least one functional annotation. In this sense, between two models that suggest different sets of interactions as reliable, our evaluation assigns a higher score MMB (K=50; MIPS de-noised with Zs & B) MMB (K=50; MIPS summarized with Πs & B) Recall (unnormalized) Precision Figure 3: Functional content of the MIPS collection of protein interactions (yellow diamond) on a precision-recall plot, compared against other published collections of interactions and microarray data, and to the posterior estimates of the MMB models—computed as described in the text. 7 to the model that reveals functionally relevant interactions according to SGD. Figure 3 shows the functional content of the original MIPS collection of physical interactions (point no.2), and of the collections of interactions computed using (B, Πs), the light blue (−×) line, and using (B, Zs), the dark blue (−+) line, thresholded at ten different levels—precision-recall curves. The posterior means of Πs provide a parsimonious representation for the MIPS collection, and lead to precise protein interaction estimates, in moderate amount (−× line). The posterior means of Zs provide a richer representation for the data, and describe most of the functional content of the MIPS collection with high precision (−+ line). Importantly, the estimated networks corresponding to lower levels of recall for both model variants (i.e., × and +) feature a more precise functional content than the original network. This means that the proposed latent block structure is helpful in effectively denoising the collection of interactions—by ranking them properly. On closer inspection, dense blocks of predicted interactions contain known functional predictions that were not in the MIPS collection, thus effectively improving the quality of the protein binding data that instantiate cellular activity of specific biological contexts, such as biopolymer catabolism and homeostasis. In conclusion, our results suggest that MMB successfully reduces the dimensionality of the data, while discovering information about the multiple functionality of proteins that can be used to inform follow-up analyses. Remarks. A. In the relational setting, cross-validation is feasible if the blockmodel estimated on training data can be expected to hold on test data; for this to happen the network must be of reasonable size, so that we can expect members of each block to be in both training and test sets. In this setting, scheduling of variational updates is important; nested variational scheduling leads to efficient and parallelizable inference. B. MMB includes two sources of variability, B, Πs, that are apparently in competition for explaining the data, possibly raising an identifiability issue. This is not the case, however, as the blockmodel B captures global/asymmetric relations, while the mixed membership vectors Πs capture local/symmetric relations. This difference practically eliminates the issue, unless there is no signal in the data to begin with. C. MMB generalizes to two important cases. First, multiple data collections Y1:M on the same objects can be generated by the same latent vectors. This might be useful, for instance, for analyzing multivariate sociometric relations simultaneously. Second, in the MMSB the data generating distribution is a Bernoulli, but B can be a matrix of parameterizes for any kind of distribution. For instance, technologies for measuring interactions between pairs of proteins, such as mass spectrometry and tandem affinity purification, which return a probabilistic assessment about the presence of interactions, thus setting the range of Y ∈[0, 1]. References [1] D. M. Blei, A. Ng, and M. I. Jordan. Latent Dirichlet allocation. Journal of Machine Learning Research, 3:993–1022, 2003. [2] P. Doreian, V. Batagelj, and A. Ferligoj. Discussion of “Model-based clustering for social networks”. Journal of the Royal Statistical Society, Series A, 170, 2007. [3] M. S. Handcock, A. E. Raftery, and J. M. Tantrum. Model-based clustering for social networks. Journal of the Royal Statistical Society, Series A, 170:1–22, 2007. [4] P. D. Hoff, A. E. Raftery, and M. S. Handcock. Latent space approaches to social network analysis. Journal of the American Statistical Association, 97:1090–1098, 2002. [5] M. Jordan, Z. Ghahramani, T. Jaakkola, and L. Saul. Introduction to variational methods for graphical models. Machine Learning, 37:183–233, 1999. [6] C. Kemp, J. B. Tenenbaum, T. L. Griffiths, T. Yamada, and N. Ueda. Learning systems of concepts with an infinite relational model. In Proc. of the 21st National Conference on Artificial Intelligence, 2006. [7] F.-F. Li and P. Perona. A Bayesian hierarchical model for learning natural scene categories. IEEE Computer Vision and Pattern Recognition, 2005. [8] K. Nowicki and T. A. B. Snijders. Estimation and prediction for stochastic blockstructures. Journal of the American Statistical Association, 96:1077–1087, 2001. [9] J. K. Pritchard, M. Stephens, N. A. Rosenberg, and P. Donnelly. Association mapping in structured populations. American Journal of Human Genetics, 67:170–181, 2000. [10] F. S. Sampson. A Novitiate in a period of change: An experimental and case study of social relationships. PhD thesis, Cornell University, 1968. [11] S. Wasserman, G. Robins, and D. Steinley. 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Shared Segmentation of Natural Scenes Using Dependent Pitman-Yor Processes Erik B. Sudderth and Michael I. Jordan Electrical Engineering & Computer Science, University of California, Berkeley sudderth@cs.berkeley.edu, jordan@cs.berkeley.edu Abstract We develop a statistical framework for the simultaneous, unsupervised segmentation and discovery of visual object categories from image databases. Examining a large set of manually segmented scenes, we show that object frequencies and segment sizes both follow power law distributions, which are well modeled by the Pitman–Yor (PY) process. This nonparametric prior distribution leads to learning algorithms which discover an unknown set of objects, and segmentation methods which automatically adapt their resolution to each image. Generalizing previous applications of PY processes, we use Gaussian processes to discover spatially contiguous segments which respect image boundaries. Using a novel family of variational approximations, our approach produces segmentations which compare favorably to state-of-the-art methods, while simultaneously discovering categories shared among natural scenes. 1 Introduction Images of natural environments contain a rich diversity of spatial structure at both coarse and fine scales. We would like to build systems which can automatically discover the visual categories (e.g., foliage, mountains, buildings, oceans) which compose such scenes. Because the “objects” of interest lack rigid forms, they are poorly suited to traditional, fixed aspect detectors. In simple cases, topic models can be used to cluster local textural elements, coarsely representing categories via a bag of visual features [1, 2]. However, spatial structure plays a crucial role in general scene interpretation [3], particularly when few labeled training examples are available. One approach to modeling additional spatial dependence begins by precomputing one, or several, segmentations of each input image [4–6]. However, low-level grouping cues are often ambiguous, and fixed partitions may improperly split or merge objects. Markov random fields (MRFs) have been used to segment images into one of several known object classes [7, 8], but these approaches require manual segmentations to train category-specific appearance models. In this paper, we instead develop a statistical framework for the unsupervised discovery and segmentation of visual object categories. We approach this problem by considering sets of images depicting related natural scenes (see Fig. 1(a)). Using color and texture cues, our method simultaneously groups dense features into spatially coherent segments, and refines these partitions using shared appearance models. This extends the cosegmentation framework [9], which matches two views of a single object instance, to simultaneously segment multiple object categories across a large image database. Some recent work has pursued similar goals [6, 10], but robust object discovery remains an open challenge. Our models are based on the Pitman–Yor (PY) process [11], a nonparametric Bayesian prior on infinite partitions. This generalization of the Dirichlet process (DP) leads to heavier-tailed, power law distributions for the frequencies of observed objects or topics. Using a large database of manual scene segmentations, Sec. 2 demonstrates that PY priors closely match the true distributions of natural segment sizes, and frequencies with which object categories are observed. Generalizing the hierarchical DP [12], Sec. 3 then describes a hierarchical Pitman–Yor (HPY) mixture model which shares “bag of features” appearance models among related scenes. Importantly, this approach coherently models uncertainty in the number of object categories and instances. 10 0 10 1 10 2 10 −4 10 −3 10 −2 10 −1 10 0 Segment Labels (sorted by frequency) Proportion of forest Segments Segment Labels PY(0.39,3.70) DP(11.40) 10 −2 10 −1 10 0 10 0 10 1 10 2 10 3 Proportion of Image Area Number of forest Segments Segment Areas PY(0.02,2.20) DP(2.40) 1 2 3 4 5 6 7 8 0 20 40 60 80 100 120 Number of Segments per Image Number of forest Images Segment Counts PY(0.02,2.20) DP(2.40) 10 0 10 1 10 2 10 −4 10 −3 10 −2 10 −1 10 0 Segment Labels (sorted by frequency) Proportion of insidecity Segments Segment Labels PY(0.47,6.90) DP(33.00) 10 −2 10 −1 10 0 10 0 10 1 10 2 10 3 Proportion of Image Area Number of insidecity Segments Segment Areas PY(0.32,0.80) DP(2.90) 1 2 3 4 5 6 7 8 0 20 40 60 80 100 120 Number of Segments per Image Number of insidecity Images Segment Counts PY(0.32,0.80) DP(2.90) (a) (b) (c) (d) Figure 1: Validation of stick-breaking priors for the statistics of human segmentations of the forest (top) and insidecity (bottom) scene categories. We compare observed frequencies (black) to those predicted by Pitman– Yor process (PY, red circles) and Dirichlet process (DP, green squares) models. For each model, we also display 95% confidence intervals (dashed). (a) Example human segmentations, where each segment has a text label such as sky, tree trunk, car, or person walking. The full segmented database is available from LabelMe [14]. (b) Frequency with which different semantic text labels, sorted from most to least frequent on a log-log scale, are associated with segments. (c) Number of segments occupying varying proportions of the image area, on a log-log scale. (d) Counts of segments of size at least 5,000 pixels in 256 × 256 images of natural scenes. As described in Sec. 4, we use thresholded Gaussian processes to link assignments of features to regions, and thereby produce smooth, coherent segments. Simulations show that our use of continuous latent variables captures long-range dependencies neglected by MRFs, including intervening contour cues derived from image boundaries [13]. Furthermore, our formulation naturally leads to an efficient variational learning algorithm, which automatically searches over segmentations of varying resolution. Sec. 5 concludes by demonstrating accurate segmentation of complex images, and discovery of appearance patterns shared across natural scenes. 2 Statistics of Natural Scene Categories To better understand the statistical relationships underlying natural scenes, we analyze manual segmentations of Oliva and Torralba’s eight categories [3]. A non-expert user partitioned each image into a variable number of polygonal segments corresponding to distinctive objects or scene elements (see Fig. 1(a)). Each segment has a semantic text label, allowing study of object co-occurrence frequencies across related scenes. There are over 29,000 segments in the collection of 2,688 images.1 2.1 Stick Breaking and Pitman–Yor Processes The relative frequencies of different object categories, as well as the image areas they occupy, can be statistically modeled via distributions on potentially infinite partitions. Let ϕ = (ϕ1, ϕ2, ϕ3, . . .), P∞ k=1 ϕk = 1, denote the probability mass associated with each subset. In nonparametric Bayesian statistics, prior models for partitions are often defined via a stick-breaking construction: ϕk = wk k−1 Y ℓ=1 (1 −wℓ) = wk 1 − k−1 X ℓ=1 ϕℓ wk ∼Beta(1 −γa, γb + kγa) (1) This Pitman–Yor (PY) process [11], denoted by ϕ ∼GEM(γa, γb), is defined by two hyperparameters satisfying 0 ≤γa < 1, γb > −γa. When γa = 0, we recover a Dirichlet process (DP) with concentration parameter γb. This construction induces a distribution on ϕ such that subsets with more mass ϕk typically have smaller indexes k. When γa > 0, E[wk] decreases with k, and the resulting partition frequencies follow heavier-tailed, power law distributions. While the sequences of beta variables underlying PY processes lead to infinite partitions, only a random, finite subset of size Kε = {k | ϕk > ε} will have probability greater than any threshold ε. Implicitly, nonparametric models thus also place priors on the number of latent classes or objects. 1See LabelMe [14]: http://labelme.csail.mit.edu/browseLabelMe/spatial envelope 256x256 static 8outdoorcategories.html 2.2 Object Label Frequencies Pitman–Yor processes have been previously used to model the well-known power law behavior of text sequences [15, 16]. Intuitively, the labels assigned to segments in the natural scene database have similar properties: some (like sky, trees, and building) occur frequently, while others (rainbow, lichen, scaffolding, obelisk, etc.) are more rare. Fig. 1(b) plots the observed frequencies with which unique text labels, sorted from most to least frequent, occur in two scene categories. The overlaid quantiles correspond to the best fitting DP and PY processes, with parameters (ˆγa, ˆγb) estimated via maximum likelihood. When ˆγa > 0, log E[eϕk | ˆγ] ≈−ˆγ−1 a log(k) + ∆(ˆγa, ˆγb) for large k [11], producing power law behavior which accurately predicts observed object frequencies. In contrast, the closest fitting DP model (ˆγa = 0) significantly underestimates the number of rare labels. We have quantitatively assessed the accuracy of these models using bootstrap significance tests [17]. The PY process provides a good fit for all categories, while there is significant evidence against the DP in most cases. By varying PY hyperparameters, we also capture interesting differences among scene types: urban, man-made environments have many more unique objects than natural ones. 2.3 Segment Counts and Size Distributions We have also used the natural scene database to quantitatively validate PY priors for image partitions [17]. For natural environments, the DP and PY processes both provide accurate fits. However, some urban environments have many more small objects, producing power law area distributions (see Fig. 1(c)) better captured by PY processes. As illustrated in Fig. 1(d), PY priors also model uncertainty in the number of segments at various resolutions. While power laws are often used simply as a descriptive summary of observed statistics, PY processes provide a consistent generative model which we use to develop effective segmentation algorithms. We do not claim that PY processes are the only valid prior for image areas; for example, log-normal distributions have similar properties, and may also provide a good model [18]. However, PY priors lead to efficient variational inference algorithms, avoiding the costly MCMC search required by other segmentation methods with region size priors [18, 19]. 3 A Hierarchical Model for Bags of Image Features We now develop hierarchical Pitman–Yor (HPY) process models for visual scenes. We first describe a “bag of features” model [1, 2] capturing prior knowledge about region counts and sizes, and then extend it to model spatially coherent shapes in Sec. 4. Our baseline bag of features model directly generalizes the stick-breaking representation of the hierarchical DP developed by Teh et al. [12]. N-gram language models based on HPY processes [15, 16] have somewhat different forms. 3.1 Hierarchical Pitman–Yor Processes Each image is first divided into roughly 1,000 superpixels [18] using a variant of the normalized cuts spectral clustering algorithm [13]. We describe the texture of each superpixel via a local texton histogram [20], using band-pass filter responses quantized to Wt = 128 bins. Similarly, a color histogram is computed by quantizing the HSV color space into Wc = 120 bins. Superpixel i in image j is then represented by histograms xji = (xt ji, xc ji) indicating its texture xt ji and color xc ji. Figure 2 contains a directed graphical model summarizing our HPY model for collections of local image features. Each of the potentially infinite set of global object categories occurs with frequency ϕk, where ϕ ∼GEM(γa, γb) as motivated in Sec. 2.2. Each category k also has an associated appearance model θk = (θt k, θc k), where θt k and θc k parameterize multinomial distributions on the Wt texture and Wc color bins, respectively. These parameters are regularized by Dirichlet priors θt k ∼Dir(ρt), θc k ∼Dir(ρc), with hyperparameters chosen to encourage sparse distributions. Consider a dataset containing J images of related scenes, each of which is allocated an infinite set of potential segments or regions. As in Sec. 2.3, region t occupies a random proportion πjt of the area in image j, where πj ∼GEM(αa, αb). Each region is also associated with a particular global object category kjt ∼ϕ. For each superpixel i, we then independently select a region tji ∼πj, and sample features using parameters determined by that segment’s global object category: p xt ji, xc ji | tji, kj, θ = Mult xt ji | θt zji ·Mult xc ji | θc zji zji ≜kjtji (2) As in other adaptations of topic models to visual data [8], we assume that different feature channels vary independently within individual object categories and segments. xji T tji kjt D J k J Nj f f U vjt wk 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 Stick−Breaking Proportion Probability Density 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 Stick−Breaking Proportion Probability Density 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 Stick−Breaking Proportion Probability Density GEM(0, 10) GEM(0.1, 2) GEM(0.5, 5) −4 −2 0 2 4 0 0.1 0.2 0.3 0.4 0.5 0.6 Stick−Breaking Threshold Probability Density −4 −2 0 2 4 0 0.1 0.2 0.3 0.4 0.5 0.6 Stick−Breaking Threshold Probability Density −4 −2 0 2 4 0 0.1 0.2 0.3 0.4 0.5 0.6 Stick−Breaking Threshold Probability Density Figure 2: Stick-breaking representation of a hierarchical Pitman–Yor (HPY) model for J groups of features. Left: Directed graphical model in which global category frequencies ϕ ∼GEM(γ) are constructed from stickbreaking proportions wk ∼Beta(1 −γa, γb + kγa), as in Eq. (1). Similarly, vjt ∼Beta(1 −αa, αb + tαa) define region areas πj ∼GEM(α) for image j. Each of the Nj features xji is independently sampled as in Eq. (2). Upper right: Beta distributions from which stick proportions wk are sampled for three different PY processes: k = 1 (blue), k = 10 (red), k = 20 (green). Lower right: Corresponding distributions on thresholds for an equivalent generative model employing zero mean, unit variance Gaussians (dashed black). See Sec. 4.1. 3.2 Variational Learning for HPY Mixture Models To allow efficient learning of HPY model parameters from large image databases, we have developed a mean field variational method which combines and extends previous approaches for DP mixtures [21, 22] and finite topic models. Using the stick-breaking representation of Fig. 2, and a factorized variational posterior, we optimize the following lower bound on the marginal likelihood: log p(x | α, γ, ρ) ≥H(q) + Eq[log p(x, k, t, v, w, θ | α, γ, ρ)] (3) q(k, t, v, w, θ) = " K Y k=1 q(wk | ωk)q(θk | ηk) # · J Y j=1 " T Y t=1 q(vjt | νjt)q(kjt | κjt) # Nj Y i=1 q(tji | τji) Here, H(q) is the entropy. We truncate the variational posterior [21] by setting q(vjT = 1) = 1 for each image or group, and q(wK = 1) = 1 for the shared global clusters. Multinomial assignments q(kjt | κjt), q(tji | τji), and beta stick proportions q(wk | ωk), q(vjt | νjt), then have closed form update equations. To avoid bias, we sort the current sets of image segments, and global categories, in order of decreasing aggregate assignment probability after each iteration [22]. 4 Segmentation with Spatially Dependent Pitman–Yor Processes We now generalize the HPY image segmentation model of Fig. 2 to capture spatial dependencies. For simplicity, we consider a single-image model in which features xi are assigned to regions by indicator variables zi, and each segment k has its own appearance parameters θk (see Fig. 3). As in Sec. 3.1, however, this model is easily extended to share appearance parameters among images. 4.1 Coupling Assignments using Thresholded Gaussian Processes Consider a generative model which partitions data into two clusters via assignments zi ∈{0, 1} sampled such that P[zi = 1] = v. One representation of this sampling process first generates a Gaussian auxiliary variable ui ∼N(0, 1), and then chooses zi according to the following rule: zi = 1 if ui < Φ−1(v) 0 otherwise Φ(u) ≜ 1 √ 2π Z u −∞ e−s2/2 ds (4) Here, Φ(u) is the standard normal cumulative distribution function (CDF). Since Φ(ui) is uniformly distributed on [0, 1], we immediately have P[zi = 1] = P ui < Φ−1(v) = P[Φ(ui) < v] = v. We adapt this idea to PY processes using the stick-breaking representation of Eq. (1). In particular, we note that if zi ∼π where πk = vk Qk−1 ℓ=1 (1 −vℓ), a simple induction argument shows that vk = P[zi = k | zi ̸= k −1, . . . , 1]. The stick-breaking proportion vk is thus the conditional probability of choosing cluster k, given that clusters with indexes ℓ< k have been rejected. Combining x1 T D k f f U vk x2 x3 x4 z1 z2 z3 z4 uk3 uk4 uk1 uk2 S1 S2 S3 S4 S1 S2 S3 S4 u1 u2 u3 S1 S2 S3 S4 Figure 3: A nonparametric Bayesian approach to image segmentation in which thresholded Gaussian processes generate spatially dependent Pitman–Yor processes. Left: Directed graphical model in which expected segment areas π ∼GEM(α) are constructed from stick-breaking proportions vk ∼Beta(1 −αa, αb + kαa). Zero mean Gaussian processes (uki ∼N(0, 1)) are cut by thresholds Φ−1(vk) to produce segment assignments zi, and thereby features xi. Right: Three randomly sampled image partitions (columns), where assignments (bottom, color-coded) are determined by the first of the ordered Gaussian processes uk to cross Φ−1(vk). this insight with Eq. (4), we can generate samples zi ∼π as follows: zi = min k | uki < Φ−1(vk) where uki ∼N(0, 1) and uki ⊥uℓi, k ̸= ℓ (5) As illustrated in Fig. 3, each cluster k is now associated with a zero mean Gaussian process (GP) uk, and assignments are determined by the sequence of thresholds in Eq. (5). If the GPs have identity covariance functions, we recover the basic HPY model of Sec. 3.1. More general covariances can be used to encode the prior probability that each feature pair occupies the same segment. Intuitively, the ordering of segments underlying this dependent PY model is analogous to layered appearance models [23], in which foreground layers occlude those that are farther from the camera. To retain the power law prior on segment sizes justified in Sec. 2.3, we transform priors on stick proportions vk ∼Beta(1 −αa, αb + kαa) into corresponding random thresholds: p(¯vk | α) = N(¯vk | 0, 1) · Beta(Φ(¯vk) | 1 −αa, αb + kαa) ¯vk ≜Φ−1(vk) (6) Fig. 2 illustrates the threshold distributions corresponding to several different PY stick-breaking priors. As the number of features N becomes large relative to the GP covariance length-scale, the proportion assigned to segment k approaches πk, where π ∼GEM(αa, αb) as desired. 4.2 Variational Learning for Dependent PY Processes Substantial innovations are required to extend the variational method of Sec. 3.2 to the Gaussian processes underlying our dependent PY processes. Complications arise due to the threshold assignment process of Eq. (5), which is “stronger” than the likelihoods typically used in probit models for GP classification, as well as the non-standard threshold prior of Eq. (6). In the simplest case, we place factorized Gaussian variational posteriors on thresholds q(¯vk) = N(¯vk | νk, δk) and assignment surfaces q(uki) = N(uki | µki, λki), and exploit the following key identities: Pq[uki < ¯vk] = Φ νk −µki √δk + λki Eq[log Φ(¯vk)] ≤log Eq[Φ(¯vk)] = log Φ νk √1 + δk (7) The first expression leads to closed form updates for Dirichlet appearance parameters q(θk | ηk), while the second evaluates the beta normalization constants in Eq. (6). We then jointly optimize each layer’s threshold q(¯vk) and assignment surface q(uk), fixing all other layers, via backtracking conjugate gradient (CG) with line search. For details and further refinements, see [17]. Figure 4: Five samples from each of four prior models for image partitions (color coded). Top Left: Nearest neighbor Potts MRF with K = 10 states. Top Right: Potts MRF with potentials biased by DP samples [28]. Bottom Left: Softmax model in which spatially varying assignment probabilities are coupled by logistically transformed GPs [25–27]. Bottom Right: PY process assignments coupled by thresholded GPs (as in Fig. 3). 4.3 Related Work Recently, Duan et. al. [24] proposed a generalized spatial Dirichlet process which links assignments via thresholded GPs, as in Sec. 4.1. However, their focus is on modeling spatial random effects for prediction tasks, as opposed to the segmentation tasks which motivate our generalization to PY processes. Unlike our HPY extension, they do not consider approaches to sharing parameters among related groups or images. Moreover, their basic Gibbs sampler takes 12 hours on a toy dataset with 2,000 observations; our variational method jointly segments 200 scenes in comparable time. Several authors have independently proposed a spatial model based on pointwise, multinomial logistic transformations of K latent GPs [25–27]. This produces a field of smoothly varying multinomial distributions ˇπi, from which segment assignments are independently sampled as zi ∼ˇπi. As shown in Fig. 4, this softmax construction produces noisy, less spatially coherent partitions. Moreover, its bias towards partitions with K segments of similar size is a poor fit for natural scenes. A previous nonparametric image segmentation method defined its prior as a normalized product of a DP sample π ∼GEM(0, α) and a nearest neighbor MRF with Potts potentials [28]. This construction effectively treats log π as the canonical, rather than moment, parameters of the MRF, and does not produce partitions whose size distribution matches GEM(0, α). Due to the phase transition which occurs with increasing potential strength, Potts models assign low probability to realistic image partitions [29]. Empirically, the DP-Potts product construction seems to have similar issues (see Fig. 4), although it can still be effective with strongly informative likelihoods [28]. 5 Results Figure 5 shows segmentation results for images from the scene categories considered in Sec. 2. We compare the bag of features PY model (PY-BOF), dependent PY with distance-based squared exponential covariance (PY-Dist), and dependent PY with covariance that incorporates intervening contour cues (PY-Edge) based on the Pb detector [20]. The conditionally specified PY-Edge model scales the covariance between superpixels i and j by p 1 −bij, where bij is the largest Pb response on the straight line connecting them. We convert these local covariance estimates into a globally consistent, positive definite matrix via an eigendecomposition. For the results in Figs. 5 and 6, we independently segment each image, without sharing appearance models or supervised training. We compare our results to the normalized cuts spectral clustering method with varying numbers of segments (NCut(K)), and a high-quality affinity function based on color, texture, and intervening contour cues [13]. Our PY models consistently capture variability in the number of true segments, and detect both large and small regions. In contrast, normalized cuts is implicitly biased towards regions of equal size, which produces distortions. To quantitatively evaluate results, we measure overlap with held-out human segments via the Rand index [30]. As summarized in Fig. 6, PY-BOF performs well for some images with unambiguous features, but PY-Edge is often substantially better. We have also experimented with our hierarchical PY extension, in which color and texture distributions are shared between images. As shown in Fig. 7, many of the inferred global visual categories align reasonably with semantic categories (e.g., sky, foliage, mountains, or buildings). 6 Discussion We have developed a nonparametric framework for image segmentation which uses thresholded Gaussian processes to produce spatially coupled Pitman–Yor processes. This approach produces empirically justified power law priors for region areas and object frequencies, allows visual appearFigure 5: Segmentation results for two images (rows) from each of the coast, mountain, and tallbuilding scene categories. From left to right, columns show LabelMe human segments, image with boundaries inferred by PY-Edge, and segments for PY-Edge, PY-Dist, PY-BOF, NCut(3), NCut(4), and NCut(6). Best viewed in color. 0.2 0.4 0.6 0.8 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized Cuts PY Gaussian (Edge Covar) 2 4 6 8 10 0.5 0.6 0.7 0.8 0.9 1 Number of Normalized Cuts Regions Average Rand Index Normalized Cuts PY Gaussian (Edge Covar) PY Gaussian (Distance Covar) PY Bag of Features 0.2 0.4 0.6 0.8 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized Cuts PY Gaussian (Edge Covar) 2 4 6 8 10 0.5 0.6 0.7 0.8 0.9 1 Number of Normalized Cuts Regions Average Rand Index Normalized Cuts PY Gaussian (Edge Covar) PY Gaussian (Distance Covar) PY Bag of Features (a) (b) (c) (d) Figure 6: Quantitative comparison of segmentation results to human segments, using the Rand index. (a) Scatter plot of PY-Edge and NCut(4) Rand indexes for 200 mountain images. (b) Average Rand indexes for mountain images. We plot the performance of NCut(K) versus the number of segments K, compared to the variable resolution segmentations of PY-Edge, PY-Dist, and PY-BOF. (c) Scatter plot of PY-Edge and NCut(6) Rand indexes for 200 tallbuilding images. (d) Average Rand indexes for tallbuilding images. ance models to be flexibly shared among natural scenes, and leads to efficient variational inference algorithms which automatically search over segmentations of varying resolution. We believe this provides a promising starting point for discovery of shape-based visual appearance models, as well as weakly supervised nonparametric learning in other, non-visual application domains. Acknowledgments We thank Charless Fowlkes and David Martin for the Pb boundary estimation and segmentation code, Antonio Torralba for helpful conversations, and Sra. Barriuso for her image labeling expertise. This research supported by ONR Grant N00014-06-1-0734, and DARPA IPTO Contract FA8750-05-2-0249. References [1] L. Fei-Fei and P. Perona. A Bayesian hierarchical model for learning natural scene categories. In CVPR, volume 2, pages 524–531, 2005. [2] J. Sivic, B. C. 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Carreira-Perpi˜n´an. Multiscale conditional random fields for image labeling. In CVPR, volume 2, pages 695–702, 2004. [8] J. Verbeek and B. Triggs. Region classification with Markov field aspect models. In CVPR, 2007. [9] C. Rother, V. Kolmogorov, T. Minka, and A. Blake. Cosegmentation of image pairs by histogram matching: Incorporating a global constraint into MRFs. In CVPR, volume 1, pages 993–1000, 2006. [10] M. Andreetto, L. Zelnik-Manor, and P. Perona. Non-parametric probabilistic image segmentation. In ICCV, 2007. [11] J. Pitman and M. Yor. The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Prob., 25(2):855–900, 1997. [12] Y. W. Teh, M. I. Jordan, M. J. Beal, and D. M. Blei. Hierarchical Dirichlet processes. J. Amer. Stat. Assoc., 101(476):1566–1581, December 2006. [13] C. Fowlkes, D. Martin, and J. Malik. Learning affinity functions for image segmentation: Combining patch-based and gradient-based approaches. In CVPR, volume 2, pages 54–61, 2003. [14] B. C. Russell, A. Torralba, K. P. Murphy, and W. T. Freeman. LabelMe: A database and web-based tool for image annotation. IJCV, 77:157–173, 2008. [15] S. Goldwater, T. L. Griffiths, and M. Johnson. Interpolating between types and tokens by estimating power-law generators. In NIPS 18, pages 459–466. MIT Press, 2006. [16] Y. W. Teh. A hierarchical Bayesian language model based on Pitman–Yor processes. In Coling/ACL, 2006. [17] E. B. Sudderth and M. I. Jordan. Shared segmentation of natural scenes using dependent Pitman-Yor processes. Technical report, Dept. of Statistics, University of California, Berkeley. In preparation, 2009. [18] X. Ren and J. Malik. Learning a classification model for segmentation. In ICCV, 2003. [19] Z. Tu and S. C. Zhu. Image segmentation by data-driven Markov chain Monte Carlo. IEEE Trans. PAMI, 24(5):657–673, May 2002. [20] D. R. Martin, C. C. Fowlkes, and J. Malik. Learning to detect natural image boundaries using local brightness, color, and texture cues. IEEE Trans. PAMI, 26(5):530–549, May 2004. [21] D. M. Blei and M. I. Jordan. Variational inference for Dirichlet process mixtures. Bayes. Anal., 1(1):121– 144, 2006. [22] K. Kurihara, M. Welling, and Y. W. Teh. Collapsed variational Dirichlet process mixture models. In IJCAI 20, pages 2796–2801, 2007. [23] J. Y. A. Wang and E. H. Adelson. Representing moving images with layers. IEEE Trans. IP, 3(5):625– 638, September 1994. [24] J. A. Duan, M. Guindani, and A. E. Gelfand. Generalized spatial Dirichlet process models. Biometrika, 94(4):809–825, 2007. [25] C. Fern´andez and P. J. Green. Modelling spatially correlated data via mixtures: A Bayesian approach. J. R. Stat. Soc. B, 64(4):805–826, 2002. [26] M. A. T. Figueiredo. Bayesian image segmentation using Gaussian field priors. In CVPR Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition, 2005. [27] M. W. Woolrich and T. E. Behrens. Variational Bayes inference of spatial mixture models for segmentation. IEEE Trans. MI, 25(10):1380–1391, October 2006. [28] P. Orbanz and J. M. Buhmann. Smooth image segmentation by nonparametric Bayesian inference. In ECCV, volume 1, pages 444–457, 2006. [29] R. D. Morris, X. Descombes, and J. Zerubia. The Ising/Potts model is not well suited to segmentation tasks. In IEEE DSP Workshop, 1996. [30] R. Unnikrishnan, C. Pantofaru, and M. Hebert. Toward objective evaluation of image segmentation algorithms. IEEE Trans. PAMI, 29(6):929–944, June 2007.
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Shape-Based Object Localization for Descriptive Classification Geremy Heitz1,∗ Gal Elidan2,3,∗ Ben Packer2,∗ Daphne Koller2 1Department of Electrical Engineering, Stanford University 2Department of Computer Science, Stanford University 3Department of Statistics, Hebrew University, Jerusalem {gaheitz,bpacker,koller}@cs.stanford.edu galel@huji.ac.il Abstract Discriminative tasks, including object categorization and detection, are central components of high-level computer vision. Sometimes, however, we are interested in more refined aspects of the object in an image, such as pose or particular regions. In this paper we develop a method (LOOPS) for learning a shape and image feature model that can be trained on a particular object class, and used to outline instances of the class in novel images. Furthermore, while the training data consists of uncorresponded outlines, the resulting LOOPS model contains a set of landmark points that appear consistently across instances, and can be accurately localized in an image. Our model achieves state-of-the-art results in precisely outlining objects that exhibit large deformations and articulations in cluttered natural images. These localizations can then be used to address a range of tasks, including descriptive classification, search, and clustering. 1 Introduction Discriminative questions such as “What is it?” (categorization) and “Where is it?” (detection) are central to machine vision and have received much attention in recent years. In many cases, we are also interested in more refined descriptive questions with regards to an object such as “What is it doing?”, “What is its pose?”, or “What color is its tail?”. For example, we may wish to determine whether a cheetah is running, or whether a giraffe is bending over to drink. In a shopping scenario, we might be interested in searching for lamps that have a particular type of lampshade. In theory it is possible to convert some descriptive questions into discriminative classification tasks given the appropriate labels. Nevertheless, it is preferable to have a single framework in which we can answer a range of questions, some of which may not be known at training time, or may not be discriminative in nature. Intuitively, if we have a good model of what objects in a particular class “look like” and the range of variation that they exhibit, we can make these descriptive distinctions more readily, with a small number of training instances. Furthermore, such a model allows us the flexibility to perform clustering, search, and other forms of exploration of the data. In this paper, we address the goal of finding precise, corresponded localizations of object classes in cluttered images while allowing for large deformations. The Localizing Object Outlines using Probabilistic Shape (LOOPS) method constructs a unified probabilistic model that combines global shape with appearance-based boosted detectors to define a joint distribution over the location of the constituent elements on the object. We can then leverage the object’s shape, an important characteristic that can be used for many descriptive distinctions [9], to address our descriptive tasks. The main challenge is to correspond this model to a novel image while accounting for the possibility of object deformation and articulation. Contour-based methods such as active shape/appearance models (AAMs) [4] were developed with this goal in mind, but typically require good initial guesses and are applied to images with significantly less clutter than real-life photographs. As a result, AAMs have not been successfully used for ∗These authors contributed equally to this manuscript 1 Figure 1: The stages of LOOPS. The shape model is depicted via principal components corresponding to the neck and legs, and the ellipse marks one standard deviation from the mean. Red circles show the location of sample instances in this space. Descriptive tasks other than classification (right box) are described in Section 4. class-level object recognition/analysis. Some works use geometry as a means toward object classification or detection [11, 2, 17, 21]). Since, for example, a misplaced leg has a negligible effect on classification, these works do not attempt to optimize localization. Other works (e.g., [3, 12]) do attempt to accurately localize objects in photographs but only allow for relatively rigid configurations, and cannot capture large deformations such as the articulation of the giraffe’s neck. To the best of our knowledge, no work uses the consistent localization of parts for descriptive tasks. Having a representation of the constituent elements of an object should aid in answering descriptive questions. For example, to decide whether a giraffe is standing upright or bending down to drink, we can use a specific representation of the head, neck, body, and legs in order to consider their relative location. We adopt the AAM-like strategy of representing the shape of an object class via an ordered set of N landmark points that together constitute a piecewise linear contour. Obtaining corresponded training outlines, however, requires painstaking supervision and we would like to be able to use readily available simple outlines such as those in the LabelMe dataset. Therefore, before we begin, we need to automatically augment the simple training outlines with a corresponded labeling. That is, we want to transform arbitrary outlines into useful training instances with consistent elements as depicted in the pipeline of our LOOPS method (Figure 1, first two boxes). The method we use for this step is reminiscent of Hill and Taylor [14]; we omit the details for lack of space. Once we have corresponded training outlines, each with N consistent landmarks, we can construct a distribution of the geometry of the objects’ outline as depicted in Figure 1(middle) and augment this with appearance based features to form a LOOPS model, as described in Section 2. Given a model, we face the computational challenge of localizing the landmarks in test images in the face of clutter, large deformations, and articulations (Figure 1, fourth box). In order to overcome the problem of local maxima faced by contour propagation methods (e.g., [4, 20]), we develop a twostage scheme. We first consider a tractable global search space, consisting of candidate landmark assignments. This allows a discrete probabilistic inference technique to achieve rough but accurate localization that robustly explores the multimodal set of solutions allowed by our large deformation model. We then refine our localization using a continuous hill-climbing approach. This hybrid approach allows LOOPS to deal effectively with complex images of natural scenes, without requiring a good initialization. Preliminary investigations showed that a simpler approach that does a purely local search, similar to the AAMs of Cootes et al. [4], was unable to deal with the challenges of our data. The localization of outlines in test images is described in detail in Section 3. We demonstrate in Section 4 that this localization achieves state-of-the-art results for objects with significant deformation and articulation in natural images. Finally, with the localized outlines in hand, we can readily perform a range of descriptive tasks (classification, ranking, clustering), based on the predicted location of landmarks in test images as well as appearance characteristics in the vicinity of those landmarks. We demonstrate how this is carried out for several descriptive tasks in Section 4. We explore the space of applications facilitated by the LOOPS model across two principal axes. The first concerns the machine learning application: we present results for classification, search (ranking), and clustering. The second axis varies the components that are extracted from the LOOPS outlines for these tasks: we show examples that use the entire object shape, a subcomponent of the object shape, and the appearance of a specific part of the object. The LOOPS framework allows us to approach any of these tasks with a single model without the need for retraining. 2 2 The LOOPS Model Given a set of training instances, each with N corresponded landmarks, the LOOPS object class model combines two components: an explicit representation of the object’s shape (2D silhouette), and a set of image-based features. We define the shape of a class of objects via the locations of the N object landmarks, each of which is assigned to one of the image pixels. We represent such an assignment as a 2N vector of image coordinates which we denote by L. Using the language of Markov random fields [18], the LOOPS model defines a conditional probability distribution over L: P(L | I, Θ) = 1 Z(I)PShape(L; µ, Σ) Y i exp ¡ wiF det i (li; I) ¢ Y i,j exp ³ wijF grad ij (li, lj; I) ´ (1) where Θ = {µ, Σ, w} are the model parameters, and i and j index the model landmarks. PShape encodes the (unnormalized) distribution over the object shape (outline), F det(li) is a landmark specific detector, and F grad ij (li, lj; I) encodes a preference for aligning outline segments along image edges. Below we describe how the shape model and the detector features are learned. We found that our results are quite robust to the choice of weights and that learning them provides no clear benefit. We note that our MRF formulation is quite general, and allows for both the incorporation of (possibly weighted) additional features. For instance, we might want to capture the notion that internal line segments (lines entirely contained within the object) should have low color variability. This can naturally be posed as a pairwise feature over landmarks on opposite sides of the object. We model the shape component of Eq. (1) as a multivariate Gaussian distribution over landmark locations with mean µ and covariance Σ. The Gaussian parametric form has many attractive properties, and has been used successfully to model shape distributions in a variety of applications (e.g., [4, 1]). In our context, one particularly useful property is that the Gaussian distribution decomposes into a product of quadratic terms over pairs of variables: PShape(L | µ, Σ) = 1 Z Y i,j exp µ −1 2(xi −µi)Σ−1 ij (xj −µj) ¶ = 1 Z Y i,j φi,j(xi, xj; µ, Σ), where Z is the normalization term. As this equation illustrates, we can specify potentials φi,j over only singletons and pairs of variables and still manage to represent the full shape distribution. This allows Eq. (1) to take an appealing form in which all terms are defined over at most a two variables. As we discuss below in Section 3, the procedure to locate the model landmarks in an image first involves discrete global inference using the LOOPS model, followed by a local refinement stage. Even if we limit ourselves to pairwise terms, performing discrete inference in a densely connected MRF may be computationally impractical. Unfortunately, a general multivariate Gaussian includes pairwise terms between all landmarks. Thus, during the discrete inference stage, we limit the number of pairwise elements by approximating the shape distribution with a sparse multivariate Gaussian. (During the final refinement stage, we use the full distribution.) To obtain the sparsity pattern, we choose a linear number of landmark pairs whose relative locations have the lowest variance across the training instances (and require that neighbor pairs be included), promoting shape stability. The sparse Gaussian is then obtained by using a gradient method to minimize the KL distance to the full distribution subject to the entries corresponding to the chosen pairs being 0. To construct detector features F det, we build on the success of boosting in state-of-the-art object detection methods [17, 22]. Specifically, we use boosting to learn a strong detector (classifier), Hi for each landmark i. We then define the feature value in the conditional MRF for the assignment of landmark i to pixel li to be F det i (li; I) = Hi(li). For weak detectors we use features that are based on our shape model as well as other features that have proven useful for the task of object detection: shape templates [5], boundary fragments [17], filter response patches [22], and SIFT descriptors [16]. The weak detector ht i(li) is one of these features chosen at round t of boosting that best predicts whether landmark i is at a particular pixel li. Boosting yields a strong detector of the form Hi(li) = PT t=1 αtht i(li). The pairwise feature F grad ij (li, lj; I) = P r∈lilj |g(r)T n(li, lj)| sums over the segment between adjacent landmarks, where g(r) is the image gradient at point r, and n(li, lj) is the segment normal. 3 Figure 2: Example outlines predicted using (candidate) the top detection for each landmark independently, (discrete) inference, (c) a continuous refinement of (b). Candidate ⇒ Discrete ⇒ Refinement 3 Localization of Object Outlines We now address our central computational challenge: assigning the landmarks of a LOOPS model to test image pixels while allowing for large deformations and articulations. Recall that the conditional MRF defines a distribution (Eq. (1)) over assignments of model landmarks to pixels. This allows us to outline objects by using probabilistic inference to find the most probable such assignment: L∗= argmaxLP(L | I, w) Because, in principle, each landmark can be assigned to any pixel, finding L∗is computationally prohibitive. One option is to use an approach analogous to active shape models, using a greedy method to deform the model from a fixed starting point. However, unlike most applications of active shape/appearance models (e.g., [4]), our images have significant clutter, and such an approach will quickly get trapped in an inferior local maxima. A possible solution to this problem is to consider a series of starting points. Preliminary experiments along these lines (not shown for lack of space), however, showed that such an approach requires a computationally prohibitive number of starting points to effectively localize even rigid objects. Furthermore, large articulations were not captured even with the “correct” starting point (placing the mean shape in the center of the true location). To overcome these limitations, we propose an alternative two step method, depicted in Figure 2: we first approximate our problem and find a coarse solution using discrete inference; we then refine our solution using continuous optimization and the full objective defined by Eq. (1). We cannot directly perform inference over the entire seach space of N P assigments (for N model landmarks and P pixels). To prune this space, we first assume that landmarks will fall on “interesting” points, and consider only candidate pixels (typically 1000-2000 per image) found by the SIFT interest operator [16]. We then use the appearance based features F det i to rank the pixel candidates and choose the top K (25) candidate pixels for each landmark. Even with this pruned space, the inference problem is quite daunting, so we further approximate our objective by sparsifying the multivariate Gaussian shape distribution, as mentioned in Section 2. The only pairwise feature functions we use are over neighboring pairs of landmarks (as described in Section 2), which does not add to the density of the MRF construction, thus allowing the inference procedure to be tractable. We perform approximate max-product inference using the Residual Belief Propagation (RBP) algorithm [6] to find the most likely assignment of landmarks to pixels L∗in the pruned space. Given the best assignment L∗predicted in the discrete stage, we perform a refinement stage in which we reintroduce the entire pixel domain and use the full shape distribution. Refinement involves a greedy hill-climbing algorithm in which we iterate across each landmark, moving it to the best candidate location using one of two types of moves, while holding the other landmarks fixed. In a local move, each landmark picks the best pixel in a small window around its current location. In a global move, each landmark can move to its mean location given all the other landmark assignments; this location is the mean of the conditional Gaussian PShape(li | L \ li), easily computed from the joint shape Gaussian. In a typical refinement, the global moves dominate the early iterations, correcting large mistakes made by the discrete stage and that resulted in an unlikely shape. In the later iterations, local moves do most of the work by carefully adapting to the local image characteristics. 4 Experimental Results Our experimental evaluation is aimed at demonstrating the ability of a single LOOPS model to perform a range of tasks based on corresponded localization of objects. In the experiments in the following sections, we train on 20 instances of each class and test on the rest, and report results averaged over 5 random train/test partitions. For the “airplane” image class, we selected examples from the Caltech airplanes image set [8]; the other classes were gathered for this paper. More detailed results, including more object classes and scenes, and an analysis of outline accuracy, appear in [13]. Full image results appear at http://ai.stanford.edu/∼gaheitz/Research/Loops. 4 LOOPS OBJ CUT kAS Detector Figure 3: Randomly selected outlines produced by LOOPS and its two competitors, displaying the variation in the four classes considered in our descriptive classification experiments. Class LOOPS OBJ CUT kAS Airplane 2.0 6.0 3.9 Cheetah 5.2 12.7 11.9 Giraffe 2.9 11.7 8.9 Lamp 2.9 7.5 5.8 Table 1: Normalized symmetric root mean squared (rms) outline error. We report the rms of the distance from each point on the outline to the nearest point on the groundtruth (and vice versa), as a percentage of the groundtruth bounding box diagonal. Accurate Outline Localization In order for a LOOPS model to achieve its goals of classification, search and clustering based on characteristics of the shape or shape-localized appearance, it is necessary for our localization to be accurate at a more refined level than the bounding box prediction that is typical in the literature. We first evaluate the ability of our model to produce accurate outlines in which the model’s landmarks are positioned consistently across test images. We compare LOOPS to two state-of-the-art methods that seek to produce accurate object outlines in cluttered images: the OBJ CUT model of Prasad and Fitzgibbon [19] and the kAS Detector of Ferrari et al. [12]. Both methods were updated to fit our data with help from the authors (P. Kumar, V. Ferrari; personal communications). Unlike both OBJ CUT and LOOPS, the kAS Detector only requires bounding box supervision for the training images rather than full outlines. To provide a quantitative evaluation of the outlines, we measured the symmetric root mean squared (rms) distance between the produced outlines and the hand-labeled groundtruth. As we can see both qualitatively in Figure 3 and quantitatively in Table 1, LOOPS produces significantly more accurate outlines than its competitors. Figure 3 shows two example test images with the outlines for each of the four classes we considered here. While in some cases the LOOPS outline is not perfect at the pixel level, it usually captures the correct articulation, pose, and shape of the object. Descriptive Classification with LOOPS Outlines Our goal is to use the predicted LOOPS outlines for distinguishing between two configurations of an object. To accomplish this, we first train the joint shape and appearance model and perform inference to localize outlines in the test images, all without knowledge of the classification task or any labels. Representing each instance as a corresponded outline provides information that can be leveraged much more easily than the pixel-based representation. We then incorporate the labels to train a descriptive classifier given a corresponded localization. To classify a test image, we used a nearest neighbor classifier, based on chamfer distance. The distance is computed efficiently by converting the training contours into an “ideal” edge image and computing the distance transform of this edge image. The LOOPS outlines are then classified based on their mean distance to each training contour. In addition, we include a GROUND measure that uses the landmark coordinates of manually corresponded groundtruth outlines as features in a logistic regression classifier. This serves as a rough upper bound on the performance achievable by 5 Figure 4: Descriptive classification results. LOOPS is compared to the Naive Bayes and boosted Centroid classifier baselines as well as the state-of-the-art OBJ CUT and kAS Detector methods. GROUND uses manually labeled outlines and approximately upper bounds the performance achievable from outlines. For both lamp tasks, the same LOOPS, OBJ CUT, and kAS Detector models and localizations are used. Note that unlike the other methods, the kAS Detector requires only bounding box supervision rather than full outlines. relying on outlines. In practice, LOOPS can outperform GROUND if the classifier picks up on signals from the automatically chosen landmarks. In addition to the kAS Detector and OBJ CUT competitors, we introduce to two baseline techniques for comparison. The first is a Naive Bayes classifier that uses a codebook of SIFT features as in [7]. The second uses a discriminative approach based on the Centroid detector described above, which is similar to the detector used by [22]; we train the descriptive classifier based on the vector of feature responses at the predicted object centroid. Figure 4 (top) shows the classification results for three tasks: giraffes standing vs. bending down; cheetahs running vs. standing; and airplanes taking off vs. flying horizontally. The first two tasks depend on the articulation of the object, while the third depends on its pose. (In this last task, where rotation is the key feature, we only normalize for translation and scale when performing Procrustes alignment.) Classification performance is shown as a function of the number of labeled instances. For all three tasks, LOOPS (solid blue) outperforms both baselines as well as the state-of-the-art competitors. Importantly, by making use of the outline predicted in a cluttered image, we surpass the fully supervised baselines (rightmost on the graphs) with as little as a single supervised instance (leftmost on the graphs). Once we have outlined instances, an important benefit of the LOOPS method is that we can in fact perform multiple descriptive tasks with the same object model. We demonstrate this with a pair of classification tasks for the lamp object class, presented in Figure 4(bottom). The tasks differ in which “part” of the object we consider for classification: triangular vs. rectangular lamp shade; and thin vs. fat lamp base. By including a few examples in the labeled set, our classifier can learn to consider only the relevant portion of the shape. We stress that both the learned lamp model and the test localizations predicted by LOOPS are the same for both tasks. Only the label set and the resulting nearest-neighbor classifier change. The consequences of this result are promising: we can do most of the work once, and then readily perform a range of descriptive classification tasks. Shape Similarity Search The second descriptive application area that we consider is similarity search, which involves the ranking of test instances based on their similarity to a search query. A shopping website, for example, might wish to allow a user to organize the examples in a database according to similarity to a query product. The similarity measure can be any feature that is easily extracted from the image while leveraging the predicted LOOPS outline. The experimental setup is as follows. Offline, we train a LOOPS model for the object class and localize corresponded outlines in the test images. On6 (a) Figure 5: Clustering airplanes. (a) sample cluster using color features of the entire airplane. (b,c) clusters containing the first two instances of (a) when using only the colors of the tail as predicted by LOOPS. (b) (c) Figure 6: (left) Object similarity search using the LOOPS output to determine the location of the lamp landmarks. (top row) searching the test database using full shape similarity to the query object on the left; (second row) evaluating similarity only using the landmarks that correspond to the lamp shade; (third row) search focused only on the lamp base; (bottom row) using color similarity of the lamp shade to rank the search results. line, a user chooses a test instance to serve as a “query” image and a similarity metric to use. We search for the test images that are most similar to the query, and return the ranked list of images. Figure 6 shows an example from the lamp dataset. Users select a query lamp instance, a subset of landmarks (possibly all), and whether to use shape or color. Each instance in the dataset is then ranked based on Euclidean distance to the query in shape PCA space or LAB color space as appropriate. The top row shows a full-shape search, where the left-most image is the query instance and the others are ordered by decreasing similarity. The second row shows the ranking when the user decides to focus on the lampshade landmarks, yielding only triangular lamp shades, and the third row focuses on the lamp base, returning only wide bases. Finally, the bottom row shows a search based on the color of the shade. In all of these examples, by projecting the images into LOOPS outlines, similarity search desiderata were easily specified and effectively taken into account. The similarity of interest in all of these cases is hard to specify without a predicted outline. Descriptive Clustering Finally, we consider clustering a database by leveraging on the LOOPS predicted outlines. As an example, we consider a large database of airplane images, and wish to group our images into “similar looking” sets of airplanes. Clustering based on shape might produce clusters corresponding to passenger jets, fighter jets, and small propeller airplanes. In this section, we consider an outline and appearance based clustering where the feature vector for each airplane includes the mean color values in the LAB color space for all pixels inside the airplane boundary (or in a region bounded by a user-selected set of landmarks). To cluster images based on this vector, we use standard K-means. Figure 5(left) shows 12 examples from one cluster that results from clustering using the entire plane, for a database of 770 images from the Caltech airplanes image set [8]. Despite the fact that the cluster is coherent when considering the whole plane (not shown), zooming in on the tails reveals that the tails are quite heterogeneous in appearance. Figure 5(middle) and (right) show the tails for the two clusters that contain the first two instances from Figure 5(left), when using only the tail region for clustering. The coherence of the tail appearance is apparent in this case, and both clusters group many tails from the same airlines. In order to perform such coherent clustering of airplane tails, one needs first to accurately localize the tail in test images. Even more than the table lamp ranking task presented above, this example highlights the ability of LOOPS to leverage localize appearance, opening the door for many additional shape and appearance based descriptive tasks. 5 Discussion and Future Work In this work we presented the Localizing Object Outlines using Probabilistic Shape (LOOPS) approach for obtaining accurate, corresponded outlines of objects in test images, with the goal of performing a variety of descriptive tasks. Our approach relies on a coherent probabilistic model in which shape is combined with discriminative detectors. We showed how the produced outlines can 7 be used to perform descriptive classification, search, and clustering based on shape and localized appearance, and we evaluated the error of our outlines compared to two state-of-the-art competitors. For the classification tasks, we showed that our method is superior to fully supervised competitors with as little as a single labeled example. Our contribution is threefold. First, we introduce a model that combines both generative and discriminative elements, allowing us to localize precise outlines of highly articulated objected in cluttered natural images. Second, in order to achieve this localization, we present a hybrid global-discrete then local-continuous optimization approach to the model-to-image correspondence problem. Third, we demonstrate that precise localization is of value for a range of descriptive tasks, including those that are based on appearance. Several existing methods produce outlines either as a by-product of detection (e.g., [3, 17, 21]) or as a targeted goal (e.g., [12, 19]). In experiments above, we compared LOOPS to two state-ofthe-art methods. We showed that LOOPS produces far more accurate outlines when dealing with significant object deformation and articulation, and demonstrated that it is able to translate this into superior classification rates for descriptive tasks. No other work that considers object classes in natural images has demonstrated a combination of accurate localization and shape analysis that has solved these problems. There are further directions to pursue. We would like to automatically learn coherent parts of objects (e.g., the neck of the giraffe) as a set of landmarks that articulate together, and achieve better localization by estimating a distribution over part articulation (e.g., synchronized legs). A natural extension of our model is a scene-level variant in which each object is treated as a “landmark.” The geometry of such a model will then capture relative spatial location and orientations so that we can answer questions such as whether a man is walking the dog, or whether the dog is chasing the man. Acknowledgements This work was supported by the DARPA Transfer Learning program under contract number FA8750-05-2-0249 and the Multidisciplinary University Research Initiative (MURI), contract number N000140710747, managed by the Office of Naval Research. We would also like to thank Vittorio Ferrari and Pawan Kumar for providing us code and helping us to get their methods working on our data. References [1] D. Anguelov, P. Srinivasan, D. Koller, S. Thrun, J. Rodgers, and J. Davis. Scape: shape completion and animation of people. SIGGRAPH, ’05. 3 [2] A. Bar-Hillel, T. Hertz, D. Weinshall. Efficient learning of relational object class models. ICCV, ’05. 2 [3] A. Berg, T. Berg, and J. Malik. Shape matching and object recognition using low distortion correspondence. CVPR, ’05. 2, 8 [4] T. Cootes, G. Edwards, and C. Taylor. Active appearance models. ECCV, ’98. 1, 2, 3, 4 [5] G. Elidan, G. Heitz, and D. Koller. Learning object shape: From cartoons to images. CVPR, ’06. 3 [6] G. Elidan, I. McGraw, and D. Koller. Residual belief propagation: Informed scheduling for async. message passing. UAI, ’06. 4 [7] L. Fei-Fei and P. Perona. A bayesian hier. model for learning natural scene categories. CVPR, ’05. 6 [8] L. Fei-Fei, R. Fergus, and P. Perona. Learning generative visual models from few training examples: an incremental bayesian approach tested on 101 object categories. CVPR, ’04. 4, 7 [9] P. Felzenszwalb and D. Huttenlocher. Efficient matching of pictorial structures. CVPR, ’00. 1 [10] P. Felzenszwalb and J. Schwartz. Hierarchical matching of deformable shapes. CVPR, ’07. [11] R. Fergus, P. Perona, and A. Zisserman. Object class recognition by unsupervised scale-invariant learning. CVPR, ’03. 2 [12] V. Ferrari, F. Jurie, and C. Schmid. Accurate object detection with deformable shape models learnt from images. CVPR, ’07. 2, 5, 8 [13] G. Heitz, G. Elidan, B. Packer, and D. Koller. Shape-based object localization for descriptive classification. Technical report, available at http://ai.stanford.edu/˜gaheitz/Research/Loops/TR.pdf 4 [14] A. Hill and C. Taylor. Non-rigid corresp. for automatic landmark identification. BMVC, ’96. 2 [15] A. Holub and P. Perona. A discriminative framework for modeling object class. CVPR, ’05. [16] D. Lowe. Distinctive image features from scale-invariant keypoints. IJCV, ’03. 3, 4 [17] A. Opelt, A. Pinz, and A. Zisserman. Incremental learning of object detectors using a visual shape alphabet. CVPR, ’06. 2, 3, 8 [18] J. Pearl. Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, ’88. 3 [19] M. Prasad and A. Fitzgibbon. Single view reconstruction of curved surfaces. CVPR 2006, ’06. 5, 8 [20] J. Sethian. Level Set Methods and Fast Marching Methods. Cambridge, ’98. 2 [21] J. Shotton, A. Blake, and R. Cipolla. Contour-based learning for object detection. ICCV, ’05. 2, 8 [22] A. Torralba, K. Murphy, and W. Freeman. Contextual models for object detection using boosted random fields. 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Covariance Estimation for High Dimensional Data Vectors Using the Sparse Matrix Transform Guangzhi Cao Charles A. Bouman School of Electrical and Computer Enigneering Purdue University West Lafayette, IN 47907 {gcao, bouman}@purdue.edu Abstract Covariance estimation for high dimensional vectors is a classically difficult problem in statistical analysis and machine learning. In this paper, we propose a maximum likelihood (ML) approach to covariance estimation, which employs a novel sparsity constraint. More specifically, the covariance is constrained to have an eigen decomposition which can be represented as a sparse matrix transform (SMT). The SMT is formed by a product of pairwise coordinate rotations known as Givens rotations. Using this framework, the covariance can be efficiently estimated using greedy minimization of the log likelihood function, and the number of Givens rotations can be efficiently computed using a cross-validation procedure. The resulting estimator is positive definite and well-conditioned even when the sample size is limited. Experiments on standard hyperspectral data sets show that the SMT covariance estimate is consistently more accurate than both traditional shrinkage estimates and recently proposed graphical lasso estimates for a variety of different classes and sample sizes. 1 Introduction Many problems in statistical pattern recognition and analysis require the classification and analysis of high dimensional data vectors. However, covariance estimation for high dimensional vectors is a classically difficult problem because the number of coefficients in the covariance grows as the dimension squared [1, 2]. This problem, sometimes referred to as the curse of dimensionality [3], presents a classic dilemma in statistical pattern analysis and machine learning. In a typical application, one measures n versions of a p dimensional vector. If n < p, then the sample covariance matrix will be singular with p −n eigenvalues equal to zero. Over the years, a variety of techniques have been proposed for computing a nonsingular estimate of the covariance. For example, regularized and shrinkage covariance estimators [4, 5, 6] are examples of such techniques. In this paper, we propose a new approach to covariance estimation, which is based on constrained maximum likelihood (ML) estimation of the covariance [7]. In particular, the covariance is constrained to have an eigen decomposition which can be represented as a sparse matrix transform (SMT) [8, 9]. The SMT is formed by a product of pairwise coordinate rotations known as Givens rotations [10]. Using this framework, the covariance can be efficiently estimated using greedy minimization of the log likelihood function, and the number of Givens rotations can be efficiently computed using a cross-validation procedure. The estimator obtained using this method is always positive definite and well-conditioned even when the sample size is limited. In order to validate our model, we perform experiments using a standard set of hyperspectral data [11], and we compare against both traditional shrinkage estimates and recently proposed graphical lasso estimates [12] for a variety of different classes and sample sizes. Our experiments show that, 1 for this example, the SMT covariance estimate is consistently more accurate. The SMT method also has a number of other advantages. It seems to be particularly good when estimating small eigenvalues and their associated eigenvectors. The cross-validation procedure used to estimate the SMT model order requires little additional computation, and the resulting eigen decomposition can be computed with very little computation (i.e. ≪p2 operations). 2 Covariance estimation for high dimensional vectors In the general case, we observe a set of n vectors, y1, y2, · · · , yn, where each vector, yi, is p dimensional. Without loss of generality, we assume yi has zero mean. We can represent this data as the following p × n matrix Y = [y1, y2, · · · , yn] . (1) If the vectors yi are identically distributed, then the sample covariance is given by S = 1 nY Y t , (2) and S is an unbiased estimate of the true covariance matrix with R = E [yiyt i] = E[S]. While S is an unbiased estimate of R it is also singular when n < p. This is a serious deficiency since as the dimension p grows, the number of vectors needed to estimate R also grows. In practical applications, n may be much smaller than p which means that most of the eigenvalues of R are erroneously estimated as zero. A variety of methods have been proposed to regularize the estimate of R so that it is not singular. Shrinkage estimators are a widely used class of estimators which regularize the covariance matrix by shrinking it toward some target structures [4, 5, 13]. Shrinkage estimators generally have the form ˆR = αD +(1−α)S, where D is some positive definite matrix. Some popular choices for D are the identity matrix (or its scaled version) [5, 13] and the diagonal entries of S, i.e. diag(S) [5, 14]. In both cases, the shrinkage intensity α can be estimated using cross-validation or boot-strap methods. Recently, a number of methods have been proposed for regularizing the estimate by making either the covariance or its inverse sparse [6, 12]. For example, the graphical lasso method enforces sparsity by imposing an L1 norm constraint on the inverse covariance [12]. Banding or thresholding can also be used to obtain a sparse estimate of the covariance [15]. 2.1 Maximum likelihood covariance estimation Our approach will be to compute a constrained maximum likelihood (ML) estimate of the covariance R, under the modeling assumption that eigenvectors of R may be represented as a sparse matrix transform (SMT) [8, 9]. To do this, we first decompose R as R = EΛEt , (3) where E is the orthonormal matrix of eigenvectors and Λ is the diagonal matrix of eigenvalues. Then we will estimate the covariance by maximizing the likelihood of the data Y subject to the constraint that E is an SMT. By varying the order, K, of the SMT, we may then reduce or increase the regularizing constraint on the covariance. If we assume that the columns of Y are independent and identically distributed Gaussian random vectors with mean zero and positive-definite covariance R, then the likelihood of Y given R is given by pR(Y ) = 1 (2π) np 2 |R|−n 2 exp −1 2tr{Y tR−1Y } . (4) The log-likelihood of Y is then given by [7] log p(E,Λ)(Y ) = −n 2 tr{diag(EtSE)Λ−1} −n 2 log |Λ| −np 2 log(2π) , (5) where R = EΛEt is specified by the orthonormal eigenvector matrix E and diagonal eigenvalue matrix Λ. Jointly maximizing the likelihood with respect to E and Λ then results in the ML estimates 2 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 y0 y1 y2 y3 y4 y5 y6 y7 ˜y0 ˜y1 ˜y2 ˜y3 ˜y4 ˜y5 ˜y6 ˜y7 W 0 8 W 2 8 W 0 8 W 1 8 W 2 8 W 3 8 W 0 8 W 2 8 W 0 8 W 0 8 W 0 8 W 0 8 (a) FFT y0 ˜y0 ˜y1 ˜yp−2 ˜yp−1 ˜y2 ˜y3 ˜yp−3 ˜yp−4 E5 E6 y1 y2 y3 E2 E1 E0 E3 E4 EK−1 yp−4 yp−3 yp−2 yp−1 (b) SMT Figure 1: (a) 8-point FFT. (b) The SMT implementation of ˜y = Ey. The SMT can be viewed as a generalization of FFT and orthonormal wavelet transforms. of E and Λ given by [7] ˆE = arg min E∈Ω diag(EtSE) (6) ˆΛ = diag( ˆEtS ˆE) , (7) where Ωis the set of allowed orthonormal transforms. So we may compute the ML estimate by first solving the constrained optimization of (6), and then computing the eigenvalue estimates from (7). 2.2 ML estimation of eigenvectors using SMT model The ML estimate of E can be improved if the feasible set of eigenvector transforms, Ω, can be constrained to a subset of all possible orthonormal transforms. By constraining Ω, we effectively regularize the ML estimate by imposing a model. However, as with any model-based approach, the key is to select a feasible set, Ω, which is as small as possible while still accurately modeling the behavior of the data. Our approach is to select Ωto be the set of all orthonormal transforms that can be represented as an SMT of order K [9]. More specifically, a matrix E is an SMT of order K if it can be written as a product of K sparse orthornormal matrices, so that E = k=K−1 Y 0 Ek = E0E1 · · · EK−1 , (8) where every sparse matrix, Ek, is a Givens rotation operating on a pair of coordinate indices (ik, jk) [10]. Every Givens rotation Ek is an orthonormal rotation in the plane of the two coordinates, ik and jk, which has the form Ek = I + Θ(ik, jk, θk) , (9) where Θ(ik, jk, θk) is defined as [Θ]ij = cos(θk) −1 if i = j = ik or i = j = jk sin(θk) if i = ik and j = jk −sin(θk) if i = jk and j = ik 0 otherwise . (10) Figure 1(b) shows the flow diagram for the application of an SMT to a data vector y. Notice that each 2D rotation, Ek, plays a role analogous to a “butterfly” used in a traditional fast Fourier transform (FFT) [16] in Fig. 1(a). However, unlike an FFT, the organization of the butterflies in an SMT is unstructured, and each butterfly can have an arbitrary rotation angle θk. This more general structure allows an SMT to implement a larger set of orthonormal transformations. In fact, the SMT can be used to represent any orthonormal wavelet transform because, using the theory of paraunitary wavelets, orthonormal wavelets can be represented as a product of Givens rotations and delays [17]. More generally, when K = p 2 , the SMT can be used to exactly represent any p × p orthonormal transformation [7]. 3 Using the SMT model constraint, the ML estimate of E is given by ˆE = arg min E=Qk=K−1 0 Ek diag(EtSE) . (11) Unfortunately, evaluating the constrained ML estimate of (11) requires the solution of an optimization problem with a nonconvex constraint. So evaluation of the globally optimal solutions is difficult. Therefore, our approach will be to use greedy minimization to compute a locally optimal solution to (11). The greedy minimization approach works by selecting each new butterfly Ek to minimize the cost, while fixing the previous butterflies, El for l < k. This greedy optimization algorithm can be implemented with the following simple recursive procedure. We start by setting S0 = S to be the sample covariance, and initialize k = 0. Then we apply the following two steps for k = 0 to K −1. E∗ k = arg min Ek diag Et kSkEk (12) Sk+1 = E∗t k SkE∗ k . (13) The resulting values of E∗ k are the butterflies of the SMT. The problem remains of how to compute the solution to (12). In fact, this can be done quite easily by first determining the two coordinates, ik and jk, that are most correlated, (ik, jk) ←arg min (i,j) 1 − [Sk]2 ij [Sk]ii[Sk]jj ! . (14) It can be shown that this coordinate pair, (ik, jk), can most reduce the cost in (12) among all possible coordinate pairs [7]. Once ik and jk are determined, we apply the Givens rotation E∗ k to minimize the cost in (12), which is given by E∗ k = I + Θ(ik, jk, θk) , (15) where θk = 1 2atan(−2[Sk]ikjk, [Sk]ikik −[Sk]jkjk) . (16) By iterating the (12) and (13) K times, we obtain the constrained ML estimate of E given by ˆE = k=K−1 Y 0 E∗ k . (17) The model order, K, can be determined by a simple cross-validation procedure. For example, we can partition the data into three subsets, and K is chosen to maximize the average likelihood of the left-out subsets given the estimated covariance using the other two subsets. Once K is determined, the proposed covariance estimator is re-computed using all the data and the estimated model order. The SMT covariance estimator obtained as above has some interesting properties. First, it is positive definite even for the limited sample size n < p. Also, it is permutation invariant, that is, the covariance estimator does not depend on the ordering of the data. Finally, the eigen decomposition Ety can be computed very efficiently by applying the K sparse rotations in sequence. 2.3 SMT Shrinkage Estimator In some cases, the accuracy of the SMT estimator can be improved by shrinking it towards the sample covariance. Let ˆRSMT represent the SMT covariance estimator. Then the SMT shrinkage estimate (SMT-S) can be obtained as ˆRSMT −S = α ˆRSMT + (1 −α)S , (18) where the parameter α can be computed using cross validation. Notice that p ˆ RSMT −S(Y ) = p ˆ E ˆ RSMT −S ˆ Et( ˆEY ) = pαˆΛ+(1−α) ˆ ES ˆ Et( ˆEY ) . (19) So cross validation can be efficiently implemented as in [5]. 4 3 Experimental results The effectiveness of the SMT covariance estimation depends on how well the SMT model can capture the behavior of real data vectors. Therefore in this section, we compare the performance of the SMT covariance estimator to commonly used shrinkage and graphical lasso estimators. We do this comparison using hyperspectral remotely sensed data as our high dimensional data vectors. The hyperspectral data we use is available with the recently published book [11]. Figure 2(a) shows a simulated color IR view of an airborne hyperspectral data flightline over the Washington DC Mall. The sensor system measured the pixel response in 191 effective bands in the 0.4 to 2.4 µm region of the visible and infrared spectrum. The data set contains 1208 scan lines with 307 pixels in each scan line. The image was made using bands 60, 27 and 17 for the red, green and blue colors, respectively. The data set also provides ground truth pixels for five classes designated as grass, water, roof, street, and tree. In Fig. 2(a), the ground-truth pixels of the grass class are outlined with a white rectangle. Figure 2(b) shows the spectrum of the grass pixels, and Fig. 2(c) shows multivariate Gaussian vectors that were generated using the measured sample covariance for the grass class. For each class, we computed the “true” covariance by using all the ground truth pixels to calculate the sample covariance. The covariance is computed by first subtracting the sample mean vector for each class, and then computing the sample covariance for the zero mean vectors. The number of pixels for the ground-truth classes of grass, water, roof, street, and tree are 1928, 1224, 3579, 416, and 388, respectively. In each case, the number of ground truth pixels was much larger than 191, so the true covariance matrices are nonsingular, and accurately represent the covariance of the hyperspectral data for that class. 3.1 Review of alternative estimators A popular choice of the shrinkage target is the diagonal of S [5, 14]. In this case, the shrinkage estimator is given by ˆR = αdiag (S) + (1 −α) S . (20) We use an efficient algorithm implementation of the leave-one-out likelihood (LOOL) crossvalidation method to choose α as suggested in [5]. An alternative estimator is the graphic lasso (glasso) estimate recently proposed in [12] which is an L1 regularized maximum likelihood estimate, such that ˆR = arg max R∈Ψ log(Y | R) −ρ ∥R−1 ∥1 , (21) where Ψ denotes the set of p × p positive definite matrices and ρ the regularization parameter. We used the R code for glasso that is publically available online. We found cross-validation estimation of ρ to be difficult, so in each case we manually selected the value of ρ to minimize the KullbackLeibler distance to the known covariance. 3.2 Gaussian case First, we compare how different estimators perform when the data vectors are samples from an ideal multivariate Gaussian distribution. To do this, we first generated zero mean multivariate vectors with the true covariance for each of the five classes. Next we estimated the covariance using the four methods, the shrinkage estimator, glasso, SMT and SMT shrinkage estimation. In order to determine the effect of sample size, we also performed each experiment for a sample size of n = 80, 40, and 20, respectively. Every experiment was repeated 10 times. In order to get an aggregate accessment of the effectiveness of SMT covariance estimation, we compared the estimated covariance for each method to the true covariance using the Kullback-Leibler (KL) distance [7]. The KL distance is a measure of the error between the estimated and true distribution. Figure 3(a)(b) and (c) show plots of the KL distances as a function of sample size for the four estimators. The error bars indicate the standard deviation of the KL distance due to random variation in the sample statistics. Notice that the SMT shrinkage (SMT-S) estimator is consistently the best of the four. 5 (a) (b) (c) Figure 2: (a) Simulated color IR view of an airborne hyperspectral data over the Washington DC Mall [11]. (b) Ground-truth pixel spectrum of grass that are outlined with the white rectangles in (a). (c) Synthesized data spectrum using the Gaussian distribution. Figure 4(a) shows the estimated eigenvalues for the grass class with n = 80. Notice that the eigenvalues of the SMT and SMT-S estimators are much closer to the true values than the shrinkage and glasso methods. Notice that the SMT estimators generate good estimates especially for the small eigenvalues. Table 1 compares the computational complexity, CPU time and model order for the four estimators. The CPU time and model order were measured for the Guassian case of the grass class with n = 80. Notice that even with the cross validation, the SMT and SMT-S estimators are much faster than glasso. This is because the SMT transform is a sparse operator. In this case, the SMT uses an average of K = 495 rotations, which is equal to K/p = 495/191 = 2.59 rotations (or equivalently multiplies) per spectral sample. 3.3 Non-Gaussian case In practice, the sample vectors may not be from an ideal multivariate Gaussian distribution. In order to see the effect of the non-Gaussian statistics on the accuracy of the covariance estimate, we performed a set of experiments which used random samples from the ground truth pixels as input. Since these samples are from the actual measured data, their distribution is not precisely Gaussian. Using these samples, we computed the covariance estimates for the five classes using the four different methods with sample sizes of n = 80, 40, and 20. Plots of the KL distances for the non-Gaussian grass case1are shown in Fig. 3(d)(e) and (f); and Figure 4(b) shows the estimated eigenvalues for grass with n = 80. Note that the results are similar to those found for the ideal Guassian case. 4 Conclusion We have proposed a novel method for covariance estimation of high dimensional data. The new method is based on constrained maximum likelihood (ML) estimation in which the eigenvector transformation is constrained to be the composition of K Givens rotations. This model seems to capture the essential behavior of the data with a relatively small number of parameters. The constraint set is a K dimensional manifold in the space of orthonormal transforms, but since it is not a linear space, the resulting ML estimation optimization problem does not yield a closed form global optimum. However, we show that a recursive local optimization procedure is simple, intuitive, and yields good results. We also demonstrate that the proposed SMT covariance estimation methods substantially reduce the error in the covariance estimate as compared to current state-of-the-art estimates for a standard hyperspectral data set. The MATLAB code for SMT covariance estimation is available at: https://engineering.purdue.edu/∼bouman/publications/pub smt.html. 1In fact, these are the KL distances between the estimated covariance and the sample covariance computed from the full set of training data, under the assumption of a multivariate Gaussian distribution. 6 10 20 30 40 50 60 70 80 90 60 80 100 120 140 160 180 200 220 240 260 KL distance Sample size Shrinkage Estimator Glasso Estimator SMT Estimator SMT−S Estimator (a) Grass 10 20 30 40 50 60 70 80 90 20 40 60 80 100 120 140 160 KL distance Sample size Shrinkage Estimator Glasso Estimator SMT Estimator SMT−S Estimator (b) Water 10 20 30 40 50 60 70 80 90 40 60 80 100 120 140 160 180 200 220 240 KL distance Sample size Shrinkage Estimator Glasso Estimator SMT Estimator SMT−S Estimator (c) Street 10 20 30 40 50 60 70 80 90 60 80 100 120 140 160 180 200 220 240 260 KL distance Sample size Shrinkage Estimator Glasso Estimator SMT Estimator SMT−S Estimator (d) Grass 10 20 30 40 50 60 70 80 90 20 40 60 80 100 120 140 160 180 200 220 KL distance Sample size Shrinkage Estimator Glasso Estimator SMT Estimator SMT−S Estimator (e) Water 10 20 30 40 50 60 70 80 90 40 60 80 100 120 140 160 180 200 220 240 KL distance Sample size Shrinkage Estimator Glasso Estimator SMT Estimator SMT−S Estimator (f) Street Figure 3: Kullback-Leibler distance from true distribution versus sample size for various classes: (a) (b) (c) Gaussian case (d) (e) (f) non-Gaussian case. 0 50 100 150 200 10 −2 10 0 10 2 10 4 10 6 10 8 index eigenvalues True Eigenvalues Shrinkage Estimator Glasso Estimator SMT Estimator SMT−S Estimator (a) 0 50 100 150 200 10 −2 10 0 10 2 10 4 10 6 10 8 index eigenvalues True Eigenvalues Shrinkage Estimator Glasso Estimator SMT Estimator SMT−S Estimator (b) Figure 4: The distribution of estimated eigenvalues for the grass class with n = 80: (a) Gaussian case (b) Non-Gaussian case. Complexity CPU time Model order (without cross-validation) (seconds) Shrinkage Est. p 8.6 (with cross-validation) 1 glasso p3I 422.6 (without cross-validation) 4939 SMT p2 + Kp 6.5 (with cross-validation) 495 SMT-S p2 + Kp 7.2 (with cross-validation) 496 Table 1: Comparison of computational complexity, CPU time and model order for various covariance estimators. The complexity is without cross validation and does not include the computation of the sample covariance (order of np2). The CPU time and model order were measured for the Guassian case of the grass class with n = 80. I is the number of cycles used in glasso. 7 Acknowledgments This work was supported by the National Science Foundation under Contract CCR-0431024. We would also like to thank James Theiler (J.T.) and Mark Bell for their insightful comments and suggestions. References [1] C. Stein, B. Efron, and C. Morris, “Improving the usual estimator of a normal covariance matrix,” Dept. of Statistics, Stanford University, Report 37, 1972. [2] K. Fukunaga, Introduction to Statistical Pattern Recognition. Boston, MA: Academic Press, 1990, 2nd Ed. [3] A. K. Jain, R. P. Duin, and J. Mao, “Statistical pattern recognition: A review,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 22, no. 1, pp. 4–37, 2000. [4] J. H. Friedman, “Regularized discriminant analysis,” Journal of the American Statistical Association, vol. 84, no. 405, pp. 165–175, 1989. [5] J. P. Hoffbeck and D. A. Landgrebe, “Covariance matrix estimation and classification with limited training data,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 18, no. 7, pp. 763–767, 1996. [6] P. J. Bickel and E. Levina, “Regularized estimation of large covariance matrices,” Annals of Statistics, vol. 36, no. 1, pp. 199–227, 2008. [7] G. Cao and C. A. Bouman, “Covariance estimation for high dimensional data vectors using the sparse matrix transform,” Purdue University, Technical Report ECE 08-05, 2008. [8] G. Cao, C. A. Bouman, and K. J. Webb, “Fast reconstruction algorithms for optical tomography using sparse matrix representations,” in Proceedings of 2007 IEEE International Symposium on Biomedical Imaging, April 2007. [9] ——, “Non-iterative MAP reconstruction using sparse matrix representations,” (submitted to) IEEE Trans. on Image Processing. [10] W. Givens, “Computation of plane unitary rotations transforming a general matrix to triangular form,” Journal of the Society for Industrial and Applied Mathematics, vol. 6, no. 1, pp. 26–50, March 1958. [11] D. A. Landgrebe, Signal Theory Methods in Multispectral Remote Sensing. New York: WileyInterscience, 2005. [12] J. Friedman, T. Hastie, and R. Tibshirani, “Sparse inverse covariance estimation with the graphical lasso,” Biostatistics, vol. 9, no. 3, pp. 432–441, Jul. 2008. [13] M. J. Daniels and R. E. Kass, “Shrinkage estimators for covariance matrices,” Biometrics, vol. 57, no. 4, pp. 1173–1184, 2001. [14] J. Schafer and K. Strimmer, “A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics,” Statistical Applications in Genetics and Molecular Biology, vol. 4, no. 1, 2005. [15] P. J. Bickel and E. Levina, “Covariance regularization by thresholding,” Department of Statistics, UC Berkeley, Technical Report 744, 2007. [16] J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Mathematics of Computation, vol. 19, no. 90, pp. 297–301, April 1965. [17] A. Soman and P. Vaidyanathan, “Paraunitary filter banks and wavelet packets,” Acoustics, Speech, and Signal Processing, 1992. ICASSP-92., 1992 IEEE International Conference on, vol. 4, pp. 397–400 vol.4, Mar 1992. 8
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Characterizing neural dependencies with copula models Pietro Berkes Volen Center for Complex Systems Brandeis University, Waltham, MA 02454 berkes@brandeis.edu Frank Wood and Jonathan Pillow Gatsby Computational Neuroscience Unit, UCL London WC1N 3AR, UK {fwood,pillow}@gatsby.ucl.ac.uk Abstract The coding of information by neural populations depends critically on the statistical dependencies between neuronal responses. However, there is no simple model that can simultaneously account for (1) marginal distributions over single-neuron spike counts that are discrete and non-negative; and (2) joint distributions over the responses of multiple neurons that are often strongly dependent. Here, we show that both marginal and joint properties of neural responses can be captured using copula models. Copulas are joint distributions that allow random variables with arbitrary marginals to be combined while incorporating arbitrary dependencies between them. Different copulas capture different kinds of dependencies, allowing for a richer and more detailed description of dependencies than traditional summary statistics, such as correlation coefficients. We explore a variety of copula models for joint neural response distributions, and derive an efficient maximum likelihood procedure for estimating them. We apply these models to neuronal data collected in macaque pre-motor cortex, and quantify the improvement in coding accuracy afforded by incorporating the dependency structure between pairs of neurons. We find that more than one third of neuron pairs shows dependency concentrated in the lower or upper tails for their firing rate distribution. 1 Introduction An important problem in systems neuroscience is to develop flexible, statistically accurate models of neural responses. The stochastic spiking activity of individual neurons in cortex is often well described by a Poisson distribution. Responses from multiple neurons also exhibit strong dependencies (i.e., correlations) due to shared input noise and lateral network interactions. However, there is no natural multivariate generalization of the Poisson distribution. For this reason, much of the literature on population coding has tended either to ignore correlations entirely, treating neural responses as independent Poisson random variables [1, 2], or to adopt a Gaussian model of joint responses [3, 4], assuming a parametric form for dependencies but ignoring key features (e.g., discreteness, non-negativity) of the marginal distribution. Recent work has focused on the construction of large parametric models that capture inter-neuronal dependencies using generalized linear point-process models [5, 6, 7, 8, 9] and binary second-order maximum-entropy models [10, 11, 12]. Although these approaches are quite powerful, they model spike trains only in very fine time bins, and thus describe the dependencies in neural spike count distributions only implicitly. Modeling the joint distribution of neural activities is therefore an important open problem. Here we show how to construct non-independent joint distributions over firing rates using copulas. In particular, this approach can be used to combine arbitrary marginal firing rate distributions. The development of the paper is as follows: in Section 2, we provide a basic introduction to copulas; in Section 3, we derive a maximum likelihood estimation procedure for neural copula models, in Sections 4 and 5, we apply these models to physiological data collected in macaque pre-motor 1 Figure 1: Samples drawn from a joint distribution defined using the dependency structure of a bivariate Gaussian distribution and changing the marginal distributions. Top row: The marginal distributions (the leftmost marginal is uniform, by definition of copula). Bottom row: The log-density function of a Gaussian copula, and samples from the joint distribution defined as in Eq. 2. cortex; finally, in Section 6 we review the insights provided by neural copula models and discuss several extensions and future directions. 2 Copulas A copula C(u1, . . . , un) : [0, 1]n →[0, 1] is a multivariate distribution function on the unit cube with uniform marginals [13, 14]. The basic idea behind copulas is quite simple, and is closely related to that of histogram equalization: for a random variable yi with continuous cumulative distribution function (cdf) Fi, the random variable ui := Fi(yi) is uniformly distributed on the interval [0, 1]. One can use this basic property to separate the marginals from the dependency structure in a multivariate distribution: the full multivariate distribution is standardized by projecting each marginal onto one axis of the unit hyper-cube, and leaving one with a distribution on the hyper-cube (the copula, by definition) that represent dependencies in the marginals’ quantiles. This intuition has been formalized in Sklar’s Theorem [15]: Theorem 1 (Sklar, 1959) Given u1, . . . , un random variables with continuous distribution functions F1, . . . , Fn and joint distribution F, there exist a unique copula C such that for all ui: C(u1, . . . , un) = F(F −1 1 (u1), . . . , F −1 n (un)) (1) Conversely, given any distribution functions F1, . . . , Fn and copula C, F(y1, . . . , yn) = C(F1(y1), . . . , Fn(yn)) (2) is a n-variate distribution function with marginal distribution functions F1, . . . , Fn. This result gives a way to derive a copula given the joint and marginal distributions (using Eq. 1), and also, more importantly here, to construct a joint distribution by specifying the marginal distributions and the dependency structure separately (Eq. 2). For example, one can keep the dependency structure fixed and vary the marginals (Fig. 1), or vice versa given fixed marginal distributions define new joint distributions using parametrized copula families (Fig. 2). For illustration, in this paper we are going to consider only the bivariate case. All the methods, however, generalize straightforwardly to the multivariate case. Since copulas do not depend on the marginals, one can define in this way dependency measures that are insensitive to non-linear transformations of the individual variables [14] and generalize correlation coefficients, which are only appropriate for elliptic distributions. The copula representation has also been used to estimate the conditional entropy of neural latencies by separating the contribution of the individual latencies from that coming from their correlations [16]. Dependencies structures are specified by parametric copula families. One notable example is the Gaussian copula, which generalizes the dependency structure of the multivariate Gaussian distribution to arbitrary marginal distribution (Fig. 1), and is defined as C(u1, u2; Σ) = ΦΣ ! φ−1(u1), φ−1(u2) " , (3) 2 Figure 2: Samples drawn from a joint distribution with fixed Gaussian marginals and dependency structure defined by parametric copula families, as indicated by the labels. Top row: log-density function for three copula families. Bottom row: Samples from the joint distribution (Eq. 2). Gaussian CN Σ (u1, u2) = ΦΣ ! φ−1(u1), φ−1(u2) " Frank CF r θ (u1, u2) = −1 θ log # 1 + (e−θu1−1)(e−θu2−1) e−θ−1 $ Clayton CCl θ (u1, u2) = (u−θ 1 + u−θ 2 −1)−1/θ, θ > 0 Clayton negative CNeg θ (u1, u2) = max % (u−θ 1 + u−θ 2 −1), 0 &−1/θ, −1 ≤θ < 0 Gumbel CGu θ (u1, u2) = exp ! −(˜uθ 1 + ˜uθ 2)1/θ" , ˜uj = −log uj, θ ≥1 Table 1: Definition of families of copula distribution functions. where φ(u) is the cdf of the univariate Gaussian with mean 0 and variance 1, and ΦΣ is the cdf of a standard multivariate Gaussian with mean 0 and covariance matrix Σ. Other families derive from the economics literature, and are typically one-parameter families that capture various possible dependencies, for example dependencies only in one of the tails of the distribution. Table 1 shows the definition of the copula distributions used in this paper (see [14], for an overview of known copulas and copula construction methods). 3 Maximum Likelihood estimation for discrete marginal distributions In the case where the random variables have discrete distribution functions, as in the case of neural firing rates, only a weaker version of Theorem 1 is valid: there always exists a copula that satisfies Eq. 2, but it is no longer guaranteed to be unique [17]. With discrete data, the probability of a particular outcome is determined by an integral over the region of [0, 1]n corresponding to that outcome; any two copulas that integrate to the same values on all such regions produce the same joint distribution. We can derive a Maximum Likelihood (ML) estimation of the parameters θ by considering a generative model where uniform marginals are generated from the copula density, and in turn use these to generate the discrete variables deterministically using the inverse (marginal) distribution functions, as in Fig. 3. These marginals can be given by the empirical cumulative distribution of firing rates (as in this paper) or by any parametrized family of univariate distributions (such as Poisson). The ML equation then becomes argmax θ p(y|θ) = argmax θ ' p(y|u)p(u|θ)du (4) = argmax θ ' F1(y1) F1(y1−1) · · · ' Fn(yn) Fn(yn−1) cθ(u1, . . . , un) du , (5) 3 θ u y λ p(u|θ) = cθ(u1, . . . , un) p(yi|u, λ) = ! 1, yi = F −1 i (ui; λi) 0, otherwise Figure 3: Graphical representation of the copula model with discrete marginals. Uniform marginals u are drawn from the copula density function cθ(u1, . . . , un), parametrized by θ. The discrete marginals are then generated deterministically using the inverse cdf of the marginals, which are parametrized by λ. −1 −0.5 0 0.5 1 −0.3 0 0.3 Gaussian 0 1 2 3 4 5 −0.3 0 0.3 Clayton 1 2 3 4 5 6 7 8 9 10 −0.3 0 0.3 Gumbel −10 −5 0 5 10 −0.3 0 0.3 Frank Figure 4: Distribution of the maximum likelihood estimation of the parameters of four copula families, for various setting of their parameter (x-axis). On the y-axis, estimates are centered such that 0 corresponds to an unbiased estimate. Error bars are one standard deviation of the estimate. where Fi can depend on additional parameters λi. The last equation is the copula probability mass inside the volume defined by the vertices Fi(yi) and Fi(yi −1), and can be readily computed using the copula distribution Cθ(u1, . . . , un). For example, in the bivariate case one obtains argmax θ p(y1, y2|θ) = argmax θ ( Cθ(u1, u2) + Cθ(u− 1 , u− 2 ) −Cθ(u− 1 , u2) −Cθ(u1, u− 2 ) ) , (6) where ui = Fi(yi) and u− i = Fi(yi −1). ML optimization can be performed using standard methods, like gradient descent. In the bivariate case, we find that optimization using the standard MATLAB optimization routines is relatively efficient. Given neural data in the form of firing rates y1, y2 from a pair of neurons, we collect the empirical cumulative histogram of responses, Fi(k) = P(yi ≤k). The data is then transformed through the cdfs ui = Fi(yi), and the copula model is fit according to Eq. 6. If a parametric distribution family is used for the marginals, the parameters of the copula θ and those of the marginals λ can be estimated simultaneously, or alternatively λ can be fitted first, followed by θ. In our experience, the second method is much faster and the quality of the fit is typically unchanged. We checked for biases in ML estimation due to a limited amount of data and low firing rate by generating data from the discrete copula model (Fig. 3), for a number of copula families and Poisson marginals with parameters λ1 = 2, λ2 = 3. The estimate is based on 3500 observations generated from the models (1000 for the Gaussian copula). The estimation is repeated 200 times (100 for the Gaussian copula) in order to compute the mean and standard deviation of the ML estimate. Figure 4 shows that the estimate is unbiased and accurate for a wide range of parameters. Inaccuracy in the estimation becomes larger as the copulas approach functional dependency (i.e., u2 = f(u1) for a deterministic function f), as it is the case for the Gaussian copula when ρ tends to 1, and for the Gumbel copula as θ goes to infinity. This is an effect due to low firing rates. Given our formulation of the estimation problem as a generative model, one could use more sophisticated Bayesian methods in place of the ML estimation, in order to infer a whole distribution over parameters given the data. This would allow to put error bars on the estimated parameters, and would avoid overfitting at the cost of computational time. 4 Figure 5: Empirical joint distribution and copula fit for two neuron pairs. The top row shows two neurons that have dependencies mainly in the upper tails of their marginal distribution. The pair in the bottom row has negative dependency. a,d) Histogram of the firing rate of the two neurons. Colors correspond to the logarithm of the normalized frequency. b,e) Empirical copula. The color intensity has been cut off at 2.0 to improve visibility. c,f) Density of the copula fit. 4 Results To demonstrate the ability of copula models to fit joint firing rate distribution, we model neural data recorded using a multi-electrode array implanted in the pre-motor cortex (PMd) area of a macaque monkey [18, 19]. The array consisted in 10 × 10 electrodes separated by 400µm. Firing times were recorded while the monkey executed a center-out reaching task. See [19] for a description of the task and general experimental setup. We fit the copula model using the marginal distribution of neural activity over the entire recording session, including data recorded between trials (i.e., while the monkey was freely behaving). Although one might also like to consider data collected during a single task condition (i.e., the stimulus-conditional response distribution), the marginal response distribution is an important statistical object in its own right, and has been the focus of recent much literature [10, 11]. For example, the joint activity across neurons, averaged over stimuli, is the only distribution the brain has access to, and must be sufficient for learning to construct representations of the external world. We collected spike responses in 100ms bins, and selected at random, without repetition, a training set of 4000 bins and a test set of 2000 bins. Out of a total of 194 neurons we select a subset of 33 neurons that fired a minimum of 2500 spikes over the whole data set. For every pair of neurons in this subset (528 pairs), we fit the parameters of several copula families to the joint firing rate. Figure 5 shows two examples of the kind of the dependencies present in the data set and how they are fit by different copula families. The neuron pair in the top row shows dependency in the upper tails of their distribution, as can be seen in the histogram of joint firing rates (colors represent the logarithm of the frequency): The two neurons have the tendency to fire strongly together, but are relatively independent at low firing rates. This is confirmed by the empirical copula, which shows the probability mass in the regions defined by the cdfs of the marginal distribution. Since the marginal cdfs are discrete, the data is projected on a discrete set of points on the unit cube; the colors in the empirical copula plots represent the probability mass in the region where the marginal cdfs are constant. The axis in the empirical copula should be interpreted as the quantiles of the marginal distributions – for example, 0.5 on the x-axis corresponds to the median of the distribution of y1. The higher probability mass in the upper right corner of the plot thus means that the two neurons tend to be in the upper tails of the distributions simultaneously, and thus to have higher firing rates together. On the right, one can see that this dependency structure is well captured by the Gumbel copula fit. The second pair of neuron in the bottom row have negative dependency, in the sense that when one of them has high firing rate the other tends to be silent. Although this is not readily visible in the joint histogram, the dependency becomes clear in the empirical copula plot. This structure is captured by the Frank copula fit. 5 −0.5 0 0.5 1 −4 −2 0 2 4 6 Gauss parameter Frank parameter < 0 0 10 < 0 0 10 Gauss gain (bits/sec) Frank gain (bits/sec) Figure 6: In the pairs where their fit improves over the independence model, the parameters (left) and the score (right) of the Gaussian and Frank models are highly correlated. The goodness-of-fit of the copula families is evaluated by cross-validation: We fit different models on training data, and compute the log-likelihood of test data under the fitted model. The models are scored according to the difference between the log-likelihood of a model that assumes independent neurons and the log-likelihood of the copula model. This measure (appropriately renormalized) can be interpreted as the number of bits per second that can be saved when coding the firing rate by taking into account the dependencies encoded by the copula family. This is because this quantity can be expressed as an estimation of the difference in the Kullback-Leibler divergence of the independent (pindep) and copula model (pθ) to the real distribution p∗ ⟨log pθ(y)⟩y∼p∗−⟨log pindep(y)⟩y∼p∗ (7) ≈ ' p∗(y) log pθ(y)dy − ' p∗(y) log pindep(y) (8) = KL(p∗||pindep) −KL(p∗||pθ). (9) We took particular care in selecting a small set of copula families that would be able to capture the dependencies occurring in the data. Some of the families that we considered at first capture similar kind of dependencies, and their scores are highly correlated. For example, the Frank and Gaussian copulas are able to represent both positive and negative dependencies in the data, and simultaneously in lower and upper tails, although the dependencies in the tails are less strong for the Frank family (compare the copula densities in Figs. 1 and 5f). Fig. 6 (left) shows that both the parameter fits and their performance are highly correlated. An advantage of the Frank copula is that it is much more efficient to fit, since the Gaussian copula requires multiple evaluations of the bivariate Gaussian cdf, which requires expensive numerical calculations. In addition, The Gaussian copula was also found to be more prone to overfitting on this data set (Fig. 6, right). For these reasons, we decided to use the Frank family only for the rest of the analysis. With similar procedures we shortlisted a total 3 families that cover the vast majority of dependencies in our data set: Frank, Clayton, and Gumbel copulas. Examples of the copula density of these families can be found in Figs. 2, and 5. The Clayton and Gumbel copulas describe dependencies in the lower and upper tails of the distributions, respectively. We didn’t find any example of neuron pairs where the dependency would be in the upper tail of the distribution for one and in the lower tail for the other distribution, or more complicated dependencies. Out of all 528 neuron pairs, 393 had a significant improvement (P<0.05 on test data) over a model with independent neurons1 and for 102 pairs the improvement was larger than 1 bit/sec. Dependencies in the data set seem thus to be widespread, despite the fact that individual neurons are recorded from electrodes that are up to 4.4 mm apart. Fig. 7 shows the histogram of improvement in bits/sec. The most common dependencies structures over all neuron pairs are given by the Gaussian-like dependencies of the Frank copula (54% of the pairs). Interestingly, a large proportion of the neurons showed dependencies concentrated in the upper tails (Gumbel copula, 22%) or lower tails (Clayton copula, 16%) of the distributions (Fig. 7). 1We computed the significance level by generating an artificial data set using independent neurons with the same empirical pdf as the monkey data. We analyzed the generated data and computed the maximal improvement over an independent model (due to the limited number of samples) on artificial test data. The resulting distribution is very narrowly distributed around zero. We took the 95th percentile of the distribution (0.02 bits/sec) as the threshold for significance. 6 0 5 10 15 0 20 40 60 80 100 bits/sec number of pairs Improvement over independent model <− 371 Independent 8% Clayton 15% Gumbel 23% Frank 54% Figure 7: For every pair of neurons, we select the copula family that shows the largest improvement over a model with independent neurons, in bits/sec. Left: histogram of the gain in bits/sec over the independent model. Right: Pie chart of the copula families that best fit the neuron pairs. 5 Discussion The results presented here show that it is possible to represent neuronal spike responses using a model that preserves discrete, non-negative marginals while incorporating various types of dependencies between neurons. Mathematically, it is straightforward to generalize these methods to the n-variate case (i.e., distributions over the responses of n neurons). However, many copula families have only one or two parameters, regardless of the copula dimensionality. If the dependency structure across a neural population is relatively homogeneous, then these copulas may be useful in that they can be estimated using far less data than required, e.g., for a full covariance matrix (which has O(n2) parameters). On the other hand, if the dependencies within a population vary markedly for different pairs of neurons (as in the data set examined here), such copulas will lack the flexibility to capture the complicated dependencies within a full population. In such cases, we can still apply the Gaussian copula (and other copulas derived from elliptically symmetric distributions), since it is parametrized by the same covariance matrix as a n-dimensional Gaussian. However, the Gaussian copula becomes prohibitively expensive to fit in high dimensions, since evaluating the likelihood requires an exponential number of evaluations of the multivariate Gaussian cdf, which itself must be computed numerically. One challenge for future work will therefore be to design new parametric families of copulas whose parameters grow with the number of neurons, but remain tractable enough for maximum-likelihood estimation. Recently, Kirshner [20] proposed a copula-based representation for multivariate distributions using a model that averages over tree-structured copula distributions. The basic idea is that pairwise copulas can be easily combined to produce a tree-structured representation of a multivariate distribution, and that averaging over such trees gives an even more flexible class of multivariate distributions. We plan to examine this approach using neural population data in future work. Another future challenge is to combine explicit models of the stimulus-dependence underlying neural responses with models capable of capturing their joint response dependencies. The data set analyzed here concerned the distribution over spike responses during all all stimulus conditions (i.e., the marginal distribution over responses, as opposed to the the conditional response distribution given a stimulus). Although this marginal response distribution is interesting in its own right, for many applications one is interested in separating correlations that are induced by external stimuli from internal correlations due to the network interactions. One obvious approach is to consider a hybrid model with a Linear-Nonlinear-Poisson model [21] capturing stimulus-induced correlation, adjoined to a copula distribution that models the residual dependencies between neurons (Fig. 8). This is an important avenue for future exploration. Acknowledgments We’d like to thank Matthew Fellows for providing the data used in this study. This work was supported by the Gatsby Charitable Foundation. 7 Figure 8: Hybrid LNP-copula model. The LNP part of the model removes stimulus-induced correlations from the neural data, so that the copula model can take into account residual network-related dependencies. References [1] R. Zemel, P. Dayan, and A. Pouget. Probabilistic interpretation of population codes. Neural Computation, 10:403–430, 1998. [2] A. Pouget, K. Zhang, S. Deneve, and P.E. Latham. Statistically efficient estimation using population coding. Neural Computation, 10(2):373–401, 1998. [3] L. Abbott and P. Dayan. The effect of correlated variability on the accuracy of a population code. Neural Computation, 11:91–101, 1999. [4] E. Maynard, N. Hatsopoulos, C. Ojakangas, B. Acuna, J. Sanes, R. Normann, and J. Donoghue. Neuronal interactions improve cortical population coding of movement direction. 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Learning with Consistency between Inductive Functions and Kernels Haixuan Yang1,2 Irwin King1 Michael R. Lyu1 1Department of Computer Science & Engineering The Chinese University of Hong Kong {hxyang,king,lyu}@cse.cuhk.edu.hk 2Department of Computer Science Royal Holloway University of London haixuan@cs.rhul.ac.hk Abstract Regularized Least Squares (RLS) algorithms have the ability to avoid over-fitting problems and to express solutions as kernel expansions. However, we observe that the current RLS algorithms cannot provide a satisfactory interpretation even on the penalty of a constant function. Based on the intuition that a good kernelbased inductive function should be consistent with both the data and the kernel, a novel learning scheme is proposed. The advantages of this scheme lie in its corresponding Representer Theorem, its strong interpretation ability about what kind of functions should not be penalized, and its promising accuracy improvements shown in a number of experiments. Furthermore, we provide a detailed technical description about heat kernels, which serves as an example for the readers to apply similar techniques for other kernels. Our work provides a preliminary step in a new direction to explore the varying consistency between inductive functions and kernels under various distributions. 1 Introduction Regularized Least Squares (RLS) algorithms have been drawing people’s attention since they were proposed due to their ability to avoid over-fitting problems and to express solutions as kernel expansions in terms of the training data [4, 9, 12, 13]. Various modifications of RLS are made to improve its performance either from the viewpoint of manifold [1] or in a more generalized form [7, 11]. However, despite these modifications, problems still remain. We observe that the previous RLS-related work has the following problem: Over Penalization. For a constant function f = c, a nonzero term ||f||K is penalized in both RLS and LapRLS [1]. As a result, for a distribution generalized by a nonzero constant function, the resulting regression function by both RLS and LapRLS is not a constant as illustrated in the left diagram in Fig. 1. For such situations, there is an over-penalization. In this work, we aim to provide a new viewpoint for supervised or semi-supervised learning problems. By such a viewpoint we can provide a general condition under which constant functions should not be penalized. The basic idea is that, if a learning algorithm can learn an inductive function f(x) from examples generated by a joint probability distribution P on X × R, then the learned function f(x) and the marginal PX represents a new distribution on X × R, from which there is a re-learned function r(x). The re-learned function should be consistent with the learned function in the sense that the expected difference on distribution PX is small. Because the re-learned function depends on the underlying kernel, the difference f(x) −r(x) depends on f(x) and the kernel, and from this point of view, we name this work. 0 0.5 1 0.96 0.97 0.98 0.99 1 1.01 1.02 x y RLS 0 0.5 1 0.998 0.999 1 1.001 1.002 1.003 x y RLS vs PRLS −2 0 2 −1 −0.5 0 0.5 1 The Re−learned function and the Residual x y The Ideal Function Labeled Data RLS−γ=0.1 RLS−γ=0.01 RLS−γA=0 RLS−γ=0.005 RLS−γ=0.005 PRLS−γ=1000 PRLS−γ=1 PRLS−γ=0.001 PRLS−γ=0 f(x) r(x) f(x)−r(x) Figure 1: Illustration for over penalization. Left diagram: The training set contains 20 points, whose x is randomly drawn from the interval [0 1], whereas the test set contains another 20 points, and y is generated by 1 + 0.005ε, ε ∼N(0, 1). The over penalized constant functions in the term ||f||K cause the phenomena that smaller γ can achieve better results. On the other hand, the overfitting phenomenon when γ = 0 suggests the necessity of the regularization term. Based on these observations, an appropriate penalization on a function is expected. Middle diagram: r(x) is very smooth, and f(x)−r(x) remains the uneven part of f(x); therefore f(x)−r(x) should be penalized while f is over penalized in ||f||K. Right diagram: the proposed model has a stable property so that a large variant of γ results in small changes of the curves, suggesting a right way of penalizing functions. 2 Background The RKHS Theory enables us to express solutions of RLS as kernel expansions in terms of the training data. Here we give a brief description of the concepts. For a complete discussion, see [2]. Let X be a compact domain or manifold, ν be a Borel measure on X, and K : X × X →R be a Mercer kernel, then there is an associated Hilbert space RKHS HK of functions X →R with the corresponding norm || · ||K. HK satisfies the reproducing property, i.e., for all f ∈HK, f(x) = ⟨Kx, f⟩, where Kx is the function K(x, ·). Moreover, an operator LK can be defined on HK as: (LKf)(x) = R X f(y)K(x, y)dν(y), where L2 ν(X) is the Hilbert space of square integrable functions on X with the scalar product ⟨f, g⟩ν = R X f(x)g(x)dν(x). Given a Mercer kernel and a set of labeled examples (xi, yi) (i = 1, ..., l), there are two popular inductive learning algorithms: RLS [12, 13] and the Nadaraya-Watson Formula [5, 8, 14]. By the standard Tikhonov regularization, RLS is a special case of the following functional extreme problem: f ∗= arg min f∈HK 1 l l X i=1 V (xi, yi, f) + γ||f||2 K (1) where V is some loss function. The Classical Representer Theorem states that the solution to this minimization problem exists in HK and can be written as f ∗(x) = l X i=1 αiK(xi, x). (2) Such a Representer Theorem is general because it plays an important role in both RLS in the case when V (x, y, f) = (y −f(x))2, and SVM in the case when V (x, y, f) = max(0, 1 −yf(x)). The Nadaraya-Watson Formula is based on local weighted averaging, and it comes with a closed form: r(x) = l X i=1 yiK(x, xi)/ l X i=1 K(x, xi). (3) The formula has a similar appearance as Eq. (2), but it plays an important role in this paper because we can write it in an integral form which makes our idea technically feasible as follows. Let p(x) be a probability density function over X, P(x) be the corresponding cumulative distribution function, and f(x) be an inductive function. We observe that, if (xi, f(xi))(i = 1, 2, . . . , l) are sampled from the function y = f(x), then A Re-learned Function can be expressed as r(x) = lim l→∞ Pl i=1 f(xi)K(x, xi) Pl i=1 K(x, xi) = R X f(α)K(x, α)dP(α) R X K(x, α)dP(α) = LK(f) R X K(x, α)dP(α), (4) based on f(x) and P(x). From this form, we show two points: (1) If r(x) = f(x), then f(x) is completely predicted by itself through the Nadaraya-Watson Formula, and so f(x) is considered to be completely consistent with the kernel K(x, y); if r(x) ̸= f(x), then the difference ||f(x) − r(x)||K can measure how badly f(x) is consistent with the kernel K(x, y) and (2) Intuitively r(x) can also be understood as the smoothed function of f(x) through a kernel K. Consequently, f(x)− r(x) represents the intrinsically uneven part of f(x), which we will penalize. This intuition is illustrated in the middle diagram in Fig. 1. Throughout this paper, we assume that R X K(x, α)dP(α) is a constant, and for simplicity all kernels are normalized by K/ R X K(x, α)dP(α) so that r(x) = LK(f). Moreover, we assume that X is compact, and the measure ν is specified as P(x). 3 Partially-penalized Regularization For a given kernel K and an inductive function f, LK(f) is the prediction function produced by K through the Nadaraya-Watson Formula. Based on Eq. (1), penalizing the inconsistent part f(x) − LK(f) leads to the following Partially-penalized Regularization problem: f ∗= arg min f∈HK 1 l l X i=1 V (xi, yi, f) + γ||f −LK(f)||2 K. (5) To obtain a Representer Theorem, we need one assumption. Assumption 1 Let f1, f2 ∈HK. If ⟨f1, f2⟩K = 0, then ||f1 −LK(f1) + f2 −LK(f2)||2 K = ||f1 −LK(f1)||2 K + ||f2 −LK(f2)||2 K . It is well-known that the operator LK is compact, self-adjoint, and positive with respect to L2 ν(X), and by the Spectral Theorem [2, 3], its eigenfunctions e1(x), e2(x), . . . form an orthogonal basis of L2 ν(X) and the corresponding eigenvalues λ1 ≥λ2, . . . are either finitely many that are nonzero, or there are infinitely many, in which case λk →0. Let f1 = P i aiei(x), f2 = P i biei(x), then f1−LK(f1) = P i aiei(x)−LK(P i aiei(x)) = P i aiei(x)−P i λiaiei(x) = P i(1−λi)aiei(x), and similarly, f2 −LK(f2) = P i(1 −λi)biei(x). By the discussions in [1], we have ⟨ei, ej⟩ν = 0 if i ̸= j, and ⟨ei, ei⟩ν = 1; ⟨ei, ej⟩K = 0 if i ̸= j, and ⟨ei, ei⟩K = 1 λi . If we consider the situation that ai, bi ≥0 for all i ≥1, then ⟨f1, f2⟩K = 0 implies that aibi = 0 for all i ≥1, and consequently ⟨f1 −LK(f1), f2 −LK(f2)⟩K = P i(1 −λi)2aibi⟨ei(x), ei(x)⟩K = 0. Therefore, under some constrains, this assumption is a fact. Under this assumption, we have a Representer Theorem. Theorem 2 Let µj(x) be a basis in H0 of the operator I −LK, i.e., H0 = {f ∈HK|f −LK(f) = 0}. Under Assumption 1, the minimizer of the optimization problem in Eq. (5) is f ∗(x) = o X j=1 βjµj(x) + l X i=1 αiK(xi, x) (6) Proof of the Representer Theorem. Any function f ∈HK can be uniquely decomposed into a component f|| in the linear subspace spanned by the kernel functions {K(xi, ·)}l i=1, and a component f⊥orthogonal to it. Thus, f = f|| + f⊥= lP i=1 αiK(xi, ·) + f⊥. By the reproducing property and the fact that ⟨f⊥, K(xi, ·)⟩= 0 for 1 ≤i ≤l, we have f(xj) = ⟨f, K(xj, ·)⟩= ⟨ l X i=1 αiK(xi, ·), K(xj, ·)⟩+⟨f⊥, K(xj, ·)⟩= ⟨ l X i=1 αiK(xi, ·), K(xj, ·)⟩. Thus the empirical terms involving the loss function in Eq. (5) depend only on the value of the coefficients {αi}l i=1 and the gram matrix of the kernel function. By Assumption 1, we have ||f −LK(f)||2 K = || lP i=1 αiK(xi, ·) −LK( lP i=1 αiK(xi, ·))||2 K + ||f⊥−LK(f⊥)||2 K ≥ || lP i=1 αiK(xi, ·) −LK( lP i=1 αiK(xi, ·))||2 K. It follows that the minimizer of Eq. (5) must have ||f⊥−LK(f⊥)||2 K = 0, and therefore admits a representation f ∗(x) = f⊥+ lP i=1 αiK(xi, x) = oP j=1 βjµj(x) + lP i=1 αiK(xi, x). 3.1 Partially-penalized Regularized Least Squares (PRLS) Algorithm In this section, we focus our attention in the case that V (xi, yi, f) = (yi −f(xi))2, i.e, the Regularized Least Squares algorithm. In our setting, we aim to solve: min f∈HK 1 l X (yi −f(xi))2 + γ||f −LK(f)||2 K. (7) By the Representer Theorem, the solution to Eq. (7) is of the following form: f ∗(x) = o X j=1 βjµj(x) + l X i=1 αiK(xi, x). (8) By the proof of Theorem 2, we have f⊥= oP j=1 βjµj(x) and ⟨f⊥, lP i=1 αiK(xi, x)⟩K = 0. By Assumption 1 and the fact that f⊥belongs to the null space H0 of the operator I −LK, we have ||f ∗−LK(f ∗)||2 K = ||f⊥−LK(f⊥)||2 K + || Pl i=1 αiK(xi, x) −LK(Pl i=1 αiK(xi, x))||2 K = || Pl i=1 αiK(xi, x) −Pl i=1 αiLK(K(xi, x))||2 K = αT (K −2K′ + K′′)α, (9) where α = [α1, α2, . . . , αl]T , K is the l × l gram matrix Kij = K(xi, xj), K′ and K′′ are reconstructed l × l matrices K′ ij = ⟨K(xi, x), LK(K(xj, x))⟩K, and K′′ ij = ⟨LK(K(xi, x)), LK(K(xj, x))⟩K. Substituting Eq. (8) and Eq. (9) to the problem in Eq. (7), we arrive at the following quadratic objective function of the l-dimensional variable α and o-dimensional variable β = [β1, β2, . . . , βo]T : [α∗, β∗] = arg min 1 l (Y −Kα −Ψβ)T (Y −Kα −Ψβ) + γαT (K −2K′ + K′′)α, (10) where Ψ is an l×o matrix Ψij = µj(xi), and Y = [y1, y2, . . . , yl]T . Taking derivatives with respect to α and β, since the derivative of the objective function vanishes at the minimizer, we obtain (γl(K −2K′ + K′′) + K2)α + KΨβ = KY, ΨT (Y −Kα −Ψβ) = 0. (11) In the term ||f −LK(f)||, f is subtracted by LK(f), and so it partially penalized. For this reason, the resulting algorithm is referred as Partially-penalized Regularized Least Squares algorithm (PRLS). 3.2 The PLapRLS Algorithm The idea in the previous section can also be extended to LapRLS in the manifold regularization framework [1]. In the manifold setting, the smoothness on the data adjacency graph should be considered, and Eq. (5) is modified as f ∗= arg min f∈HK 1 l l X i=1 V (xi, yi, f)+γA||f−LK(f)||2 K+ γI (u + l)2 l+u X i,j=1 (f(xi)−f(xj))2Wij, (12) where Wij are edge weights in the data adjacency. From W, the graph Laplacian L is given by L = D−W, where D is the diagonal matrix with Dii = Pl+u j=1 Wij. For this optimization problem, the result in Theorem 2 can be modified slightly as: Theorem 3 Under Assumption 1, the minimizer of the optimization problem in Eq. (12) admits an expansion f ∗(x) = o X j=1 βjµj(x) + l+u X i=1 αiK(xi, x). (13) Following Eq. (13), we continue to optimize the (l + u)-dimensional variable α = [α1, α2, . . . , αl+u]α and the o-dimensional variable β = [β1, β2, . . . , βo]T . In a similar way as the previous section and LapRLS in [1], α and β are determined by the following linear systems: (KJK + λ1(K −2K′ + K′′) + λ2KLK)α + (KJΨ + λ2KLΨ)β = KJY, (Ψ′JK −λ2Ψ′LK)α + (Ψ′Ψ −λ2Ψ′LΨ)β = Ψ′ ∗Y, (14) where K, K′, K′′ are the (l +u)×(l +u) Gram matrices over labeled and unlabeled points; Y is an (l + u) dimensional label vector given by: Y = [y1, y2, . . . , yl, 0, . . . , 0], J is an (l + u) × (l + u) diagonal matrix given by J = diag(1, 1, . . . , 1, 0, . . . , 0) with the first l diagonal entries as 1 and the rest 0, and Ψ is an (l + u) × o matrix Ψij = µj(xi). 4 Discussions 4.1 Heat Kernels and the Computation of K′ and K′′ In this section we will illustrate the computation of K′ and K′′ in the case of heat kernels. The basic facts about heat kernels are excerpted from [6], and for more materials, see [10]. Given a manifold M and points x and y, the heat kernel Kt(x, y) is a special solution to the heat equation with a special initial condition called the delta function δ(x−y). More specifically, δ(x−y) describes a unit heat source at position y with no heat in other positions. Namely, δ(x −y) = 0 for x ̸= y and R +∞ −∞δ(x −y)dx = 1. If we let f0(x, 0) = δ(x −y), then Kt(x, y) is a solution to the following differential equation on a manifold M: ∂f ∂t −Lf = 0, f(x, 0) = f0(x), (15) where f(x, t) is the temperature at location x at time t, beginning with an initial distribution f0(x) at time zero, and L is the Laplace-Beltrami operator. Equation (15) describes the heat flow throughout a geometric manifold with initial conditions. Theorem 4 Let M be a complete Riemannian manifold. Then there exists a function K ∈ C∞(R+ × M × M), called the heat kernel, which satisfies the following properties for all x, y ∈ M, with Kt(x, y) = K(t, x, y): (1) Kt(x, y) defines a Mercer kernel. (2) Kt(x, y) = R M Kt−s(x, z)Ks(z, y)dz for any s > 0. (3) The solution to Eq. (15) is f(x, t) = R M Kt(x, y)f0(y)dy. (4) 1 = R M Kt(x, y)1dy and (5) When M = Rm, Lf is simplified as P i ∂2f ∂x2 i , and the heat kernel takes the Gaussian RBF form Kt(x, y) = (4πt)−m 2 e−||x−y||2 4t . K′ and K′′ can be computed as follows: K′ ij = ⟨Kt(xi, x), LK(Kt(xj, x))⟩K (by definition) = LK(Kt(xj, x))|x=xi (by the reproducing property of a Mercer kernel) = R X Kt(xj, y)Kt(xi, y)dν(y) (by the definition of LK) = K2t(xi, xj) (by Property 2 in Theorem 4) (16) Based on the fact that LK is self-adjoint, we can similarly derive K′′ ij = K3t(xi, xj). For other kernels, K′ and K′′ can also be computed. 4.2 What should not be penalized? From Theorem 2, we know that the functions in the null space H0 = {f ∈HK|f −LK(f) = 0} should not be penalized. Although there may be looser assumptions that can guarantee the validity of the result in Theorem 2, there are two assumptions in this work: X is compact and R X K(x, α)dP(α) in Eq. (4) is a constant. Next we discuss the constant functions and the linear functions. Should constant functions be penalized? Under the two assumptions, a constant function c should not be penalized, because c = R X cK(x, α)p(α)dα/ R X K(x, α)p(α)dα, i.e., c ∈H0. For heat kernels, if P(x) is uniformly distributed on M, then by Property 4 in Theorem 4, R X K(x, α)dP(α) is a constant, and so c should not be penalized. For polynomial kernels, the theory cannot guarantee that constant functions should not be penalized even with a uniform distribution P(x). For example, considering the polynomial kernel xy+1 in the interval X = [0 1] and the uniform distribution on X, R X(xy+1)dP(y) = R 1 0 (xy+1)dy = x/2+1 is not a constant. As a counter example, we will show in Section 5.3 that not penalizing constant functions in polynomial kernels will result in much worse accuracy. The reason for this phenomenon is that constant functions may not be smooth in the feature space produced by the polynomial kernel under some distributions. The readers can deduce an example for p(x) such that R 1 0 (xy + 1)dP(y) happens to be a constant. Should linear function aT x be penalized? In the case when X is a closed ball Br with radius r when P(x) is uniformly distributed over Br and when K is the Gaussian RBF kernel, then aT x should not be penalized when r is big enough. 1 Since r is big enough, we have R Rn ·dx ≈ R Br ·dx and R Br Kt(x, y)dy ≈1, and so aT x = R Rn Kt(x, y)aT ydy ≈ R Br Kt(x, y)aT ydy ≈LK(aT x). Consequently ||aT x −LK(aT x)||K will be small enough, and so the linear function aT x needs not be penalized. For other kernels, other spaces, or other PX, the conclusion may not be true. 5 Experiments In this section, we evaluate the proposed algorithms PRLS and PLapRLS on a toy dataset (size: 40), a medium-sized dataset (size: 3,119), and a large-sized dataset (size: 20,000), and provide a counter example for constant functions on another dataset (size: 9,298). We use the Gaussian RBF kernels in the first three datasets, and use polynomial kernels to provide a counter example on the last dataset. Without any prior knowledge about the data distribution, we assume that the examples are uniformly distributed, and so constant functions are considered to be in H0 for the Gaussian RBF kernel, but linear functions are not considered to be in H0 since it is rare for data to be distributed uniformly on a large ball. The data and results for the toy dataset are illustrated in the left diagram and the right diagram in Fig. 1. 5.1 UCI Dataset Isolet about Spoken Letter Recognition We follow the same semi-supervised settings as that in [1] to compare RLS with PRLS, and compare LapRLS with PLapRLS on the Isolet database. The dataset contains utterances of 150 subjects who 1Note that a subset of Rn is compact if and only if it is closed and bounded. Since Rn is not bounded, it is not compact, and so the Representer Theorem cannot be established. This is the reason why we cannot talk about Rn directly. 0 5 10 15 20 25 30 10 12 14 16 18 20 22 24 26 28 Labeled Speaker # Error Rate (unlabeled set) RLS vs PRLS 0 5 10 15 20 25 30 10 15 20 25 Labeled Speaker # Error Rates (unlabeled set) LapRLS vs PLapRLS 0 5 10 15 20 25 30 15 20 25 30 35 Labeled Speaker # Error Rates (test set) RLS vs PPLS 0 5 10 15 20 25 30 14 16 18 20 22 24 26 28 30 32 Labeled Speaker # Error Rates (test set) LapRLS vs PLapRLS RLS PRLS LapRLS PLapRLS RLS PRLS LapRLS PLapRLS Figure 2: Isolet Experiment pronounced the name of each letter of the English alphabet twice. The speakers were grouped into 5 sets of 30 speakers each. The data of the first 30 speakers forms a training set of 1,560 examples, and that of the last 29 speakers forms the test set. The task is to distinguish the first 13 letters from the last 13. To simulate a real-world situation, 30 binary classification problems corresponding to 30 splits of the training data where all 52 utterances of one speaker were labeled and all the rest were left unlabeled. All the algorithms use Gaussian RBF kernels. For RLS and LapRLS, the results were obtained with width σ = 10, γl = 0.05, γAl = γIl/(u + l)2 = 0.005. For PRLS and PLapRLS, the results were obtained with width σ = 4, γl = 0.01, and γAl = γIl/(u + l)2 = 0.01. In Fig. 2, we can see that both PRLS and PLapRLS make significant performance improvements over their corresponding counterparts on both unlabeled data and test set. 5.2 UCI Dataset Letter about Printed Letter Recognition In Dataset Letter, there are 16 features for each example, and there are 26 classes representing the upper case printed letters. The first 400 examples were taken to form the training set. The remaining 19,600 examples form the test set. The parameters are set as follows: σ = 1, γl = γA(l+u) = 0.25, and γIl/(u + l)2 = 0.05. For each of the four algorithms RLS, PRLS, LapRLS, and PLapRLS, for each of the 26 one-versus-all binary classification tasks, and for each of 10 runs, two examples for each class were randomly labeled. For each algorithm, the averages over all the 260 one-versus-all binary classification error rates for unlabeled 398 examples and test set are listed respectively as follows: (5.79%, 5.23%) for RLS, (5.12%, 4.77%) for PRLS, (0%, 2.96%) for LapRLS, and (0%, 3.15%) for PLapRLS respectively. From the results, we can see that RLS is improved on both unlabeled examples and test set. The fact that there is no error in the total 260 tasks for LapRLS and PLapRLS on unlabeled examples suggests that the data is distributed in a curved manifold. On a curved manifold, the heat kernels do not take the Gaussian RBF form, and so PLapRLS using the Gaussian RBF form cannot achieve its best. This is the reason why we can observe that PLapRLS is slightly worse than LapRLS on the test set. This suggests the need for a vast of investigations on heat kernels on a manifold. 5.3 A Counter Example in Handwritten Digit Recognition Note that, polynomial kernels with degree 3 were used on USPS dataset in [1], and 2 images for each class were randomly labeled. We follow the same experimental setting as that in [1]. For RLS, if we use Eq. (2), then the averages of 45 pairwise binary classification error rates are 8.83% and 8.41% for unlabeled 398 images and 8,898 images in the test set respectively. If constant functions are not penalized, then we should use f ∗(x) = Pl i=1 αiK(xi, x) + a, and the corresponding error rates are 9.75% and 9.09% respectively. By this example, we show that leaving constant functions outside the regularization term is dangerous; however, it is fortunate that we have a theory to guide this in Section 4: if X is compact and R X K(x, α)dP(α) in Eq. (4) is a constant, then constant functions should not be penalized. 6 Conclusion A novel learning scheme is proposed based on a new viewpoint of penalizing the inconsistent part between inductive functions and kernels. In theoretical aspects, we have three important claims: (1) On a compact domain or manifold, if the denominator in Eq. (4) is a constant, then there is a new Representer Theorem; (2) The same conditions become a sufficient condition under which constant functions should not be penalized; and (3) under the same conditions, a function belongs to the null space if and only if the function should not be penalized. Empirically, we claim that the novel learning scheme can achieve accuracy improvement in practical applications. Acknowledgments The work described in this paper was supported by two grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CUHK4150/07E) and Project No. CUHK4235/04E). 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Syntactic Topic Models Jordan Boyd-Graber Department of Computer Science 35 Olden Street Princeton University Princeton, NJ 08540 jbg@cs.princeton.edu David Blei Department of Computer Science 35 Olden Street Princeton University Princeton, NJ 08540 blei@cs.princeton.edu Abstract We develop the syntactic topic model (STM), a nonparametric Bayesian model of parsed documents. The STM generates words that are both thematically and syntactically constrained, which combines the semantic insights of topic models with the syntactic information available from parse trees. Each word of a sentence is generated by a distribution that combines document-specific topic weights and parse-tree-specific syntactic transitions. Words are assumed to be generated in an order that respects the parse tree. We derive an approximate posterior inference method based on variational methods for hierarchical Dirichlet processes, and we report qualitative and quantitative results on both synthetic data and hand-parsed documents. 1 Introduction Probabilistic topic models provide a suite of algorithms for finding low dimensional structure in a corpus of documents. When fit to a corpus, the underlying representation often corresponds to the “topics” or “themes” that run through it. Topic models have improved information retrieval [1], word sense disambiguation [2], and have additionally been applied to non-text data, such as for computer vision and collaborative filtering [3, 4]. Topic models are widely applied to text despite a willful ignorance of the underlying linguistic structures that exist in natural language. In a topic model, the words of each document are assumed to be exchangeable; their probability is invariant to permutation. This simplification has proved useful for deriving efficient inference techniques and quickly analyzing very large corpora [5]. However, exchangeable word models are limited. While useful for classification or information retrieval, where a coarse statistical footprint of the themes of a document is sufficient for success, exchangeable word models are ill-equipped for problems relying on more fine-grained qualities of language. For instance, although a topic model can suggest documents relevant to a query, it cannot find particularly relevant phrases for question answering. Similarly, while a topic model might discover a pattern such as “eat” occurring with “cheesecake,” it lacks the representation to describe selectional preferences, the process where certain words restrict the choice of the words that follow. It is in this spirit that we develop the syntactic topic model, a nonparametric Bayesian topic model that can infer both syntactically and thematically coherent topics. Rather than treating words as the exchangeable unit within a document, the words of the sentences must conform to the structure of a parse tree. In the generative process, the words arise from a distribution that has both a documentspecific thematic component and a parse-tree-specific syntactic component. We illustrate this idea with a concrete example. Consider a travel brochure with the sentence “In the near future, you could find yourself in .” Both the low-level syntactic context of a word and its document context constrain the possibilities of the word that can appear next. Syntactically, it 1 α αT β πk τk ∞ M θd αD σ Parse trees grouped into M documents (a) Overall Graphical Model w1:laid w2:phrases w6:for w5:his w4:some w5:mind w7:years w3:in z1 z2 z3 z4 z5 z5 z6 z7 (b) Sentence Graphical Model Figure 1: In the graphical model of the STM, a document is made up of a number of sentences, represented by a tree of latent topics z which in turn generate words w. These words’ topics are chosen by the topic of their parent (as encoded by the tree), the topic weights for a document θ, and the node’s parent’s successor weights π. (For clarity, not all dependencies of sentence nodes are shown.) The structure of variables for sentences within the document plate is on the right, as demonstrated by an automatic parse of the sentence “Some phrases laid in his mind for years.” The STM assumes that the tree structure and words are given, but the latent topics z are not. is going to be a noun consistent as the object of the preposition “of.” Thematically, because it is in a travel brochure, we would expect to see words such as “Acapulco,” “Costa Rica,” or “Australia” more than “kitchen,” “debt,” or “pocket.” Our model can capture these kinds of regularities and exploit them in predictive problems. Previous efforts to capture local syntactic context include semantic space models [6] and similarity functions derived from dependency parses [7]. These methods successfully determine words that share similar contexts, but do not account for thematic consistency. They have difficulty with polysemous words such as “fly,” which can be either an insect or a term from baseball. With a sense of document context, i.e., a representation of whether a document is about sports or animals, the meaning of such terms can be distinguished. Other techniques have attempted to combine local context with document coherence using linear sequence models [8, 9]. While these models are powerful, ordering words sequentially removes the important connections that are preserved in a syntactic parse. Moreover, these models generate words either from the syntactic or thematic context. In the syntactic topic model, words are constrained to be consistent with both. The remainder of this paper is organized as follows. We describe the syntactic topic model, and develop an approximate posterior inference technique based on variational methods. We study its performance both on synthetic data and hand parsed data [10]. We show that the STM captures relationships missed by other models and achieves lower held-out perplexity. 2 The syntactic topic model We describe the syntactic topic model (STM), a document model that combines observed syntactic structure and latent thematic structure. To motivate this model, we return to the travel brochure sentence “In the near future, you could find yourself in .”. The word that fills in the blank is constrained by its syntactic context and its document context. The syntactic context tells us that it is an object of a preposition, and the document context tells us that it is a travel-related word. The STM attempts to capture these joint influences on words. It models a document corpus as exchangeable collections of sentences, each of which is associated with a tree structure such as a 2 parse tree (Figure 1(b)). The words of each sentence are assumed to be generated from a distribution influenced both by their observed role in that tree and by the latent topics inherent in the document. The latent variables that comprise the model are topics, topic transition vectors, topic weights, topic assignments, and top-level weights. Topics are distributions over a fixed vocabulary (τk in Figure 1). Each is further associated with a topic transition vector (πk), which weights changes in topics between parent and child nodes. Topic weights (θd) are per-document vectors indicating the degree to which each document is “about” each topic. Topic assignments (zn, associated with each internal node of 1(b)) are per-word indicator variables that refer to the topic from which the corresponding word is assumed to be drawn. The STM is a nonparametric Bayesian model. The number of topics is not fixed, and indeed can grow with the observed data. The STM assumes the following generative process of a document collection. 1. Choose global topic weights β ∼GEM(α) 2. For each topic index k = {1, . . . }: (a) Choose topic τk ∼Dir(σ) (b) Choose topic transition distribution πk ∼DP(αT , β) 3. For each document d = {1, . . . M}: (a) Choose topic weights θd ∼DP(αD, β) (b) For each sentence in the document: i. Choose topic assignment z0 ∝θdπstart ii. Choose root word w0 ∼mult(1, τz0) iii. For each additional word wn and parent pn, n ∈{1, . . . dn} • Choose topic assignment zn ∝θdπzp(n) • Choose word wn ∼mult(1, τzn) The distinguishing feature of the STM is that the topic assignment is drawn from a distribution that combines two vectors: the per-document topic weights and the transition probabilities of the topic assignment from its parent node in the parse tree. By merging these vectors, the STM models both the local syntactic context and corpus-level semantics of the words in the documents. Because they depend on their parents, the topic assignments and words are generated by traversing the tree. A natural alternative model would be to traverse the tree and choose the topic assignment from either the parental topic transition πzp(n) or document topic weights θd, based on a binary selector variable. This would be an extension of [8] to parse trees, but it does not enforce words to be syntactically consistent with their parent nodes and be thematically consistent with a topic of the document. Only one of the two conditions must be true. Rather, this approach draws on the idea behind the product of experts [11], multiplying two vectors and renormalizing to obtain a new distribution. Taking the point-wise product can be thought of as viewing one distribution through the “lens” of another, effectively choosing only words whose appearance can be explained by both. The STM is closely related to the hierarchical Dirichlet process (HDP). The HDP is an extension of Dirichlet process mixtures to grouped data [12]. Applied to text, the HDP is a probabilistic topic model that allows each document to exhibit multiple topics. It can be thought of as the “infinite” topic version of latent Dirichlet allocation (LDA) [13]. The difference between the STM and the HDP is in how the per-word topic assignment is drawn. In the HDP, this topic assignment is drawn directly from the topic weights and, thus, the HDP assumes that words within a document are exchangeable. In the STM, the words are generated conditioned on their parents in the parse tree. The exchangeable unit is a sentence. The STM is also closely related to the infinite tree with independent children [14]. The infinite tree models syntax by basing the latent syntactic category of children on the syntactic category of the parent. The STM reduces to the Infinite Tree when θd is fixed to a vector of ones. 3 Approximate posterior inference The central computational problem in topic modeling is to compute the posterior distribution of the latent structure conditioned on an observed collection of documents. Specifically, our goal is to compute the posterior topics, topic transitions, per-document topic weights, per-word topic assign3 ments, and top-level weights conditioned on a set of documents, each of which is a collection of parse trees. This posterior distribution is intractable to compute. In typical topic modeling applications, it is approximated with either variational inference or collapsed Gibbs sampling. Fast Gibbs sampling relies on the conjugacy between the topic assignment and the prior over the distribution that generates it. The syntactic topic model does not enjoy such conjugacy because the topic assignment is drawn from a multiplicative combination of two Dirichlet distributed vectors. We appeal to variational inference. In variational inference, the posterior is approximated by positing a simpler family of distributions, indexed by free variational parameters. The variational parameters are fit to be close in relative entropy to the true posterior. This is equivalent to maximizing the Jensen’s bound on the marginal probability of the observed data [15]. We use a fully-factorized variational distribution, q(β, z, θ, π, τ|β∗, φ, γ, ν) = q(β|β∗) Q d q(θd|γd) Q k q(πk|νk) Q n q(zn|φn). (1) Following [16], q(β|β∗) is not a full distribution, but is a degenerate point estimate truncated so that all weights whose index is greater than K are zero in the variational distribution. The variational parameters γd and νz index Dirichlet distributions, and φn is a topic multinomial for the nth word. From this distribution, the Jensen’s lower bound on the log probability of the corpus is L(γ, ν, φ; β, θ, π, τ) = Eq [log p(β|α) + log p(θ|αD, β) + log p(π|αP , β) + log p(z|θ, π)+ log p(w|z, τ) + log p(τ|σ)] −Eq [log q(θ) + log q(π) + log q(z)] . (2) Expanding Eq [log p(z|θ, π)] is difficult, so we add an additional slack parameter, ωn to approximate the expression. This derivation and the complete likelihood bound is given in the supplement. We use coordinate ascent to optimize the variational parameters to be close to the true posterior. Per-word variational updates The variational update for the topic assignment of the nth word is φni ∝ exp n Ψ(γi) −Ψ(PK j=1 γj) + PK j=1 φp(n),j Ψ(νj,i) −Ψ PK k=1 νj,k + P c∈c(n) PK j=1 φc,j Ψ(νi,j) −Ψ PK k=1 νi,k −P c∈c(n) ω−1 c PK j γjνi,j P k γk P k νi,k + log τi,wn o . (3) The influences on estimating the posterior of a topic assignment are: the document’s topic γ, the topic of the node’s parent p(n), the topic of the node’s children c(n), the expected transitions between topics ν, and the probability of the word within a topic τi,wn. Most terms in Equation 3 are familiar from variational inference for probabilistic topic models, as the digamma functions appear in the expectations of multinomial distributions. The second to last term is new, however, because we cannot assume that the point-wise product of πk and θd will sum to one. We approximate the normalizer for their produce by introducing ω; its update is ωn = X i=1 X j=1 φp(n),j γiνj,i PK k=1 γk PK k=1 νj,k . Variational Dirichlet distributions and topic composition This normalizer term also appears in the derivative of the likelihood function for γ and ν (the parameters to the variational distributions on θ and π, respectively), which cannot be solved in a closed form. We use conjugate gradient optimization to determine the appropriate updates for these parameters [17]. Top-level weights Finally, we consider the top-level weights. The first K −1 stick-breaking proportions are drawn from a Beta distribution with parameters (1, α), but we assume that the final stick-breaking proportion is unity (thus implying β∗is non-zero only from 1 . . . K). Thus, we only optimize the first K −1 positions and implicitly take β∗ K = 1 −PK−1 i β∗ i . This constrained optimization is performed using the barrier method [17]. 4 4 Empirical results Before considering real-world data, we demonstrate the STM on synthetic natural language data. We generated synthetic sentences composed of verbs, nouns, prepositions, adjectives, and determiners. Verbs were only in the head position; prepositions could appear below nouns or verbs; nouns only appeared below verbs; prepositions or determiners and adjectives could appear below nouns. Each of the parts of speech except for prepositions and determiners were sub-grouped into themes, and a document contains a single theme for each part of speech. For example, a document can only contain nouns from a single “economic,” “academic,” or “livestock” theme. Using a truncation level of 16, we fit three different nonparametric Bayesian language models to the synthetic data (Figure 2).1 The infinite tree model is aware of the tree structure but not documents [14] It is able to separate parts of speech successfully except for adjectives and determiners (Figure 2(a)). However, it ignored the thematic distinctions that actually divided the terms between documents. The HDP is aware of document groupings and treats the words exchangeably within them [12]. It is able to recover the thematic topics, but has missed the connections between the parts of speech, and has conflated multiple parts of speech (Figure 2(b)). The STM is able to capture the the topical themes and recover parts of speech (with the exception of prepositions that were placed in the same topic as nouns with a self loop). Moreover, it was able to identify the same interconnections between latent classes that were apparent from the infinite tree. Nouns are dominated by verbs and prepositions, and verbs are the root (head) of sentences. Qualitative description of topics learned from hand-annotated data The same general properties, but with greater variation, are exhibited in real data. We converted the Penn Treebank [10], a corpus of manually curated parse trees, into a dependency parse [18]. The vocabulary was pruned to terms that appeared in at least ten documents. Figure 3 shows a subset of topics learned by the STM with truncation level 32. Many of the resulting topics illustrate both syntactic and thematic consistency. A few nonspecific function topics emerged (pronoun, possessive pronoun, general verbs, etc.). Many of the noun categories were more specialized. For instance, Figure 3 shows clusters of nouns relating to media, individuals associated with companies (“mr,” “president,” “chairman”), and abstract nouns related to stock prices (“shares,” “quarter,” “earnings,” “interest”), all of which feed into a topic that modifies nouns (“his,” “their,” “other,” “last”). Thematically related topics are separated by both function and theme. This division between functional and topical uses for the latent classes can also been seen in the values for the per-document multinomial over topics. A number of topics in Figure 3(b), such as 17, 15, 10, and 3, appear to some degree in nearly every document, while other topics are used more sparingly to denote specialized content. With α = 0.1, this plot also shows that the nonparametric Bayesian framework is ignoring many later topics. Perplexity To study the performance of the STM on new data, we estimated the held out probability of previously unseen documents with an STM trained on a portion of the Penn Treebank. For each position in the parse trees, we estimate the probability the observed word. We compute the perplexity as the exponent of the inverse of the per-word average log probability. The lower the perplexity, the better the model has captured the patterns in the data. We also computed perplexity for individual parts of speech to study the differences in predictive power between content words, such as nouns and verbs, and function words, such as prepositions and determiners. This illustrates how different algorithms better capture aspects of context. We expect function words to be dominated by local context and content words to be determined more by the themes of the document. This trend is seen not only in the synthetic data (Figure 4(a)), where parsing models better predict functional categories like prepositions and document only models fail to account for patterns of verbs and determiners, but also in real data. Figure 4(b) shows that HDP and STM both perform better than parsing models in capturing the patterns behind nouns, while both the STM and the infinite tree have lower perplexity for verbs. Like parsing models, our model was better able to 1In Figure 2 and Figure 3, we mark topics which represent a single part of speech and are essentially the lone representative of that part of speech in the model. This is a subjective determination of the authors, does not reflect any specialization or special treatment of topics by the model, and is done merely for didactic purposes. 5 PROFESSOR, PHD_CANDIDATE, SHEEP, GRAD_STUDENT, PONY on, about, over, with START ponders, discusses, falls queries, runs that, evil, the, this, a 0.28 0.46 0.20 0.98 0.61 1.00 (a) Parse transition only PHD_CANDIDATE, GRAD_STUDENT, with, over, PROFESSOR, on SHEEP, COW, PONY, over, about, on STOCK, MUTUAL_FUND, SHARE, with, about PROFESSOR, PHD_CANDIDATE, evil, GRAD_STUDENT, ponders. this PROFESSOR, GRAD_STUDENT, over, PHD_CANDIDATE, on SHARE, STOCK, MUTUAL_FUND, on, over (b) Document multinomial only 0.31 0.26 0.30 0.24 0.40 0.22 0.30 0.35 0.35 0.22 0.36 0.33 0.23 0.27 0.36 0.30 0.33 0.30 0.13 0.26 0.26 START 0.30 0.33 0.37 SHEEP, PONY, COW, over, with evil, this, the, that stupid, that, the, insolent hates, dreads, mourns, fears, despairs evil, that, this, the bucks, surges, climbs, falls, runs ponders, discusses, queries, falls, runs runs, falls, walks, sits, climbs PROFESSOR, PHD_CANDIDATE, GRAD_STUDENT, over, on 0.38 STOCK SHARE, MUTUAL_FUND, on, with Parts of Speech Themes (c) Combination of parse transition and document multinomial Figure 2: Three models were fit to the synthetic data described in Section 4. Each box illustrates the top five words of a topic; boxes that represent homogenous parts of speech have rounded edges and are shaded. Edges between topics are labeled with estimates of their transition weight π. While the infinite tree model (a) is able to reconstruct the parts of speech used to generate the data, it lumps all topics into the same categories. Although the HDP (b) can discover themes of recurring words, it cannot determine the interactions between topics or separate out ubiquitous words that occur in all documents. The STM (c) is able to recover the structure. predict the appearance of prepositions, but also remained competitive with HDP on content words. On the whole, the STM had lower perplexity than HDP and the infinite tree. 5 Discussion We have introduced and evaluated the syntactic topic model, a nonparametric Bayesian model of parsed documents. The STM achieves better perplexity than the infinite tree or the hierarchical Dirichlet process and uncovers patterns in text that are both syntactically and thematically consistent. This dual relevance is useful for work in natural language processing. For example, recent work [19, 20] in the domain of word sense disambiguation has attempted to combine syntactic similarity with topical information in an ad hoc manner to improve the predominant sense algorithm [21]. The syntactic topic model offers a principled way to learn both simultaneously rather than combining two heterogenous methods. The STM is not a full parsing model, but it could be used as a means of integrating document context into parsing models. This work’s central premise is consistent with the direction of recent improvements in parsing technology in that it provides a method for refining the parts of speech present in a corpus. For example, lexicalized parsers [22] create rules specific to individual terms, and grammar refinement [23] divides general roles into multiple, specialized ones. The syntactic topic model offers an alternative method of finding more specific rules by grouping words together that appear in similar documents and could be extended to a full parser. 6 his, their, other, us, its, last, one, all 0.42 0.10 0.57 0.06 0.26 0.29 0.08 0.31 0.67 0.06 0.28 policy, gorbachev, mikhail, leader, soviet, restructuring, software 0.95 START garden, visit, having, aid, prime, despite, minister, especially 0.37 television, public, australia, cable, host, franchise, service 0.34 says, could, can, did, do, may, does, say 0.11 they, who, he, there, one, we, also, if 0.11 mr, inc, co, president, corp, chairman, vice, analyst, europe, eastern, protection, corp, poland, hungary, chapter, aid 0.52 shares, quarter, market, sales, earnings, interest, months, yield 0.22 0.25 0.09 (a) Sinks and sources 1 3 5 7 9 12 15 18 21 24 27 30 sales market junk fund bonds his her their other one says could can did do Topic Document (b) Topic usage Figure 3: Selected topics (along with strong links) after a run of the syntactic topic model with a truncation level of 32. As in Figure 2, parts of speech that aren’t subdivided across themes are indicated. In the Treebank corpus (left), head words (verbs) are shared, but the nouns split off into many separate specialized categories before feeding into pronoun sinks. The specialization of topics is also visible in plots of the variational parameter γ normalized for the first 300 documents of the Treebank (right), where three topics columns have been identified. Many topics are used to some extent in every document, showing that they are performing a functional role, while others are used more sparingly for semantic content. STM Infinite Tree, independent children HDP 10 20 25 30 PREP DET ADJ VERB NOUN ALL Perplexity 0 5 15 (a) Synthetic STM Infinite Tree, independent children HDP 1,000 2,000 2,500 3,000 3,500 4,000 4,500 VERB NOUN PREP ALL Perplexity 0 500 1,500 (b) Treebank Figure 4: After fitting three models on synthetic data, the syntactic topic model has better (lower) perplexity on all word classes except for adjectives. HDP is better able to capture document-level patterns of adjectives. The infinite tree captures prepositions best, which have no cross-document variation. On real data 4(b), the syntactic topic model was able to combine the strengths of the infinite tree on functional categories like prepositions with the strengths of the HDP on content categories like nouns to attain lower overall perplexity. While traditional topic models reveal groups of words that are used in similar documents, the STM uncovers groups that are used the same way in similar documents. This decomposition is useful for tasks that require a more fine-grained representation of language than the bag of words can offer or for tasks that require a broader context than parsing models. References [1] Wei, X., B. Croft. LDA-based document models for ad-hoc retrieval. In Proceedings of the ACM SIGIR Conference on Research and Development in Information Retrieval. 2006. [2] Cai, J. F., W. S. Lee, Y. W. Teh. NUS-ML:Improving word sense disambiguation using topic features. In Proceedings of SemEval-2007. Association for Computational Linguistics, 2007. 7 [3] Fei-Fei Li, P. Perona. A Bayesian hierarchical model for learning natural scene categories. In CVPR ’05 - Volume 2, pages 524–531. IEEE Computer Society, Washington, DC, USA, 2005. [4] Marlin, B. Modeling user rating profiles for collaborative filtering. In S. Thrun, L. Saul, B. Sch¨olkopf, eds., Advances in Neural Information Processing Systems. MIT Press, Cambridge, MA, 2004. [5] Griffiths, T., M. Steyvers. 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A general framework for investigating how far the decoding process in the brain can be simplified Masafumi Oizumi1, Toshiyuki Ishii2, Kazuya Ishibashi1 Toshihiko Hosoya2, Masato Okada1,2 oizumi@mns.k.u-tokyo.ac.jp tishii@brain.riken.jp,kazuya@mns.k.u-tokyo.ac.jp hosoya@brain.riken.jp, okada@k.u-tokyo.ac.jp 1 University of Tokyo, Kashiwa-shi, Chiba, JAPAN 2 RIKEN Brain Science Institute, Wako-shi, Saitama, JAPAN Abstract “How is information decoded in the brain?” is one of the most difficult and important questions in neuroscience. Whether neural correlation is important or not in decoding neural activities is of special interest. We have developed a general framework for investigating how far the decoding process in the brain can be simplified. First, we hierarchically construct simplified probabilistic models of neural responses that ignore more than Kth-order correlations by using a maximum entropy principle. Then, we compute how much information is lost when information is decoded using the simplified models, i.e., “mismatched decoders”. We introduce an information theoretically correct quantity for evaluating the information obtained by mismatched decoders. We applied our proposed framework to spike data for vertebrate retina. We used 100-ms natural movies as stimuli and computed the information contained in neural activities about these movies. We found that the information loss is negligibly small in population activities of ganglion cells even if all orders of correlation are ignored in decoding. We also found that if we assume stationarity for long durations in the information analysis of dynamically changing stimuli like natural movies, pseudo correlations seem to carry a large portion of the information. 1 Introduction An ultimate goal of neuroscience is to elucidate how information is encoded and decoded by neural activities. To investigate what information is encoded by neurons in certain area of the brain, the mutual information between stimuli and neural responses is often calculated. In the analysis of mutual information, it is implicitly assumed that encoded information is decoded by an optimal decoder, which exactly matches the encoder. In other words, the brain is assumed to have full knowledge of the encoding process. Generally, if the neural activities are correlated, the amount of data needed for the optimal decoding scales exponentially with the number of neurons. Since a large amount of data and many complex computations are needed for optimal decoding, the assumption of an optimal decoder in the brain is doubtful. The reason mutual information is widely used in neuroscience despite the doubtfulness of the optimal decoder is that we are completely ignorant of how information is decoded in the brain. Thus, we simply evaluate the maximal amount of information that can be extracted from neural activities by calculating the mutual information. To address this lack of knowledge, we can ask a different question: “How much information can be obtained by a decoder that has partial knowledge of the encoding process?” [10, 14] We call this type of a decoder “simplified decoder” or a “mismatched decoder”. For example, an independent decoder is a simplified decoder; it takes only the marginal 1 distribution of the neural responses into consideration and ignores the correlations between neuronal activities. The independent decoder is of particular importance because several studies have shown that maximum likelihood estimation can be implemented by a biologically plausible network [2, 4]. If it is experimentally shown that a sufficiently large portion of information is obtained by the independent decoder, we can say that the brain may function in a manner similar to the independent decoder. In this context, Nirenberg et al. computed the amount of information obtained by the independent decoder in pairs of retinal ganglion cells activities [10]. They showed that no pair of cells showed a loss of information greater than 11%. Because only pairs of cells were considered in their analysis, it has not been still elucidated whether correlations are not important in population activities. To elucidate whether correlations are important or not in population activities, we have developed a general framework for investigating the importance of correlation in decoding neural activities. When population activities are analyzed, we have to deal with not only second-order correlations but also higher-order correlations in general. Therefore, we need to hierarchically construct simplified decoders that account of up to Kth-order correlations, where K = 1, 2, ..., N. By computing how much information is obtained by the simplified decoders, we investigate how many orders of correlation should be taken into account to extract enough information. To compute the information obtained by the mismatched decoders, we introduce a information theoretically correct quantity derived by Merhav et al. [8]. Information for mismatched decoders previously proposed by Nirenberg and Latham is the lower bound on the correct information [5, 11]. Because this lower bound can be very loose and their proposed information can be negative when many cells are analyzed as is shown in the paper, we need to accurately evaluate the information obtained by mismatched decoders. The plan of the paper is as follows. In Section 2, we describe a way of computing the information that can be extracted from neural activities by mismatched decoders using the information derived by Merhav et al.. Using analytical computation, we demonstrate how information for mismatched decoders previously proposed by Nirenberg and Latham differs from the correct information derived by Merhav et al., especially when many cells are analyzed. In Section 3, we apply our framework to spike data for ganglion cells in the salamander retina. We first describe the method of hierarchically constructing simplified decoders by using the maximum entropy principle [12]. We then compute the information obtained with the simplified decoders. We find that more than 90% of the information can be extracted from the population activities of ganglion cells even if all orders of correlations are ignored in decoding. We also describe the problem of previous studies [10, 12] in which the stationarity of stimuli is assumed for a duration that is too long. Using a toy model, we demonstrate that pseudo correlations seem to carry a large portion of the information because of the stationarity assumption. 2 Information for mismatched decoders Let us consider how much information about stimuli can be extracted from neural responses. We assume that we experimentally obtain the conditional probability distribution p(r|s) that neural responses r are evoked by stimulus s. We can say that the stimulus is encoded by neural response r, which obeys the distribution p(r|s). We call p(r|s) the “encoding model”. The maximal amount of information obtained with the optimal decoder can be evaluated by using the mutual information: I = − Z drp(r) log2 p(r) + Z dr X s p(s)p(r|s) log2 p(r|s), (1) where p(r) = P s p(r|s)p(s) and p(s) is the prior probability of stimuli. In the optimal decoder, the probability distribution q(r|s) that exactly matches the encoding model p(r|s) is used for decoding; that is, q(r|s) = p(r|s). We call q(r|s) the “decoding model”. We can also compute the maximal amount of information obtained by a decoder using a decoding model q(r|s) that does not match the encoding model p(r|s) by using an equation derived by Merhav et al. [8]: I∗(β) = − Z drp(r) log2 X s p(s)q(r|s)β + Z dr X s p(s)p(r|s) log2 q(r|s)β, (2) where β takes the value that maximizes I∗(β). Thus, β is the value that satisfies ∂I∗/∂β = 0. We call a decoder using the mismatched decoding model a “mismatched decoder”. 2 ̂ ̂ 㪠㪁㪆㪠 㪠㪥㪣㪆㪠 㪥㫌㫄㪹㪼㫉㩷㫆㪽㩷㪺㪼㫃㫃㫊 㪘㫄㫆㫌㫅㫋㩷㫆㪽㩷㫀㫅㪽㫆㫉㫄㪸㫋㫀㫆㫅㩷㫆㪹㫋㪸㫀㫅㪼㪻㩷 㩷㩷㩷㩷㪹㫐㩷㫄㫀㫊㫄㪸㫋㪺㪿㪼㪻㩷㪻㪼㪺㫆㪻㪼㫉㩷㩿㩼㪀 㪘 㪠㪁 㪈㪆㪠 㪠㪥㪣㪆㪠 㪙 㪥㫌㫄㪹㪼㫉㩷㫆㪽㩷㪺㪼㫃㫃㫊 㪘㫄㫆㫌㫅㫋㩷㫆㪽㩷㫀㫅㪽㫆㫉㫄㪸㫋㫀㫆㫅㩷㫆㪹㫋㪸㫀㫅㪼㪻㩷 㩷㩷㩷㩷㪹㫐㩷㫄㫀㫊㫄㪸㫋㪺㪿㪼㪻㩷㪻㪼㪺㫆㪻㪼㫉㩷㩿㩼㪀 Figure 1: Comparison between correct information I∗derived by Merhav et al. and NirenbergLatham information INL. A: Difference between I∗/I (solid line) and INL/I (dotted line) in Gaussian model where correlations and derivatives of mean firing rates are uniform. Correlation parameter c = 0.01. B: Difference between I∗ 1/I (solid line) and INL 1 /I (dotted line) when spike data in Figure 3A are used. For this spike data and other spike data analyzed, Nirenberg-Latham information provides a tight lower bound on the correct information, possibly because the number of cells is small. Previously, Nirenberg and Latham proposed that the information obtained by mismatched decoders can be evaluated by using [11] INL = − Z drp(r) log2 X s p(s)q(r|s) + Z dr X s p(s)p(r|s) log2 q(r|s). (3) We call their proposed information “Nirenberg-Latham information”. If we set β = 1 in Eq. 2, we obtain Nirenberg-Latham information, I∗(1) = INL. Thus, Nirenberg-Latham information does not give correct information; instead, it simply provides the lower bound on the correct information, I∗(β), which is the maximum value with respect to β [5, 8]. The lower bound provided by Nirenberg-Latham information can be very loose and the Nirenberg-Latham information can be negative when many cells are analyzed. Theoretical evaluation of information I, I∗, and INL We consider the problem where mutual information is computed when stimulus s, which is a single variable, and slightly different stimulus s + ∆s are presented. We assume the prior probability of stimuli, p(s) and p(s + ∆s), are equal: p(s) = p(s + ∆s) = 1/2. Neural responses evoked by the stimuli are denoted by r, which is considered here to be the neuron firing rate. When the difference between two stimuli is small, the conditional probability p(r|s + ∆s) can be expanded with respect to ∆s as p(r|s+∆s) = p(r|s)+p′(r|s)∆s+ 1 2p′′(r|s)(∆s)2+..., where ′ represents differentiation with respect to s. Using the expansion, to leading order of ∆s, we can write mutual information I as I = ∆s2 8 Z dr(p′(r|s))2 p(r|s) , (4) where R dr p′(r|s)2 p(r|s) is the Fisher information. Thus, we can see that the mutual information is proportional to the Fisher information when ∆s is small. Similarly, the correct information I∗for the mismatched decoders and the Nirenberg-Latham information INL can be written as I∗= ∆s2 8 µZ drp′(r|s)q′(r|s) q(r|s) ¶2 µZ drp(r|s)(q′(r|s))2 q(r|s)2 ¶−1 , (5) INL = ∆s2 8 Ã − Z drp(r|s) µq′(r|s) q(r|s) ¶2 + 2 Z drp′(r|s)q(r|s) q(r|s) ! . (6) Taking into consideration the proportionality of the mutual information to the Fisher information, we can interpret that ³R dr p′(r|s)q′(r|s) q(r|s) ´2 ³R dr p(r|s)(q′(r|s))2 q(r|s)2 ´−1 in Eq. 5 is a Fisher information-like quantity for mismatched decoders. 3 Let us consider the case in which the encoding model p(r|s) obeys the Gaussian distribution p(r|s) = 1 Z exp µ −1 2(r −f(s))T C−1(r −f(s)) ¶ , (7) where T stands for the transpose operation, f(s) is the mean firing rates given stimulus s, and C is the covariance matrix. We consider an independent decoding model q(r|s) that ignores correlations: q(r|s) = 1 ZD exp µ −1 2(r −f(s))T C−1 D (r −f(s)) ¶ , (8) where CD is the diagonal covariance matrix obtained by setting the off-diagonal elements of C to 0. If the Gaussian integral is performed for Eqs. 4-5, I, I∗, and INL can be written as I = ∆s2 8 f ′T (s)C−1f ′(s), (9) I∗= ∆s2 8 (f ′T (s)C−1 D f ′(s))2 f ′T (s)C−1 D CC−1 D f ′(s), (10) INL = ∆s2 8 ¡ −f ′T (s)C−1 D CC−1 D f ′(s) + 2f ′T (s)C−1 D f ′(s) ¢ . (11) The correct information obtained by the independent decoder for the Gaussian model (Eq. 10) is inversely proportional to the decoding error of s when the independent decoder is applied, which was computed from the generalized Cram´er Rao bound by Wu et al. [14]. As a simple example, we consider a uniform correlation model [1, 14] in which covariance matrix C is given by Cij = σ2[δij +c(1−δij)] and assume that the derivatives of the firing rates are uniform: that is f ′ i = f ′. In this case, I, I∗, and INL can be computed using I = ∆s2 8 Nf ′2 σ2(Nc + 1 −c), (12) I∗= ∆s2 8 Nf ′2 σ2(Nc + 1 −c), (13) INL = ∆s2 8 (−c(N −1) + 1)Nf ′2 σ2 , (14) where N is the number of cells. We can see that I∗is equal to I, which means that information is not lost even if correlation is ignored in the decoding process. Figure 1A shows INL/I and I∗/I when the degree of correlation c is 0.01. As shown in Figure 1A, the difference between the correct information I∗and Nirenberg-Latham information INL is very large when the number of cells N is large. When N > c+1 c , INL is negative. Analysis showed that using Nirenberg-Latham information INL as a lower bound on the correct information I∗can lead to wrong conclusions, especially when many cells are analyzed. 3 Analysis of information in population activities of ganglion cells 3.1 Methods We analyzed the data obtained when N = 7 retinal ganglion cells were simultaneously recorded using a multielectrode array. The stimulus was a natural movie, which was 200 s long and repeated 45 times. We divided the movie into many short natural movies and considered them as stimuli over which information contained in neural activities is computed. For instance, when it was divided into 10-s-long natural movies, there were 20 stimuli. Figure 2A shows the response of the seven retinal ganglion cells to natural movies from 0 to 10 s in length. To apply information theoretic techniques, we first discretized the time into small time bins ∆τ and indicated whether a spike was emitted or not in each time bin with a binary variable: σi = 1 means that the cell i spiked and σi = 0 means that it did not spike. We set the length of the time, ∆τ, to 5 ms so that it was short enough to avoid two spikes falling into the same bin. In this way, the spike pattern of ganglion cells was transformed into an N-letter binary word, σ = {σ1, σ2, ..., σN}, as shown in Figure 2B. Then, we determined the 4 㪇㪇㪇㪇㪇㪇㪇 㪇㪇㪇㪇㪇㪇㪇 㪇㪇㪇㪇㪇㪇㪇 㪇㪈㪇㪇㪇㪇㪇 㪇㪈㪇㪇㪇㪇㪇 㪇㪇㪇㪇㪈㪇㪇 㪇㪈㪇㪇㪇㪇㪇 㪇㪈㪇㪇㪇㪇㪇 㪇㪈㪈㪇㪇㪇㪇 㪇㪈㪇㪇㪈㪇㪇 㪇㪈㪇㪇㪈㪇㪇 㪇㪈㪈㪈㪇㪇㪇 㪇㪇㪇㪈㪈㪇㪇 㪇㪇㪇㪇㪇㪇㪇 㪇㪈㪇㪇㪇㪇㪇 㪇㪈㪇㪇㪇㪇㪇 㪈㪇㪈㪈㪇㪇㪇 㪇㪈㪇㪈㪇㪇㪇 㪈㪇㪇㪇㪈㪇㪇 㪇㪇㪇㪇㪇㪇㪇 㪚㪼㫃㫃㩷㫅㫌㫄㪹㪼㫉 㪫㫀㫄㪼㩷㩿㫊㪀 㪙 㪘 㪚㪼㫃㫃㩷㫅㫌㫄㪹㪼㫉 㪫㫀㫄㪼㩷㩿㫊㪀 Figure 2: A: Raster plot of seven retinal ganglion cells responding to a natural movie. B: Transformation of spike trains into binary words. frequency with which a particular spike pattern, σ, was observed during each stimulus and estimated the conditional probability distribution pdata(σ|s) from experimental data. Using these conditional probabilities, we evaluated the information contained in N-letter binary words σ. Generally, the joint probability of N binary variables can be written as [9] pN(σ) = 1 Z exp X i θiσi + X i<j θijσiσj + · · · + θ12...Nσ1σ2...σN . (15) This type of probability distribution is called a log-linear model. Because the number of parameters in a log-linear model is equal to the number of all possible configurations of an N-letter binary word σ, we can determine the values of parameters so that the log-linear model pN(σ) exactly matches empirical probability distribution pdata(σ): that is, pN(σ) = pdata(σ). To compute the information for mismatched decoders, we construct simplified models of neural responses that partially match the empirical distribution, pdata(σ). The simplest model is an “independent model” p1(σ), where only the average of each σi agrees with the experimental data: that is, ⟨σi⟩p1(σ) = ⟨σi⟩pdata(σ). There are many possible probability distributions that satisfy these constraints. In accordance with the maximum entropy principle [12], we choose the one that maximizes entropy H, H = −P σ p1(σ) log p1(σ). The resulting maximum entropy distribution is p1(σ) = 1 Z1 exp "X i θ(1) i σi # . (16) in which model parameters θ(1) are determined so that the constraints are satisfied. This model corresponds to a log-linear model in which all orders of correlation parameters {θij, θijk, ..., θ12...N} are omitted. If we perform maximum likelihood estimation of model parameters θ(1) in the loglinear model, the result is that the average σi under the log-linear model equals the average σi found in the data: that is, ⟨σi⟩p1(σ) = ⟨σi⟩pdata(σ). This result is identical to the constraints of the maximum entropy model. Generally, the maximum entropy method is equivalent to maximum likelihood fitting of a log-linear model [6]. Similarly, we can consider a “second-order correlation model” p2(σ), which is consistent with not only the averages of σi but also the averages of all products σiσj found in the data. Maximizing the entropy with constraints ⟨σi⟩p2(σ) = ⟨σi⟩pdata(σ) and ⟨σiσj⟩p2(σ) = ⟨σiσj⟩pdata(σ), we obtain p2(σ) = 1 Z2 exp X i θ(2) i σi + X i,j θ(2) ij σiσj , (17) in which model parameters θ(2) are determined so that the constraints are satisfied. The procedure described above can also be used to construct a “Kth-order correlation model” pK(σ). If we substitute the simplified models of neural responses pK(σ|s) into mismatched decoding models q(σ|s) in 5 㪥㫌㫄㪹㪼㫉㩷㫆㪽㩷㪺㪼㫃㫃㫊 㪠㪁 㪈㪆㪠 㪠㪁 㪉㪆㪠 㪘 㪥㫌㫄㪹㪼㫉㩷㫆㪽㩷㪺㪼㫃㫃㫊 㪙 㪠㪁 㪈㪆㪠 㪠㪁 㪉㪆㪠 㪘㫄㫆㫌㫅㫋㩷㫆㪽㩷㫀㫅㪽㫆㫉㫄㪸㫋㫀㫆㫅㩷㫆㪹㫋㪸㫀㫅㪼㪻㩷 㩷㩷㩷㩷㪹㫐㩷㫄㫀㫊㫄㪸㫋㪺㪿㪼㪻㩷㪻㪼㪺㫆㪻㪼㫉㩷㩿㩼㪀 㪘㫄㫆㫌㫅㫋㩷㫆㪽㩷㫀㫅㪽㫆㫉㫄㪸㫋㫀㫆㫅㩷㫆㪹㫋㪸㫀㫅㪼㪻㩷 㩷㩷㩷㩷㪹㫐㩷㫄㫀㫊㫄㪸㫋㪺㪿㪼㪻㩷㪻㪼㪺㫆㪻㪼㫉㩷㩿㩼㪀 Figure 3: Dependence of amount of information obtained by simplified decoders on number of ganglion cells analyzed. Same spike data obtained from retinal ganglion cells responding to a natural movie were used to obtain analysis results shown in panels A and B. A: 10-s-long natural movie B: 100-ms-long natural movie Eq. 2, we can compute the amount of information that can be obtained when more than Kth-order correlations are ignored in the decoding, I∗ K(β) = − X σ pN(σ) log2 X s p(s)pK(σ|s)β + X s p(s) X σ pN(σ|s) log2 pK(σ|s)β. (18) By evaluating the ratio of information, I∗ K/I, we can infer how many orders of correlation should be taken into account to extract enough information. 3.2 Results First, we investigated how the ratio of information obtained by an independent model, I∗ 1/I, and that obtained by a second-order correlation model, I∗ 2/I, changed when the number of cells analyzed was changed. We set the length of the stimulus to 10 s. We could obtain 20 kinds of stimuli from a 200-slong natural movie (see Methods). In previous studies, comparable length stimuli (7 s for Nirenberg et al.’s study [10] and 20 s for Schneidman et al.’s study [12]) were used. When two neurons were analyzed, there were 21 possible combinations for choosing 2 cells out of 7 cells, which is the total number of cells simultaneously recorded. We computed the average value of I∗ K/I for K = 1, 2 over all possible combinations of cells. Figure 3A shows that I∗ 1/I and I∗ 2/I monotonically decreased when the number of cells was increased. A comparison between the correct information, I∗ 1/I, and Nirenberg-Latham information, INL 1 /I where INL 1 = I∗ 1(β = 1), is shown in Figure 1B. When only two cells were considered, I∗ 1/I exceeded 90%, which means that ignoring correlation leads to only a small loss of information. This is consistent with the result obtained by Nirenberg et al. [10]. However, when all cells (N = 7) were used in the analysis, I∗ 1/I becomes only about 60%. Thus, correlation seems to be much more important for decoding when population activities are considered than when only two cells are considered. At least, we can say that qualitatively different things occur when large populations of cells are analyzed, as Schneidman et al. pointed out [12]. We should be careful about concluding from the results shown in Figure 3A that correlation is important for decoding. In this analysis, we considered a 10-s-long stimuli and assumed stationarity during each stimulus. By stationarity we mean that we assumed spikes are generated by a single process that can be described by a single conditional distribution p(σ|s). Because the natural movies change much more rapidly and our visual system has much higher time resolution than 10 s [13], we also considered shorter stimuli. In Figure 3B, we computed I∗ 1/I and I∗ 2/I over 100-ms-long natural movies. In this case, we could obtain 2000 stimuli from the 200-s-long natural movie. When the length of each stimulus was 100 ms, no spikes occurred while some stimuli were presented. We removed those stimuli and used the remaining stimuli for the analysis. In this case, the amount of information obtained by independent model I∗ 1 was more than 90% even when all cells (N = 7) were considered. Although 100 ms may still be too long to be considered as a single process, the result shown in Figure 3B reflects a situation that our brain has to deal with, that is more realistic than that reflected in Figure 3A. Figure 4A shows the dependence of information obtained by simplified decoders on the length of stimulus. In this analysis, we changed the length of the stimulus from 100 ms to 10 s and computed I∗ 1/I and I∗ 2/I for activities of N = 7 cells. We also analyzed additional experimental data obtained when N = 6 retinal ganglion cells were simultaneously recorded from 6 ̂ 㪣㪼㫅㪾㫋㪿㩷㫆㪽㩷㫊㫋㫀㫄㫌㫃㫌㫊㩷㩿㫊㪀 㪠㪁 㪈㪆㪠 㪠㪁 㪉㪆㪠 㪘 ̂ 㪣㪼㫅㪾㫋㪿㩷㫆㪽㩷㫊㫋㫀㫄㫌㫃㫌㫊㩷㩿㫊㪀 㪠㪁 㪈㪆㪠 㪠㪁 㪉㪆㪠 㪙 㪣㪼㫅㪾㫋㪿㩷㫆㪽㩷㫊㫋㫀㫄㫌㫃㫌㫊㩷㩿㫊㪀 㪚 㪠㪁 㪈㪆㪠 㪘㫄㫆㫌㫅㫋㩷㫆㪽㩷㫀㫅㪽㫆㫉㫄㪸㫋㫀㫆㫅㩷㫆㪹㫋㪸㫀㫅㪼㪻㩷 㩷㩷㩷㩷㪹㫐㩷㫄㫀㫊㫄㪸㫋㪺㪿㪼㪻㩷㪻㪼㪺㫆㪻㪼㫉㩷㩿㩼㪀 㪘㫄㫆㫌㫅㫋㩷㫆㪽㩷㫀㫅㪽㫆㫉㫄㪸㫋㫀㫆㫅㩷㫆㪹㫋㪸㫀㫅㪼㪻㩷 㩷㩷㩷㩷㪹㫐㩷㫄㫀㫊㫄㪸㫋㪺㪿㪼㪻㩷㪻㪼㪺㫆㪻㪼㫉㩷㩿㩼㪀 㪘㫄㫆㫌㫅㫋㩷㫆㪽㩷㫀㫅㪽㫆㫉㫄㪸㫋㫀㫆㫅㩷㫆㪹㫋㪸㫀㫅㪼㪻㩷 㩷㩷㩷㩷㪹㫐㩷㫄㫀㫊㫄㪸㫋㪺㪿㪼㪻㩷㪻㪼㪺㫆㪻㪼㫉㩷㩿㩼㪀 Figure 4: Dependence of amount of information obtained by simplified decoders on length of stimuli. Stimulus was same natural movie for both panels, but spike data obtained from retinas of different salamander were used in panels A and B. A: Seven simultaneously recorded ganglion cells B: Six simultaneously recorded ganglion cells C: Artificial spike data generated according to the firing rates shown in Figure 5A 㪘 㪙 㪚 㪫㫀㫄㪼㩷㩿㫊㪀 㪝㫀㫉㫀㫅㪾㩷㫉㪸㫋㪼㩷㩿㫊㫇㫀㫂㪼㫊㪆㫊㪀 㩺㪈 㩺㪉 㪫㫀㫄㪼㩷㩿㫊㪀 㪫㫀㫄㪼㩷㩿㫊㪀 㫊㪈 㫊㪉 㫊㪈 㫊㪉 㫊㪊 㫊㪋 Figure 5: Firing rates of two model cells. Rate of cell #1 shown in top panel; rate of cell #2 is shown in bottom panel. A: Firing rates from 0 to 2 s. B: Firing rates (solid line) and mean firing rates (dashed line) when stimulus was 1 s long. C: Firing rates (solid line) and mean firing rates (dashed line) when stimulus was 500 ms long. another salamander retina. The same 200-s-long natural movie was used as a stimulus for Figure 4B as for Figure 4A, and the activities of N = 6 cells were analyzed. Figure 4B shows the result. We can clearly see the same tendency as shown in Figures 4A and B: the amount of information decoded by the simplified decoders monotonically increased as the length of the stimulus was shortened. To clarify the reason the correlation becomes less important as the stimulus is shortened, we used the toy model shown in Figure 5. We considered the case in which two cells fire independently in accordance with a Poisson process and performed an analysis similar to the one we did for the actual spike data. We used simulated spike data for the two cells generated in accordance with the firing rates shown in Figure 5A. The firing rates with a 2-s stimulus sinusoidally change with time. We divided the 2-s-long stimulus into two 1-s-long stimulus, s1 and s2, as shown in Figure 5B. Then, we computed mutual information I and the information obtained by independent model I∗ 1 over s1 and s2. Because the two cells fired independently, there were no correlations between two cells essentially. However, there was pseudo correlation due to the assumption of stationarity for the dynamically changing stimulus. The pseudo correlation was high for s1 and low for s2. This means that “correlation” plays an important role in discriminating two stimuli, s1 and s2. In contrast, the mean firing rates of the two cells during each stimulus were equal for s1 and s2. Therefore, if the stimulus is 1 s long, we cannot discriminate two stimuli by using the independent model, that is, I∗ 1 = 0. We also considered the case in which the stimulus was 0.5 s long, as shown in Figure 5C. In this case, pseudo correlations again appeared but there was a significant difference in the mean firing rates between the stimuli. Thus, the independent model can be used to extract almost all the information. The dependence of I∗ 1/I on the stimulus length is shown in Figure 4C. Behaviors similar to those represented in Figure 4C were also observed in the analysis of the actual spike data for retinal ganglion cells (Figure 4A and 4B). Even if we observe that correlation carries a significant large portion of information for longer stimuli compared with the speed of change in the firing rates, 7 it may simply be caused by meaningless pseudo correlation. To assess the role of correlation in information processing, the stimuli used should be short enough to think neural responses to these stimuli generated by a single process. 4 Summary and Discussion We described a general framework for investigating how far the decoding process in the brain can be simplified. We computed the amount of information that can be extracted by using simplified decoders constructed using a maximum entropy model, i.e., mismatched decoders. We showed that more than 90% of the information encoded in retinal ganglion cells activities can be decoded by using an independent model that ignores correlation. Our results imply that the brain uses a simplified decoding strategy in which correlation is ignored. When we computed the information obtained by the independent model, we regarded a 100-ms-long natural movie as one stimulus. However, when we considered stimuli that were long compared with the speed of the change in the firing rates as one stimulus, correlation carried a large portion of information. This is due to pseudo correlation, which is observed if stationarity is assumed for long durations. The human visual system can process visual information in less than 150 ms [13]. We should set the length of the stimulus appropriately by taking the time resolution of our visual system into account. Our results do not imply that any kind of correlation does not carry much information because we dealt only with correlated spikes within a 5-ms time bin. In our analysis, we did not analyze the correlation on a longer time scale, which can be observed in the activities of retinal ganglion cells [7]. We also did not investigate the information carried by the relative timing of spikes [3]. Further investigations are needed for these types of correlation. Our approach of comparing the mutual information with the information obtained by simplified decoders can also be used for studying other types of correlations. References [1] Abbott, L. F., & Dayan, P. (1999). Neural Comput., 11, 91-101. [2] Deneve, S., Latham, P. E., & Pouget, A. (1999). Nature Neurosci., 2, 740-745. [3] Gollish, S., & Meister, M. (2008). Science, 319, 1108-1111. [4] Jazayeri, M. & Movshon, J. A. (2006). Nature Neurosci., 9, 690-696. [5] Latham, P. E., & Nirenberg, S. (2005). J. Neurosci., 25, 5195-5206. [6] MacKay, D. (2003). Information Theory, Inference and Learning Algorithms (Cambridge Univ. Press, Cambridge, England). [7] Meister, M., & Berry, M. J. II (1999). Neuron, 22, 435-450. [8] Merhav, N., Kaplan, G., Lapidoth, A., & Shamai Shitz, S. (1994). IEEE Trans. Inform. Theory, 40, 1953-1967. [9] Nakahara, H., & Amari, S. (2002). Neural Comput., 14, 2269-2316. [10] Nirenberg, S., Carcieri, S. M., Jacobs, A. L., & Latham, P. E. (2001). Nature, 411, 698-701. [11] Nirenberg, S., & Latham, P. (2003). Proc. Natl. Acad. Sci. USA, 100, 7348-7353. [12] Schneidman, E., Berry, M. J. II, Segev, R., & Bialek. W. (2006). Nature, 440, 1007-1012. [13] Thorpe, S., Fize, D., & Marlot, C. (1996). Nature, 381, 520-522. [14] Wu, S., Nakahara, H., & Amari, S. (2001). Neural Comput., 13, 775-797. 8
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Signal-to-Noise Ratio Analysis of Policy Gradient Algorithms John W. Roberts and Russ Tedrake Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology Cambridge, MA 02139 Abstract Policy gradient (PG) reinforcement learning algorithms have strong (local) convergence guarantees, but their learning performance is typically limited by a large variance in the estimate of the gradient. In this paper, we formulate the variance reduction problem by describing a signal-to-noise ratio (SNR) for policy gradient algorithms, and evaluate this SNR carefully for the popular Weight Perturbation (WP) algorithm. We confirm that SNR is a good predictor of long-term learning performance, and that in our episodic formulation, the cost-to-go function is indeed the optimal baseline. We then propose two modifications to traditional model-free policy gradient algorithms in order to optimize the SNR. First, we examine WP using anisotropic sampling distributions, which introduces a bias into the update but increases the SNR; this bias can be interpreted as following the natural gradient of the cost function. Second, we show that non-Gaussian distributions can also increase the SNR, and argue that the optimal isotropic distribution is a ‘shell’ distribution with a constant magnitude and uniform distribution in direction. We demonstrate that both modifications produce substantial improvements in learning performance in challenging policy gradient experiments. 1 Introduction Model-free policy gradient algorithms allow for the optimization of control policies on systems which are impractical to model effectively, whether due to cost, complexity or uncertainty in the very structure and dynamics of the system (Kohl & Stone, 2004; Tedrake et al., 2004). However, these algorithms often suffer from high variance and relatively slow convergence times (Greensmith et al., 2004). As the same systems on which one wishes to use these algorithms tend to have a high cost of policy evaluation, much work has been done on maximizing the policy improvement from any individual evaluation (Meuleau et al., 2000; Williams et al., 2006). Techniques such as Natural Gradient (Amari, 1998; Peters et al., 2003a) and GPOMDP (Baxter & Bartlett, 2001) have become popular through their ability to match the performance gains of more basic model-free policy gradient algorithms while using fewer policy evaluations. As practitioners of policy gradient algorithms in complicated mechanical systems, our group has a vested interest in making practical and substantial improvements to the performance of these algorithms. Variance reduction, in itself, is not a sufficient metric for optimizing the performance of PG algorithms - of greater significance is the magnitude of the variance relative to the magnitude of the gradient update. Here we formulate a signal-to-noise ratio (SNR) which facilitates simple and fast evaluations of a PG algorithm’s average performance, and facilitates algorithmic performance improvements. Though the SNR does not capture all facets of a policy gradient algorithm’s capability to learn, we show that achieving a high SNR will often result in a superior convergence rate with less violent variations in the policy. 1 Through a close analysis of the SNR, and the means by which it is maximized, we find several modifications to traditional model-free policy gradient updates that improve learning performance. The first of these is the reshaping of distributions such that they are different on different parameters, a modification which introduces a bias to the update. We show that this reshaping can improve performance, and that the introduced bias results in following the natural gradient of the cost function, rather than the true point gradient. The second improvement is the use of non-Gaussian distributions for sampling, and through the SNR we find a simple distribution which improves performance without increasing the complexity of implementation. 2 The weight perturbation update Consider minimizing a scalar function J(⃗w) with respect to the parameters ⃗w (note that it is possible that J(⃗w) is a long-term cost and results from running a system with the parameters ⃗w until conclusion). The weight perturbation algorithm (Jabri & Flower, 1992) performs this minimization with the update: ∆⃗w = −η (J(⃗w + ⃗z) −J(⃗w))⃗z, (1) where the components of the ‘perturbation’, ⃗z, are drawn independently from a mean-zero distribution, and η is a positive scalar controlling the magnitude of the update (the “learning rate”). Performing a first-order Taylor expansion of J(⃗w + ⃗z) yields: ∆⃗w = −η J(⃗w) + X i ∂J ∂⃗w izi −J(⃗w) ! ⃗z = −η X i ∂J ∂⃗w izi · ⃗z. (2) In expectation, this becomes the gradient times a (diagonal) covariance matrix, and reduces to E[∆⃗w] = −ησ2 ∂J ∂⃗w, (3) an unbiased estimate of the gradient, scaled by the learning rate and σ2, the variance of the perturbation. However, this unbiasedness comes with a very high variance, as the direction of an update is uniformly distributed. It is only the fact that updates near the direction of the true gradient have a larger magnitude than do those nearly perpendicular to the gradient that allows for the true gradient to be achieved in expectation. Note also that all samples parallel to the gradient are equally useful, whether they be in the same or opposite direction, as the sign does not affect the resulting update. The WP algorithm is one of the simplest examples of a policy gradient reinforcement learning algorithm, and thus is well suited for analysis. In the special case when ⃗z is drawn from a Gaussian distribution, weight perturbation can be interpreted as a REINFORCE update(Williams, 1992). 3 SNR for policy gradient algorithms The SNR is the expected power of the signal (update in the direction of the true gradient) divided by the expected power of the noise (update perpendicular to the true gradient). Taking care to ensure that the magnitude of the true gradient does not effect the SNR, we have: SNR = E h ∆⃗wT ∥∆⃗w∥ i E ∆⃗wT ⊥∆⃗w⊥ , (4) ∆⃗w∥= ∆⃗wT ⃗Jw
⃗Jw
⃗Jw
⃗Jw
, ∆⃗w⊥= ∆⃗w −⃗w∥, (5) and using ⃗Jw(⃗w0) = ∂J(⃗w) ∂⃗w (⃗w=⃗w0) for convenience. Intuitively, this expression measures how large a proportion of the update is “useful”. If the update is purely in the direction of the gradient the SNR would be infinite, while if the update moved perpendicular to the true gradient, it would be zero. As such, all else being equal, a higher SNR should generally perform as well or better than a lower SNR, and result in less violent swings in cost and policy for the same improvement in performance. 2 3.1 Weight perturbation with Gaussian distributions Evaluating the SNR for the WP update in Equation 1 with a deterministic J(⃗w) and ⃗z drawn from a Gaussian distribution yields a surprisingly simple result. If one first considers the numerator: E h ∆⃗wT ∥∆⃗w∥ i = E η2
⃗Jw
4 X i,j JwiJwjzizj ⃗Jw T · X k,p JwkJwpzkzp ⃗Jw = E η2
⃗Jw
2 X i,j,k,p JwiJwjJwkJwpzizjzkzp = Q, (6) where we have named this term Q for convenience as it occurs several times in the expansion of the SNR. We now expand the denominator as follows: E ∆⃗wT ⊥∆⃗w⊥ = E h ∆⃗wT ∆⃗w −2∆⃗wT ∥(∆⃗w∥+ ∆⃗w⊥) + ∆⃗wT ∥∆⃗w∥ i = E ∆⃗wT ∆⃗w −2Q+Q (7) Substituting Equation (1) into Equation (7) and simplifying results in: E ∆⃗wT ⊥∆⃗w⊥ = η2
⃗Jw
2 E X i,j,k JwiJwjzizjz2 k −Q. (8) We now assume that each component zi is drawn from a Gaussian distribution with variance σ2. Taking the expected value, it may be further simplified to: Q = η2
⃗Jw
4 3σ4 X i Jwi 4 + 3σ4 X i Jwi 2 X j̸=i Jwj 2 = 3σ4
⃗Jw
4 X i,j Jwi 2Jwj 2 = 3σ4, (9) E ∆⃗wT ⊥∆⃗w⊥ = η2σ4
⃗Jw
2 2 X i Jwi 2 + X i,j Jwi 2 −Q = σ4(2+N)−3σ4 = σ4(N −1), (10) where N is the number of parameters. Canceling σ results in: SNR = 3 N −1. (11) Thus, for small noises and constant σ the SNR and the parameter number have a simple inverse relationship. This is a particularly concise model for performance scaling in PG algorithms. 3.2 Relationship of the SNR to learning performance To evaluate the degree to which the SNR is correlated with actual learning performance, we ran a number of experiments on a simple quadratic bowl cost function, which may be written as: J(⃗w) = ⃗wT A⃗w, (12) where the optimal is always at the point ⃗0. The SNR suggests a simple inverse relationship between the number of parameters and the learning performance. To evalute this claim we performed three tests: 1) true gradient descent on the identity cost function (A set to the identity matrix) as a benchmark, 2) WP on the identity cost function and 3) WP on 150 randomly generated cost functions (each component drawn from a Gaussian distribution), all of the form given in Equation (12), and for values of N between 2 and 10. For each trial ⃗w was intially set to be ⃗1. As can be seen in Figure 1a, both the SNR and the reduction in cost after running WP for 100 iterations decrease monotonically as the number of parameters N increases. The fact that this occurs in the case of randomly generated cost functions demonstrates that this effect is not related to the simple form of the identity cost function, but is in fact related to the number of dimensions. 3 Figure 1: Two comparisons of SNR and learning performance: (A) Relationship as dimension N is increased (Section 3.2). The curves are 15,000 averaged runs, each run 100 iterations. For randomly generated cost functions, 150 A matrices were tested. True gradient descent was run on the identity cost function. The SNR for each case was computed in with Equation (11). (B) Relationship as Gaussian is reshaped by changing variances for case of 2D anisotropic cost function(ratio of gradients in different directions is 5) cost function (Section 4.1.1). The constraint σ2 1 + σ2 2 = 0.1 is imposed, while σ2 1 is between 0 and .1. For each value of σ1 15,000 updates were averaged to produce the curve plotted. The plot shows that variances which increase the SNR also improve the performance of the update. 3.3 SNR with parameter-independent additive noise In many real world systems, the evaluation of the cost J(⃗w) is not deterministic, a property which can significantly affect learning performance. In this section we investigate how additive ‘noise’ in the function evaluation affects the analytical expression for the SNR. We demonstrate that for very high noise WP begins to behave like a random walk, and we find in the SNR the motivation for an improvement in the WP algorithm that will be examined in Section 4.2. Consider modifying the update seen in Equation (1) to allow for a parameter-independent additive noise term v and a more general baseline b(⃗w), and again perform the Taylor expansion. Writing the update with these terms gives: ∆⃗w = −η J(⃗w) + X i Jwizi −b(⃗w) + v ! ⃗z = −η X i Jwizi + ξ(⃗w) ! ⃗z. (13) where we have combined the terms J(⃗w), b(⃗w) and v into a single random variable ξ(⃗w). The new variable ξ(⃗w) has two important properties: its mean can be controlled through the value of b(⃗w), and its distribution is independent of parameters ⃗w, thus ξ(⃗w) is independent of all the zi. We now essentially repeat the calculation seen in Section 3.1, with the small modification of including the noise term. When we again assume independent zi, each drawn from identical Gaussian distributions with standard deviation σ, we obtain the expression: SNR = φ + 3 (N −1)(φ + 1), φ = (J(⃗w) −b(⃗w))2 + σ2 v σ2∥⃗Jw∥2 (14) where σv is the standard deviation of the noise v and we have termed the error component φ. This expression depends upon the fact that the noise v is mean-zero and independent of the parameters, although as stated earlier, the assumption that v is mean-zero is not limiting. It is clear that in the limit of small φ the expression reduces to that seen in Equation (11), while in the limit of very large φ it becomes the expression for the SNR of a random walk (see Section 3.4). This expression makes it clear that minimizing φ is desirable, a result that suggests two things: (1) the optimal baseline (from the perspective of the SNR) is the value function (i.e. b∗(⃗w) = J(⃗w)) and (2) higher values of σ are desirable, as they reduce φ by increasing the size of its denominator. However, there is clearly a limit on the size of σ due to higher order terms in the Taylor expansion; very large σ will result in samples which do not represent the local gradient. Thus, in the case of noisy measurements, there is some optimal sampling distance that is as large as possible without resulting in poor sampling of the local gradient. This is explored in Section 4.2.1. 4 3.4 SNR of a Random Walk Due to the fact that the update is squared in the SNR, only its degree of parallelity to the true gradient is relevant, not its direction. In the case of WP on a deterministic function, this is not a concern as the update is always within 90◦of the gradient, and thus the parallel component is always in the correct direction. For a system with noise, however, components of the update parallel to the gradient can in fact be in the incorrect direction, contributing to the SNR even though they do not actually result in learning. This effect only becomes significant when the noise is particularly large, and reaches its extreme in the case of a true random walk (a strong bias in the “wrong” direction is in fact a good update with an incorrect sign). If one considers moving by a vector drawn from a multivariate Gaussian distribution without any correlation to the cost function, the SNR is particularly easy to compute, taking the form: SNR = 1 ∥⃗Jw∥4 X i Jwizi ⃗Jw T X j Jwjzj ⃗Jw (⃗z − 1 ∥⃗Jw∥2 X i Jwizi ⃗Jw)T (⃗z − 1 ∥⃗Jw∥2 X i Jwizi ⃗Jw) = σ2 Nσ2 −2σ2 + σ2 = 1 N −1 (15) As was discussed in Section 3.3, this value of the SNR is the limiting case of very high measurement noise, a situation which will in fact produce a random walk. 4 Applications of SNR 4.1 Reshaping the Gaussian Distribution Consider a generalized WP algorithm, in which we allow each component zi to be drawn independently from separate mean-zero distributions. Returning to the derivation in Section 3.1, we no longer assume each zi is drawn from an identical distribution, but rather associate each with its own σi (the vector of the σi will be referred to as ⃗σ). Removing this assumption results in the SNR: SNR(⃗σ, ⃗Jw) =
⃗Jw
2 2 X i Jwi 2σ4 i + X i,j Jwi 2σ2 i σ2 j 3 X i,j Jwi 2σ2 i Jwj 2σ2 j −1 −1 . (16) An important property of this SNR is that it depends only upon the direction of ⃗Jw and the relative magnitude of the σi (as opposed to parameters such as the learning rate η and the absolute magnitudes ∥⃗σ∥and ∥⃗Jw∥). 4.1.1 Effect of reshaping on performance While the absolute magnitudes of the variance and true gradient do not affect the SNR given in Equation (16), the relative magnitudes of the different σi and their relationship to the true gradient can affect it. To study this property, we investigate a cost function with a significant degree of anisotropy. Using a cost function of the form given in Equation (12) and N = 2, we choose an A matrix whose first diagonal component is five times that of the second. We then investigate a series of possible variances σ2 1 and σ2 2 constrained such that their sum is a constant (σ2 1 + σ2 2 = C). We observe the performance of the first update (rather than the full trial) as the true gradient can vary significantly over the course of a trial, thereby having major effects on the SNR even as the variances are unchanged. As is clear in Figure 1b, as the SNR is increased through the choice of variances the performance of this update is improved. The variation of the SNR is much more significant than the change in performance, however this is not surprising as the SNR is infinite if the update is exactly along the correct direction, while the improvement from this update will eventually saturate. 5 4.1.2 Demonstration in simulation The improved performance of the previous section suggests the possibility of a modification to the WP algorithm in which an estimate of the true gradient is used before each update to select new variances which are more likely to learn effectively. Changing the shape of the distribution does add a bias to the update direction, but the resulting biased update is in fact descending the natural gradient of the cost function. To make use of this opportunity, some knowledge of the likely gradient direction is required. This knowledge can be provided via a momentum estimate (an average of previous updates) or through an inaccurate model that is able to capture some facets of the geometry of the cost function. With this estimated gradient the expression given in Equation (16) can be optimized over the σi numerically using a method such as Sequential Quadratic Programming (SQP). Care must be taken to avoid converging to very narrow distributions (e.g. placing some small minimum noise on all parameters regardless of the optimization), but ultimately this reshaping of the Gaussian can provide real performance benefits. f p x l m c g θ m (a) (b) Figure 2: (a) The cart-pole system. The task is to apply a horizontal force f to the cart such that the pole swings to the vertical position. (b) The average of 200 curves showing reduction in cost versus trial number for both a symmetric Gaussian distribution and a distribution reshaped using the SNR. The blue shaded region marks the area within one standard deviation for a symmetric Gaussian distribution, the red region marks one standard deviation for the reshaped distribution and the purple is within one standard deviation of both. The reshaping began on the eighth trial to give time for the momentum-based gradient estimate to stabilize. To demonstrate the improvement in convergence time this reshaping can achieve, weight perturbation was used to develop a barycentric feedback policy for the cart-pole swingup task, where the cost was defined as a weighted sum of the actuation used and the squared distance from the upright position. A gradient estimate was obtained through averaging previous updates, and SQP was used to optimize the SNR prior to each trial. Figure 2 demonstrates the superior performance of the reshaped distribution over a symmetric Guassian using the same total variance (i.e. the traces of the covariance matrices for both distributions were the same). 4.1.3 WP with Gaussian distributions follow the natural gradient The natural gradient for a policy that samples with a mean-zero Gaussian of covariance Σ may be written (see (Peters et al., 2003b)): ˜⃗Jw = F −1 ⃗Jw, F = Eπ(⃗ξ;⃗w) " ∂log π(⃗ξ; ⃗w) ∂wi ∂log π(⃗ξ; ⃗w) ∂wj # . (17) where F is the Fisher Information matrix, π is the sampling distribution, and ⃗ξ = ⃗w + ⃗z. Using the Gaussian form of the sampling, F may be evaluated easily, and becomes as Σ−1, thus: ˜⃗Jw = Σ ⃗Jw. (18) This is true for all mean-zero multivariate Gaussian distributions, thus the biased update, while no longer following the local point gradient, does follow the natural gradient. It is important to note that the natural gradient is a function of the shape of the sampling distribution, and it is because of this that all sampling distributions of this form can follow the natural gradient. 6 4.2 Non-Gaussian Distributions Figure 3: SNR vs. update magnitude for a 2D quadratic cost function. Mean-zero measurement noise is included with variances from 0 to .65. As the noise is increased, the sampling magnitude producing the maximum SNR is larger and the SNR achieved is lower. Note that the highest SNR achieved is for the smallest sampling magnitude with no noise where it approaches the theoretical value (for 2D) of 3. Also note that for small sampling magnitudes and large noises the SNR approaches the random walk value. The analysis in Section 3.3 suggests that for a function with noisy measurements there is an optimal sampling distance which depends upon the local noise and gradient as well as the strength of higher-order terms in that region. For a two-dimensional cost function of the form given in Equation (12), Figure 3 shows the SNR’s dependence upon the radius of the shell distribution (i.e. the magnitude of the sampling). For various levels of additive mean-zero noise the SNR was computed for a distribution uniform in angle and fixed in its distance from the mean (this distance is the “sampling magnitude”). The fact that there is a unique maximum for each case suggests the possibility of sampling only at that maximal magnitude, rather than over all magnitudes as is done with a Gaussian, and thus improving SNR and performance. While determining the exact magnitude of maximum SNR may be impractical, choosing a distribution with uniformly distributed direction and a constant magnitude close to this optimal value, performance can be improved. This idea was tested on the benchmark proposed in (Riedmiller et al., 2007), where comparisons showed it was able to learn at rates similar to optimized RPROP from reasonable initial policies, and was capable of learning from a zero initial policy. 4.2.1 Experimental Demonstration To provide compelling evidence of improved performance, the shell distribution was implemented on a laboratory experimental system with actuator limitations and innate stochasticity. We have recently been exploring the use of PG algorithms in an incredibly difficult and exciting control domain -fluid dynamics - and as such applied the shell distribution to a fluid dynamical system. Specifically, we applied learning to a system used to sudy the dynamics of flapping flight via a wing submerged in water (see Figure 4 for a description of the system (Vandenberghe et al., 2004)). The task is to determine the vertical motion producing the highest ratio of rotational displacement to energy input. Model-free methods are particularly exciting in this domain because direct numerical simulation can take days(Shelley et al., 2005) - in contrast optimizationg on the experimental physical flapping wing can be done in real-time, at the cost of dealing with noise in the evaluation of the cost function; success here would be enabling for experimental fluid dynamics. We explored the idea of using a “shell” distribution to improve the performance of our PG learning on this real-world system. (a) (b) Figure 4: (a) Schematic of the flapping setup. The plate rotates freely about its vertical axis, while the vertical motion is prescribed by the learnt policy. This vertical motion is coupled with the plate’s rotation through hydrodynamic effects. (b) 5 averaged runs on the flapping plate using Gaussian or Shell distributions for sampling. The error bars represent one standard deviation in the performance of different runs at that trial. 7 Representing the vertical position as a function of time with a 13-point periodic cubic spline, a 5D space was searched (points 1, 7 and 13 were fixed at zero, while points 2 and 8, 3 and 9 etc. were set to equal and opposite values determined by the control parameters). Beginning with a smoothed square wave, WP was run for 20 updates using shell distributions and Gaussians. Both forms of distributions were run 5 times and averaged to produce the curves in Figure 4. The sampling magnitude of the shell distribution was set to be the expected value of the length of a sample from the Gaussian distribution, while all other parameters were set equal. With optimized sampling, we acquired locally optimal policies in as little as 15 minutes, with repeated optimizations from very different initial policies converging to the same waveform. The result deepened our understanding of this fluid system and suggests promising applications to other fluid systems of similar complexity. 5 Conclusion In this paper we present an expression for the SNR of PG algorithms, and looked in detail at the common case of WP. This expression gives us a quantitative means of evaluating the expected performance of a PG algorithm, although the SNR does not completely capture an algorithm’s capacity to learn. SNR analysis revealed two distinct mechanisms for improving the WP update - perturbing different parameters with different distributions, and using non-Gaussian distributions. Both of them showed real improvement on highly nonlinear problems (the cart-pole example used a very high-dimensional policy), without knowledge of the problem’s dynamics and structure. We believe that SNR-optimized PG algorithms show promise for many complicated, real-world applications. 6 Acknowledgements The authors thank Drs. Lionel Moret and Jun Zhang for valuable assistance with the heaving foil. References Amari, S. (1998). Natural gradient works efficiently in learning. Neural Computation, 10, 251–276. Baxter, J., & Bartlett, P. (2001). Infinite-horizon policy-gradient estimation. Journal of Artificial Intelligence Research, 15, 319–350. Greensmith, E., Bartlett, P. L., & Baxter, J. (2004). Variance reduction techniques for gradient estimates in reinforcement learning. Journal of Machine Learning Research, 5, 1471–1530. Jabri, M., & Flower, B. (1992). Weight perturbation: An optimal architecture and learning technique for analog VLSI feedforward and recurrent multilayer networks. IEEE Trans. Neural Netw., 3, 154–157. Kohl, N., & Stone, P. (2004). Policy gradient reinforcement learning for fast quadrupedal locomotion. Proceedings of the IEEE International Conference on Robotics and Automation (ICRA). Meuleau, N., Peshkin, L., Kaelbling, L. P., & Kim, K.-E. (2000). Off-policy policy search. NIPS. Peters, J., Vijayakumar, S., & Schaal, S. (2003a). Policy gradient methods for robot control (Technical Report CS-03-787). University of Southern California. Peters, J., Vijayakumar, S., & Schaal, S. (2003b). Reinforcement learning for humanoid robotics. Proceedings of the Third IEEE-RAS International Conference on Humanoid Robots. Riedmiller, M., Peters, J., & Schaal, S. (2007). Evaluation of policy gradient methods and variants on the cart-pole benchmark. Symposium on Approximate Dynamic Programming and Reinforcement Learning (pp. 254–261). Shelley, M., Vandenberghe, N., & Zhang, J. (2005). Heavy flags undergo spontaneous oscillations in flowing water. Physical Review Letters, 94. Tedrake, R., Zhang, T. W., & Seung, H. S. (2004). Stochastic policy gradient reinforcement learning on a simple 3D biped. Proceedings of the IEEE International Conference on Intelligent Robots and Systems (IROS) (pp. 2849–2854). Sendai, Japan. Vandenberghe, N., Zhang, J., & Childress, S. (2004). Symmetry breaking leads to forward flapping flight. Journal of Fluid Mechanics, 506, 147–155. Williams, J. L., III, J. W. F., & Willsky, A. S. (2006). Importance sampling actor-critic algorithms. Proceedings of the 2006 American Control Conference. Williams, R. (1992). Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8, 229–256. 8
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Artificial Olfactory Brain for Mixture Identification Mehmet K. Muezzinoglu1 Alexander Vergara1 Ramon Huerta1 Thomas Nowotny2 Nikolai F. Rulkov1 Heny D. I. Abarbanel1 Allen Selverston1 Mikhail I. Rabinovich1 1 Institute for Nonlinear Science 2 Centre for Computational Neuroscience and Robotics University of California San Diego Department of Informatics, University of Sussex 9500 Gilman Dr., La Jolla, CA, 92093-0402 Falmer, Brighton, BN1 9QJ, UK Abstract The odor transduction process has a large time constant and is susceptible to various types of noise. Therefore, the olfactory code at the sensor/receptor level is in general a slow and highly variable indicator of the input odor in both natural and artificial situations. Insects overcome this problem by using a neuronal device in their Antennal Lobe (AL), which transforms the identity code of olfactory receptors to a spatio-temporal code. This transformation improves the decision of the Mushroom Bodies (MBs), the subsequent classifier, in both speed and accuracy. Here we propose a rate model based on two intrinsic mechanisms in the insect AL, namely integration and inhibition. Then we present a MB classifier model that resembles the sparse and random structure of insect MB. A local Hebbian learning procedure governs the plasticity in the model. These formulations not only help to understand the signal conditioning and classification methods of insect olfactory systems, but also can be leveraged in synthetic problems. Among them, we consider here the discrimination of odor mixtures from pure odors. We show on a set of records from metal-oxide gas sensors that the cascade of these two new models facilitates fast and accurate discrimination of even highly imbalanced mixtures from pure odors. 1 Introduction Odor sensors are diverse in terms of their sensitivity to odor identity and concentrations. When arranged in parallel arrays, they may provide a rich representation of the odor space. Biological olfactory systems owe the bulk of their success to employing a large number of olfactory receptor neurons (ORNs) of various phenotypes. However, chemo-diversity comes at the expense of two pressing factors, namely response time and reproducibility, while fast and accurate processing of chemo-sensory information is vital for survival not only in natural, but also in many artificial situations, including security applications. Identifying and quantifying an odor accurately in a short time is an impressive characteristic of insect olfaction. Given that there are approximately tens of thousands of ORNs sending slow and noisy messages in parallel to downstream olfactory layers, in order to account for the observed recognition performance, a computationally non-trivial process must be taking place along the insect olfactory pathway following the transduction. The two stations responsible for this processing are the Antennal Lobe (AL) and the Mushroom Bodies (MBs). The former acts as a signal conditioning / feature extraction device and the latter as an algebraic classifier. Our motivation in this study is the potential for skillful feature extraction and classification methods by insect olfactory systems in synthetic applications, which also deal with slow and noisy sensory data. The particular problem we address is the discrimination of two-component odor mixtures from 1 Mushroom Body Classifier Sensor Array 1 2 3 ... 16 Odor Identity Odor Snapshot Dynamical Antennal Lobe Model Model Figure 1: The considered biomimetic framework to identify whether an applied gas is a pure odor or a mixture. The input is transduced by 16 parallel metal-oxide gas sensors of different type generating slow and noisy resistance time series. The signal conditioning in the antennal lobe is achieved by the interaction of an excitatory Projection Neuron (PN) population (white nodes) with an inhibitory Local Neurons (LNs, black nodes). The outcomes of AL processing is read from the PNs and classified in the Mushroom Body, which is trained by a local Hebbian rule. pure odors in a three-class classification setting. The problem is nontrivial when concentrations of mixture components are imbalanced. It becomes particularly challenging when the overall mixture concentration is small. We treat the problem on two mixture datasets recorded from metal-oxide gas sensors (included in the supplementary material). We propose in the next section a dynamical rate model mimicking the AL’s signal conditioning function. By testing the model first with a generic Support Vector Machine (SVM) classifier, we validate the substantial improvement that AL adds on the classificatory value of raw sensory signal (Section 2). Then, we introduce a MB-like classifier to substitute for the SVM and complete the biomimetic framework, as outlined in Fig. 1. The model MB exploits the structural organization of the insect MB. Its plasticity is adjusted by a local Hebbian learning procedure, which is gated by a binary learning signal (Section 3). Some concluding remarks are given in Section 4. 2 The Antennal Lobe 2.1 Insect Antennal Lobe Outline The Antennal Lobe is a spatio-temporal encoder for ORN signals that include time in coding space. Some of its qualitative properties are apparent from the input-output perspective, without requiring much insight into its physiology. A direct analysis of spiking rates in raw ORN responses and in the AL output [1] shows that in fruit fly AL maps ORN output to a low dimensional feature space while providing lower variability in responses to the same odor type (reducing within-class scatter) and longer average distance between responses for different odors (boosting between-class scatter). These observations constitute sufficient evidence that a realistic AL model should be sought within the class of nonlinear filters. Another remarkable achievement of the AL shows itself in terms of recognition time. When subjected to a constant odor concentration, the settling time of ORN activity is on the order of hundreds of milliseconds to seconds [3], whereas recognition is known to occur earlier [7]. This is a clear indicator that the AL makes extensive use of the ORN transient, since instantaneous activity is less odor-specific in transient than it is in during the steady state. To provide high accuracy under such a temporal constraint, the classificatory information during this period must be somehow accumulated, which means that AL has to be a dynamical system, utilizing memory. It is the cooperation of these filtering and memory mechanisms in the AL that expedites and consolidates the decision made in the subsequent classifier. Strong experimental evidence suggests that the insect AL representation of odors is a transient, yet reproducible, spatio-temporal encoding [8]. The AL is a dynamical network that is formed by the coupling of an excitatory neuron population (projection neurons, PNs) with an inhibitory one 2 (local neurons, LNs). It receives input from glomeruli, junctions of synapses that group the ORNs according to the receptor gene they express. The fruit fly has about 50 glomeruli as chemotopic clusters of synapses from nearly 50, 000 ORNs. There is no consensus on the functional role of this convergence beyond serving as an input terminal to AL, which is certainly an active processing layer. In the analogy we are building here (c.f. Fig. 1), the 16 artificial gas sensors actually correspond to glomeruli (rather than individual ORNs) so that the AL has direct access to sensor resistances. We suggest that the two key principles underlying the AL’s information processing are decorrelation (filtering) and integration (memory), which can be unified on a dynamical system. The filter property provides selectivity, while the integrator accumulates the refined information on trajectories. This setting is capable of evaluating the transient portion of the sensory signal effectively. An instantaneous value read from a receptor early in the transduction process is considered as immature, failing to convey a consistently high classificatory value by its own. Nevertheless, the ORN transient as an interval indeed offers unique features to expedite the classification. In particular, the novelty gained due to observing consecutive samples during the transient is on average greater than the informational gain obtained during the steady-state. Hence, newly observed samples of the receptor transients are likely to contribute to the cumulative classificatory information base formed so far, whereas the informational entropy vanishes as the signal reaches the steady-state. As a device that extracts and integrates odor-specific information in ORN signals, the AL provides an enriched transient to the subsequent MB so that it can achieve accurate classification early in the odor period. We also note that there have been efforts, e.g., [9, 10] to illustrate the sharpening effect of inhibition in the olfactory system. However, to the best of our knowledge, the approach we present here is the first to formulate the temporal gain due to AL processing. 2.2 The Model The model AL is comprised of a population of PNs that project from the AL to downstream processing. The neural activity corresponding to the rate of action potential generation of the biological neurons is given by xi(t), i = 1, 2, ..., NE, for the NE neurons in the PN population. There are also NI interneurons or LNs whose activity is yi(t); i = 1, 2, ..., NI. The rate of change in these activities is stimulated by a weighted sum over both populations and a set of input signals SE i (t) and SI i (t) indicating the activity in the glomeruli stimulating the PNs and the LNs, respectively. In addition, each population receives noise from the AL environment. Our formulation of these ideas is through a Wilson-Cowan-like population model [11] βE i dxi(t) dt = KE i · Θ − NI X j=1 wEI ij yj(t) + gE inpSE i (t) −xi(t) + µE i (t), i ∈1, . . . , NE, βI i dyi(t) dt = KI i · Θ NE X j=1 wIE ij xj(t) + gI inpSI i (t) −yi(t) + µI i (t), i ∈1, . . . , NI. The superscripts E and I stand for excitatory and inhibitory populations. The matrix elements wXY ij , X, Y ∈{E, I} are time-independent weights quantifying the effect from units of type Y to units of type X. SX i (t) is the external input to i-th unit from a glomerulus (odor sensor) weighted by coupling strength gX inp. µY i is an additive noise process and Θ(·), the unit-ramp activation function: Θ(u) = 0 for u < 0, and Θ(u) = u, otherwise. The gains KE i , KI i and time constants βE i , βI i are fixed for an individual unit but vary across PN and LN populations. The network topology is formed through a random process of Bernoulli type: wXY ij = gY · 1 , with probability pXY 0 , with probability 1 −pXY where gY is a fixed coupling strength. pXY is a design parameter to be chosen by us. Each unit, regardless of its type, accepts external input from exactly one sensor in the form of raw resistance time series. This sensor is assigned randomly among all 16 available sensors, ensuring that all sensors are covered1. 1It is assumed that NE + NI > 16. 3 0 20 40 60 80 100 0 5 10 15x 10 4 Time (s) Sensor response (Ω) TGS 2602 TGS 2600 TGS 2010 TGS 2620 0 0.5 1 1.5 2 0 2 4 6 8 10 12x 10 5 Time (s) xi(t) i=1,...,75 (a) (b) Figure 2: (a) A record from Dataset 1, where 100ppm acetaldehyde was applied to the sensor array for 0 ≤t ≤100s. Offsets are removed from the time-series. Curve labels indicate the sensor types. (b) Activity of NE = 75 excitatory PN units of the sample AL model in response to the (time-scaled version of) record shown on panel (a). The conductances are selected as (gE, gI) = (10−6, 9·10−6) and other parameters as given in text. Bar indicates the odor period. For the mixture identification problem of this study, we consider a network with NE = NI = 75 and gE inp = gI inp = 10−2. The probabilities used in the generative Bernoulli process are fixed at pIE = pEI = 0.5. The synaptic conductances gE and gI are optimized for the particular classification instance through the brute force search described below. The gains KE i , KI j and the time-scales βE i , βI j , i = 1, . . . , NE, j = 1, . . . , NI are drawn independently from exponential distributions with λK = 7.5 and λβ = 0.5, respectively. Following construction, the initial condition of each unit is taken as zero and µ is taken as a white noise process with variance 10−4 independently for each unit. We perform the simulation of the 150-dimensional Wilson-Cowan dynamics by 5/6 Runge-Kutta integration with variable step size where the error tolerance is set to 10−15. Although the considered network structure can accommodate limit cycles and strange attractors, the selected range of parameters yield a fixed point behavior. We confirm this in all simulations with the selected parameter values, both during and after the sensory input (odor) period (see Fig. 2(b)). 2.3 Validation We consider the activity in PN population as the only piece of information regarding the input odor that is passed on to higher-order layers of the olfactory system. Access to this activity by those layers can be modeled as instantaneous sampling of a selected brief window of temporal behavior of PNs [7]. Therefore, the recognition system in our model utilizes such snapshots from the spiking activity in the excitatory population xi(t). A snapshot is passed as the feature vector to the classifier; it is comprised of an NE-dimensional fixed vector taken as a sample from the states x1, . . . , xNE at a particular time ts. 2.3.1 Dataset The model is driven by responses recorded from 16 metal-oxide gas sensors in parallel. We have made 80 recordings and grouped them into two sets based on vapor concentration: records for 100ppm vapor in Dataset 1 and 50ppm in Dataset 2. Each dataset contains 40 records from three classes: 10 pure acetaldehyde, 10 pure toluene, and 20 mixture records. The mixture class contains records from imbalanced acetaldehyde-toluene mixtures with 96%-4%, 98%-2%, 2%-98%, and 4%-96% partial concentrations, five from each. Hence, we have two instances of the mixture identification problem in the form of three-class classification. See the supplementary material for details on measurement process. We removed the offset from each sensor record and scaled the odor period to 1s. This was done by mapping the odor period, which has fixed length of 100s in the original records, to 1s by reindexing the time series. These one-second long raw time series, included in the supplementary material, constitute the pool of raw inputs to be applied to the AL network during the time interval 0.5 ≤t ≤1.5s. The input is set to zero outside of this odor period. See Fig. 2 for a sample record and the AL network’s response to it. Note that, although we apply the network to pre-recorded data in simulations, the general scheme is causal, thus can be applied in real-time. 4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 20 40 60 80 100 Snapshot time ts (s) Success rate (%) without AL network with optimized AL network 0 0.5 1 x 10 −5 0 0.5 1 x 10 −5 94 96 98 100 Success Rate (%) at ts=1.5s 95 96 97 98 99 No connectivity Max 99.7% at (1e−6,9e−6) gI gE (a) (b) 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 20 40 60 80 100 Success rate (%) Snapshot time ts (s) without AL network using SVM with optimized AL network using SVM with optimized AL network using MB 0 0.5 1 x 10 −5 0 0.5 1 x 10 −5 80 90 100 Success Rate (%) at ts=1.5s 85 90 95 gI gE Max 95.5% at (6e−6,9e−6) No connectivity (c) (d) Figure 3: (a) Estimated correct classification profile versus snapshot time ts during the normalized odor period for Dataset 1. The red curve is the classification profile due to the proposed AL network, which has the fixed sample topology with (gE, gI) = (10−6, 9 · 10−6). The black baseline profile is obtained by discarding AL and directly classifying snapshots from raw sensor responses by SVM. (b) Correct classification rates extracted by a sweep through gE, gI using Dataset 1. Panels (c) & (d) show the results for Dataset 2, where the best pair is determined as (gE, gI) = (6 · 10−6, 9 · 10−6). 2.3.2 Adjustment of AL Network and Performance Evaluation To reveal the signal conditioning performance of the stand-alone AL model, we first interface it with an established classifier. We use a Support Vector Machine (SVM) classifier with linear kernel to map the snapshots from PN activity to odor identity. This choice is due to the parameter-free design that rules out the possibility of over-fitting. The classifier is realized by the publicly available software LibSVM [2]. Due to the wide diversity of PNs and LNs in terms of their time scales β and gains K, the performance of the network is highly sensitive to the agreement between the outcome of the generative process and the choice of parameters gE and gI. Therefore, it is not possible to give a one-sizefits-all value for these. Instead, we have generated one sample network topology via the Bernoulli process described above and customized gE and gI for it on each problem. For reproducibility, this topology is provided in the supplementary material. Comparable results can be obtained with other topologies but possibly with different gE, gI values than the ones reported below. The validation is carried out in the following way: First we set the classification problem (i.e., select Dataset 1 or 2) and fix gE = gI = 0 (suppress the connectivity). We present each record in the dataset to the network and then log the network response from excitatory population in the form of NE simultaneous time series (see Fig. 2). Then, at each percentile of the odor period ts ∈{0.5 + k/100}100 k=0, we take a snapshot from each NE-dimensional time series and label it by the odor identity (pure acetaldehyde, pure toluene, or mixture). We use randomly selected 80% of the resulting data in training the SVM classifier and keep the remaining 20% for testing it. We record the rate of correct classification on the test data. The train-test stage is repeated 1000 times with different random splits of labelled data. The average correct classification rate is assigned as the performance of the AL model at that ts. The classification profile versus time is extracted when the ts sweep through the odor period is complete. To maximize the performance over conductances gE and gI, we further perform a sweep through a range of these parameters by repeating the above procedure for each combination of gE, gI. Fig5 ure 3 (a) shows the classification profile for the best pair encountered along the parameter sweep gE, gI ∈{k/100}100 k=0. This pair is determined as the one maximizing classification success rate when samples from the end of odor period is used ts = 1.5. Note that these optimum values are problem-specific. For the two instances considered in this work, we mark them by the peaks of the surfaces in Fig. 3 (b) and (d). Dataset 1 induces an easier instance of the identification problem toward the end of odor period, which can be resolved reasonably well using raw sensor data at the steady state. Therefore, the gain over baseline due to AL processing is not so significant in later portions of the odor period for Dataset 1. Also observe from panels (b) and (d) that, when dealing with Dataset 1, the conductance values are less decisive than they are for Dataset 2. Again, this is because the former is an easier problem when the sensors reach the steady-state at ts = 1.5s, where almost all conductance within the swept range ensures > 95% performance. The relative difficulty of the problem in Dataset 2 manifests itself as the fluctuations in the baseline performance. We see in Fig. 3(c) that there are actually periods early in the period where the raw sensor data can be fairly indicative of the class information; however, it is not possible to predict these intervals in advance. It should also be noted that some of these peaks in baseline performance, at least the very first one near ts = 0.55s, are artifacts (due to classification of pure noise) since we know that there is hardly any vapor in the measurement chamber during that period (see Fig. 2(a) and other records in supplementary material). In any case, in both problems, the suggested AL dynamics (with adjusted parameters) contributes substantially to the classification performance during the transient of the sensory signal. This makes early decisions of the classifier substantially and consistently more accurate with respect to the baseline classification. Having established the contribution of the AL network to classification, our goal in the remainder of the paper is to replace the unbiased SVM classifier by a biologically plausible MB model, while preserving the performance gain seen in Fig. 3. 3 Mushroom Body Classifier The MBs of insects employ a large number of identical small intrinsic cells, the so-called Kenyon cells, and fewer output neurons in the MB lobes. It has been observed that, unlike in the AL, the activity in the KCs is very sparse, both across the population and for individual cells over time. Theoretical work suggests that a large number of cells with sparse activity enables efficient classification with random connectivity [4]. The power of this architecture lies in its versatility: The connectivity is not optimized for any specific task and can, therefore, accommodate a variety of input types. 3.1 The Model The insect MB consists of four crucial elements (see Fig. 4): i) a nonlinear expansion from the AL representation at the final stage, x, that resembles the connectivity from the Antennal Lobe to the MBs, ii) a gain control in the MB to achieve a uniform level of sparse activity the KCs, y, iii) a classification phase, where the connections from the KCs to the output neurons, z, are modified according to a Hebbian learning rule, and iv) a learning signal that determines when and which output neuron’s synapses are reinforced. It has been shown in locusts that the activity patterns in the AL are practically discretized by a periodic feedforward inhibition onto the MB calyces and that the activity levels in KCs are very low [7]. Based on the observed discrete and sparse activity pattern in insect MB, we choose to represent KC units as simple algebraic McCulloch-Pitts ‘neurons.’ The neural activity values taken by this neural model are binary (0 = no spike and 1 = spike): µj = Φ PNE i=1 cjixi −θKC j = 1, 2, ..., NKC. The vector x is the representation of the odor that is received as a snapshot from the excitatory PN units of AL model. The components of the vector x = (x1, x2, ..., xNE) are the direct values obtained by integration of the ODE of the AL model described above. The KC layer vector µ is NKC dimensional. cij ∈{0, 1} are the components of the connectivity matrix which is NE×NKC in size. The firing threshold θKC is integer number and Φ(·) is the Heaviside function. The connectivity matrix [cji] is determined randomly by an independent Bernoulli process. Since the degree of connectivity from the input neurons to the KC neurons did not appear to be critical for the performance of the system, we made it uniform by setting the connection probability as pc = 0.1. It, nevertheless, seems advisable to ensure in the construction that the input-to-KC layer mapping is bijective to avoid loss of information. We performed this check during network construction. All other parameters of the KC layer are then assigned admissible values uniformly randomly and fixed. 6 ji lj [ w ] Antennal Lobe PNs x Calyx µ [ c ] Output Nodes z Acetaldehyde Toluene + Mixture Acetaldehyde Pure Pure Toluene Figure 4: Suggested MB model for classifying the AL output. The first layer of connections from AL to calyx are set randomly and fixed. The plasticity of the output layer is due to a binary learning signal that rewards the weights of output units responding to the correct stimulus. Although the basic system described so far implements the divergent (and static) input layer observed in insect calyx, it is very unstable against fluctuations in the total number of active input neurons due to the divergence of connectivity. This is an obstacle for inducing sparse activity at KC level. One mechanism suggested to remove this instability is gain control by feedforward inhibition. For our purposes, we impose a number nKC of simultaneously active KCs, and admit the firing of only the top nKC = NKC/5 neurons that receive the most excitation in the first layer. The fan-in stage of projections from the KCs to the extrinsic MB cells in the MB lobes is the hypothesized locus of learning. In our model, the output units in the MB lobes are again McCullochPitts neurons: zl = Φ PNKC j=1 wlj · µj −θLB , l = 1, 2, ..., NLB. Here, the index LB denotes the MB lobes. The output vector z of the MB lobes has dimension NLB (equals 3 in our problem) and θLB is the threshold for the decision neurons in the MB lobes. The NLB × NKC connectivity matrix wlj has integer entries. Similar to the above-mentioned gain control, we allow only the decision neuron that receives the highest synaptic input to fire. These synaptic strengths wlj are subject to changes during learning according to a Hebbian type plasticity rule described next. 3.2 Training The hypothesis of locating reinforcement learning in mushroom bodies goes back to Montague and collaborators [6]. Every odor class is associated with an output neuron of the MB, so there are three output nodes firing for either pure toluene, pure acetaldehyde, or mixture type of input. The plasticity rule is applied on the connectivity matrix W, whose entries are randomly and independently initialized within [0, 10]. The exact initial distribution of weights have no significant impact on the resulting performance nor on the learning speed. During learning, the inputs are presented to the system in an arbitrary order. The entries of the connectivity matrix at the time of the nth input are denoted by wlj(n). When the next training input with label ℓis applied, then the weight wℓj is updated by the rule wℓj(n + 1) = H (zℓ, µj, wℓj(n)), where H(z, µ, w) = w + 1 when z = 1, µ = 1; and 0, otherwise. This learning rule strenghtens a synaptic connection with probability p+ if presynaptic activity is accompanied by postsynaptic activity. To facilitate learning during the training phase, the ‘correct’ output neuron ℓis forced to fire for an input with label ℓ, while the rest are kept silent. This is provided by pulling down the threshold θLB ℓ, unless neuron ℓis already firing for such input. Learning is terminated when the performance (correct classification rate) converges. 3.3 Validation Using Dataset 2, we applied the proposed MB model with NKC = 10, 000 KCs at the output of the sample AL topology having the same parameters reported in Section 2. For p+ = p−= 1, we trained the output layer of MB using the labelled AL outputs sampled at 10 points in the odor period. The mean correct classification rate over 20 splits of the labelled snapshots (five-fold crossvalidation) are shown in Fig.3(c) as blue dots. With respect to the red curve on the same panel, which was obtained by the (maximum-margin) SVM classifier, a slight reduction in the generalization capability is visible. Nevertheless, the MB classifier in its current form still exploits the superior job of AL over baseline classification during transient, while mimicking two essential features of the biological MB, namely sparsity in KC-layer and incremental local learning in MB lobes. The implementation details and parameters of the MB model are provided in the supplementary material. 7 4 Conclusions We have presented a complete odor identification scheme based on the key principles of insect olfaction, and demonstrated its validity in discriminating mixtures of odors from pure odors using actual records from metal-oxide gas sensors. The bulk of the observed performance is due to the AL, which is a dynamical feature extractor for slow and noisy chemo-sensory time series. The cooperation of integration (accumulation) mechanism and sharpening filter enabled by inhibition leave an almost linearly separable problem for the subsequent classifier. The proposed signal conditioning scheme can be considered as a mathematical image of reservior computing [5]. For this simplified classification task, we have also suggested a bio-inspired MB classifier with local Hebbian plasticity. By exploiting the dynamical nature of the AL stage and the sparsity in MB representation, the overall model provides an explanation for the high speed and accuracy of odor identification in insect olfactory processing. For future study, we envision an improvement on the MB classification performance, which has been explored here to be slightly worse than linear SVM. We think that this can be done without compromising biological plausibility, by imposing mild constraints on the KC-level generative process. The mixture identification problem investigated here is in general more difficult than the traditional problem of discriminating pure odors, since the mixture class can be made arbitrarily close to the pure odor classes. The classification performance attained here is promising for other mixturerelated problems that are among the hardest in the field of artificial olfaction. Acknowledgments This work was supported by the MURI grant ONR N00014-07-1-0741. References [1] V. 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Bee foraging in uncertain environments using predictive Hebbian learning. Nature, 337:725–728, 1995. [7] J. Perez-Orive, O. Mazor, G. C. Turner, S. Cassenaer, R. I. Wilson, and G. Laurent. Oscillations and sparsening of odor representations in the mushroom body. Science, 297:359–365, 2002. [8] M. I. Rabinovich, R. Huerta, and G. Laurent. Transient dynamics for neural processing. Science, 321:48– 50, 2008. [9] B. Raman and R. Gutierrez-Osuna. Chemosensory processing in a spiking model of the olfactory bulb: Chemotopic convergence and center surround inhibition. In L. K. Saul, Y. Weiss, and L. Bottou, editors, NIPS 17, pages 1105–1112. MIT Press, Cambridge, MA, 2005. [10] M. Schmuker and G. Schneider. Processing and classification of chemical data inspired by insect olfaction. Proc. Nat. Acad. Sci., 104:20285–20289, 2007. [11] H. R. Wilson and J. D. Cowan. A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik, 13:55–80, 1973. 8
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On the Efficient Minimization of Classification Calibrated Surrogates Richard Nock CEREGMIA — Univ. Antilles-Guyane 97275 Schoelcher Cedex, Martinique, France rnock@martinique.univ-ag.fr Frank Nielsen LIX - Ecole Polytechnique 91128 Palaiseau Cedex, France nielsen@lix.polytechnique.fr Abstract Bartlett et al (2006) recently proved that a ground condition for convex surrogates, classification calibration, ties up the minimization of the surrogates and classification risks, and left as an important problem the algorithmic questions about the minimization of these surrogates. In this paper, we propose an algorithm which provably minimizes any classification calibrated surrogate strictly convex and differentiable — a set whose losses span the exponential, logistic and squared losses —, with boosting-type guaranteed convergence rates under a weak learning assumption. A particular subclass of these surrogates, that we call balanced convex surrogates, has a key rationale that ties it to maximum likelihood estimation, zerosum games and the set of losses that satisfy some of the most common requirements for losses in supervised learning. We report experiments on more than 50 readily available domains of 11 flavors of the algorithm, that shed light on new surrogates, and the potential of data dependent strategies to tune surrogates. 1 Introduction A very active supervised learning trend has been flourishing over the last decade: it studies functions known as surrogates — upperbounds of the empirical risk, generally with particular convexity properties —, whose minimization remarkably impacts on empirical / true risks minimization [3, 4, 10]. Surrogates play fundamental roles in some of the most successful supervised learning algorithms, including AdaBoost, additive logistic regression, decision tree induction, Support Vector Machines [13, 7, 10]. As their popularity has been rapidly spreading, authors have begun to stress the need to set in order the huge set of surrogates, and better understand their properties. Statistical consistency properties have been shown for a wide set containing most of the surrogates relevant to learning, classification calibrated surrogates (CCS) [3]; other important properties, like the algorithmic questions about minimization, have been explicitly left as important problems to settle [3]. In this paper, we address and solve this problem for all strictly convex differentiable CCS, a set referred to as strictly convex surrogates (SCS). We propose a minimization algorithm, ULS, which outputs linear separators, with two key properties: it provably achieves the optimum of the surrogate, and meets Boosting-type convergence under a weak learning assumption. There is more, as we show that SCS strictly contains another set of surrogates of important rationale, balanced convex surrogates (BCS). This set, which contains the logistic and squared losses but not the exponential loss, coincides with the set of losses satisfying three common requirements about losses in learning. In fact, BCS spans a large subset of the expected losses for zero-sum games of [9], by which ULS may also be viewed as an efficient learner for decision making (in simple environments, though). Section 2 gives preliminary definitions; section 3 presents surrogates losses and risks; sections 4 and 5 present ULS and its properties; section 6 discusses experiments with ULS; section 7 concludes. 2 Preliminary definitions Unless otherwise stated, bold-faced variables like w denote vectors (components are wi, i = 1, 2, ...), calligraphic upper-cases like S denote sets, and blackboard faces like O denote subsets of R, the set of real numbers. We let set O denote a domain (Rn, [0, 1]n, etc., where n is the number of description variables), whose elements are observations. An example is an ordered pair (o, c) ∈O × {c−, c+}, where {c−, c+} denotes the set of classes (or labels), and c+ (resp. c−) is the positive class (resp. negative class). Classes are abstracted by a bijective mapping to one of two other sets: c ∈{c−, c+} ⇋y∗∈{−1, +1} ⇋y ∈{0, 1} . (1) The convention is c+ ⇌+1 ⇌1 and c−⇌−1 ⇌0. We thus have three distinct notations for an example: (o, c), (o, y∗), (o, y), that shall be used without ambiguity. We suppose given a set of m examples, S = {(oi, ci), i = 1, 2, ..., m}. We wish to build a classifier H, which can either be a function H : O →O ⊆R (hereafter, O is assumed to be symmetric with respect to 0), or a function H : O →[0, 1]. Following a convention of [6], we compute to which extent the outputs of H and the labels in S disagree, ε(S, H), by summing a loss which quantifies pointwise disagreements: ε(S, H) .= X i ℓ(ci, H(oi)) . (2) The fundamental loss is the 0/1 loss, ℓ0/1(c, H) (to ease readability, the second argument is written H instead of H(o)). It takes on two forms depending on im(H): ℓ0/1 R (y∗, H) .= 1y∗̸=σ◦H if im(H) = O , or ℓ0/1 [0,1](y, H) .= 1y̸=τ◦H if im(H) = [0, 1] . (3) The following notations are introduced in (3): for a clear distinction of the output of H, we put in index to ℓand ε an indication of the loss’ domain of parameters: R, meaning it is actually some O ⊆R, or [0, 1]. The exponent to ℓgives the indication of the loss name. Finally, 1π is the indicator variable that takes value 1 iff predicate π is true, and 0 otherwise; σ : R →{−1, +1} is +1 iff x ≥0 and −1 otherwise; τ : [0, 1] →{0, 1} is 1 iff x ≥1/2, and 0 otherwise. Both losses ℓR and ℓ[0,1] are defined simultaneously via popular transforms on H, such as the logit transform logit(p) .= log(p/(1 −p)), ∀p ∈[0, 1] [7]. We have indeed ℓ0/1 [0,1](y, H) = ℓ0/1 R (y∗, logit(H)) and ℓ0/1 R (y∗, H) = ℓ0/1 [0,1](y, logit−1(H)). We have implicitly closed the domain of the logit, adding two symbols ±∞to ensure that the eventual infinite values for H can be mapped back to [0, 1]. In supervised learning, the objective is to carry out the minimization of the expectation of the 0/1 loss in generalization, the so-called true risk. Very often however, this task can be relaxed to the minimization of the empirical risk of H, which is (2) with the 0/1 loss [6]: ε0/1(S, H) .= X i ℓ0/1(ci, H(oi)) . (4) The main classifiers we investigate are linear separators (LS). In this case, H(o) .= P t αtht(o) for features ht with im(ht) ⊆R and leveraging coefficients αt ∈R. 3 Losses and surrogates A serious alternative to directly minimizing (4) is to rather focus on the minimization of a surrogate risk [3]. This is a function ε(S, H) as in (2) whose surrogate loss ℓ(c, H(o)) satisfies ℓ0/1(c, H(o)) ≤ℓ(c, H(o)). Four are particularly important in supervised learning, defined via the following surrogate losses: ℓexp R (y∗, H) .= exp(−y∗H) , (5) ℓlog R (y∗, H) .= log(1 + exp(−y∗H)) , (6) ℓsqr R (y∗, H) .= (1 −y∗H)2 , (7) ℓhinge R (y∗, H) .= max{0, 1 −y∗H} . (8) (5) is the exponential loss, (6) is the logistic loss, (7) is the squared loss and (8) is hinge loss. Definition 1 A Strictly Convex Loss (SCL) is a strictly convex function ψ : X →R+ differentiable on int(X) with X symmetric interval with respect to zero, s. t. ∇ψ(0) < 0. φ(x) aφ im(∇φ) Fφ(y∗H) ˆPr[c = c+|H; o] ⊇im(H) = (φ ⋆(−y∗H) −aφ)/bφ = ∇−1 φ (H) φµ,µ∈(0,1)(x) .= µ + (1 −µ) p x(1 −x) µ R −y∗H+√ (1−µ)2+(y∗H)2 1−µ 1 2 + H 2√ (1−µ)2+H2 φM(x) .= p x(1 −x) 0 R −y∗H + p 1 + (y∗H)2 1 2 + H 2√ 1+H2 φQ(x) .= −x log x −(1 −x) log(1 −x) 0 R log(1 + exp(−y∗H)) exp(H) 1+exp(H) φB(x) .= x(1 −x) 0 [−1, 1] (1 −y∗H)2 1 2 + H 2 Table 1: permissible functions, their corresponding BCLs and the matching [0, 1] predictions. ∇. is the gradient notation (here, the derivative). Any surrogate risk built from a SCL is called a Strictly Convex Surrogate (SCS). From Theorem 4 in [3], it comes that SCL contains all classification calibrated losses (CCL) that are strictly convex and differentiable, such as (5), (6), (7). Fix ψ ∈SCL. The Legendre conjugate ψ⋆of ψ is ψ⋆(x) .= supx′∈int(X){xx′ −ψ(x′)}. Because of the strict convexity of ψ, the analytic expression of the Legendre conjugate becomes ψ⋆(x) .= x∇−1 ψ (x) −ψ(∇−1 ψ (x)). ψ⋆is also strictly convex and differentiable. A function φ : [0, 1] →R+ is called permissible iff it is differentiable on (0, 1), strictly concave, symmetric about x = 1/2, and with φ(0) = φ(1) = aφ ≥0. We let bφ .= φ(1/2) −aφ > 0. Permissible functions with aφ = 0 span a very large subset of generalized entropies [9]. Permissible functions are useful to define the following subclass of SCL, of particular interest (here, φ .= −φ). 0 2 4 6 8 10 12 -3 -2 -1 0 1 2 3 (φ = φB) (φ = φM) (φ = φµ = 1/3) (φ = φQ) Figure 1: Bold curves depict plots of φ ⋆(−x) for the φ in Table 1; thin dotted half-lines are its asymptotes. Definition 2 Let φ permissible. The Balanced Convex Loss (BCL) with signature φ, Fφ, is: Fφ(x) .= (φ ⋆(−x) −aφ)/bφ . (9) Balanced Convex Surrogates (BCS) are defined accordingly. All BCL share a common shape. Indeed, φ ⋆(x) satisfies the following relationships: φ ⋆(x) = φ ⋆(−x) + x , (10) lim x→infim(∇φ) φ ⋆(x) = aφ . (11) Noting that Fφ(0) = 1 and ∇Fφ(0) = −(1/bφ)∇−1 φ (0) < 0, it follows that BCS ⊂ SCS, where the strict inequality comes from the fact that (5) is a SCL but not a BCL. It also follows limx→supim(∇φ) Fφ(x) = 0 from (11), and limx→infim(∇φ) Fφ(x) = −x/bφ from (10). We get that the asymptotes of any BCL can be summarized as ℓ(x) .= x(σ(x) −1)/(2bφ). When bφ = 1, this is the linear hinge loss [8], a generalization of (8) for which x .= y∗H −1. Thus, while hinge loss is not a BCL, it is the limit behavior of any BCL (see Figure 1). Table 1 (left column) gives some examples of permissible φ. When scaled so that φ(1/2) = 1, some confound with popular choices: φB with Gini index, φQ with the Bit-entropy, and φM with Matsushita’s error [10, 11]. Table 1 also gives the expressions of Fφ along with the im(H) = O ⊆R allowed by the BCL, for the corresponding permissible function. It is interesting to note the constraint on im(H) for the squared loss to be a BCL, which makes it monotonous in the interval, but implies to rescale the outputs of classifiers like linear separators to remain in [−1, 1]. 4 ULS: the efficient minimization of any SCS For any strictly convex function ψ : X →R differentiable on int(X), the Bregman Loss Function (BLF) Dψ with generator ψ is [5]: Dψ(x||x′) .= ψ(x) −ψ(x′) −(x −x′)∇ψ(x′) . (12) The following Lemma states some relationships that are easy to check using ψ⋆⋆= ψ. They are particularly interesting when im(H) = O ⊆R. Algorithm 1: Algorithm ULS(M, ψ) Input: M ∈Rm×T , SCL ψ with dom(ψ) = R; Let α1 ←0; Let w0 ←∇−1 ˜ ψ (0)1; for j = 1, 2, ...J do [WU] (weight update) wj ←(Mαj) ⋄w0 ; Let Tj ⊆{1, 2, ..., T}; let δj ←0; [LC] (leveraging coefficients) ∀t ∈Tj, pick δj,t such that: Pm i=1 mit((Mδj) ⋄wj)i = 0 ; Let αj+1 ←αj + δj; Output: H(x) .= PT t=1 αJ+1,tht(x) ∈LS Lemma 1 For any SCL ψ, ψ(y∗H) = Dψ⋆(0||∇−1 ψ⋆(y∗H))−ψ⋆(0). Furthermore, for any BCL Fφ, Dφ(y||∇−1 φ (H)) = bφFφ(y∗H) and Dφ(y||∇−1 φ (H)) = Dφ(1||∇−1 φ (y∗H)). The second equality is important because it ties real predictions (right) with [0, 1] predictions (left). It also separates SCL and BCL, as for any ψ in SCL, it can be shown that there exists a functions ϕ such that Dϕ(y||∇−1 ϕ (H)) = ψ(y∗H) iff ψ ∈BCL. We now focus on the minimization of any SCS. We show that there exists an algorithm, ULS, which fits a linear separator H to the minimization of any SCS εψ R .= P i ψ(y∗ i H(oi)) for any SCL ψ with dom(ψ) = R, in order not to restrict the LS built. To simplify notations, we let: ˜ψ(x) .= ψ⋆(−x) . (13) With this notation, the first equality in Lemma 1 becomes: ψ(y∗H) = D ˜ ψ(0||∇−1 ˜ ψ (−y∗H)) −˜ψ(0) . (14) We let W .= dom(∇˜ ψ) = −im(∇ψ), where this latter equality comes from ∇˜ ψ(x) = −∇ψ⋆(−x) = −∇−1 ψ (−x). It also comes im(∇˜ ψ) = R. Because any BLF is strictly convex in its first argument, we can compute its Legendre conjugate. In fact, we shall essentially need the argument that realizes the supremum: for any x ∈R, for any p ∈W, we let: x ⋄p .= argp′∈W sup{xp′ −D ˜ ψ(p′||p)} . (15) We do not make reference to ˜ψ in the ⋄notation, as it shall be clear from context. We name x ⋄p the Legendre dual of the ordered pair (x, p), closely following a notation by [6]. The Legendre dual is unique and satisfies: ∇˜ ψ(x ⋄p)=x + ∇˜ ψ(p) , (16) ∀x, x′ ∈R, ∀p ∈W, x ⋄(x′ ⋄p)=(x + x′) ⋄p . (17) To state ULS, we follow the setting of [6] and suppose that we have T features ht (t = 1, 2, ..., T) known in advance, the problem thus reducing to the computation of the leveraging coefficients. We define m × T matrix M with: mit .= −y∗ i ht(oi) . (18) Given leveraging coefficients vector α ∈RT , we get: −y∗ i H(oi) = (Mα)i . (19) We can specialize this setting to classical greedy induction frameworks for LS: in classical boosting, at step j, we would fit a single αt [6]; in totally corrective boosting, we would rather fit {αt, 1 ≤t ≤ j} [14]. Intermediate schemes may be used as well for Tj, provided they ensure that, at each step j of the algorithm and for any feature ht, it may be chosen at some j′ > j. ULS is displayed in Algorithm 1. In Algorithm 1, notations are vector-based: the Legendre duals are computed component-wise; furthermore, Tj may be chosen according to whichever scheme underlined above. The following Theorem provides a first general convergence property for ULS. Theorem 1 ULS(M, ψ) converges to a classifier H realizing the minimum of εψ R. Proof sketch: In step [WU] in ULS, (17) brings wj+1 = (Mαj+1) ⋄w0 = (Mδj) ⋄wj. After few derivations involving the choice of δj and step [LC] in ULS, we obtain (with vector notations, BLFs are the sum of the component-wise BLFs): D ˜ ψ(0||wj+1) −D ˜ ψ(0||wj) = −D ˜ ψ(wj+1||wj) (20) Let A ˜ ψ(wj+1, wj) .= −D ˜ ψ(wj+1||wj), which is just, from (20) and (14), the difference between two successive SCL in Algorithm 1. Thus, A ˜ ψ(wj+1, wj) < 0 whenever wj+1 ̸= wj. Should we be able to prove that when ULS has converged, w. ∈KerM ⊤, this would make A ˜ ψ(wj+1, wj) an auxiliary function for ULS, which is enough to prove the convergence of ULS towards the optimum [6]. Thus, suppose that wj+1 = wj (ULS has converged). Suppose that Tj is a singleton (e.g. classical boosting scheme). In this case, δj = 0 and so ∀t = 1, 2, ..., T, Pm i=1 mit(0 ⋄wj)i = Pm i=1 mitwj,i = 0, i.e. w⊤ j M = w⊤ j+1M = 0⊤, and wj, wj+1 ∈KerM ⊤. The case of totally corrective boosting is simpler, as after the last iteration we would have wJ+1 ∈KerM ⊤. Intermediate choices for Tj ⊂{1, 2, ..., T} are handled in the same way. We emphasize the fact that Theorem 1 proves the convergence towards the global optimum of εψ R, regardless of ψ. The optimum is defined by the LS with features in M that realizes the smallest εψ R. Notice that in practice, it may be a tedious task to satisfy exactly (20), in particular for totally corrective boosting [14]. ULS has the flavor of boosting algorithms, repeatedly modifying a set of weights w over the examples. In fact, this similarity is more than syntactical, as ULS satisfies two first popular algorithmic boosting properties, the first of which being that step [LC] in ULS is equivalent to saying that this LS has zero edge on wj+1 [14]. The following Lemma shows that this edge conditions is sound. Lemma 2 Suppose that there does not exist some ht with all mit of the same sign, ∀i = 1, 2, ..., m. Then, for any choice of Tj, step [LC] in ULS has always a finite solution. Proof: Let: Z .= D ˜ ψ(0||(Mαj+1) ⋄w0) . (21) We have Z = m ˜ψ(0) + Pm i=1 ˜ψ(−(M(δj + αj))i) from (14), a function convex in all leveraging coefficients. Define |Tj| × |Tj| matrix E with euv .= ∂2Z/(∂δj,uδj,v) (for the sake of simplicity, Tj = {1, 2, ..., |Tj|}, where |.| denotes the cardinal). We have euv = Pm i=1 miumiv/ϕ(((Mδj) ⋄ wj)i), with ϕ(x) .= d2 ˜ψ(x)/dx2 a function strictly positive in int(W) since ˜ψ is strictly convex. Let qi,j .= 1/ϕ(((Mδj)⋄wj)i) > 0. It is easy to show that x⊤Ex = Pm i=1 qi,j⟨x, ˜mi⟩2 ≥0, ∀x ∈ R|Tj|, with ˜mi ∈R|Tj| the vector with ˜mit .= mit. Thus, E is positive semidefinite; as such, step [LC] in ULS, which is the same as solving ∂Z/∂δj,u = 0, ∀u ∈Tj (i.e. minimizing Z) has always a solution. The condition for the Lemma to work is absolutely not restrictive, as if such an ht were to exist, we would not need to run ULS: indeed, we would have either ε0/1(S, ht) = 0, or ε0/1(S, −ht) = 0. The second property met by ULS is illustrated in the second example below. x 0 p 1 1/2 ∇φ x ⋄p Figure 2: A typical ∇φ (red: strictly increasing, symmetric wrt point (1/2, 0)), with Legendre dual x ⋄p computed from x and p. We give two examples of specializations of ULS. Take for example ψ(x) = exp(−x) (5). In this case, W = R+, w0 = 1 and it is not hard to see that ULS matches real AdaBoost with unnormalized weights [13]. The difference is syntactical: the LS output by ULS and real AdaBoost are the same. Now, take any BCL. In this case, ˜ψ = φ, W = [0, 1] (scaling issues underlined for the logit in Section 2 make it desirable to close W), and w0 = 1/21. In all these cases, where W ⊆R+, wj is always a distribution up to a normalization factor, and this would also be the case for any strictly monotonous SCS ψ. The BCL case brings an appealing display of how the weights behave. Figure 2 displays a typical Legendre dual for a BCL. Consider example (oi, yi), and its weight update, wj,i ←(Mαj)i ⋄w0,i = (−y∗ i H(oi)) ⋄w0,i for the current classifier H. Fix p = w0,i and x = −y∗ i H(oi) in Figure 2. We see that the new weight of the example gets larger iff x > 0, i.e. iff the example is given the wrong class by H, which is the second boosting property met by ULS. ULS turns out to meet a third boosting property, and the most important as it contributes to root the algorithm in the seminal boosting theory of the early nineties: we have guarantees on its convergence rate under a generalization of the well-known “Weak Learning Assumption” (WLA) [13]. To state the WLA, we plug the iteration in the index of the distribution normalization coefficient in (21), and define Zj .= ||wj||1 (||.||k is the Lk norm). The WLA is: (WLA)∀j, ∃γj > 0 : |(1/|Tj|) X t∈Tj (1/Zj) m X i=1 mitwj,i| ≥ γj . (22) This is indeed a generalization of the usual WLA for boosting algorithms, that we obtain taking |Tj| = 1, ht ∈{−1, +1} [12]. Few algorithms are known that formally boost WLA in the sense that requiring only WLA implies guaranteed rates for the minimization of εψ R. We show that ULS meets this property ∀ψ ∈SCL. To state this, we need few more definitions. Let mt denote the tth column vector of M, am .= maxt ||mt||2 and aZ .= minj Zj. Let aγ denote the average of γj (∀j), and aϕ .= minx∈int(W) ϕ(x) (ϕ defined in the proof of Lemma 2). Theorem 2 Under the WLA, ULS terminates in at most J = O(ma2 m/(aϕa2 Za2 γ)) iterations. Proof sketch: We use Taylor expansions with Lagrange remainder for ˜ψ, and then the mean-value theorem, and obtain that ∀w, w + ∆∈W, ∃w⋆∈[min{w + ∆, w}, max{w + ∆, w}] such that D ˜ ψ(w + ∆||w) = ∆2ϕ(w⋆)/2 ≥(∆2/2)aϕ ≥0. We use m times this inequality with w = wj,i and ∆= (wj+1,i −wj,i), sum the inequalities, combine with Cauchy - Schwartz and Jensen’s inequalities, and obtain: D ˜ ψ(wj+1||wj) ≥ aϕ(aZγj/(2am))2 . (23) Using (20), we obtain that D ˜ ψ(0||wJ+1) −m ˜ψ(0) equals: −m ˜ψ(0) + D ˜ ψ(0||w1) + J X j=1 (D ˜ ψ(0||wj+1) −D ˜ ψ(0||wj))=mψ(0) − J X j=1 D ˜ ψ(wj+1||wj) . (24) But, (14) together with the definition of wj in [WU] (see ULS) yields D ˜ ψ(0||wJ+1,i) = ˜ψ(0) + ψ(y∗ i H(oi)), ∀i = 1, 2, ..., m, which ties up the SCS to (24); the guaranteed decrease in the rhs of (24) by (23) makes that there remains to check when the rhs becomes negative to conclude that ULS has terminated. This gives the bound of the Theorem. The bound in Theorem 2 is mainly useful to prove that the WLA guarantees a convergence rate of order O(m/a2 γ) for ULS, but not the best possible as it is in some cases far from being optimal. 5 ULS, BCL, maximum likelihood and zero-sum games BCL matches through the second equality in Lemma 1 the set of losses that satisfy the main requirements about losses used in machine learning. This is a strong rationale for its use. Suppose im(H) ⊆[0, 1], and consider the following requirements about some loss ℓ[0,1](y, H): (R1) The loss is lower-bounded. ∃z ∈R such that infy,H ℓ[0,1](y, H) = z. (R2) The loss is a proper scoring rule. Consider a singleton domain O = {o}. Then, the best (constant) prediction is arg minx∈[0,1] ε[0,1](S, x) = p .= ˆPr[c = c+|o] ∈[0, 1], where p is the relative proportion of positive examples with observation o. (R3) The loss is symmetric in the following sense: ℓ[0,1](y, H) = ℓ[0,1](1 −y, 1 −H). R1 is standard. For R2, we can write ε[0,1](S, x) = pℓ[0,1](1, x) + (1 −p)ℓ[0,1](0, x) = L(p, x), which is just the expected loss of zero-sum games used in [9] (eq. (8)) with Nature states reduced to the class labels. The fact that the minimum is achieved at x = p makes the loss a proper scoring rule. R3 implies ℓ[0,1](1, 1) = ℓ[0,1](0, 0), which is virtually assumed for any domain; otherwise, it scales to H ∈[0, 1] a well-known symmetry in the cost matrix that holds for domains without class dependent misclassification costs. For these domains indeed, it is assumed ℓ[0,1](1, 0) = ℓ[0,1](0, 1). Finally, we say that loss ℓ[0,1] is properly defined iff dom(ℓ[0,1]) = [0, 1]2 and it is twice differentiable on (0, 1)2. This is only a technical convenience: even the 0/1 loss coincides on {0, 1} with properly defined losses. In addition, the differentiability condition would be satisfied by many popular losses. The proof of the following Lemma involves Theorem 3 in [1] and additional facts to handle R3. Lemma 3 Assume im(H) ⊆[0, 1]. Loss ℓ[0,1](., .) is properly defined and meets requirements R1, R2, R3 iff ℓ[0,1](y, H) = z + Dφ(y||H) for some permissible φ. Thus, φ maybe viewed as the “signature” of the loss. The second equality in Lemma 1 makes a tight connection between the predictions of H in [0, 1] and R. Let it be more formal: the matching [0, 1] prediction for some H with im(H) = O is: ˆPrφ[c = c+|H; o] .= ∇−1 φ (H(o)) , (25) With this definition, illustrated in Table 1, Lemma 3 and the second equality in Lemma 1 show that BCL matches the set of losses of Lemma 3. This definition also brings the true nature of the minimization of any BCS with real valued hypotheses like linear separators (in ULS). From Lemma 3 and [2], there exists a bijection between BCL and a subclass of the exponential families whose members’ pdfs may be written as: Prφ[y|θ] = exp(−Dφ(y||∇−1 φ (θ)) + φ(y) −ν(y)), where θ ∈R is the natural parameter and ν(.) is used for normalization. Plugging θ = H(o), using (25) and the second equality in Lemma 1, we obtain that any BCS can be rewritten as εφ R = U +P i −log Prφ[yi|H(oi)], where U does not play a role in its minimization. We obtain the following Lemma, in which we suppose im(H) = O. Lemma 4 Minimizing any BCS with classifier H yields the maximum likelihood estimation, for each observation, of the natural parameter θ = H(o) of an exponential family defined by signature φ. In fact, one exponential family is concerned in fine. To see this, we can factor the pdf as Pr[y|θ] .= exp (θλ(y) −ψ(θ)) /z, with ψ = φ ⋆the cumulant function, λ(y) the sufficient statistic and z the normalization function. Since y ∈{0, 1}, we easily end up with Prφ[y|θ] = 1/(1 + exp(−θ)), the logistic prediction for a Bernoulli prior. To summarize, minimizing any loss that meets R1, R2 and R3 (i.e. any BCL) amounts to the same ultimate goal; Since ULS works for any of the corresponding surrogate risks, the crux of the choice of the BCL relies on data-dependent considerations. Finally, we can go further in the parallel with game theory developed above for R2: using notations in [9], the loss function of the decision maker can be written L(X, q) = Dφ(1||q(X)). R3 makes it easy to recover losses like the log loss or the Brier score [9] respectively from φQ and φB (Table 1). In this sense, ULS is also a sound learner for decision making in the zero-sum game of [9]. Notice however that, to work, it requires that Nature has a restricted sample space size ({0, 1}). 6 Experiments We have compared against each other 11 flavors of ULS, including real AdaBoost [13], on a benchmark of 52 domains (49 from the UCI repository). True risks are estimated via stratified 10-fold cross validation; ULS is ran for r (fixed) features ht, each of which is a Boolean rule: If Monomial then Class= ±1 else Class = ∓1, with at most l (fixed) literals, induced following the greedy minimization of the BCS at hand. Leveraging coefficients ([LC] in ULS) are approximated up to 10−10 precision. Figure 3 summarizes the results for two values of the couple (l, r). Histograms are ordered from left to right in increasing average true risk over all domains (shown below histograms). The italic numbers give, for each algorithm, the number of algorithms it beats according to a Student paired t-test over all domains with .1 threshold probability. Out of the 10 flavors of ULS, the first four flavors pick φ in Table 1. The fifth uses another permissible function: φυ(x) .= (x(1 −x))υ , ∀υ ∈(0, 1). The last five adaptively tune the BCS at hand out-of-a-bag of BCS. The first four fit the BCS at each stage of the inner loop (for j ...) of ULS. Two (noted “F.”) pick the BCS which minimizes the empirical risk in the bag; two others (noted “E.”) pick the BCS which maximizes the current edge. There are two different bags corresponding to four permissible functions each: the first (index “1”) contains the φ in Table 1, the second (index “2”) replaces φB by φυ. We wanted to evaluate φB because it forces to renormalize the leveraging coefficients in H each time it is selected, to ensure that the output of H lies in [−1, 1]. The last adaptive flavor, F ∗, “externalizes” the choice of the BCS: it selects for each fold the BCS which yields the smallest empirical risk in a bag corresponding to five φ: those of Table 1 plus φυ. 0 5 10 15 20 25 11 10 9 8 7 6 5 4 3 2 1 F∗ 14.18 (10) 11 10 9 8 7 6 5 4 3 2 1 φM 14.70 (5) 11 10 9 8 7 6 5 4 3 2 1 φυ 14.71 (3) 11 10 9 8 7 6 5 4 3 2 1 φµ 14.83 (2) 11 10 9 8 7 6 5 4 3 2 1 F2 15.03 (1) 11 10 9 8 7 6 5 4 3 2 1 φQ 15.06 (1) 11 10 9 8 7 6 5 4 3 2 1 E1 15.22 (1) 11 10 9 8 7 6 5 4 3 2 1 φB 15.25 (1) 11 10 9 8 7 6 5 4 3 2 1 AdaBoost 15.35 (1) 11 10 9 8 7 6 5 4 3 2 1 E2 15.36 (1) 11 10 9 8 7 6 5 4 3 2 1 F1 17.37 (0) 0 5 10 15 20 25 11 10 9 8 7 6 5 4 3 2 1 F∗ 12.15 (10) 11 10 9 8 7 6 5 4 3 2 1 φQ 12.39 (3) 11 10 9 8 7 6 5 4 3 2 1 AdaBoost 12.56 (3) 11 10 9 8 7 6 5 4 3 2 1 φM 12.59 (3) 11 10 9 8 7 6 5 4 3 2 1 φB 12.62 (3) 11 10 9 8 7 6 5 4 3 2 1 E2 12.63 (3) 11 10 9 8 7 6 5 4 3 2 1 φυ 12.74 (2) 11 10 9 8 7 6 5 4 3 2 1 φµ 12.79 (2) 11 10 9 8 7 6 5 4 3 2 1 F2 13.10 (2) 11 10 9 8 7 6 5 4 3 2 1 F1 17.57 (1) 11 10 9 8 7 6 5 4 3 2 1 E1 23.60 (0) Figure 3: Summary of our results over the 52 domains for the 11 algorithms (top: l = 2, r = 10; bottom: l = 3, r = 100). Vertical (red) bars show the average rank over all domains (see text). Three main conclusions emerge from Figure 3. First, F ∗appears to be superior to all other approaches, but slightly more sophisticated choices for the SCS (i.e. E., F.) fail at improving the results; this is a strong advocacy for a particular treatment of this surrogate tuning problem. Second, Matsushita’s BCL, built from φM, appears to be a serious alternative to the logistic loss. Third and last, a remark previously made by [10] for decision trees seems to hold as well for linear separators, as stronger concave regimes for φ in BCLs tend to improve performances at least for small r. Conclusion In this paper, we have shown the existence of a supervised learning algorithm which minimizes any strictly convex, differentiable classification calibrated surrogate [3], inducing linear separators. Since the surrogate is now in the input of the algorithm, along with the learning sample, it opens the interesting problem of the tuning of this surrogate to the data at hand to further reduce the true risk. While the strategies we have experimentally tested are, with this respect, a simple primer for eventual solutions, they probably display the potential and the non triviality of these solutions. References [1] A. Banerjee, X. Guo, and H. Wang. On the optimality of conditional expectation as a bregman predictor. IEEE Trans. on Information Theory, 51:2664–2669, 2005. [2] A. Banerjee, S. Merugu, I. Dhillon, and J. Ghosh. Clustering with Bregman divergences. Journal of Machine Learning Research, 6:1705–1749, 2005. [3] P. Bartlett, M. Jordan, and J. D. McAuliffe. Convexity, classification, and risk bounds. Journal of the Am. Stat. Assoc., 101:138–156, 2006. [4] P. Bartlett and M. Traskin. Adaboost is consistent. In NIPS*19, 2006. [5] L. M. Bregman. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comp. Math. and Math. Phys., 7:200–217, 1967. [6] M. Collins, R. Schapire, and Y. Singer. Logistic regression, adaboost and Bregman distances. In COLT’00, pages 158–169, 2000. [7] J. Friedman, T. Hastie, and R. Tibshirani. Additive Logistic Regression : a Statistical View of Boosting. Ann. of Stat., 28:337–374, 2000. [8] C. Gentile and M. Warmuth. Linear hinge loss and average margin. In NIPS*11, pages 225–231, 1998. [9] P. Gr¨unwald and P. Dawid. Game theory, maximum entropy, minimum discrepancy and robust Bayesian decision theory. Ann. of Statistics, 32:1367–1433, 2004. [10] M.J. Kearns and Y. Mansour. On the boosting ability of top-down decision tree learning algorithms. Journal of Comp. Syst. Sci., 58:109–128, 1999. [11] K. Matsushita. Decision rule, based on distance, for the classification problem. Ann. of the Inst. for Stat. Math., 8:67–77, 1956. [12] R. Nock and F. Nielsen. A Real Generalization of discrete AdaBoost. Artif. Intell., 171:25–41, 2007. [13] R. E. Schapire and Y. Singer. Improved boosting algorithms using confidence-rated predictions. In COLT’98, pages 80–91, 1998. [14] M. Warmuth, J. Liao, and G. R¨atsch. Totally corrective boosting algorithms that maximize the margin. In ICML’06, pages 1001–1008, 2006.
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Robust Kernel Principal Component Analysis Minh Hoai Nguyen & Fernando De la Torre Carnegie Mellon University, Pittsburgh, PA 15213, USA. Abstract Kernel Principal Component Analysis (KPCA) is a popular generalization of linear PCA that allows non-linear feature extraction. In KPCA, data in the input space is mapped to higher (usually) dimensional feature space where the data can be linearly modeled. The feature space is typically induced implicitly by a kernel function, and linear PCA in the feature space is performed via the kernel trick. However, due to the implicitness of the feature space, some extensions of PCA such as robust PCA cannot be directly generalized to KPCA. This paper presents a technique to overcome this problem, and extends it to a unified framework for treating noise, missing data, and outliers in KPCA. Our method is based on a novel cost function to perform inference in KPCA. Extensive experiments, in both synthetic and real data, show that our algorithm outperforms existing methods. 1 Introduction Principal Component Analysis (PCA) [9] is one of the primary statistical techniques for feature extraction and data modeling. One drawback of PCA is its limited ability to model non-linear structures that exist in many computing applications. Kernel methods [18] enable us to extend PCA to model non-linearities while retaining its computational efficiency. In particular, Kernel PCA (KPCA) [19] has repeatedly outperformed PCA in many image modeling tasks [19, 14]. Unfortunately, realistic visual data is often corrupted by undesirable artifacts due to occlusion (e.g. a hand in front of a face, Fig. 1.d), illumination (e.g. specular refection, Fig. 1.e), noise (e.g. from capturing device, Fig.1.b), or from the underlying data generation method (e.g. missing data due to transmission, Fig. 1.c). Therefore, robustness to noise, missing data, and outliers is a desired property to have for algorithms in computer vision. a b c d e f g h i Figure 1: Several types of data corruption and results of our method. a) original image, b) corruption by additive Gaussian noise, c) missing data, d) hand occlusion, e) specular reflection. f) to i) are the results of our method for recovering uncorrupted data from b) to e) respectively. Input Space Feature Space x z? Principal Subspace Figure 2: Using KPCA principal subspace to find z, a clean version of corrupted sample x. Throughout the years, several extensions of PCA have been proposed to address the problems of outliers and missing data, see [6] for a review. However, it still remains unclear how to generalize those extensions to KPCA; since directly migrating robust PCA techniques to KPCA is not possible 1 due to the implicitness of the feature space. To overcome this problem, in this paper, we propose Robust KPCA (RKPCA), a unified framework for denoising images, recovering missing data, and handling intra-sample outliers. Robust computation in RKPCA does not suffer from the implicitness of the feature space because of a novel cost function for reconstructing “clean” images from corrupted data. The proposed cost function is composed of two terms, requiring the reconstructed image to be close to the KPCA principal subspace as well as to the input sample. We show that robustness can be naturally achieved by using robust functions to measure the closeness between the reconstructed and the input data. 2 Previous work 2.1 KPCA and pre-image KPCA [19, 18, 20] is a non-linear extension of principal component analysis (PCA) using kernel methods. The kernel represents an implicit mapping of the data to a (usually) higher dimensional space where linear PCA is performed. Let X denote the input space and H the feature space. The mapping function ϕ : X →H is implicitly induced by a kernel function k : X × X →ℜthat defines the similarity between data in the input space. One can show that if k(·, ·) is a kernel then the function ϕ(·) and the feature space H exist; furthermore k(x, y) = ⟨ϕ(x), ϕ(y)⟩[18]. However, directly performing linear PCA in the feature space might not be feasible because the feature space typically has very high dimensionality (including infinity). Thus KPCA is often done via the kernel trick. Let D = [d1 d2 ... dn], see notation1, be a training data matrix, such that di ∈X ∀i = 1, n. Let k(·, ·) denote a kernel function, and K denote the kernel matrix (element ij of K is kij = k(di, dj)). KPCA is computed in closed form by finding first m eigenvectors (ai’s) corresponding to the largest eigenvalues (λi’s) of the kernel matrix K (i.e. KA = AΛ). The eigenvectors in the feature space V can be computed as V = ΓA, where Γ = [ϕ(d1)...ϕ(dn)]. To ensure orthonormality of {vi}m i=1, KPCA imposes that λi⟨ai, ai⟩= 1. It can be shown that {vi}m i=1 form an orthonormal basis of size m that best preserves the variance of data in the feature space [19]. Assume x is a data point in the input space, and let Pϕ(x) denote the projection of ϕ(x) onto the principal subspace {vi}m 1 . Because {vi}m 1 is a set of orthonormal vectors, we have Pϕ(x) = Pm i=1 ⟨ϕ(x), vi⟩vi. The reconstruction error (in feature space) is given by: Eproj(x) = ||ϕ(x) −Pϕ(x)||2 2 = ⟨ϕ(x), ϕ(x)⟩− X ⟨ϕ(x), vi⟩2 = k(x, x) −r(x)T Mr(x), where r(x) = ΓT ϕ(x), and M = X aiaT i . (1) The pre-image of the projection is the z ∈X that satisfies ϕ(z) = Pϕ(x); z is also referred to as the KPCA reconstruction of x. However, the pre-image of Pϕ(x) usually does not exist, so finding the KPCA reconstruction of x means finding z such that ϕ(z) is as close to Pϕ(x) as possible. It should be noted that the closeness between ϕ(z) and Pϕ(x) can be defined in many ways, and different cost functions lead to different optimization problems. Sch¨olkopf et al [17] and Mika et al [13] propose to approximate the reconstruction of x by arg minz ||ϕ(z) −Pϕ(x)||2 2. Two other objective functions have been proposed by Kwok & Tsang [10] and Bakir et al [2]. 2.2 KPCA-based algorithms for dealing with noise, outliers and missing data Over the years, several methods extending KPCA algorithms to deal with noise, outliers, or missing data have been proposed. Mika et al [13], Kwok & Tsang [10], and Bakir et al [2] show how denoising can be achieved by using the pre-image. While these papers present promising denoising results for handwritten digits, there are at least two problems with these approaches. Firstly, because the input image x is noisy, the similarity measurement between x and other data point di (i.e. k(x, di) the kernel) might be adversely affected, biasing the KPCA reconstruction of x. Secondly, 1Bold uppercase letters denote matrices (e.g. D), bold lowercase letters denote column vectors (e.g. d). dj represents the jth column of the matrix D. dij denotes the scalar in the row ith and column jth of the matrix D and the ith element of the column vector dj. Non-bold letters represent scalar variables. 1k ∈Rk×1 is a column vector of ones. Ik ∈Rk×k is the identity matrix. 2 b Input Space Feature Space x z? Principal Subspace a Input Space Feature Space x z? Principal Subspace Figure 3: Key difference between previous work (a) and ours (b). In (a), one seeks z such that ϕ(z) is close to Pϕ(x). In (b), we seek z such that ϕ(z) is close to both ϕ(x) and the principal subspace. current KPCA reconstruction methods equally weigh all the features (i.e. pixels); it is impossible to weigh the importance of some features over the others. Other existing methods also have limitations. Some [7, 22, 1] only consider robustness of the principal subspace; they do not address robust fitting. Lu et al [12] present an iterative approach to handle outliers in training data. At each iteration, the KPCA model is built, and the data points that have the highest reconstruction errors are regarded as outliers and discarded from the training set. However, this approach does not handle intra-sample outliers (outliers that occur at a pixel level [6]). Several other approaches also considering Berar et al [3] propose to use KPCA with polynomial kernels to handle missing data. However, it is not clear how to extend this approach to other kernels. Furthermore, with polynomial kernels of high degree, the objective function is hard to optimize. Sanguinetti & Lawrence [16] propose an elegant framework to handle missing data. The framework is based on the probabilistic interpretation inherited from Probabilistic PCA [15, 21, 11]. However, Sanguinetti & Lawrence [16] do not address the problem of outliers. This paper presents a novel cost function that unifies the treatment of noise, missing data and outliers in KPCA. Experiments show that our algorithm outperforms existing approaches [6, 10, 13, 16]. 3 Robust KPCA 3.1 KPCA reconstruction revisited Given an image x ∈X, Fig. 2 describes the task of finding the KPCA-reconstructed image of x (uncorrupted version of x to which we will refer as KPCA reconstruction). Mathematically, the task is to find a point z ∈X such that ϕ(z) is in the principal subspace (denote PS) and ϕ(z) is as close to ϕ(x) as possible. In other words, finding the KPCA reconstruction of x is to optimize: arg min z ||ϕ(z) −ϕ(x)||2 s.t. ϕ(z) ∈PS . (2) However, since there might not exist z ∈X such that ϕ(z) ∈PS, the above optimization problem needs to be relaxed. There is a common relaxation approach used by existing methods for computing the KPCA reconstruction of x. This approach conceptually involves two steps:(i) finding Pϕ(x) which is the closest point to ϕ(x) among all the points in the principal subspace, (ii) finding z such that ϕ(z) is as close to Pϕ(x) as possible. This relaxation is depicted in Fig. 3a. If L2 norm is used to measure the closeness between ϕ(z) and Pϕ(x), the resulting KPCA reconstruction is arg minz ||ϕ(z) −Pϕ(x)||2 2 . This approach for KPCA reconstruction is not robust. For example, if x is corrupted with intrasample outliers (e.g. occlusion), ϕ(x) and Pϕ(x) will also be adversely affected. As a consequence, finding z that minimizes ||ϕ(z) −Pϕ(x)||2 2 does not always produce a “clean” version of x. Furthermore, it is unclear how to incorporate robustness to the above formulation. Here, we propose a novel relaxation of (2) that enables the incorporation of robustness. The KPCA reconstruction of x is taken as: arg min z ||ϕ(x) −ϕ(z)||2 2 + C ||ϕ(z) −Pϕ(z)||2 2 | {z } Eproj(z) . (3) 3 Algorithm 1 RKPCA for missing attribute values in training data Input: training data D, number of iterations m, number of partitions k. Initialize: missing values by the means of known values. for iter = 1 to m do Randomly divide D into k equal partitions D1, ..., Dk for i = 1 to k do Train RKPCA using data D \ Di Run RKPCA fitting for Di with known missing attributes. end for Update missing values of D end for Intuitively, the above cost function requires the KPCA reconstruction of x is a point z that ϕ(z) is close to both ϕ(x) and the principal subspace. C is a user-defined parameter that controls the relative importance of these two terms. This approach is depicted in Fig. 3b. It is possible to generalize the above cost function further. The first term of Eq. 3 is not necessarily ||ϕ(x) −ϕ(z)||2 2. In fact, for the sake of robustness, it is preferable that ||ϕ(x) −ϕ(z)||2 2 is replaced by a robust function E0 : X × X →ℜfor measuring similarity between x and z. Furthermore, there is no reason why E0 should be restricted to the metric of the feature space. In short, the KPCA reconstruction of x can be taken as: arg min z E0(x, z) + CEproj(z) . (4) By choosing appropriate forms for E0, one can use KPCA to handle noise, missing data, and intrasample outliers. We will show that in the following sections. 3.2 Dealing with missing data in testing samples Assume the KPCA has been learned from complete and noise free data. Given a new image x with missing values, a logical function E0 that does not depend on the the missing values could be: E0(x, z) = −exp(−γ2||W(x −z)||2 2), where W is a diagonal matrix; the elements of its diagonal are 0 or 1 depending on whether the corresponding attributes of x have missing values or not. 3.3 Dealing with intra-sample outliers in testing samples To handle intra-sample outliers, we could use a robust function for E0. For instance: E0(x, z) = −exp(−γ2 Pd i=1 ρ(xi −zi, σ)), where ρ(·, ·) is the Geman-McClure function, ρ(y, σ) = y2 y2+σ2 , and σ is a parameter of the function. This function is also used in [6] for Robust PCA. 3.4 Dealing with missing data and intra-sample outliers in training data Previous sections have shown how to deal with outliers and missing data in the testing set (assuming KPCA has been learned from a clean training set). If we have missing data in the training samples [6], a simple approach is to iteratively alternate between estimating the missing values and updating the KPCA principal subspace until convergence. Algorithm 1 outlines the main steps of this approach. An algorithm for handling intra-sample outliers in training data could be constructed similarly. Alternatively, a kernel matrix could be computed ignoring the missing values, that is, each kij = exp(−γ2||WiWj(xi −xj)||2 2), where γ2 = 1 trace(WiWj). However, the positive definiteness of the resulting kernel matrix cannot be guaranteed. 3.5 Optimization In general, the objective function in Eq. 4 is not concave, hence non-convex optimization techniques are required. In this section, we restrict our attention to the Gaussian kernel (k(x, y) = exp(−γ||x −y||2)) that is the most widely used kernel. If E0 takes the form of Sec.3.2, we need to maximize E(z) = exp(−γ2||W(x −z)||2) | {z } E1(z) +C ∗r(z)T Mr(z) | {z } E2(z) , (5) 4 where r(·), and M are defined in Eq.1. Note that optimizing this function is not harder than optimizing the objective function used by Mika et al [13]. Here, we also derive a fixed-point optimization algorithm. The necessary condition for a minimum has to satisfy the following equation: ∇zE(z) = ∇zE1(z) + ∇zE2(z) = 0 . The expression for the gradients are given by: ∇zE1(z) = −2γ2 exp(−γ2||W(x −z)||2)W2 | {z } W2 (z −x), ∇zE2(z) = −4γ[(1T nQ1n)z −DQ1n] , where Q is a matrix such that qij = mijexp(−γ||z −di||2 −γ||z −dj||2). A fixed-point update is: z = [ 1 2C γ2 γ W2 + (1T nQ1n)In | {z } W3 ]−1( 1 2C γ2 γ W2x + DQ1n | {z } u ) (6) The above equation is the update rule for z at every iteration of the algorithm. The algorithm stops when the difference between two successive z’s is smaller than a threshold. Note that W3 is a diagonal matrix with non-negative entries since Q is a positive semi-definite matrix. Therefore, W3 is not invertible only if there are some zero elements on the diagonal. This only happens if some elements of the diagonal of W are 0 and 1T nQ1n = 0. It can be shown that 1T nQ1n = Pm i=1(vT i ϕ(z))2, so 1T nQ1n = 0 when ϕ(z)⊥vi, ∀i. However, this rarely occurs in practice; moreover, if this happens we can restart the algorithm from a different initial point. Consider the update rule given in Eq.6: z = W−1 3 u. The diagonal matrix W−1 3 acts as a normalization factor of u. Vector u is a weighted combination of two terms, the training data D and x. Furthermore, each element of x is weighted differently by W2 which is proportional to W. In the case of missing data (some entries in the diagonal of W, and therefore W2, will be zero), missing components of x would not affect the computation of u and z. Entries corresponding to the missing components of the resulting z will be pixel-weighted combinations of the training data. The contribution of x also depends on the ratio γ2/γ, C, and the distance from the current z to x. Similar to the observation of Mika et al [13], the second term of vector u pulls z towards a single Gaussian cluster. The attraction force generated by a training data point di reflects the correlation between ϕ(z) and ϕ(di), the correlation between ϕ(z) and eigenvectors vj’s, and the contributions of ϕ(di) to the eigenvectors. The forces from the training data, together with the attraction force by x, draw z towards a Gaussian cluster that is close to x. One can derive a similar update rule for z if E0 takes the form in Sec.3.3. z = [ 1 2C γ2 γ W4 + (1T nQ1n)In]−1( 1 2C γ2 γ W4x + DQ1n), with W4 = exp(−γ2 Pd i=1 ρ(xi −zi, σ))W2 5, where W5 is a diagonal matrix; the ith entry of the diagonal is σ/((zi −xi)2 +σ2). The parameter σ is updated at every iteration as follows: σ = 1.4826 × median({|zi −xi|}d i=1) [5]. 4 Experiments 4.1 RKPCA for intra-sample outliers In this section, we compare RKPCA with three approaches for handling intra-sample outliers: (i) Robust Linear PCA [6], (ii) Mika et al’s KPCA reconstruction [13], and (iii) Kwok & Tsang’s KPCA reconstruction [10]. The experiments are done on the CMU Multi-PIE database [8]. The Multi-PIE database consists of facial images of 337 subjects taken under different illuminations, expressions and poses, at four different sessions. We only make use of the directly-illuminated frontal face images under five different expressions (smile, disgust, squint, surprise and scream), see Fig. 4. Our dataset contains 1100 images, 700 are randomly selected for training, 100 are used for validation, and the rest is reserved for testing. Note that no subject in the testing set appears in the training set. Each face is manually labeled with 68 landmarks, as shown in Fig. 4a. A shape-normalized face is generated for every face by warping it towards the mean shape using affine transformation. Fig. 4b shows an example of such a shape-normalized face. The mean shape is used as the face mask and the values inside the mask are vectorized. To quantitatively compare different methods, we introduce synthetic occlusions of different sizes (20, 30, and 40 pixels) into the test images. For each occlusion size and test image pair, we generate 5 Figure 4: a) 68 landmarks, b) a shape-normalized face, c) synthetic occlusion. Occ.Sz Region Type Base Line Mika et al Kwok&Tsang Robust PCA Ours Energy 80% 20 Whole face 14.0±5.5 13.5±3.3 14.1±3.4 10.8±2.4 8.1±2.3 Occ. Reg. 71.5±5.5 22.6±7.9 17.3±6.6 13.3±5.5 16.1±6.1 Non-occ Reg. 0.0±0.0 11.3±2.3 13.2±2.9 10.1±2.2 6.0±1.7 30 Whole face 27.7±10.2 17.5±4.8 16.6±4.6 12.2±3.2 10.9±4.2 Occ. Reg. 70.4±3.9 24.2±7.1 19.3±6.6 15.4±5.1 18.4±5.8 Non-occ Reg. 0.0±0.0 13.3±3.0 14.7±3.8 9.6±2.3 5.7±4.3 40 Whole face 40.2±12.7 20.9±5.9 18.8±5.8 16.4±7.1 14.3±6.3 Occ. Reg. 70.6±3.6 25.7±7.2 21.1±7.1 20.1±8.0 19.8±6.3 Non-occ Reg. 0.0±0.0 15.2±4.2 16.1±5.3 9.4±2.3 8.8±8.1 Energy 95% 20 Whole face 14.2±5.3 12.6±3.1 13.8±3.2 9.1±2.3 7.0±2.1 Occ. Reg. 71.2±5.4 29.2±8.4 17.3±6.4 18.6±7.1 18.1±6.1 Non-occ Reg. 0.0±0.0 8.6±1.6 12.9±2.9 6.5±1.4 4.1±1.6 30 Whole face 26.8±9.5 17.4±4.4 16.2±4.1 13.4±5.0 10.2±3.7 Occ. Reg. 70.9±4.4 30.0±7.6 19.5±6.5 23.8±7.8 21.0±6.3 Non-occ Reg. 0.0±0.0 10.1±1.9 14.1±3.2 6.3±1.4 3.1±1.7 40 Whole face 40.0±11.9 22.0±5.9 18.9±6.0 22.7±11.7 14.3±5.8 Occ. Reg. 70.7±3.6 30.1±7.2 21.4±7.4 32.4±11.9 22.4±7.0 Non-occ Reg. 0.0±0.0 12.1±3.3 15.9±5.2 7.0±2.5 5.0±6.7 Figure 5: Results of several methods on MPIE database. This shows the means and standard deviations of the absolute differences between reconstructed images and the ground-truths. The statistics are available for three types of face regions (whole face, occluded region, and non-occluded region), different occlusion sizes, and different energy settings. Our method consistently outperforms other methods for different occlusion sizes and energy levels. a square occlusion window of that size, drawing the pixel values randomly from 0 to 255. A synthetic testing image is then created by pasting the occlusion window at a random position. Fig. 4c displays such an image with occlusion size of 20. For every synthetic testing image and each of the four algorithms, we compute the mean (at pixel level) of the absolute differences between the reconstructed image and the original test image without occlusion. We record these statistics for occluded region, non-occluded region and the whole face. The average statistics together with standard deviations are then calculated over the set of all testing images. These results are displayed in Fig. 5. We also experiment with several settings for the energy levels for PCA and KPCA. The energy level essentially means the number of components of PCA/KPCA subspace. In the interest of space, we only display results for two settings 80% and 95%. Base Line is the method that does nothing; the reconstructed images are exactly the same as the input testing images. As can be seen from Fig.5, our method consistently outperforms others for all energy levels and occlusion sizes (using the whole-face statistics). Notably, the performance of our method with the best parameter settings is also better than the performances of other methods with their best parameter settings. The experimental results for Mika et al, Kwok & Tsang, Robust PCA [6] are generated using our own implementations. For Mika et al, Kwok & Tsang’s methods, we use Gaussian kernels with γ = 10−7. For our method, we use E0 defined in Sec. 3.3. The kernel is Gaussian with γ = 10−7, γ2 = 10−6, and C = 0.1. The parameters are tuned using validation data. 4.2 RKPCA for incomplete training data To compare the ability to handle missing attributes in training data of our algorithm with other methods, we perform some experiments on the well known Oil Flow dataset [4]. This dataset is also used by Sanguinetti & Lawrence [16]. This dataset contains 3000 12-dimensional synthetically generated data points modeling the flow of a mixture of oil, water and gas in a transporting pipe-line. We test our algorithm with different amount of missing data (from 5% to 50%) and repeat each experiment for 50 times. For each experiment, we randomly choose 100 data points and randomly remove some attribute values at some certain rate. We run Algorithm 1 to recover the values of the missing attributes, with m = 25, k = 10, γ = 0.0375 (same as [16]), γ2 = 0.0375, C = 107. The squared difference between the reconstructed data and the original groundtruth data is measured, and the mean and standard deviation for 50 runs are calculated. Note that this experiment setting is exactly the same as the setting by [16]. 6 Table 1: Reconstruction errors for 5 different methods and 10 probabilities of missing values for the Oil Flow dataset. Our method outperforms other methods for all levels of missing data. p(del) 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 mean 13 ± 4 28 ± 4 43 ± 7 53 ± 8 70 ± 9 81 ± 9 97 ± 9 109 ± 8 124 ± 7 139 ± 7 1-NN 5 ± 3 14 ± 5 30 ± 10 60 ± 20 90 ± 20 NA NA NA NA NA PPCA 3.7 ± .6 9 ± 2 17 ± 5 25 ± 9 50 ± 10 90 ± 30 110 ± 30 110 ± 20 120 ± 30 140 ± 30 PKPCA 5 ± 1 12 ± 3 19 ± 5 24 ± 6 32 ± 6 40 ± 7 45 ± 4 61 ± 8 70 ± 10 100 ± 20 Ours 3.2 ± 1.9 8 ± 4 12 ± 4 19 ± 6 27 ± 8 34 ± 10 44 ± 9 53 ± 12 69 ± 13 83 ± 15 Experimental results are summarized in Tab. 1. The results of our method are shown in the last column. The results of other methods are copied verbatim from [16]. The mean method is a widely used heuristic where the missing value of an attribute of a data point is filled by the mean of known values of the same attribute of other data points. The 1-NN method is another widely used heuristic in which the missing values are replaced by the values of the nearest point, where the pairwise distance is calculated using only the attributes with known values. PPCA is the probabilistic PCA method [11], and PKPCA is the method proposed by [16]. As can be seen from Tab. 1, our method outperforms other methods for all levels of missing data. 4.3 RKPCA for denoising This section describes denoising experiments on the Multi-PIE database with Gaussian additive noise. For a fair evaluation, we only compare our algorithm with Mika et al’s, Kwok & Tsang’s and Linear PCA. These are the methods that perform denoising based on subspaces and do not rely explicitly on the statistics of natural images. Quantitative evaluations show that the denoising ability of our algorithm is comparable with those of other methods. Figure 6: Example of denoised images. a) original image, b) corrupted by Gaussian noise, c) denoised using PCA, d) using Mika et al, e) using Kwok & Tsang method, f) result of our method. The set of images used in these experiments is exactly the same as those in the occlusion experiments described in Sec. 4.1. For every testing image, we synthetically corrupt it with Gaussian additive noise with standard deviation of 0.04. An example for a pair of clean and corrupted images are shown in Fig. 6a and 6b. For every synthetic testing image, we compute the mean (at pixel level) of the absolute difference between the denoised image and the ground-truth. The results of different methods with different energy settings are summarized in Tab. 2. For these experiments, we use E0 defined in Sec. 3.2 with W being the identity matrix. We use Gaussian kernel with γ = γ2 = 10−7, and C = 1. These parameters are tuned using validation data. Table 2: Results of image denoising on the Multi-PIE database. Base Line is the method that does nothing. The best energy setting for all methods is 100%. Our method is better than the others. Energy Base Line Mika Kwok& Tsang PCA Ours 80% 8.14±0.16 9.07±1.86 11.79±2.56 10.04±1.99 7.01±1.27 95% 8.14±0.16 6.37±1.30 11.55±2.52 6.70±1.20 5.70±0.96 100% 8.14±0.16 5.55±0.97 11.52±2.52 6.44±0.39 5.43±0.78 Tab. 2 and Fig. 6 show the performance of our method is comparable with others. In fact, the quantitative results show that our method is marginally better than Mika et al’s method and substantially better than the other two. In terms of visual appearance (Fig. 6c-f), the reconstruction image of our method preserves much more fine details than the others. 7 5 Conclusion In this paper, we have proposed Robust Kernel PCA, a unified framework for handling noise, occlusion and missing data. Our method is based on a novel cost function for Kernel PCA reconstruction. The cost function requires the reconstructed data point to be close to the original data point as well as to the principal subspace. Notably, the distance function between the reconstructed data point and the original data point can take various forms. This distance function needs not to depend on the kernel function and can be evaluated easily. Therefore, the implicitness of the feature space is avoided and optimization is possible. Extensive experiments, in two well known data sets, show that our algorithm outperforms existing methods. References [1] Alzate, C. & Syukens, J.A. (2005) ‘Robust Kernel Principal Component Analysis uisng Huber’s Loss Function.’ 24th Benelux Meeting on Systems and Control. [2] Bakir, G.H., Weston, J. & Sch¨olkopf, B. (2004) ‘Learning to Find Pre-Images.’ in Thrun, S., Saul, L. & Sch¨olkopf, B. (Eds) Advances in Neural Information Processing Systems. [3] Berar, M., Desvignes, M., Bailly, G., Payan, Y. & Romaniuk, B. (2005) ‘Missing Data Estimation using Polynomial Kernels.’ Proceedings of International Conference on Advances in Pattern Recognition. [4] Bishop, C.M., Svens´en, M. & Williams, C.K.I. (1998) ‘GTM: The Generative Topographic Mapping.’ Neural Computation, 10(1), 215–234. [5] Black, M.J. & Anandan, P. (1996) ‘The Robust Estimation of Multiple Motions: Parametric and Piecewise-smooth Flow Fields.’ Computer Vision and Image Understanding, 63(1), 75–104. [6] de la Torre, F. & Black, M.J. (2003) ‘A Framework for Robust Subspace Learning.’ International Journal of Computer Vision, 54(1–3), 117–142. [7] Deng, X., Yuan, M. & Sudijanto, A. (2007) ‘A Note on Robust Principal Component Analysis.’ Contemporary Mathematics, 443, 21–33. [8] Gross, R., Matthews, I., Cohn, J., Kanade, T. & Baker, S. (2007) ‘The CMU Multi-pose, Illumination, and Expression (Multi-PIE) Face Database.’ Technical report, Carnegie Mellon University.TR-07-08. [9] Jolliffe, I. (2002) Principal Component Analysis. 2 edn. Springer-Verlag, New York. [10] Kwok, J.T.Y. & Tsang, I.W.H. (2004) ‘The Pre-Image Problem in Kernel Methods.’ IEEE Transactions on Neural Networks, 15(6), 1517–1525. [11] Lawrence, N.D. (2004) ‘Gaussian Process Latent Variable Models for Visualization of High Dimensional Data.’ in Thrun, S., Saul, L. & Sch¨olkopf, B. (Eds) Advances in Neural Information Processing Systems. [12] Lu, C., Zhang, T., Zhang, R. & Zhang, C. (2003) ‘Adaptive Robust Kernel PCA Algorithm.’ Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing. [13] Mika, S., Sch¨olkopf, B., Smola, A., M¨uller, K.R., Scholz, M. & R¨atsch, G. (1999) ‘Kernel PCA and De-Noising in Feature Spaces.’ Advances in Neural Information Processing Systems. [14] Romdhani, S., Gong, S. & Psarrou, A. (1999) ‘Multi-view Nonlinear Active Shape Model Using Kernel PCA.’ British Machine Vision Conference, 483–492. [15] Roweis, S. (1998) ‘EM Algorithms for PCA and SPCA.’ in Jordan, M., Kearns, M. & Solla, S. (Eds) Advances in Neural Information Processing Systems 10. [16] Sanguinetti, G. & Lawrence, N.D. (2006) ‘Missing Data in Kernel PCA.’ Proceedings of European Conference on Machine Learning. [17] Sch¨olkopf, B., Mika, S., Smola, A., R¨atsch, G. & M¨uller, K.R. (1998) ‘Kernel PCA Pattern Reconstruction via Approximate Pre-Images.’ International Conference on Artificial Neural Networks. [18] Sch¨olkopf, B. & Smola, A. (2002) Learning with Kernels: Support Vector Machines, Regularization, Optimization, and beyond. MIT Press, Cambridge, MA. [19] Sch¨olkopf, B., Smola, A. & Mller, K. (1998) ‘Nonlinear Component Analysis as a Kernel Eigenvalue Problem.’ Neural Computation, 10, 1299–1319. [20] Shawe-Taylor, J. & Cristianini, N. (2004) Kernel Methods for Pattern Analysis. Cambridge Uni. Press. [21] Tipping, M. & Bishop, C.M. (1999) ‘Probabilistic Principal Component Analysis.’ Journal of the Royal Statistical Society B, 61, 611–622. [22] Wang, L., Pang, Y.W., Shen, D.Y. & Yu, N.H. (2007) ‘An Iterative Algorithm for Robust Kernel Principal Component Analysis.’ Conference on Machine Learning and Cybernetics. 8
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Relative Margin Machines Pannagadatta K Shivaswamy and Tony Jebara Department of Computer Science, Columbia University, New York, NY pks2103,jebara@cs.columbia.edu Abstract In classification problems, Support Vector Machines maximize the margin of separation between two classes. While the paradigm has been successful, the solution obtained by SVMs is dominated by the directions with large data spread and biased to separate the classes by cutting along large spread directions. This article proposes a novel formulation to overcome such sensitivity and maximizes the margin relative to the spread of the data. The proposed formulation can be efficiently solved and experiments on digit datasets show drastic performance improvements over SVMs. 1 Introduction The goal of most machine learning problems is to generalize from a limited number of training examples. For example, in support vector machines [10] (SVMs) a hyperplane 1 of the form w⊤x + b = 0, w ∈Rm, x ∈Rm, b ∈R is recovered as a decision boundary after observing a limited number of training examples. The parameters of the hyperplane (w, b) are estimated by maximizing the margin (the distance between w⊤x + b = 1 and w⊤x + b = −1) while minimizing a weighted upper bound on the misclassification rate on the training data (the so called slack variables). In practice, the margin is maximized by minimizing 1 2w⊤w. While this works well in practice, we point out that merely changing the scale of the data can give a different solution. On one hand, an adversary can exploit this shortcoming to transform the data so as to give bad performance. More distressingly, this shortcoming can naturally lead to a bad performance especially in high dimensional settings. The key problem is that SVMs simply find a large margin solution giving no attention to the spread of the data. An excellent discriminator lying in a dimension with relatively small data spread may be easily overlooked by the SVM solution. In this paper, we propose novel formulations to overcome such a limitation. The crux here is to find the maximum margin solution with respect to the spread of the data in a relative sense rather than finding the absolute large margin solution. Linear discriminant analysis finds a projection of the data so that the inter-class separation is large while within class scatter is small. However, it only makes use of the first and the second order statistics of the data. Feature selection with SVMs [12] remove that have low discriminative value. Ellipsoidal kernel machines [9] normalize data in feature space by estimating bounding ellipsoids. While these previous methods showed performance improvements, both relied on multiple-step locally optimal algorithms for interleaving spread information with margin estimation. Recently, additional examples were used to improve the generalization of the SVMs with so called “Universum” samples [11]. Instead of leveraging additional data or additional model assumptions such as axis-aligned feature selection, 1In this paper we use the dot product w⊤x with the understanding that it can be replaced with an inner product. 1 the proposed method overcomes what seems to be a fundamental limitation of the SVMs and subsequently yield improvements in the same supervised setting. In addition, the formulations derived in this paper are convex, can be efficiently solved and admit some useful generalization bounds. Notation Boldface letters indicate vectors/matrices. For two vectors u ∈Rm and v ∈Rm, u ≤v indicates that ui ≤vi for all i from 1 to m. 1, 0 and I denote the vectors of all ones, all zeros and the identity matrix respectively. Their dimensions are clear from the context. −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 Figure 1: Top: As the data is scaled along the x-axis, the SVM solution (red or dark shade) deviates from the maximum relative margin solution (green or light shade). Bottom: The projections of the examples in the top row on the real line for the SVM solution (red or dark shade) and the proposed classifier (green or light shade) in each case. 2 Motivation with a two dimensional example Let us start with a simple two dimensional toy dataset to illustrate a problem with the SVM solution. Consider the binary classification example shown in the top row of Figure 1 where squares denote examples from one class and triangles denote examples from the other class. Consider the leftmost plot in the top row of Figure 1. One possible decision boundary separating the two classes is shown in green (or light shade). The solution shown in red (or dark shade) is the SVM estimate; it achieves the largest margin possible while still separating both the classes. Is this necessarily “the best” solution? Let us now consider the same set of points after scaling the x-axis in the second and the third plots. With progressive scaling, the SVM increasingly deviates from the green solution, clearly indicating that the SVM decision boundary is sensitive to affine transformations of the data and produces a family of different solutions as a result. This sensitivity to scaling and affine transformations is worrisome. If there is a best and a worst solution in the family of SVM estimates, there is always the possibility that an adversary exploits this scaling such that the SVM solution we recover is poor. Meanwhile, an algorithm producing the green decision boundary remains resilient to such adversarial scalings. In the previous example, a direction with a small spread in the data produced a good discriminator. Merely finding a large margin solution, on the other hand, does not recover the best possible discriminator. This particular weakness in large margin estimation has only received limited attention in previous work. In the above example, suppose each class is generated from a one dimensional distribution on a line with the two classes on two parallel lines. In this case, the green decision boundary should obtain zero test error even if it is estimated from a finite number of samples. However, for finite training data, the SVM solution will make errors and will do so increasingly as the data is scaled along the x-axis. Using kernels and nonlinear mappings may help in some cases but might also exacerbate such problems. Similarly, simple prepossessing of the data (affine “whitening” to make the 2 dataset zero mean and unit covariance or scaling to place the data into a zero-one box) may fail to resolve such problems. For more insight, consider the uni-dimensional projections of the data given by the green and red solutions in the bottom row of Figure 1. In the green solution, all points in the first class are mapped to a single coordinate and all points in the other class are mapped to another (distinct) coordinate. Meanwhile, the red solution produces more dispersed projections of the two classes. As the adversarial scaling is increased, the spread of the projection in the SVM solution increases correspondingly. Large margins are not sufficient on their own and what is needed is a way to also control the spread of the data after projection. Therefore, rather than just maximizing the margin, a trade-offregularizer should also be used to minimize the spread of the projected data. In other words, we will couple large margin estimation with regularization which seeks to bound the spread |w⊤x + b| of the data. This will allow the linear classifier to recover large margin solutions not in the absolute sense but rather relative to the spread of the data in that projection direction. 3 Formulations Given (xi, yi)n i=1 where xi ∈Rm and yi ∈{±1} drawn independent and identically distributed from a distribution Pr(x, y), the Support Vector Machine primal formulation 2 is as follows: min w,b,ξ≥0 1 2∥w∥2 + Cξ⊤1 s.t. yi(w⊤xi + b) ≥1 −ξi, ∀1 ≤i ≤n. (1) The above formulation minimizes an upper bound on the misclassification while maximizing the margin (the two quantities are traded offby C). In practice, the following dual of the formulation (1) is solved: max 0≤α≤C1 −1 2 n X i=1 n X j=1 αiαjyiyjx⊤ i xj + n X i=1 αi s.t. α⊤y = 0. (2) It is easy to see that the above formulation (2) is rotation invariant; if all the xi are replaced by Axi where A ∈Rm×m, A⊤A = I, then the solution remains the same. However, the solution is not guaranteed to be the same when A is not a rotation matrix. In addition, the solution is sensitive to translations as well. Typically, the dot product between the examples is replaced by a kernel function k : Rm × Rm →R such that k(xi, xj) = φ(xi)⊤φ(xj), where φ : Rm →H is a mapping to a Hilbert space to obtain non-linear decision boundaries in the input space. Thus, in (2), x⊤ i xj is replaced by k(xi, xj) to obtain non-linear solutions. In rest of this paper, we denote by K ∈Rn×n the Gram matrix, whose individual entries are given by Kij = k(xi, xj). Next, we consider the formulation which corresponds to whitening the data with the covariance matrix. Denote by Σ = 1 n Pn i=1 xix⊤ i − 1 n2 Pn i=1 xi Pn j=1 x⊤ j , and µ = 1 n Pn i=1 xi, the sample covariance and mean respectively. Consider the following formulation which we call Σ-SVM: min w,b,ξ≥0 1 −D 2 ∥w∥2 + D 2 ∥Σ 1 2 w∥2 + Cξ⊤1 s.t. yi(w⊤(xi −µ) + b) ≥1 −ξi, (3) where 0 ≤D ≤1 is an additional parameter that trades offbetween the two regularization terms. The dual of (3) can be shown to be: max 0≤α≤C1,y⊤α=0 n X i=1 αi −1 2 n X i=1 αiyi(xi −µ)⊤((1 −D)I + DΣ)−1 n X j=1 αjyj(xj −µ). (4) 2After this formulation, we stop explicitly writing ∀1 ≤i ≤n since it will be obvious from the context. 3 It is easy to see that the above formulation (4) is translation invariant and tends to an affine invariant solution when D tends to one. When 0 < D < 1, it can be shown, by using the Woodbury matrix inversion formula, that the above formulation can be “kernelized” simply by replacing the dot products x⊤ i xj in (2) by: 1 1 −D k(xi, xj) −K⊤ i 1 n −K⊤ j 1 n + 1⊤K1 n2 ! − 1 1 −D Ki −K1 n ⊤ I n −11⊤ n2 1 −D D I + K I n −11⊤ n2 −1 Kj −K1 n ! , where Ki is the ith column of K. For D = 0 and D = 1, it is much easier to obtain the kernelized formulations. Note that the above formula involves a matrix inversion of size n, making the kernel computation alone O(n3). 3.1 RMM and its geometrical interpretation From Section 2, it is clear that large margin in the absolute sense might be deceptive and could merely be a by product of bad scaling of the data. To overcome this limitation, as we pointed out earlier, we need to bound the projections of the training examples as well. As in the two dimensional example, it is necessary to trade offbetween the margin and the spread of the data. We propose a slightly modified formulation in the next section that can be solved efficiently. For now, we write the following formulation, mainly to show how it compares with the Σ-SVM. In addition, writing the dual of the following formulation gives some geometric intuition. Since we trade offbetween the projections and the margin, implicitly, we find large relative margin. Thus we call the following formulation the Relative Margin Machine (RMM): min w,b,ξ≥0 1 2∥w∥2 + Cξ⊤1 s.t. yi(w⊤xi + b) ≥1 −ξi, 1 2(w⊤xi + b)2 ≤B2 2 . (5) This is a quadratically constrained quadratic problem (QCQP). This formulation has one extra parameter B in addition to the SVM parameter. Note that B ≥1 since having a B less than one would mean none of the examples would satisfy yi(w⊤xi + b) ≥1. Let wC and bC be the solutions obtained by solving the SVM (1) for a particular value of C, then B > maxi |w⊤ Cxi + bC|, makes the constraint on the second line in the formulation (5) inactive for each i and the solution obtained is the same as the SVM estimate. For smaller B values, we start getting different solutions. Specifically, with a smaller B, we still find a large margin solution such that all the projections of the training examples are bounded by B. Thus by trying out different B values, we explore different large margin solutions with respect to the projection and spread of the data. In the following, we assume that the value of B is smaller than the threshold mentioned above. The Lagrangian of (5) is given by: 1 2∥w∥2 + Cξ⊤1 − n X i=1 αi yi(w⊤xi + b) −1 + ξi −β⊤ξ + n X i=1 λi 1 2(w⊤xi + b)2 −1 2B2 , where α, β, λ ≥0 are the Lagrange multipliers corresponding to the constraints. Differentiating with respect to the primal variables and equating them to zero, it can be shown that: (I+ n X i=1 λixix⊤ i )w−b n X i=1 λixi = n X i=1 αiyixi, b = 1 λ⊤1( n X i=1 αiyi− n X i=1 λiw⊤xi), C1 = α+β. Denoting by Σλ = Pn i=1 λixix⊤ i − 1 λ⊤1 Pn i=1 λixi Pn j=1 λjx⊤ j , and by µλ = 1 λ⊤1 Pn j=1 λjxj the dual of (5) can be shown to be: max 0≤α≤C1,λ≥0 n X i=1 αi −1 2 n X i=1 αiyi(xi −µλ)⊤(I + Σλ)−1 n X j=1 αjyj(xj −µλ) −1 2B2λ⊤1 (6) 4 Note that the above formulation is translation invariant since µλ is subtracted from each xi. Σλ corresponds to a “shape matrix” (potentially low rank) determined by xi’s that have non-zero λi. From the KKT conditions of (5), λi( 1 2(w⊤xi + b)2 −B2 2 ) = 0. Consequently λi > 0 implies ( 1 2(w⊤xi + b)2 −B2 2 ) = 0. Geometrically, in the above formulation (6), the data is whitened with the matrix (I + Σλ) while solving SVM. While this is similar to what is done by the Σ-SVM, the matrix (I+Σλ) is determined jointly considering both the margin of the data and the spread. In contrast, in Σ-SVM, whitening is simply a prepossessing step which can be done independently of the margin. Note that the constraint 1 2(w⊤xi+b)2 ≤1 2B2 can be relaxed with slack variables at the expense of one additional parameter however this will not be investigated in this paper. The proposed formulation is of limited use unless it can be solved efficiently. Solving (6) amounts to solving a semi-definite program; it cannot scale beyond a few hundred data points. Thus, for efficient solution, we consider a different but equivalent formulation. Note that the constraint 1 2(w⊤xi + b)2 ≤ 1 2B2 can be equivalently posed as two linear constraints : (w⊤xi + b) ≤B and −(w⊤xi + b) ≤B. With these constraints replacing the quadratic constraint, we have a quadratic program to solve. In the primal, we have 4n constraints (including ξ ≥0 ) instead of the 2n constraints in the SVM. Thus, solving RMM as a standard QP has the same order of complexity as the SVM. In the next section, we briefly explain how the RMM can be solved efficiently from the dual. 3.2 Fast algorithm The main idea for the fast algorithm is to have linear constraints bounding the projections rather than quadratic constraints. The fast algorithm that we developed is based on SVMlight [5]. We first write the equivalent of (5) with linear constraints: min w,b,ξ≥0 1 2∥w∥2 + Cξ⊤1 s.t. yi(w⊤xi + b) ≥1 −ξi, w⊤xi + b ≤B, −w⊤xi −b ≤B. (7) The dual of (7) can be shown to be the following: max α,λ,λ∗−1 2 (α ⊗y −λ + λ∗)⊤K (α ⊗y −λ + λ∗) + α⊤1 −Bλ⊤1 −Bλ∗⊤1 (8) s.t. α⊤y −λ⊤1 + λ∗⊤1 = 0, 0 ≤α ≤C1, λ, λ∗≥0, where, the operator ⊗denotes the element-wise product of two vectors. The above QP (8) is solved in an iterative way. In each step, only a subset of the dual variables are optimized. Let us say, q, r and s (˜q, ˜r and ˜s) are the indices to the free (fixed) variables in α, λ and λ∗respectively (such that q ∪˜q = {1, 2, · · ·n} and q ∩˜q = ∅, similarly for the other two indices) in a particular iteration. Then the optimization over the free variables in that step can be expressed as: max αq,λr,λ∗s −1 2 " αq ⊗yq λr λ∗ s #⊤" Kqq −Kqr Kqs −Krq Krr −Krs Ksq −Ksr Kss # " αq ⊗yq λr λ∗ s # (9) −1 2 " αq ⊗yq λr λ∗ s #⊤" Kq˜q −Kq˜r Kq˜s −Kr˜q Kr˜r −Kr˜s Ks˜q −Ks˜r Ks˜s # " α˜q ⊗y˜q λ˜r λ∗ ˜s # + α⊤ q 1 −Bλ⊤ r 1 −Bλ∗⊤ s 1 s.t. α⊤ q yq −λ⊤ r 1 + λ∗⊤ s 1 = −α⊤ ˜q y˜q + λ⊤ ˜r 1 −λ∗⊤ ˜s 1, 0 ≤αq ≤C1, λr, λ∗ s ≥0. Note that while the first term in the objective above is quadratic in the free variables (over which it is optimized), the second term is only linear. The algorithm, solves a small sub-problem like (9) in each step until the KKT conditions of the formulation (8) are satisfied to a given tolerance. In each step, the free variables are selected using heuristics similar to those in SVMlight but slightly adapted to our formulation. 5 We omit the details due to lack of space. Since only a small subset of the variables is optimized, book-keeping can be done efficiently in each step. Moreover, the algorithm can be warm-started with a previous solution just like SVMlight. 4 Experiments Experiments were carried out on three sets of digits - optical digits from the UCI machine learning repository [1], USPS digits [6] and MNIST digits [7]. These datasets have different number of features (64 in optical digits, 256 in USPS and 784 in MNIST) and training examples (3823 in optical digits, 7291 in USPS and 60000 in MNIST). In all these multiclass experiments one versus one classification strategy was used. We start by noting that, on the MNIST test set, an improvement of 0.1% is statistically significant [3, 4]. This corresponds to 10 or fewer errors by one method over another on the MNIST test set. All the parameters were tuned by splitting the training data in each case in the ratio 80:20 and using the smaller split for validation and the larger split for training. The process was repeated five times over random splits to pick best parameters (C for SVM, C and D for Σ-SVM and C and B for RMM). A final classifier was trained for each of the 45 classification problems with the best parameters found from cross validation using all the training examples in those classes. In the case of MNIST digits, training Σ-SVM and KLDA are prohibitive since they involve inverting a matrix. So, to compare all the methods, we conducted an experiment with 1000 examples per training. For the larger experiments we simply excluded Σ-SVM and KLDA. The larger experiment on MNIST consisted of training with two thirds of the digits (note that this amounts to training with 8000 examples on an average for each pair of digits) for each binary classification task. In both the experiments, the remaining training data was used as a validation set. The classifier that performed the best on the validation set was used for testing. Once we had 45 classifiers for each pair of digits, testing was done on the separate test set available in each of these three datasets (1797 examples in the case of optical digits, 2007 examples in USPS and 10000 examples in MNIST). The final prediction given for each test example was based on the majority of predictions made by the 45 classifiers on the test example with ties broken uniformly at random. Table 1 shows the result on all the three datasets for polynomial kernel with various degrees and the RBF kernel. For each dataset, we report the number of misclassified examples using the majority voting scheme mentioned above. It can be seen that while Σ-SVM usually performs much better compared to SVM, RMM performs even better than Σ-SVM in most cases. Interestingly, with higher degree kernels, Σ-SVM seems to match the performance of the RMM, but in most of the lower degree kernels, RMM outperforms both SVM and Σ-SVM convincingly. Since, Σ-SVM is prohibitive to run on large scale datasets, the RMM was clearly the most competitive method in these experiments. Training with entire MNIST We used the best parameters found by crossvalidation in the previous experiments on MNIST and trained 45 classifiers for both SVM and RMM with all the training examples for each class in MNIST for various kernels. The test results are reported in Table 1; the advantage still carries over to the full MNIST dataset. 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 0 0.5 1 1.5 2 2.5 3 3.5 4 SVM RMM B1 RMM B2 RMM B3 Figure 2: Log run time versus log number of examples from 1000 to 10000 in steps of 1000. 6 1 2 3 4 5 6 7 RBF OPT SVM 71 57 54 47 40 46 46 51 Σ-SVM 61 48 41 36 35 31 29 47 KLDA 71 57 54 47 40 46 46 45 RMM 71 36 32 31 33 30 29 51 USPS SVM 145 109 109 103 100 95 93 104 Σ-SVM 132 108 99 94 89 87 90 97 KLDA 132 119 121 117 114 118 117 101 RMM 153 109 94 91 91 90 90 98 1000-MNIST SVM 696 511 422 380 362 338 332 670 Σ-SVM 671 470 373 341 322 309 303 673 KLDA 1663 848 591 481 430 419 405 1597 RMM 689 342 319 301 298 290 296 613 2/3-MNIST SVM 552 237 200 183 178 177 164 166 RMM 534 164 148 140 123 129 129 144 Full MNIST SVM 536 198 170 156 157 141 136 146 RMM 521 146 140 130 119 116 115 129 Table 1: Number of digits misclassified with various kernels by SVM, Σ-SVM and RMM for three different datasets. Run time comparison We studied the empirical run times using the MNIST digits 3 vs 8 and polynomial kernel with degree two. The tolerance was set to 0.001 in both the cases. The size of the sub-problem (9) solved was 500 in all the cases. The number of training examples were increased in steps of 1000 and the training time was noted. C value was set at 1000. SVM was first run on the training examples. The value of maximum absolute prediction θ was noted. We then tried three different values of B for RMM, B1 = 1+(θ−1)/2, B2 = 1 + (θ −1)/4 B3 = 1 + (θ −1)/10. In all the cases, the run time was noted. We show a log-log plot comparing the number of examples to the run time in Figure 2. Both SVM and RMM have similar asymptotic behavior. However, in many cases, warm starting RMM with previous solution significantly helped in reducing the run times. 5 Conclusions We identified a sensitivity of Support Vector Machines and maximum absolute margin criteria to affine scalings. These classifiers are biased towards producing decision boundaries that separate data along directions with large data spread. The Relative Margin Machine was proposed to overcome such a problem and optimizes the projection direction such that the margin is large only relative to the spread of the data. By deriving the dual with quadratic constraints, a geometrical interpretation was also formulated for RMMs. An implementation for RMMs requiring only additional linear constraints in the SVM quadratic program leads to a competitively fast implementation. Experiments showed that while affine transformations can improve over the SVMs, RMM performs even better in practice. The maximization of relative margin is fairly promising as it is compatible with other popular problems handled by the SVM framework such as ordinal regression, structured prediction etc. These are valuable future extensions for the RMM. Furthermore, the constraints that bound the projection are unsupervised; thus RMMs can readily work in semi-supervised and transduction problems. We will study these extensions in detail in an extended version of this paper. References [1] A. Asuncion and D.J. Newman. UCI machine learning repository, 2007. [2] P. L. Bartlett and S. Mendelson. Rademacher and Gaussian complexities: Risk bounds and structural results. Journal of Machine Learning Research, 3:463–482, 2002. [3] Y. Bengio, P. Lamblin, D. Popovici, and H. Larochelle. Greedy layer-wise training of deep networks. In Advances in Neural Information Processing Systems 19, pages 153–160. MIT Press, Cambridge, MA, 2007. 7 [4] D. Decoste and B. Sch¨olkopf. Training invariant support vector machines. Machine Learning, pages 161–190, 2002. [5] T. Joachims. Making large-scale support vector machine learning practical. In Advances in Kernel Methods: Support Vector Machines. MIT Press, Cambridge, MA, 1998. [6] Y. LeCun, B. Boser, J.S. Denker, D. Henderson, R.E. Howard, W. Hubbard, and L. Jackel. Back-propagation applied to handwritten zip code recognition. Neural Computation, 1:541– 551, 1989. [7] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998. [8] J. Shawe-Taylor and N. Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, 2004. [9] P. K. Shivaswamy and T. Jebara. Ellipsoidal kernel machines. In Proceedings of the Artificial Intelligence and Statistics, 2007. [10] V. Vapnik. The Nature of Statistical Learning Theory. Springer Verlag, New York, 1995. [11] J. Weston, R. Collobert, F. H. Sinz, L. Bottou, and V. Vapnik. Inference with the universum. In Proceedings of the International Conference on Machine Learning, pages 1009–1016, 2006. [12] J. Weston, S. Mukherjee, O. Chapelle, M. Pontil, T. Poggio, and V. Vapnik. Feature selection for SVMs. In Neural Information Processing Systems, pages 668–674, 2000. A Generalization Bound In this section, we give the empirical Rademacher complexity [2, 8] for function classes used by the SVM, and modified versions of RMM and Σ-SVM which can be plugged into a generalization bound. Maximizing the margin can be seen as choosing a function f(x) = w⊤x from a bounded class of functions FE := {x →w⊤x| 1 2∥w∥2 ≤E}. For a technical reason, instead of bounding the projection on the training examples as in (5), we consider bounding the projections on an independent set of examples drawn from Pr(x), that is, a set U = {u1, u2, . . . unu}. Note that if we have an iid training set, it can be split into two parts and one part can be used exclusively to bound the projections and the other part can be used exclusively for classification constraints. Since the labels of the examples used to bound the projections do not matter, we denote this set by U and the other part of the set by (xi, yi)n i=1 We now consider the following function class which is closely related to RMM: HE,D := {x → w⊤x| 1 2w⊤w + D 2 (w⊤ui)2 ≤E ∀1 ≤i ≤nu} where D > 0 trades offbetween large margin and small bound on the projections. Similarly, consider: GE,D := {x →w⊤x| 1 2w⊤w + D 2nu Pnu i=1(w⊤ui)2 ≤E}, which is closely related to the class of functions considered by Σ-SVM. The empirical Rademacher complexities of the three classes of functions are as below: ˆR(FE) ≤UFE := 2 √ 2E n v u u t n X i=1 x⊤ i xi, ˆR(GE,D) ≤UGE,D := 2 √ 2E n v u u t n X i=1 x⊤ i Σ−1 D xi, ˆR(HE,D) ≤UHE,D := min λ≥0 1 n n X i=1 x⊤ i Σ−1 λ,Dxi + 2 nE nu X i=1 λi, where ΣD = I + D nu Pnu i=1 uiu⊤ i and Σλ,D = Pnu i=1 λiI + D Pnu i=1 λiuiu⊤ i . Note that the last upper bound is not a closed form expression, but a semi-definite optimization. Now, the upper bounds UFE, UGE,D and UHE,D can be plugged in the following theorem in place of ˆR(F) to obtain Rademacher type generalization bounds. Theorem 1 Fix γ > 0, let F be the class of functions from Rm × {±1} →R given by f(x, y) = −yg(x). Let {(x1, y1), . . . , (xn, yn)} be drawn iid from a probability distribution D. Then, with probability at least 1−δ over the samples of size n, the following bound holds: PrD[y ̸= sign(g(x))] ≤ξ⊤1/n + 2 ˆR(F)/γ + 3 p (ln(2/δ))/2n, where ξi = max(0, 1 −yig(xi)) are the so-called slack variables. 8
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Bayesian Kernel Shaping for Learning Control Jo-Anne Ting1, Mrinal Kalakrishnan1, Sethu Vijayakumar2 and Stefan Schaal1,3 1Computer Science, U. of Southern California, Los Angeles, CA 90089, USA 2School of Informatics, University of Edinburgh, Edinburgh, EH9 3JZ, UK 3ATR Computational Neuroscience Labs, Kyoto 619-02, Japan Abstract In kernel-based regression learning, optimizing each kernel individually is useful when the data density, curvature of regression surfaces (or decision boundaries) or magnitude of output noise varies spatially. Previous work has suggested gradient descent techniques or complex statistical hypothesis methods for local kernel shaping, typically requiring some amount of manual tuning of meta parameters. We introduce a Bayesian formulation of nonparametric regression that, with the help of variational approximations, results in an EM-like algorithm for simultaneous estimation of regression and kernel parameters. The algorithm is computationally efficient, requires no sampling, automatically rejects outliers and has only one prior to be specified. It can be used for nonparametric regression with local polynomials or as a novel method to achieve nonstationary regression with Gaussian processes. Our methods are particularly useful for learning control, where reliable estimation of local tangent planes is essential for adaptive controllers and reinforcement learning. We evaluate our methods on several synthetic data sets and on an actual robot which learns a task-level control law. 1 Introduction Kernel-based methods have been highly popular in statistical learning, starting with Parzen windows, kernel regression, locally weighted regression and radial basis function networks, and leading to newer formulations such as Reproducing Kernel Hilbert Spaces, Support Vector Machines, and Gaussian process regression [1]. Most algorithms start with parameterizations that are the same for all kernels, independent of where in data space the kernel is used, but later recognize the advantage of locally adaptive kernels [2, 3, 4]. Such locally adaptive kernels are useful in scenarios where the data characteristics vary greatly in different parts of the workspace (e.g., in terms of data density, curvature and output noise). For instance, in Gaussian process (GP) regression, using a nonstationary covariance function, e.g., [5], allows for such a treatment. Performing optimizations individually for every kernel, however, becomes rather complex and is prone to overfitting due to a flood of open parameters. Previous work has suggested gradient descent techniques with cross-validation methods or involved statistical hypothesis testing for optimizing the shape and size of a kernel in a learning system [6, 7]. In this paper, we consider local kernel shaping by averaging over data samples with the help of locally polynomial models and formulate this approach, in a Bayesian framework, for both function approximation with piecewise linear models and nonstationary GP regression. Our local kernel shaping algorithm is computationally efficient (capable of handling large data sets), can deal with functions of strongly varying curvature, data density and output noise, and even rejects outliers automatically. Our approach to nonstationary GP regression differs from previous work by avoiding Markov Chain Monte Carlo (MCMC) sampling [8, 9] and by exploiting the full nonparametric characteristics of GPs in order to accommodate nonstationary data. One of the core application domains for our work is learning control, where computationally efficient function approximation and highly accurate local linearizations from data are crucial for deriving controllers and for optimizing control along trajectories [10]. The high variations from fitting noise, seen in Fig. 3, are harmful to the learning system, potentially causing the controller to be unstable. Our final evaluations illustrate such a scenario by learning an inverse kinematics model for a real robot arm. 2 Bayesian Local Kernel Shaping We develop our approach in the context of nonparametric locally weighted regression with locally linear polynomials [11], assuming, for notational simplicity, only a one-dimensional output— extensions to multi-output settings are straightforward. We assume a training set of N samples, D = {xi, yi}N i=1, drawn from a nonlinear function y = f(x) + ϵ that is contaminated with meanzero (but potentially heteroscedastic) noise ϵ. Each data sample consists of a d-dimensional input vector xi and an output yi. We wish to approximate a locally linear model of this function at a query point xq ∈ℜd×1 in order to make a prediction yq = bT xq, where b ∈ℜd×1. We assume the existence of a spatially localized weighting kernel wi = K (xi, xq, h) that assigns a weight to every {xi, yi} according to its Euclidean distance in input space from the query point xq. A popular choice for K is the Gaussian kernel, but other kernels may be used as well [11]. The bandwidth h ∈ℜd×1 of the kernel is the crucial parameter that determines the local model’s quality of fit. Our goal is to find a Bayesian formulation of determining b and h simultaneously. 2.1 Model yi hd É bd h1 b2 i = 1,..,N xi1 É É b1 zi1 zid zi2 wi1 wi2 wid xi2 xid !z2 !zd !z1 h2 ! 2 Figure 1: Graphical model. Random variables are in circles, and observed random variables are in shaded double circles. For the locally linear model at the query point xq, we can introduce hidden random variables z [12] and modify the linear model yi = bT xi so that yi = Pd m=1zim+ ϵ, where zim = bT mxim + ϵzm and ϵzm ∼ Normal (0, ψzm), ϵ ∼Normal 0, σ2 are both additive noise terms. Note that xim = [xim 1]T and bm = [bm bm0]T , where xim is the mth coefficient of xi, bm is the mth coefficient of b and bm0 is the offset value. The z variables allow us to derive computationally efficient O(d) EM-like updates, as we will see later. The prediction at the query point xq is then Pd m bT mxqm. We assume the following prior distributions for our model, shown graphically in Fig. 1: p(yi|zi) ∼Normal 1T zi, σ2 p(bm|ψzm) ∼Normal (0, ψzmΣbm,0) p(zim|xim) ∼Normal bT mxim, ψzm p(ψzm) ∼Scaled-Inv-χ2 (nm0, ψzm,0) where 1 is a vector of 1s, zi ∈ℜd×1, zim is the mth coefficient of zi, and Σbm,0 is the prior covariance matrix of bm and a 2 × 2 diagonal matrix. nm0 and σ2 mN0 are the prior parameters of the Scaled-inverse-χ2 distribution (nm0 is the number of degrees of freedom parameter and σ2 mN0 is the scale parameter). The Scaled-Inverse-χ2 distribution was used for ψzm since it is the conjugate prior for the variance parameter of a Gaussian distribution. In contrast to classical treatments of Bayesian weighted regression [13] where the weights enter as a heteroscedastic correction on the noise variance of each data sample, we associate a scalar indicator-like weight, wi ∈{0, 1}, with each sample {xi, yi} in D. The sample is fully included in the local model if wi = 1 and excluded if wi = 0. We define the weight wi to be wi = Qd m=1 wim, where wim is the weight component in the mth input dimension. While previous methods model the weighting kernel K as some explicit function, we model the weights wim as Bernoulli-distributed random variables, i.e., p(wim) ∼Bernoulli(qim), choosing a symmetric bell-shaped function for the parameter qim: qim = 1/(1 + (xim −xqm)2rhm). xqm is the mth coefficient of xq, hm is the mth coefficient of h, and r > 0 is a positive integer1. As pointed out in [11], the particular mathematical formulation of a weighting kernel is largely computationally irrelevant for locally weighted learning. Our choice of function for qim was dominated by the desire to obtain analytically tractable learning updates. We place a Gamma prior over the bandwidth hm, i.e., p(hm) ∼Gamma (ahm0, bhm0) where ahm0 and bhm0 are parameters of the Gamma distribution, to ensure that a positive weighting kernel width. 2.2 Inference We can treat the entire regression problem as an EM learning problem [14, 15] and maximize the log likelihood log p(y|X) for generating the observed data. We can maximize this incomplete log likelihood by maximizing the expected value of the complete log likelihood p(y, Z, b, w, h, σ2, ψz|X) = QN i=1 p(yi, zi, b, wi, h, σ2, ψz|xi). In our model, each data sample i has an indicator-like scalar weight wi associated with it, allowing us to express the complete log likelihood L, in a similar fashion to mixture models, as: L = log " N Y i=1 " p(yi|zi, σ2)p(zi|xi, b, ψz) wi d Y m=1 p(wim) # d Y m=1 p(bm|ψzm)p(ψzm)p(hm)p(σ2) # Expanding the log p(wim) term from the expression above results in a problematic −log(1 + (xim −xqm)2r) term that prevents us from deriving an analytically tractable expression for the posterior of hm. To address this, we use a variational approach on concave/convex functions suggested by [16] to produce analytically tractable expressions. We can find a lower bound on the term so that −log(1 + xim −xqm)2r ≥−λim (xim −xqm)2r hm, where λim is a variational parameter to be optimized in the M-step of our final EM-like algorithm. Our choice of weighting kernel allows us to find a lower bound to L in this manner. We explored the use of other weighting kernels (e.g., a quadratic negative exponential), but had issues with finding a lower bound to the problematic terms in log p(wim) such that analytically tractable inference for hm could be done. The resulting lower bound to L is ˆL; due to lack of space, we give the expression for ˆL in the appendix. The expectation of ˆL should be taken with respect to the true posterior distribution of all hidden variables Q(b, ψz, z, h). Since this is an analytically tractable expression, a lower bound can be formulated using a technique from variational calculus where we make a factorial approximation of the true posterior, e.g., Q(b, ψz, z, h) = Q(b, ψz)Q(h)Q(z) [15], that allows resulting posterior distributions over hidden variables to become analytically tractable. The posterior of wim, p(wim = 1|yi, zi, xi, θ, wi,k̸=m), is inferred using Bayes’ rule: p(yi, zi|xi, θ, wi,k̸=m, wim = 1) Qd t=1,t̸=m⟨wit⟩p(wim = 1) p(yi, zi|xi, θ, wi,k̸=m, wim = 1) Qd t=1,t̸=m⟨wit⟩p(wim = 1) + p(wim = 0) (1) where θ = {b, ψz, h} and wi,k̸=m denotes the set of weights {wik}d k=1,k̸=m. For the dimension m, we account for the effect of weights in the other d −1 dimensions. This is a result of wi being defined as the product of weights in all dimensions. The posterior mean of wim is then ⟨p(wim = 1|yi, zi, xi, θ, wi,k̸=m)⟩, and ⟨wi⟩= Qd m=1 ⟨wim⟩, where ⟨.⟩denotes the expectation operator. We omit the full set of posterior EM update equations (please refer to the appendix for this) and list only the posterior updates for hm, wim, bm and zi: Σbm = Σ−1 bm,0 + N X i=1 ⟨wi⟩ximxT im !−1 Σzi|yi,xi = ΨzN ⟨wi⟩−1 si ΨzN ⟨wi⟩11T ΨzN ⟨wi⟩ ⟨bm⟩= Σbm N X i=1 ⟨wi⟩⟨zim⟩xim ! ⟨zi⟩= ΨzN1 si ⟨wi⟩+ Id,d −ΨzN11T si ⟨wi⟩ bxi ⟨wim⟩= qimA Qd k=1,k̸=m⟨wik⟩ i qimA Qd k=1,k̸=m⟨wik⟩ i + 1 −qim ⟨hm⟩= ahm0 + N −PN i=1 ⟨wim⟩ bhm0 + PN i=1 λim (xim −xqm)2r 1(xim −xqm) is taken to the power 2r in order to ensure that the resulting expression is positive. Adjusting r affects how long the tails of the kernel are. We use r = 2 for all our experiments. where Id,d is a d × d identity matrix, bxi is a d by 1 vector with coefficients ⟨bm⟩T xim, ⟨wi⟩= Qd m=1 ⟨wim⟩, ΨzN is a diagonal matrix with ψzN on its diagonal, si = σ2 +1T ΨzN ⟨wi⟩1 (to avoid division by zero, ⟨wi⟩needs to be capped to some small non-zero value), qim = λim = 1/(1+(xim− xqm)2r ⟨hm⟩), and Ai = N(yi; 1T ⟨zi⟩, σ2) Qd m=1 N(zim; ⟨bm⟩T xim, ψzm). Closer examination of the expression for ⟨bm⟩shows that it is a standard Bayesian weighted regression update [13], i.e., a data sample i with lower weight wi will be downweighted in the regression. Since the weights are influenced by the residual error at each data point (see posterior update for ⟨wim⟩), an outlier will be downweighted appropriately and eliminated from the local model. Fig. 2 shows how local kernel shaping is able to ignore outliers that a classical GP fits. −1 0 1 −0.5 0 0.5 1 1.5 2 2.5 3 x y Training data Stationary GP Kernel Shaping Figure 2: Effect of outliers (in black circles) A few remarks should be made regarding the initialization of priors used in the posterior EM updates. Σbm,0 can be set to 106I to reflect a large uncertainty associated with the prior distribution of b. The initial noise variance, ψzm,0, should be set to the best guess on the noise variance. To adjust the strength of this prior, nm0 can be set to the number of samples one believes to have seen with noise variance ψzm,0 Finally, the initial h of the weighting kernel should be set so that the kernel is broad and wide. We use values of ahm0 = bhm0 = 10−6 so that hm0 = 1 with high uncertainty. Note that some sort of initial belief about the noise level is needed to distinguish between noise and structure in the training data. Aside from the initial prior on ψzm, we used the same priors for all data sets in our evaluations. 2.3 Computational Complexity For one local model, the EM update equations have a computational complexity of O(Nd) per EM iteration, where d is the number input dimensions and N is the size of the training set. This efficiency arises from the introduction of the hidden random variables zi, which allows ⟨zi⟩and Σzi|yi,xi to be computed in O(d) and avoids a d × d matrix inversion which would typically require O(d3). Some nonstationary GP methods, e.g., [5], require O(N 3) + O(N 2) for training and prediction, while other more efficient stationary GP methods, e.g., [17], require O(M 2N) + O(M 2) training and prediction costs (where M << N is the number of pseudoinputs used in [17]). Our algorithm requires O(NdIEM), where IEM is the number of EM iterations—with a maximal cap of 1000 iterations used. Our algorithm also does not require any MCMC sampling as in [8, 9], making it more appealing to real-time applications. 3 Extension to Gaussian Processes We can apply the algorithm in section 2 not only to locally weighted learning with linear models, but also to derive a nonstationary GP method. A GP is defined by a mean and and a covariance function, where the covariance function K captures dependencies between any two points as a function of the corresponding inputs, i.e., k (xi, xj) = cov f(xi), f(x′ j) , where i, j = 1, .., N. Standard GP models use a stationary covariance function, where the covariance between any two points in the training data is a function of the distances |xi −xj|, not of their locations. Stationary GPs perform suboptimally for functions that have different properties in various parts of the input space (e.g., discontinuous functions) where the stationary assumption fails to hold. Various methods have been proposed to specify nonstationary GPs. These include defining a nonstationary Mat´ern covariance function [5], adopting a mixture of local experts approach [18, 8, 9] to use independent GPs to cover data in different regions of the input space, and using multidimensional scaling to map a nonstationary spatial GP into a latent space [19]. Given the data set D drawn from the function y = f(x)+ϵ, as previously introduced in section 2, we propose an approach to specify a nonstationary covariance function. Assuming the use of a quadratic negative exponential covariance function, the covariance function of a stationary GP is k(xi, xj) = v2 1 exp(−0.5 Pd m=1 hm(xim −x′ jm)2) + v0, where the hyperparameters {h1, h2, ..., hd, v0, v1} are optimized. In a nonstationary GP, the covariance function could then take the form2 k(xi, xj) = v2 1 exp −0.5 Pd m=1(xim −xjm)2 himhjm (him+hjm) +v0, where him is the bandwidth of the local model centered at xim and hjm is the bandwidth of the local model centered at xjm. We learn first the values of {him}d m=1 for all training data samples i = 1, ..., N using our proposed local kernel shaping algorithm and then optimize the hyperparameters v0 and v1. To make a prediction for a test sample xq, we learn also the values of {hqm}d m=1, i.e., the bandwidth of the local model centered at xq. Importantly, since the covariance function of the GP is derived from locally constant models, we learn with locally constant, instead of locally linear, polynomials. We use r = 1 for the weighting kernel in order keep the degree of nonlinearity consistent with that in the covariance function (i.e., quadratic). Even though the weighting kernel used in the local kernel shaping algorithm is not a quadratic negative exponential, it has a similar bell shape, but with a flatter top and shorter tails. Because of this, our augmented GP is an approximated form of a nonstationary GP. Nonetheless, it is able to capture nonstationary properties of the function f without needing MCMC sampling, unlike previously proposed nonstationary GP methods [8, 9]. 4 Experimental Results 4.1 Synthetic Data First, we show our local kernel shaping algorithm’s bandwidth adaptation abilities on several synthetic data sets, comparing it to a stationary GP and our proposed augmented nonstationary GP. For the ease of visualization, we consider the following one-dimensional functions, similar to those in [5]: i) a function with a discontinuity, ii) a spatially inhomogeneous function, and iii) a straight line function. The data set for function i) consists of 250 training samples, 201 test inputs (evenly spaced across the input space) and output noise with σ2 = 0.3025; the data set for function ii) consists of 250 training samples, 101 test inputs and an output signal-to-noise ratio (SNR) of 10; and the data set for function iii) has 50 training samples, 21 test inputs and an output SNR of 100. Fig. 3 shows the predicted outputs of a stationary GP, augmented nonstationary GP and the local kernel shaping algorithm for data sets i)-iii). The local kernel shaping algorithm smoothes over regions where a stationary GP overfits and yet, it still manages to capture regions of highly varying curvature, as seen in Figs. 3(a) and 3(b). It correctly adjusts the bandwidths h with the curvature of the function. When the data looks linear, the algorithm opens up the weighting kernel so that all data samples are considered, as Fig. 3(c) shows. Our proposed augmented nonstationary GP also can handle the nonstationary nature of the data sets as well, and its performance is quantified in Table 1. Returning to our motivation to use these algorithms to obtain linearizations for learning control, it is important to realize that the high variations from fitting noise, as shown by the stationary GP in Fig. 3, are detrimental for learning algorithms, as the slope (or tangent hyperplane, for highdimensional data) would be wrong. Table 1 reports the normalized mean squared prediction error (nMSE) values for function i) and function ii) data sets, averaged over 20 random data sets. Fig. 4 shows results of the local kernel shaping algorithm and the proposed augmented nonstationary GP on the “real-world” motorcycle data set [20] consisting of 133 samples (with 80 equally spaced input query points used for prediction). We also show results from a previously proposed MCMC-based nonstationary GP method: an alternate infinite mixture of GP experts [9]. We can see that the augmented nonstationary GP and the local kernel shaping algorithm both capture the leftmost flatter region of the function, as well as some of the more nonlinear and noisier regions after 30msec. 4.2 Robot Data Next, we move on to an example application: learning an inverse kinematics model for a 3 degree-offreedom (DOF) haptic robot arm (manufactured by SensAble, shown in Fig. 5(a)) in order to control the end-effector along a desired trajectory. This will allow us to verify that the kernel shaping algo2This is derived from the definition of K as a positive semi-definite matrix, i.e. where the integral is the product of two quadratic negative exponentials—one with parameter him and the other with parameter hjm. −2 −1 0 1 2 −4 −2 0 2 x y −2 −1 0 1 2 −1 0 1 2 x y Training data Stationary GP Aug GP Kernel Shaping −2 −1 0 1 2 −2 −1 0 1 2 x y 0 1 w −2 −1 0 1 210 0 10 3 10 7 x h w xq (a) Function i) 0 1 w −2 −1 0 1 210 0 10 6 x h (b) Function ii) 0 1 w −2 −1 0 1 2 10 −6 10 0 10 6 x h (c) Function iii) Figure 3: Predicted outputs using a stationary GP, our augmented nonstationary GP and local kernel shaping. Figures on the bottom show the bandwidths learnt by local kernel shaping and the corresponding weighting kernels (in dotted black lines) for input query points (shown in red circles). rithm can successfully deal with a large, noisy real-world data set with outliers and non-stationary properties—typical characteristics of most control learning problems. We collected 60, 000 data samples from the arm while it performed random sinusoidal movements within a constrained box volume of Cartesian space. Each sample consists of the arm’s joint angles q, joint velocities ˙q, end-effector position in Cartesian space x, and end-effector velocities ˙x. From this data, we first learn a forward kinematics model: ˙x = J(q) ˙q, where J(q) is the Jacobian matrix. The transformation from ˙q to ˙x can be assumed to be locally linear at a particular configuration q of the robot arm. We learn the forward model using kernel shaping, building a local model around each training point only if that point is not already sufficiently covered by an existing local model (e.g., having an activation weight of less than 0.2). Using insights into robot geometry, we localize the models only with respect to q while the regression of each model is trained only on a mapping from ˙q to ˙x—these geometric insights are easily incorporated as priors in the Bayesian model. This procedure resulted in 56 models being built to cover the entire space of training data. We artificially introduce a redundancy in our inverse kinematics problem on the 3-DOF arm by specifying the desired trajectory (x, ˙x) only in terms of x, z positions and velocities, i.e., the movement is supposed to be in a vertical plane in front of the robot. Analytically, the inverse kinematics equation is ˙q = J#(q) ˙x −α(I −J#J) ∂g ∂q, where J#(q) is the pseudo-inverse of the Jacobian. The second term is an optimal solution to the redundancy problem, specified here by a cost function g in terms of joint angles q. To learn a model for J#, we can reuse the local regions of q from the forward model, where J# is also locally linear. The redundancy issue can be solved by applying an additional weight to each data point according to a reward function [21]. In our case, the task is specified in terms of { ˙x, ˙z}, so we define a reward based on a desired y coordinate, ydes, that we would like to enforce as a soft constraint. Our reward function is g = e−1 2 h(k(ydes−y)−˙y)2, where k is a gain and h specifies the steepness of the reward. This ensures that the learnt inverse model chooses a solution which produces a ˙y that pushes the y coordinate toward ydes. We invert each forward local model using a weighted linear regression, where each data point is weighted by the weight from the forward model and additionally weighted by the reward. We test the performance of this inverse model (Learnt IK) in a figure-eight tracking task as shown in Fig. 5(b). As seen, the learnt model performs as well as the analytical inverse kinematics solution (IK), with root mean squared tracking errors in positions and velocities very close to that of the Table 1: Average normalized mean squared prediction error values, for a stationary GP model, our augmented nonstationary GP, local kernel shaping—averaged over 20 random data sets. Method Function i) Function ii) Stationary GP 0.1251 ± 0.013 0.0230 ± 0.0047 Augmented nonstationary GP 0.0110 ± 0.0078 0.0212 ± 0.0067 Local Kernel Shaping 0.0092 ± 0.0068 0.0217 ± 0.0058 0 10 20 30 40 50 60 −150 −100 −50 0 50 100 Time (ms) Acceleration (g) Training Data AiMoGPE SingleGP (A) (a) Alternate infinite mix. of GPs 0 10 20 30 40 50 60 −150 −100 −50 0 50 100 Time (ms) Acceleration (g) Training data Aug GP Stationary GP (b) Augmented nonstationary GP 0 10 20 30 40 50 60 −150 −100 −50 0 50 100 Time (ms) Acceleration (g) Training data Kernel Shaping Stationary GP (c) Local Kernel Shaping Figure 4: Motorcycle impact data set from [20], with predicted results shown for our augmented GP and local kernel shaping algorithms. Results from the alternate infinite mixture of GP experts (AiMoGPE) are taken from [9]. analytical solution. This demonstrates that kernel shaping is an effective learning algorithm for use in robot control learning applications. Applying any arbitrary nonlinear regression method (such as a GP) to the inverse kinematics problem would, in fact, lead to unpredictably bad performance. The inverse kinematics problem is a one-tomany mapping and requires careful design of a learning problem to avoid problems with non-convex solution spaces [22]. Our suggested method of learning linearizations with a forward mapping (which is a proper function), followed by learning an inverse mapping within the local region of the forward mapping, is one of the few clean approaches to the problem. Instead of using locally linear methods, one could also use density-based estimation techniques like mixture models [23]. However, these methods must select the correct mode in order to arrive at a valid solution, and this final step may be computationally intensive or involve heuristics. For these reasons, applying a MCMC-type approach or GP-based method to the inverse kinematics problem was omitted as a comparison. 5 Discussion We presented a full Bayesian treatment of nonparametric local multi-dimensional kernel adaptation that simultaneously estimates the regression and kernel parameters. The algorithm can also be integrated into nonlinear algorithms, offering a valuable and flexible tool for learning. We show that our local kernel shaping method is particularly useful for learning control, demonstrating results on an inverse kinematics problem, and envision extensions to more complex problems with redundancy, (a) Robot arm −0.1 −0.05 0 0.05 0.1 −0.1 0 0.1 0.2 x (m) z (m) Desired Analytical IK −0.1 −0.05 0 0.05 0.1 −0.1 0 0.1 0.2 x (m) z (m) Desired Learnt IK (b) Desired versus actual trajectories Figure 5: Desired versus actual trajectories for SensAble Phantom robot arm e.g., learning inverse dynamics models of complete humanoid robots. Note that our algorithm requires only one prior be set by the user, i.e., the prior on the output noise. All other biases are initialized the same for all data sets and kept uninformative. In its current form, our Bayesian kernel shaping algorithm is built for high-dimensional inputs due to its low computational complexity— it scales linearly with the number of input dimensions. However, numerical problems may arise in case of redundant and irrelevant input dimensions. Future work will address this issue through the use of an automatic relevant determination feature. Other future extensions include an online implementation of the local kernel shaping algorithm. References [1] C. K. I. Williams and C. E. Rasmussen. Gaussian processes for regression. In David S. Touretzky, Michael C. Mozer, and Michael E. Hasselmo, editors, In Advances in Neural Information Processing Systems 8, volume 8. MIT Press, 1995. [2] J. H. Friedman. A variable span smoother. Technical report, Stanford University, 1984. [3] T. Poggio and F. Girosi. Regularization algorithms for learning that are equivalent to multilayer networks. Science, 247:213–225, 1990. [4] J. Fan and I. Gijbels. Local polynomial modeling and its applications. Chapman and Hall, 1996. [5] C. J. Paciorek and M. J. Schervish. Nonstationary covariance functions for Gaussian process regression. In Advances in Neural Information Processing Systems 16. MIT Press, 2004. [6] J. Fan and I. Gijbels. Data-driven bandwidth selection in local polynomial fitting: Variable bandwidth and spatial adaptation. Journal of the Royal Statistical Society B, 57:371–395, 1995. [7] S. Schaal and C.G. Atkeson. Assessing the quality of learned local models. In G. Tesauro J. Cowan and J. Alspector, editors, Advances in Neural Information Processing Systems, pages 160–167. Morgan Kaufmann, 1994. [8] C. E. Rasmussen and Z. Ghahramani. Infinite mixtures of Gaussian processes. In Advances in Neural Information Processing Systems 14. MIT Press, 2002. [9] E. Meeds and S. Osindero. An alternative infinite mixture of Gaussian process experts. In Advances in Neural Information Processing Systems 17. MIT Press, 2005. [10] C. Atkeson and S. Schaal. Robot learning from demonstration. In Proceedings of the 14th international conference on Machine learning, pages 12–20. Morgan Kaufmann, 1997. [11] C. Atkeson, A. Moore, and S. Schaal. Locally weighted learning. AI Review, 11:11–73, April 1997. [12] A. D’Souza, S. Vijayakumar, and S. Schaal. The Bayesian backfitting relevance vector machine. In Proceedings of the 21st International Conference on Machine Learning. ACM Press, 2004. [13] A. Gelman, J. Carlin, H.S. Stern, and D.B. Rubin. Bayesian Data Analysis. Chapman and Hall, 2000. [14] A. Dempster, N. Laird, and D. Rubin. Maximum likelihood from incomplete data via the EM algorithm. Journal of Royal Statistical Society. Series B, 39(1):1–38, 1977. [15] Z. Ghahramani and M.J. Beal. Graphical models and variational methods. In D. Saad and M. Opper, editors, Advanced Mean Field Methods - Theory and Practice. MIT Press, 2000. [16] T. S. Jaakkola and M. I. Jordan. Bayesian parameter estimation via variational methods. Statistics and Computing, 10:25–37, 2000. [17] E. Snelson and Z. Ghahramani. Sparse Gaussian processes using pseudo-inputs. In Advances in Neural Information Processing Systems 18. MIT Press, 2006. [18] V. Tresp. Mixtures of Gaussian processes. In Advances in Neural Information Processing Systems 13. MIT Press, 2000. [19] A. M. Schmidt and A. O’Hagan. Bayesian inference for nonstationary spatial covariance structure via spatial deformations. Journal of Royal Statistical Society. Series B, 65:745–758, 2003. [20] B. W. Silverman. Some aspects of the spline smoothing approach to non-parametric regression curve fitting. Journal of Royal Statistical Society. Series B, 47:1–52, 1985. [21] J. Peters and S. Schaal. Learning to control in operational space. International Journal of Robotics Research, 27:197–212, 2008. [22] M. I. Jordan and D. E. Rumelhart. Internal world models and supervised learning. In Machine Learning: Proceedings of Eighth Internatinoal Workshop, pages 70–85. Morgan Kaufmann, 1991. [23] Z. Ghahramani. Solving inverse problems using an EM approach to density estimation. In Proceedings of the 1993 Connectionist Models summer school, pages 316–323. Erlbaum Associates, 1994.
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An Extended Level Method for Efficient Multiple Kernel Learning Zenglin Xu† Rong Jin‡ † Dept. of Computer Science & Engineering The Chinese University of Hong Kong Shatin, N.T., Hong Kong {zlxu, king, lyu}@cse.cuhk.edu.hk Irwin King† Michael R. Lyu† ‡Dept. of Computer Science & Engineering Michigan State University East Lansing, MI, 48824 rongjin@cse.msu.edu Abstract We consider the problem of multiple kernel learning (MKL), which can be formulated as a convex-concave problem. In the past, two efficient methods, i.e., Semi-Infinite Linear Programming (SILP) and Subgradient Descent (SD), have been proposed for large-scale multiple kernel learning. Despite their success, both methods have their own shortcomings: (a) the SD method utilizes the gradient of only the current solution, and (b) the SILP method does not regularize the approximate solution obtained from the cutting plane model. In this work, we extend the level method, which was originally designed for optimizing non-smooth objective functions, to convex-concave optimization, and apply it to multiple kernel learning. The extended level method overcomes the drawbacks of SILP and SD by exploiting all the gradients computed in past iterations and by regularizing the solution via a projection to a level set. Empirical study with eight UCI datasets shows that the extended level method can significantly improve efficiency by saving on average 91.9% of computational time over the SILP method and 70.3% over the SD method. 1 Introduction Kernel learning [5, 9, 7] has received a lot of attention in recent studies of machine learning. This is due to the importance of kernel methods in that kernel functions define a generalized similarity measure among data. A generic approach to learning a kernel function is known as multiple kernel learning (MKL) [5]: given a list of base kernel functions/matrices, MKL searches for the linear combination of base kernel functions which maximizes a generalized performance measure. Previous studies [5, 14, 13, 4, 1] have shown that MKL is usually able to identify appropriate combination of kernel functions, and as a result to improve the performance. A variety of methods have been used to create base kernels. For instance, base kernels can be created by using different kernel functions; they can also be created by using a single kernel function but with different subsets of features. As for the performance measures needed to find the optimal kernel function, several measures have been studied for multiple kernel learning, including maximum margin classification errors [5], kernel-target alignment [4], and Fisher discriminative analysis [13]. The multiple kernel learning problem was first formulated as a semi-definite programming (SDP) problem by [5]. An SMO-like algorithm was proposed in [2] in order to solve medium-scale problems. More recently, a Semi-Infinite Linear Programming (SILP) approach was developed for MKL [12]. SILP is an iterative algorithm that alternates between the optimization of kernel weights and the optimization of the SVM classifier. In each step, given the current solution of kernel weights, it solves a classical SVM with the combined kernel; it then constructs a cutting plane model for the objective function and updates the kernel weights by solving a corresponding linear programming problem. Although the SILP approach can be employed for large scale MKL problems, it often suffers from slow convergence. One shortcoming of the SILP method is that it updates kernel weights solely based on the cutting plane model. Given that a cutting plane model usually differs significantly from the original objective function when the solution is far away from the points where the cutting plane model is constructed, the optimal solution to the cutting plane model could be significantly off target. In [10], the authors addressed the MKL problems by a simple Subgradient Descent (SD) method. However, since the SD method is memoryless, it does not utilize the gradients computed in previous iterations, which could be very useful in boosting the efficiency of the search. To further improve the computational efficiency of MKL, we extended the level method [6], which was originally designed for optimizing non-smooth functions, to the optimization of convex-concave problems. In particular, we regard the MKL problem as a saddle point problem. In the present work, similar to the SILP method, we construct in each iteration a cutting plane model for the target objective function using the solutions to the intermediate SVM problems. A new solution for kernel weights is obtained by solving the cutting plane model. We furthermore adjust the new solution via a projection to a level set. This adjustment is critical in that it ensures on one hand the new solution is sufficiently close to the current solution, and on the other hand the new solution significantly reduces the objective function. We show that the extended level method has a convergence rate of O(1/ε2) for a ε-accurate solution. Although this is similar to that of the SD method, the extended level method is advantageous in that it utilizes all the gradients that have been computed so far. Empirical results with eight UCI datasets show that the extended level method is able to greatly improve the efficiency of multiple kernel learning in comparison with the SILP method and the SD method. The rest of this paper is organized as follows. In section 2, we review the efficient algorithms that have been designed for multiple kernel learning. In section 3, we describe the details of the extended level method for MKL, including a study of its convergence rate. In section 4, we present experimental results by comparing both the effectiveness and the efficiency of the extended level method with the corresponding measures of SILP and SD. We conclude this work in section 5. 2 Related Work Let X = (x1, . . . , xn) ∈Rn×d denote the collection of n training samples that are in a ddimensional space. We further denote by y = (y1, y2, . . . , yn) ∈{−1, +1}n the binary class labels for the data points in X. We employ the maximum margin classification error, an objective used in SVM, as the generalized performance measure. Following [5], the problem of multiple kernel learning for classification in the primal form is defined as follows: min p∈P max α∈Q f(p, α) = α⊤e −1 2(α ◦y)⊤ m X i=1 piKi ! (α ◦y), (1) where P = {p ∈Rm : p⊤e = 1, 0 ≤p ≤1} and Q = {α ∈Rn : α⊤y = 0, 0 ≤α ≤C} are two solid convex regions, denoting the set of kernel weights and the set of SVM dual variables, respectively. Here, e is a vector of all ones, C is the trade-off parameter in SVM, {Ki}m i=1 is a group of base kernel matrices, and ◦defines the element-wise product between two vectors. It is easy to verify that f(p, α) is convex on p and concave on α. Thus the above optimization problem is indeed a convex-concave problem. It is important to note that the block-minimization formulation of MKL presented in [10, 2] is equivalent to (1). A straightforward approach toward solving the convex-concave problem in (1) is to transform it into a Semi-definite Programming (SDP) or a Quadratically Constrained Quadratic Programming (QCQP) [5, 2]. However, given their computational complexity, they cannot be applied to largescale MKL problems. Recently, Semi-infinite Linear Programming (SILP) [12] and Subgradient Descent (SD) [10] have been applied to handle large-scale MKL problems. We summarize them into a unified framework in Algorithm 1. Note that a superscript is used to indicate the index of iteration, a convention that is used throughout this paper. We use [x]t to denote x to the power of t in the case of ambiguity. As indicated in Algorithm 1, both methods divide the MKL problem into two cycles: the inner cycle solves a standard SVM problem to update α, and the outer cycle updates the kernel weight vector Algorithm 1 A general framework for solving MKL 1: Initialize p0 = e/m and i = 0 2: repeat 3: Solve the dual of SVM with kernel K = Pm j=1 pi jKj and obtain optimal solution αi 4: Update kernel weights by pi+1 = arg min{ϕi(p; α) : p ∈P} 5: Update i = i + 1 and calculate stopping criterion ∆i 6: until ∆i ≤ε p. They differ in the 4th step in Algorithm 1: the SILP method updates p by solving a cutting plane model, while the SD method updates p using the subgradient of the current solution. More specifically, ϕi(p; α) for SILP and SD are defined as follows: ϕi SILP (p; α) = min ν {ν : ν ≥f(p, αj), j = 0, . . . , i}, (2) ϕi SD(p; α) = 1 2∥p −pi∥2 2 + γi(p −pi)⊤∇pf(pi, αi), (3) where γi is the step size that needs to be decided dynamically (e.g., by a line search). ∇pf(pi, αi) = −1 2[(αi◦y)⊤K1(αi◦y), . . . , (αi◦y)⊤Km(αi◦y)]⊤denotes the subgradient of f(·, ·) with respect to p at (pi, αi). Comparing the two methods, we observe • In SILP, the cutting plane model ϕSILP (p) utilizes all the {αj}i j=1 obtained in past iterations. In contrast, SD only utilizes αi of the current solution pi. • SILP updates the solution for p based on the cutting plane model ϕSILP (p). Since the cutting plane model is usually inaccurate when p is far away from {pj}i j=1, the updated solution p could be significantly off target [3]. In contrast, a regularization term ∥p − pi∥2 2/2 is introduced in SD to prevent the new solution being far from the current one, pi. The proposed level method combines the strengths of both methods. Similar to SILP, it utilizes the gradient information of all the iterations; similar to SD, a regularization scheme is introduced to prevent the updated solution from being too far from the current solution. 3 Extended Level Method for MKL We first introduce the basic steps of the level method, followed by the extension of the level method to convex-concave problems and its application to MKL. 3.1 Introduction to the Level Method The level method [6] is from the family of bundle methods, which have recently been employed to efficiently solve regularized risk minimization problems [11]. It is an iterative approach designed for optimizing a non-smooth objective function. Let f(x) denote the convex objective function to be minimized over a convex domain G. In the ith iteration, the level method first constructs a lower bound for f(x) by a cutting plane model, denoted by gi(x). The optimal solution, denoted by ˆxi, that minimizes the cutting plane model gi(x) is then computed. An upper bound f i and a lower bound f i are computed for the optimal value of the target optimization problem based on ˆxi. Next, a level set for the cutting plane model gi(x) is constructed, denoted by Li = {x ∈G : gi(x) ≤λf i + (1 −λ)f i} where λ ∈(0, 1) is a tradeoff constant. Finally, a new solution xi+1 is computed by projecting xi onto the level set Li. It is important to note that the projection step, serving a similar purpose to the regularization term in SD, prevents the new solution xi+1 from being too far away from the old one xi. To demonstrate this point, consider a simple example minx{f(x) = [x]2 : x ∈[−4, 4]}. Assume x0 = −3 is the initial solution. The cutting plane model at x0 is g0(x) = 9 −6(x + 3). The optimal solution minimizing g0(x) is ˆx1 = 4. If we directly take ˆx1 as the new solution, as SILP does, we found it is significantly worse than x0 in terms of [x]2. The level method alleviates this problem by projecting x0 = −3 to the level set L0 = {x : g0(x) ≤0.9[x0]2 + 0.1g0(ˆx1), −4 ≤x ≤4} where λ = 0.9. It is easy to verify that the projection of x0 to L0 is x1 = −2.3, which significantly reduces the objective function f(x) compared with x0. 3.2 Extension of the Level Method to MKL We now extend the level method, which was originally designed for optimizing non-smooth functions, to convex-concave optimization. First, since f(p, α) is convex in p and concave in α, according to van Neuman Lemma, for any optimal solution (p∗, α∗) we have f(p, α∗) = max α∈Q f(p, α) ≥f(p∗, α∗) ≥f(p∗, α) = min p∈P f(p, α). (4) This observation motivates us to design an MKL algorithm which iteratively updates both the lower and the upper bounds for f(p, α) in order to find the saddle point. To apply the level method, we first construct the cutting plane model. Let {pj}i j=1 denote the solutions for p obtained in the last i iterations. Let αj = arg maxα∈Q f(pj, α) denote the optimal solution that maximizes f(pj, α). We construct a cutting plane model gi(p) as follows: gi(p) = max 1≤j≤i f(p, αj). (5) We have the following proposition for the cutting plane model gi(x) Proposition 1. For any p ∈P, we have (a) gi+1(p) ≥gi(p), and (b) gi(p) ≤maxα∈Q f(p, α). Next, we construct both the lower and the upper bounds for the optimal value f(p∗, α∗). We define two quantities f i and f i as follows: f i = min p∈P gi(p) and f i = min 1≤j≤i f(pj, αj). (6) The following theorem shows that {f j}i j=1 and {f j}i j=1 provide a series of increasingly tight bounds for f(p∗, α∗). Theorem 1. We have the following properties for {f j}i j=1 and {f j}i j=1: (a) f i ≤f(p∗, α∗) ≤f i, (b) f 1 ≥f 2 ≥. . . ≥f i, and (c) f 1 ≤f 2 ≤. . . ≤f i. Proof. First, since gi(p) ≤maxα∈Q f(p, α) for any p ∈P, we have f i = min p∈P gi(p) ≤min p∈P max α∈Q f(p, α). Second, since f(pj, αj) = max α∈Q f(pj, α), we have f i = min 1≤j≤i f(pj, αj) = min p∈{p1,...,pi} max α∈Q f(p, α) ≥min p∈P max α∈Q f(p, α) = f(p∗, α∗). Combining the above results, we have (a) in the theorem. It is easy to verify (b) and (c). We furthermore define the gap ∆i as ∆i = f i −f i. The following corollary indicates that the gap ∆i can be used to measure the sub-optimality for solution pi and αi. Corollary 2. (a) ∆j ≥0, j = 1, . . . , i, (b) ∆1 ≥∆2 ≥. . . ≥∆i, (c) |f(pj, αj)−f(p∗, α∗)| ≤∆i It is easy to verify these three properties of ∆i in the above corollary using the results of Theorem 1. In the third step, we construct the level set Li using the estimated bounds f i and f i as follows: Li = {p ∈P : gi(p) ≤ℓi = λf i + (1 −λ)f i}, (7) where λ ∈(0, 1) is a predefined constant. The new solution, denoted by pi+1, is computed as the projection of pi onto the level set Li, which is equivalent to solving the following optimization problem: pi+1 = arg min p ∥p −pi∥2 2 : p ∈P, f(p, αj) ≤ℓi, j = 1, . . . , i . (8) Although the projection is regarded as a quadratic programming problem, it can often be solved efficiently because its solution is likely to be the projection onto one of the hyperplanes of polyhedron Li. In other words, only very few linear constraints of L are active; most of them are inactive. This sparse nature usually leads to significant speedup of QP, similar to the solver of SVM. As we argue in the last subsection, by means of the projection, we on the one hand ensure pi+1 is not very far away from pi, and on the other hand ensure significant progress is made in terms of gi(p) when the solution is updated from pi to pi+1. Note that the projection step in the level method saves the effort of searching for the optimal step size in SD, which is computationally expensive as will be revealed later. We summarize the steps of the extended level method in Algorithm 2. Algorithm 2 The Level Method for Multiple Kernel Learning 1: Initialize p0 = e/m and i = 0 2: repeat 3: Solve the dual problem of SVM with K = Pm j=1 pi jKj to obtain the optimal solution αi 4: Construct the cutting plane model gi(p) in (5) 5: Calculate the lower bound f i and the upper bound f i in (6), and the gap ∆i in (3.2) 6: Compute the projection of pi onto the level set Li by solving the optimization problem in (8) 7: Update i = i + 1 8: until ∆i ≤ε Finally, we discuss the convergence behavior of the level method. In general, convergence is guaranteed because the gap ∆i, which bounds the absolute difference between f(p∗, α∗) and f(pi, αi), monotonically decreases through iterations. The following theorem shows the convergence rate of the level method when applied to multiple kernel learning. Theorem 3. To obtain a solution p that satisfies the stopping criterion, i.e., | maxα∈Q f(p, α) − f(p∗, α∗)| ≤ε, the maximum number of iterations N that the level method requires is bounded as follows N ≤2c(λ)L2 ε2 , where c(λ) = 1 (1−λ)2λ(2−λ) and L = 1 2 √mnC2 max 1≤i≤m Λmax(Ki). The operator Λmax(M) computes the maximum eigenvalue of matrix M. Due to space limitation, the proof of Theorem 3 can be found in the long version of this paper. Theorem 3 tells us that the convergence rate of the level method is O(1/ε2). It is important to note that according to Information Based Complexity (IBC) theory, given a function family F(L) with a fixed Lipschitz constant L, O(1/ε2) is almost the optimal convergence rate that can be achieved for any optimization method based on the black box first order oracle. In other words, no matter which optimization method is used, there always exists an function f(·) ∈F(L) such that the convergence rate is O(1/ε2) as long as the optimization method is based on a black box first order oracle. More details can be found in [8, 6]. 4 Experiments We conduct experiments to evaluate the efficiency of the proposed algorithm for MKL in constrast with SILP and SD, the two state-of-the-art algorithms for MKL. 4.1 Experimental Setup We follow the settings in [10] to construct the base kernel matrices, i.e., • Gaussian kernels with 10 different widths ({2−3, 2−2, . . . , 26}) on all features and on each single feature • Polynomial kernels of degree 1 to 3 on all features and on each single feature. Table 1: The performance comparison of three MKL algorithms. Here n and m denote the size of training samples and the number of kernels, respectively. SD SILP Level SD SILP Level Iono n = 175 m = 442 Breast n = 342 m = 117 Time(s) 33.5 ±11.6 1161.0 ±344.2 7.1 ±4.3 47.4 ±8.9 54.2 ±9.4 4.6 ±1.0 Accuracy (%) 92.1 ±2.0 92.0 ±1.9 92.1±1.9 96.6 ±0.9 96.6 ±0.8 96.6±0.8 #Kernel 26.9 ±4.0 24.4 ±3.4 25.4±3.9 13.1 ±1.7 10.6 ±1.1 13.3±1.5 Pima n = 384 m = 117 Sonar n = 104 m = 793 Time(s) 39.4 ±8.8 62.0 ±15.2 9.1 ±1.6 60.1 ±29.6 1964.3±68.4 24.9±10.6 Accuracy (%) 76.9 ±1.9 76.9 ±2.1 76.9±2.1 79.1 ±4.5 79.3 ±4.2 79.0±4.7 #Kernel 16.6 ±2.2 12.0 ±1.8 17.6±2.6 39.8 ±3.9 34.2 ±2.6 38.6±4.1 Wpbc n = 198 m = 442 Heart n = 135 m = 182 Time(s) 7.8 ±2.4 142.0 ±122.3 5.3 ±1.3 4.7 ±2.8 79.2 ±38.1 2.1 ±0.4 Accuracy (%) 77.0 ±2.9 76.9 ±2.8 76.9±2.9 82.2 ±2.2 82.2 ±2.0 82.2±2.1 #Kernel 19.5 ±2.8 17.2 ±2.2 20.3±2.6 17.5 ±1.8 15.2 ±1.5 18.6±1.9 Vote n = 218 m = 205 Wdbc n = 285 m = 403 Time(s) 23.7 ±9.7 26.3 ±12.4 4.1 ±1.3 122.9±38.2 146.3 ±48.3 15.5±7.5 Accuracy (%) 95.7 ±1.0 95.7 ±1.0 95.7±1.0 96.7 ±0.8 96.5 ±0.9 96.7±0.8 #Kernel 14.0 ±3.6 10.6 ±2.6 13.8±2.6 16.6 ±3.2 12.9 ±2.3 15.6±3.0 Each base kernel matrix is normalized to unit trace. The experiments are conducted on a PC with 3.2GHz CPU and 2GB memory. According to the above scheme of constructing base kernel matrices, we select a batch of UCI data sets, with the cardinality and dimension allowed by the memory limit of the PC, from the UCI repository for evaluation. We repeat all the algorithms 20 times for each data set. In each run, 50% of the examples are randomly selected as the training data and the remaining data are used for testing. The training data are normalized to have zero mean and unit variance, and the test data are then normalized using the mean and variance of the training data. The regularization parameter C in SVM is set to 100 as our focus is to evaluate the computational time, as justified in [10]. For a fair comparison among the MKL algorithms, we adopt the same stopping criterion for all three algorithms under comparison: we adopt the duality gap criterion used in [10], i.e., max 1≤i≤m(α◦y)⊤Ki(α◦y)−(α◦y)⊤Pm j=1 pjKj (α◦y), and stop the algorithm when the criterion is less than 0.01 or the number of iterations larger than 500. We empirically initialize the parameter λ to 0.9 and increase it to 0.99 when the ratio ∆i/ℓi is less than 0.01 for all experiments, since a larger λ accelerates the projection when the solution is close to the optimal one. We use the SimpleMKL toolbox [10] to implement the SILP and SD methods. The linear programming in the SILP method and the auxiliary subproblems in the level method are solved using a general optimization toolbox MOSEK (http://www.mosek.com). The toolbox for the level method can be downloaded from http://www.cse.cuhk.edu.hk/˜zlxu/toolbox/level_mkl.html. 4.2 Experimental Results We report the following performance measures: prediction accuracy, training time, and the averaged number of kernels selected. From Table 1, we observe that all algorithms achieve almost the same prediction accuracy under the same stopping criterion. This is not surprising because all algorithms are essentially trying to solve the same optimization problem. Regarding the computational efficiency, we observe that the time cost of the SILP approach is the highest among all the three MKL algorithms. For datasets “Iono” and “Sonar”, the SILP method consumes more than 30 times the computational cycles of the other two methods for MKL. We also observe that the level method is the most efficient among three methods in comparison. To obtain a better picture of the computational efficiency of the proposed level method, we compute the time-saving ratio, as shown in Table 2. We observe that the level method saves 91.9% of computational time on average when compared with the SILP method, and 70.3% of computational time when compared with the SD method. In order to see more details of each optimization algorithm, we plot the logarithm values of the MKL objective function to base 10 against time in Figure 1. Due to space limitation, we randomly choose only three datasets, “Iono”, “Breast”, and “Pima”, as examples. It is interesting to find that the level method converges overwhelmingly faster than the other two methods. The efficiency of the level method arises from two aspects: (a) the cutting plane model utilizes the computational results of all iterations and therefore boosts the search efficiency, and (b) the projection to the level sets ensures the stability of the new solution. A detailed analysis of the SD method reveals that a large number of function evaluations are consumed in order to compute the optimal stepsize via a line search. Note that in convex-concave optimization, every function evaluation in the line search of SD requires solving an SVM problem. As an example, we found that for dataset “Iono”, although SD and the level method require similar numbers of iterations, SD calls the SVM solver 1231 times on average, while the level method only calls it 47 times. For the SILP method, the high computational cost is mainly due to the oscillation of solutions. This instability leads to very slow convergence when the solution is close to the optimal one, as indicated by the long tail of SILP in Figure 1. The instability of SILP is further confirmed by the examination of kernel weights, as shown below. To understand the evolution of kernel weights (i.e., p), we plot the evolution curves of the five largest kernel weights for datasets “Iono”, “Breast”, and “Pima” in Figure 2. We observe that the values of p computed by the SILP method are the most unstable due to oscillation of the solutions to the cutting plane models. Although the unstable-solution problem is to some degree improved by the SD method, we still clearly observe that p fluctuates significantly through iterations. In contrast, for the proposed level method, the values of p change smoothly through iterations. We believe that the stability of the level method is mainly due to the accurate estimation of bounds as well as the regularization of the projection to the level sets. This observation also sheds light on why the level method can be more efficient than the SILP and the SD methods. Table 2: Time-saving ratio of the level method over the SILP and the SD method Iono Breast Pima Sonar Wpbc Heart Vote Wdbc Average Level/SD (%) 78.9 90.4 77.0 58.7 32.5 54.7 82.8 87.4 70.3 Level/SILP (%) 99.4 91.6 85.4 98.7 88.7 97.3 84.5 89.4 91.9 0 20 40 60 80 100 3.4 3.45 3.5 3.55 3.6 3.65 3.7 3.75 3.8 log of objective time (s) Evolution of objective values with time SD SILP Level 0 10 20 30 40 50 60 3.5 3.55 3.6 3.65 3.7 3.75 log of objective time (s) Evolution of objective values with time SD SILP Level 0 10 20 30 40 50 60 70 4.2 4.22 4.24 4.26 4.28 4.3 4.32 4.34 4.36 4.38 4.4 log of objective time (s) Evolution of objective values with time SD SILP Level (a) Iono (b) Breast (c) Pima Figure 1: Evolution of objective values over time (seconds) for datasets “Iono”, “Breast”, and “Pima”. The objective values are plotted on a logarithm scale (base 10) for better comparison. Only parts of the evolution curves are plotted for SILP due to their long tails. 5 Conclusion and Future Work In this paper, we propose an extended level method to efficiently solve the multiple kernel learning problem. In particular, the level method overcomes the drawbacks of both the SILP method and the SD method for MKL. Unlike the SD method that only utilizes the gradient information of the current solution, the level method utilizes the gradients of all the solutions that are obtained in past iterations; meanwhile, unlike the SILP method that updates the solution only based on the cutting plane model, the level method introduces a projection step to regularize the updated solution. It is the employment of the projection step that guarantees finding an updated solution that, on the one hand, is close to the existing one, and one the other hand, significantly reduces the objective function. Our experimental results have shown that the level method is able to greatly reduce the computational time of MKL over both the SD method and the SILP method. For future work, we plan to find a scheme to adaptively set the value of λ in the level method and apply the level method to other tasks, such as one-class classification, multi-class classification, and regression. Acknowledgement The work was supported by the National Science Foundation (IIS-0643494), National Institute of Health (1R01GM079688-01) and Research Grants Council of Hong Kong (CUHK4150/07E and CUHK4125/07). References [1] F. R. Bach. Consistency of the group Lasso and multiple kernel learning. Journal of Machine Learning Research, 9:1179–1225, 2008. 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 iteration p values Evolution of the kernel weight values in SD 0 100 200 300 400 500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 iteration p values Evolution of the kernel weight values in SILP 0 5 10 15 20 25 30 35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 iteration p values Evolution of the kernel weight values in Level method (a) Iono/SD (b) Iono/SILP (c) Iono/Level 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 iteration p values Evolution of the kernel weight values in SD 0 20 40 60 80 100 120 140 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 iteration p values Evolution of the kernel weight values in SILP 0 5 10 15 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 iteration p values Evolution of the kernel weight values in Level method (d) Breast/SD (e) Breast/SILP (f) Breast/Level 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 iteration p values Evolution of the kernel weight values in SD 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 iteration p values Evolution of the kernel weight values in SILP 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 iteration p values Evolution of the kernel weight values in Level method (g) Pima/SD (h) Pima/SILP (i) Pima/Level Figure 2: The evolution curves of the five largest kernel weights for datasets “Iono”, “Breast” and “Pima” computed by the three MKL algorithms [2] F. R. Bach, G. R. G. Lanckriet, and M. I. Jordan. Multiple kernel learning, conic duality, and the SMO algorithm. In ICML, 2004. [3] J. Bonnans, J. Gilbert, C. Lemar´echal, and C. Sagastiz´abal. Numerical Optimization, Theoretical and Practical Aspects. Springer-Verlag, Berlin, 2nd ed., 2006. [4] N. Cristianini, J. Shawe-Taylor, A. Elisseeff, and J. S. Kandola. On kernel-target alignment. In NIPS 13, pages 367–373, 2001. [5] G. R. G. Lanckriet, N. Cristianini, P. Bartlett, L. E. Ghaoui, and M. I. Jordan. Learning the kernel matrix with semidefinite programming. Journal of Machine Learning Research, 5, 2004. [6] C. Lemar´echal, A. Nemirovski, and Y. Nesterov. New variants of bundle methods. Mathematical Programming, 69(1), 1995. [7] C. A. Micchelli and M. Pontil. Learning the kernel function via regularization. Journal of Machine Learning Research, 6, 2005. [8] A. Nemirovski and D. Yudin. Problem Complexity and Method Efficiency in Optimization. John Wiley and Sons Ltd, 1983. [9] C. S. Ong, A. J. Smola, and R. C. Williamson. Learning the kernel with hyperkernels. Journal of Machine Learning Research, 6, 2005. [10] A. Rakotomamonjy, F. R. Bach, S. Canu, and Y. Grandvalet. SimpleMKL. Technical Report HAL00218338, INRIA, 2008. [11] A. Smola, S. V. N. Vishwanathan, and Q. Le. Bundle methods for machine learning. In NIPS 20, pages 1377–1384, 2007. [12] S. Sonnenburg, G. R¨atsch, C. Sch¨afer, and B. Sch¨olkopf. Large scale multiple kernel learning. Journal of Machine Learning Research, 7, 2006. [13] J. Ye, J. Chen, and S. Ji. Discriminant kernel and regularization parameter learning via semidefinite programming. In ICML, 2007. [14] A. Zien and C. S. Ong. Multiclass multiple kernel learning. In ICML, 2007.
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Sparse Online Learning via Truncated Gradient John Langford Yahoo! Research jl@yahoo-inc.com Lihong Li Department of Computer Science Rutgers University lihong@cs.rutgers.edu Tong Zhang Department of Statistics Rutgers University tongz@rci.rutgers.edu Abstract We propose a general method called truncated gradient to induce sparsity in the weights of online-learning algorithms with convex loss. This method has several essential properties. First, the degree of sparsity is continuous—a parameter controls the rate of sparsification from no sparsification to total sparsification. Second, the approach is theoretically motivated, and an instance of it can be regarded as an online counterpart of the popular L1-regularization method in the batch setting. We prove small rates of sparsification result in only small additional regret with respect to typical online-learning guarantees. Finally, the approach works well empirically. We apply it to several datasets and find for datasets with large numbers of features, substantial sparsity is discoverable. 1 Introduction We are concerned with machine learning over large datasets. As an example, the largest dataset we use in this paper has over 107 sparse examples and 109 features using about 1011 bytes. In this setting, many common approaches fail, simply because they cannot load the dataset into memory or they are not sufficiently efficient. There are roughly two approaches which can work: one is to parallelize a batch learning algorithm over many machines (e.g., [3]), the other is to stream the examples to an online-learning algorithm (e.g., [2, 6]). This paper focuses on the second approach. Typical online-learning algorithms have at least one weight for every feature, which is too expensive in some applications for a couple reasons. The first is space constraints: if the state of the onlinelearning algorithm overflows RAM it can not efficiently run. A similar problem occurs if the state overflows the L2 cache. The second is test-time constraints: reducing the number of features can significantly reduce the computational time to evaluate a new sample. This paper addresses the problem of inducing sparsity in learned weights while using an onlinelearning algorithm. Natural solutions do not work for our problem. For example, either simply adding L1 regularization to the gradient of an online weight update or simply rounding small weights to zero are problematic. However, these two ideas are closely related to the algorithm we propose and more detailed discussions are found in section 3. A third solution is black-box wrapper approaches which eliminate features and test the impact of the elimination. These approaches typically run an algorithm many times which is particularly undesirable with large datasets. Similar problems have been considered in various settings before. The Lasso algorithm [12] is commonly used to achieve L1 regularization for linear regression. This algorithm does not work automatically in an online fashion. There are two formulations of L1 regularization. Consider a loss function L(w, zi) which is convex in w, where zi = (xi, yi) is an input–output pair. One is the convex-constraint formulation ˆw = arg min w n X i=1 L(w, zi) subject to ∥w∥1 ≤s, (1) where s is a tunable parameter. The other is soft-regularization with a tunable parameter g: ˆw = arg min w n X i=1 L(w, zi) + g∥w∥1. (2) With appropriately chosen g, the two formulations are equivalent. The convex-constraint formulation has a simple online version using the projection idea in [14]. It requires the projection of weight w into an L1 ball at every online step. This operation is difficult to implement efficiently for large-scale data with many features even if all features are sparse, although important progress was made recently so the complexity is logarithmic in the number of features [5]. In contrast, the soft-regularization formulation (2) is efficient for a batch setting[8] so we pursue it here in an online setting where it has complexity independent of the number of features. In addition to the L1 regularization formulation (2), the family of online-learning algorithms we consider also includes some non-convex sparsification techniques. The Forgetron [4] is an online-learning algorithm that manages memory use. It operates by decaying the weights on previous examples and then rounding these weights to zero when they become small. The Forgetron is stated for kernelized online algorithms, while we are concerned with the simple linear setting. When applied to a linear kernel, the Forgetron is not computationally or space competitive with approaches operating directly on feature weights. At a high level, our approach is weight decay to a default value. This simple method enjoys strong performance guarantees (section 3). For instance, the algorithm never performs much worse than a standard online-learning algorithm, and the additional loss due to sparsification is controlled continuously by a single real-valued parameter. The theory gives a family of algorithms with convex loss functions for inducing sparsity—one per online-learning algorithm. We instantiate this for square loss in section 4 and show how this algorithm can be implemented efficiently in large-scale problems with sparse features. For such problems, truncated gradient enjoys the following properties: (i) It is computationally efficient: the number of operations per online step is linear in the number of nonzero features, and independent of the total number of features; (ii) It is memory efficient: it maintains a list of active features, and can insert (when the corresponding weight becomes nonzero) and delete (when the corresponding weight becomes zero) features dynamically. Theoretical results stating how much sparsity is achieved using this method generally require additional assumptions which may or may not be met in practice. Consequently, we rely on experiments in section 5 to show truncated gradient achieves good sparsity in practice. We compare truncated gradient to a few others on small datasets, including the Lasso, online rounding of coefficients to zero, and L1-regularized subgradient descent. Details of these algorithms are given in section 3. 2 Online Learning with Stochastic Gradient Descent We are interested in the standard sequential prediction problems where for i = 1, 2, . . .: 1. An unlabeled example xi arrives. 2. We make a prediction ˆyi based on the current weights wi = [w1 i , . . . , wd i ] ∈Rd. 3. We observe yi, let zi = (xi, yi), and incur some known loss L(wi, zi) convex in wi. 4. We update weights according to some rule: wi+1 ←f(wi). We want an update rule f allows us to bound the sum of losses, Pt i=1 L(wi, zi), as well as achieving sparsity. For this purpose, we start with the standard stochastic gradient descent (SGD) rule, which is of the form: f(wi) = wi −η∇1L(wi, zi), (3) where ∇1L(a, b) is a subgradient of L(a, b) with respect to the first variable a. The parameter η > 0 is often referred to as the learning rate. In the analysis, we only consider constant learning rate for simplicity. In theory, it might be desirable to have a decaying learning rate ηi which becomes smaller when i increases to get the so-called no-regret bound without knowing T in advance. However, if T is known in advance, one can select a constant η accordingly so the regret vanishes as T →∞. Since the focus of the present paper is on weight sparsity, rather than choosing the learning rate, we use a constant learning rate in the analysis because it leads to simpler bounds. The above method has been widely used in online learning (e.g., [2, 6]). Moreover, it is argued to be efficient even for solving batch problems where we repeatedly run the online algorithm over training data multiple times. For example, the idea has been successfully applied to solve large-scale standard SVM formulations [10, 13]. In the scenario outlined in the introduction, online-learning methods are more suitable than some traditional batch learning methods. However, the learning rule (3) itself does not achieve sparsity in the weights, which we address in this paper. Note that variants of SGD exist in the literature, such as exponentiated gradient descent (EG) [6]. Since our focus is sparsity, not SGD vs. EG, we shall only consider modifications of (3) for simplicity. 3 Sparse Online Learning In this section, we first examine three methods for achieving sparsity in online learning, including a novel algorithm called truncated gradient. As we shall see, all these ideas are closely related. Then, we provide theoretical justifications for this algorithm, including a general regret bound and a fundamental connection to the Lasso. 3.1 Simple Coefficient Rounding In order to achieve sparsity, the most natural method is to round small coefficients (whose magnitudes are below a threshold θ > 0) to zero after every K online steps. That is, if i/K is not an integer, we use the standard SGD rule (3); if i/K is an integer, we modify the rule as: f(wi) = T0(wi −η∇1L(wi, zi), θ), (4) where: θ ≥0 is a threshold, T0(v, θ) = [T0(v1, θ), . . . , T0(vd, θ)] for vector v = [v1, . . . , vd] ∈Rd, T0(vj, θ) = vjI(|vj| < θ), and I(·) is the set-indicator function. In other words, we first perform a standard stochastic gradient descent, and then round the coefficients to zero. The effect is to remove nonzero and small weights. In general, we should not take K = 1, especially when η is small, since in each step wi is modified by only a small amount. If a coefficient is zero, it remains small after one online update, and the rounding operation pulls it back to zero. Consequently, rounding can be done only after every K steps (with a reasonably large K); in this case, nonzero coefficients have sufficient time to go above the threshold θ. However, if K is too large, then in the training stage, we must keep many more nonzero features in the intermediate steps before they are rounded to zero. In the extreme case, we may simply round the coefficients in the end, which does not solve the storage problem in the training phase at all. The sensitivity in choosing appropriate K is a main drawback of this method; another drawback is the lack of theoretical guarantee for its online performance. These issues motivate us to consider more principled solutions. 3.2 L1-Regularized Subgradient In the experiments, we also combined rounding-in-the-end-of-training with a simple online subgradient method for L1 regularization with a regularization parameter g > 0: f(wi) = wi −η∇1L(wi, zi) −ηg sgn(wi), (5) where for a vector v = [v1, . . . , vd], sgn(v) = [sgn(v1), . . . , sgn(vd)], and sgn(vj) = 1 if vj > 0, sgn(vj) = −1 if vj < 0, and sgn(vj) = 0 if vj = 0. In the experiments, the online method (5) with rounding in the end is used as a simple baseline. Notice this method does not produce sparse weights online simply because only in very rare cases do two floats add up to 0. Therefore, it is not feasible in large-scale problems for which we cannot keep all features in memory. 3.3 Truncated Gradient In order to obtain an online version of the simple rounding rule in (4), we observe that direct rounding to zero is too aggressive. A less aggressive version is to shrink the coefficient to zero by a smaller amount. We call this idea truncated gradient, where the amount of shrinkage is controlled by a gravity parameter gi > 0: f(wi) = T1(wi −η∇1L(wi, zi), ηgi, θ), (6) where for a vector v = [v1, . . . , vd] ∈ Rd, and a scalar g ≥ 0, T1(v, α, θ) = [T1(v1, α, θ), . . . , T1(vd, α, θ)], with T1(vj, α, θ) = max(0, vj −α) if vj ∈[0, θ] min(0, vj + α) if vj ∈[−θ, 0] vj otherwise . Again, the truncation can be performed every K online steps. That is, if i/K is not an integer, we let gi = 0; if i/K is an integer, we let gi = Kg for a gravity parameter g > 0. The reason for doing so (instead of a constant g) is that we can perform a more aggressive truncation with gravity parameter Kg after each K steps. This can potentially lead to better sparsity. We also note that when ηKg ≥θ, truncate gradient coincides with (4). But in practice, as is also verified by the theory, one should adopt a small g; hence, the new learning rule (6) is expected to differ from (4). In general, the larger the parameters g and θ are, the more sparsity is expected. Due to the extra truncation T1, this method can lead to sparse solutions, which is confirmed empirically in section 5. A special case, which we use in the experiment, is to let g = θ in (6). In this case, we can use only one parameter g to control sparsity. Since ηKg ≪θ when ηK is small, the truncation operation is less aggressive than the rounding in (4). At first sight, the procedure appears to be an ad-hoc way to fix (4). However, we can establish a regret bound (in the next subsection) for this method, showing it is theoretically sound. Therefore, it can be regarded as a principled variant of rounding. Another important special case of (6) is setting θ = ∞, in which all weight components shrink in every online step. The method is a modification of the L1-regularized subgradient descent rule (5). The parameter gi ≥0 controls the sparsity achieved with the algorithm, and setting gi = 0 gives exactly the standard SGD rule (3). As we show in section 3.5, this special case of truncated gradient can be regarded as an online counterpart of L1 regularization since it approximately solves an L1 regularization problem in the limit of η →0. We also show the prediction performance of truncated gradient, measured by total loss, is comparable to standard stochastic gradient descent while introducing sparse weight vectors. 3.4 Regret Analysis Throughout the paper, we use ∥· ∥1 for 1-norm, and ∥· ∥for 2-norm. For reference, we make the following assumption regarding the loss function: Assumption 3.1 We assume L(w, z) is convex in w, and there exist non-negative constants A and B such that (∇1L(w, z))2 ≤AL(w, z) + B for all w ∈Rd and z ∈Rd+1. For linear prediction problems, we have a general loss function of the form L(w, z) = φ(wT x, y). The following are some common loss functions φ(·, ·) with corresponding choices of parameters A and B (which are not unique), under the assumption supx ∥x∥≤C. All of them can be used for binary classification where y ∈±1, but the last one is more often used in regression where y ∈R: Logistic: φ(p, y) = ln(1 + exp(−py)), with A = 0 and B = C2; SVM (hinge loss): φ(p, y) = max(0, 1 −py), with A = 0 and B = C2; Least squares (square loss): φ(p, y) = (p −y)2, with A = 4C2 and B = 0. The main result is Theorem 3.1 which is parameterized by A and B. The proof will be provided in a longer paper. Theorem 3.1 (Sparse Online Regret) Consider sparse online update rule (6) with w1 = [0, . . . , 0] and η > 0. If Assumption 3.1 holds, then for all ¯w ∈Rd we have 1 −0.5Aη T T X i=1 L(wi, zi) + gi 1 −0.5Aη ∥wi+1 · I(wi+1 ≤θ)∥1 ≤ η 2B + ∥¯w∥2 2ηT + 1 T T X i=1 [L( ¯w, zi) + gi∥¯w · I(wi+1 ≤θ)∥1], where ∥v·I(|v′| ≤θ)∥1 = Pd j=1 |vj|I(|v′ j| ≤θ) for vectors v = [v1, . . . , vd] and v′ = [v′ 1, . . . , v′ d]. The theorem is stated with a constant learning rate η. As mentioned earlier, it is possible to obtain a result with variable learning rate where η = ηi decays as i increases. Although this may lead to a no-regret bound without knowing T in advance, it introduces extra complexity to the presentation of the main idea. Since the focus is on sparsity rather than optimizing learning rate, we do not include such a result for clarity. If T is known in advance, then in the above bound, one can simply take η = O(1/ √ T) and the regret is of order O(1/ √ T). In the above theorem, the right-hand side involves a term gi∥¯w · I(wi+1 ≤θ)∥1 that depends on wi+1 which is not easily estimated. To remove this dependency, a trivial upper bound of θ = ∞ can be used, leading to L1 penalty gi∥¯w∥1. In the general case of θ < ∞, we cannot remove the wi+1 dependency because the effective regularization condition (as shown on the left-hand side) is the non-convex penalty gi∥w · I(|w| ≤θ)∥1. Solving such a non-convex formulation is hard both in the online and batch settings. In general, we only know how to efficiently discover a local minimum which is difficult to characterize. Without a good characterization of the local minimum, it is not possible for us to replace gi∥¯w · I(wi+1 ≤θ)∥1 on the right-hand side by gi∥¯w · I( ¯w ≤ θ)∥1 because such a formulation implies we could efficiently solve a non-convex problem with a simple online update rule. Still, when θ < ∞, one naturally expects the right-hand side penalty gi∥¯w · I(wi+1 ≤θ)∥1 is much smaller than the corresponding L1 penalty gi∥¯w∥1, especially when wj has many components close to 0. Therefore the situation with θ < ∞can potentially yield better performance on some data. Theorem 3.1 also implies a tradeoff between sparsity and regret performance. We may simply consider the case where gi = g is a constant. When g is small, we have less sparsity but the regret term g∥¯w · I(wi+1 ≤θ)∥1 ≤g∥¯w∥1 on the right-hand side is also small. When g is large, we are able to achieve more sparsity but the regret g∥¯w·I(wi+1 ≤θ)∥1 on the right-hand side also becomes large. Such a tradeoff between sparsity and prediction accuracy is empirically studied in section 5, where we achieve significant sparsity with only a small g (and thus small decrease of performance). Now consider the case θ = ∞and gi = g. When T →∞, if we let η →0 and ηT →∞, then 1 T T X i=1 [L(wi, zi) + g∥wi∥1] ≤inf ¯ w∈Rd " 1 T T X i=1 L( ¯w, zi) + 2g∥¯w∥1 # + o(1). follows from Theorem 3.1. In other words, if we let L′(w, z) = L(w, z) + g∥w∥1 be the L1regularized loss, then the L1-regularized regret is small when η →0 and T →∞. This implies truncated gradient can be regarded as the online counterpart of L1-regularization methods. In the stochastic setting where the examples are drawn iid from some underlying distribution, the sparse online gradient method proposed in this paper solves the L1 regularization problem. 3.5 Stochastic Setting SGD-based online-learning methods can be used to solve large-scale batch optimization problems. In this setting, we can go through training examples one-by-one in an online fashion, and repeat multiple times over the training data. To simplify the analysis, instead of assuming we go through example one by one, we assume each additional example is drawn from the training data randomly with equal probability. This corresponds to the standard stochastic optimization setting, in which observed samples are iid from some underlying distributions. The following result is a simple consequence of Theorem 3.1. For simplicity, we only consider the case with θ = ∞and constant gravity gi = g. The expectation E is taken over sequences of indices i1, . . . , iT . Theorem 3.2 (Stochastic Setting) Consider a set of training data zi = (xi, yi) for 1 ≤i ≤n. Let R(w, g) = 1 n n X i=1 L(w, zi) + g∥w∥1 be the L1-regularized loss over training data. Let ˆw1 = w1 = 0, and define recursively for t ≥1: wt+1 = T(wt −η∇1(wt, zit), gη), ˆwt+1 = ˆwt + (wt+1 −ˆwt)/(t + 1), where each it is drawn from {1, . . . , n} uniformly at random. If Assumption 3.1 holds, then for all T and ¯w ∈Rd: E » (1 −0.5Aη)R( ˆwT , g 1 −0.5Aη ) – ≤E " 1 −0.5Aη T T X i=1 R(wi, g 1 −0.5Aη ) # ≤η 2 B+ ∥¯w∥2 2ηT +R( ¯w, g). Observe that if we let η →0 and ηT →∞, the bound in Theorem 3.2 becomes E [R( ˆwT , g)] ≤ E h 1 T PT t=1 R(wt, g) i ≤inf ¯ w R( ¯w, g)+o(1). In other words, on average ˆwT approximately solves the batch L1-regularization problem infw 1 n Pn i=1 L(w, zi) + g∥w∥1 when T is large. If we choose a random stopping time T, then the above inequalities say that on average wT also solves this L1-regularization problem approximately. Thus, we use the last solution wT instead of the aggregated solution ˆwT in experiments. Since L1 regularization is often used to achieve sparsity in the batch learning setting, the connection of truncated gradient to L1 regularization can be regarded as an alternative justification for the sparsity ability of this algorithm. 4 Efficient Implementation of Truncated Gradient for Square Loss The truncated descent update rule (6) can be applied to least-squares regression using square loss, leading to f(wi) = T1(wi + 2η(yi −ˆyi)xi, ηgi, θ), where the prediction is given by ˆyi = P j wj i xj i. We altered an efficient SGD implementation, Vowpal Wabbit [7], for least-squares regression according to truncated gradient. The program operates in an entirely online fashion. Features are hashed instead of being stored explicitly, and weights can be easily inserted into or deleted from the table dynamically. So the memory footprint is essentially just the number of nonzero weights, even when the total numbers of data and features are astronomically large. In many online-learning situations such as web applications, only a small subset of the features have nonzero values for any example x. It is thus desirable to deal with sparsity only in this small subset rather than in all features, while simultaneously inducing sparsity on all feature weights. The approach we take is to store a time-stamp τj for each feature j. The time-stamp is initialized to the index of the example where feature j becomes nonzero for the first time. During online learning, at each step i, we only go through the nonzero features j of example i, and calculate the un-performed shrinkage of wj between τj and the current time i. These weights are then updated, and their time stamps are reset to i. This lazy-update idea of delaying the shrinkage calculation until needed is the key to efficient implementation of truncated gradient. The implementation satisfies efficiency requirements outlined at the end of the introduction section. A similar time-stamp trick can be applied to the other two algorithms given in section 3. 5 Empirical Results We applied the algorithm, with the efficiently implemented sparsify option, as described in the previous section, to a selection of datasets, including eleven datasets from the UCI repository [1], the much larger dataset rcv1 [9], and a private large-scale dataset Big_Ads related to ad interest prediction. While UCI datasets are useful for benchmark purposes, rcv1 and Big_Ads are more interesting since they embody real-world datasets with large numbers of features, many of which are less informative for making predictions than others. The UCI datasets we used do not have many features, and it seems a large fraction of these features are useful for making predictions. For comparison purposes and to better demonstrate the behavior of our algorithm, we also added 1000 random binary features to those datasets. Each feature has value 1 with prob. 0.05 and 0 otherwise. In the first set of experiments, we are interested in how much reduction in the number of features is possible without affecting learning performance significantly; specifically, we require the accuracy be reduced by no more than 1% for classification tasks, and the total square loss be increased by no more than 1% for regression tasks. As common practice, we allowed the algorithm to run on the training data set for multiple passes with decaying learning rate. For each dataset, we performed 10fold cross validation over the training set to identify the best set of parameters, including the learning rate η, the gravity g, number of passes of the training set, and the decay of learning rate across these passes. This set of parameters was then used on the whole training set. Finally, the learned classifier/regressor was evaluated on the test set. We fixed K = 1 and θ = ∞in this set of experiments. The effects of K and θ are included in an extended version of this paper. Figure 1 shows the fraction of reduced features after sparsification is applied to each dataset. For UCI datasets with randomly added features, truncated gradient was able to reduce the number of features by a fraction of more than 90%, except for the ad dataset in which only 71% reduction was observed. This less satisfying result might be improved by a more extensive parameter search in cross validation. However, if 0 0.2 0.4 0.6 0.8 1 Big_Ads rcv1 zoo wpbc wdbc wbc spam shroom magic04 krvskp housing crx ad Fraction Left Dataset Fraction of Features Left Base data 1000 extra 0 0.2 0.4 0.6 0.8 1 1.2 rcv1 wpbc wdbc wbc spam shroom magic04 krvskp crx ad Ratio Dataset Ratio of AUC Base data 1000 extra Figure 1: Left: the amount of features left after sparsification for each dataset without 1% performance loss. Right: the ratio of AUC with and without sparsification. we tolerated 1.3% decrease in accuracy (instead of 1%) during cross validation, truncated gradient was able to achieve 91.4% reduction, indicating a large reduction is still possible at the tiny additional accuracy loss of 0.3%. Even for the original UCI datasets without artificially added features, some of the less useful features were removed while the same level of performance was maintained. For classification tasks, we also studied how truncated gradient affects AUC (Area Under the ROC Curve), a standard metric for classification. We use AUC here because it is insensitive to threshold, unlike accuracy. Using the same sets of parameters from 10-fold cross validation described above, we found the criterion was not affected significantly by sparsification and in some cases, it was actually improved, due to removal of some irrelevant features. The ratios of the AUC with and without sparsification for all classification tasks are plotted in Figure 1. Often these ratios are above 98%. The previous results do not exercise the full power of the approach presented here because they are applied to datasets where the standard Lasso is computationally viable. We have also applied this approach to a large non-public dataset Big_Ads where the goal is predicting which of two ads was clicked on given context information (the content of ads and query information). Here, accepting a 0.9% increase in classification error allows us to reduce the number of features from about 3 × 109 to about 24 × 106—a factor of 125 decrease in the number of features. The next set of experiments compares truncated gradient to other algorithms regarding their abilities to tradeoff feature sparsification and performance. Again, we focus on the AUC metric in UCI classification tasks. The algorithms for comparison include: (i) the truncated gradient algorithm with K = 10 and θ = ∞; (ii) the truncated gradient algorithm with K = 10 and θ = g; (iii) the rounding algorithm with K = 10; (iv) the L1-regularized subgradient algorithm with K = 10; and (v) the Lasso [12] for batch L1 regularization (a publicly available implementation [11] was used). We have chosen K = 10 since it worked better than K = 1, and this choice was especially important for the coefficient rounding algorithm. All unspecified parameters were identified using cross validation. Note that we do not attempt to compare these algorithms on rcv1 and Big_Ads simply because their sizes are too large for the Lasso and subgradient descent. Figure 2 gives the results on datasets ad and spambase. Results on other datasets were qualitatively similar. On all datasets, truncated gradient (with θ = ∞) is consistently competitive with the other online algorithms and significantly outperformed them in some problems, implying truncated gradient is generally effective. Moreover, truncated gradient with θ = g behaves similarly to rounding (and sometimes better). This was expected as truncated gradient with θ = g can be regarded as a principled variant of rounding with valid theoretical justification. It is also interesting to observe the qualitative behavior of truncated gradient was often similar to LASSO, especially when very sparse weight vectors were allowed (the left sides in the graphs). This is consistent with theorem 3.2 showing the relation between these two algorithms. However, LASSO usually performed worse when the allowed number of nonzero weights was large (the right side of the graphs). In this case, LASSO seemed to overfit while truncated gradient was more robust to overfitting. The robustness of online learning is often attributed to early stopping, which has been extensively studied (e.g., in [13]). Finally, it is worth emphasizing that these comparison experiments shed some light on the relative strengths of these algorithms in terms of feature sparsification, without considering which one can be efficiently implemented. For large datasets with sparse features, only truncated gradient and the ad hoc coefficient rounding algorithm are applicable. 10 0 10 1 10 2 10 3 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ad Number of Features AUC Trunc. Grad. (θ=∞) Trunc. Grad. (θ=g) Rounding Sub−gradient Lasso 10 0 10 1 10 2 10 3 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 spambase Number of Features AUC Trunc. Grad. (θ=∞) Trunc. Grad. (θ=g) Rounding Sub−gradient Lasso Figure 2: Comparison of the five algorithms in two sample UCI datasets. 6 Conclusion This paper covers the first efficient sparsification technique for large-scale online learning with strong theoretical guarantees. The algorithm, truncated gradient, is the natural extension of Lassostyle regression to the online-learning setting. Theorem 3.1 proves the technique is sound: it never harms performance much compared to standard stochastic gradient descent in adversarial situations. Furthermore, we show the asymptotic solution of one instance of the algorithm is essentially equivalent to Lasso regression, thus justifying the algorithm’s ability to produce sparse weight vectors when the number of features is intractably large. The theorem is verified experimentally in a number of problems. In some cases, especially for problems with many irrelevant features, this approach achieves a one or two orders of magnitude reduction in the number of features. References [1] A. Asuncion and D.J. Newman. UCI machine learning repository, 2007. UC Irvine. [2] N. Cesa-Bianchi, P.M. Long, and M. Warmuth. Worst-case quadratic loss bounds for prediction using linear functions and gradient descent. IEEE Transactions on Neural Networks, 7(3):604–619, 1996. [3] C.-T. Chu, S.K. Kim, Y.-A. Lin, Y. Yu, G. Bradski, A.Y. Ng, and K. Olukotun. Map-reduce for machine learning on multicore. In Advances in Neural Information Processing Systems 20, pages 281–288, 2008. [4] O. Dekel, S. Shalev-Schwartz, and Y. Singer. The Forgetron: A kernel-based perceptron on a fixed budget. In Advances in Neural Information Processing Systems 18, pages 259–266, 2006. [5] J. Duchi, S. Shalev-Shwartz, Y. Singer, and T. Chandra. Efficient projections onto the ℓ1-ball for learning in high dimensions. In Proceedings of ICML-08, pages 272–279, 2008. [6] J. Kivinen and M.K. Warmuth. Exponentiated gradient versus gradient descent for linear predictors. Information and Computation, 132(1):1–63, 1997. [7] J. Langford, L. Li, and A.L. Strehl. Vowpal Wabbit (fast online learning), 2007. http://hunch.net/∼vw/. [8] Honglak Lee, Alexis Batle, Rajat Raina, and Andrew Y. Ng. Efficient sparse coding algorithms. In Advances in Neural Information Processing Systems 19 (NIPS-07), 2007. [9] D.D. Lewis, Y. Yang, T.G. Rose, and F. Li. RCV1: A new benchmark collection for text categorization research. Journal of Machine Learning Research, 5:361–397, 2004. [10] S. Shalev-Shwartz, Y. Singer, and N. Srebro. Pegasos: Primal Estimated sub-GrAdient SOlver for SVM. In Proceedings of ICML-07, pages 807–814, 2007. [11] K. Sjöstrand. Matlab implementation of LASSO, LARS, the elastic net and SPCA, June 2005. Version 2.0, http://www2.imm.dtu.dk/pubdb/p.php?3897. [12] R. Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, B., 58(1):267–288, 1996. [13] T. Zhang. Solving large scale linear prediction problems using stochastic gradient descent algorithms. In Proceedings of ICML-04, pages 919–926, 2004. [14] M. Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. In Proceedings of ICML-03, pages 928–936, 2003.
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Bayesian Model of Behaviour in Economic Games Debajyoti Ray Computation and Neural Systems California Institute of Technology Pasadena, CA 91125. USA dray@caltech.edu Brooks King-Casas Computational Psychiatry Unit Baylor College of Medicine. Houston, TX 77030. USA bkcasas@cpu.bcm.tmc.edu P. Read Montague Human NeuroImaging Lab Baylor College of Medicine. Houston, TX 77030. USA montague@hnl.bcm.tmc.edu Peter Dayan Gatsby Computational Neuroscience Unit University College London London. WC1N 3AR. UK dayan@gatsby.ucl.ac.uk Abstract Classical game theoretic approaches that make strong rationality assumptions have difficulty modeling human behaviour in economic games. We investigate the role of finite levels of iterated reasoning and non-selfish utility functions in a Partially Observable Markov Decision Process model that incorporates game theoretic notions of interactivity. Our generative model captures a broad class of characteristic behaviours in a multi-round Investor-Trustee game. We invert the generative process for a recognition model that is used to classify 200 subjects playing this game against randomly matched opponents. 1 Introduction Trust tasks such as the Dictator, Ultimatum and Investor-Trustee games provide an empirical basis for investigating social cooperation and reciprocity [11]. Even in completely anonymous settings, human subjects show rich patterns of behavior that can be seen in terms of such personality concepts as charity, envy and guilt. Subjects also behave as if they model these aspects of their partners in games, for instance acting to avoid being taken advantage of. Different subjects express quite different personalities, or types, and also have varying abilities at modelling their opponents. The burgeoning interaction between economic psychology and neuroscience requires formal treatments of these issues. From the perspective of neuroscience, such treatments can provide a precise quantitative window into neural structures involved in assessing utilties of outcomes, capturing risk and probabilities associated with interpersonal interactions, and imputing intentions and beliefs to others. In turn, evidence from brain responses associated with these factors should elucidate the neural algorithms of complex interpersonal choices, and thereby illuminate economic decision-making. Here, we consider a sequence of paradigmatic trust tasks that have been used to motivate a variety of behaviorally-based economic models. In brief, we provide a formalization in terms of partially observable Markov decision processes, approximating type-theoretic Bayes-Nash equilibria [8] using finite hierarchies of belief, where subjects’ private types are construed as parameters of their inequity averse utility functions [2]. Our inference methods are drawn from machine learning. Figure 1a shows a simple one-round trust game. In this, an Investor is paired against a randomly assigned Trustee. The Investor can either choose a safe option with a low payoff for both, or take a risk and pass the decision to the Trustee who can either choose to defect (and thus keep more for herself) or choose the fair option that leads to more gains for both players (though less profitable 1 Figure 1: (a) In a simple Trust game, the Investor can take a safe option with a payoff of $[Investor=20,Trustee=20] (i.e. the Investor gets $20 and the Trustee gets $20). The game ends if the Investor chooses the safe option; alternatively, he can pass the decision to the Trustee. The Trustee can now choose a fair option $[25,25] or choose to defect $[15,30]. (b) In the multi-round version of the Trust game, the Investor gets $20 dollars at every round. He can invest any (integer) part; this quantity is trebled on the way to the Trustee. In turn, she has the option of repaying any (integer) amount of her resulting allocation to the Investor. The game continues for 10 rounds. for herself alone than if she defected). Figure 1b shows the more sophisticated game we consider, namely a multi-round, sequential, version of the Trust game [15]. The fact that even in a purely anonymized setting, Investors invest at all, and Trustees reciprocate at all in games such as that of figure 1a, is a challenge to standard, money-maximizing doctrines (which expect to find the Nash equilibrium where neither happens), and pose a problem for modeling. One popular strategy is to retain the notion that subjects attempt to optimize their utilities, but to include in these utilities social factors that penalize cases in which opponents win either more (crudely envy, parameterized by α) or less (guilt, parameterized by β) than themselves [2]. One popular InequityAversion utility function [2] characterizes player i by the type Ti = (αi, βi) of her utility function: U(αi, βi) = xi −αi max{(xj −xi), 0} −βi max{(xi −xj), 0} (1) where xi, xj are the amounts received by players i and j respectively. In the multi-round version of figure 1b, reputation formation comes into play [15]. Investors have the possibility of gaining higher rewards from giving money to the Trustee; and, at least until the final round, the Trustee has an incentive to maintain a reputation of trustworthiness in order to coax the Investor to offer more (against any Nash tendencies associated with solipsistic utility functions). Social utility functions such as that of equation 1 mandate probing, belief manipulation and the like. We cast such tasks as Bayesian Games. As in the standard formulation [8], players know their own types but not those of their opponents; dyads are thus playing games of incomplete information. A player also has prior beliefs about their opponent that are updated in a Bayesian manner after observing the opponent’s actions. Their own actions also influence their opponent’s beliefs. This leads to an infinite hierarchy of beliefs: what the Trustee thinks of the Investor; what the Trustee thinks the Investor thinks of him; what the Trustee thinks the Investor thinks the Trustee thinks of her; and so on. If players have common prior beliefs over the possible types in the game, and this prior is common knowledge, then (at least one) subjective equilibrium known as the Bayes-Nash Equilibrium (BNE), exists [8]. Algorithms to compute BNE solutions have been developed but, in the general case, are NP-hard [6] and thus infeasible for complex multi-round games [9]. One obvious approach to this complexity is to consider finite rather than infinite belief hierarchies. This has both theoretical and empirical support. First, a finite hierarchy of beliefs can provably approximate the equilibrium solution that arises in an infinite belief hierarchy arbitrarily closely [10], an idea that has indeed been employed in practice to compute equilibria in a multi-agent setting [5]. Second, based on a whole wealth of games such as the p-Beauty game [11], it has been suggested that human subjects only employ a very restricted number of steps of strategic thinking. According to cognitive hierarchy theory, a celebrated account of this, this number is on average a mere 1.5 [13]. In order to capture the range of behavior exhibited by subjects in these games, we built a finite belief hierarchy model, using inequity averse utility functions in the context of a partially observable hidden Markov model of the ignorance each subject has about its opponent’s type and in the light of sequential choice. We used inference strategies from machine learning to find approximate solutions to this model. In this paper, we use this generative model to investigate the qualitative classes of behaviour that can emerge in these games. 2 Figure 2: Each player’s decision-making requires solving a POMDP, which involves solving the opponent’s POMDP. Higher order beliefs are required as each player’s action influences the opponent’s beliefs which in turn influence their policy. 2 Partially Observable Markov Games As in the framework of Bayesian games, player i’s inequity aversion type Ti = (αi, βi) is known to it, but not to the opponent. Player i does have a prior distribution over the type of the other player j, b(0) i (Tj); and, if suitably sophisticated, can also have higher-order priors over the whole hierarchy of recursive beliefs about types. We denote the collection of priors as ⃗b(0) i = {b(0) i , b(0)′ i , b(0)′′ i , ...}. Play proceeds sequentially, with player i choosing action a(t) i at time t according to the expected future value of this choice. In this (hidden) Markovian setting, this value, called a Q-value depends on the stage (given the finite horizon), the current beliefs of the player ⃗b(t) i (which are sufficient statistics for the past observations), and the policies P(a(t) i =a|D(t)) (which depend on the observations D(t)) of both players up to time t: Q(t) i (⃗b(t) i , a(t) i ) = U (t) i (⃗b(t) i , a(t) i )+ X a(t) j ∈A(t) j P(a(t) j |{D(t), a(t) i }) X a(t+1) i ∈A(t+1) i Q(t+1) i (⃗b(t+1) i , a(t+1) i )P(a(t+1) i+1 |{D(t), a(t) i , a(t) j }) (2) where we arbitrarily define the softmax policy, P(a(t) i =a|D(t)) = exp φQ(t) i (⃗b(t) i , a) / X b exp φQ(t) i (⃗b(t) i , b) (3) akin to Quantal Response Equilibrium [12], which depends on player i’s beliefs about player j, which are, in turn, updated using Bayes’ rule based on the likelihood function P(a(t) j |{D(t), a(t) i }) b(t+1) i (Tj) = P(Tj|a(t) j , a(t) i , b(t) i ) = P(Tj, a(t) i , a(t) j ) P T ′ j b(t) i (T ′ j)/P(a(t) j |a(t) i , b(t) i ) (4) switching between history-based (Dt) and belief-based (b(t) i (Tj)) representations. Given the interdependence of beliefs and actions, we expect to see probing (to find out the type and beliefs of one’s opponent) and belief manipulation (being nice now to take advantage of one’s opponent later). If the other player’s decisions are assumed to emerge from equivalent softmax choices, then for the subject to calculate this likelihood, they must also solve their opponent’s POMDP. This leads to an infinite recursion (illustrated in fig. 2). In order to break this, we assume that each player has k levels of strategic thinking as in the Cognitive Hierarchy framework [13]. Thus each k-level player assumes that his opponent is a k −1-level player. At the lowest level of the recursion, the 0-level player uses a simple likelihood to update their opponent’s beliefs. The utility U (t) i (a(t) i ) is calculated at every round for each player i for action a(t) i by marginalizing over the current beliefs b(t) i . It is extremely challenging to compute with belief states, since they are probability distributions, and are therefore continuous-valued rather than discrete. To make this computationally reasonable, we discretize the values of the types. As an example, if there are only two types for a player the belief state, which is a continuous probability distribution over the interval 3 [0, 1] is discretized to take K values bi1 = 0, . . . , biK = 1. The utility of an action is obtained by marginalizing over the beliefs as: U (t) i (a(t) i ) = X k=1:K bikQ(t) i (b(t) ik , a(t) i ) (5) Furthermore, we solve the resulting POMDP using a mixture of explicit expansion of the tree from the current start point to three stages ahead, and a stochastic, particle-filter-based scheme (as in [7]), from four stages ahead to the end of the game. One characteristic of this explicit process model, or algorithmic approach, is that it is possible to consider what happens when the priors of the players differ. In this case, as indeed also for the case of only a finite belief hierarchy, there is typically no formal Bayes-Nash equilibrium. We also verified our algorithm against the QRE and BNE solutions provided by GAMBIT ([14]) on a 1 and 2 round Trust game for k = 1, 2 respectively. However unlike the BNE solution in the extensive form game, our algorithm gives rise to belief manipulation and effects at the end of the game. 3 Generative Model for Investor-Trustee Game Reputation-formation plays a particularly critical role in the Investor-Trustee game, with even the most selfish players trying to benefit from cooperation, at least in the initial rounds. In order to reduce complexity in analyzing this, we set αI = βI = 0 (i.e., a purely selfish Investor) and consider 2 values of βT (0.3 and 0.7) such that in the last round the Trustee with type βT = 0.3 will not return any amount to the Investor and will choose fair outcome if βT = 0.7. We generate a rich tapestry of behavior by varying the prior expectations as to βT and the values of strategic (k) level (0,1,2) for the players. 3.1 Factors Affecting Behaviour As an example, fig. 3 shows the evolution of the Players’ Q-values and 1st-order beliefs of the Investor and 2nd-order beliefs of the Trustee (i.e., her beliefs as to the Investor’s beliefs about her value of βT ) over the course of a single game. Here, both players have kI = kT = 1 (i.e. they are strategic players), but the Trustee is actually less guilty βT = 0.3. In the first round, the Investor gives $15, and receives back $30 from the Trustee. This makes the Investor’s beliefs about βT go from being uniform to being about 0.75 for βT = 0.7 and 0.25 for βT = 0.3 (showing the success in the Trustee’s exercise in belief manipulation). This causes the Q-value for the action corresponding to giving $20 dollars to be highest, inspiring the Investor’s generosity in round 2. Equally, the Trustee’s (2nd-order) beliefs after receiving $15 in the first round peak for the value βT = 0.7, corresponding to thinking that the Investor believes the Trustee is Nice. In subsequent rounds, the Trustee’s nastiness limits what she returns, and so the Investor ceases giving high amounts. In response, in rounds 5 and 7, the Trustee tries to coax the Investor. We find this “reciprocal give and take” to be a characteristic behaviour of strategic Investors and Trustees (with k = 1). For naive Players with k = 0, a return of a very low amount for a high amount invested would lead to a complete breakdown of Trust formation. Fig. 4 shows the statistics of dyadic interactions between Investors and Trustees with Uniform priors. The amount given by the Investor varies significantly depending on whether or not he is strategic, and also on his priors. In round 1, Investors with kI = 0 and 1 offer $20 first (the optimal probing action based on uniform prior beliefs) and for kI = 2 offers $15 dollars. The corresponding amount returned by the Trustee depends significantly on kT . A Trustee with kT = 0 and low βT will return nothing whereas an unconditionally cooperative Trustee (high βT ) returns roughly the same amount as received. Irrespective of the Trustee’s βT type, the amount returned by strategic Trustees with kT = 1, 2 is higher (between 1.5 and 2 times the amount received). In round 2 we find that the low amount received causes trust to break down for Investors with kI = 0. In fact, naive Investors and Trustees do not form Trust in this game. Strategic Trustees return more initially and are able to coax naive Investors to give higher amounts in the game. Generally unconditionally cooperative Trustees return more, and form Trust throughout the game if they are strategic or if they are playing against strategic Investors. Trustees with low βT defect towards the end of the game but coax more investment in the beginning of the game. 4 Figure 3: The generated game shows the amount given by an Investor with kI = 1 and a Trustee with βT = 0.3 and kT = 1. The red bar indicates amount given by the Investor and the blue bar is the amount returned by the Trustee (after receiving 3 times amount given by the Investor). The figures on the right reveal the inner workings of the algorithm: Q-values through the rounds of the game for 5 different actions of the Investor (0, 5, 10, 15, 20) and 5 actions of the Trustee between values 0 and 3 times amount given by Investor. Also shown are the Investor’s 1st-order beliefs (left bar for βT = 0.3 and right bar for βT = 0.7) and Trustee’s 2nd-order beliefs over the rounds. Figure 4: The dyadic interactions between the Investor and Trustee across the 10 rounds of the game. The top half shows Investor playing against Trustee with low βT (= 0.3) and the bottom half is the Trustee with high βT (= 0.7): unconditionally cooperative. The top dyad shows the amount given the Investor and the bottom dyad shows the amount returned by Trustee. Within each dyad the rows represent the strategic (kI) levels of Investor (0, 1 or 2) and the columns represent kT level of the Trustee (0, 1 or 2). The dyads are shown here for the first 2 and final 2 rounds. Two particular examples are highlighted within the dyads: Investor with kI = 0 and Trustee with kT = 2, uncooperative (βlow T ) and Investor kI = 1 and Trustee kT = 2, cooperative (βhigh T ). Lighter colours reveal higher amounts (with amount given by Investor in first round being 15 dollars). The effect of strategic level is more dramatic for the Investor, since his ability to defect at any point places him in effective charge of the interaction. Strategic Investors give more money in the game than naive Investors. Consequently they also get more return on their investment because of the beneficial effects of this on their reputations. A further observation is that strategic Investors are more immune to the Trustee’s actions. While this means that break-downs in the game due to 5 mistakes of the Trustee (or unfortunate choices from her softmax) are more easily corrected by the strategic Investor, he is also more likely to continue investing even if the Trustee doesn’t reciprocate. It is also worth noting the differences between k = 1 and k = 2 players. The latter typically offer less in the game and are also less susceptible to the actions of their opponent. Overall in this game, the Investors with kI = 1 make the most amount of money playing against a cooperative Trustee while kI = 0 Investors make the least. The best dyad consists of a kI = 1 Investor playing with a cooperative Trustee with kT = 0 or 1. A very wide range of patterns of dyadic interaction, including the main observations of [15], can thus be captured by varying just the limited collection of parameters of our model 4 Recognition and Classification One of the main reasons to build this generative model for play is to have a refined method for classifying individual players on the basis of the dyadic behaviour. We do this by considering the statistical inverse of the generative model as a recognition model. Denote the sequence of plays in the 10-round Investor-Trustee game as D = {[a(1) 1 , a(1) 2 ], .., [a(10) 1 , a10 2 ]}. Since the game is Markovian we can calculate the probability of player i taking the action sequence {a(t) i , t = 1, ..., 10} given his Type Ti and prior beliefs⃗b(0) i as: P({at i}|Ti,⃗b(0) i ) = P(a(1) 1 |Ti,⃗b(0) i ) 10 Y t=2 P(a(t) i |D(t), Ti) (6) where P(a(1) 1 |Ti,⃗b(0) i ) is the probability of initial action a(1) i given by the softmax distribution and prior beliefs ⃗b(0) i , and P(a(t) i |D(t), Ti) is the probability of action a(t) i after updating beliefs ⃗b(t) i from previous beliefs ⃗b(t−1) i upon the observation of the past sequence of moves D(t). This is a likelihood function for Ti,⃗b(0) i , and so can be used for posterior inference about type given D. We classify the players for their utility function (βT value for the Trustee), strategic (ToM) levels and prior beliefs using the MAP value (T ∗ i ,⃗b(0) i ∗) = maxTi,⃗b(0) i P(D|Ti,⃗b(0) i ). We used our recognition model to classify subject pairs playing the 10-round Investor-Trustee game [15]. The data included 48 student pairs playing an Impersonal task for which the opponents’ identities were hidden and 54 student pairs playing a Personal task for which partners met. Each Investor-Trustee pair was classified for their level of strategic thinking k and the Trustee’s βT type (cooperative/uncooperative; see the table in Figure 5). We are able to capture some characteristic behaviours with our model. The highlighted interactions reveal that many of the pairs in the Impersonal task consisted of strategic Investors and cooperative Trustees, who formed trust in the game with the levels of investment decreasing towards the end of the game. We also highlight the difference between strategic and non-strategic Investors. An Investor with kI = 0 will not form trust if the Trustee does not return a significant amount initially whilst an Investor with kI = 2 will continue offering money in the game even if the Trustee gives back less than fair amounts in return. There is also a strong correlation between the proportion of Trustees classified as being cooperative: estimated as 48%, 30%, on the Impersonal and Personal tasks respectively and the corresponding Return on Investment (how much the Investor receives for the amount Invested): 120%, 109%. Although the recognition model captures key characteristics, we do not expect the Trustees to have the specified values of βlow T = 0.3 and βhigh T = 0.7. To test the robustness of the recognition model we generated behaviours (450 dyads) with different values of βT (βlow T = [0, 0.1, 0.2, 0.3, 0.4] and βhigh T = [0.6, 0.7, 0.8, 0.9, 1.0]), that were classified using the recognition model. Figure 5 shows how confidently players of the given type were classified to have that type. We find that the recognition model tends to misclassify Trustees with low βT as having kT = 2. This is because the Trustees with those characteristics will offer high amounts to coax the Investor. Investor are shown to be correctly classified in most cases. Overall the recognition model has a tendency to assign higher kT to the Trustees than their true type, though the model correctly assigns the right cooperative/uncooperative type to the Trustee. 6 Figure 5: Subject pairs are classified into levels of Theory of Mind for the Investor (rows) and Trustee (columns). The number of subject-pairs with the classification are shown in each entry along with whether the Trustee was classified as uncooperative / cooperative (βlow T , βhigh T ). The subjects play an Impersonal game where they do not know the identities of the opponent and a Personal game where identities are revealed. We reveal the dominant or unique behavioural classification within tables (highlighted): Impersonal (kI = 1, kT = 2, cooperative) group averaged over 10 subjects, Personal group (kI = 0, kT = 0, uncooperative) averaged over 3 subjects, and Personal group with (kI = 2, kT = 0, uncooperative) averaged over 11 subjects. We also show the classification confidence for the types given the behaviour was generated from our model with other values of βT for the Trustee, as well as the type that the player is most likely to be classified as in brackets. (A Trustee with low βT and kT = 1 is very likely to be misclassified as a player with kT = 2, while a player with kT = 2 will mostly be classified with kT = 2) 5 Discussion We built a generative model that captures classes of observed behavior in multi-round trust tasks. The critical features of the model are a social utility function, with parameters covering different types of subjects; partial observability, accounting for subjects’ ignorance about their opponents; an explicit and finite cognitive hierarchy to make approximate equilibrium calculations marginally tractable; and partly deterministic and partly sample-based evaluation methods. Despite its descriptive adequacy, we do not claim that it is uniquely competent. We also do not suggest a normative rationale for pieces of the model such as the social utility function. Nevertheless, the separation between the vagaries of utility and the exactness of inference is attractive, not the least by providing clearly distinct signals as to the inner workings of the algorithm that can be extremely useful to capture neural findings. Indeed, the model is relevant to a number of experimental findings, including those due to [15], [18], [19]. The underlying foundation in reinforcement learning is congenial, given the substantial studies of the neural bases of this [20]. The model does directly license some conclusions. For instance, we postulate that higher activation will be observed in regions of the brain associated with theory of mind for Investors that give more in the game, and for Trustees that can coax more. However, unlike [13] our Naive players still build models, albeit unsophisticated ones, of the other player (in contrast to level 0 players who assume the opponent to play a random strategy). So this might lead to an investigation of how sophisticated and naive theory of mind models are built by subjects in the game. We also constructed the recognition model, which is the statistical inverse to this generative model. While we showed this to capture a broad class of behaviours, it only explains the coarse features of the behaviour. We need to incorporate some of the other parameters of our model, such as the Investor’s envy and the temperature parameter of the softmax distribution in order to capture the nuances in the interactions. Further it would be interesting to use the recognition model in pathological populations, looking at such conditions as autism and borderline personality disorder. 7 Finally, this computational model provides a guide for designing experiments to probe aspects of social utility, strategic thinking levels and prior beliefs, as well as inviting ready extensions to related tasks such as Public Goods games. The inference method may also have wider application, for instance to identifying which of a collection of Bayes-Nash equilibria is most likely to arise, given psychological factors about human utilities. Acknowledgments We thank Wako Yoshida, Karl Friston and Terry Lohrenz for useful discussions. References [1] K.A. McCabe, M.L. Rigdon and V.L. Smith. Positive Reciprocity and Intentions in Trust Games (2003). Journal of Economic Behaviour and Organization. [2] E. Fehr and K.M. Schmidt. A Theory of Fairness, Competition and Cooperation (1999). The Quarterly Journal of Economics. [3] E. Fehr and S. Gachter. Fairness and Retaliation: The Economics of Reciprocity (2000). Journal of Economic Perspectives. [4] E. Fehr and U. Fischbacher. Social norms and human cooperation (2004). TRENDS in Cog. Sci. 8:4. [5] P.J. Gmytrasiewicz and P. Doshi. A Framework for Sequential Planning in Multi-Agent Settings (2005). Journal of Artificial Intelligence Research. [6] V. Conitzer and T. Sandholm (2002). Complexity Results about Nash Equilibria. Technical Report CMUCS-02-135, School of Computer Science, Carnegie-Mellon University. [7] S. Thrun. Monte Carlo POMDPs (2000). Advances in Neural Information Processing Systems 12. [8] JC Harsanyi (1967). Games with Incomplete Information Played by “Bayesian” Players, I-III. Management Science. [9] J.F. Mertens and S. Zamir. Formulation of Bayesian analysis for games with incomplete information (1985). International Journal of Game Theory. [10] Y. Nyarko. Convergence in Economic Models with Bayesian Hierarchies of Beliefs (1997). Journal of Economic Theory. [11] C. Camerer. Behavioural Game Theory: Experiments in Strategic Interaction (2003). Princeton Univ. [12] R. McKelvey and T. Palfrey. Quantal Response Equilibria for Extensive Form Games (1998). Experimental Economics 1:9-41. [13] C. Camerer, T-H. Ho and J-K. Chong. A Cognitive Hierarchy Model of Games (2004). The Quarterly Journal of Economics. [14] R.D. McKelvey, A.M. McLennan and T.L. Turocy (2007). Gambit: Software Tools for Game Theory. [15] B. King-Casas, D. Tomlin, C. Anen, C.F. Camerer, S.R. Quartz and P.R. Montague (2005). Getting to know you: Reputation and Trust in a two-person economic exchange. Science 308:78-83. [16] D. Tomlin, M.A. Kayali, B. King-Casas, C. Anen, C.F. Camerer, S.R. Quartz and P.R. Montague (2006). Agent-specific responses in cingulate cortex during economic exchanges. Science 312:1047-1050. [17] L.P. Kaelbling, M.L. Littman and A.R. Cassandra. Planning and acting in partially observable stochastic domains (1998). Artificial Intelligence. [18] K. McCabe, D. Houser, L. Ryan, V. Smith, T. Trouard. A functional imaging study of cooperation in two-person reciprocal exchange. Proc. Natl. Acad. Sci. USA 98:11832-35. [19] K. Fliessbach, B. Weber, P. Trautner, T. Dohmen, U. Sunde, C.E. Elger and A. Falk. Social Comparison Affects Reward-Related Brain Activity in the Human Ventral Striatum (2007). Science 318:1302-1305. [20] B. Lau and P. W. Glimcher (2008). Representations in the Primate Striatum during Matching Behaviour. Neuron 58. 8
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Skill characterization based on betweenness ¨Ozg¨ur S¸ims¸ek∗ Andrew G. Barto Department of Computer Science University of Massachusetts Amherst, MA 01003 {ozgur|barto}@cs.umass.edu Abstract We present a characterization of a useful class of skills based on a graphical representation of an agent’s interaction with its environment. Our characterization uses betweenness, a measure of centrality on graphs. It captures and generalizes (at least intuitively) the bottleneck concept, which has inspired many of the existing skill-discovery algorithms. Our characterization may be used directly to form a set of skills suitable for a given task. More importantly, it serves as a useful guide for developing incremental skill-discovery algorithms that do not rely on knowing or representing the interaction graph in its entirety. 1 Introduction The broad problem we consider is how to equip artificial agents with the ability to form useful high-level behaviors, or skills, from available primitives. For example, for a robot performing tasks that require manipulating objects, grasping is a useful skill that employs lower-level sensory and motor primitives. In approaching this problem, we distinguish between two related questions: What constitutes a useful skill? And, how can an agent identify such skills autonomously? Here, we address the former question with the objective of guiding research on the latter. Our main contribution is a characterization of a useful class of skills based on a graphical representation of the agent’s interaction with its environment. Specifically, we use betweenness, a measure of centrality on graphs [1, 2], to define a set of skills that allows efficient navigation on the interaction graph. In the game of Tic-Tac-Toe, these skills translate into setting up a fork, creating an opportunity to win the game. In the Towers of Hanoi puzzle, they include clearing the stack above the largest disk and clearing one peg entirely, making it possible to move the largest disk. Our characterization may be used directly to form a set of skills suitable for a given task if the interaction graph is readily available. More importantly, this characterization is a useful guide for developing low-cost, incremental algorithms for skill discovery that do not rely on complete representation of the interaction graph. We present one such algorithm here and perform preliminary analysis. Our characterization captures and generalizes (at least intuitively) the bottleneck concept, which has inspired many of the existing skill-discovery algorithms [3, 4, 5, 6, 7, 8, 9]. Bottlenecks have been described as regions that the agent tends to visit frequently on successful trajectories but not on unsuccessful ones [3], border states of strongly connected areas [6], and states that allow transitions to a different part of the environment [7]. The canonical example is a doorway connecting two rooms. We hope that our explicit and concrete description of what makes a useful skill will lead to further development of these existing algorithms and inspire alternative methods. ∗Now at the Max Planck Institute for Human Development, Center for Adaptive Behavior and Cognition, Berlin, Germany. 1 Figure 1: A visual representation of betweenness on two sample graphs. 2 Skill Definition We assume that the agent’s interaction with its environment may be represented as a Markov Decision Process (MDP). The interaction graph is a directed graph in which the vertices represent the states of the MDP and the edges represent possible state transitions brought about by available actions. Specifically, the edge u →v is present in the graph if and only if the corresponding state transition has a strictly positive probability through the execution of at least one action. The weight on each edge is the expected cost of the transition, or expected negative reward. Our claim is that states that have a pivotal role in efficiently navigating the interaction graph are useful subgoals to reach and that a useful measure for evaluating how pivotal a vertex v is X s̸=t̸=v σst(v) σst wst, where σst is the number of shortest paths from vertex s to vertex t, σst(v) is the number of such paths that pass through vertex v, and wst is the weight assigned to paths from vertex s to vertex t. With uniform path weights, the above expression equals betweenness, a measure of centrality on graphs [1, 2]. It gives the fraction of shortest paths on the graph (between all possible sources and destinations) that pass through the vertex of interest. If there are multiple shortest paths from a given source to a given destination, they are given equal weights that sum to one. Betweenness may be computed in O(nm) time and O(n + m) space on unweighted graphs with n nodes and m edges [10]. On weighted graphs, the space requirement remains the same, but the time requirement increases to O(nm + n2logn). In our use of betweenness, we include path weights to take into account the reward function. Depending on the reward function—or a probability distribution over possible reward functions—some parts of the interaction graph may be given more weight than others, depending on how well they serve the agent’s needs. We define as subgoals those states that correspond to local maxima of betweenness on the interaction graph, in other words, states that have a higher betweenness than other states in their neighborhood. Here, we use a simple definition of neighborhood, including in it only the states that are one hop away, which may be revised in the future. Skills for efficiently reaching the local maxima of betweenness represent a set of behaviors that may be combined in different ways to efficiently reach different regions, serving as useful building blocks for navigating the graph. Figure 1 is a visual representation of betweenness on two sample graphs, computed using uniform edge and path weights. The gray-scale shading on the vertices corresponds to the relative values of betweenness, with black representing the highest betweenness on the graph and white representing the lowest. The graph on the left corresponds to a gridworld in which a doorway connects two rooms. The graph on the right has a doorway of a different type: an edge connecting two otherwise distant nodes. In both graphs, states that are local maxima of betweenness correspond to our intuitive choice of subgoals. 2 Figure 2: Betweenness in Taxi, Playroom, and Tic-Tac-Toe (from left to right). Edge directions are omitted in the figure. 3 Examples We appled the skill definition of Section 2 to various domains in the literature: Taxi [11], Playroom [12, 13], and the game of Tic-Tac-Toe. Interaction graphs of these domains, displaying betweenness values as gray-scale shading on the vertices, are shown in Figure 2. In Taxi and Playroom, graph layouts were determined by a force-directed algorithm that models the edges as springs and minimizes the total force on the system. We considered a node to be a local maximum if its betweenness was higher than or equal to those of its immediate neighbors, taking into account both incoming and outgoing edges. Unless stated otherwise, actions had uniform cost and betweenness was computed using uniform path weights. Taxi This domain includes a taxi and a passenger on the 5 × 5 grid shown in Figure 4. At each grid location, the taxi has six primitive actions: north, east, south, west, pick-up, and put-down. The navigation actions succeed in moving the taxi in the intended direction with probability 0.8; with probability 0.2, the action takes the taxi to the right or left of the intended direction. If the direction of movement is blocked, the taxi remains in the same location. Pick-up places the passenger in the taxi if the taxi is at the passenger location; otherwise it has no effect. Similarly, put-down delivers the passenger if the passenger is inside the taxi and the taxi is at the destination; otherwise it has no effect. The source and destination of all passengers are chosen uniformly at random from among the grid squares R, G, B, Y. We used a continuing version of this problem in which a new passenger appears after each successful delivery. The highest local maxima of betweenness are at the four regions of the graph that correspond to passenger delivery. Other local maxima belong to one of the following categories: (1) taxi is at the passenger location1, (2) taxi is at one of the passenger wait locations with the passenger in the taxi2, (3) taxi and passenger are both at destination, (4) the taxi is at x = 2, y = 3, a navigational bottleneck on the grid, and (5) the taxi is at x = 3, y = 3, another navigational bottleneck. The corresponding skills are (approximately) those that take the taxi to the passenger location, to the destination (having picked up the passenger), or to a navigational bottleneck. These skills closely resemble those that are hand-coded for this domain in the literature. Playroom We created a Markov version of this domain in which an agent interacts with a number of objects in its surroundings: a light switch, a ball, a bell, a button for turning music on and off, 1Except when passenger is waiting at Y, in which case, the taxi is at x = 1, y = 3. 2For wait location Y, the corresponding subgoal has the taxi at x = 1, y = 3, having picked up the passenger. 3 0 20 40 60 0 0.5 1 1.5 2x 10 4 Episodes completed Primitives Skills Random Cumulative number of steps 0 10 20 30 40 50 0 1000 2000 3000 4000 5000 6000 7000 Episodes completed Primitives Skills Random Cumulative number of steps 0 20 40 0 1 2 3 4 5 6 7x 10 4 Episodes completed Primitives Skills−300 Random−300 Skills−100 Random−100 Cumulative number of steps Rooms Shortcut Playroom Figure 3: Learning performance in Rooms, Shortcut, and Playroom. and a toy monkey. The agent has an eye, a hand, and a marker it can place on objects. Its actions consist of looking at a randomly selected object, looking at the object in its hand, holding the object it is looking at, looking at the object that the marker is placed on, placing the marker on the object it is looking at, moving the object in its hand to the location it is looking at, flipping the light switch, pressing the music button, and hitting the ball towards the marker. The first two actions succeed with probability 1, while the remaining actions succeed with probability 0.75, producing no change in the environment if they fail. In order to operate on an object, the agent must be looking at the object and holding the object in its hand. To be able to press the music button successfully, the light should be on. The toy monkey starts to make frightened sounds if the bell is rung while the music is playing; it stops only when the music is turned off. If the ball hits the bell, the bell rings for one decision stage. The MDP state consists of the object that the agent is looking at, the object that the agent is holding, the object that the marker is placed on, music (on/off), light (on/off), monkey (frightened/not), and bell (ringing/not). The six different clusters of the interaction graph in Figure 2 emerge naturally from the force-directed layout algorithm and correspond to the different settings of the music, light, and monkey variables. There are only six such clusters because not all variable combinations are possible. Betweenness peaks at regions that immediately connect neighboring clusters, corresponding to skills that change the setting of the music, light, or monkey variables. Tic-Tac-Toe In the interaction graph, the node at the center of the interaction graph is the empty board, with other board configurations forming rings around it with respect to their distance from this initial configuration. The innermost ring shows states in which both players have played a single turn. The agent played first. The opponent followed a policy that (1) placed the third mark in a row, whenever possible, winning the game, (2) blocked the agent from completing a row, and (3) placed its mark on a random empty square, with decreasing priority. Our state representation was invariant with respect to rotational and reflective symmetries of the board. We assigned a weight of +1 to paths that terminate at a win for the agent and 0 to all other paths. The state with the highest betweenness is the one shown in Figure 4. The agent is the X player and will go next. This state gives the agent two possibilities for setting up a fork (board locations marked with *), creating an opportunity to win on the next turn. There were nine other local maxima that similarly allowed the agent to immediately create a fork. In addition, there were a number of “trivial” local maxima that allowed the agent to immediately win the game. 4 Empirical Performance We evaluated the impact of our skills on the agent’s learning performance in Taxi, Playroom, TicTac-Toe, and two additional domains, called Rooms and Shortcut, whose interaction graphs are those presented in Figure 1. Rooms is a gridworld in which a doorway connects two rooms. At each state, the available actions are north, south, east, and west. They move the agent in the intended direction with probability 0.8 and in a uniform random direction with probability 0.2. The local 4
X X O O * *
R G Y B 5 4 3 2 1 2 y x 1 3 4 5 Figure 4: Learning performance in Taxi and Tic-Tac-Toe. maxima of betweenness are the two states that surround the doorway, which have a slightly higher betweenness than the doorway itself. The transition dynamics of Shortcut is identical, except there is one additional long-range action, connecting two particular states, which are the local maxima of betweenness in this domain. We represented skills using the options framework [14, 15]. The initiation set was restricted to include a certain number of states and included those states with the least distance to the subgoal on the interaction graph. The skills terminated with probability one outside the initiation set and at the subgoal, with probability zero at all other states. The skill policy was the optimal policy for reaching the subgoal. We compared three agents: one that used only the primitive actions of the domain, one that used primitives and our skills, and one that used primitives and a control group of skills whose subgoals were selected randomly. The number of subgoals used and the size of the initiation sets were identical in the two skill conditions. The agent used Q-learning with -greedy exploration with = 0.05. When using skills, it performed both intra-option and macro-Q updates [16]. The learning rate ( ) was kept constant at 0.1. Initial Q-values were 0. Discount rate was set to 1 in episodic tasks, to 0.99 in continuing tasks. Figure 3 shows performance results in Rooms, Shortcut, and PlayRoom, where we had the agent perform 100 different episodic tasks, choosing a single goal state uniformly randomly in each task. The reward was 0.001 for each transition and an additional +1 for transitions into the goal state. The initial state was selected randomly. The labels in the figure indicate the size of the initiation sets. If no number is present, the skills were made available everywhere in the domain. The availability of our skills—those that were identified using local maxima of betweenness—revealed a big improvement compared to using primitive actions only. In some cases, random skills improved performance as well, but this improvement was much smaller than that obtained by our skills. Figure 4 shows similar results in Taxi and Tic-Tac-Toe. The figure shows mean performance in 100 trials. In Taxi, we examined performance on the single continuing task that rewarded the agent for delivering passengers. Reward was 1 for each action, an additional +50 for passenger delivery, and an additional 10 for an unsuccessful pick-up or put-down. In Tic-Tac-Toe, the agent received a reward of 0.001 for each action, an additional +1 for winning the game, and an additional 1 for losing. Creating an individual skill for reaching each of the identified subgoals (which is what we have done in other domains) generates skills that are not of much use in Tic-Tac-Toe because reaching any particular board configuration is usually not possible. Instead, we defined a single skill with multiple subgoals—the ten local maxima of betweenness that allow the agent to setup a fork. We set the initial Q-value of this skill to 1 at the start state to ensure that the skill got executed frequently enough. It is not clear what this single skill can be meaningfully compared to, so we do not provide a control condition with randomly-selected subgoals. 5 Our analysis shows that, in a diverse set of domains, the skill definition of Section 2 gives rise to skills that are consistent with common sense, are similar to skills people handcraft for these domains, and improve learning performance. The improvements in performance are greater than those observed when using a control group of randomly-generated skills, suggesting that they should not be attributed to the presence of skills alone but to the presence of the specific skills that are formed based on betweenness. 5 Related Work A graphical approach to forming high-level behavioral units was first suggested by Amarel in his classic analysis of the missionaries and cannibals problem [17]. Amarel advocated representing action consequences in the environment as a graph and forming skills that correspond to navigating this graph by exploiting its structural regularities. He did not, however, propose any general mechanism that can be used for this purpose. Our skill definition captures the “bottleneck” concept, which has inspired many of the existing skill discovery algorithms [3, 6, 4, 5, 7, 8, 9]. There is clearly an overlap between our skills and the skills that are generated by these algorithms. Here, we review these algorithms, with a focus on the extent of this overlap and sample efficiency. McGovern & Barto [3] examine past trajectories to identify states that are common in successful trajectories but not in unsuccessful ones. An important concern with their method is its need for excessive exploration of the environment. It can be applied only after the agent has successfully performed the task at least once. Typically, it requires many additional successful trajectories. Furthermore, a fundamental property of this algorithm prevents it from identifying a large portion of our subgoals. It examines different paths that reach the same destination, while we look for the most efficient ways of navigating between different source and destination pairs. Bottlenecks that are not on the path to the goal state would not be identified by this algorithm, while we consider such states to be useful subgoals. Stolle & Precup [4] and Stolle [5] address this last concern by obtaining their trajectories from multiple tasks that start and terminate at different states. As the number of tasks increases, the subgoals identified by their algorithms become more similar to ours. Unfortunately, however, sample efficiency is even a larger concern with these algorithms, because they require the agent to have already identified the optimal policy—not for only a single task, but for many different tasks in the domain. Menache et al. [6] and Mannor et al. [8] take a graphical approach and use the MDP state-transition graph to identify subgoals. They apply a clustering algorithm to partition the graph into blocks and create skills that efficiently take the agent to states that connect the different blocks. The objective is to identify blocks that are highly connected within themselves but weakly connected to each other. Different clustering techniques and cut metrics may be used towards this end. Rooms and Playroom are examples of where these algorithms can succeed. Tic-Tac-Toe and Shortcut are examples of where they fail. S¸ims¸ek, Wolfe & Barto [9] address certain shortcomings of global graph partitioning by constructing their graphs from short trajectories. S¸ims¸ek & Barto [7] take a different approach and search for states that introduce short-term novelty. Although their algorithm does not explicitly use the connectivity structure of the domain, it shares some of the limitations of graph partitioning as we discuss more fully in the next section. We claim that the more fundamental property that makes a doorway a useful subgoal is that it is between many source-destination pairs and that graph partitioning can not directly tap into this property, although it can sometimes do it indirectly. 6 An Incremental Discovery Algorithm Our skill definition may be used directly to form a set of skills suitable for a given environment. Because of its reliance on complete knowledge of the interaction graph and the computational cost of betweenness, the use of our approach as a skill-discovery method is limited, although there are conditions under which it would be useful. An important research question is whether approximate methods may be developed that do not require complete representation of the interaction graph. 6 Although betweenness of a given vertex is a global graph property that can not be estimated reliably without knowledge of the entire graph, it should be possible to reliably determine the local maxima of betweenness using limited information. Here, we investigate this possibility by combining the descriptive contributions of the present paper with algorithmic insights of earlier work. In particular, we apply the statistical approach from S¸ims¸ek & Barto [7] and S¸ims¸ek, Wolfe & Barto [9] using the skill description in the present paper. The resulting algorithm is founded on the premise that local maxima of betweenness of the interaction graph are likely to be local maxima on its subgraphs. While any single subgraph would not be particularly useful to identify such vertices, a collection of subgraphs may allow us to correctly identify them. The algorithm proceeds as follows. The agent uses short trajectories to construct subgraphs of the interaction graph and identifies the local maxima of betweenness on these subgraphs. From each subgraph, it obtains a new observation for every state represented on it. This is a positive observation if the state is a local maximum, a negative observation otherwise. We use the decision rule from S¸ims¸ek, Wolfe & Barto [9], making a particular state a subgoal if there are at least no observations on this state and if the proportion of positive observations is at least p+. The agent continues this incremental process indefinitely. Figure 5 shows the results of applying this algorithm on two domains. The first is a gridworld with six rooms. The second is also a gridworld, but the grid squares are one of two types with different rewards. The lightly colored squares produce a reward of 0.001 for actions that originate on them, while the darker squares produce 0.1. The reward structure creates two local maxima of betweenness on the graph. These are the regions that look like doorways in the figure—they are useful subgoals for the same reasons that doorways are. Graph partitioning does not succeed in identifying these states because the structure is not created through node connectivity. Similarly, the algorithms by S¸ims¸ek & Barto [7] and S¸ims¸ek, Wolfe & Barto [9] are also not suitable for this domain. We applied them and found that they identified very few subgoals (<0.05/trial) randomly distributed in the domain. In both domains, we had the agent perform a random walk of 40,000 steps. Every 1000 transitions, the agent created a new interaction graph using the last 1000 transitions. Figure 5 shows the number of times each state was identified as a subgoal in 100 trials, using no = 10, p+ = 0.2. The individual graphs had on average 156 nodes in the six-room gridworld and 224 nodes in the other one. We present this algorithm here as a proof of concept, to demonstrate the feasibility of incremental algorithms. An interesting direction is to develop algorithms that actively explore to discover local maxima of betweenness rather than only passively mining available trajectories. 10 20 30 40 50 5 10 15 20 25 Figure 5: Subgoal frequency in 100 trials using the incremental discovery algorithm. 7 Acknowledgments This work is supported in part by the National Science Foundation under grant CNS-0619337 and by the Air Force Office of Scientific Research under grant FA9550-08-1-0418. Any opinions, findings, conclusions or recommendations expressed here are those of the authors and do not necessarily reflect the views of the sponsors. References [1] L. C. Freeman. A set of measures of centrality based upon betweenness. Sociometry, 40:35–41, 1977. [2] L. C. Freeman. Centrality in social networks: Conceptual clarification. Social Networks, 1:215–239, 1979. [3] A. McGovern and A. G. Barto. Automatic discovery of subgoals in reinforcement learning using diverse density. In Proceedings of the Eighteenth International Conference on Machine Learning, 2001. [4] M. Stolle and D. Precup. Learning options in reinforcement learning. Lecture Notes in Computer Science, 2371:212–223, 2002. [5] M. Stolle. Automated discovery of options in reinforcement learning. Master’s thesis, McGill University, 2004. [6] I. Menache, S. Mannor, and N. Shimkin. Q-Cut - Dynamic discovery of sub-goals in reinforcement learning. In Proceedings of the Thirteenth European Conference on Machine Learning, 2002. [7] ¨O. S¸ims¸ek and A. G. Barto. Using relative novelty to identify useful temporal abstractions in reinforcement learning. In Proceedings of the Twenty-First International Conference on Machine Learning, 2004. [8] S. Mannor, I. Menache, A. Hoze, and U. Klein. Dynamic abstraction in reinforcement learning via clustering. In Proceedings of the Twenty-First International Conference on Machine Learning, 2004. [9] ¨O. S¸ims¸ek, A. P. Wolfe, and A. G. Barto. Identifying useful subgoals in reinforcement learning by local graph partitioning. In Proceedings of the Twenty-Second International Conference on Machine Learning, 2005. [10] U. Brandes. A faster algorithm for betweenness centrality. Journal of Mathematical Sociology, 25(2):163–177, 2001. [11] T. G. Dietterich. Hierarchical reinforcement learning with the MAXQ value function decomposition. Journal of Artificial Intelligence Research, 13:227–303, 2000. [12] A. G. Barto, S. Singh, and N. Chentanez. Intrinsically motivated learning of hierarchical collections of skills. In Proceedings of the Third International Conference on Developmental Learning, 2004. [13] S. Singh, A. G. Barto, and N. Chentanez. Intrinsically motivated reinforcement learning. In Advances in Neural Information Processing Systems, 2005. [14] R. S. Sutton, D. Precup, and S. P. Singh. Between MDPs and Semi-MDPs: A framework for temporal abstraction in reinforcement learning. Artificial Intelligence, 112(1-2):181–211, 1999. [15] D. Precup. Temporal abstraction in reinforcement learning. PhD thesis, University of Massachusetts Amherst, 2000. [16] A. McGovern, R. S. Sutton, and A. H. Fagg. Roles of macro-actions in accelerating reinforcement learning. In Grace Hopper Celebration of Women in Computing, 1997. [17] S. Amarel. On representations of problems of reasoning about actions. In Machine Intelligence 3, pages 131–171. Edinburgh University Press, 1968. 8
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Improved Moves for Truncated Convex Models M. Pawan Kumar P.H.S. Torr Dept. of Engineering Science Dept. of Computing University of Oxford Oxford Brookes University pawan@robots.ox.ac.uk philiptorr@brookes.ac.uk Abstract We consider the problem of obtaining the approximate maximum a posteriori estimate of a discrete random field characterized by pairwise potentials that form a truncated convex model. For this problem, we propose an improved st-MINCUT based move making algorithm. Unlike previous move making approaches, which either provide a loose bound or no bound on the quality of the solution (in terms of the corresponding Gibbs energy), our algorithm achieves the same guarantees as the standard linear programming (LP) relaxation. Compared to previous approaches based on the LP relaxation, e.g. interior-point algorithms or treereweighted message passing (TRW), our method is faster as it uses only the efficient st-MINCUT algorithm in its design. Furthermore, it directly provides us with a primal solution (unlike TRW and other related methods which solve the dual of the LP). We demonstrate the effectiveness of the proposed approach on both synthetic and standard real data problems. Our analysis also opens up an interesting question regarding the relationship between move making algorithms (such as α-expansion and the algorithms presented in this paper) and the randomized rounding schemes used with convex relaxations. We believe that further explorations in this direction would help design efficient algorithms for more complex relaxations. 1 Introduction Discrete random fields are a powerful tool for formulating several problems in Computer Vision such as stereo reconstruction, segmentation, image stitching and image denoising [22]. Given data D (e.g. an image or a video), random fields model the probability of a set of random variables v, i.e. either the joint distribution of v and D as in the case of Markov random fields (MRF) [2] or the conditional distribution of v given D as in the case of conditional random fields (CRF) [18]. The word ‘discrete’ refers to the fact that each of the random variables va ∈v = {v0, · · · , vn−1} can take one label from a discrete set l = {l0, · · · , lh−1}. Throughout this paper, we will assume a MRF framework while noting that our results are equally applicable for an CRF. An MRF defines a neighbourhood relationship (denoted by E) over the random variables such that (a, b) ∈E if, and only if, va and vb are neighbouring random variables. Given an MRF, a labelling refers to a function f such that f : {0, · · · , n −1} −→{0, · · · , h −1}. In other words, the function f assigns to each random variable va ∈v, a label lf(a) ∈l. The probability of the labelling is given by the following Gibbs distribution: Pr(f, D|θ) = exp(−Q(f, D; θ))/Z(θ), where θ is the parameter of the MRF and Z(θ) is the normalization constant (i.e. the partition function). Assuming a pairwise MRF, the Gibbs energy is given by: Q(f, D; θ) = X va∈v θ1 a;f(a) + X (a,b)∈E θ2 ab;f(a)f(b), (1) where θ1 a;f(a) and θ2 ab;f(a)f(b) are the unary and pairwise potentials respectively. The superscripts ‘1’ and ‘2’ indicate that the unary potential depends on the labelling of one random variable at a time, while the pairwise potential depends on the labelling of two neighbouring random variables. Clearly, the labelling f which maximizes the posterior Pr(f, D|θ) can be obtained by minimizing the Gibbs energy. The problem of obtaining such a labelling f is known as maximum a posteriori 1 (MAP) estimation. In this paper, we consider the problem of MAP estimation of random fields where the pairwise potentials are defined by truncated convex models [4]. Formally speaking, the pairwise potentials are of the form θ2 ab;f(a)f(b) = wab min{d(f(a) −f(b)), M} (2) where wab ≥0 for all (a, b) ∈E, d(·) is a convex function and M > 0 is the truncation factor. Recall that, by the definition of Ishikawa [9], a function d(·) defined at discrete points (specifically over integers) is convex if, and only if, d(x+1)−2d(x)+d(x−1) ≥0, for all x ∈Z. It is assumed that d(x) = d(−x). Otherwise, it can be replaced by (d(x)+d(−x))/2 without changing the Gibbs energy of any of the possible labellings of the random field [23]. Examples of pairwise potentials of this form include the truncated linear metric and the truncated quadratic semi-metric, i.e. θ2 ab;f(a)f(b) = wab min{|f(a) −f(b)|, M}, θ2 ab;f(a)f(b) = wab min{(f(a) −f(b))2, M}. (3) Before proceeding further, we would like to note here that the method presented in this paper can be trivially extended to truncated submodular models (a generalization of truncated convex models). However, we will restrict our discussion to truncated convex models for two reasons: (i) it makes the analysis of our approach easier; and (ii) truncated convex pairwise potentials are commonly used in several problems such as stereo reconstruction, image denoising and inpainting [22]. Note that in the absence of a truncation factor (i.e. when we only have convex pairwise potentials) the exact MAP estimation can be obtained efficiently using the methods of Ishikawa [9] or Veksler [23]. However, minimizing the Gibbs energy in the presence of a truncation factor is well-known to be NP-hard. Given their widespread use, it is not surprising that several approximate MAP estimation algorithms have been proposed in the literature for the truncated convex model. Below, we review such algorithms. 1.1 Related Work Given a random field with truncated convex pairwise potentials, Felzenszwalb and Huttenlocher [6] improved the efficiency of the popular max-product belief propagation (BP) algorithm [19] to obtain the MAP estimate. BP provides the exact MAP estimate when the neighbourhood structure E of the MRF defines a tree (i.e. it contains no loops). However, for a general MRF, BP provides no bounds on the quality of the approximate MAP labelling obtained. In fact, it is not even guaranteed to converge. The results of [6] can be used directly to speed-up the tree-reweighted message passing algorithm (TRW) [24] and its sequential variant TRW-S [10]. Both TRW and TRW-S attempt to optimize the Lagrangian dual of the standard linear programming (LP) relaxation of the MAP estimation problem [5, 15, 21, 24]. Unlike BP and TRW, TRW-S is guaranteed to converge. However, it is wellknown that TRW-S and other related algorithms [7, 13, 25] suffer from the following problems: (i) they are slower than algorithms based on efficient graph-cuts [22]; and (ii) they only provide a dual solution [10]. The primal solution (i.e. the labelling f) is often obtained from the dual solution in an unprincipled manner1. Furthermore, it was also observed that, unlike graph-cuts based approaches, TRW-S does not work well when the random field models long range interactions (i.e. when the neighbourhood relationship E is highly connected) [11]. However, due to the lack of experimental results, it is not clear whether this observation applies to the methods described in [7, 13, 25]. Another way of solving the LP relaxation is to resort to interior point algorithms [3]. Although interior point algorithms are much slower in practice than TRW-S, they have the advantage of providing the primal (possibly fractional) solution of the LP relaxation. Chekuri et al. [5] showed that when using certain randomized rounding schemes on the primal solution (to get the final labelling f), the following guarantees hold true: (i) for Potts model (i.e. d(f(a) −f(b)) = |f(a) −f(b)| and M = 1), we obtain a multiplicative bound2 of 2; (ii) for the truncated linear metric (i.e. 1We note here that the recently proposed algorithm in [20] directly provides the primal solution. However, it is much slower than the methods which solve the dual. 2Let f be the labelling obtained by an algorithm A (e.g. in this case the LP relaxation followed by the rounding scheme) for a class of MAP estimation problems (e.g. in this case when the pairwise potentials form a Potts model). Let f ∗be the optimal labelling. The algorithm A is said to achieve a multiplicative bound of σ, if for every instance in the class of MAP estimation problems the following holds true: E „ Q(f, D; θ) Q(f ∗, D; θ) « ≤σ, where E(·) denotes the expectation of its argument under the rounding scheme. 2 Initialization - Initialize the labelling to some function f1. For example, f1(a) = 0 for all va ∈v. Iteration - Choose an interval Im = [im + 1, jm] where (jm −im) = L such that d(L) ≥M. - Move from current labelling fm to a new labelling fm+1 such that fm+1(a) = fm(a) or fm+1(a) ∈Im, ∀va ∈v. The new labelling is obtained by solving the st-MINCUT problem on a graph described in § 2.1. Termination - Stop when there is no further decrease in the Gibbs energy for any interval Im. Table 1: Our Algorithm. As is typical with move making methods, our approach iteratively goes from one labelling to the next by solving an st-MINCUT problem. It converges when there remain no moves which reduce the Gibbs energy further. d(f(a) −f(b)) = |f(a) −f(b)| and a general M > 0), we obtain a multiplicative bound of 2 + √ 2; and (iii) for the truncated quadratic semi-metric (i.e. d(f(a) −f(b)) = (f(a) −f(b))2 and a general M > 0), we obtain a multiplicative bound of O( √ M). The algorithms most related to our approach are the so-called move making methods which rely on solving a series of graph-cut (specifically st-MINCUT) problems. Move making algorithms start with an initial labelling f0 and iteratively minimize the Gibbs energy by moving to a better labelling. At each iteration, (a subset of) random variables have the option of either retaining their old label or taking a new label from a subset of the labels l. For example, in the αβ-swap algorithm [4] the variables currently labelled lα or lβ can either retain their labels or swap them (i.e. some variables labelled lα can be relabelled as lβ and vice versa). The recently proposed range move algorithm [23] modifies this approach such that any variable currently labelled li where i ∈[α, β] can be assigned any label lj where j ∈[α, β]. Note that the new label lj can be different from the old label li, i.e. i ̸= j. Both these algorithms (i.e. αβ-swap and range move) do not provide any guarantees on the quality of the solution. In contrast, the α-expansion algorithm [4] (where each variable can either retain its label or get assigned the label lα at an iteration) provides a multiplicative bound of 2 for the Potts model and 2M for the truncated linear metric. Gupta and Tardos [8] generalized the α-expansion algorithm for the truncated linear metric and obtained a multiplicative bound of 4. Komodakis and Tziritas [14] designed a primal-dual algorithm which provides a bound of 2M for the truncated quadratic semimetric. Note that these bounds are inferior to the bounds obtained by the LP relaxation. However, all the above move making algorithms use only a single st-MINCUT at each iteration and are hence, much faster than interior point algorithms, TRW, TRW-S and BP. 1.2 Our Results We further extend the approach of Gupta and Tardos [8] in two ways (section 2). The first extension allows us to handle any truncated convex model (and not just truncated linear). The second extension allows us to consider a potentially larger subset of labels at each iteration compared to [8]. As will be seen in the subsequent analysis (§2.2), these two extensions allow us to solve the MAP estimation problem efficiently using st-MINCUT whilst obtaining the same guarantees as the LP relaxation [5]. Furthermore, our approach does not suffer from the problems of TRW-S mentioned above. In order to demonstrate its practical use, we provide a favourable comparison of our method with several state of the art MAP estimation algorithms (section 3). 2 Description of the Algorithm Table 1 describes the main steps of our approach. Note that unlike the methods described in [4, 23] we will not be able to obtain the optimal move at each iteration. In other words, if in the mth iteration we move from label fm to fm+1 then it is possible that there exists another labelling f ′ m+1 such that f ′ m+1(a) = fm(a) or f ′ m+1(a) ∈Im for all va ∈v and Q(f ′ m+1, D; θ) < Q(fm+1, D; θ). However, our analysis in the next section shows that we are still able to reduce the Gibbs energy sufficiently at each iteration so as to obtain the guarantees of the LP relaxation. We now turn our attention to designing a method of moving from labelling fm to fm+1. Our approach relies on constructing a graph such that every st-cut on the graph corresponds to a labelling f ′ of the random variables which satisfies: f ′(a) = fm(a) or f ′(a) ∈Im, for all va ∈v. The new labelling fm+1 is obtained in two steps: (i) we obtain a labelling f ′ which corresponds to the 3 st-MINCUT on our graph; and (ii) we choose the new labelling fm+1 as fm+1 = f ′ if Q(f ′, D; θ) ≤Q(fm, D; θ), fm otherwise. (4) Below, we provide the details of the graph construction. 2.1 Graph Construction At each iteration of our algorithm, we are given an interval Im = [im +1, jm] of L labels (i.e. (jm − im) = L) where d(L) ≥M. We also have the current labelling fm for all the random variables. We construct a directed weighted graph (with non-negative weights) Gm = {Vm, Em, cm(·, ·)} such that for each va ∈v, we define vertices {aim+1, aim+2, · · · , ajm} ∈Vm. In addition, as is the case with every st-MINCUT problem, there are two additional vertices called terminals which we denote by s (the source) and t (the sink). The edges e ∈Em with capacity (i.e. weight) cm(e) are of two types: (i) those that represent the unary potentials of a labelling corresponding to an st-cut in the graph and; (ii) those that represent the pairwise potentials of the labelling. Figure 1: Part of the graph Gm containing the terminals and the vertices corresponding to the variable va. The edges which represent the unary potential of the new labelling are also shown. Representing Unary Potentials For all random variables va ∈v, we define the following edges which belong to the set Em: (i) For all k ∈[im + 1, jm), edges (ak, ak+1) have capacity cm(ak, ak+1) = θ1 a;k; (ii) For all k ∈[im + 1, jm), edges (ak+1, ak) have capacity cm(ak+1, ak) = ∞; (iii) Edges (ajm, t) have capacity cm(ajm, t) = θ1 a;jm; (iv) Edges (t, ajm) have capacity cm(t, ajm) = ∞; (v) Edges (s, aim+1) have capacity cm(s, aim+1) = θ1 a;fm(a) if fm(a) /∈Im and ∞otherwise; and (vi) Edges (aim+1, s) have capacity cm(aim+1, s) = ∞. Fig. 1 shows the above edges together with their capacities for one random variable va. Note that there are two types of edges in the above set: (i) with finite capacity; and (ii) with infinite capacity. Any st-cut with finite cost3 contains only one of the finite capacity edges for each random variable va. This is because if an st-cut included more than one finite capacity edge, then by construction it must include at least one infinite capacity edge thereby making its cost infinite [9, 23]. We interpret a finite cost st-cut as a relabelling of the random variables as follows: f ′(a) = ( k if st-cut includes edge (ak, ak+1) where k ∈[im + 1, jm), jm if st-cut includes edge (ajm, t), fm(a) if st-cut includes edge (s, aim+1). (5) Note that the sum of the unary potentials for the labelling f ′ is exactly equal to the cost of the st-cut over the edges defined above. However, the Gibbs energy of the labelling also includes the sum of the pairwise potentials (as shown in equation (1)). Unlike the unary potentials we will not be able to model the sum of pairwise potentials exactly. However, we will be able to obtain its upper bound using the cost of the st-cut over the following edges. Representing Pairwise Potentials For all neighbouring random variables va and vb, i.e. (a, b) ∈ E, we define edges (ak, bk′) ∈Em where either one or both of k and k′ belong to the set (im+1, jm] (i.e. at least one of them is different from im + 1). The capacity of these edges is given by cm(ak, bk′) = wab 2 (d(k −k′ + 1) −2d(k −k′) + d(k −k′ −1)) . (6) The above capacity is non-negative due to the fact that wab ≥0 and d(·) is convex. Furthermore, we also add the following edges: cm(ak, ak+1) = wab 2 (d(L −k + im) + d(k −im)) , ∀(a, b) ∈E, k ∈[im + 1, jm) cm(bk′, bk′+1) = wab 2 (d(L −k′ + im) + d(k′ −im)) , ∀(a, b) ∈E, k′ ∈[im + 1, jm) cm(ajm, t) = cm(bjm, t) = wab 2 d(L), ∀(a, b) ∈E. (7) 3Recall that the cost of an st-cut is the sum of the capacities of the edges whose starting point lies in the set of vertices containing the source s and whose ending point lies in the set of vertices containing the sink t. 4 (a) (b) (c) (d) Figure 2: (a) Edges that are used to represent the pairwise potentials of two neighbouring random variables va and vb are shown. Undirected edges indicate that there are opposing edges in both directions with equal capacity (as given by equation 6). Directed dashed edges, with capacities shown in equation (7), are added to ensure that the graph models the convex pairwise potentials correctly. (b) An additional edge is added when fm(a) ∈Im and fm(b) /∈Im. The term κab = wabd(L). (c) A similar additional edge is added when fm(a) /∈Im and fm(b) ∈Im. (d) Five edges, with capacities as shown in equation (8), are added when fm(a) /∈Im and fm(b) /∈Im. Undirected edges indicate the presence of opposing edges with equal capacity. Note that in [23] the graph obtained by the edges in equations (6) and (7) was used to find the exact MAP estimate for convex pairwise potentials. A proof that the above edges exactly model convex pairwise potentials up to an additive constant κab = wabd(L) can be found in [17]. However, we are concerned with the NP-hard case where the pairwise potentials are truncated. In order to model this case, we incorporate some additional edges to the above set. These additional edges are best described by considering the following three cases for all (a, b) ∈E. • If fm(a) ∈Im and fm(b) ∈Im then we do not add any more edges in the graph (see Fig. 2(a)). • If fm(a) ∈Im and fm(b) /∈Im then we add an edge (aim+1, bim+1) with capacity wabM +κab/2, where κab = wabd(L) is a constant for a given pair of neighbouring random variables (a, b) ∈E (see Fig. 2(b)). Similarly, if fm(a) /∈Im and fm(b) ∈Im then we add an edge (bim+1, aim+1) with capacity wabM + κab/2 (see Fig. 2(c)). • If fm(a) /∈Im and fm(b) /∈Im, we introduce a new vertex pab. Using this vertex pab, five edges are defined with the following capacities (see Fig. 2(d)): cm(aim+1, pab) = cm(pab, aim+1) = cm(bim+1, pab) = cm(pab, bim+1) = wabM + κab/2, cm(s, pab) = θ2 ab;fm(a),fm(b) + κab. (8) This completes our graph construction. Given the graph Gm we solve the st-MINCUT problem which provides us with a labelling f ′ as described in equation (5). The new labelling fm+1 is obtained using equation (4). Note that our graph construction is similar to that of Gupta and Tardos [8] with two notable exceptions: (i) we can handle any general truncated convex model and not just truncated linear as in the case of [8]. This is achieved in part by using the graph construction of [23]; and (ii) we have the freedom to choose the value of L, while [8] fixed this value to M. A logical choice would be to use that value of L which minimizes the worst case multiplicative bound for a particular class of problems. The following properties provide such a value of L for both the truncated linear and the truncated quadratic models. Our worst case multiplicative bounds are exactly those achieved by the LP relaxation (see [5]). 5 2.2 Properties of the Algorithm For the above graph construction, the following properties hold true: • The cost of the st-MINCUT provides an upper bound on the Gibbs energy of the labelling f ′ and hence, on the Gibbs energy of fm+1 (see section 2.2 of [17]). • For the truncated linear metric, our algorithm obtains a multiplicative bound of 2 + √ 2 using L = √ 2M (see section 3, Theorem 1, of [17]). Note that this bound is better than those obtained by α-expansion [4] (i.e. 2M) and its generalization [8] (i.e. 4). • For the truncated quadratic semi-metric, our algorithm obtains a multiplicative bound of O( √ M) using L = √ M (see section 3, Theorem 2, of [17]). Note that both α-expansion and the approach of Gupta and Tardos provide no bounds for the above case. The primal-dual method of [14] obtains a bound of 2M which is clearly inferior to our guarantees. 3 Experiments We tested our approach using both synthetic and standard real data. Below, we describe the experimental setup and the results obtained in detail. 3.1 Synthetic Data Experimental Setup We used 100 random fields for both the truncated linear and truncated quadratic models. The variables v and neighbourhood relationship E of the random fields described a 4-connected grid graph of size 50 × 50. Note that 4-connected grid graphs are widely used to model several problems in Computer Vision [22]. Each variable was allowed to take one of 20 possible labels, i.e. l = {l0, l1, · · · , l19}. The parameters of the random field were generated randomly. Specifically, the unary potentials θ1 a;i were sampled uniformly from the interval [0, 10] while the weights wab, which determine the pairwise potentials, were sampled uniformly from [0, 5]. The parameter M was also chosen randomly while taking care that d(5) ≤M ≤d(10). Results Fig. 3 shows the results obtained by our approach and five other state of the art algorithms: αβ-swap, α-expansion, BP, TRW-S and the range move algorithm of [23]. We used publicly available code for all previously proposed approaches with the exception of the range move algorithm4. As can be seen from the figure, the most accurate approach is the method proposed in this paper, followed closely by the range move algorithm. Recall that, unlike range move, our algorithm is guaranteed to provide the same worst case multiplicative bounds as the LP relaxation. As expected, both the range move algorithm and our method are slower than αβ-swap and α-expansion (since each iteration computes an st-MINCUT on a larger graph). However, they are faster than TRW-S, which attempts to minimize the LP relaxation, and BP. We note here that our implementation does not use any clever tricks to speed up the max-flow algorithm (such as those described in [1]) which can potentially decrease the running time by orders of magnitude. 3.2 Real Data - Stereo Reconstruction Given two epipolar rectified images D1 and D2 of the same scene, the problem of stereo reconstruction is to obtain a correspondence between the pixels of the images. This problem can be modelled using a random field whose variables correspond to pixels of one image (say D1) and take labels from a set of disparities l = {0, 1, · · · , h −1}. A disparity value i for a random variable a denoting pixel (x, y) in D1 indicates that its corresponding pixel lies in (x + i, y) in the second image. For the above random field formulation, the unary potentials were defined as in [22] and were truncated at 15. As is typically the case, we chose the neighbourhood relationship E to define a 4neighbourhood grid graph. The number of disparities h was set to 20. We experimented using the following truncated convex potentials: θ2 ab;ij = 50 min{|i −j|, 10}, θ2 ab;ij = 50 min{(i −j)2, 100}. (9) The above form of pairwise potentials encourage neighbouring pixels to take similar disparity values which corresponds to our expectations of finding smooth surfaces in natural images. Truncation of pairwise potentials is essential to avoid oversmoothing, as observed in [4, 23]. Note that using spatially varying weights wab provides better results. However, the main aim of this experiment is to demonstrate the accuracy and speed of our approach and not to design the best possible Gibbs 4When using α-expansion with the truncated quadratic semi-metric, all edges with negative capacities in the graph construction were removed, similar to the experiments in [22]. 6 (a) (b) Figure 3: Results of the synthetic experiment. (a) Truncated linear metric. (b) Truncated quadratic semi-metric. The x-axis shows the time taken in seconds. The y-axis shows the average Gibbs energy obtained over all 100 random fields using the six algorithms. The lower blue curve is the value of the dual obtained by TRW-S. In both the cases, our method and the range move algorithm provide the most accurate solution and are faster than TRW-S and BP. energy. Table 2 provides the value of the Gibbs energy and the total time taken by all the approaches for a standard stereo pair (Teddy). As in the case of the synthetic experiments, the range move algorithm and our method provide the most accurate solutions while taking less time than TRW-S and BP. Additional experiments on other stereo pairs with similar observations about the performances of the various algorithms can be found in [17]. However, we would again like to emphasize that unlike our method the range move algorithm provides no theoretical guarantees about the quality of the solution. Algorithm Energy-1 Time-1(s) Energy-2 Time-2(s) αβ-swap 3678200 18.48 3707268 20.25 α-expansion 3677950 11.73 3687874 8.79 TRW-S 3677578 131.65 3679563 332.94 BP 3789486 272.06 5180705 331.36 Range Move 3686844 97.23 3679552 141.78 Our Approach 3613003 120.14 3679552 191.20 Table 2: The energy obtained and the time taken by the algorithms used in the stereo reconstruction experiment with the Teddy image pair. Columns 2 and 3 : truncated linear metric. Columns 4 and 5: truncated quadratic semi-metric. 4 Discussion We have presented an st-MINCUT based algorithm for obtaining the approximate MAP estimate of discrete random fields with truncated convex pairwise potentials. Our method improves the multiplicative bound for the truncated linear metric compared to [4, 8] and provides the best known bound for the truncated quadratic semi-metric. Due to the use of only the st-MINCUT problem in its design, it is faster than previous approaches based on the LP relaxation. In fact, its speed can be further improved by a large factor using clever techniques such as those described in [12] (for convex unary potentials) and/or [1] (for general unary potentials). Furthermore, it overcomes the well-known deficiencies of TRW and its variants. Experiments on synthetic and real data problems demonstrate its effectiveness compared to several state of the art algorithms. The analysis in §2.2 shows that, for the truncated linear and truncated quadratic models, the bound achieved by our move making algorithm over intervals of any length L is equal to that of rounding the LP relaxation’s optimal solution using the same intervals [5]. This equivalence also extends to the Potts model (in which case α-expansion provides the same bound as the LP relaxation). A natural question would be to ask about the relationship between move making algorithms and the rounding schemes used in convex relaxations. Note that despite recent efforts [14] which analyze certain move making algorithms in the context of primal-dual approaches for the LP relaxation, not many results 7 are known about their connection with randomized rounding schemes. Although the discussion in §2.2 cannot be trivially generalized to all random fields, it offers a first step towards answering this question. We believe that further exploration in this direction would help improve the understanding of the nature of the MAP estimation problem, e.g. how to derandomize approaches based on convex relaxations. Furthermore, it would also help design efficient move making algorithms for more complex relaxations such as those described in [16]. Acknowledgments The first author was supported by the EU CLASS project and EPSRC grant EP/C006631/1(P). The second author is in receipt of a Royal Society Wolfson Research Merit Award, and would like to acknowledge support from the Royal Society and Wolfson foundation. References [1] K. Alahari, P. Kohli, and P. H. S. Torr. Reduce, reuse & recycle: Efficiently solving multi-label MRFs. In CVPR, 2008. [2] J. Besag. On the statistical analysis of dirty pictures. Journal of the Royal Statistical Society, Series B, 48:259–302, 1986. [3] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004. [4] Y. Boykov, O. Veksler, and R. Zabih. Fast approximate energy minimization via graph cuts. PAMI, 23(11):1222–1239, 2001. [5] C. Chekuri, S. Khanna, J. Naor, and L. Zosin. A linear programming formulation and approximation algorithms for the metric labelling problem. SIAM Journal on Disc. Math., 18(3):606–635, 2005. [6] P. Felzenszwalb and D. Huttenlocher. Efficient belief propagation for early vision. In CVPR, 2004. [7] A. Globerson and T. Jaakkola. Fixing max-product: Convergent message passing for MAP LPrelaxations. In NIPS, 2007. [8] A. Gupta and E. Tardos. A constant factor approximation algorithm for a class of classification problems. In STOC, 2000. [9] H. Ishikawa. Exact optimization for Markov random fields with convex priors. PAMI, 25(10):1333–1336, October 2003. [10] V. Kolmogorov. Convergent tree-reweighted message passing for energy minimization. PAMI, 28(10):1568–1583, 2006. [11] V. Kolmogorov and C. Rother. Comparison of energy minimization algorithms for highly connected graphs. In ECCV, pages II: 1–15, 2006. [12] V. Kolmogorov and A. Shioura. New algorithms for the dual of the convex cost network flow problem with applications to computer vision. Technical report, University College London, 2007. [13] N. Komodakis, N. Paragios, and G. Tziritas. MRF optimization via dual decomposition: Message-passing revisited. In ICCV, 2007. [14] N. Komodakis and G. Tziritas. Approximate labeling via graph-cuts based on linear programming. PAMI, 2007. [15] A. Koster, C. van Hoesel, and A. Kolen. The partial constraint satisfaction problem: Facets and lifting theorems. Operations Research Letters, 23(3-5):89–97, 1998. [16] M. P. Kumar, V. Kolmogorov, and P. H. S. Torr. An analysis of convex relaxations for MAP estimation. In NIPS, 2007. [17] M. P. Kumar and P. H. S. Torr. Improved moves for truncated convex models. Technical report, University of Oxford, 2008. [18] J. Lafferty, A. McCallum, and F. Pereira. Conditional random fields: Probabilistic models for segmenting and labelling sequence data. In ICML, 2001. [19] J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kauffman, 1988. [20] P. Ravikumar, A. Agarwal, and M. Wainwright. Message-passing for graph-structured linear programs: Proximal projections, convergence and rounding schemes. In ICML, 2008. [21] M. Schlesinger. Sintaksicheskiy analiz dvumernykh zritelnikh singnalov v usloviyakh pomekh (syntactic analysis of two-dimensional visual signals in noisy conditions). Kibernetika, 4:113–130, 1976. [22] R. Szeliski, R. Zabih, D. Scharstein, O. Veksler, V. Kolmogorov, A. Agarwala, M. Tappen, and C. Rother. A comparative study of energy minimization methods for Markov random fields with smoothness-based priors. PAMI, 2008. [23] O. Veksler. Graph cut based optimization for MRFs with truncated convex priors. In CVPR, 2007. [24] M. Wainwright, T. Jaakkola, and A. Willsky. MAP estimation via agreement on trees: Message passing and linear programming. IEEE Trans. on Information Theory, 51(11):3697–3717, 2005. [25] Y. Weiss, C. Yanover, and T. Meltzer. MAP estimation, linear programming and belief propagation with convex free energies. In UAI, 2007. 8
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Kernelized Sorting Novi Quadrianto RSISE, ANU & SML, NICTA Canberra, ACT, Australia novi.quad@gmail.com Le Song SCS, CMU Pittsburgh, PA, USA lesong@cs.cmu.edu Alex J. Smola Yahoo! Research Santa Clara, CA, USA alex@smola.org Abstract Object matching is a fundamental operation in data analysis. It typically requires the definition of a similarity measure between the classes of objects to be matched. Instead, we develop an approach which is able to perform matching by requiring a similarity measure only within each of the classes. This is achieved by maximizing the dependency between matched pairs of observations by means of the Hilbert Schmidt Independence Criterion. This problem can be cast as one of maximizing a quadratic assignment problem with special structure and we present a simple algorithm for finding a locally optimal solution. 1 Introduction Matching pairs of objects is a fundamental operation of unsupervised learning. For instance, we might want to match a photo with a textual description of a person, a map with a satellite image, or a music score with a music performance. In those cases it is desirable to have a compatibility function which determines how one set may be translated into the other. For many such instances we may be able to design a compatibility score based on prior knowledge or to observe one based on the co-occurrence of such objects. In some cases, however, such a match may not exist or it may not be given to us beforehand. That is, while we may have a good understanding of two sources of observations, say X and Y, we may not understand the mapping between the two spaces. For instance, we might have two collections of documents purportedly covering the same content, written in two different languages. Here it should be our goal to determine the correspondence between both sets and to identify a mapping between the two domains. In the following we present a method which is able to perform such matching without the need of a cross-domain similarity measure. Our method relies on the fact that one may estimate the dependence between sets of random variables even without knowing the cross-domain mapping. Various criteria are available. We choose the Hilbert Schmidt Independence Criterion between two sets and we maximize over the permutation group to find a good match. As a side-effect we obtain an explicit representation of the covariance. We show that our method generalizes sorting. When using a different measure of dependence, namely an approximation of the mutual information, our method is related to an algorithm of [1]. Finally, we give a simple approximation algorithm for kernelized sorting. 1.1 Sorting and Matching The basic idea underlying our algorithm is simple. Denote by X = {x1, . . . , xm} ⊆X and Y = {y1, . . . , ym} ⊆Y two sets of observations between which we would like to find a correspondence. That is, we would like to find some permutation π ∈Πm on m terms, that is Πm := n π|π ∈{0, 1}m×m and π1m = 1m and π⊤1m = 1m o , (1) such that the pairs Z(π) := (xi, yπ(i)) for 1 ≤i ≤m correspond to dependent random variables. Here 1m ∈Rm is the vector of all ones. We seek a permutation π such that the mapping xi →yπ(i) and its converse mapping from y to x are simple. Denote by D(Z(π)) a measure of the dependence between x and y. Then we define nonparametric sorting of X and Y as follows π∗:= argmaxπ∈Πm D(Z(π)). (2) This paper is concerned with measures of D and approximate algorithms for (2). In particular we will investigate the Hilbert Schmidt Independence Criterion and the Mutual Information. 2 Hilbert Schmidt Independence Criterion Let sets of observations X and Y be drawn jointly from some probability distribution Prxy. The Hilbert Schmidt Independence Criterion (HSIC) [2] measures the dependence between x and y by computing the norm of the cross-covariance operator over the domain X×Y in Hilbert Space. It can be shown, provided the Hilbert Space is universal, that this norm vanishes if and only if x and y are independent. A large value suggests strong dependence with respect to the choice of kernels. Formally, let F be the Reproducing Kernel Hilbert Space (RKHS) on X with associated kernel k : X × X →R and feature map φ : X →F. Let G be the RKHS on Y with kernel l and feature map ψ. The cross-covariance operator Cxy : G 7→F is defined by [3] as Cxy = Exy[(φ(x) −µx) ⊗(ψ(y) −µy)], (3) where µx = E[φ(x)], µy = E[ψ(y)], and ⊗is the tensor product. HSIC, denoted as D, is then defined as the square of the Hilbert-Schmidt norm of Cxy [2] via D(F, G, Prxy) := ∥Cxy∥2 HS. In term of kernels HSIC can be expressed as Exx′yy′[k(x, x′)l(y, y′)] + Exx′[k(x, x′)]Eyy′[l(y, y′)] −2Exy[Ex′[k(x, x′)]Ey′[l(y, y′)]], (4) where Exx′yy′ is the expectation over both (x, y) ∼Prxy and an additional pair of variables (x′, y′) ∼Prxy drawn independently according to the same law. Given a sample Z = {(x1, y1), . . . , (xm, ym)} of size m drawn from Prxy an empirical estimate of HSIC is D(F, G, Z) = (m −1)−2 tr HKHL = (m −1)−2 tr ¯K ¯L. (5) where K, L ∈Rm×m are the kernel matrices for the data and the labels respectively, i.e. Kij = k(xi, xj) and Lij = l(yi, yj). Moreover, Hij = δij −m−1 centers the data and the labels in feature space. Finally, ¯K := HKH and ¯L := HLH denote the centered versions of K and L respectively. Note that (5) is a biased estimate where the expectations with respect to x, x′, y, y′ have all been replaced by empirical averages over the set of observations. 2.1 Kernelized Sorting Previous work used HSIC to measure independence between given random variables [2]. Here we use it to construct a mapping between X and Y by permuting Y to maximize dependence. There are several advantages in using HSIC as a dependence criterion. First, HSIC satisfies concentration of measure conditions [2]. That is, for random draws of observation from Prxy, HSIC provides values which are very similar. This is desirable, as we want our mapping to be robust to small changes. Second, HSIC is easy to compute, since only the kernel matrices are required and no density estimation is needed. The freedom of choosing a kernel allows us to incorporate prior knowledge into the dependence estimation process. The consequence is that we are able to generate a family of methods by simply choosing appropriate kernels for X and Y . Lemma 1 The nonparametric sorting problem is given by π∗= argmaxπ∈Πm tr ¯Kπ⊤¯Lπ. Proof We only need to establish that Hπ = πH since the rest follows from the definition of (5). Note that since H is a centering matrix, it has the eigenvalue 0 for the vector of all ones and the eigenvalue 1 for all vectors orthogonal to that. Next note that the vector of all ones is also an eigenvector of any permutation matrix π with π1 = 1. Hence H and π matrices commute. Next we show that the objective function is indeed reasonable: for this we need the following inequality due to Polya, Littlewood and Hardy: Lemma 2 Let a, b ∈Rm where a is sorted ascendingly. Then a⊤πb is maximized for π = argsort b. Lemma 3 Let X = Y = R and let k(x, x′) = xx′ and l(y, y′) = yy′. Moreover, assume that x is sorted ascendingly. In this case (5) is maximized by either π = argsort y or by π = argsort −y. Proof Under the assumptions we have that ¯K = Hxx⊤H and ¯L = Hyy⊤H. Hence we may rewrite the objective as (Hx)⊤π(Hy) 2. This is maximized by sorting Hy ascendingly. Since the centering matrix H only changes the offset but not the order this is equivalent to sorting y. We have two alternatives, since the objective function is insensitive to sign reversal of y. This means that sorting is a special case of kernelized sorting, hence the name. In fact, when solving the general problem, it turns out that a projection onto the principal eigenvectors of ¯K and ¯L is a good initialization of an optimization procedure. 2.2 Diagonal Dominance In some cases the biased estimate of HSIC as given in (5) leads to very undesirable results, in particular in the case of document analysis. This is the case since kernel matrices on texts tend to be diagonally dominant: a document tends to be much more similar to itself than to others. In this case the O(1/m) bias of (5) is significant. Unfortunately, the minimum variance unbiased estimator [2] does not have a computationally appealing form. This can be addressed as follows at the expense of a slightly less efficient estimator with a considerably reduced bias: we replace the expectations (4) by sums where no pairwise summation indices are identical. This leads to the objective function 1 m(m−1) X i̸=j KijLij + 1 m2(m−1)2 X i̸=j,u̸=v KijLuv − 2 m(m−1)2 X i,j̸=i,v̸=i KijLiv. (6) This estimator still has a small degree of bias, albeit significantly reduced since it only arises from the product of expectations over (potentially) independent random variables. Using the shorthand ˜Kij = Kij(1−δij) and ˜Lij = Lij(1−δij) for kernel matrices where the main diagonal terms have been removed we arrive at the expression (m −1)−2 tr H ˜LH ˜K. The advantage of this term is that it can be used as a drop-in replacement in Lemma 1. 2.3 Mutual Information An alternative, natural means of studying the dependence between random variables is to compute the mutual information between the random variables xi and yπ(i). In general, this is difficult, since it requires density estimation. However, if we assume that x and y are jointly normal in the Reproducing Kernel Hilbert Spaces spanned by the kernels k, l and k · l we can devise an effective approximation of the mutual information. Our reasoning relies on the fact that the differential entropy of a normal distribution with covariance Σ is given by h(p) = 1 2 log |Σ| + constant. (7) Since the mutual information between random variables X and Y is I(X, Y ) = h(X) + h(Y ) − h(X, Y ) we will obtain maximum mutual information by minimizing the joint entropy h(X, Y ). Using the Gaussian upper bound on the joint entropy we can maximize a lower bound on the mutual information by minimizing the joint entropy of J(π) := h(X, Y ). By defining a joint kernel on X × Y via k((x, y), (x′, y′)) = k(x, x′)l(y, y′) we arrive at the optimization problem argminπ∈Πm log |HJ(π)H| where Jij = KijLπ(i),π(j). (8) Note that this is related to the optimization criterion proposed by Jebara [1] in the context of sorting via minimum volume PCA. What we have obtained here is an alternative derivation of Jebara’s criterion based on information theoretic considerations. The main difference is that [1] uses the setting to align bags of observations by optimizing log |HJ(π)H| with respect to re-ordering within each of the bags. We will discuss multi-variable alignment at a later stage. In terms of computation (8) is considerably more expensive to optimize. As we shall see, for the optimization in Lemma 1 a simple iteration over linear assignment problems will lead to desirable solutions, whereas in (8) even computing derivatives is a computational challenge. 3 Optimization DC Programming To find a local maximum of the matching problem we may take recourse to a well-known algorithm, namely DC Programming [4] which in machine learning is also known as the Concave Convex Procedure [5]. It works as follows: for a given function f(x) = g(x) −h(x), where g is convex and h is concave, a lower bound can be found by f(x) ≥g(x0) + ⟨x −x0, ∂xg(x0)⟩−h(x). (9) This lower bound is convex and it can be maximized effectively over a convex domain. Subsequently one finds a new location x0 and the entire procedure is repeated. Lemma 4 The function tr ¯Kπ⊤¯Lπ is convex in π. Since ¯K, ¯L ⪰0 we may factorize them as ¯K = U ⊤U and ¯L = V ⊤V we may rewrite the objective function as
V πU ⊤
2 which is clearly a convex quadratic function in π. Note that the set of feasible permutations π is constrained in a unimodular fashion, that is, the set Pm := n M ∈Rm×m where Mij ≥0 and X i Mij = 1 and X j Mij = 1 o (10) has only integral vertices, namely admissible permutation matrices. This means that the following procedure will generate a succession of permutation matrices which will yield a local maximum for the assignment problem: πi+1 = (1 −λ)πi + λ argmaxπ∈Pm tr ¯Kπ⊤¯Lπi (11) Here we may choose λ = 1 in the last step to ensure integrality. This optimization problem is well known as a Linear Assignment Problem and effective solvers exist for it [6]. Lemma 5 The algorithm described in (11) for λ = 1 terminates in a finite number of steps. We know that the objective function may only increase for each step of (11). Moreover, the solution set of the linear assignment problem is finite. Hence the algorithm does not cycle. Nonconvex Maximization When using the bias corrected version of the objective function the problem is no longer guaranteed to be convex. In this case we need to add a line-search procedure along λ which maximizes tr H ˜KH[(1 −λ)πi + λˆπi]⊤H ˜LH[(1 −λ)πi + λˆπi]. Since the function is quadratic in λ we only need to check whether the search direction remains convex in λ; otherwise we may maximize the term by solving a simple linear equation. Initialization Since quadratic assignment problems are in general NP hard we may obviously not hope to achieve an optimal solution. That said, a good initialization is critical for good estimation performance. This can be achieved by using Lemma 3. That is, if ¯K and ¯L only had rank-1, the problem could be solved by sorting X and Y in matching fashion. Instead, we use the projections onto the first principal vectors as initialization in our experiments. Relaxation to a constrained eigenvalue problem Yet another alternative is to find an approximate solution of the problem in Lemma 1 by solving maximizeη η⊤Mη subject to Aη = b (12) Here the matrix M = ¯K ⊗¯L ∈Rm2×m2 is given by the outer product of the constituting kernel matrices, η ∈Rm2 is a vectorized version of the permutation matrix π, and the constraints imposed by A and b amount to the polytope constraints imposed by Πm. This is essentially the approach proposed by [7] in the context of balanced graph matching, albeit with a suboptimal optimization procedure. Instead, one may use the exact algorithm proposed by [8]. The problem with the relaxation (12) is that it does not scale well to large estimation problems (the size of the optimization problem scales O(m4)) and that the relaxation does not guarantee a feasible solution which means that subsequent projection heuristics need to be found. Hence we did not pursue this approach in our experiments. 4 Multivariate Extensions A natural extension is to align several sets of observations. For this purpose we need to introduce a multivariate version of the Hilbert Schmidt Independence Criterion. One way of achieving this goal is to compute the Hilbert Space norm of the difference between the expectation operator for the joint distribution and the expectation operator for the product of the marginal distributions. Formally, let there be T random variables xi ∈Xi which are jointly drawn from some distribution p(x1, . . . , xm). Moreover, denote by ki : Xi × Xi →R the corresponding kernels. In this case we can define a kernel on X1 ⊗. . . ⊗XT by k1 · . . . kT . The expectation operator with respect to the joint distribution and with respect to the product of the marginals is given by [2] Ex1,...,xT " T Y i=1 ki(xi, ·) # and T Y i=1 Exi [ki(xi, ·)] (13) respectively. Both terms are equal if and only if all random variables are independent. The squared difference between both is given by ExT i=1,x′T i=1 " T Y i=1 ki(xi, x′ i) # + T Y i=1 Exi,x′ i[ki(xi, x′ i)] −2ExT i=1 " T Y i=1 Ex′ i[k(xi, x′ i)] # . (14) which we refer to as multiway HSIC. A biased empirical estimate of the above is obtained by replacing sums by empirical averages. Denote by Ki the kernel matrix obtained from the kernel ki on the set of observations Xi := {xi1, . . . , xim}. In this case the empirical estimate of (14) is given by HSIC[X1, . . . , XT ] := 1⊤ m T K i=1 Ki ! 1m + T Y i=1 1⊤ mKi1m −2 · 1⊤ m T K i=1 Ki1m ! (15) where ⊙T t=1∗denotes elementwise product of its arguments (the ’.*’ notation of Matlab). To apply this to sorting we only need to define T permutation matrices πi ∈Πm and replace the kernel matrices Ki by π⊤ i Kiπi. Without loss of generality we may set π1 = 1, since we always have the freedom to fix the order of one of the T sets with respect to which the other sets are to be ordered. In terms of optimization the same considerations as presented in Section 3 apply. That is, the objective function is convex in the permutation matrices πi and we may apply DC programming to find a locally optimal solution. The experimental results for multiway HSIC can be found in the appendix. 5 Applications To investigate the performance of our algorithm (it is a fairly nonstandard unsupervised method) we applied it to a variety of different problems ranging from visualization to matching and estimation. In all our experiments, the maximum number of iterations used in the updates of π is 100 and we terminate early if progress is less than 0.001% of the objective function. 5.1 Data Visualization In many cases we may want to visualize data according to the metric structure inherent in it. In particular, we want to align it according to a given template, such as a grid, a torus, or any other fixed structure. Such problems occur when presenting images or documents to a user. While there is a large number of algorithms for low dimensional object layout (self organizing maps, maximum variance unfolding, local-linear embedding, generative topographic map, ...), most of them suffer from the problem that the low dimensional presentation is nonuniform. This has the advantage of revealing cluster structure but given limited screen size the presentation is undesirable. Instead, we may use kernelized sorting to align objects. Here the kernel matrix L is given by the similarity measure between the objects xi that are to be aligned. The kernel K, on the other hand, denotes the similarity between the locations where objects are to be aligned to. For the sake of simplicity we used a Gaussian RBF kernel between the objects to laid out and also between the Figure 1: Layout of 284 images into a ‘NIPS 2008’ letter grid using kernelized sorting. positions of the grid, i.e. k(x, x′) = exp(−γ ∥x −x′∥2). The kernel width γ was adjusted to the inverse median of ∥x −x′∥2 such that the argument of the exponential is O(1). Our choice of the Gaussian RBF kernel is likely not optimal for the specific set of observations (e.g. SIFT feature extraction followed by a set kernel would be much more appropriate for images). That said we want to emphasize that the gains arise from the algorithm rather than a specific choice of a function class. We obtained 284 images from http://www.flickr.com which were resized and downsampled to 40 × 40 pixels. We converted the images from RGB into Lab color space, yielding 40 × 40 × 3 dimensional objects. The grid, corresponding to X is a ‘NIPS 2008’ letters on which the images are to be laid out. After sorting we display the images according to their matching coordinates (Figure 1). We can see images with similar color composition are found at proximal locations. We also lay out the images (we add 36 images to make the number 320) into a 2D grid of 16 × 20 mesh using kernelized sorting. For comparison we use a Self-Organizing Map (SOM) and a Generative Topographic Mapping (GTM) and the results are shown in the appendix. Although the images are also arranged according to the color grading, the drawback of SOM (and GTM) is that it creates blank spaces in the layout. This is because SOM maps several images into the same neuron. Hence some neurons may not have data associated with them. While SOM is excellent in grouping similar images together, it falls short in exactly arranging the images into 2D grid. 5.2 Matching To obtain more quantifiable results rather than just generally aesthetically pleasing pictures we apply our algorithm to matching problems where the correct match is known. Image matching: Our first test was to match image halves. For this purpose we used the data from the layout experiment and we cut the images into two 20 × 40 pixel patches. The aim was to find an alignment between both halves such that the dependence between them is maximized. In other words, given xi being the left half of the image and yi being the right half, we want to find a permutation π which lines up xi and yi. This would be a trivial undertaking when being able to compare the two image halves xi and yi. While such comparison is clearly feasible for images where we know the compatibility function, it may not be possible for generic objects. The figure is presented in the appendix. For a total of 320 images we recovered 140 pairs. This is quite respectable given that chance level would be 1 correct pair (a random permutation matrix has on expectation one nonzero diagonal entry). Estimation In a next experiment we aim to determine how well the overall quality of the matches is. That is, whether the objects matched share similar properties. For this purpose we used binary, multiTable 1: Error rate for matching problems Type Data set m Kernelized Sorting Baseline Reference Binary australian 690 0.29±0.02 0.49 0.21±0.04 breastcancer 683 0.06±0.01 0.46 0.06±0.03 derm 358 0.08±0.01 0.43 0.00±0.00 optdigits 765 0.01±0.00 0.49 0.01±0.00 wdbc 569 0.11±0.04 0.47 0.05±0.02 Multiclass satimage 620 0.20±0.01 0.80 0.13±0.04 segment 693 0.58±0.02 0.86 0.05±0.02 vehicle 423 0.58±0.08 0.75 0.24±0.07 Regression abalone 417 13.9±1.70 18.7 6.44±3.14 bodyfat 252 4.5±0.37 7.20 3.80±0.76 Table 2: Number of correct matches (out of 300) for English aligned documents. Source language Pt Es Fr Sv Da It Nl De Kernelized Sorting 252 218 246 150 230 237 223 95 Baseline (length match) 9 12 8 6 6 11 7 4 Reference (dictionary) 298 298 298 296 297 300 298 284 class, and regression datasets from the UCI repository http://archive.ics.uci.edu/ml and the LibSVM site http://www.csie.ntu.edu.tw/˜cjlin/libsvmtools. In our setup we split the dimensions of the data into two sets and permute the data in the second set. The so-generated two datasets are then matched and we use the estimation error to quantify the quality of the match. That is, assume that yi is associated with the observation xi. In this case we compare yi and yπ(i) using binary classification, multiclass, or regression loss accordingly. To ensure good dependence between the subsets of variables we choose a split which ensures correlation. This is achieved as follows: we pick the dimension with the largest correlation coefficient as a reference. We then choose the coordinates that have at least 0.5 correlation with the reference and split those equally into two sets, set A and set B. We also split the remainder coordinates equally into the two existing sets and finally put the reference coordinate into set A. This ensures that the set B of dimensions will have strong correlation with at least one dimension in the set A. The listing of the set members for different datasets can be found in the appendix. The results are summarized in Table 1. As before, we use a Gaussian RBF kernel with median adjustment of the kernel width. To obtain statistically meaningful results we subsample 80% of the data 10 times and compute the error of the match on the subset (this is done in lieu of cross-validation since the latter is meaningless for matching). As baseline we compute the expected performance of random permutations which can be done exactly. Finally, as reference we use SVM classification / regression with results obtained by 10-fold cross-validation. Matching is able to retrieve significant information about the labels of the corresponding classes, in some cases performing as well as a full classification approach. Multilingual Document Matching To illustrate that kernelized sorting is able to recover nontrivial similarity relations we applied our algorithm to the matching of multilingual documents. For this purpose we used the Europarl Parallel Corpus. It is a collection of the proceedings of the European Parliament, dating back to 1996 [9]. We select the 300 longest documents of Danish (Da), Dutch (Nl), English (En), French (Fr), German (De), Italian (It), Portuguese (Pt), Spanish (Es), and Swedish (Sv). The purpose is to match the non-English documents (source languages) to its English translations (target language). Note that our algorithm does not require a cross-language dictionary. In fact, one could use kernelized sorting to generate a dictionary after initial matching has occurred. In keeping with the choice of a simple kernel we used standard TF-IDF (term frequency - inverse document frequency) features of a bag of words kernel. As preprocessing we remove stopwords (via NLTK) and perform stemming using http://snowball.tartarus.org. Finally, the feature vectors are normalized to unit length in term of ℓ2 norm. Since kernel matrices on documents are notoriously diagonally dominant we use the bias-corrected version of our optimization problem. As baseline we used a fairly straightforward means of document matching via its length. That is, longer documents in one language will be most probably translated into longer documents in the other language. This observation has also been used in the widely adopted sentence alignment method [10]. As a dictionary-based alternative we translate the documents using Google’s translation engine http://translate.google.com to find counterparts in the source language. Smallest distance matches in combination with a linear assignment solver are used for the matching. The experimental results are summarized in Table 2. We describe a line search procedure in Section 3. In practice we find that fixing λ at a given step size and choosing the best solution in terms of the objective function for λ ∈{0.1, 0.2, . . . , 1.0} works better. Further details can be found in the appendix. Low matching performance for the document length-based method might be due to small variance in the document length after we choose the 300 longest documents. The dictionary-based method gives near-to-perfect matching performance. Further in forming the dictionary, we do not perform stemming on English words and thus the dictionary is highly customized to the problem at hand. Our method produces results consistent to the dictionary-based method with notably low performance for matching German documents to its English translations. We conclude that the difficulty of German-English document matching is inherent to this dataset [9]. Arguably the results are quite encouraging as our method uses only a within class similarity measure while still matches more than 2/3 of what is possible by a dictionary-based method. 6 Summary and Discussion In this paper, we generalized sorting by maximizing the dependency between matched pairs or observations by means of the Hilbert Schmidt Independence Criterion. This way we are able to perform matching without the need of a cross-domain similarity measure. The proposed sorting algorithm is efficient and it can be applied to a variety of different problems ranging from data visualization to image and multilingual document matching and estimation. Further examples of kernelized sorting and of reference algorithms are given in the appendix. Acknowledgments NICTA is funded through the Australian Government’s Backing Australia’s Ability initiative, in part through the ARC.This research was supported by the Pascal Network. Parts of this work were done while LS and AJS were working at NICTA. References [1] T. Jebara. Kernelizing sorting, permutation, and alignment for minimum volume PCA. In Conference on Computational Learning Theory (COLT), volume 3120 of LNAI, pages 609– 623. Springer, 2004. [2] A.J. Smola, A. Gretton, L. Song, and B. Sch¨olkopf. A hilbert space embedding for distributions. In E. Takimoto, editor, Algorithmic Learning Theory, Lecture Notes on Computer Science. Springer, 2007. [3] K. Fukumizu, F. R. Bach, and M. I. Jordan. Dimensionality reduction for supervised learning with reproducing kernel Hilbert spaces. J. Mach. Learn. Res., 5:73–99, 2004. [4] T. Pham Dinh and L. Hoai An. A D.C. optimization algorithm for solving the trust-region subproblem. SIAM Journal on Optimization, 8(2):476–505, 1988. [5] A.L. Yuille and A. Rangarajan. The concave-convex procedure. Neural Computation, 15:915– 936, 2003. [6] R. Jonker and A. Volgenant. A shortest augmenting path algorithm for dense and sparse linear assignment problems. Computing, 38:325–340, 1987. [7] T. Cour, P. Srinivasan, and J. Shi. Balanced graph matching. In B. Sch¨olkopf, J. Platt, and T. Hofmann, editors, Advances in Neural Information Processing Systems 19, pages 313–320. MIT Press, December 2006. [8] W. Gander, G.H. Golub, and U. von Matt. A constrained eigenvalue problem. In Linear Algebra Appl. 114-115, pages 815–839, 1989. [9] P. Koehn. Europarl: A parallel corpus for statistical machine translation. In Machine Translation Summit X, pages 79–86, 2005. [10] W. A. Gale and K. W. Church. A program for aligning sentences in bilingual corpora. In Meeting of the Association for Computational Linguistics, pages 177–184, 1991.
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Posterior Consistency of the Silverman g-prior in Bayesian Model Choice Zhihua Zhang School of Computer Science & Technology Zhejiang University, Hangzhou, China Michael I. Jordan Departments of EECS and Statistics University of California, Berkeley, CA, USA Dit-Yan Yeung Department of Computer Science & Engineering HKUST, Hong Kong, China Abstract Kernel supervised learning methods can be unified by utilizing the tools from regularization theory. The duality between regularization and prior leads to interpreting regularization methods in terms of maximum a posteriori estimation and has motivated Bayesian interpretations of kernel methods. In this paper we pursue a Bayesian interpretation of sparsity in the kernel setting by making use of a mixture of a point-mass distribution and prior that we refer to as “Silverman’s g-prior.” We provide a theoretical analysis of the posterior consistency of a Bayesian model choice procedure based on this prior. We also establish the asymptotic relationship between this procedure and the Bayesian information criterion. 1 Introduction We address a supervised learning problem over a set of training data {xi, yi}n i=1 where xi ∈X ⊂Rp is a p-dimensional input vector and yi is a univariate response. Using the theory of reproducing kernels, we seek to find a predictive function f(x) from the training data. Suppose f = u + h ∈({1} + HK) where HK is a reproducing kernel Hilbert space (RKHS). The estimation of f(x) is then formulated as a regularization problem of the form min f∈HK ( 1 n n X i=1 L(yi, f(xi)) + g 2∥h∥2 HK ) , (1) where L(y, f(x)) is a loss function, ∥h∥2 HK is the RKHS norm and g > 0 is the regularization parameter. By the representer theorem [7], the solution for (1) is of the form f(x) = u + n X j=1 βjK(x, xj), (2) where K(·, ·) is the kernel function. Noticing that ∥h∥2 HK = Pn i,j=1 K(xi, xj)βiβj and substituting (2) into (1), we obtain the minimization problem with respect to (w.r.t.) the βi as min u,β ½ 1 n n X i=1 L(yi, f(xi)) + g 2β′Kβ ¾ , (3) where K = [K(xi, xj)] is the n×n kernel matrix and β = (β1, . . . , βn)′ is the vector of regression coefficients. From the Bayesian standpoint, the role of the regularization term g 2β′Kβ can be captured by assigning a design-dependent prior Nn(0, g−1K−1) to the regression vector β. The prior Nn ¡ 0, K−1¢ for β was first proposed by [5] in his Bayesian formulation of spline smoothing. Here we refer to the prior β ∼Nn ¡ 0, g−1K−1¢ as the Silverman g-prior by analogy to the Zellner g-prior [9]. When K is singular, by analogy to generalized singular g-prior (gsg-prior) [8], we call Nn ¡ 0, g−1K−1¢ a generalized Silverman g-prior. Given the high dimensionality generally associated with RKHS methods, sparseness has emerged as a significant theme, particularly when computational concerns are taken into account. For example, the number of support vectors in support vector machine (SVM) is equal to the number of nonzero components of β. That is, if βj = 0, the jth input vector is excluded from the basis expansion in (2); otherwise the jth input vector is a support vector. We are thus interested in a prior for β which allows some components of β to be zero. To specify such a prior we first introduce an indicator vector γ = (γ1, . . . , γn)′ such that γj = 1 if xj is a support vector and γj = 0 if it is not. Let nγ = Pn j=1 γj be the number of support vectors, let Kγ be the n×nγ submatrix of K consisting of those columns of K for which γj = 1, and let βγ be the corresponding subvector of β. Accordingly, we let βγ ∼Nnγ ¡ 0, g−1K−1 γγ ¢ where Kγγ is the nγ×nγ submatrix of Kγ consisting of those rows of Kγ for which γj = 1. We thus have a Bayesian model choice problem in which a family of models is indexed by an indicator vector γ. Within the Bayesian framework we can use Bayes factors to choose among these models [3]. In this paper we provide a frequentist theoretical analysis of this Bayesian procedure. In particular, motivated by the work of [1] on the consistency of the Zellner g-prior, we investigate the consistency for model choice of the Silverman g-prior for sparse kernel-based regression. 2 Main Results Our analysis is based on the following regression model Mγ: y = u1n + Kγβγ + ϵ (4) ϵ ∼ Nn(0, σ2In), βγ|σ ∼Nnγ ¡ 0, σ2(gγKγγ)−1¢ , where y = (y1, . . . , yn)′. Here and later, 1m denotes the m×1 vector of ones and Im denotes the m×m identity matrix. We compare each model Mγ with the null model M0, formulating the model choice problem via the hypotheses H0 : β = 0 and Hγ : βγ ∈Rnγ. Throughout this paper, for any nγ, it is always assumed to take a finite value even though n →∞. Let eKγ = [1n, Kγ]. The following condition is also assumed: For a fixed nγ < n, 1 n eK′ γ eKγ is positive definite and converges to a positive definite matrix as n →∞. (5) Suppose that the sample y is generated by model Mν with parameter values u, βν and σ. We formalize the problem of consistency for model choice as follows [1]: plim n→∞p(Mν|y) = 1 and plim n→∞p(Mγ|y) = 0 for all Mγ ̸= Mν, (6) where “plim” denotes convergence in probability and the limit is taken w.r.t. the sampling distribution under the true model Mν. 2.1 A Noninformative Prior for (u, σ2) We first consider the case when (u, σ2) is assigned the following noninformative prior: (u, σ2) ∝1/σ2. (7) Moreover, we assume 1′ nK = 0. In this case, we have 1′ nKγ = 0 so that the intercept u may be regarded as a common parameter for both Mγ and M0. After some calculations the marginal likelihood is found to be p(y|Mγ) = Γ( n−1 2 ) π n−1 2 √n ∥y −¯y1n∥−n+1|Qγ|−1 2 (1 −F 2 γ )−n−1 2 , (8) where ¯y = 1 n Pn i=1 yi, Qγ = In + gγ−1KγK−1 γγ K′ γ and F 2 γ = y′Kγ(gγKγγ + K′ γKγ)−1K′ γy ∥y −¯y1n∥2 . Let RSSγ = (1 −R2 γ)∥y −¯y1n∥2 be the residual sum of squares. Here, R2 γ = (y −¯y1n)′Kγ(K′ γKγ)−1K′ γ(y −¯y1n) ∥y −¯y1n∥2 = y′Kγ(K′ γKγ)−1K′ γy ∥y −¯y1n∥2 . It is easily proven that for fixed n, plimgγ→0 F 2 γ = R2 γ and plimgγ→0(1−F 2 γ )∥y−¯y1n∥2 = RSSγ, and RSSγ = y′(In −eHγ)y where eHγ = eKγ( eK′ γ eKγ)−1 eK′ γ. As a special case of (8), it is also immediate to obtain the marginal distribution of the null model as p(y|M0) = Γ( n−1 2 ) π n−1 2 √n ∥y −¯y1n∥−n+1. Then the Bayes factor for Mγ versus M0 is BFγ0 = |Qγ|−1 2 (1 −F 2 γ )−n−1 2 . In the limiting case when gγ →0 and both n and nγ are fixed, BFγ0 tends to 0. This implies that a large spread of the prior forces the Bayes factor to favor the null model. Thus, as in the case of the Zellner g-prior [4], Bartlett’s paradox arises for the Silverman g-prior. The Bayes factor for Mγ versus Mκ is given by BFγκ = BFγ0 BFκ0 = |Qγ|−1 2 |Qκ|−1 2 (1 −F 2 γ )−n−1 2 (1 −F 2κ)−n−1 2 . (9) Based on the Bayes factor, we now explore the consistency of the Silverman g-prior. Suppose that the sample y is generated by model Mν with parameter values u, βν and σ2. Then the consistency property (6) is equivalent to plim n→∞BFγν = 0, for all Mγ ̸= Mν. Assume that under any model Mγ that does not contain Mν, i.e, Mγ ⊉Mν, lim n→∞ eβ ′ γ eK′ ν(In −eHγ) eKν eβγ n = cγ ∈(0, ∞), (10) where eβ ′ γ = (u, β′ γ). Note that In −eHγ is a symmetric idempotent matrix which projects onto the subspace of Rn orthogonal to the span of eKγ. Given that (In −eHγ)1n = 0 and 1′ nKν = 0, condition (10) reduces to lim n→∞ β′ νK′ ν(In −Hγ)Kνβν n = cγ ∈(0, ∞), where Hγ = Kγ(K′ γKγ)−1K′ γ. We now have the following theorem whose proof is given in Sec. 3. Theorem 1 Consider the regression model (4) with the noninformative prior for (u, σ2) in (7). Assume that conditions (5) and (10) are satisfied and assume that gγ can be written in the form gγ = w1(nγ) w2(n) with lim n→∞w2(n) = ∞and lim n→∞ w′ 2(n) w2(n) = 0 (11) for particular choices of functions w1 and w2, where w2 is differentiable and w′ 2(n) is the first derivative w.r.t. n. When the true model Mν is not the null model, i.e., Mν ̸= M0, the posterior probabilities are consistent for model choice. Theorem 1 can provide an empirical methodology for setting g. For example, it is clear that g = 1/n where w1(nγ) = 1 and w2(n) = n satisfies condition (11). It is interesting to consider the (asymptotic) relationship between the Bayes factor and Bayesian information (or Schwartz) criterion (BIC) in our setting. Given two models Mγ and Mκ, the difference between the BICs of these two models is given by Sγκ = n 2 ln RSSκ RSSγ + nκ −nγ 2 ln(n). We thus obtain the following asymptotic relationship (the proof is given in Sec. 3): Theorem 2 Under the regression model and the conditions in Theorem 1, we have plim n→∞ ln BFγν Sγν + nν−nγ 2 ln w2(n) = 1. Furthermore, if Mν is not nested within Mγ, then plimn→∞ ln BFγν Sγν = 1. Here the probability limits are taken w.r.t. the model Mν. 2.2 A Natural Conjugate Prior for (u, σ2) In this section, we analyze consistency for model choice under a different prior for (u, σ2), namely the standard conjugate prior: p(u, σ2) = N(u|0, σ2η−1)Ga(σ−2|aσ/2, bσ/2) (12) where Ga(u|a, b) is the Gamma distribution: p(u) = ba Γ(a)ua−1 exp(−bu), a > 0, b > 0. We further assume that u and βγ are independent. Then eβγ ∼Nnγ+1(0, σ2Σ−1 γ ) with Σγ = · η 0 0 gγKγγ ¸ . (13) The marginal likelihood of model Mγ is thus p(y|Mγ) = baσ/2 σ Γ( n+aσ 2 ) πn/2Γ( aσ 2 ) |Mγ|−1 2 £ bσ + y′M−1 γ y ¤−aσ+n 2 , (14) where Mγ = In + eKγΣ−1 γ eK′ γ. The Bayes factor for Mγ versus Mκ is given by BFγκ = ·|Mκ| |Mγ| ¸ 1 2 ·bσ + y′M−1 κ y bσ + y′M−1 γ y ¸ aσ+n 2 . Because M−1 γ = In −eKγΘ−1 γ eK′ γ and |Mγ| = |Θγ||Σγ|−1 = η−1g−nγ γ |Kγγ|−1|Θγ| where Θγ = eK′ γ eKγ + Σγ, we have BFγκ = gnγ/2 γ gnκ/2 κ ·|Kγγ||Θκ| |Kκκ||Θγ| ¸ 1 2 ·bσ + y′¡ In−eKκΘ−1 κ eK′ κ ¢ y bσ + y′¡ In−eKγΘ−1 γ eK′γ ¢ y ¸ aσ+n 2 . Theorem 3 Consider the regression model (4) with the conjugate prior for (u, σ2) in (12). Assume that conditions (5) and (10) are satisfied and that gγ takes the form in (11) with w1(nγ) being a decreasing function. When the true model Mν is not the null model, i.e., Mν ̸= M0, the posterior probabilities are consistent for model choice. Note the difference between Theorem 1 and Theorem 3: in the latter theorem w1(nγ) is required to be a decreasing function of nγ. Thanks to the fact that gγ = w1(nγ)/w2(n), such a condition is equivalent to assuming that gγ is a decreasing function of nγ. Again, gγ = 1/n satisfies these conditions. Similarly with Theorem 2, we also have Theorem 4 Under the regression model and the conditions in Theorem 3, we have plim n→∞ ln BFγν Sγν + nν−nγ 2 ln w2(n) = 1. Furthermore, if Mν is not nested within Mγ, then plimn→∞ ln BFγν Sγν = 1. Here the probability limits are taken w.r.t. the model Mν. 3 Proofs In order to prove these theorems, we first give the following lemmas. Lemma 1 Let A = · A11 A12 A21 A22 ¸ be symmetric and positive definite, and let B = · A−1 11 0 0 0 ¸ have the same size as A. Then A−1 −B is positive semidefinite. Proof The proof follows readily once we express A−1 and B as A−1 = · I −A−1 11 A12 0 I ¸ · A−1 11 0 0 A−1 22·1 ¸ · I 0 −A21A−1 11 I ¸ , B = · I −A−1 11 A12 0 I ¸ · A−1 11 0 0 0 ¸ · I 0 −A21A−1 11 I ¸ , where A22·1 = A22 −A21A−1 11 A12 is also positive definite. The following two lemmas were presented by [1]. Lemma 2 Under the sampling model Mν: (i) if Mν is nested within or equal to a model Mγ, i.e., Mν ⫅Mγ, then plim n→∞ RSSγ n = σ2 and (ii) for any model Mγ that does not contain Mν, if (10) satisfies, then plim n→∞ RSSγ n = σ2 + cγ. Lemma 3 Under the sampling model Mν, if Mν is nested within a model Mγ, i.e., Mν ⊂Mγ, then n ln ³ RSSν RSSγ ´ d −→χ2 nγ−nν as n →∞where d −→denotes convergence in distribution. Lemma 4 Under the regression model (4), if limn→∞gγ(n) = 0 and condition (5) is satisfied, then plim n→∞(1 −F 2 γ )∥y −¯y1n∥2 −RSSγ = 0. Proof It is easy to compute (1 −F 2 γ )∥y −¯y1n∥2 −RSSγ σ2 = y′Kγ[(K′ γKγ)−1 −(K′ γKγ + gγ(n)Kγγ)−1]K′ γy σ2 . Since both K′ γKγ/n and Kγγ are positive definite, there exists an nγ×nγ nonsingular matrix An and an nγ×nγ positive diagonal matrix Λnγ such that K′ γKγ/n = A′ nΛnγAn and Kγγ = A′ nAn. Letting z = σ−1(nΛnγ)−1/2(A′ n)−1K′ γy, we have z ∼Nnγ(σ−1(nΛnγ)1/2Anβ, Inγ) and f(z) ≜(1 −F 2 γ )∥y −¯y1n∥2 −RSSγ σ2 = z′z −z′nΛnγ £ nΛnγ + gγ(n)Inγ ¤−1z = nγ X j=1 gγ(n) nλj(n) + gγ(n)z2 j . Note that z2 j follows a noncentral chi-square distribution, χ2(1, vj), with vj = nλj(n)(aj(n)′β)2/σ2 where λj(n) > 0 is the jth diagonal element of Λnγ and aj(n) is the jth column of An. We thus have E(z2 j ) = 1 + vj and Var(z2 j ) = 2(1 + 2vj). It follows from condition (5) that lim n→∞K′ γKγ/n = lim n→∞A′ nΛnγAn = A′ΛγA, where A is nonsingular and Λγ is a diagonal matrix with positive diagonal elements, and both are independent of n. Hence, lim n→∞E ³ gγ(n) nλj(n) + gγ(n)z2 j ´ = 0 and lim n→∞Var ³ gγ(n) nλj(n) + gγ(n)z2 j ´ = 0. We thus have plimn→∞f(z) = 0. The proof is completed. Lemma 5 Assume that Mκ is nested within Mγ and gγ is a decreasing function of nγ. Then y′(In −eKκΘ−1 κ eK′ κ)y ≥y′(In −eKγΘ−1 γ eK′ γ)y. Proof Since Mκ is nested within Mγ, we express eKγ = [ eKκ, K2] without loss of generality. We now write Σγ = · Σ11 γ Σ12 γ Σ21 γ Σ22 γ ¸ where Σ11 γ is of size nκ×nκ. Hence, we have Θ−1 γ = " eK′ κ eKκ + Σ11 γ eK′ κK2 + Σ12 γ K′ 2 eKκ + Σ21 γ K′ 2K2 + Σ22 γ #−1 . Because 0 < gγ ≤gκ, eK′ κ eKκ+Σκ −( eK′ κ eKκ+Σ11 γ ) = · 0 0 0 (gκ−gγ)Kκκ ¸ is positive semidefinite. Consequently, ( eK′ κ eKκ+Σ11 γ )−1 −( eK′ κ eKκ+Σκ)−1 is positive semidefinite. It follows from Lemma 1 that Θ−1 γ − · ( eK′ κ eKκ+Σκ)−1 0 0 0 ¸ is also positive semidefinite. We thus have y′(In −eKκΘ−1 κ eK′ κ)y −y′(In −eKγΘ−1 γ eK′ γ)y = y′ eKγ µ " eK′ κ eKκ+Σ11 γ eK′ κK2+Σ12 γ K′ 2 eKκ+Σ21 γ K′ 2K2+Σ22 γ #−1 − · ( eK′ κ eKκ+Σκ)−1 0 0 0 ¸ ¶ eK′ γy ≥0. 3.1 Proof of Theorem 1 We now prove Theorem 1. Consider that ln BFγν = 1 2 ln |Qν| |Qγ| + n−1 2 ln (1 −F 2 ν ) (1 −F 2γ ). Because |Qγ|−1 2 = gγ nγ 2 |Kγγ|1/2 |gγKγγ + K′γKγ|1/2 , we have ln |Qν| |Qγ| = ln w1(nγ)nγ w1(nν)nν + ln |Kγγ| |Kνν| + ln ¯¯ w1(nν) nw2(n)Kνν + 1 nK′ νKν ¯¯ ¯¯ w1(nγ) nw2(n)Kγγ + 1 nK′γKγ ¯¯ + (nν−nγ) ln(nw2(n)). Because α = lim n→∞ln ¯¯ w1(nν) nw2(n)Kνν + 1 nK′ νKν ¯¯ ¯¯ w1(nγ) nw2(n)Kγγ + 1 nK′γKγ ¯¯ = lim n→∞ln | 1 nK′ νKν| | 1 nK′γKγ| ∈(−∞, ∞), it is easily proven that lim n→∞ 1 2 ln |Qν| |Qγ| = ( ∞ nγ < nν −∞ nγ > nν const nγ = nν, (15) where const = α 2 + 1 2 ln |Kγγ| |Kνν|. According to Lemma 4, we also have plim n→∞ n−1 2 ln (1−F 2 ν ) (1−F 2γ ) = plim n→∞ n−1 2 ln (1−F 2 ν )∥y−¯y1n∥2 (1−F 2γ )∥y−¯y1n∥2 = plim n→∞ n−1 2 ln RSSν RSSγ . Now consider the following two cases: (a) Mν is not nested within Mγ: From Lemma 2, we obtain plim n→∞ln RSSν RSSγ = plim n→∞ln RSSν/n RSSγ/n = ln σ2 σ2+cγ . Moreover, we have the following limit lim n→∞ n−1 2 h ln ³ σ2 σ2+cγ ´ + nν−nγ n−1 ln(nw2(n)) i = −∞ due to limn→∞ nν−nγ n−1 ln(nw2(n)) = limn→∞(nν−nγ) w2(n)+nw′ 2(n) nw2(n) = 0 and ln ³ σ2 σ2+cγ ´ < 1. This implies that limn→∞ln BFγν = −∞. Thus we obtain limn→∞BFγν = 0. (b) Mν is nested within Mγ: We always have nγ > nν. By Lemma 3, we have (n−1) ln(RSSν/RSSγ) d −→χ2 nγ−nν. Hence, (RSSν/RSSγ)(n−1)/2 d −→exp(χ2 nγ−nν/2). Combining this result with (15) leads to a zero limit for BFγν. 3.2 Proof of Theorem 2 Using the same notations as those in Theorem 1, we have Cγν = ln BFγν Sγν + nν−nγ 2 ln w2(n) = n−1 n ln (1−F 2 ν ) (1−F 2 γ ) + nν−nγ n ln(nw2(n)) + 2 nConst ln RSSν RSSγ + nν−nγ n ln(nw2(n)) . (a) Mν is not nested within Mγ: From Lemma 4, we obtain plim n→∞Cγν = lim n→∞ ln σ2 σ2+cγ + nν−nγ n ln(nw2(n)) ln σ2 σ2+cγ + nν−nγ n ln(nw2(n)) = 1. In this case, we also have plim n→∞ ln BFγν Sγν = lim n→∞ ln σ2 σ2+cγ + nν−nγ n ln(nw2(n)) ln σ2 σ2+cγ + nν−nγ n ln n = 1. (b) Mν is nested within Mγ: We obtain plim n→∞Cγν = plim n→∞ (n−1) ln (1−F 2 ν ) (1−F 2 γ ) + (nν−nγ) ln(nw2(n)) + 2 × Const n ln RSSν RSSγ + (nν−nγ) ln(nw2(n)) = 1 due to nγ > nν and n ln(RSSν/RSSγ) d −→χ2 nγ−nν. 3.3 Proof of Theorem 3 We now sketch the proof of Theorem 3. For the case that Mν is not nested within Mγ, the proof is similar to that of Theorem 1. When Mν is nested within Mγ, Lemma 5 shows the following relationship ln ·bσ + y′¡ In−eKνΘ−1 ν eK′ ν ¢ y bσ + y′¡ In−eKγΘ−1 γ eK′γ ¢ y ¸ ≤ln ·y′¡ In−eKνΘ−1 ν eK′ ν ¢ y y′¡ In−eKγΘ−1 γ eK′γ ¢ y ¸ . We thus have plim n→∞ aσ+n 2 ln ·bσ + y′¡ In−eKνΘ−1 ν eK′ ν ¢ y bσ + y′¡ In−eKγΘ−1 γ eK′γ ¢ y ¸ ≤plim n→∞ aσ+n 2 ln ·y′¡ In−eKνΘ−1 ν eK′ ν ¢ y y′¡ In−eKγΘ−1 γ eK′γ ¢ y ¸ = plim n→∞ aσ+n 2 ln ·y′¡ In−eHν ¢ y y′¡ In−eHγ ¢ y ¸ ∈(0, ∞). From this result the proof follows readily. 4 Conclusions In this paper we have presented a frequentist analysis of a Bayesian model choice procedure for sparse regression. We have captured sparsity by a particular choice of prior distribution which we have referred to as a “Silverman g-prior.” This prior emerges naturally from the RKHS perspective. It is similar in spirit to the Zellner g-prior, which has been widely used for Bayesian variable selection and Bayesian model selection due to its computational tractability in the evaluation of marginal likelihoods [6, 2]. Our analysis provides a theoretical foundation for the Silverman g-prior and suggests that it can play a similarly wide-ranging role in the development of fully Bayesian kernel methods. References [1] C. Fern´andez, E. Ley, and M. F. J. Steel. Benchmark priors for Bayesian model averaging. Journal of Econometrics, 100:381–427, 2001. [2] E. I. George and R. E. McCulloch. Approaches for Bayesian variable selection. Statistica Sinica, 7:339–374, 1997. [3] R. E. Kass and A. E. Raftery. Bayes factors. Journal of the American Statistical Association, 90:773–795, 1995. [4] F. Liang, R. Paulo, G. Molina, M. A. Clyde, and J. O. Berger. Mixtures of g-priors for Bayesian variable selection. Journal of the American Statistical Association, 103(481):410–423, 2008. [5] B. W. Silverman. Some aspects of the spline smoothing approach to non-parametric regression curve fitting (with discussion). Journal of the Royal Statistical Society, B, 47(1):1–52, 1985. [6] M. Smith and R. Kohn. Nonparametric regression using Bayesian variable selection. Journal of Econometrics, 75:317–344, 1996. [7] G. Wahba. Spline Models for Observational Data. SIAM, Philadelphia, 1990. [8] M. West. Bayesian factor regression models in the “large p, small n” paradigm. In J. M. Bernardo, M. J. Bayarri, J. .O Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith, and M. West, editors, Bayesian Statistics 7, pages 723–732. Oxford University Press, 2003. [9] A. Zellner. On assessing prior distributions and Bayesian regression analysis with g−prior distributions. In P. K. Goel and A. Zellner, editors, Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti, pages 233–243. North-Holland, Amsterdam, 1986.
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Unlabeled data: Now it helps, now it doesn’t Aarti Singh, Robert D. Nowak∗ Xiaojin Zhu† Department of Electrical and Computer Engineering Department of Computer Sciences University of Wisconsin - Madison University of Wisconsin - Madison Madison, WI 53706 Madison, WI 53706 {singh@cae,nowak@engr}.wisc.edu jerryzhu@cs.wisc.edu Abstract Empirical evidence shows that in favorable situations semi-supervised learning (SSL) algorithms can capitalize on the abundance of unlabeled training data to improve the performance of a learning task, in the sense that fewer labeled training data are needed to achieve a target error bound. However, in other situations unlabeled data do not seem to help. Recent attempts at theoretically characterizing SSL gains only provide a partial and sometimes apparently conflicting explanations of whether, and to what extent, unlabeled data can help. In this paper, we attempt to bridge the gap between the practice and theory of semi-supervised learning. We develop a finite sample analysis that characterizes the value of unlabeled data and quantifies the performance improvement of SSL compared to supervised learning. We show that there are large classes of problems for which SSL can significantly outperform supervised learning, in finite sample regimes and sometimes also in terms of error convergence rates. 1 Introduction Labeled data can be expensive, time-consuming and difficult to obtain in many applications. Semisupervised learning (SSL) aims to capitalize on the abundance of unlabeled data to improve learning performance. Empirical evidence suggests that in certain favorable situations unlabeled data can help, while in other situations it does not. As a result, there have been several recent attempts [1, 2, 3, 4, 5, 6] at developing a theoretical understanding of semi-supervised learning. It is wellaccepted that unlabeled data can help only if there exists a link between the marginal data distribution and the target function to be learnt. Two common types of links considered are the cluster assumption [7, 3, 4] which states that the target function is locally smooth over subsets of the feature space delineated by some property of the marginal density (but may not be globally smooth), and the manifold assumption [4, 6] which assumes that the target function lies on a low-dimensional manifold. Knowledge of these sets, which can be gleaned from unlabeled data, simplify the learning task. However, recent attempts at characterizing the amount of improvement possible under these links only provide a partial and sometimes apparently conflicting (for example, [4] vs. [6]) explanations of whether or not, and to what extent semi-supervised learning helps. In this paper, we bridge the gap between these seemingly conflicting views and develop a minimax framework based on finite sample bounds to identify situations in which unlabeled data help to improve learning. Our results quantify both the amount of improvement possible using SSL as well as the the relative value of unlabeled data. We focus on learning under a cluster assumption that is formalized in the next section, and establish that there exist nonparametric classes of distributions, denoted PXY , for which the decision sets (over which the target function is smooth) are discernable from unlabeled data. Moreover, we show that there exist clairvoyant supervised learners that, given perfect knowledge of the decision sets denoted by D, can significantly outperform any generic supervised learner fn in these ∗Supported in part by the NSF grants CCF-0353079, CCF-0350213, and CNS-0519824. †Supported in part by the Wisconsin Alumni Research Foundation. 1 (a) (b) (c) Figure 1: (a) Two separated high density sets with different labels that (b) cannot be discerned if the sample size is too small, but (c) can be estimated if sample density is high enough. classes. That is, if R denotes a risk of interest, n denotes the labeled data sample size, bfD,n denotes the clairvoyant supervised learner, and E denotes expectation with respect to training data, then supPXY E[R( bfD,n)] < inffn supPXY E[R(fn)]. Based on this, we establish that there also exist semi-supervised learners, denoted bfm,n, that use m unlabeled examples in addition to the n labeled examples in order to estimate the decision sets, which perform as well as bfD,n, provided that m grows appropriately relative to n. Specifically, if the error bound for bfD,n decays polynomially (exponentially) in n, then the number of unlabeled data m needs to grow polynomially (exponentially) with the number of labeled data n. We provide general results for a broad range of learning problems using finite sample error bounds. Then we examine a concrete instantiation of these general results in the regression setting by deriving minimax lower bounds on the performance of any supervised learner and compare that to upper bounds on the errors of bfD,n and bfm,n. In their seminal papers, Castelli and Cover [8, 9] suggested that in the classification setting the marginal distribution can be viewed as a mixture of class conditional distributions. If this mixture is identifiable, then the classification problem may reduce to a simple hypothesis testing problem for which the error converges exponentially fast in the number of labeled examples. The ideas in this paper are similar, except that we do not require identifiability of the mixture component densities, and show that it suffices to only approximately learn the decision sets over which the label is smooth. More recent attempts at theoretically characterizing SSL have been relatively pessimistic. Rigollet [3] establishes that for a fixed collection of distributions satisfying a cluster assumption, unlabeled data do not provide an improvement in convergence rate. A similar argument was made by Lafferty and Wasserman [4], based on the work of Bickel and Li [10], for the manifold case. However, in a recent paper, Niyogi [6] gives a constructive example of a class of distributions supported on a manifold whose complexity increases with the number of labeled examples, and he shows that the error of any supervised learner is bounded from below by a constant, whereas there exists a semisupervised learner that can provide an error bound of O(n−1/2), assuming infinite unlabeled data. In this paper, we bridge the gap between these seemingly conflicting views. Our arguments can be understood by the simple example shown in Fig. 1, where the distribution is supported on two component sets separated by a margin γ and the target function is smooth over each component. Given a finite sample of data, these decision sets may or may not be discernable depending on the sampling density (see Fig. 1(b), (c)). If γ is fixed (this is similar to fixing the class of cluster-based distributions in [3] or the manifold in [4, 10]), then given enough labeled data a supervised learner can achieve optimal performance (since, eventually, it operates in regime (c) of Fig. 1). Thus, in this example, there is no improvement due to unlabeled data in terms of the rate of error convergence for a fixed collection of distributions. However, since the true separation between the component sets is unknown, given a finite sample of data, there always exists a distribution for which these sets are indiscernible (e.g., γ →0). This perspective is similar in spirit to the argument in [6]. We claim that meaningful characterizations of SSL performance and quantifications of the value of unlabeled data require finite sample error bounds, and that rates of convergence and asymptotic analysis may not capture the distinctions between SSL and supervised learning. Simply stated, if the component density sets are discernable from a finite sample size m of unlabeled data but not from a finite sample size n < m of labeled data, then SSL can provide better performance than supervised learning. We also show that there are certain plausible situations in which SSL yields rates of convergence that cannot be achieved by any supervised learner. 2 γ positive γ negative γ γ x1 x2 g (x ) 2 (2) g (x ) 1 (1) g (x ) 1 (2) g (x ) 2 (1) x2 x1 g (x ) 2 (2) g (x ) 1 (1) g (x ) 2 (1) g (x ) 1 (2) 1 1 1 1 1 1 1 1 Figure 2: Margin γ measures the minimum width of a decision set or separation between the support sets of the component marginal mixture densities. The margin is positive if the component support sets are disjoint, and negative otherwise. 2 Characterization of model distributions under the cluster assumption Based on the cluster assumption [7, 3, 4], we define the following collection of joint distributions PXY (γ) = PX × PY |X indexed by a margin parameter γ. Let X, Y be bounded random variables with marginal distribution PX ∈PX and conditional label distribution PY |X ∈PY |X, supported on the domain X = [0, 1]d. The marginal density p(x) = PK k=1 akpk(x) is the mixture of a finite, but unknown, number of component densities {pk}K k=1, where K < ∞. The unknown mixing proportions ak ≥a > 0 and PK k=1 ak = 1. In addition, we place the following assumptions on the mixture component densities: 1. pk is supported on a unique compact, connected set Ck ⊆X with Lipschitz boundaries. Specifically, we assume the following form for the component support sets: (See Fig. 2 for d=2 illustration.) Ck = {x ≡(x1, . . . , xd) ∈X : g(1) k (x1, . . . , xd−1) ≤xd ≤g(2) k (x1, . . . , xd−1)}, where g(1) k (·), g(2) k (·) are d −1 dimensional Lipschitz functions with Lipschitz constant L.1 2. pk is bounded from above and below, 0 < b ≤pk ≤B. 3. pk is H¨older-α smooth on Ck with H¨older constant K1 [12, 13]. Let the conditional label density on Ck be denoted by pk(Y |X = x). Thus, a labeled training point (X, Y ) is obtained as follows. With probability ak, X is drawn from pk and Y is drawn from pk(Y |X = x). In the supervised setting, we assume access to n labeled data L = {Xi, Yi}n i=1 drawn i.i.d according to PXY ∈PXY (γ), and in the semi-supervised setting, we assume access to m additional unlabeled data U = {Xi}m i=1 drawn i.i.d according to PX ∈PX. Let D denote the collection of all non-empty sets obtained as intersections of {Ck}K k=1 or their complements {Cc k}K k=1, excluding the set ∩K k=1Cc k that does not lie in the support of the marginal density. Observe that |D| ≤2K, and in practical situations the cardinality of D is much smaller as only a few of the sets are non-empty. The cluster assumption is that the target function will be smooth on each set D ∈D, hence the sets in D are called decision sets. At this point, we do not consider a specific target function. The collection PXY is indexed by a margin parameter γ, which denotes the minimum width of a decision set or separation between the component support sets Ck. The margin γ is assigned a positive sign if there is no overlap between components, otherwise it is assigned a negative sign as illustrated in Figure 2. Formally, for j, k ∈{1, . . . , K}, let djk := min p,q∈{1,2} ∥g(p) j −g(q) k ∥∞ j ̸= k, dkk := ∥g(1) k −g(2) k ∥∞. Then the margin is defined as γ = σ · min j,k∈{1,...,K} djk, where σ = 1 if Cj ∩Ck = ∅∀j ̸= k −1 otherwise . 1This form is a slight generalization of the boundary fragment class of sets which is used as a common tool for analysis of learning problems [11]. Boundary fragment sets capture the salient characteristics of more general decision sets since, locally, the boundaries of general sets are like fragments in a certain orientation. 3 3 Learning Decision Sets Ideally, we would like to break a given learning task into separate subproblems on each D ∈D since the target function is smooth on each decision set. Note that the marginal density p is also smooth within each decision set, but exhibits jumps at the boundaries since the component densities are bounded away from zero. Hence, the collection D can be learnt from unlabeled data as follows: 1) Marginal density estimation — The procedure is based on the sup-norm kernel density estimator proposed in [14]. Consider a uniform square grid over the domain X = [0, 1]d with spacing 2hm, where hm = κ0 ((log m)2/m)1/d and κ0 > 0 is a constant. For any point x ∈X, let [x] denote the closest point on the grid. Let G denote the kernel and Hm = hmI, then the estimator of p(x) is bp(x) = 1 mhdm m X i=1 G(H−1 m (Xi −[x])). 2) Decision set estimation — Two points x1, x2 ∈X are said to be connected, denoted by x1 ↔x2, if there exists a sequence of points x1 = z1, z2, . . . , zl−1, zl = x2 such that z2, . . . , zl−1 ∈U, ∥zj−zj+1∥≤2 √ dhm, and for all points that satisfy ∥zi−zj∥≤hm log m, |bp(zi)−bp(zj)| ≤δm := (log m)−1/3. That is, there exists a sequence of 2 √ dhm-dense unlabeled data points between x1 and x2 such that the marginal density varies smoothly along the sequence. All points that are pairwise connected specify an empirical decision set. This decision set estimation procedure is similar in spirit to the semi-supervised learning algorithm proposed in [15]. In practice, these sequences only need to be evaluated for the test and labeled training points. The following lemma shows that if the margin is large relative to the average spacing m−1/d between unlabeled data points, then with high probability, two points are connected if and only if they lie in the same decision set D ∈D, provided the points are not too close to the decision boundaries. The proof sketch of the lemma and all other results are deferred to Section 7. Lemma 1. Let ∂D denote the boundary of D and define the set of boundary points as B = {x : inf z∈∪D∈D∂D ∥x −z∥≤2 √ dhm}. If |γ| > Co(m/(log m)2)−1/d, where Co = 6 √ dκ0, then for all p ∈PX, all pairs of points x1, x2 ∈supp(p) \ B and all D ∈D, with probability > 1 −1/m, x1, x2 ∈D if and only if x1 ↔x2 for large enough m ≥m0, where m0 depends only on the fixed parameters of the class PXY (γ). 4 SSL Performance and the Value of Unlabeled Data We now state our main result that characterizes the performance of SSL relative to supervised learning and follows as a corollary to the lemma stated above. Let R denote a risk of interest and E( bf) = R( bf) −R∗, where R∗is the infimum risk over all possible learners. Corollary 1. Assume that the excess risk E is bounded. Suppose there exists a clairvoyant supervised learner bfD,n, with perfect knowledge of the decision sets D, for which the following finite sample upper bound holds sup PXY (γ) E[E( bfD,n)] ≤ǫ2(n). Then there exists a semi-supervised learner bfm,n such that if |γ| > Co(m/(log m)2)−1/d, sup PXY (γ) E[E( bfm,n)] ≤ǫ2(n) + O 1 m + n m (log m)2 −1/d! . This result captures the essence of the relative characterization of semi-supervised and supervised learning for the margin based model distributions. It suggests that if the sets D are discernable using unlabeled data (the margin is large enough compared to average spacing between unlabeled data points), then there exists a semi-supervised learner that can perform as well as a supervised learner with clairvoyant knowledge of the decision sets, provided m ≫n so that (n/ǫ2(n))d = 4 O(m/(log m)2) implying that the additional term in the performace bound for bfm,n is negligible compared to ǫ2(n). This indicates that if ǫ2(n) decays polynomially (exponentially) in n, then m needs to grow polynomially (exponentially) in n. Further, suppose that the following finite sample lower bound holds for any supervised learner: inf fn sup PXY (γ) E[E(fn)] ≥ǫ1(n). If ǫ2(n) < ǫ1(n), then there exists a clairvoyant supervised learner with perfect knowledge of the decision sets that outperforms any supervised learner that does not have this knowledge. Hence, Corollary 1 implies that SSL can provide better performance than any supervised learner provided (i) m ≫n so that (n/ǫ2(n))d = O(m/(log m)2), and (ii) knowledge of the decision sets simplifies the supervised learning task, so that ǫ2(n) < ǫ1(n). In the next section, we provide a concrete application of this result in the regression setting. As a simple example in the binary classification setting, if p(x) is supported on two disjoint sets and if P(Y = 1|X = x) is strictly greater than 1/2 on one set and strictly less than 1/2 on the other, then perfect knowledge of the decision sets reduces the problem to a hypothesis testing problem for which ǫ2(n) = O(e−ζ n), for some constant ζ > 0. However, if γ is small relative to the average spacing n−1/d between labeled data points, then ǫ1(n) = cn−1/d where c > 0 is a constant. This lower bound follows from the minimax lower bound proofs for regression in the next section. Thus, an exponential improvement is possible using semi-supervised learning provided m grows exponentially in n. 5 Density-adaptive Regression Let Y denote a continuous and bounded random variable. Under squared error loss, the target function is f(x) = E[Y |X = x], and E( bf) = E[( bf(X) −f(X))2]. Recall that pk(Y |X = x) is the conditional density on the k-th component and let Ek denote expectation with respect to the corresponding conditional distribution. The regression function on each component is fk(x) = Ek[Y |X = x] and we assume that for k = 1, . . . , K 1. fk is uniformly bounded, |fk| ≤M. 2. fk is H¨older-α smooth on Ck with H¨older constant K2. This implies that the overall regression function f(x) is piecewise H¨older-α smooth; i.e., it is H¨older-α smooth on each D ∈D, except possibly at the component boundaries. 2 Since a H¨older-α smooth function can be locally well-approximated by a Taylor polynomial, we propose the following semi-supervised learner that performs local polynomial fits within each empirical decision set, that is, using training data that are connected as per the definition in Section 3. While a spatially uniform estimator suffices when the decision sets are discernable, we use the following spatially adaptive estimator proposed in Section 4.1 of [12]. This ensures that when the decision sets are indiscernible using unlabeled data, the semi-supervised learner still achieves an error bound that is, up to logarithmic factors, no worse than the minimax lower bound for supervised learners. bfm,n,x(·) = arg min f ′∈Γ n X i=1 (Yi −f ′(Xi))21x↔Xi + pen(f ′) and bfm,n(x) ≡bfm,n,x(x) Here 1x↔Xi is the indicator of x ↔Xi and Γ denotes a collection of piecewise polynomials of degree [α] (the maximal integer < α) defined over recursive dyadic partitions of the domain X = [0, 1]d with cells of sidelength between 2−⌈log(n/ log n)/(2α+d)⌉and 2−⌈log(n/ log n)/d⌉. The penalty term pen(f ′) is proportional to log(Pn i=1 1x↔Xi) #f ′, where #f ′ denotes the number of cells in the recursive dyadic partition on which f ′ is defined. It is shown in [12] that this estimator yields a finite sample error bound of n−2α/(2α+d) for H¨older-α smooth functions, and max{n−2α/(2α+d), n−1/d} for piecewise H¨older-α functions, ignoring logarithmic factors. Using these results from [12] and Corollary 1, we now state finite sample upper bounds on the semisupervised learner (SSL) described above. Also, we derive finite sample minimax lower bounds on the performance of any supervised learner (SL). Our main results are summarized in the following table, for model distributions characterized by various values of the margin parameter γ. A sketch 2If the component marginal densities and regression functions have different smoothnesses, say α and β, the same analysis holds except that f(x) is H¨older-min(α, β) smooth on each D ∈D. 5 of the derivations of the results is provided in Section 7.3. Here we assume that dimension d ≥ 2α/(2α −1). If d < 2α/(2α −1), then the supervised learning error due to to not resolving the decision sets (which behaves like n−1/d) is smaller than error incurred in estimating the target function itself (which behaves like n−2α/(2α+d)). Thus, when d < 2α/(2α −1), the supervised regression error is dominated by the error in smooth regions and there appears to be no benefit to using a semi-supervised learner. In the table, we suppress constants and log factors in the bounds, and also assume that m ≫n2d so that (n/ǫ2(n))d = O(m/(log m)2). The constants co and Co only depend on the fixed parameters of the class PXY (γ) and do not depend on γ. Margin range SSL upper bound SL lower bound SSL helps γ ǫ2(n) ǫ1(n) γ ≥γ0 n−2α/(2α+d) n−2α/(2α+d) No γ ≥con−1/d n−2α/(2α+d) n−2α/(2α+d) No con−1/d > γ ≥Co( m (log m)2 )−1/d n−2α/(2α+d) n−1/d Yes Co( m (log m)2 )−1/d > γ ≥−Co( m (log m)2 )−1/d n−1/d n−1/d No −Co( m (log m)2 )−1/d > γ n−2α/(2α+d) n−1/d Yes −γ0 > γ n−2α/(2α+d) n−1/d Yes If γ is large relative to the average spacing between labeled data points n−1/d, then a supervised learner can discern the decision sets accurately and SSL provides no gain. However, if γ > 0 is small relative to n−1/d, but large with respect to the spacing between unlabeled data points m−1/d, then the proposed semi-supervised learner provides improved error bounds compared to any supervised learner. If |γ| is smaller than m−1/d, the decision sets are not discernable with unlabeled data and SSL provides no gain. However, notice that the performance of the semi-supervised learner is no worse than the minimax lower bound for supervised learners. In the γ < 0 case, if −γ larger than m−1/d, then the semi-supervised learner can discern the decision sets and achieves smaller error bounds, whereas these sets cannot be as accurately discerned by any supervised learner. For the overlap case (γ < 0), supervised learners are always limited by the error incurred due to averaging across decision sets (n−1/d). In particular, for the collection of distributions with γ < −γ0, a faster rate of error convergence is attained by SSL compared to SL, provided m ≫n2d. 6 Conclusions In this paper, we develop a framework for evaluating the performance gains possible with semisupervised learning under a cluster assumption using finite sample error bounds. The theoretical characterization we present explains why in certain situations unlabeled data can help to improve learning, while in other situations they may not. We demonstrate that there exist general situations under which semi-supervised learning can be significantly superior to supervised learning in terms of achieving smaller finite sample error bounds than any supervised learner, and sometimes in terms of a better rate of error convergence. Moreover, our results also provide a quantification of the relative value of unlabeled to labeled data. While we focus on the cluster assumption in this paper, we conjecture that similar techniques can be applied to quantify the performance of semi-supervised learning under the manifold assumption as well. In particular, we believe that the use of minimax lower bounding techniques is essential because many of the interesting distinctions between supervised and semi-supervised learning occur only in finite sample regimes, and rates of convergence and asymptotic analyses may not capture the complete picture. 7 Proofs We sketch the main ideas behind the proofs here, please refer to [13] for details. Since the component densities are bounded from below and above, define pmin := b mink ak ≤p(x) ≤B =: pmax. 7.1 Proof of Lemma 1 First, we state two relatively straightforward results about the proposed kernel density estimator. Theorem 1 (Sup-norm density estimation of non-boundary points). Consider the kernel density estimator bp(x) proposed in Section 3. If the kernel G satisfies supp(G) = [−1, 1]d, 0 < G ≤ Gmax < ∞, R [−1,1]d G(u)du = 1 and R [−1,1]d ujG(u)du = 0 for 1 ≤j ≤[α], then for all 6 p ∈PX, with probability at least 1 −1/m, sup x∈supp(p)\B |p(x) −bp(x)| = O hmin(1,α) m + q log m/(mhdm) =: ǫm. Notice that ǫm decreases with increasing m. A detailed proof can be found in [13]. Corollary 2 (Empirical density of unlabeled data). Under the conditions of Theorem 1, for all p ∈PX and large enough m, with probability > 1 −1/m, for all x ∈supp(p) \ B, ∃Xi ∈U s.t. ∥Xi −x∥≤ √ dhm. Proof. From Theorem 1, for all x ∈supp(p) \ B, bp(x) ≥p(x) −ǫm ≥pmin −ǫm > 0, for m sufficiently large. This implies Pm i=1 G(H−1 m (Xi −x)) > 0, and ∃Xi ∈U within √ dhm of x. Using the density estimation results, we now show that if |γ| > 6 √ dhm, then for all p ∈PX, all pairs of points x1, x2 ∈supp(p)\B and all D ∈D, for large enough m, with probability > 1−1/m, we have x1, x2 ∈D if and only if x1 ↔x2. We establish this in two steps: 1. x1 ∈D, x2 ̸∈D ⇒x1 ̸↔x2 : Since x1 and x2 belong to different decision sets, all sequences connecting x1 and x2 through unlabeled data points pass through a region where either (i) the density is zero and since the region is at least |γ| > 6 √ dhm wide, there cannot exist a sequence as defined in Section 3 such that ∥zj −zj+1∥≤2 √ dhm, or (ii) the density is positive. In the latter case, the marginal density p(x) jumps by at least pmin one or more times along all sequences connecting x1 and x2. Suppose the first jump occurs where decision set D ends and another decision set D′ ̸= D begins (in the sequence). Then since D′ is at least |γ| > 6 √ dhm wide, by Corollary 2 for all sequences connecting x1 and x2 through unlabeled data points, there exist points zi, zj in the sequence that lie in D \ B, D′ \ B, respectively, and ∥zi −zj∥≤hm log m. Since the density on each decision set is H¨older-α smooth, we have |p(zi) −p(zj)| ≥pmin −O((hm log m)min(1,α)). Since zi, zj ̸∈B, using Theorem 1, |bp(zi) −bp(zj)| ≥|p(zi) −p(zj)| −2ǫm > δm for large enough m. Thus, x1 ̸↔x2. 2. x1, x2 ∈D ⇒x1 ↔x2 : Since D has width at least |γ| > 6 √ dhm, there exists a region of width > 2 √ dhm contained in D \ B, and Corollary 2 implies that with probability > 1 −1/m, there exist sequence(s) contained in D \ B connecting x1 and x2 through 2 √ dhm-dense unlabeled data points. Since the sequence is contained in D and the density on D is H¨older-α smooth, we have for all points zi, zj in the sequence that satisfy ∥zi −zj∥≤hm log m, |p(zi) −p(zj)| ≤O((hm log m)min(1,α)). Since zi, zj ̸∈B, using Theorem 1, |bp(zi) −bp(zj)| ≤|p(zi) −p(zj)| + 2ǫm ≤δm for large enough m. Thus, x1 ↔x2. □ 7.2 Proof of Corollary 1 Let Ω1 denote the event under which Lemma 1 holds. Then P(Ωc 1) ≤1/m. Let Ω2 denote the event that the test point X and training data X1, . . . , Xn ∈L don’t lie in B. Then P(Ωc 2) ≤ (n + 1)P(B) ≤(n + 1)pmaxvol(B) = O(nhm). The last step follows from the definition of the set B and since the boundaries of the support sets are Lipschitz, K is finite, and hence vol(B) = O(hm). Now observe that bfD,n essentially uses the clairvoyant knowledge of the decision sets D to discern which labeled points X1, . . . , Xn are in the same decision set as X. Conditioning on Ω1, Ω2, Lemma 1 implies that X, Xi ∈D iff X ↔Xi. Thus, we can define a semi-supervised learner bfm,n to be the same as bfD,n except that instead of using clairvoyant knowledge of whether X, Xi ∈D, bfm,n is based on whether X ↔Xi. It follows that supPXY (γ) E[E( bfm,n)|Ω1, Ω2] = supPXY (γ) E[E( bfD,n)], and since the excess risk is bounded: supPXY (γ) E[E( bfm,n)] ≤supPXY (γ) E[E( bfm,n)|Ω1, Ω2] + O (1/m + nhm) . □ 7.3 Density adaptive Regression results 1) Semi-Supervised Learning Upper Bound: The clairvoyant counterpart of bfm,n(x) is given as bfD,n(x) ≡bfD,n,x(x), where bfD,n,x(·) = arg minf ′∈Γ Pn i=1(Yi −f ′(Xi))21x,Xi∈D +pen(f ′), and is a standard supervised learner that performs piecewise polynomial fit on each decision set, where the regression function is H¨older-α smooth. Let nD = 1 n Pn i=1 1Xi∈D. It follows [12] that E[(f(X) −bfD,n(X))21X∈D|nD] ≤C (nD/ log nD)− 2α d+2α . 7 Since E[(f(X) −bfD,n(X))2] = P D∈D E[(f(X) −bfD,n(X))21X∈D]P(D), taking expectation over nD ∼Binomial(n, P(D)) and summing over all decision sets recalling that |D| is a finite constant, the overall error of bfD,n scales as n−2α/(2α+d), ignoring logarithmic factors. If |γ| > Co(m/(log m)2)−1/d, using Corollary 1, the same performance bound holds for bfm,n provided m ≫n2d. See [13] for further details. If |γ| < Co(m/(log m)2)−1/d, the decision sets are not discernable using unlabeled data. Since the regression function is piecewise H¨older-α smooth on each empirical decision set, Using Theorem 9 in [12] and similar analysis, an upper bound of max{n−2α/(2α+d), n−1/d} follows, which scales as n−1/d when d ≥2α/(2α −1). 2) Supervised Learning Lower Bound: The formal minimax proof requires construction of a finite subset of distributions in PXY (γ) that are the hardest cases to distinguish based on a finite number of labeled data n, and relies on a Hellinger version of Assouad’s Lemma (Theorem 2.10 (iii) in [16]). Complete details are given in [13]. Here we present the simple intuition behind the minimax lower bound of n−1/d when γ < con−1/d. In this case the decision boundaries can only be localized to an accuracy of n−1/d, the average spacing between labeled data points. Since the boundaries are Lipschitz, the expected volume that is incorrectly assigned to any decision set is > c1n−1/d, where c1 > 0 is a constant. Thus, if the expected excess risk at a point that is incorrectly assigned to a decision set can be greater than a constant c2 > 0, then the overall expected excess risk is > c1c2n−1/d. This is the case for both regression and binary classification. If γ > con−1/d, the decision sets can be accurately discerned from the labeled data alone. In this case, it follows that the minimax lower bound is equal to the minimax lower bound for H¨older-α smooth regression functions, which is cn−2α/(d+2α), where c > 0 is a constant [17]. References [1] Balcan, M.F., Blum, A.: A PAC-style model for learning from labeled and unlabeled data. In: 18th Annual Conference on Learning Theory, COLT. (2005) [2] K¨a¨ari¨ainen, M.: Generalization error bounds using unlabeled data. In: 18th Annual Conference on Learning Theory, COLT. (2005) [3] Rigollet, P.: Generalization error bounds in semi-supervised classification under the cluster assumption. Journal of Machine Learning Research 8 (2007) 1369–1392 [4] Lafferty, J., Wasserman, L.: Statistical analysis of semi-supervised regression. In: Advances in Neural Information Processing Systems 21, NIPS. (2007) 801–808 [5] Ben-David, S., Lu, T., Pal, D.: Does unlabeled data provably help? worst-case analysis of the sample complexity of semi-supervised learning. In: 21st Annual Conference on Learning Theory, COLT. (2008) [6] Niyogi, P.: Manifold regularization and semi-supervised learning: Some theoretical analyses. Technical Report TR-2008-01, Computer Science Department, University of Chicago. URL http://people.cs.uchicago.edu/∼niyogi/papersps/ssminimax2.pdf (2008) [7] Seeger, M.: Learning with labeled and unlabeled data. Technical report, Institute for ANC, Edinburgh, UK. URL http://www.dai.ed.ac.uk/∼seeger/papers.html (2000) [8] Castelli, V., Cover, T.M.: On the exponential value of labeled samples. Pattern Recognition Letters 16(1) (1995) 105–111 [9] Castelli, V., Cover, T.M.: The relative value of labeled and unlabeled samples in pattern recognition. IEEE Transactions on Information Theory 42(6) (1996) 2102–2117 [10] Bickel, P.J., Li, B.: Local polynomial regression on unknown manifolds. In: IMS Lecture NotesMonograph Series, Complex Datasets and Inverse Problems: Tomography, Networks and Beyond. Volume 54. (2007) 177–186 [11] Korostelev, A.P., Tsybakov, A.B.: Minimax Theory of Image Reconstruction. Springer, NY (1993) [12] Castro, R., Willett, R., Nowak, R.: Faster rates in regression via active learning. Technical Report ECE-05-03, ECE Department, University of Wisconsin Madison. URL http://www.ece.wisc.edu/∼nowak/ECE-05-03.pdf (2005) [13] Singh, A., Nowak, R., Zhu, X.: Finite sample analysis of semi-supervised learning. Technical Report ECE-08-03, ECE Department, University of Wisconsin Madison. URL http://www.ece.wisc.edu/∼nowak/SSL TR.pdf (2008) [14] Korostelev, A., Nussbaum, M.: The asymptotic minimax constant for sup-norm loss in nonparametric density estimation. Bernoulli 5(6) (1999) 1099–1118 [15] Chapelle, O., Zien, A.: Semi-supervised classification by low density separation. In: Tenth International Workshop on Artificial Intelligence and Statistics. (2005) 57–64 [16] Tsybakov, A.B.: Introduction a l’estimation non-parametrique. Springer, Berlin Heidelberg (2004) [17] Stone, C.J.: Optimal rates of convergence for nonparametric estimators. The Annals of Statistics 8(6) (1980) 1348–1360 8
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Nonparametric Bayesian Learning of Switching Linear Dynamical Systems Emily B. Fox Electrical Engineering & Computer Science, Massachusetts Institute of Technology ebfox@mit.edu Erik B. Sudderth†, Michael I. Jordan†‡ †Electrical Engineering & Computer Science and ‡Statistics, University of California, Berkeley {sudderth, jordan}@eecs.berkeley.edu Alan S. Willsky Electrical Engineering & Computer Science, Massachusetts Institute of Technology willsky@mit.edu Abstract Many nonlinear dynamical phenomena can be effectively modeled by a system that switches among a set of conditionally linear dynamical modes. We consider two such models: the switching linear dynamical system (SLDS) and the switching vector autoregressive (VAR) process. Our nonparametric Bayesian approach utilizes a hierarchical Dirichlet process prior to learn an unknown number of persistent, smooth dynamical modes. We develop a sampling algorithm that combines a truncated approximation to the Dirichlet process with efficient joint sampling of the mode and state sequences. The utility and flexibility of our model are demonstrated on synthetic data, sequences of dancing honey bees, and the IBOVESPA stock index. 1 Introduction Linear dynamical systems (LDSs) are useful in describing dynamical phenomena as diverse as human motion [9], financial time-series [4], maneuvering targets [6, 10], and the dance of honey bees [8]. However, such phenomena often exhibit structural changes over time and the LDS models which describe them must also change. For example, a coasting ballistic missile makes an evasive maneuver; a country experiences a recession, a central bank intervention, or some national or global event; a honey bee changes from a waggle to a turn right dance. Some of these changes will appear frequently, while others are only rarely observed. In addition, there is always the possibility of a new, previously unseen dynamical behavior. These considerations motivate us to develop a nonparametric Bayesian approach for learning switching LDS (SLDS) models. We also consider a special case of the SLDS—the switching vector autoregressive (VAR) process—in which direct observations of the underlying dynamical process are assumed available. Although a special case of the general linear systems framework, autoregressive models have simplifying properties that often make them a practical choice in applications. One can view switching dynamical processes as an extension of hidden Markov models (HMMs) in which each HMM state, or mode, is associated with a dynamical process. Existing methods for learning SLDSs and switching VAR processes rely on either fixing the number of HMM modes, such as in [8], or considering a change-point detection formulation where each inferred change is to a new, previously unseen dynamical mode, such as in [14]. In this paper we show how one can remain agnostic about the number of dynamical modes while still allowing for returns to previously exhibited dynamical behaviors. Hierarchical Dirichlet processes (HDP) can be used as a prior on the parameters of HMMs with unknown mode space cardinality [2, 12]. In this paper we make use of a variant of the HDPHMM—the sticky HDP-HMM of [5]—that provides improved control over the number of modes inferred by the HDP-HMM; such control is crucial for the problems we examine. Although the HDP-HMM and its sticky extension are very flexible time series models, they do make a strong Markovian assumption that observations are conditionally independent given the HMM mode. This assumption is often insufficient for capturing the temporal dependencies of the observations in real data. Our nonparametric Bayesian approach for learning switching dynamical processes extends the sticky HDP-HMM formulation to learn an unknown number of persistent, smooth dynamical modes and thereby capture a wider range of temporal dependencies. 2 Background: Switching Linear Dynamic Systems A state space (SS) model provides a general framework for analyzing many dynamical phenomena. The model consists of an underlying state, xt ∈Rn, with linear dynamics observed via yt ∈Rd. A linear time-invariant SS model, in which the dynamics do not depend on time, is given by xt = Axt−1 + et yt = Cxt + wt, (1) where et and wt are independent Gaussian noise processes with covariances Σ and R, respectively. An order r VAR process, denoted by VAR(r), with observations yt ∈Rd, can be defined as yt = r X i=1 Aiyt−i + et et ∼N(0, Σ). (2) Here, the observations depend linearly on the previous r observation vectors. Every VAR(r) process can be described in SS form by, for example, the following transformation: xt = A1 A2 . . . Ar I 0 . . . 0 ... ... ... ... 0 . . . I 0 xt−1 + I 0 ... 0 et yt = [I 0 . . . 0] xt. (3) Note that there are many such equivalent minimal SS representations that result in the same inputoutput relationship, where minimality implies that there does not exist a realization with lower state dimension. On the other hand, not every SS model may be expressed as a VAR(r) process for finite r [1]. We can thus conclude that considering a class of SS models with state dimension r · d and arbitrary dynamic matrix A subsumes the class of VAR(r) processes. The dynamical phenomena we examine in this paper exhibit behaviors better modeled as switches between a set of linear dynamical models. Due to uncertainty in the mode of the process, the overall model is nonlinear. We define a switching linear dynamical system (SLDS) by xt = A(zt)xt−1 + et(zt) yt = Cxt + wt. (4) The first-order Markov process zt indexes the mode-specific LDS at time t, which is driven by Gaussian noise et(zt) ∼N(0, Σ(zt)). We similarly define a switching VAR(r) process by yt = r X i=1 A(zt) i yt−i + et(zt) et(zt) ∼N(0, Σ(zt)). (5) Note that the underlying state dynamics of the SLDS are equivalent to a switching VAR(1) process. 3 Background: Dirichlet Processes and the Sticky HDP-HMM A Dirichlet process (DP), denoted by DP(γ, H), is a distribution on discrete measures G0 = ∞ X k=1 βkδθk θk ∼H (6) on a parameter space Θ. The weights are generated via a stick-breaking construction [11]: βk = β′ k k−1 Y ℓ=1 (1 −β′ ℓ) β′ k ∼Beta(1, γ). (7) (a) (b) (c) (d) Figure 1: For all graphs, β ∼GEM(γ) and θk ∼H(λ). (a) DP mixture model in which zi ∼β and yi ∼f(y | θzi). (b) HDP mixture model with πj ∼DP(α, β), zji ∼πj, and yji ∼f(y | θzji). (c)-(d) Sticky HDP-HMM prior on switching VAR(2) and SLDS processes with the mode evolving as zt+1 ∼πzt for πk ∼DP(α + κ, (αβ + κδk)/(α + κ)). The dynamical processes are as in Eq. (13). We denote this distribution by β ∼GEM(γ). The DP is commonly used as a prior on the parameters of a mixture model, resulting in a DP mixture model (see Fig.1(a)). To generate observations, we choose ¯θi ∼G0 and yi ∼F(¯θi). This sampling process is often described via a discrete variable zi ∼β indicating which component generates yi ∼F(θzi). The hierarchical Dirichlet process (HDP) [12] extends the DP to cases in which groups of data are produced by related, yet distinct, generative processes. Taking a hierarchical Bayesian approach, the HDP draws G0 from a Dirichlet process prior DP(γ, H), and then draws group specific distributions Gj ∼DP(α, G0). Here, the base measure G0 acts as an “average” distribution (E[Gj | G0] = G0) encoding the frequency of each shared, global parameter: Gj = ∞ X t=1 ˜πjtδ˜θjt ˜πj ∼GEM(α) (8) = ∞ X k=1 πjkδθk πj ∼DP(α, β) . (9) Because G0 is discrete, multiple ˜θjt ∼G0 may take identical values θk. Eq. (9) aggregates these probabilities, allowing an observation yji to be directly associated with the unique global parameters via an indicator random variable zji ∼πj. See Fig. 1(b). An alternative, non–constructive characterization of samples G0 ∼DP(γ, H) from a Dirichlet process states that for every finite partition {A1, . . . , AK} of Θ, (G0(A1), . . . , G0(AK)) ∼Dir(γH(A1), . . . , γH(AK)). (10) Using this expression, it can be shown that the following finite, hierarchical mixture model converges in distribution to the HDP as L →∞[7, 12]: β ∼Dir(γ/L, . . . , γ/L) πj ∼Dir(αβ1, . . . , αβL). (11) This weak limit approximation is used by the sampler of Sec. 4.2. The HDP can be used to develop an HMM with a potentially infinite mode space [2, 12]. For this HDP-HMM, each HDP group-specific distribution, πj, is a mode-specific transition distribution and, due to the infinite mode space, there are infinitely many groups. Let zt denote the mode of the Markov chain at time t. For discrete Markov processes zt ∼πzt−1, so that zt−1 indexes the group to which yt is assigned. The current HMM mode zt then indexes the parameter θzt used to generate observation yt. See Fig. 1(c), ignoring the direct correlation in the observations. By sampling πj ∼DP(α, β), the HDP prior encourages modes to have similar transition distributions (E[πjk | β] = βk). However, it does not differentiate self–transitions from moves between modes. When modeling dynamical processes with mode persistence, the flexible nature of the HDP-HMM prior allows for mode sequences with unrealistically fast dynamics to have large posterior probability. Recently, it has been shown [5] that one may mitigate this problem by instead considering a sticky HDP-HMM where πj is distributed as follows: πj ∼DP α + κ, αβ + κδj α + κ . (12) Here, (αβ +κδj) indicates that an amount κ > 0 is added to the jth component of αβ. The measure of πj over a finite partition (Z1, . . . , ZK) of the positive integers Z+, as described by Eq. (10), adds an amount κ only to the arbitrarily small partition containing j, corresponding to a self-transition. When κ = 0 the original HDP-HMM is recovered. We place a vague prior on κ and learn the self-transition bias from the data. 4 The HDP-SLDS and HDP-AR-HMM Models For greater modeling flexibility, we take a nonparametric approach in defining the mode space of our switching dynamical processes. Specifically, we develop extensions of the sticky HDP-HMM for both the SLDS and switching VAR models. For the SLDS, we consider conditionally-dependent emissions of which only noisy observations are available (see Fig. 1(d)). For this model, which we refer to as the HDP-SLDS, we place a prior on the parameters of the SLDS and infer their posterior from the data. We do, however, fix the measurement matrix, C, for reasons of identifiability. Let ˜C ∈Rd×n, n ≥d, be the measurement matrix associated with a dynamical system defined by ˜A, and assume ˜C has full row rank. Then, without loss of generality, we may consider C = [I 0] since there exists an invertible transformation T such that the pair C = ˜CT = [I 0] and A = T −1 ˜AT defines an equivalent input-output system. The dimensionality of I is determined by that of the data. Our choice of the number of columns of zeros is, in essence, a choice of model order. The previous work of Fox et al. [6] considered a related, yet simpler formulation for modeling a maneuvering target as a fixed LDS driven by a switching exogenous input. Since the number of maneuver modes was assumed unknown, the exogenous input was taken to be the emissions of a HDP-HMM. This work can be viewed as an extension of the work by Caron et. al. [3] in which the exogenous input was an independent noise process generated from a DP mixture model. The HDP-SLDS is a major departure from these works since the dynamic parameters themselves change with the mode and are learned from the data, providing a much more expressive model. The switching VAR(r) process can similarly be posed as an HDP-HMM in which the observations are modeled as conditionally VAR(r). This model is referred to as the HDP-AR-HMM and is depicted in Fig. 1(c). The generative processes for these two models are summarized as follows: HDP-AR-HMM HDP-SLDS Mode dynamics zt ∼πzt−1 zt ∼πzt−1 Observation dynamics yt = Pr i=1 A(zt) i yt−i + et(zt) xt = A(zt)xt−1 + et(zt) yt = Cxt + wt (13) Here, πj is as defined in Sec. 3 and the additive noise processes as in Sec. 2. 4.1 Posterior Inference of Dynamic Parameters In this section we focus on developing a prior to regularize the learning of different dynamical modes conditioned on a fixed mode assignment z1:T . For the SLDS, we analyze the posterior distribution of the dynamic parameters given a fixed, known state sequence x1:T . Methods for learning the number of modes and resampling the sequences x1:T and z1:T are discussed in Sec. 4.2. Conditioned on the mode sequence, one may partition the observations into K different linear regression problems, where K = |{z1, . . . , zT }|. That is, for each mode k, we may form a matrix Y(k) with Nk columns consisting of the observations yt with zt = k. Then, Y(k) = A(k) ¯Y(k) + E(k), (14) where A(k) = [A(k) 1 . . . A(k) r ], ¯Y(k) is a matrix of lagged observations, and E(k) the associated noise vectors. Let D(k) = {Y(k), ¯Y(k)}. The posterior distribution over the VAR(r) parameters associated with the kth mode decomposes as follows: p(A(k), Σ(k) | D(k)) = p(A(k) | Σ(k), D(k))p(Σ(k) | D(k)). (15) We place a conjugate matrix-normal inverse-Wishart prior on the parameters {A(k), Σ(k)} [13], providing a reasonable combination of flexibility and analytical convenience. A matrix A ∈Rd×m has a matrix-normal distribution MN(A; M, V , K) if p(A) = |K| d 2 |2πV | m 2 e−1 2 tr “ (A−M) TV −1(A−M)K ” , (16) where M is the mean matrix and V and K−1 are the covariances along the rows and columns, respectively. A vectorization of the matrix A results in p(vec(A)) = N(vec(M), K−1 ⊗V ), (17) where ⊗denotes the Kronecker product. The resulting posterior is derived as p(A(k) | Σ(k), D(k)) = MN(A(k); S(k) y¯y S−(k) ¯y¯y , Σ−(k), S(k) ¯y¯y ), (18) with B−(k) denoting (B(k))−1 for a given matrix B, and S(k) ¯y¯y = ¯Y(k) ¯Y(k)T + K S(k) y¯y = Y(k) ¯Y(k)T + MK S(k) yy = Y(k)Y(k)T + MKM T . We place an inverse-Wishart prior IW(S0, n0) on Σ(k). Then, p(Σ(k) | D(k)) = IW(S(k) y|¯y + S0, Nk + n0), (19) where S(k) y|¯y = S(k) yy −S(k) y¯y S−(k) ¯y¯y S(k)T y¯y . When A is simply a vector, the matrix-normal inverseWishart prior reduces to the normal inverse-Wishart prior with scale parameter K. For the HDP-SLDS, we additionally place an IW(R0, r0) prior on the measurement noise covariance R, which is shared between modes. The posterior distribution is given by p(R | y1:T , x1:T ) = IW(SR + R0, T + r0), (20) with SR=PT t=1(yt −Cxt)(yt −Cxt)T . Further details are provided in supplemental Appendix I. 4.2 Gibbs Sampler For the switching VAR(r) process, our sampler iterates between sampling the mode sequence, z1:T , and both the dynamic and sticky HDP-HMM parameters. The sampler for the SLDS is identical to that of a switching VAR(1) process with the additional step of sampling the state sequence, x1:T , and conditioning on the state sequence when resampling dynamic parameters. The resulting Gibbs sampler is described below and further elaborated upon in supplemental Appendix II. Sampling Dynamic Parameters Conditioned on a sample of the mode sequence, z1:T , and the observations, y1:T , or state sequence, x1:T , we can sample the dynamic parameters θ = {A(k), Σ(k)} from the posterior density described in Sec. 4.1. For the HDP-SLDS, we additionally sample R. Sampling z1:T As shown in [5], the mixing rate of the Gibbs sampler for the HDP-HMM can be dramatically improved by using a truncated approximation to the HDP, such as the weak limit approximation, and jointly sampling the mode sequence using a variant of the forward-backward algorithm. Specifically, we compute backward messages mt+1,t(zt) ∝p(yt+1:T |zt, yt−r+1:t, π, θ) and then recursively sample each zt conditioned on zt−1 from p(zt | zt−1, y1:T , π, θ) ∝p(zt | πzt−1)p(yt | yt−r:t−1, A(zt), Σ(zt))mt+1,t(zt), (21) where p(yt | yt−r:t−1, A(zt), Σ(zt)) = N(Pr i=1 A(zt) i yt−i, Σ(zt)). Joint sampling of the mode sequence is especially important when the observations are directly correlated via a dynamical process since this correlation further slows the mixing rate of the direct assignment sampler of [12]. Note that the approximation of Eq. (11) retains the HDP’s nonparametric nature by encouraging the use of fewer than L components while allowing the generation of new components, upper bounded by L, as new data are observed. Sampling x1:T (HDP-SLDS only) Conditioned on the mode sequence z1:T and the set of dynamic parameters θ, our dynamical process simplifies to a time-varying linear dynamical system. We can then block sample x1:T by first running a backward filter to compute mt+1,t(xt) ∝ p(yt+1:T |xt, zt+1:T , θ) and then recursively sampling each xt conditioned on xt−1 from p(xt | xt−1, y1:T , z1:T, θ) ∝p(xt | xt−1, A(zt), Σ(zt))p(yt | xt, R)mt+1,t(xt). (22) The messages are given in information form by mt,t−1(xt−1) ∝N −1(xt−1; θt,t−1, Λt,t−1), where the information parameters are recursively defined as θt,t−1 = A(zt)T Σ−(zt)(Σ−(zt) + CT R−1C + Λt+1,t)−1(CT R−1yt + θt+1,t) (23) Λt,t−1 = A(zt)T Σ−(zt)A(zt) −A(zt)T Σ−(zt)(Σ−(zt) + CT R−1C + Λt+1,t)−1Σ−(zt)A(zt). See supplemental Appendix II for a more numerically stable version of this recursion. 0 500 1000 1500 2000 −2 0 2 4 6 8 10 12 14 Time 200 400 600 800 1000 0 0.1 0.2 0.3 0.4 0.5 Iteration Normalized Hamming Distance 200 400 600 800 1000 0 0.1 0.2 0.3 0.4 0.5 Iteration Normalized Hamming Distance 200 400 600 800 1000 0 0.1 0.2 0.3 0.4 0.5 Iteration Normalized Hamming Distance 200 400 600 800 1000 0 0.1 0.2 0.3 0.4 0.5 Iteration Normalized Hamming Distance 0 200 400 600 800 1000 0 2 4 6 8 10 12 14 16 Time 1000 2000 3000 4000 5000 0 0.1 0.2 0.3 0.4 0.5 0.6 Iteration Normalized Hamming Distance 1000 2000 3000 4000 5000 0 0.1 0.2 0.3 0.4 0.5 0.6 Iteration Normalized Hamming Distance 1000 2000 3000 4000 5000 0 0.1 0.2 0.3 0.4 0.5 0.6 Iteration Normalized Hamming Distance 1000 2000 3000 4000 5000 0 0.1 0.2 0.3 0.4 0.5 0.6 Iteration Normalized Hamming Distance 0 200 400 600 800 1000 0 50 100 150 200 Time 1000 2000 3000 4000 5000 0 0.1 0.2 0.3 0.4 0.5 0.6 Iteration Normalized Hamming Distance 1000 2000 3000 4000 5000 0 0.1 0.2 0.3 0.4 0.5 0.6 Iteration Normalized Hamming Distance 1000 2000 3000 4000 5000 0 0.1 0.2 0.3 0.4 0.5 0.6 Iteration Normalized Hamming Distance 1000 2000 3000 4000 5000 0 0.1 0.2 0.3 0.4 0.5 0.6 Iteration Normalized Hamming Distance (a) (b) (c) (d) (e) Figure 2: (a) Observation sequence (blue, green, red) and associated mode sequence (magenta) for a 5-mode switching VAR(1) process (top), 3-mode switching AR(2) process (middle), and 3-mode SLDS (bottom). The associated 10th, 50th, and 90th Hamming distance quantiles over 100 trials are shown for the (b) HDP-VAR(1)HMM, (c) HDP-VAR(2)-HMM, (d) HDP-SLDS with C = I (top and bottom) and C = [1 0] (middle), and (e) sticky HDP-HMM using first difference observations. 5 Results Synthetic Data In Fig. 2, we compare the performance of the HDP-VAR(1)-HMM, HDP-VAR(2)HMM, HDP-SLDS, and a baseline sticky HDP-HMM on three sets of test data (see Fig. 2(a)). The Hamming distance error is calculated by first choosing the optimal mapping of indices maximizing overlap between the true and estimated mode sequences. For the first scenario, the data were generated from a 5-mode switching VAR(1) process. The three switching linear dynamical models provide comparable performance since both the HDP-VAR(2)-HMM and HDP-SLDS with C = I contain the class of HDP-VAR(1)-HMMs. Note that the HDP-SLDS sampler is slower to mix since the hidden, three-dimensional continuous state is also sampled. In the second scenario, the data were generated from a 3-mode switching AR(2) process. The HDP-AR(2)-HMM has significantly better performance than the HDP-AR(1)-HMM while the performance of the HDP-SLDS with C = [1 0] is comparable after burn-in. As shown in Sec. 2, this HDP-SLDS model encompasses the class of HDP-AR(2)-HMMs. The data in the third scenario were generated from a 3-mode SLDS model with C = I. Here, we clearly see that neither the HDP-VAR(1)-HMM nor HDP-VAR(2)-HMM is equivalent to the HDP-SLDS. Together, these results demonstrate both the differences between our models as well as the models’ ability to learn switching processes with varying numbers of modes. Finally, note that all of the switching models yielded significant improvements relative to the baseline sticky HDP-HMM, even when the latter was given first differences of the observations. This input representation, which is equivalent to an HDP-VAR(1)-HMM with random walk dynamics (A(k) = I for all k), is more effective than using raw observations for HDP-HMM learning, but still much less effective than richer models which switch among learned LDS. IBOVESPA Stock Index We test the HDP-SLDS model on the IBOVESPA stock index (Sao Paulo Stock Exchange) over the period of 01/03/1997 to 01/16/2001. There are ten key world events shown in Fig. 3 and cited in [4] as affecting the emerging Brazilian market during this time period. In [4], a 2-mode Markov switching stochastic volatility (MSSV) model is used to identify periods of higher volatility in the daily returns. The MSSV assumes that the log-volatilities follow an AR(1) process with a Markov switching mean. This underlying process is observed via conditionally independent and normally distributed daily returns. The HDP-SLDS is able to infer very similar change points to those presented in [4]. Interestingly, the HDP-SLDS consistently identifies three regimes of volatility versus the assumed 2-mode model. In Fig. 3, the overall performance of the 1/3/97 7/2/97 6/1/98 1/15/99 1/13/00 Date 1/3/97 7/2/97 6/1/98 1/15/99 1/13/00 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Probability of Change Point Date 1/3/97 7/2/97 6/1/98 1/15/99 1/13/00 0 0.2 0.4 0.6 0.8 1 Probability of Change Point Date 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Detection Rate False Alarm Rate HDP−SLDS HDP−SLDS, non−sticky HDP−AR(1)−HMM HDP−AR(2)−HMM (a) (b) (c) (d) Figure 3: (a) IBOVESPA stock index daily returns from 01/03/1997 to 01/16/2001. (b) Plot of the estimated probability of a change point on each day using 3000 Gibbs samples for the HDP-SLDS. The 10 key events are indicated with red lines. (c) Similar plot for the non-sticky HDP-SLDS with no bias towards self-transitions. (d) ROC curves for the HDP-SLDS, non-sticky HDP-SLDS, HDP-AR(1)-HMM, and HDP-AR(2)-HMM. HDP-SLDS is compared to that of the HDP-AR(1)-HMM, HDP-AR(2)-HMM, and HDP-SLDS with no bias for self-transitions (i.e., κ = 0.) The ROC curves shown in Fig. 3(d) are calculated by windowing the time axis and taking the maximum probability of a change point in each window. These probabilities are then used as the confidence of a change point in that window. We clearly see the advantage of using a SLDS model combined with the sticky HDP-HMM prior on the mode sequence. Without the sticky extension, the HDP-SLDS over-segments the data and rapidly switches between redundant states which leads to a dramatically larger number of inferred change points. Dancing Honey Bees We test the HDP-VAR(1)-HMM on a set of six dancing honey bee sequences, aiming to segment the sequences into the three dances displayed in Fig. 4. (Note that we did not see performance gains by considering the HDP-SLDS, so we omit showing results for that architecture.) The data consist of measurements yt = [cos(θt) sin(θt) xt yt]T , where (xt, yt) denotes the 2D coordinates of the bee’s body and θt its head angle. We compare our results to those of Xuan and Murphy [14], who used a change-point detection technique for inference on this dataset. As shown in Fig. 4(d)-(e), our model achieves a superior segmentation compared to the change-point formulation in almost all cases, while also identifying modes which reoccur over time. Oh et al. [8] also presented an analysis of the honey bee data, using an SLDS with a fixed number of modes. Unfortunately, that analysis is not directly comparable to ours, because [8] used their SLDS in a supervised formulation in which the ground truth labels for all but one of the sequences are employed in the inference of the labels for the remaining held-out sequence, and in which the kernels used in the MCMC procedure depend on the ground truth labels. (The authors also considered a “parameterized segmental SLDS (PS-SLDS),” which makes use of domain knowledge specific to honey bee dancing and requires additional supervision during the learning process.) Nonetheless, in Table 1 we report the performance of these methods as well as the median performance (over 100 trials) of the unsupervised HDP-VAR(1)-HMM to provide a sense of the level of performance achievable without detailed, manual supervision. As seen in Table 1, the HDP-VAR(1)-HMM yields very good performance on sequences 4 to 6 in terms of the learned segmentation and number of modes (see Fig. 4(a)-(c)); the performance approaches that of the supervised method. For sequences 1 to 3—which are much less regular than sequences 4 to 6—the performance of the unsupervised procedure is substantially worse. This motivated us to also consider a partially supervised variant of the HDP-VAR(1)-HMM in which we fix the ground truth mode sequences for five out of six of the sequences, and jointly infer both a combined set of dynamic parameters and the left-out mode sequence. As we see in Table 1, this considerably improved performance for these three sequences. Not depicted in the plots in Fig. 4 is the extreme variation in head angle during the waggle dances of sequences 1 to 3. This dramatically affects our performance since we do not use domain-specific information. Indeed, our learned segmentations consistently identify turn-right and turn-left modes, but often create a new, sequence-specific waggle dance mode. Many of our errors can be attributed to creating multiple waggle dance modes within a sequence. Overall, however, we are able to achieve reasonably good segmentations without having to manually input domain-specific knowledge. 6 Discussion In this paper, we have addressed the problem of learning switching linear dynamical models with an unknown number of modes for describing complex dynamical phenomena. We presented a non(1) (2) (3) (4) (5) (6) 0 200 400 600 1 2 3 4 Estimated mode Time 0 200 400 600 800 1 2 3 Estimated mode Time 0 200 400 600 1 2 3 4 Estimated mode Time 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Detection Rate False Alarm Rate HDP−VAR−HMM, unsupervised HDP−VAR−HMM, supervised Change−point formulation Viterbi sequence 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Detection Rate False Alarm Rate HDP−VAR−HMM, unsupervised HDP−VAR−HMM, supervised Change−point formulation Viterbi sequence (a) (b) (c) (d) (e) Figure 4: (top) Trajectories of the dancing honey bees for sequences 1 to 6, colored by waggle (red), turn right (blue), and turn left (green) dances. (a)-(c) Estimated mode sequences representing the median error for sequences 4, 5, and 6 at the 200th Gibbs iteration, with errors indicated in red. (d)-(e) ROC curves for the unsupervised HDP-VAR-HMM, partially supervised HDP-VAR-HMM, and change-point formulation of [14] using the Viterbi sequence for segmenting datasets 1-3 and 4-6, respectively. Sequence 1 2 3 4 5 6 HDP-VAR(1)-HMM unsupervised 46.5 44.1 45.6 83.2 93.2 88.7 HDP-VAR(1)-HMM partially supervised 65.9 88.5 79.2 86.9 92.3 89.1 SLDS DD-MCMC 74.0 86.1 81.3 93.4 90.2 90.4 PS-SLDS DD-MCMC 75.9 92.4 83.1 93.4 90.4 91.0 Table 1: Median label accuracy of the HDP-VAR(1)-HMM using unsupervised and partially supervised Gibbs sampling, compared to accuracy of the supervised PS-SLDS and SLDS procedures, where the latter algorithms were based on a supervised MCMC procedure (DD-MCMC) [8]. parametric Bayesian approach and demonstrated both the utility and versatility of the developed HDP-SLDS and HDP-AR-HMM on real applications. Using the same parameter settings, in one case we are able to learn changes in the volatility of the IBOVESPA stock exchange while in another case we learn segmentations of data into waggle, turn-right, and turn-left honey bee dances. An interesting direction for future research is learning models of varying order for each mode. References [1] M. Aoki and A. Havenner. State space modeling of multiple time series. Econ. Rev., 10(1):1–59, 1991. [2] M. J. Beal, Z. Ghahramani, and C. E. Rasmussen. The infinite hidden Markov model. In NIPS, 2002. [3] F. Caron, M. Davy, A. Doucet, E. Duflos, and P. Vanheeghe. Bayesian inference for dynamic models with Dirichlet process mixtures. In Int. Conf. Inf. Fusion, July 2006. [4] C. Carvalho and H. Lopes. Simulation-based sequential analysis of Markov switching stochastic volatility models. Comp. Stat. & Data Anal., 2006. [5] E. B. Fox, E. B. Sudderth, M. I. Jordan, and A. S. Willsky. An HDP-HMM for systems with state persistence. In ICML, 2008. [6] E. B. Fox, E. B. Sudderth, and A. S. Willsky. Hierarchical Dirichlet processes for tracking maneuvering targets. In Int. Conf. Inf. Fusion, July 2007. [7] H. Ishwaran and M. Zarepour. Exact and approximate sum–representations for the Dirichlet process. Can. J. Stat., 30:269–283, 2002. [8] S. Oh, J. Rehg, T. Balch, and F. Dellaert. Learning and inferring motion patterns using parametric segmental switching linear dynamic systems. IJCV, 77(1–3):103–124, 2008. [9] J. M. Pavlovi´c, V. Rehg and J. MacCormick. Learning switching linear models of human motion. In NIPS, 2000. [10] X. Rong Li and V. Jilkov. Survey of maneuvering target tracking. Part V: Multiple-model methods. IEEE Trans. Aerosp. Electron. Syst., 41(4):1255–1321, 2005. [11] J. Sethuraman. A constructive definition of Dirichlet priors. Stat. Sinica, 4:639–650, 1994. [12] Y. W. Teh, M. I. Jordan, M. J. Beal, and D. M. Blei. Hierarchical Dirichlet processes. J. Amer. Stat. Assoc., 101(476):1566–1581, 2006. [13] M. West and J. Harrison. Bayesian Forecasting and Dynamic Models. Springer, 1997. [14] X. Xuan and K. Murphy. Modeling changing dependency structure in multivariate time series. In ICML, 2007.
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Learning to use Working Memory in Partially Observable Environments through Dopaminergic Reinforcement Michael T. Todd, Yael Niv, Jonathan D. Cohen Department of Psychology & Princeton Neuroscience Institute Princeton University, Princeton, NJ 08544 {mttodd,yael,jdc}@princeton.edu Abstract Working memory is a central topic of cognitive neuroscience because it is critical for solving real-world problems in which information from multiple temporally distant sources must be combined to generate appropriate behavior. However, an often neglected fact is that learning to use working memory effectively is itself a difficult problem. The Gating framework [14] is a collection of psychological models that show how dopamine can train the basal ganglia and prefrontal cortex to form useful working memory representations in certain types of problems. We unite Gating with machine learning theory concerning the general problem of memory-based optimal control [5-6]. We present a normative model that learns, by online temporal difference methods, to use working memory to maximize discounted future reward in partially observable settings. The model successfully solves a benchmark working memory problem, and exhibits limitations similar to those observed in humans. Our purpose is to introduce a concise, normative definition of high level cognitive concepts such as working memory and cognitive control in terms of maximizing discounted future rewards. 1 Introduction Working memory is loosely defined in cognitive neuroscience as information that is (1) internally maintained on a temporary or short term basis, and (2) required for tasks in which immediate observations cannot be mapped to correct actions. It is widely assumed that prefrontal cortex (PFC) plays a role in maintaining and updating working memory. However, relatively little is known about how PFC develops useful working memory representations for a new task. Furthermore, current work focuses on describing the structure and limitations of working memory, but does not ask why, or in what general class of tasks, is it necessary. Borrowing from the theory of optimal control in partially observable Markov decision problems (POMDPs), we frame the psychological concept of working memory as an internal state representation, developed and employed to maximize future reward in partially observable environments. We combine computational insights from POMDPs and neurobiologically plausible models from cognitive neuroscience to suggest a simple reinforcement learning (RL) model of working memory function that can be implemented through dopaminergic training of the basal ganglia and PFC. The Gating framework is a series of cognitive neuroscience models developed to explain how dopaminergic RL signals can shape useful working memory representations [1-4]. Computationally this framework models working memory as a collection of past observations, each of which can occasionally be replaced with the current observation, and addresses the problem of learning when to update each memory element versus maintaining it. In the original Gating model [1-2] the PFC contained a unitary working memory representation that was updated whenever a phasic dopamine (DA) burst occurred (e.g., due to unexpected reward or novelty). That model was the first to connect working memory and RL via the temporal difference (TD) model of DA firing [7-8], and thus to suggest how working memory might serve a normative purpose. However, that model had limited computational flexibility due to the unitary nature of the working memory (i.e., a singleobservation memory controlled by a scalar DA signal). More recent work [3-4] has partially repositioned the Gating framework within the Actor/Critic model of mesostriatal RL [9-10], positing memory updating as but another cortical action controlled by the dorsal striatal "actor." This architecture increased computational flexibility by introducing multiple working memory elements, corresponding to multiple corticostriatal loops, that could be quasi-independently updated. However, that model combined a number of components (including supervised and unsupervised learning, and complex neural network dynamics), making it difficult to understand the relationship between simple RL mechanisms and working memory function. Moreover, because the model used the Rescorla-Wagner-like PVLV algorithm [4] rather than TD [7-8] as the model of phasic DA bursts, the model's behavior and working memory representations were not directly shaped by standard normative criteria for RL models (i.e., discounted future reward or reward per unit time). We present a new Gating model, synthesizing the mesostriatal Actor/Critic architecture of [4] with a normative POMDP framework, and reducing the Gating model to a fourparameter, pure RL model in the process. This produces a model very similar to previous machine learning work on "model-free" approximate POMDP solvers [5,6], which attempt to form good solutions without explicit knowledge of the environment's structure or dynamics. That is, we model working memory as a discrete memory system (a collection of recent observations) rather than a continuous "belief state" (an inferred probability distribution over hidden states). In some environments this may permit only an approximate solution. However, the strength of such a system is that it requires very little prior knowledge, and is thus potentially useful for animals, who must learn effective behavior and memorymanagement policies in completely novel environments (i.e., in the absence of a “world model”). Therefore, we retain the computational flexibility of the more recent Gating models [3-4], while re-establishing the goal of defining working memory in normative terms [1-2]. To illustrate the strengths and limitations of the model, we apply it to two representative working-memory tasks. The first is the 12-AX task proposed as a Gating benchmark in [4]. Contrary to previous claims that TD learning is not sufficient to solve this task, we show that with an eligibility trace (i.e., TD(𝜆) with 0 < < 1), the model can achieve optimal behavior. The second task highlights important limitations of the model. Since our model is a POMDP solver and POMDPs are, in general, intractable (i.e., solution algorithms require an infeasible number of computations), it is clear that our model must ultimately fail to achieve optimal performance as environments increase even to moderate complexity. However, human working memory also exhibits sharp limitations. We apply our model to an implicit artificial grammar learning task [11] and show that it indeed fails in ways reminiscent of human performance. Moreover, simulating this task with increased working memory capacity reveals diminishing returns as capacity increases beyond a small number, suggesting that the "magic number" limited working memory capacity found in humans [12] might in fact be optimal from a learning standpoint. 2 Model Architecture As with working memory tasks, a POMDP does not admit an optimal behavior policy based only on the current observation. Instead, the optimal policy generally depends on some combination of memory as well as the current observation. Although the type of memory required varies across POMDPs, in certain cases a finite memory system is a sufficient basis for an optimal policy. Peshkin, Meuleau, and Kaelbling [6] used an external finite memory device (e.g., a shopping list) to improve the performance of RL in a model-free POMDP setting. Their model's "state" variable consisted of the current observation augmented by the memory device. An augmented action space, consisting of both memory actions and motor actions, allowed the model to learn effective memory-management and motor policies simultaneously. We integrate this approach with the Gating model, altering the semantics so that the external memory device becomes internal working memory (presumed Choose motor action, 𝑎௧, and gating action, 𝑔௧, for current state, 𝑠௧ according to softmax over motor and gating action preferences, 𝑢 and 𝑣, respectively. 𝑎௧←Softmax(𝑢; 𝑠௧) 𝑔௧←Softmax(𝑣; 𝑠௧) Update motor and gating action eligibility traces, 𝑒ெ and 𝑒ீ, respectively. (Update shown for motor action eligibility trace. Gating action trace is analogous.) 𝑒ெ(𝑠, 𝑎) ←ቐ 1 −Pr(𝑎|𝑠) , 𝑠= 𝑠௧, 𝑎= 𝑎௧ −Pr(𝑎|𝑠) , 𝑠= 𝑠௧, 𝑎≠𝑎௧ 𝛾𝜆𝑒ெ(𝑠, 𝑎), 𝑠≠𝑠௧ ∀ 𝑠, 𝑎 Update (hidden) environment state, 𝜎, with motor action. Get next reward, 𝑟, and observation, 𝑜. 𝜎௧ାଵ←Environment(𝑎௧, 𝜎௧) 𝑟, 𝑜←Environment(𝜎௧ାଵ) Update internal state based on previous state, gating action, and new observation 𝑠௧ାଵ←𝑜, 𝑔௧, 𝑠௧ Compute state-value prediction error, 𝛿௧, based on critic’s state-value approximation, 𝑉(𝑠) 𝛿௧←𝑟+ 𝛾𝑉(𝑠௧ାଵ) −𝑉(𝑠௧) Update state-value eligibility traces, 𝑒. 𝑒(𝑠) = ൜𝛾𝜆𝑒(𝑠) + 1, 𝑠= 𝑠௧ 𝛾𝜆𝑒(𝑠), 𝑠≠𝑠௧ , ∀ 𝑠 Update state-values 𝑉(𝑠) = 𝑉(𝑠) + 𝛼𝛿௧𝑒(𝑠), ∀𝑠 Update motor action preferences 𝑢(𝑠, 𝑎) = 𝑢(𝑠, 𝑎) + 𝛼𝛿௧𝑒ெ(𝑠, 𝑎), ∀ 𝑠, 𝑎 Update gating action preferences 𝑣(𝑠, 𝑔) = 𝑣(𝑠, 𝑔) + 𝛼𝛿௧𝑒ீ(𝑠, 𝑔), ∀ 𝑠, 𝑔 Next trial… 𝑠௧←𝑠௧ାଵ Table 1 Pseudocode of one trial of the model, based on the Actor/Critic architecture with eligibility traces. Following [13], we substitute the critic's state-value prediction error for Williams's (𝑟−𝑏) term [14]. We describe here a single gating actor, but it is straightforward to generalize to an array of independent gating actors as we use in our simulations. 𝛾= discount rate; 𝜆= eligibility trace decay rate; 𝛼=learning rate. In all simulations, 𝛾= 0.94, 𝛼= 0.1. to be supported in PFC), and altering the Gating model so that the role of working memory is explicitly to support optimal behavior (in terms of discounted future reward) in a POMDP. Like [6], the key difference between our model and standard RL methods is that our state variable includes controlled memory elements (i.e., working memory), which augment the current observation. The action space is similarly augmented to include memory or gating actions, and the model learns by trial-and-error how to update its working memory (to resolve hidden states when such resolution leads to greater rewards) as well as its motor policy. The task for our model then, is to learn a working memory policy such that the current internal state (i.e., memory and current observation) admits an optimal behavioral policy. Our model (Table 1) consists of a critic, a motor actor, and several gating actors. As in the standard Actor/Critic architecture, the critic learns to evaluate (internal) states and, based on the ongoing temporal difference of these values, generates at each time step a prediction error (PE) signal (thought to correspond to phasic bursts and dips in DA [8]). The PE is used to train the critic's state values and the policies of the actors. The motor actor also fulfills the usual role, choosing actions to send to the environment based on its policy and the current internal state. Finally, gating actors correspond one-to-one with each memory element. At each time point, each gating actor independently chooses (via a policy based on the internal state) whether to (1) maintain its element's memory for another time step, or (2) replace (update) its element's memory with the current observation. To remain aligned with the Actor/Critic online learning framework of mesostriatal RL [910], learning in our model is based on REINFORCE [14] modified for expected discounted future reward [13], rather than the Monte-Carlo policy learning algorithm in [6] (which is more suitable for offline, episodic learning). Furthermore, because it has been shown that eligibility traces are particularly useful when applying TD to POMDPs (e.g., [15-16]), we used TD(𝜆), taking the characteristic eligibilities of the REINFORCE algorithm [14] as the impulse function for a replacing eligibility trace [17]. For simplicity of exposition and interpretation, we used tabular policy and state-value representations throughout. Figure 1 12-AX: Average performance over 40 training runs, each consisting of 2×107 timesteps. (A) As indicated by reward rate over the last 105 time steps, the model learns an optimal policy when the eligibility trace parameter, 𝝀, is between zero and one. (B) The time required for the model to reach 300 consecutive correct trials increases rapidly as 𝝀 decreases. (C) Sample sequence of the 12-AX task. 3 Benchmark Performance and Psychological Data We now describe the model's performance on the 12-AX task proposed as a benchmark for Gating models [4]. We then turn to a comparison of the model's behavior against actual psychological data. 3.1 12-AX Performance The 12-AX task was used in [4] to illustrate the problem of learning a task in which correct behavior depends on multiple previous observations. In the task (Figure 1C), subjects are presented with a sequence of observations drawn from the set {1, 2, A, B, C, X, Y, Z}. They gain rewards by responding L or R according to the following rules: Respond R if (1) the current observation is an X, the last observation from the set {A, B, C} was an A, and the last observation from the set {1, 2} was a 1; or (2) the current observation is a Y, the last observation from the set {A, B, C} was a B, and the last observation from the set {1, 2} was a 2. Respond L otherwise. In our implementation, reward is 1 for correct responses when the current observation is X or Y, 0.25 for all other correct responses, and 0 for incorrect responses. We modeled this task using two memory elements, the minimum theoretically necessary for optimal performance. The results (Figure 1A,B) show that our TD(𝜆) Gating model can indeed achieve optimal 12-AX performance. The results also demonstrate the reliance of the model on the eligibility trace parameter, 𝜆, with best performance at high intermediate values of 𝜆. When = 0, the model finds a suboptimal policy that is only slightly better than the optimal policy for a model without working memory. With = 1 performance is even worse, as can be expected for an online policy improvement method with non-decaying traces (a point of comparison with [6] to which we will return in the Discussion). These results are consistent with previous work showing that TD(0) performs poorly in partially observable (non-Markovian) settings [15], whereas TD(𝜆) (without memory) with 𝜆≈0.9 performs best [16]. Indeed, early in training, as our model learns to convert a POMDP to an MDP via its working memory, the internal state dynamics are not Markovian, and thus an eligibility trace is necessary. 3.2 Psychological data We are the first to interpret the Gating framework (and the use of working memory) as an attempt to solve POMDPs. This brings a large body of theoretical work to bear on the properties of Gating models. Importantly, it implies that, as task complexity increases, both the Gating model and humans must fail to find optimal solutions in reasonable time frames Figure 2 (A) Artificial grammar from [11]. Starting from node 0, the grammar generates a continuing sequence of observations. All nodes with two transitions (edges) make either transition with p=0.5. Edge labels mark grammatical observations. At each transition, the grammatical observation is replaced with a random, ungrammatical, observation with p=0.15. The task is to predict the next observation at each time point. (B) The model shows a gradual increase in sensitivity to sequences of length 2 and 3, but not length 4, replicating the human data. Sensitivity is measured as probability of choosing grammatical action for the true state, minus probability of choosing grammatical action for the aliased state; 0 indicates complete aliasing, 1 complete resolution. (C) Model performance (reward rate) averaged over training runs with variable numbers of time steps shows diminishing returns as the number of memory elements increases. due to the generally intractable nature of POMDPs. Given this inescapable conclusion, it is interesting to compare model failures to corresponding human failures: a pattern of failures matching human data would provide support for our model. In this subsection we describe a simulation of artificial grammar learning [11], and then offer an account of the pervasive "magic number" observations concerning limits of working memory capacity (e.g., [12]). In artificial grammar learning, subjects see a seemingly random sequence of observations, and are instructed to mimic each observation as quickly as possible (or to predict the next observation) with a corresponding action. Unknown to the subjects, the observation sequence is generated by a stochastic process called a "grammar" (Figure 2A). Artificial grammar tasks constitute POMDPs: the (recent) observation history can predict the next observation better than the current observation alone, so optimal performance requires subjects to remember information distilled from the history. Although subjects typically report no knowledge of the underlying structure, after training their reaction times (RTs) reveal implicit structural knowledge. Specifically, RTs become significantly faster for "grammatical" as compared to "ungrammatical" observations (see Figure 2). Cleeremans and McClelland [11] examined the limits of subjects' capacity to detect grammar structure. The grammar they used is shown in Figure 2A. They found that, although subjects grew increasingly sensitive to sequences of length two and three throughout training, (as measured by transient RT increases following ungrammatical observations), they remained insensitive, even after 60,000 time steps of training, to sequences of length four. This presumably reflected a failure of subjects' implicit working memory learning mechanisms, and was confirmed in a second experiment [11]. We replicated these results, as shown in Figure 2B. To simulate the task, we gave the model two memory elements (results were no different with three elements), and reward 1 for each correct prediction. We tested the model's ability to resolve states based on previous observations by contrasting its behavior across pairs of observation sequences that differed only in the first observation. State resolution based on sequences of length two, three, and four were represented by VS versus XS (leading to predictions Q vs. V/P, respectively), SQX versus XQX (S/Q vs. P/T), and XTVX versus PTVX (S/Q vs. P/T), respectively. In this task, optimal use of information from sequences of length four or more proved impossible for the model and, apparently, for humans. To understand intuitively this limitation, consider a problem of two hidden states, 1 and 2, with optimal actions L and R, respectively. The states are preceded by identical observation sequences of length . However, at + 1 time steps in the past, observation A precedes state 1, whereas observation B precedes state 2. The probability that A/B are held in memory for the required + 1 time steps decreases geometrically with , thus the probability of resolving states 1 and 2 decreases geometrically. Because the agent cannot resolve state 1 from state 2, it can never learn the appropriate 1-L, 2-R action preferences even if it explores those actions, a more insidious problem than an RL agent faces in a fully observable setting. As a result, the model can’t reinforce optimal gating policies, eventually learning an internal state space and dynamics that fail to reflect the true environment. The problem is that credit assignment (i.e., learning a mapping from working memory to actions) is only useful inasmuch as the internal state corresponds to the true hidden state of the POMDP, leading to a “chicken-and-egg” problem. Given the preceding argument, one obvious modification that might lead to improved performance is to increase the number of memory elements. As the number of memory elements increases, the probability that the model remembers observation A for the required amount of time approaches one. However, this strategy introduces the curse of dimensionality due to the rapidly increasing size of the internal state space. This intuitive analysis suggests a normative explanation for the famous "magic number" limitation observed in human working memory capacity, thought to be about four independent elements (e.g., [12]). We demonstrate this idea by again simulating the artificial grammar task, this time averaging performance over a range of training times (1 to 10 million time steps) to capture the idea that humans may practice novel tasks for a typical, but variable, amount of time. Indeed the averaged results show diminishing returns of increasing memory elements (Figure 2C). This simulation used tabular (rather than more neurally plausible) representations and a highly simplified model, so the exact number of policy parameters and state values to be estimated, time steps, and working memory elements is somewhat arbitrary in relation to human learning. Still, the model's qualitative behavior (evidenced by the shape of the resulting curve and the order of magnitude of the optimal number of working memory elements) is surprisingly reminiscent of human behavior. Based on this we suggest that the limitation on working memory capacity may be due to a limitation on learning rather than on storage: it may be impractical to learn to utilize more than a very small number (i.e., smaller than 10) of independent working memory elements, due to the curse of dimensionality. 4 Discussion We have presented a psychological model that suggests that dopaminergic PE signals can implicitly shape working memory representations in PFC. Our model synthesizes recent advances in the Gating literature [4] with normative RL theory regarding model-free, finite memory solutions to POMDPs [6]. We showed that the model learns to behave optimally in the benchmark 12-AX task. We also related the model's computational limitations to known limitations of human working memory [11-12]. 4.1 Relation to other theoretical work Other recent work in neural RL has argued that the brain applies memory-based POMDP solution mechanisms to the real-world problems faced by animals [17-20]. That work primarily considers model-based mechanisms, in which the temporary memory is a continuous belief state, and assumes that a function of cerebral cortex is to learn the required world model, and specifically that PFC should represent temporary goal- or policy-related information necessary for optimal POMDP behavior. The model that we present here is related to that line of thinking, demonstrating a model-free, rather than model-based, mechanism for learning to store policy-related information in PFC. Different learning systems may form different types of working memory representations. Future work may investigate the relationship between implicit learning (as in this Gating model) and modelfree POMDP solutions, versus other kinds of learning and model-based POMDP solutions. Irrespective of the POMDP framework, other work has assumed that there exists a gating policy that controls task-relevant working memory updating in PFC (e.g., [21]). The present work further develops a model of how this policy can be learned. It is interesting to compare our model to previous work on model-free POMDP solutions. McCallum first emphasized the importance of learning utile distinctions [5], or learning to resolve two hidden states only if they have different optimal actions. This is an emphasis that our model shares, at least in spirit. Humans must of course be extremely flexible in their behavior. Therefore there is an inherent tension between the need to focus cognitive resources on learning the immediate task, and the need to form a basis of general task knowledge [3]. It would be interesting for future work to explore how closely the working memory representations learned by our model align to McCallum's utile (and less generalizable) distinctions as opposed to more generalizable representations of the underlying hidden structure of the world, or whether our model could be modified to incorporate a mixture of both kinds of knowledge, depending on some exploration/exploitation parameter. Our model most closely follows the Gating model described in [4], and the theoretical model described in [6]. Our model is clearly more abstract and less biologically detailed than [4]. However, our intent was to ask whether the important insights and capabilities of that model could be captured using a four-parameter, pure RL model with a clear normative basis. Accordingly, we have shown that such a model is comparably equipped to simulate a range of psychological phenomena. Our model also makes equally testable (albeit different) predictions about the neural DA signal. Relative to [6], our model places biological and psychological concerns at the forefront, eliminating the episodic memory requirements of the Monte-Carlo algorithm. It is perhaps interesting, vis á vis [6], that our model performed so poorly when = 1, as this produces a nearly Monte-Carlo scheme. The difference was likely due to our model's online learning (i.e., we updated the policy at each time step rather than at the ends of episodes), which invalidates the Monte-Carlo approach. Thus it might be said that our model is a uniquely psychological variant of that previous architecture. 4.2 Implications for Working Memory and Cognitive Control Subjects in cognitive control experiments typically face situations in which correct behavior is indeterminate given only the immediate observation. Working memory is often thought of as the repository of temporary information that augments the immediate observation to permit correct behavior, sometimes called goals, context, task set, or decision categories. These concepts are difficult to define. Here we have proposed a formal theoretical definition for the cognitive control and working memory constructs. Due to the importance of temporally distant goals and of information that is not immediately observable, the canonical cognitive control environment is well captured by a POMDP. Working memory is then the temporary information, defined and updated by a memory control policy, that the animal uses to solve these POMDPs. Model-based research might identify working memory with continuous belief states, whereas our model-free framework identifies working memory with a discrete collection of recent observations. These may correspond to the products of different learning systems, but the outcome is the same in either case: cognitive control is defined as an animal's memory-based POMDP solver, and working memory is defined as the information, derived from recent history, that the solver requires. 4.3 Psychological and neural validity Although the intractability of solving a POMDP means that all models such as the one we present here must ultimately fail to find an optimal solution in a practical amount of time (if at all), the particular manifestation of computational limitations in our model aligns qualitatively with that observed in humans. Working memory, the psychological construct that the Gating model addresses, is famously limited (see [12] for a review). Beyond canonical working memory capacity limitations, other work has shown subtler limitations arising in learning contexts (e.g., [11]). The results that we presented here are promising, but it remains for future work to more fully explore the relation between the failures exhibited by this model and those exhibited by humans. In conclusion, we have shown that the Gating framework provides a connection between high level cognitive concepts such as working memory and cognitive control, systems neuroscience, and current neural RL theory. The framework's trial-and-error method for solving POMDPs gives rise to particular limitations that are reminiscent of observed psychological limits. It remains for future work to further investigate the model's ability to capture a range of specific psychological and neural phenomena. Our hope is that this link between working memory and POMDPs will be fruitful in generating new insights, and suggesting further experimental and theoretical work. Acknowledgments We thank Peter Dayan, Randy O'Reilly, and Michael Frank for productive discussions, and three anonymous reviewers for helpful comments. This work was supported by NIH grant 5R01MH052864 (MT & JDC) and a Human Frontiers Science Program Fellowship (YN) References [1] Braver, T. S., & Cohen, J. D. (1999). Dopamine, cognitive control, and schizophrenia: The gating model. In J. A. Reggia, E. Ruppin, & D. Glanzman (Eds.), Progress in Brain Research (pp. 327-349). Amsterdam, North-Holland: Elsevier Science. [2] Braver, T. S., & Cohen, J. D. (2000). On the Control of Control: The Role of Dopamine in Regulating Prefrontal Function and Working Memory. In S. Monsell, & J. S. Driver (Eds.), Control of Cognitive Processes: Attention and Performance XVIII (pp. 713-737). Cambridge, MA: MIT Press. [3] Rougier, A., Noelle, D., Braver, T., Cohen, J., & O'Reilly, R. (2005). Prefrontal Cortex and Flexible Cognitive Control: Rules Without Symbols. Proceedings of the National Academy of Sciences , 102 (20), 7338-7343. [4] O'Reilly, R. C., & Frank, M. J. (2006). Making Working Memory Work: A Computational Model of Learning in the Prefrontal Cortex and Basal Ganglia. Neural Computation , 18, 283-328. [5] McCallum, A. (1995). Instance-Based Utile Distinctions for Reinforcement Learning with Hidden State. International Conference on Machine Learning, (pp. 387-395). [6] Peshkin, L., Meuleau, N., & Kaelbling, L. (1999). Learning Policies with External Memory. Sixteenth International Conference on Machine Learning, (pp. 307-314). [7] Montague, P. R., Dayan, P., & Sejnowski, T. J. (1996). A Framework for Mesencephalic Dopamine Systems Based on Predictive Hebbian Learning. The Journal of Neuroscience , 16 (5), 1936-1947. [8] Schultz, W., Dayan, P., & Montague, P. R. (1997). A Neural Substrate of Prediction and Reward. Science , 275, 1593-1599. [9] Houk, J., Adams, J., & Barto, A. (1995). A Model of how the Basal Ganglia Generate and use Neural Signals that Predict Reinforcement. In J. Houk, J. Davis, & D. Beiser, Models of Information Processing in the Basal Ganglia. MIT Press. [10] Joel, D., Niv, Y., & Ruppin, E. (2002). Actor-critic Models of the Basal Ganglia: New Anatomical and Computational Perspectives. Neural Networks , 15, 535-547. [11] Cleeremans, A., & McClelland, J. (1991). Learning the Structure of Event Sequences. Journal of Experimental Psychology: General , 120 (3), 235-253. [12] Cowan, N. (2000). The Magical Number 4 in Short-term Memory: A Reconsideration of Mental Storage Capacity. Behavioral and Brain Sciences , 24, 87-114. [13] Dayan, P., & Abbott, L. (2001). Theoretical Neuroscience. Cambridge, MA: MIT Press. [14] Williams, R. (1992). Simple Statistical Gradient-Following Algorithms for Connectionist Reinforcement Learning. Machine Learning , 8, 229-256. [15] Singh, S., Jaakkola, T., & Jordan, M. I. (1994). Learning Without State-Estimation in Partially Observable Markovian Decision Processes. Eleventh International Conference on Machine Learning, (pp. 284-292). [16] Loch, J., & Singh, S. (1998). Using Eligibility Traces to Find the Best Memoryless Policy in Partially Observable Markov Decision Processes. Fifteenth International Conference on Machine Learning, (pp. 323331). [17] Sutton, R., & Barto, A. (1998). Reinforcement Learning: An Introduction. Cambridge, MA: The MIT Press. [18] Daw, N., Courville, A., & Touretzky, D. (2006). Representation and Timing in Theories of the Dopamine System. Neural Computation , 18, 1637-1677. [19] Samejima, K., & Doya, K. (2007). Multiple Representations of Belief States and Action Values in Corticobasal Ganglia Loops. Annals of the New York Academy of Sciences , 213-228. [20] Yoshida, W., & Ishii, S. (2006). Resolution of Uncertainty in Prefrontal Cortex. Neuron , 50, 781-789. [21] Dayan, P. (2007). Bilinearity, Rules, and Prefrontal Cortex. Frontiers in Computational Neuroscience , 1, 1-14.
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