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Reducing Spike Train Variability: A Computational Theory Of Spike-Timing Dependent Plasticity Sander M. Bohte1,2 S.M.Bohte@cwi.nl 1Dept. Software Engineering CWI, Amsterdam, The Netherlands Michael C. Mozer2 mozer@cs.colorado.edu 2Dept. of Computer Science University of Colorado, Boulder, USA Abstract Experimental studies have observed synaptic potentiation when a presynaptic neuron fires shortly before a postsynaptic neuron, and synaptic depression when the presynaptic neuron fires shortly after. The dependence of synaptic modulation on the precise timing of the two action potentials is known as spike-timing dependent plasticity or STDP. We derive STDP from a simple computational principle: synapses adapt so as to minimize the postsynaptic neuron’s variability to a given presynaptic input, causing the neuron’s output to become more reliable in the face of noise. Using an entropy-minimization objective function and the biophysically realistic spike-response model of Gerstner (2001), we simulate neurophysiological experiments and obtain the characteristic STDP curve along with other phenomena including the reduction in synaptic plasticity as synaptic efficacy increases. We compare our account to other efforts to derive STDP from computational principles, and argue that our account provides the most comprehensive coverage of the phenomena. Thus, reliability of neural response in the face of noise may be a key goal of cortical adaptation. 1 Introduction Experimental studies have observed synaptic potentiation when a presynaptic neuron fires shortly before a postsynaptic neuron, and synaptic depression when the presynaptic neuron fires shortly after. The dependence of synaptic modulation on the precise timing of the two action potentials, known as spike-timing dependent plasticity or STDP, is depicted in Figure 1. Typically, plasticity is observed only when the presynaptic and postsynaptic spikes (hereafter, pre and post) occur within a 20–30 ms time window, and the transition from potentiation to depression is very rapid. Another important observation is that synaptic plasticity decreases with increased synaptic efficacy. The effects are long lasting, and are therefore referred to as long-term potentiation (LTP) and depression (LTD). For detailed reviews of the evidence for STDP, see [1, 2]. Because these intriguing findings appear to describe a fundamental learning mechanism in the brain, a flurry of models have been developed that focus on different aspects of STDP, from biochemical models that explain the underlying mechanisms giving rise to STDP [3], to models that explore the consequences of a STDP-like learning rules in an ensemble of spiking neurons [4, 5, 6, 7], to models that propose fundamental computational justifications for STDP. Most commonly, STDP Figure 1: (a) Measuring STDP experimentally: pre-post spike pairs are repeatedly induced at a fixed interval ∆tpre−post, and the resulting change to the strength of the synapse is assessed; (b) change in synaptic strength after repeated spike pairing as a function of the difference in time between the pre and post spikes (data from Zhang et al., 1998). We have superimposed an exponential fit of LTP and LTD. is viewed as a type of asymmetric Hebbian learning with a temporal dimension. However, this perspective is hardly a fundamental computational rationale, and one would hope that such an intuitively sensible learning rule would emerge from a first-principle computational justification. Several researchers have tried to derive a learning rule yielding STDP from first principles. Rao and Sejnowski [8] show that STDP emerges when a neuron attempts to predict its membrane potential at some time t from the potential at time t −∆t. However, STDP emerges only for a narrow range of ∆t values, and the qualitative nature of the modeling makes it unclear whether a quantitative fit can be obtained. Dayan and H¨ausser [9] show that STDP can be viewed as an optimal noise-removal filter for certain noise distributions. However, even small variation from these noise distributions yield quite different learning rules, and the noise statistics of biological neurons are unknown. Eisele (private communication) has shown that an STDP-like learning rule can be derived from the goal of maintaining the relevant connections in a network. Chechik [10] is most closely related to the present work. He relates STDP to information theory via maximization of mutual information between input and output spike trains. This approach derives the LTP portion of STDP, but fails to yield the LTD portion. The computational approach of Chechik (as well as Dayan and H¨ausser) is premised on a rate-coding neuron model that disregards the relative timing of spikes. It seems quite odd to argue for STDP using rate codes: if spike timing is irrelevant to information transmission, then STDP is likely an artifact and is not central to understanding mechanisms of neural computation. Further, as noted in [9], because STDP is not quite additive in the case of multiple input or output spikes that are near in time [11], one should consider interpretations that are based on individual spikes, not aggregates over spike trains. Here, we present an alternative computational motivation for STDP. We conjecture that a fundamental objective of cortical computation is to achieve reliable neural responses, that is, neurons should produce the identical response—both in the number and timing of spikes—given a fixed input spike train. Reliability is an issue if neurons are affected by noise influences, because noise leads to variability in a neuron’s dynamics and therefore in its response. Minimizing this variability will reduce the effect of noise and will therefore increase the informativeness of the neuron’s output signal. The source of the noise is not important; it could be intrinsic to a neuron (e.g., a noisy threshold) or it could originate in unmodeled external sources causing fluctuations in the membrane potential uncorrelated with a particular input. We are not suggesting that increasing neural reliability is the only learning objective. If it were, a neuron would do well to give no response regardless of the input. Rather, reliability is but one of many objectives that learning tries to achieve. This form of unsupervised learning must, of course, be complemented by supervised and reinforcement learning that allow an organism to achieve its goals and satisfy drives. We derive STDP from the following computational principle: synapses adapt so as to minimize the entropy of the postsynaptic neuron’s output in response to a given presynaptic input. In our simulations, we follow the methodology of neurophysiological experiments. This approach leads to a detailed fit to key experimental results. We model not only the shape (sign and time course) of the STDP curve, but also the fact that potentiation of a synapse depends on the efficacy of the synapse—it decreases with increased efficacy. In addition to fitting these key STDP phenomena, the model allows us to make predictions regarding the relationship between properties of the neuron and the shape of the STDP curve. Before delving into the details of our approach, we attempt to give a basic intuition about the approach. Noise in spiking neuron dynamics leads to variability in the number and timing of spikes. Given a particular input, one spike train might be more likely than others, but the output is nondeterministic. By the entropyminimization principle, adaptation should reduce the likelihood of these other possibilities. To be concrete, consider a particular experimental paradigm. In [12], a pre neuron is identified with a weak synapse to a post neuron, such that the pre is unlikely to cause the post to fire. However, the post can be induced to fire via a second presynaptic connection. In a typical trial, the pre is induced to fire a single spike, and with a variable delay, the post is also induced to fire (typically) a single spike. To increase the likelihood of the observed post response, other response possibilities must be suppressed. With presynaptic input preceding the postsynaptic spike, the most likely alternative response is no output spikes at all. Increasing the synaptic connection weight should then reduce the possibility of this alternative response. With presynaptic input following the postsynaptic spike, the most likely alternative response is a second output spike. Decreasing the synaptic connection weight should reduce the possibility of this alternative response. Because both of these alternatives become less likely as the lag between pre and post spikes is increased, one would expect that the magnitude of synaptic plasticity diminishes with the lag, as is observed in the STDP curve. Our approach to reducing response variability given a particular input pattern involves computing the gradient of synaptic weights with respect to a differentiable model of spiking neuron behavior. We use the Spike Response Model (SRM) of [13] with a stochastic threshold, where the stochastic threshold models fluctuations of the membrane potential or the threshold outside of experimental control. For the stochastic SRM, the response probability is differentiable with respect to the synaptic weights, allowing us to calculate the entropy gradient with respect to the weights conditional on the presented input. Learning is presumed to take a gradient step to reduce this conditional entropy. In modeling neurophysiological experiments, we demonstrate that this learning rule yields the typical STDP curve. We can predict the relationship between the exact shape of the STDP curve and physiologically measurable parameters, and we show that our results are robust to the choice of the few free parameters of the model. Two papers in these proceedings are closely related to our work. They also find STDP-like curves when attempting to maximize an information-theoretic measure— the mutual information between input and output—for a Spike Response Model [14, 15]. Bell & Parra [14] use a deterministic SRM model which does not model the LTD component of STDP properly. The derivation by Toyoizumi et al. [15] is valid only for an essentially constant membrane potential with small fluctuations. Neither of these approaches has succeeded in quantitatively modeling specific experimental data with neurobiologically-realistic timing parameters, and neither explains the saturation of LTD/LTP with increasing weights as we do. Nonetheless, these models make an interesting contrast to ours by suggesting a computational principle of optimization of information transmission, as contrasted with our principle of neural noise reduction. Perhaps experimental tests can be devised to distinguish between these competing theories. 2 The Stochastic Spike Response Model The Spike Response Model (SRM), defined by Gerstner [13], is a generic integrateand-fire model of a spiking neuron that closely corresponds to the behavior of a biological spiking neuron and is characterized in terms of a small set of easily interpretable parameters [16]. The standard SRM formulation describes the temporal evolution of the membrane potential based on past neuronal events, specifically as a weighted sum of postsynaptic potentials (PSPs) modulated by reset and threshold effects of previous postsynaptic spiking events. Following [13], the membrane potential of cell i at time t, ui(t), is defined as: ui(t) = η(t −ˆfi) + X j∈Γi wij X fj∈Ft j ε(t −ˆfi, t −fj), (1) where Γi is the set of inputs connected to neuron i, Ft j is the set of times prior to t that neuron j has spiked, ˆfi is the time of the last spike of neuron i, wij is the synaptic weight from neuron j to neuron i, ε(t −ˆfi, t −fj) is the PSP in neuron i due to an input spike from neuron j at time fj, and η(t −ˆfi) is the refractory response due to the postsynaptic spike at time ˆfi. Neuron i fires when the potential ui(t) exceeds a threshold (ϑ) from below. The postsynaptic potential ε is modeled as the differential alpha function in [13], defined with respect to two variables: the time since the most recent postsynaptic spike, x, and the time since the presynaptic spike, s: ε(x, s) = 1 1 −τs τm exp −s τm −exp −s τs H(s)H(x −s)+ (2) +exp −s −x τs exp −x τm −exp −x τs H(x)H(s −x) , where τs and τm are the rise and decay time-constants of the PSP, and H is the Heaviside function. The refractory reset function is defined to be [13]: η(x) = uabsH(∆abs −x)H(−x) + uabsexp −x + ∆abs τ f r + us rexp −x τ sr , (3) where uabs is a large negative contribution to the potential to model the absolute refractory period, with duration ∆abs. We smooth this refractory response by a fast decaying exponential with time constant τ f r . The third term in the sum represents the slow decaying exponential recovery of an elevated threshold, us r, with time constant τ s r . (Graphs of these ε and η functions can be found in [13].) We made a minor modification to the SRM described in [13] by relaxing the constraint that τ s r = τm; smoothing the absolute refractory function is mentioned in [13] but not explicitly defined as we do here. In all simulations presented, ∆abs = 2ms, τ s r = 4τm, and τ f r = 0.1τm. The SRM we just described is deterministic. Gerstner [13] introduces a stochastic variant of the SRM (sSRM) by incorporating the notion of a stochastic firing threshold: given membrane potential ui(t), the probability density of the neuron firing at time t is specified by ρ(ui(t)). Herrmann & Gerstner [17] find that then for a realistic escape-rate noise model the firing probability density as a function of the potential is initially small and constant, transitioning to asymptotically linear increasing around threshold ϑ. In our simulations, we use such a function: ρ(v) = β α(ln[1 + exp(α(ϑ −v))] −α(ϑ −v)), (4) where ϑ is the firing threshold in the absence of noise, α determines the abruptness of the constant-to-linear probability density transition around ϑ, and β determines the slope of the increasing part. Experiments with sigmoidal and exponential density functions were found to not qualitatively affect the results. 3 Minimizing Conditional Entropy We now derive the rule for adjusting the weight from a presynaptic neuron j to a postsynaptic sSRM neuron i, so as to minimize the entropy of i’s response given a particular spike sequence from j. A spike sequence is described by the set of all times at which spikes have occurred within some interval between 0 and T, denoted FT j for neuron j. We assume the interval is wide enough that spikes outside the interval do not influence the state of the neuron within the interval (e.g., through threshold reset effects). We can then treat intervals as independent of each other. Let the postsynaptic neuron i produce a response ξ ∈Ωi, where Ωi is the set of all possible responses given the input, ξ ≡FT i , and g(ξ) is the probability density over responses. The differential conditional entropy h(Ωi) of neuron i’s response is then defined as: h(Ωi) = − Z Ωi g(ξ)log g(ξ) dξ. (5) To minimize the differential conditional entropy by adjusting the neuron’s weights, we compute the gradient of the conditional entropy with respect to the weights: ∂h(Ωi) ∂wij = − Z Ωi g(ξ)∂log(g(ξ)) ∂wij log(g(ξ)) + 1 dξ. (6) For a differentiable neuron model, ∂log(g(ξ))/∂wij can be expressed as follows when neuron i fires once at time ˆfi [18]: ∂log(g(ξ)) ∂wij = Z T t=0 ∂ρ(ui(t)) ∂ui(t) ∂ui(t) ∂wij δ(t −ˆfi) −ρ(ui(t)) ρ(ui(t)) dt, (7) where δ(.) is the Dirac delta, and ρ(ui(t)) is the firing probability-density of neuron i at time t. (See [18] for the generalization to multiple postsynaptic spikes.) With the sSRM we can compute the partial derivatives ∂ρ(ui(t))/∂ui(t) and ∂ui(t)/∂wij. Given the density function (4), ∂ρ(ui(t)) ∂ui(t) = β 1 + exp(α(ϑ −ui(t)), ∂ui(t) ∂wij = ε(t −ˆfi, t −fj). To perform gradient descent in the conditional entropy, we use the weight update ∆wij ∝−∂h(Ωi) ∂wij (8) ∝ Z Ωi g(ξ) log(g(ξ)) + 1 Z T t=0 βε(t −ˆfi, t −fj) δ(t −ˆfi) −ρ(ui(t)) (1 + exp(α(ϑ −ui(t)))ρ(ui(t)) dt dξ. We can use numerical methods to evaluate Equation (8). However, it seems biologically unrealistic to suppose a neuron can integrate over all possible responses ξ. This dilemma can be circumvented in two ways. First, the resulting learning rule might be cached in some form through evolution so that the full computation is not necessary (e.g., in an STDP curve). Second, the specific response produced by a neuron on a single trial might be considered to be a sample from the distribution g(ξ), and the integration is performed by a sampling process over repeated trials; Figure 2: (a) Experimental setup of Zhang et al. and (b) their experimental STDP curve (small squares) vs. our model (solid line). Model parameters: τs = 1.5ms, τm = 12.25ms. each trial would produce a stochastic gradient step. 4 Simulation Methodology We model in detail the experiment of Zhang et al. [12] (Figure 2a). In this experiment, a post neuron is identified that has two neurons projecting to it, call them the pre and the driver. The pre is subthreshold: it produces depolarization but no spike. The driver is suprathreshold: it induces a spike in the post. Plasticity of the pre-post synapse is measured as a function of the timing between pre and post spikes (∆tpre−post) by varying the timing between induced spikes in the pre and the driver (∆tpre−driver). This measurement yields the well-known STDP curve (Figure 1b).1 The experiment imposes several constraints on a simulation: The driver alone causes spiking > 70% of the time, the pre alone causes spiking < 10% of the time, synchronous firing of driver and pre cause LTP if and only if the post fires, and the time constants of the EPSPs—τs and τm in the sSRM—are in the range of 1–3ms and 10–15ms respectively. These constraints remove many free parameters from our simulation. We do not explicitly model the two input cells; instead, we model the EPSPs they produce. The magnitude of these EPSPs are picked to satisfy the experimental constraints: the driver EPSP alone causes a spike in the post on 77.4% of trials, and the pre EPSP alone causes a spike on fewer than 0.1% of trials. Free parameters of the simulation are ϑ and β in the spike-probability function (α can be folded into ϑ), and the magnitude (us r, uabs) and reset time constants (τ s r , τ f r , ∆abs). The dependent variable of the simulation is ∆tpre−driver, and we measure the time of the post spike to determine ∆tpre−post. We estimate the weight update for a given ∆tpre−driver using Equation 8, approximating the integral by a summation over all time-discretized output responses consisting of 0, 1, or 2 spikes. Three or more spikes have a probability that is vanishingly small. 5 Results Figure 2b shows a typical STDP curve obtained from the model by plotting the estimated weight update of Equation 8 against ∆tpre−post. The model also explains a key finding that has not been explained by any other account, namely, that the magnitude of LTP or LTD decreases as the efficacy of the synapse between the pre and the post increases [2]. Further, the dependence is stronger for LTP than LTD. Figure 3a plots the magnitude of LTP for ∆tpre−post = −5 ms and the magnitude of LTD for ∆tpre−post = 7 ms as the amplitude of the pre’s EPSP is increased. The magnitude of the weight change decreases as the weight increases, and this 1In most experimental studies of STDP, the driver neuron is not used: the post is induced to spike by a direct depolarizing current injection. Modeling current injections requires additional assumptions. Consequently, we focus on the Zhang et al. experiment. Figure 3: (a) LTP and LTD plasticity as a function of synaptic efficacy of the subthreshold input. (b)-(d) STDP curves predicted by model as τm, us r, and ϑ are manipulated. effect is stronger for LTP than LTD. The model’s explanation for this phenomenon is simple: As the weight increases, its effect saturates, and a small change to the weight does little to alter its influence. Consequently, the gradient of the entropy with respect to the weight goes toward zero. The qualitative shape of the STDP curve is robust to settings of the model’s parameters, e.g., the EPSP decay time constant τm (Figure 3b), the strength of the threshold reset us r (Figure 3c), and the spiking threshold ϑ (Figure 3d). Additionally, the spike-probability function (exponential, sigmoidal, or linear) is not critical. The model makes two predictions relating the shape of the STDP curve to properties of a neuron. These predictions are empirically testable if a diverse population of cells can be studied: (1) the width of the LTD and LTP windows should depend on the EPSP decay time constant (Figure 3b), (2) the strength of LTP to LTD should depend on the strength of the threshold reset (Figure 3c), because stronger resets lead to reduced LTD by reducing the probability of a second spike. 6 Discussion In this paper, we explored a fundamental computational principle, that synapses adapt so as to minimize the variability of a neuron’s response in the face of noisy inputs, yielding more reliable neural representations. From this principle— instantiated as conditional entropy minimization—we derived the STDP learning curve. Importantly, the simulation methodology we used to derive the curve closely follows the procedure used in neurophysiological experiments [12]. Our simulations obtain an STDP curve that is robust to model parameters and details of the noise distribution. Our results are critically dependent on the use of Gerstner’s stochastic Spike Response Model, whose dynamics are a good approximation to those of a biological spiking neuron. The sSRM has the virtue of being characterized by parameters that are readily related to neural dynamics, and its dynamics are differentiable, allowing us to derive a gradient-descent learning rule. Our simulations are based on the classical STDP experiment in which a single presynaptic spike is paired with a single postsynaptic spike. The same methodology can be applied to the situation in which there are multiple presynaptic and/or postsynaptic spikes, although the computation involved becomes nontrivial. We are currently modeling the data from multi-spike experiments. We modeled the Zhang et al. experiment in which a driver neuron is used to induce the post to fire. To induce the post to fire, most other studies use a depolarizing current injection. We are not aware of any established model for current injection within the SRM framework, and we are currently elaborating such a model. We expect to then be able to simulate experiments in which current injections are used, allowing us to investigate the interesting issue of whether the two experimental techniques produce different forms of STDP. Acknowledgement Work of SMB supported by the Netherlands Organization for Scientific Research (NWO), TALENT grant S-62 588. References [1] G-q. Bi and M-m. Poo. Synaptic modification by correlated activity: Hebb’s postulate revisited. Ann. Rev. Neurosci., 24:139–166, 2001. [2] A. Kepecs, M.C.W. van Rossum, S. Song, and J. Tegner. Spike-timing-dependent plasticity: common themes and divergent vistas. Biol. Cybern., 87:446–458, 2002. [3] A. Saudargiene, B. Porr, and F. W¨org¨otter. How the shape of pre- and postsynaptic signals can influence stdp: A biophysical model. Neural Comp., 16:595–625, 2004. [4] W. Gerstner, R. Kempter, J. L. van Hemmen, and H. Wagner. A neural learning rule for sub-millisecond temporal coding. Nature, 383:76–78, 1996. [5] S. Song, K. Miller, and L. Abbott. Competitive hebbian learning through spiketime -dependent synaptic plasticity. Nat. Neurosci., 3:919–926, 2000. [6] R. van Rossum, G.-q. Bi, and G.G. Turrigiano. Stable hebbian learning from spike time dependent plasticity. J. Neurosci., 20:8812–8821, 2000. [7] L.F. Abbott and W. Gerstner. Homeostasis and Learning through STDP. In D. Hansel et al(eds), Methods and Models in Neurophysics, 2004. [8] R.P.N. Rao and T.J. Sejnowski. Spike-timing-dependent plasticity as temporal difference learning. Neural Comp., 13:2221–2237, 2001. [9] P. Dayan and M. H¨ausser. Plasticity kernels and temporal statistics. In S. Thrun, L. Saul, and B. Sch¨olkopf, editors, NIPS 16. 2004. [10] G. Chechik. Spike-timing-dependent plascticity and relevant mutual information maximization. Neural Comp., 15:1481–1510, 2003. [11] R.C. Froemke and Y. Dan. Spike-timing-dependent synaptic modification induced by natural spike trains. Nature, 416:433–438, 2002. [12] L.l. Zhang, H.W. Tao, C.E. Holt, W.A. Harris, and M-m. Poo. A critical window for cooperation and competition among developing retinotectal synapses. Nature, 395:37–44, 1998. [13] W. Gerstner. A framework for spiking neuron models: The spike response model. In F. Moss & S. Gielen (eds), The Handbook of Biol. Physics, vol 4, pp 469–516, 2001. [14] A.J. Bell and L.C. Parra. Maximizing information yields spike timing dependent plasticity. NIPS 17. 2005. [15] T. Toyoizumi, J-P. Pfister, K. Aihara, and W. Gerstner. Spike-timing dependent plasticity and mutual information maximization for a spiking neuron model. NIPS 17. 2005. [16] R. Jolivet, T.J. Lewis, and W. Gerstner. The spike response model: a framework to predict neuronal spike trains. In Kaynak et al.(eds), Proc. ICANN/ICONIP 2003, pp 846–853. 2003. [17] A. Herrmann and W. Gerstner. Noise and the PSTH response to current transients: I. J. Comp. Neurosci., 11:135–151, 2001. [18] X. Xie and H.S. Seung. Learning in neural networks by reinforcement of irregular spiking. Physical Review E, 69(041909), 2004.
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Maximum Likelihood Estimation of Intrinsic Dimension Elizaveta Levina Department of Statistics University of Michigan Ann Arbor MI 48109-1092 elevina@umich.edu Peter J. Bickel Department of Statistics University of California Berkeley CA 94720-3860 bickel@stat.berkeley.edu Abstract We propose a new method for estimating intrinsic dimension of a dataset derived by applying the principle of maximum likelihood to the distances between close neighbors. We derive the estimator by a Poisson process approximation, assess its bias and variance theoretically and by simulations, and apply it to a number of simulated and real datasets. We also show it has the best overall performance compared with two other intrinsic dimension estimators. 1 Introduction There is a consensus in the high-dimensional data analysis community that the only reason any methods work in very high dimensions is that, in fact, the data are not truly high-dimensional. Rather, they are embedded in a high-dimensional space, but can be efficiently summarized in a space of a much lower dimension, such as a nonlinear manifold. Then one can reduce dimension without losing much information for many types of real-life high-dimensional data, such as images, and avoid many of the “curses of dimensionality”. Learning these data manifolds can improve performance in classification and other applications, but if the data structure is complex and nonlinear, dimensionality reduction can be a hard problem. Traditional methods for dimensionality reduction include principal component analysis (PCA), which only deals with linear projections of the data, and multidimensional scaling (MDS), which aims at preserving pairwise distances and traditionally is used for visualizing data. Recently, there has been a surge of interest in manifold projection methods (Locally Linear Embedding (LLE) [1], Isomap [2], Laplacian and Hessian Eigenmaps [3, 4], and others), which focus on finding a nonlinear low-dimensional embedding of high-dimensional data. So far, these methods have mostly been used for exploratory tasks such as visualization, but they have also been successfully applied to classification problems [5, 6]. The dimension of the embedding is a key parameter for manifold projection methods: if the dimension is too small, important data features are “collapsed” onto the same dimension, and if the dimension is too large, the projections become noisy and, in some cases, unstable. There is no consensus, however, on how this dimension should be determined. LLE [1] and its variants assume the manifold dimension is provided by the user. Isomap [2] provides error curves that can be “eyeballed” to estimate dimension. The charting algorithm, a recent LLE variant [7], uses a heuristic estimate of dimension which is essentially equivalent to the regression estimator of [8] discussed below. Constructing a reliable estimator of intrinsic dimension and understanding its statistical properties will clearly facilitate further applications of manifold projection methods and improve their performance. We note that for applications such as classification, cross-validation is in principle the simplest solution – just pick the dimension which gives the lowest classification error. However, in practice the computational cost of cross-validating for the dimension is prohibitive, and an estimate of the intrinsic dimension will still be helpful, either to be used directly or to narrow down the range for cross-validation. In this paper, we present a new estimator of intrinsic dimension, study its statistical properties, and compare it to other estimators on both simulated and real datasets. Section 2 reviews previous work on intrinsic dimension. In Section 3 we derive the estimator and give its approximate asymptotic bias and variance. Section 4 presents results on datasets and compares our estimator to two other estimators of intrinsic dimension. Section 5 concludes with discussion. 2 Previous Work on Intrinsic Dimension Estimation The existing approaches to estimating the intrinsic dimension can be roughly divided into two groups: eigenvalue or projection methods, and geometric methods. Eigenvalue methods, from the early proposal of [9] to a recent variant [10] are based on a global or local PCA, with intrinsic dimension determined by the number of eigenvalues greater than a given threshold. Global PCA methods fail on nonlinear manifolds, and local methods depend heavily on the precise choice of local regions and thresholds [11]. The eigenvalue methods may be a good tool for exploratory data analysis, where one might plot the eigenvalues and look for a clear-cut boundary, but not for providing reliable estimates of intrinsic dimension. The geometric methods exploit the intrinsic geometry of the dataset and are most often based on fractal dimensions or nearest neighbor (NN) distances. Perhaps the most popular fractal dimension is the correlation dimension [12, 13]: given a set Sn = {x1, . . . , xn} in a metric space, define Cn(r) = 2 n(n −1) n X i=1 n X j=i+1 1{∥xi −xj∥< r}. (1) The correlation dimension is then estimated by plotting log Cn(r) against log r and estimating the slope of the linear part [12]. A recent variant [13] proposed plotting this estimate against the true dimension for some simulated data and then using this calibrating curve to estimate the dimension of a new dataset. This requires a different curve for each n, and the choice of calibration data may affect performance. The capacity dimension and packing numbers have also been used [14]. While the fractal methods successfully exploit certain geometric aspects of the data, the statistical properties of these methods have not been studied. The correlation dimension (1) implicitly uses NN distances, and there are methods that focus on them explicitly. The use of NN distances relies on the following fact: if X1, . . . , Xn are an independent identically distributed (i.i.d.) sample from a density f(x) in Rm, and Tk(x) is the Euclidean distance from a fixed point x to its k-th NN in the sample, then k n ≈f(x)V (m)[Tk(x)]m, (2) where V (m) = πm/2[Γ(m/2+1)]−1 is the volume of the unit sphere in Rm. That is, the proportion of sample points falling into a ball around x is roughly f(x) times the volume of the ball. The relationship (2) can be used to estimate the dimension by regressing log ¯Tk on log k over a suitable range of k, where ¯Tk = n−1 Pn i=1 Tk(Xi) is the average of distances from each point to its k-th NN [8, 11]. A comparison of this method to a local eigenvalue method [11] found that the NN method suffered more from underestimating dimension for high-dimensional datasets, but the eigenvalue method was sensitive to noise and parameter settings. A more sophisticated NN approach was recently proposed in [15], where the dimension is estimated from the length of the minimal spanning tree on the geodesic NN distances computed by Isomap. While there are certainly existing methods available for estimating intrinsic dimension, there are some issues that have not been adequately addressed. The behavior of the estimators as a function of sample size and dimension is not well understood or studied beyond the obvious “curse of dimensionality”; the statistical properties of the estimators, such as bias and variance, have not been looked at (with the exception of [15]); and comparisons between methods are not always presented. 3 A Maximum Likelihood Estimator of Intrinsic Dimension Here we derive the maximum likelihood estimator (MLE) of the dimension m from i.i.d. observations X1, . . . , Xn in Rp. The observations represent an embedding of a lower-dimensional sample, i.e., Xi = g(Yi), where Yi are sampled from an unknown smooth density f on Rm, with unknown m ≤p, and g is a continuous and sufficiently smooth (but not necessarily globally isometric) mapping. This assumption ensures that close neighbors in Rm are mapped to close neighbors in the embedding. The basic idea is to fix a point x, assume f(x) ≈const in a small sphere Sx(R) of radius R around x, and treat the observations as a homogeneous Poisson process in Sx(R). Consider the inhomogeneous process {N(t, x), 0 ≤t ≤R}, N(t, x) = n X i=1 1{Xi ∈Sx(t)} (3) which counts observations within distance t from x. Approximating this binomial (fixed n) process by a Poisson process and suppressing the dependence on x for now, we can write the rate λ(t) of the process N(t) as λ(t) = f(x)V (m)mtm−1 (4) This follows immediately from the Poisson process properties since V (m)mtm−1 = d dt[V (m)tm] is the surface area of the sphere Sx(t). Letting θ = log f(x), we can write the log-likelihood of the observed process N(t) as (see e.g., [16]) L(m, θ) = Z R 0 log λ(t) dN(t) − Z R 0 λ(t) dt This is an exponential family for which MLEs exist with probability →1 as n →∞ and are unique. The MLEs must satisfy the likelihood equations ∂L ∂θ = Z R 0 dN(t) − Z R 0 λ(t)dt = N(R) −eθV (m)Rm = 0, (5) ∂L ∂m = µ 1 m + V ′(m) V (m) ¶ N(R) + Z R 0 log t dN(t) − −eθV (m)Rm µ log R + V ′(m) V (m) ¶ = 0. (6) Substituting (5) into (6) gives the MLE for m: ˆmR(x) = 1 N(R, x) N(R,x) X j=1 log R Tj(x) −1 . (7) In practice, it may be more convenient to fix the number of neighbors k rather than the radius of the sphere R. Then the estimate in (7) becomes ˆmk(x) = 1 k −1 k−1 X j=1 log Tk(x) Tj(x) −1 . (8) Note that we omit the last (zero) term in the sum in (7). One could divide by k −2 rather than k −1 to make the estimator asymptotically unbiased, as we show below. Also note that the MLE of θ can be used to obtain an instant estimate of the entropy of f, which was also provided by the method used in [15]. For some applications, one may want to evaluate local dimension estimates at every data point, or average estimated dimensions within data clusters. We will, however, assume that all the data points come from the same “manifold”, and therefore average over all observations. The choice of k clearly affects the estimate. It can be the case that a dataset has different intrinsic dimensions at different scales, e.g., a line with noise added to it can be viewed as either 1-d or 2-d (this is discussed in detail in [14]). In such a case, it is informative to have different estimates at different scales. In general, for our estimator to work well the sphere should be small and contain sufficiently many points, and we have work in progress on choosing such a k automatically. For this paper, though, we simply average over a range of small to moderate values k = k1 . . . k2 to get the final estimates ˆmk = 1 n n X i=1 ˆmk(Xi) , ˆm = 1 k2 −k1 + 1 k2 X k=k1 ˆmk . (9) The choice of k1 and k2 and behavior of ˆmk as a function of k are discussed further in Section 4. The only parameters to set for this method are k1 and k2, and the computational cost is essentially the cost of finding k2 nearest neighbors for every point, which has to be done for most manifold projection methods anyway. 3.1 Asymptotic behavior of the estimator for m fixed, n →∞. Here we give a sketchy discussion of the asymptotic bias and variance of our estimator, to be elaborated elsewhere. The computations here are under the assumption that m is fixed, n →∞, k →∞, and k/n →0. As we remarked, for a given x if n →∞and R →0, the inhomogeneous binomial process N(t, x) in (3) converges weakly to the inhomogeneous Poisson process with rate λ(t) given by (4). If we condition on the distance Tk(x) and assume the Poisson approximation is exact, then © m−1 log(Tk/Tj) : 1 ≤j ≤k −1 ª are distributed as the order statistics of a sample of size k−1 from a standard exponential distribution. Hence U = m−1 Pk−1 j=1 log(Tk/Tj) has a Gamma(k−1, 1) distribution, and EU −1 = 1/(k −2). If we use k −2 to normalize, then under these assumptions, to a first order approximation E ( ˆmk(x)) = m, Var ( ˆmk(x)) = m2 k −3 (10) As this analysis is asymptotic in both k and n, the factor (k −1)/(k −2) makes no difference. There are, of course, higher order terms since N(t, x) is in fact a binomial process with EN(t, x) = λ(t) ¡ 1 + O(t2) ¢ , where O(t2) depends on m. With approximations (10), we have E ˆm = E ˆmk = m, but the computation of Var( ˆm) is complicated by the dependence among ˆmk(Xi). We have a heuristic argument (omitted for lack of space) that, by dividing ˆmk(Xi) into n/k roughly independent groups of size k each, the variance can be shown to be of order n−1, as it would if the estimators were independent. Our simulations confirm that this approximation is reasonable – for instance, for m-d Gaussians the ratio of the theoretical SD = C(k1, k2)m/√n (where C(k1, k2) is calculated as if all the terms in (9) were independent) to the actual SD of ˆm was between 0.7 and 1.3 for the range of values of m and n considered in Section 4. The bias, however, behaves worse than the asymptotics predict, as we discuss further in Section 5. 4 Numerical Results (a) (b) 0 10 20 30 40 50 60 70 80 90 100 3 3.5 4 4.5 5 5.5 6 6.5 7 k Dimension estimate mk n=2000 n=1000 n=500 n=200 0 10 20 30 40 50 60 70 80 90 100 5 10 15 20 25 k Dimension estimate mk m=20 m=10 m=5 m=2 Figure 1: The estimator ˆmk as a function of k. (a) 5-dimensional normal for several sample sizes. (b) Various m-dimensional normals with sample size n = 1000. We first investigate the properties of our estimator in detail by simulations, and then apply it to real datasets. The first issue is the behavior of ˆmk as a function of k. The results shown in Fig. 1 are for m-d Gaussians Nm(0, I), and a similar pattern holds for observations in a unit cube, on a hypersphere, and on the popular “Swiss roll” manifold. Fig. 1(a) shows ˆmk for a 5-d Gaussian as a function of k for several sample sizes n. For very small k the approximation does not work yet and ˆmk is unreasonably high, but for k as small as 10, the estimate is near the true value m = 5. The estimate shows some negative bias for large k, which decreases with growing sample size n, and, as Fig. 1(b) shows, increases with dimension. Note, however, that it is the intrinsic dimension m rather than the embedding dimension p ≥m that matters; and as our examples below and many examples elsewhere show, the intrinsic dimension for real data is frequently low. The plots in Fig. 1 show that the “ideal” range k1 . . . k2 is different for every combination of m and n, but the estimator is fairly stable as a function of k, apart from the first few values. While fine-tuning the range k1 . . . k2 for different n is possible and would reduce the bias, for simplicity and reproducibility of our results we fix k1 = 10, k2 = 20 throughout this paper. In this range, the estimates are not affected much by sample size or the positive bias for very small k, at least for the range of m and n under consideration. Next, we investigate an important and often overlooked issue of what happens when the data are near a manifold as opposed to exactly on a manifold. Fig. 2(a) shows simulation results for a 5-d correlated Gaussian with mean 0, and covariance matrix [σij] = [ρ + (1 −ρ)δij], with δij = 1{i = j}. As ρ changes from 0 to 1, the dimension changes from 5 (full spherical Gaussian) to 1 (a line in R5), with intermediate values of ρ providing noisy versions. (a) (b) 0 10 −4 10 −3 10 −2 10 −1 1 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 1−ρ (log scale) MLE of dimension n=2000 n=1000 n=500 n=100 0 5 10 15 20 25 30 0 5 10 15 20 25 30 True dimension Estimated dimension MLE Regression Corr.dim. Figure 2: (a) Data near a manifold: estimated dimension for correlated 5-d normal as a function of 1 −ρ. (b) The MLE, regression, and correlation dimension for uniform distributions on spheres with n = 1000. The three lines for each method show the mean ±2 SD (95% confidence intervals) over 1000 replications. The plots in Fig. 2(a) show that the MLE of dimension does not drop unless ρ is very close to 1, so the estimate is not affected by whether the data cloud is spherical or elongated. For ρ close to 1, when the dimension really drops, the estimate depends significantly on the sample size, which is to be expected: n = 100 highly correlated points look like a line, but n = 2000 points fill out the space around the line. This highlights the fundamental dependence of intrinsic dimension on the neighborhood scale, particularly when the data may be observed with noise. The MLE of dimension, while reflecting this dependence, behaves reasonably and robustly as a function of both ρ and n. A comparison of the MLE, the regression estimator (regressing log T k on log k), and the correlation dimension is shown in Fig. 2(b). The comparison is shown on uniformly distributed points on the surface of an m−dimensional sphere, but a similar pattern held in all our simulations. The regression range was held at k = 10 . . . 20 (the same as the MLE) for fair comparison, and the regression for correlation dimension was based on the first 10 . . . 100 distinct values of log Cn(r), to reflect the fact there are many more points for the log Cn(r) regression than for the log T k regression. We found in general that the correlation dimension graph can have more than one linear part, and is more sensitive to the choice of range than either the MLE or the regression estimator, but we tried to set the parameters for all methods in a way that does not give an unfair advantage to any and is easily reproducible. The comparison shows that, while all methods suffer from negative bias for higher dimensions, the correlation dimension has the smallest bias, with the MLE coming in close second. However, the variance of correlation dimension is much higher than that of the MLE (the SD is at least 10 times higher for all dimensions). The regression estimator, on the other hand, has relatively low variance (though always higher than the MLE) but the largest negative bias. On the balance of bias and variance, MLE is clearly the best choice. Figure 3: Two image datasets: hand rotation and Isomap faces (example images). Table 1: Estimated dimensions for popular manifold datasets. For the Swiss roll, the table gives mean(SD) over 1000 uniform samples. Dataset Data dim. Sample size MLE Regression Corr. dim. Swiss roll 3 1000 2.1(0.02) 1.8(0.03) 2.0(0.24) Faces 64 × 64 698 4.3 4.0 3.5 Hands 480 × 512 481 3.1 2.5 3.91 Finally, we compare the estimators on three popular manifold datasets (Table 1): the Swiss roll, and two image datasets shown on Fig. 3: the Isomap face database2, and the hand rotation sequence3 used in [14]. For the Swiss roll, the MLE again provides the best combination of bias and variance. The face database consists of images of an artificial face under three changing conditions: illumination, and vertical and horizontal orientation. Hence the intrinsic dimension of the dataset should be 3, but only if we had the full 3-d images of the face. All we have, however, are 2-d projections of the face, and it is clear that one needs more than one “basis” image to represent different poses (from casual inspection, front view and profile seem sufficient). The estimated dimension of about 4 is therefore very reasonable. The hand image data is a real video sequence of a hand rotating along a 1-d curve in space, but again several basis 2-d images are needed to represent different poses (in this case, front, back, and profile seem sufficient). The estimated dimension around 3 therefore seems reasonable. We note that the correlation dimension provides two completely different answers for this dataset, depending on which linear part of the curve is used; this is further evidence of its high variance, which makes it a less reliable estimate that the MLE. 5 Discussion In this paper, we have derived a maximum likelihood estimator of intrinsic dimension and some asymptotic approximations to its bias and variance. We have shown 1This estimate is obtained from the range 500...1000. For this dataset, the correlation dimension curve has two distinct linear parts, with the first part over the range we would normally use, 10...100, producing dimension 19.7, which is clearly unreasonable. 2http://isomap.stanford.edu/datasets.html 3http://vasc.ri.cmu.edu//idb/html/motion/hand/index.html that the MLE produces good results on a range of simulated and real datasets and outperforms two other dimension estimators. It does, however, suffer from a negative bias for high dimensions, which is a problem shared by all dimension estimators. One reason for this is that our approximation is based on sufficiently many observations falling into a small sphere, and that requires very large sample sizes in high dimensions (we shall elaborate and quantify this further elsewhere). For some datasets, such as points in a unit cube, there is also the issue of edge effects, which generally become more severe in high dimensions. One can potentially reduce the negative bias by removing the edge points by some criterion, but we found that the edge effects are small compared to the sample size problem, and we have been unable to achieve significant improvement in this manner. Another option used by [13] is calibration on simulated datasets with known dimension, but since the bias depends on the sampling distribution, and a different curve would be needed for every sample size, calibration does not solve the problem either. One should keep in mind, however, that for most interesting applications intrinsic dimension will not be very high – otherwise there is not much benefit in dimensionality reduction; hence in practice the MLE will provide a good estimate of dimension most of the time. References [1] S. T. Roweis and L. K. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290:2323–2326, 2000. [2] J. B. Tenenbaum, V. de Silva, and J. C. Landford. A global geometric framework for nonlinear dimensionality reduction. Science, 290:2319–2323, 2000. [3] M. Belkin and P. Niyogi. Laplacian eigenmaps and spectral techniques for embedding and clustering. In Advances in NIPS, volume 14. MIT Press, 2002. [4] D. L. Donoho and C. Grimes. Hessian eigenmaps: New locally linear embedding techniques for high-dimensional data. Technical Report TR 2003-08, Department of Statistics, Stanford University, 2003. [5] M. Belkin and P. Niyogi. Using manifold structure for partially labelled classification. In Advances in NIPS, volume 15. MIT Press, 2003. [6] M. Vlachos, C. Domeniconi, D. Gunopulos, G. Kollios, and N. Koudas. Non-linear dimensionality reduction techniques for classification and visualization. In Proceedings of 8th SIGKDD, pages 645–651. Edmonton, Canada, 2002. [7] M. Brand. Charting a manifold. In Advances in NIPS, volume 14. MIT Press, 2002. [8] K.W. Pettis, T.A. Bailey, A.K. Jain, and R.C. Dubes. An intrinsic dimensionality estimator from near-neighbor information. IEEE Trans. on PAMI, 1:25–37, 1979. [9] K. Fukunaga and D.R. Olsen. An algorithm for finding intrinsic dimensionality of data. IEEE Trans. on Computers, C-20:176–183, 1971. [10] J. Bruske and G. Sommer. Intrinsic dimensionality estimation with optimally topology preserving maps. IEEE Trans. on PAMI, 20(5):572–575, 1998. [11] P. Verveer and R. Duin. An evaluation of intrinsic dimensionality estimators. IEEE Trans. on PAMI, 17(1):81–86, 1995. [12] P. Grassberger and I. Procaccia. Measuring the strangeness of strange attractors. Physica, D9:189–208, 1983. [13] F. Camastra and A. Vinciarelli. Estimating the intrinsic dimension of data with a fractal-based approach. IEEE Trans. on PAMI, 24(10):1404–1407, 2002. [14] B. Kegl. Intrinsic dimension estimation using packing numbers. In Advances in NIPS, volume 14. MIT Press, 2002. [15] J. Costa and A. O. Hero. Geodesic entropic graphs for dimension and entropy estimation in manifold learning. IEEE Trans. on Signal Processing, 2004. To appear. [16] D. L. Snyder. Random Point Processes. Wiley, New York, 1975.
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Synergies between Intrinsic and Synaptic Plasticity in Individual Model Neurons Jochen Triesch Dept. of Cognitive Science, UC San Diego, La Jolla, CA, 92093-0515, USA Frankfurt Institute for Advanced Studies, Frankfurt am Main, Germany triesch@ucsd.edu Abstract This paper explores the computational consequences of simultaneous intrinsic and synaptic plasticity in individual model neurons. It proposes a new intrinsic plasticity mechanism for a continuous activation model neuron based on low order moments of the neuron’s firing rate distribution. The goal of the intrinsic plasticity mechanism is to enforce a sparse distribution of the neuron’s activity level. In conjunction with Hebbian learning at the neuron’s synapses, the neuron is shown to discover sparse directions in the input. 1 Introduction Neurons in the primate visual system exhibit a sparse distribution of firing rates. In particular, neurons in different visual cortical areas show an approximately exponential distribution of their firing rates in response to stimulation with natural video sequences [1]. The brain may do this because the exponential distribution maximizes entropy under the constraint of a fixed mean firing rate. The fixed mean firing rate constraint is often considered to reflect a desired level of metabolic costs. This view is theoretically appealing. However, it is currently not clear how neurons adjust their firing rate distribution to become sparse. Several different mechanisms seem to play a role: First, synaptic learning can change a neuron’s response to a distribution of inputs. Second, intrinsic learning may change conductances in the dendrites and soma to adapt the distribution of firing rates [7]. Third, non-linear lateral interactions in a network can make a neuron’s responses more sparse [8]. In the extreme case this leads to winner-take-all networks, which form a code where only a single unit is active for any given stimulus. Such ultra-sparse codes are considered inefficient, however. This paper investigates the interaction of intrinsic and synaptic learning processes in individual model neurons in the learning of sparse codes. We consider an individual continuous activation model neuron with a non-linear transfer function that has adjustable parameters. We are proposing a simple intrinsic learning mechanism based on estimates of low-order moments of the activity distribution that allows the model neuron to adjust the parameters of its non-linear transfer function to obtain an approximately exponential distribution of its activity. We then show that if combined with a standard Hebbian learning rule employing multiplicative weight normalization, this leads to the extraction of sparse features from the input. This is in sharp contrast to standard Hebbian learning in linear units with multiplicative weight normalization, which leads to the extraction of the principal Eigenvector of the input correlation matrix. We demonstrate the behavior of the combined intrinsic and synaptic learning mechanisms on the classic bars problem [4], a non-linear independent component analysis problem. The remainder of this paper is organized as follows. Section 2 introduces our scheme for intrinsic plasticity and presents experiments demonstrating the effectiveness of the proposed mechanism for inducing a sparse firing rate distribution. Section 3 studies the combination of intrinsic plasticity with Hebbian learning at the synapses and demonstrates how it gives rise to the discovery of sparse directions in the input. Finally, Sect. 4 discusses our findings in the context of related work. 2 Intrinsic Plasticity Mechanism Biological neurons do not only adapt synaptic properties but also change their excitability through the modification of voltage gated channels. Such intrinsic plasticity has been observed across many species and brain areas [9]. Although our understanding of these processes and their underlying mechanisms remains quite unclear, it has been hypothesized that this form of plasticity contributes to a neuron’s homeostasis of its mean firing rate level. Our basic hypothesis is that the goal of intrinsic plasticity is to ensure an approximately exponential distribution of firing rate levels in individual neurons. To our knowledge, this idea was first investigated in [7], where a Hodgkin-Huxley style model with a number of voltage gated conductances was considered. A learning rule was derived that adapts the properties of voltage gated channels to match the firing rate distribution of the unit to a desired distribution. In order to facilitate the simulation of potentially large networks we choose a different, more abstract level of modeling employing a continuous activation unit with a non-linear transfer function. Our model neuron is described by: Y = Sθ(X) , X = wT u , (1) where Y is the neuron’s output (firing rate), X is the neuron’s total synaptic current, w is the neuron’s weight vector representing synaptic strengths, the vector u represents the pre-synaptic input, and Sθ(.) is the neuron’s non-linear transfer function (activation function), parameterized by a vector of parameters θ. In this section we will not be concerned with synaptic mechanism changing the weight vector w, so we will just consider a particular distribution p(X = x) ≡p(x) of the net synaptic current and consider the resulting distribution of firing rates p(Y = y) ≡p(y). Intrinsic plasticity is modeled as inducing changes to the non-linear transfer function with the goal of bringing the distribution of activity levels p(y) close to an exponential distribution. In general terms, the problem is that of matching a distribution to another. Given a signal with a certain distribution, find a non-linear transfer function that converts the signal to one with a desired distribution. In image processing, this is typically called histogram matching. If there are no restrictions on the non-linearity then a solution can always be found. The standard example is histogram equalization, where a signal is passed through its own cumulative density function to give a uniform distribution over the interval [0, 1]. While this approach offers a general solution, it is unclear how individual neurons could achieve this goal. In particular, it requires that the individual neuron can change its nonlinear transfer function arbitrarily, i.e. it requires infinitely many degrees of freedom. 2.1 Intrinsic Plasticity Based on Low Order Moments of Firing Rate In contrast to the general scheme outlined above the approach proposed here utilizes a simple sigmoid non-linearity with only two adjustable parameters a and b: Sab(X) = 1 1 + exp (−(X −b) /a) . (2) Parameter a > 0 changes the steepness of the sigmoid, while parameter b shifts it left/right1. Qualitatively similar changes in spike threshold and slope of the activation function have been observed in cortical neurons. Since the non-linearity has only two degrees of freedom it is generally not possible to ascertain an exponential activity distribution for an arbitrary input distribution. A plausible alternative goal is to just match low order moments of the activity distribution to those of a specific target distribution. Since our sigmoid non-linearity has two parameters, we consider the first and second moments. For a random variable T following an exponential distribution with mean µ we have: p(T = t) = 1 µ exp (−t/µ) ; M 1 T ≡⟨T⟩= µ ; M 2 T ≡ T 2 = 2µ2 , (3) where ⟨.⟩denotes the expected value operator. Our intrinsic plasticity rule is formulated as a set of simple proportional control laws for a and b that drive the first and second moments ⟨Y ⟩and Y 2 of the output distributions to the values of the corresponding moments of an exponential distribution M 1 T and M 2 T : ˙a = γ Y 2 −2µ2 , ˙b = η (⟨Y ⟩−µ) , (4) where γ and η are learning rates. The mean µ of the desired exponential distribution is a free parameter which may vary across cortical areas. Equations (4) describe a system of coupled integro-differential equations where the integration is implicit in the expected value operations. Note that both ⟨Y ⟩and Y 2 depend on the sigmoid parameters a and b. From (4) it is obvious that there is a stationary point of these dynamics if the first and second moment of Y equal the desired values of µ and 2µ2, respectively. The first and second moments of Y need to be estimated online. In our model, we calculate estimates ˆ M 1 Y and ˆ M 2 Y of ⟨Y ⟩and Y 2 according to: ˙ˆ M 1 Y = λ(y −ˆ M 1 Y ) , ˙ˆ M 2 Y = λ(y2 −ˆ M 2 Y ) , (5) where λ is a small learning rate. 2.2 Experiments with Intrinsic Plasticity Mechanism We tested the proposed intrinsic plasticity mechanism for a number of distributions of the synaptic current X (Fig. 1). Consider the case where this current follows a Gaussian distribution with zero mean and unit variance: X ∼N(0, 1). Under this assumption we can calculate the moments ⟨Y ⟩and Y 2 (although only numerically) for any particular values of a and b. Panel a in Fig. 1 shows a phase diagram of this system. Its flow field is sketched and two sample trajectories converging to a stationary point are given. The stationary point is at the intersection of the nullclines where ⟨Y ⟩= µ and Y 2 = 2µ2. Its coordinates are a∞≈0.90, b∞≈2.38. Panel b compares the theoretically optimal transfer function (dotted), which would lead to an exactly exponential distribution of Y , with the learned sigmoidal transfer function (solid). The learned transfer function gives a very good fit. The resulting distribution of Y is in fact very close to the desired exponential distribution. For the general case of a Gaussian input distribution with mean µG and standard deviation σG, the sigmoid parameters will converge to a →a∞σG and b →b∞σG + µG under the intrinsic plasticity rule. If the input to the unit can be assumed to be Gaussian, this relation can be used to calculate the desired parameters of the sigmoid non-linearity directly. 1Note that while we view adjusting a and b as changing the shape of the sigmoid non-linearity, an equivalent view is that a and b are used to linearly rescale the signal X before it is passed through a “standard” logistic function. In general, however, intrinsic plasticity may give rise to non-linear changes that cannot be captured by such a linear re-scaling of all weights. 0 1 2 3 4 5 0 2 4 6 8 10 a b 0 1 2 3 4 5 0 2 4 6 8 10 a b −4 −2 0 2 4 0 0.2 0.4 0.6 0.8 1 input distribution optimal transfer fct. learned transfer fct. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 input distribution optimal transfer fct. learned transfer fct. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 input distribution optimal transfer fct. learned transfer fct. b a c d Figure 1: Dynamics of intrinsic plasticity mechanism for various input distributions. a,b: Gaussian input distribution. Panel a shows the phase plane diagram. Arrows indicate the flow field of the system. Dotted lines indicate approximate locations of the nullclines (found numerically).Two example trajectories are exhibited which converge to the stationary point (marked with a circle). Panel b shows the optimal (dotted) and learned transfer function (solid). The Gaussian input distribution (dashed, not drawn to scale) is also shown. c,d: same as b but for uniform and exponential input distribution. Parameters were µ = 0.1, λ = 5 × 10−4, η = 2 × 10−3, γ = 10−3. Panels c and d show the result of intrinsic plasticity for two other input distributions. In the case of a uniform input distribution in the interval [0, 1] (panel c) the optimal transfer function becomes infinitely steep for x →1. For an exponentially distributed input (panel d), the ideal transfer function would simply be the identity function. In both cases the intrinsic plasticity mechanism adjusts the sigmoid non-linearity in a sensible fashion and the output distribution is a fair approximation of the desired exponential distribution. 2.3 Discussion of the Intrinsic Plasticity Mechanism The proposed mechanism for intrinsic plasticity is effective in driving a neuron to exhibit an approximately exponential distribution of firing rates as observed in biological neurons in the visual system. The general idea is not restricted to the use of a sigmoid non-linearity. The same adaptation mechanism can also be used in conjunction with, say, an adjustable threshold-linear activation function. An interesting alternative to the proposed mechanism can be derived by directly minimizing the KL divergence between the output distribution and the desired exponential distribution through stochastic gradient descent. The resulting learning rule, which is closely related to a rule for adapting a sigmoid nonlinearity to maximize the output entropy derived by Bell and Sejnowski[2], will be discussed elsewhere. It leads to very similar results to the ones presented here. A biological implementation of the proposed mechanism is plausible. All that is needed are estimates of the first and second moment of the firing rate distribution. A specific, testable prediction of the simple model is that changes to the distribution of a neuron’s firing rate levels that keep the average firing rate of the neuron unchanged but alter the second moment of the firing rate distribution should lead to measurable changes in the neuron’s excitability. 3 Combination of Intrinsic and Synaptic Plasticity In this Section we want to study the effects of simultaneous intrinsic and synaptic learning for an individual model neuron. Synaptic learning is typically modeled with Hebbian learning rules, of which a large number are being used in the literature. In principle, any Hebbian learning rule can be combined with our scheme for intrinsic plasticity. Due to space limitations, we only consider the simplest of all Hebbian learning rules: ∆w = αuY (u) = αuSab(wT u) , (6) where the notation is identical to that of Sec. 2 and α is a learning rate. This learning rule is unstable and needs to be accompanied by a scheme limiting weight growth. We simply adopt a multiplicative normalization scheme that after each update re-scales the weight vector to unit length: w ←w/|| w ||. 3.1 Analysis for the Limiting Case of Fast Intrinsic Plasticity Under a few assumptions, an interesting intuition about the simultaneous intrinsic and Hebbian learning can be gained. Consider the limit of intrinsic plasticity being much faster than Hebbian plasticity. This may not be very plausible biologically, but it allows for an interesting analysis. In this case we may assume that the non-linearity has adapted to give an approximately exponential distribution of the firing rate Y before w can change much. Thus, from (6), ∆w can be seen as a weighted sum of the inputs u, with the activities Y acting as weights that follow an approximately exponential distribution. Since similar inputs u will produce similar outputs Y , the expected value of the weight update ⟨∆w⟩ will be dominated by a small set of inputs that produce the highest output activities. The remainder of the inputs will “pull” the weight vector back to the average input ⟨u⟩. Due to the multiplicative weight normalization, the stationary states of the weight vector are reached if ∆w is parallel to w, i.e., if ⟨∆w⟩= kw for some constant k. A simple example shall illustrate the effect of intrinsic plasticity on Hebbian learning in more detail. Consider the case where there are only two clusters of inputs at the locations c1 and c2. Let us also assume that both clusters account for exactly half of the inputs. If the weight vector is slightly closer to one of the two clusters, inputs from this cluster will activate the unit more strongly and will exert a stronger “pull” on the weight vector. Let m = µ ln(2) denote the median of the exponential firing rate distribution with mean µ. Then inputs from the closer cluster, say c1, will be responsible for all activities above m while the inputs from the other cluster will be responsible for all activities below m. Hence, the expected value of the weight update ⟨∆w⟩will be given by: ⟨∆w⟩ ≈ αc1 Z ∞ m y µ exp(−y/µ)dy + αc2 Z m 0 y µ exp(−y/µ)dy (7) = αµ 2 ((1 + ln 2) c1 + (1 −ln 2) c2) . (8) Taking the multiplicative weight normalization into account, we see that the weight vector 0 200 400 600 800 1000 0 2 4 6 8x 10 −3 cluster number i contribution to weight vector fi 0 2 4 6 8 x 10 −3 10 −3 10 −2 10 −1 10 0 contribution to weight vector fi frequency Figure 2: Left: relative contributions to the weight vector fi for N = 1000 input clusters (sorted). Right: the distribution of the fi is approximately exponential. will converge to either of the following two stationary states: w = (1 ± ln 2)c1 + (1 ∓ln 2)c2 || (1 ± ln 2)c1 + (1 ∓ln 2)c2 || . (9) The weight vector moves close to one of the two clusters but does not fully commit to it. For the general case of N input clusters, only a few clusters will strongly contribute to the final weight vector. Generalizing the result from above, it is not difficult to derive that the weight vector w will be proportional to a weighted sum of the cluster centers: w ∝ N X i=1 fici ; with fi = 1 + log(N) −i log(i) + (i −1) log(i −1) , (10) where we define 0 log(0) ≡0. Here, fi denotes the relative contribution of the i-th closest input cluster to the final weight vector. There can be at most N! resulting weight vectors owing to the number of possible assignments of the fi to the clusters. Note that the final weight vector does not depend on the desired mean activity level µ. Fig. 2 plots (10) for N = 1000 (left) and shows that the resulting distribution of the fi is approximately exponential (right). We can see why such a weight vector may correspond to a sparse direction in the input space as follows: consider the case where the input cluster centers are random vectors of unit length in a high-dimensional space. It is a property of high-dimensional spaces that random vectors are approximately orthogonal, so that cT i cj ≈δij, where δij is the Kronecker delta. If we consider the projection of an input from an arbitrary cluster, say cj, onto the weight vector, we see that wT cj ∝ P i ficT i cj ≈fj. The distribution of X = wT u follows the distribution of the fi, which is approximately exponential. Thus, the projection of all inputs onto the weight vector has an approximately exponential distribution. Note that this behavior is markedly different from Hebbian learning in a linear unit which leads to the extraction of the principal eigenvector of the input correlation matrix. It is interesting to note that in this situation the optimal transfer function S∗that will make the unit’s activity Y have an exponential distribution of a desired mean µ is simply a multiplication with a constant k, i.e. S∗(X) = kX. Thus, depending on the initial weight vector and the resulting distribution of X, the neuron’s activation function may transiently adapt to enforce an approximately exponential firing rate distribution, but the simultaneous Hebbian learning drives it back to a linear form. In the end, a simple linear activation function may result from this interplay of intrinsic and synaptic plasticity. In fact, the observation of approximately linear activation functions in cortical neurons is not uncommon. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10 4 0 0.5 1 input patterns/10000 activity Figure 3: Left: example stimuli from the “bars” problem for a 10 by 10 pixel retina. Right: the activity record shows the unit’s response to every 10th input pattern. Below, we show the learned weight vector after presentation of 10,000, 20,000, and 30,000 training patterns. 3.2 Application to the “Bars” Problem The “bars” problem is a standard problem for unsupervised learning architectures [4]. It is a non-linear ICA problem for which traditional ICA approaches have been shown to fail [5]. The input domain consists of an N-by-N retina. On this retina, all horizontal and vertical bars (2N in total) can be displayed. The presence or absence of each bar is determined independently, with every bar occurring with the same probability p (in our case p = 1/N). If a horizontal and a vertical bar overlap, the pixel at the intersection point will be just as bright as any other pixels on the bars, rather than twice as bright. This makes the problem a non-linear ICA problem. Example stimuli from the bars dataset are shown in Fig. 3 (left). Note that we normalize input vectors to unit length. The goal of learning in the bars problem is to find the independent sources of the images, i.e., the individual bars. Thus, the neural learning system should develop filters that represent the individual bars. We have trained an individual sigmoidal model neuron on the bars input domain. The theoretical analysis above assumed that intrinsic plasticity is much faster than synaptic plasticity. Here, we set the intrinsic plasticity to be slower than the synaptic plasticity, which is more plausible biologically, to see if this may still allow the discovery of sparse directions in the input. As illustrated in Fig. 3 (right) the unit’s weight vector aligns with one of the individual bars as soon as the intrinsic plasticity has pushed the model neuron into a regime where its responses are sparse: the unit has discovered one of the independent sources of the input domain. This result is robust if the desired mean activity µ of the unit is changed over a wide range. If µ is reduced from its default value (1/2N = 0.05) over several orders of magnitude (we tried down to 10−5) the result remains unchanged. However, if µ is increased above about 0.15, the unit will fail to represent an individual bar but will learn a mixture of two or more bars, with different bars being represented with different strengths. Thus, in this example — in contrast to the theoretical result above — the desired mean activity µ does influence the weight vector that is being learned. The reason for this is that the intrinsic plasticity only imperfectly adjusts the output distribution to the desired exponential shape. As can be seen in Fig. 3 the output has a multimodal structure. For low µ, only the highest mode, which corresponds to a specific single bar presented in isolation, contributes strongly to the weight vector. 4 Discussion Biological neurons are highly adaptive computation devices. While the plasticity of a neuron’s synapses has always been a core topic of neural computation research, there has been little work investigating the computational properties of intrinsic plasticity mechanisms and the relation between intrinsic and synaptic learning. This paper has investigated the potential role of intrinsic learning mechanisms operating at the soma when used in conjunction with Hebbian learning at the synapses. To this end, we have proposed a new intrinsic plasticity mechanism that adjusts the parameters of a sigmoid nonlinearity to move the neuron’s firing rate distribution to a sparse regime. The learning mechanism is effective in producing approximately exponential firing rate distributions as observed in neurons in the visual system of cats and primates. Studying simultaneous intrinsic and synaptic learning, we found a synergistic relation between the two. We demonstrated how the two mechanisms may cooperate to discover sparse directions in the input. When applied to the classic “bars” problem, a single unit was shown to discover one of the independent sources as soon as the intrinsic plasticity moved the unit’s activity distribution into a sparse regime. Thus, this research is related to other work in the area of Hebbian projection pursuit and Hebbian ICA, e.g., [3, 6]. In such approaches, the “standard” Hebbian weight update rule is modified to allow the discovery of non-gaussian directions in the input. We have shown that the combination of intrinsic plasticity with the standard Hebbian learning rule can be sufficient for the discovery of sparse directions in the input. Future work will analyze the combination of intrinsic plasticity with other Hebbian learning rules. Further, we would like to consider networks of such units and the formation of map-like representations. The nonlinear nature of the transfer function may facilitate the construction of hierarchical networks for unsupervised learning. It will also be interesting to study the effects of intrinsic plasticity in the context of recurrent networks, where it may contribute to keeping the network in a certain desired dynamic regime. Acknowledgments The author is supported by the National Science Foundation under grants NSF 0208451 and NSF 0233200. I thank Erik Murphy-Chutorian and Emanuel Todorov for discussions and comments on earlier drafts. References [1] R. Baddeley, L. F. Abbott, M.C. Booth, F. Sengpiel, and T. Freeman. Responses of neurons in primary and inferior temporal visual cortices to natural scenes. Proc. R. Soc. London, Ser. B, 264:1775–1783, 1998. [2] A. J. Bell and T. J. Sejnowski. An information-maximization approach to blind separation and blind deconvolution. Neural Computation, 7:1129–1159, 1995. [3] B. S. Blais, N. Intrator, H. Shouval, and L. N. Cooper. Receptive field formation in natural scene environments. Neural Computation, 10:1797–1813, 1998. [4] P. F¨oldi´ak. Forming sparse representations by local anti-hebbian learning. Biological Cybernetics, 64:165–170, 1990. [5] S. Hochreiter and J. Schmidhuber. Feature extraction through LOCOCODE. Neural Computation, 11(3):679–714, 1999. [6] A. Hyv¨arinen and E. Oja. Independent component analysis by general nonlinear hebbian-like learning rules. Signal Processing, 64(3):301–313, 1998. [7] M. Stemmler and C. Koch. How voltage-dependent conductances can adapt to maximize the information encoded by neuronal firing rate. Nature Neuroscience, 2(6):521–527, 1999. [8] W. E. Vinje and J. L. Gallant. Sparse coding and decorrelation in primary visual cortex during natural vision. Science, 287:1273–1276, 2000. [9] W. Zhang and D. J. Linden. The other side of the engram: Experience-driven changes in neuronal intrinsic excitability. Nature Reviews Neuroscience, 4:885–900, 2003.
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Distributed Information Regularization on Graphs Adrian Corduneanu CSAIL MIT Cambridge, MA 02139 adrianc@csail.mit.edu Tommi Jaakkola CSAIL MIT Cambridge, MA 02139 tommi@csail.mit.edu Abstract We provide a principle for semi-supervised learning based on optimizing the rate of communicating labels for unlabeled points with side information. The side information is expressed in terms of identities of sets of points or regions with the purpose of biasing the labels in each region to be the same. The resulting regularization objective is convex, has a unique solution, and the solution can be found with a pair of local propagation operations on graphs induced by the regions. We analyze the properties of the algorithm and demonstrate its performance on document classification tasks. 1 Introduction A number of approaches and algorithms have been proposed for semi-supervised learning including parametric models [1], random field/walk models [2, 3], or discriminative (kernel based) approaches [4]. The basic intuition underlying these methods is that the labels should not change within clusters of points, where the definition of a cluster may vary from one method to another. We provide here an alternative information theoretic criterion and associated algorithms for solving semi-supervised learning problems. Our formulation, an extension of [5, 6], is based on the idea of minimizing the number of bits required to communicate labels for unlabeled points, and involves no parametric assumptions. The communication scheme inherent to the approach is defined in terms of regions, weighted sets of points, that are shared between the sender and the receiver. The regions are important in capturing the topology over the points to be labeled, and, through the communication criterion, bias the labels to be the same within each region. We start by defining the communication game and the associated regularization problem, analyze properties of the regularizer, derive distributed algorithms for finding the unique solution to the regularization problem, and demonstrate the method on a document classification task. … … P (R) P (x|R) Q(y|x) R1 Rm x1 x2 xn−1 xn Figure 1: The topology imposed by the set of regions (squares) on unlabeled points (circles) 2 The communication problem Let S = {x1, . . . , xn} be the set of unlabeled points and Y the set of possible labels. We assume that target labels are available only for a small subset Sl ⊂S of the unlabeled points. The objective here is to find a conditional distribution Q(y|x) over the labels at each unlabeled point x ∈S. The estimation is made possible by a regularization criterion over the conditionals which we define here through a communication problem. The communication scheme relies on a set of regions R = {R1, . . . , Rm}, where each region R ∈R is a subset of the unlabeled points S (cf. Figure 1). The weights of points within each region are expressed in terms of a conditional distribution P(x|R), P x∈R P(x|R) = 1, and each region has an a priori probability P(R). We require only that P R∈R P(x|R)P(R) = 1/n for all x ∈S. (Note: in our overloaded notation “R” stands both for the set of points and its identity as a set). The regions and the membership probabilities are set in an application specific manner. For example, in a document classification setting we might define regions as sets of documents containing each word. The probabilities P(R) and P(x|R) could be subsequently derived from a word frequency representation of documents: if f(w|x) is the frequency of word w in document x, then for each pair of w and the corresponding region R we can set P(R) = P x∈S f(w|x)/n and P(x|R) = f(w|x)/(nP(R)). For any fixed conditionals {Q(y|x)} we define the communication problem as follows. The sender selects a region R ∈R with probability P(R) and a point within the region according to P(x|R). Since P R∈R P(x|R)P(R) = 1/n, each point x is overall equally likely to be selected. The label y is sampled from Q(y|x) and communicated to the receiver optimally using a coding scheme tailored to the region R (based on knowing P(x|R) and Q(y|x), x ∈R). The receiver has access to x, R, and the region specific coding scheme to reproduce y. The rate of information needed to be sent to the receiver in this scheme is given by Jc(Q; R) = X R∈R P(R)IR(x; y) = X R∈R P(R) X x∈R X y∈Y P(x|R)Q(y|x) log Q(y|x) Q(y|R) (1) where Q(y|R) = P x∈R P(x|R)Q(y|x) is the overall probability of y within the region. 3 The regularization problem We use Jc(Q; R) to regularize the conditionals. This regularizer biases the conditional distributions to be constant within each region so as to minimize the communication cost IR(x; y). Put another way, by introducing a region R we bias the points in the region to be labeled the same. By adding the cost of encoding the few available labeled points, expressed here in terms of the empirical distribution ˆP(y, x) whose support lies in Sl, the overall regularization criterion is given by J(Q; λ) = − X x∈Sl X y∈Y ˆP(y, x) log Q(y|x) + λJc(Q; R) (2) where λ > 0 is a regularization parameter. The following lemma guarantees that the solution is always unique: Lemma 1 J(Q; λ) for λ > 0 is a strictly convex function of the conditionals {Q(y|x)} provided that 1) each point x ∈S belongs to at least one region containing at least two points, and 2) the membership probabilities P(x|R) and P(R) are all non-zero. The proof follows immediately from the strict convexity of mutual information [7] and the fact that the two conditions guarantee that each Q(y|x) appears non-trivially in at least one mutual information term. 4 Regularizer and the number of labelings We consider here a simple setting where the labels are hard and binary, Q(y|x) ∈{0, 1}, and seek to bound the number of possible binary labelings consistent with a cap on the regularizer. We assume for simplicity that points in a region have uniform weights P(x|R). Let N(I) be the number of labelings of S consistent with an upper bound I on the regularizer Jc(Q, R). The goal is to show that N(I) is significantly less than 2n and N(I) →2 as I →0. Theorem 2 log2 N(I) ≤C(I) + I · n · t(R)/ minR P(R), where C(I) →1 as I →0, and t(R) is a property of R. Proof Let f(R) be the fraction of positive samples in region R. Because the labels are binary IR(x; y) is given by H(f(R)), where H is the entropy. If P R P(R)H(f(R)) ≤I then certainly H(f(R)) ≤I/P(R). Since the binary entropy is concave and symmetric w.r.t. 0.5, this is equivalent to f(R) ≤gR(I) or f(R) >= 1 −gR(I), where gR(I) is the inverse of H at I/P(R). We say that a region is mainly negative if the former condition holds, or mainly positive if the latter. If two regions R1 and R2 overlap by a large amount, they must be mainly positive or mainly negative together. Specifically this is the case if |R1 ∩R2| > gR1(I)|R1| + gR2(I)|R2| Consider a graph with vertices the regions, and edges whenever the above condition holds. Then regions in a connected component must be all mainly positive or mainly negative together. Let C(I) be the number of connected components in this graph, and note that C(I) →1 as I →0. We upper bound the number of labelings of the points spanned by a given connected component C, and subsequently combine the bounds. Consider the case in which all regions in C are mainly negative. For any subset C′ of C that still covers all the points spanned by C, f(C) ≤1 |C| X R∈C′ gI(R)|R| ≤max R gI(R) · P R∈C′ |R| |C′| (3) Thus f(C) ≤t(C) maxR gI(R) where t(C) = minC′∈C, C′ cover P R∈C′ |R| |C′| is the minimum average number of times a point in C is necessarily covered. There at most 2nf(R) log2(2/f(R)) labelings of a set of points of which at most nf(R) are positive. 1. Thus the number of feasible labelings of the connected component C is upper bounded by 21+nt(C) maxR gI(R) log2(2/(t(C) maxR gI(R))) where 1 is because C can be either mainly positive or mainly negative. By cumulating the bounds over all connected components and upper bounding the entropy-like term with I/P(R) we achieve the stated result. 2 Note that t(R), the average number of times a point is covered by a minimal subcovering of R normally does not scale with |R| and is a covering dependent constant. 5 Distributed propagation algorithm We introduce here a local propagation algorithm for minimizing J(Q; λ) that is both easy to implement and provably convergent. The algorithm can be seen as a variant of the BlahutArimoto algorithm in rate-distortion theory [8], adapted to the more structured context here. We begin by rewriting each mutual information term IR(x; y) in the criterion IR(x; y) = X x∈R X y∈Y P(x|R)Q(y|x) log Q(y|x) Q(y|R) (4) = min QR(·) X x∈R X y∈Y P(x|R)Q(y|x) log Q(y|x) QR(y) (5) where the variational distribution QR(y) can be chosen independently from Q(y|x) but the unique minimum is attained when QR(y) = Q(y|R) = P x∈R P(x|R)Q(y|x). We can extend the regularizer over both {Q(y|x)} and {QR(y)} by defining Jc(Q, QR; R) = X R∈R P(R) X x∈R X y∈Y P(x|R)Q(y|x) log Q(y|x) QR(y) (6) so that Jc(Q; R) = min{QR(·),R∈R} Jc(Q, QR; R) recovers the original regularizer. The local propagation algorithm follows from optimizing each Q(y|x) based on fixed {QR(y)} and subsequently finding each QR(y) given fixed {Q(y|x)}. We omit the straightforward derivation and provide only the resulting algorithm: for all points x ∈ S ∩Sl (not labeled) and for all regions R ∈R we perform the following complementary averaging updates Q(y|x) ← 1 Zx exp( X R:x∈R [nP(R)P(x|R)] log QR(y) ) (7) QR(y) ← X x∈R P(x|R)Q(y|x) (8) where Zx is a normalization constant. In other words, Q(y|x) is obtained by taking a weighted geometric average of the distributions associated with the regions, whereas QR(y) is (as before) a weighted arithmetic average of the conditionals within each region. In terms of the document classification example discussed earlier, the weight [nP(R)P(x|R)] appearing in the geometric average reduces to f(w|x), the frequency of word w identified with region R in document x. 1The result follows from Pk i=0 n i ≤ 2n k k Updating Q(y|x) for each labeled point x ∈Sl involves minimizing X y∈Y ˆP(y, x) log Q(y|x) −λ nH(Q(·|x)) − −λ X y∈Y Q(y|x) X R:x∈R P(R)P(x|R) log QR(y) (9) where H(Q(·|x)) is the Shannon entropy of the conditional. While the objective is strictly convex, the solution cannot be written in closed form and have to be found iteratively (e.g., via Newton-Raphson or simple bracketing when the labels are binary). A much simpler update Q(y|x) = δ(y, ˆyx), where ˆyx is the observed label for x, may suffice in practice. This update results from taking the limit of small λ and approximates the iterative solution. 6 Extensions 6.1 Structured labels and generalized propagation steps Here we extend the regularization framework to the case where the labels represent more structured annotations of objects. Let y be a vector of elementary labels y = [y1, . . . , yk]′ associated with a single object x. We assume that the distribution Q(y|x) = Q(y1, . . . , yk|x), for any x, can be represented as a tree structured graphical model, where the structure is the same for all x ∈S. The model is appropriate, e.g., in the context of assigning topics to documents. While the regularization principle applies directly if we leave Q(y|x) unconstrained, the calculations would be potentially infeasible due to the number of elementary labels involved, and inefficient as we would not explicitly make use of the assumed structure. Consequently, we seek to extend the regularization framework to handle distributions of the form QT (y|x) = k Y i=1 Qi(yi|x) Y (i,j)∈T Qij(yi, yj|x) Qi(yi|x)Qj(yj|x) (10) where T defines the edge set of the tree. The regularization problem will be formulated over {Qi(yi|x), Qij(yi, yj|x)} rather than unconstrained Q(y|x). The difficulty in this case arises from the fact that the arithmetic average (mixing) in eq (8) is not structure preserving (tree structured models are not mean flat). We can, however, also constrain QR(y) to factor according to the same tree structure. By restricting the class of variational distributions QR(y) that we consider, we necessarily obtain an upper bound on the original information criterion. If we minimize this upper bound with respect to {QR(y)}, under the factorization constraint QR,T (y) = k Y i=1 QR,i(yi) Y (i,j)∈T QR,ij(yi, yj) QR,i(yi|x)QR,j(yj), (11) given fixed {QT (y|x)}, we can replace eq (8) with simple “moment matching” updates QR,ij(yi, yj) ← X x∈R P(x|R)Qij(yi, yj|x) (12) The geometric update of Q(y|x) in eq (7) is structure preserving in the sense that if QR,T (y), R ∈R share the same tree structure, then so will the resulting conditional. The new updates will result in a monotonically decreasing bound on the original criterion. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Figure 2: Clusters correctly separated by information regularization given one label from each class 6.2 Complementary sets of regions In many cases the points to be labeled may have alternative feature representations, each leading to a different set of natural regions R(k). For example, in web page classification both the content of the page, and the type of documents that link to that page should be correlated with its topic. The relationship between these heterogeneous features may be complex, with some features more relevant to the classification task than others. Let Jc(Q; R(k)) denote the regularizer from the kth feature representation. Since the regularizers are on a common scale we can combine them linearly: Jc(Q; K, α) = K X k=1 αkJc(Q; R(k)) = K X k=1 X R∈R(k) αkPk(R)IR(x; y) (13) where αk ≥0 and P k αk = 1. The result is a regularizer with regions K = ∪kR(k) and adjusted a priori weights αkPk(R) over the regions. The solution can therefore be found as before provided that {αk} are known. When {αk} are unknown, we set them competitively. In other words, we minimize the worst information rate across the available representations. This gives rise to the following regularization problem: max αk≥0,P αk=1 min Q(y|x) J(Q; λ, α) (14) where J(Q; λ, α) is the overall objective that uses Jc(Q; K, α) as the regularizer. The maximum is well-defined since the objective is concave in {αk}. This follows immediately as the objective is a minimum of a collection of linear functions J(Q; λ, α) (linear in {αk}). At the optimum all Jc(Q; R(k)) for which αk > 0 have the same value (the same information rate). Other feature sets, those with αk = 0, do not contribute to the overall solution as their information rates are dominated by others. 7 Experiments We first illustrate the performance of information regularization on two generated binary classification tasks in the plane. Here we can derive a region covering from the Euclidean metric as spheres of a certain radius centered at each data point. On the data set in Figure 2 inspired from [3] the method correctly propagates the labels to the clusters starting 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 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 Figure 3: Ability of information regularization to correct the output of a prior classifier (left: before, right: after) from a single labeled point in each class. In the example in Figure 3 we demonstrate that information regularization can be used as a post-processing to supervised classification and improve error rates by taking advantage of the topology of the space. All points are a priori labeled by a linear classifier that is non-optimal and places a decision boundary through the negative and positive clusters. Information regularization (on a Euclidean region covering defined as circles around each data point) is able to correct the mislabeling of the clusters. Next we test the algorithm on a web document classification task, the WebKB data set of [1]. The data consists of 1051 pages collected from the websites of four universities. This particular subset of WebKB is a binary classification task into ’course’ and ’non-course’ pages. 22% of the documents are positive (’course’). The dataset is interesting because apart from the documents contents we have information about the link structure of the documents. The two sources of information can illustrate the capability of information regularization of combining heterogeneous unlabeled representations. Both ’text’ and ’link’ features used here are a bag-of-words representation of documents. To obtain ’link’ features we collect text that appears under all links that link to that page from other pages, and produce its bag-of-words representation. We employ no stemming, or stop-word processing, but restrict the vocabulary to 2000 text words and 500 link words. The experimental setup consists of 100 random selections of 3 positive labeled, 9 negative labeled, and the rest unlabeled. The test set includes all unlabeled documents. We report a na¨ıve Bayes baseline based on the model that features of different words are independent given the document class. The na¨ıve Bayes algorithm can be run on text features, link features, or combine the two feature sets by assuming independence. We also quote the performance of the semi-supervised method obtained by combining na¨ıve Bayes with the EM algorithm as in [9]. We measure the performance of the algorithms by the F-score equal to 2pr/(p+r), where p and r are the precision and recall. A high F-score indicates that the precision and recall are high and also close to each other. To compare algorithms independently of the probability threshold that decides between positive and negative samples, the results reported are the best F-scores for all possible settings of the threshold. The key issue in applying information regularization is the derivation of a sound region covering R. For document classification we obtained the best results by grouping all documents that share a certain word into the same region; thus each region is in fact a word, and there are as many regions as the size of the vocabulary. Regions are weighted equally, as well as the words belonging to the same region. The choice of λ is also task dependent. Here cross-validation selected a optimal value λ = 90. When running information reguTable 1: Web page classification comparison between na¨ıve Bayes and information regularization and semi-supervised na¨ı ve Bayes+EM on text, link, and joint features na¨ıve Bayes inforeg na¨ıve Bayes+EM text 82.85 85.10 93.69 link 65.64 82.85 67.18 both 83.33 86.15 91.01 larization with both text and link features we combined the coverings with a weight of 0.5 rather than optimizing it in a min-max fashion. All results are reported in Table 1. We observe that information regularization performs better than na¨ıve Bayes on all types of features, that combining text and link features improves performance of the regularization method, and that on link features the method performs better than the semi-supervised na¨ı ve Bayes+EM. Most likely the results do not reflect the full potential of information regularization due to the ad-hoc choice of regions based on the vocabulary used by na¨ıve Bayes. 8 Discussion The regularization principle introduced here provides a general information theoretic approach to exploiting unlabeled points. The solution implied by the principle is unique and can be found efficiently with distributed algorithms, performing complementary averages, on the graph induced by the regions. The propagation algorithms also extend to more structured settings. Our preliminary theoretical analysis concerning the number of possible labelings with bounded regularizer is suggestive but rather loose (tighter results can be found). The effect of the choice of the regions (sets of points that ought to be labeled the same) is critical in practice but not yet well-understood. References [1] A. Blum and T. Mitchell. Combining labeled and unlabeled data with co-training. In Proceedings of the 1998 Conference on Computational Learning Theory, 1998. [2] X. Zhu, Z. Ghahramani, and J. Lafferty. Semi-supervised learning using gaussian fields and harmonic functions. In Machine Learning: Proceedings of the Twentieth International Conference, 2003. [3] M. Szummer and T. Jaakkola. Partially labeled classification with markov random walks. In Advances in Neural Information Processing Systems 14, 2001. [4] O. Chapelle, J. Weston, and B. Schoelkopf. Cluster kernels for semi-supervised learning. In Advances in Neural Information Processing Systems 15, 2002. [5] M. Szummer and T. Jaakkola. Information regularization with partially labeled data. In NIPS’2002, volume 15, 2003. [6] A. Corduneanu and T. Jaakkola. On information regularization. In Proceedings of the 19th UAI, 2003. [7] T. M. Cover and J. A. Thomas. Elements of Information Theory. Wiley & Sons, New York, 1991. [8] R. E. Blahut. Computation of channel capacity and rate distortion functions. In IEEE Trans. Inform. Theory, volume 18, pages 460–473, July 1972. [9] K. Nigam, A.K. McCallum, S. Thrun, and T. Mitchell. Text classification from labeled and unlabeled documents using EM. Machine Learning, 39:103–134, 2000.
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The Convergence of Contrastive Divergences Alan Yuille Department of Statistics University of California at Los Angeles Los Angeles, CA 90095 yuille@stat.ucla.edu Abstract This paper analyses the Contrastive Divergence algorithm for learning statistical parameters. We relate the algorithm to the stochastic approximation literature. This enables us to specify conditions under which the algorithm is guaranteed to converge to the optimal solution (with probability 1). This includes necessary and sufficient conditions for the solution to be unbiased. 1 Introduction Many learning problems can be reduced to statistical inference of parameters. But inference algorithms for this task tend to be very slow. Recently Hinton proposed a new algorithm called contrastive divergences (CD) [1]. Computer simulations show that this algorithm tends to converge, and to converge rapidly, although not always to the correct solution [2]. Theoretical analysis shows that CD can fail but does not give conditions which guarantee convergence [3,4]. This paper relates CD to the stochastic approximation literature [5,6] and hence derives elementary conditions which ensure convergence (with probability 1). We conjecture that far stronger results can be obtained by applying more advanced techniques such as those described by Younes [7]. We also give necessary and sufficient conditions for the solution of CD to be unbiased. Section (2) describes CD and shows that it is closely related to a class of stochastic approximation algorithms for which convergence results exist. In section (3) we state and give a proof of a simple convergence theorem for stochastic approximation algorithms. Section (4) applies the theorem to give sufficient conditions for convergence of CD. 2 Contrastive Divergence and its Relations The task of statistical inference is to estimate the model parameters ω∗which minimize the Kullback-Leibler divergence D(P0(x)||P(x|ω)) between the empirical distribution function of the observed data P0(x) and the model P(x|ω). It is assumed that the model distribution is of the form P(x|ω) = e−E(x;ω)/Z(ω). Estimating the model parameters is difficult. For example, it is natural to try performing steepest descent on D(P0(x)||P(x|ω)). The steepest descent algorithm can be expressed as: ωt+1 −ωt = γt{− X x P0(x)∂E(x; ω) ∂ω + X x P(x|ω)∂E(x; ω) ∂ω }, (1) where the {γt} are constants. Unfortunately steepest descent is usually computationally intractable because of the need to compute the second term on the right hand side of equation (1). This is extremely difficult because of the need to evaluate the normalization term Z(ω) of P(x|ω). Moreover, steepest descent also risks getting stuck in a local minimum. There is, however, an important exception if we can express E(x; ω) in the special form E(x; ω) = ω · φ(x), for some function φ(x). In this case D(P0(x)||P(x|ω)) is convex and so steepest descent is guaranteed to converge to the global minimum. But the difficulty of evaluating Z(ω) remains. The CD algorithm is formally similar to steepest descent. But it avoids the need to evaluate Z(ω). Instead it approximates the second term on the right hand side of the steepest descent equation (1) by a stochastic term. This approximation is done by defining, for each ω, a Markov Chain Monte Carlo (MCMC) transition kernel Kω(x, y) whose invariant distribution is P(x|ω) (i.e. P x P(x|ω)Kω(x, y) = P(y|ω)). Then the CD algorithm can be expressed as: ωt+1 −ωt = γt{− X x P0(x)∂E(x; ω) ∂ω + X x Qω(x)∂E(x; ω) ∂ω }, (2) where Qω(x) is the empirical distribution function on the samples obtained by initializing the chain at the data samples P0(x) and running the Markov chain forward for m steps (the value of m is a design choice). We now observe that CD is similar to a class of stochastic approximation algorithms which also use MCMC methods to stochastically approximate the second term on the right hand side of the steepest descent equation (1). These algorithms are reviewed in [7] and have been used, for example, to learn probability distributions for modelling image texture [8]. A typical algorithm of this type introduces a state vector St(x) which is initialized by setting St=0(x) = P0(x). Then St(x) and ωt are updated sequentially as follows. St(x) is obtained by sampling with the transition kernel Kωt(x, y) using St−1(x) as the initial state for the chain. Then ωt+1 is computed by replacing the second term in equation (1) by the expectation with respect to St(x). From this perspective, we can obtain CD by having a state vector St(x) (= Qω(x)) which gets re-initialized to P0(x) at each time step. This stochastic approximation algorithm, and its many variants, have been extensively studied and convergence results have been obtained (see [7]). The convergence results are based on stochastic approximation theorems [6] whose history starts with the analysis of the Robbins-Monro algorithm [5]. Precise conditions can be specified which guarantee convergence in probability. In particular, Kushner [9] has proven convergence to global optima. Within the NIPS community, Orr and Leen [10] have studied the ability of these algorithms to escape from local minima by basin hopping. 3 Stochastic Approximation Algorithms and Convergence The general stochastic approximation algorithm is of the form: ωt+1 = ωt −γtS(ωt, Nt), (3) where Nt is a random variable sampled from a distribution Pn(N), γt is the damping coefficient, and S(., .) is an arbitrary function. We now state a theorem which gives sufficient conditions to ensure that the stochastic approximation algorithm (3) converges to a (solution) state ω∗. The theorem is chosen because of the simplicity of its proof and we point out that a large variety of alternative results are available, see [6,7,9] and the references they cite. The theorem involves three basic concepts. The first is a function L(ω) = (1/2)|ω −ω∗|2 which is a measure of the distance of the current state ω from the solution state ω∗(in the next section we will require ω∗= arg minω D(P0(x)||P(x|ω))). The second is the expected value P N Pn(N)S(ω, N) of the update term in the stochastic approximation algorithm (3). The third is the expected squared magnitude ⟨|S(ω, N)|2⟩of the update term. The theorem states that the algorithm will converge provided three conditions are satisfied. These conditions are fairly intuitive. The first condition requires that the expected update P N Pn(N)S(ω, N) has a large component towards the solution ω∗(i.e. in the direction of the negative gradient of L(ω)). The second condition requires that the expected squared magnitude ⟨|S(ω, N)|2⟩is bounded, so that the “noise” in the update is not too large. The third condition requires that the damping coefficients γt decrease with time t, so that the algorithm eventually settles down into a fixed state. This condition is satisfied by setting γt = 1/t, ∀t (which is the fastest fall off rate consistent with the SAC theorem). We now state the theorem and briefly sketch the proof which is based on martingale theory (for an introduction, see [11]). Stochastic Approximation Convergence (SAC) Theorem. Consider the stochastic approximation algorithm, equation (3), and let L(ω) = (1/2)|ω −ω∗|2. Then the algorithm will converge to ω∗with probability 1 provided: (1) −∇L(ω) · P N Pn(N)S(ω, N) ≥ K1L(ω) for some constant K1, (2) ⟨|S(ω, N)|2⟩t ≤K2(1 + L(ω)), where K2 is some constant and the expectation ⟨.⟩t is taken with respect to all the data prior to time t, and (3) P∞ t=1 γt = ∞and P∞ t=1 γ2 t < ∞. Proof. The proof [12] is a consequence of the supermartingale convergence theorem [11]. This theorem states that if Xt, Yt, Zt are positive random variables obeying P∞ t=0 Yt ≤∞ with probability one and ⟨Xt+1⟩≤Xt +Yt −Zt, ∀t, then Xt converges with probability 1 and P∞ t=0 Zt < ∞. To apply the theorem, set Xt = (1/2)|ωt −ω∗|2, set Yt = (1/2)K2γ2 t and Zt = −Xt(K2γ2 t −K1γt) (Zt is positive for sufficiently large t). Conditions 1 and 2 imply that Xt can only converge to 0. The result follows after some algebra. 4 CD and SAC The CD algorithm can be expressed as a stochastic approximation algorithm by setting: S(ωt, Nt) = − X x P0(x)∂E(x; ω) ∂ω + X x Qω(x)∂E(x; ω) ∂ω , (4) where the random variable Nt corresponds to the MCMC sampling used to obtain Qω(x). We can now apply the SAC to give three conditions which guarantee convergence of the CD algorithm. The third condition can be satisfied by setting γt = 1/t, ∀t. We can satisfy the second condition by requiring that the gradient of E(x; ω) with respect to ω is bounded, see equation (4). We conjecture that weaker conditions, such as requiring only that the gradient of E(x; ω) be bounded by a function linear in ω, can be obtained using the more sophisticated martingale analysis described in [7]. It remains to understand the first condition and to determine whether the solution is unbiased. These require studying the expected CD update: X Nt Pn(Nt)S(ωt, Nt) = − X x P0(x)∂E(x; ω) ∂ω + X y,x P0(y)Km ω (y, x)∂E(x; ω) ∂ω , (5) which is derived using the fact that the expected value of Qω(x) is P y P0(y)Km ω (y, x) (where the superscript m indicates running the transition kernel m times). We now re-express this expected CD update in two different ways, Results 1 and 2, which give alternative ways of understanding it. We then proceed to Results 3 and 4 which give conditions for convergence and unbiasedness of CD. But we must first introduce some background material from Markov Chain theory [13]. We choose the transition kernel Kω(x, y) to satisfy detailed balance so that P(x|ω)Kω(x, y) = P(y|ω)Kω(y, x). Detailed balance is obeyed by many MCMC algorithms and, in particular, is always satisfied by Metropolis-Hasting algorithms. It implies that P(x|ω) is the invariant kernel of Kω(x, y) so that P x P(x|ω)Kω(x, y) = P(y|ω) (all transition kernels satisfy P y Kω(x, y) = 1, ∀x). Detailed balance implies that the matrix Qω(x, y) = P(x|ω)1/2Kω(x, y)P(y|ω)−1/2 is symmetric and hence has orthogonal eigenvectors and eigenvalues {eµ ω(x), λµ ω}. The eigenvalues are ordered by magnitude (largest to smallest). The first eigenvalue is λ1 = 1 (so |λµ| < 1, µ ≥2). By standard linear algebra, we can write Qω(x, y) in terms of its eigenvectors and eigenvalues Qω(x, y) = P µ λµ ωeµ ω(x)eµ ω(y), which implies that we can express the transition kernel applied m times by: Km ω (x, y) = X µ {λµ ω}m{P(x|ω)}−1/2eµ ω(x){P(y|ω)}1/2eµ ω(y) = X µ {λµ ω}muµ ω(x)vµ ω(y), (6) where the {vµ ω(x)} and {uµ ω(x)} are the left and right eigenvectors of the transition kernel Kω(x, y). They are defined by: vµ ω(x) = eµ ω(x){P(x|ω)}1/2, uµ ω(x) = eµ ω(x){P(x|ω)}−1/2, ∀µ, (7) and it can be verified that P x vµ ω(x)Kω(x, y) = λµ ωvµ ω(y), ∀µ and P y Kω(x, y)uµ ω(y) = λµ ωuµ ω(x), ∀µ. In addition, the left and right eigenvectors are mutually orthonormal so that P x vµ ω(x)uν ω(x) = δµν, where δµν is the Kronecker delta function. This implies that we can express any function f(x) in equivalent expansions, f(x) = X µ { X y f(y)uµ ω(y)}vµ ω(x), f(x) = X µ { X y f(y)vµ ω(y)}uµ ω(x). (8) Moreover, the first left and right eigenvectors can be calculated explicitly to give: v1 ω(x) = P(x|ω), u1 ω(x) ∝1, λ1 ω = 1, (9) which follows because P(x|ω) is the (unique) invariant distribution of the transition kernel Kω(x, y) and hence is the first left eigenvector. We now have sufficient background to state and prove our first result. Result 1. The expected CD update corresponds to replacing the update term P x P(x|ω) ∂E(x;ω) ∂ω in the steepest descent equation (1) by: X x ∂E(x; ω) ∂ω P(x|ω) + X µ=2 {λµ ω}m{ X y P0(y)uµ ω(y)}{ X x vµ ω(x)∂E(x; ω) ∂ω }, (10) where {vµ ω(x), uµ ω(x)} are the left and right eigenvectors of Kω(x, y) with eigenvalues {λµ}. Proof. The expected CD update replaces P x P(x|ω) ∂E(x;ω) ∂ω by P y,x P0(y)Km ω (y, x) ∂E(x;ω) ∂ω , see equation (5). We use the eigenvector expansion of the transition kernel, equation (6), to express this as P y,x,µ P0(y){λµ ω}muµ ω(y)vµ ω(x) ∂E(x;ω) ∂ω . The result follows using the specific forms of the first eigenvectors, see equation (9). Result 1 demonstrates that the expected update of CD is similar to the steepest descent rule, see equations (1,10), but with an additional term P µ=2{λµ ω}m{P y P0(y)uµ ω(y)} {P x vµ ω(x) ∂E(x;ω) ∂ω } which will be small provided the magnitudes of the eigenvalues {λµ ω} are small for µ ≥2 (or if the transition kernel can be chosen so that P y P0(y)uµ ω is small for µ ≥2). We now give a second form for the expected update rule. To do this, we define a new variable g(x; ω). This is chosen so that P x P(x|ω)g(x; ω) = 0, ∀ω and the extrema of the Kullback-Leibler divergence occurs when P x P0(x)g(x; ω) = 0. Result 2. Let g(x; ω) = ∂E(x;ω) ∂ω −P x P(x|ω) ∂E(x;ω) ∂ω , then P x P(x|ω)g(x; ω) = 0, the extrema of the Kullback-Leibler divergence occur when P x P0(x)g(x; ω) = 0, and the expected update rule can be written as: ωt+1 = ωt −γt{ X x P0(x)g(x; ω) − X y,x P0(y)Km ω (y, x)g(x; ω)}. (11) Proof. The first result follows directly. The second follows because P x P0(x)g(x; ω) = P x P0(x) ∂E(x;ω) ∂ω −P x P(x|ω) ∂E(x;ω) ∂ω . To get the third we substitute the definition of g(x; ω) into the expected update equation (5). The result follows using the standard property of transition kernels that P y Km ω (x, y) = 1, ∀x. We now use Results 1 and 2 to understand the fixed points of the CD algorithm and determine whether it is biased. Result 3. The fixed points ω∗of the CD algorithm are true (unbiased) extrema of the KL divergence (i.e. P x P0(x)g(x; ω∗) = 0) if, and only if, we also have P y,x P0(y)Km ω∗(y, x)g(x; ω∗) = 0. A sufficient condition is that P0(y) and g(x; ω) lie in orthogonal eigenspaces of Kω∗(y, x). This includes the (known) special case when there exists ω∗such that P(x|ω∗) = P0(x) (see [2]). Proof. The first part follows directly from equation (11) in Result 2. The second part can be obtained by the eigenspace analysis in Result 1. Suppose P0(x) = P(x|ω∗). Recall that v1 ω∗(x) = P(x|ω∗), and so P y P0(y)uµ ωast(y) = 0, µ ̸= 1. Moreover, P x v1 ω∗g(x; ω∗) = 0. Hence P0(x) and g(x; ω∗) lie in orthogonal eigenspaces of Kω∗(y, x). Result 3 shows that whether CD converges to an unbiased estimate usually depends on the specific form of the MCMC transition matrix Kω(y, x). But there is an intuitive argument why the bias term P y,x P0(y)Km ω∗(y, x)g(x; ω∗) may tend to be small at places where P x P0(x)g(x; ω∗) = 0. This is because for small m, P y P0(y)Km ω∗(y, x) ≈P0(x) which satisfies P x P0(x)g(x; ω∗) = 0. Moreover, for large m, P y P0(y)Km ω∗(y, x) ≈P(x|ω∗) and we also have P x P(x|ω∗)g(x; ω∗) = 0. Alternatively, using Result 1, the bias term P y,x P0(y)Km ω∗(y, x)g(x; ω∗) can be expressed as P µ=2{λµ ω∗}m{P y P0(y)uµ ω∗(y)}{P x vµ ω∗(x) ∂E(x;ω∗) ∂ω }. This will tend to be small provided the eigenvalue moduli |λµ ω∗| are small for µ ≥2 (i.e. the standard conditions for a well defined Markov Chain). In general the bias term should decrease exponentially as |λ2 ω∗|m. Clearly it is also desirable to define the transition kernels Kω(x, y) so that the right eigenvectors {uµ ω(y) : µ ≥2} are as orthogonal as possible to the observed data P0(y). The practicality of CD depends on whether we can find an MCMC sampler such that the bias term P y,x P0(y)Km ω∗(y, x)g(x; ω∗) = 0 is small for most ω. If not, then the alternative stochastic algorithms may be preferable. Finally we give convergence conditions for the CD algorithm. Result 4 CD will converge with probability 1 to state ω∗provided γt = 1/t, ∂E ∂ω is bounded, and (ω −ω∗) · { X x P0(x)g(x; ω) − X y,x P0(y)Km ω (y, x)g(x; ω)} ≥K1|ω −ω∗|2, (12) for some K1. Proof. This follows from the SAC theorem and Result 2. The boundedness of ∂E ∂ω is required to ensure that the “update noise” is bounded in order to satisfy the second condition of the SAC theorem. Results 3 and 4 can be combined to ensure that CD converges (with probability 1) to the correct (unbiased) solution. This requires specifying that ω∗in Result 4 also satisfies the conditions P x P0(x)g(x; ω∗) = 0 and P y,x P0(y)Km ω∗(y, x)g(x; ω∗) = 0. 5 Conclusion The goal of this paper was to relate the Contrastive Divergence (CD) algorithm to the stochastic approximation literature. This enables us to give convergence conditions which ensure that CD will converge to the parameters ω∗that minimize the Kullback-Leibler divergence D(P0(x)||P(x|ω)). The analysis also gives necessary and sufficient conditions to determine whether the solution is unbiased. For more recent results, see Carreira-Perpignan and Hinton (in preparation). The results in this paper are elementary and preliminary. We conjecture that far more powerful results can be obtained by adapting the convergence theorems in the literature [6,7,9]. In particular, Younes [7] gives convergence results when the gradient of the energy ∂E(x; ω)/∂ω is bounded by a term that is linear in ω (and hence unbounded). He is also able to analyze the asymptotic behaviour of these algorithms. But adapting his mathematical techniques to Contrastive Divergence is beyond the scope of this paper. Finally, the analysis in this paper does not seem to capture many of the intuitions behind Contrastive Divergence [1]. But we hope that the techniques described in this paper may also stimulate research in this direction. Acknowledgements I thank Geoff Hinton, Max Welling and Yingnian Wu for stimulating conversations and feedback. Yingnian provided guidance to the stochastic approximation literature and Max gave useful comments on an early draft. This work was partially supported by an NSF SLC catalyst grant “Perceptual Learning and Brain Plasticity” NSF SBE-0350356. References [1]. G. Hinton. “Training Products of Experts by Minimizing Contrastive Divergence””. Neural Computation. 14, pp 1771-1800. 2002. [2]. Y.W. Teh, M. Welling, S. Osindero and G.E. Hinton. “Energy-Based Models for Sparse Overcomplete Representations”. Journal of Machine Learning Research. To appear. 2003. [3]. D. MacKay. “Failures of the one-step learning algorithm”. Available electronically at http://www.inference.phy.cam.ac.uk/mackay/abstracts/gbm.html. 2001. [4]. C.K.I. Williams and F.V. Agakov. “An Analysis of Contrastive Divergence Learning in Gaussian Boltzmann Machines”. Technical Report EDI-INF-RR-0120. Institute for Adaptive and Neural Computation. University of Edinburgh. 2002. [5]. H. Robbins and S. Monro. “A Stochastic Approximation Method”. Annals of Mathematical Sciences. Vol. 22, pp 400-407. 1951. [6]. H.J. Kushner and D.S. Clark. Stochastic Approximation for Constrained and Unconstrained Systems. New York. Springer-Verlag. 1978. [7]. L. Younes. “On the Convergence of Markovian Stochastic Algorithms with Rapidly Decreasing Ergodicity rates.” Stochastics and Stochastic Reports, 65, 177-228. 1999. [8]. S.C. Zhu and X. Liu. “Learning in Gibbsian Fields: How Accurate and How Fast Can It Be?”. IEEE Trans. Pattern Analysis and Machine Intelligence. Vol. 24, No. 7, July 2002. [9]. H.J. Kushner. “Asymptotic Global Behaviour for Stochastic Approximation and Diffusions with Slowly Decreasing Noise Effects: Global Minimization via Monte Carlo”. SIAM J. Appl. Math. 47:169-185. 1987. [10]. G.B. Orr and T.K. Leen. “Weight Space Probability Densities on Stochastic Learning: II Transients and Basin Hopping Times”. Advances in Neural Information Processing Systems, 5. Eds. Giles, Hanson, and Cowan. Morgan Kaufmann, San Mateo, CA. 1993. [11]. G.R. Grimmett and D. Stirzaker. Probability and Random Processes. Oxford University Press. 2001. [12]. B. Van Roy. Course notes. Prof. B. Van Roy. Stanford. (www.stanford.edu/class/msande339/notes/lecture6.ps). [13]. P. Bremaud. Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. Springer. New York. 1999.
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Log-concavity results on Gaussian process methods for supervised and unsupervised learning Liam Paninski Gatsby Computational Neuroscience Unit University College London liam@gatsby.ucl.ac.uk http://www.gatsby.ucl.ac.uk/∼liam Abstract Log-concavity is an important property in the context of optimization, Laplace approximation, and sampling; Bayesian methods based on Gaussian process priors have become quite popular recently for classification, regression, density estimation, and point process intensity estimation. Here we prove that the predictive densities corresponding to each of these applications are log-concave, given any observed data. We also prove that the likelihood is log-concave in the hyperparameters controlling the mean function of the Gaussian prior in the density and point process intensity estimation cases, and the mean, covariance, and observation noise parameters in the classification and regression cases; this result leads to a useful parameterization of these hyperparameters, indicating a suitably large class of priors for which the corresponding maximum a posteriori problem is log-concave. Introduction Bayesian methods based on Gaussian process priors have recently become quite popular for machine learning tasks (1). These techniques have enjoyed a good deal of theoretical examination, documenting their learning-theoretic (generalization) properties (2), and developing a variety of efficient computational schemes (e.g., (3–5), and references therein). We contribute to this theoretical literature here by presenting results on the log-concavity of the predictive densities and likelihood associated with several of these methods, specifically techniques for classification, regression, density estimation, and point process intensity estimation. These results, in turn, imply that it is relatively easy to tune the hyperparameters for, approximate the posterior distributions of, and sample from these models. Our results are based on methods which we believe will be applicable more widely in machine learning contexts, and so we give all necessary details of the (fairly straightforward) proof techniques used here. Log-concavity background We begin by discussing the log-concavity property: its uses, some examples of log-concave (l.c.) functions, and the key theorem on which our results are based. Log-concavity is perhaps most important in a maximization context: given a real function f ofsome vector parameter ⃗θ, if g(f(⃗θ)) is concave for some invertible function g, and the parameters ⃗θ live in some convex set, then f is unimodal, with no non-global local maxima. (Note that in this case a global maximum, if one exists, is not necessarily unique, but maximizers of f do form a convex set, and hence maxima are essentially unique in a sense.) Thus ascent procedures for maximization can be applied without fear of being trapped in local maxima; this is extremely useful when the space to be optimized over is high-dimensional. This logic clearly holds for any arbitrary rescaling g; of course, we are specifically interested in g(t) = log t, since logarithms are useful in the context of taking products (in a probabilistic context, read conditional independence): log-concavity is preserved under multiplication, since the logarithm converts multiplication into addition and concavity is preserved under addition. Log-concavity is also useful in the context of Laplace (central limit theorem - type) approximations (3), in which the logarithm of a function (typically a probability density or likelihood function) is approximated via a second-order (quadratic) expansion about its maximum or mean (6); this log-quadratic approximation is a reasonable approach for functions whose logs are known to be concave. Finally, l.c. distributions are in general easier to sample from than arbitrary distributions, as discussed in the context of adaptive rejection and slice sampling (7,8) and the randomwalk-based samplers analyzed in (9). We should note that log-concavity is not a generic property: l.c. probability densities necessarily have exponential tails (ruling out power law tails, and more generally distributions with any infinite moments). Log-concavity also induces a certain degree of smoothness; for example, l.c. densities must be continuous on the interior of their support. See, e.g., (9) for more detailed information on the various special properties implied by log-concavity. A few simple examples of l.c. functions are as follows: the Gaussian density in any dimension; the indicator of any convex set (e.g., the uniform density over any convex, compact set); the exponential density; the linear half-rectifier. More interesting well-known examples include the determinant of a matrix, or the inverse partition function of an energybased probabilistic model (e.g., an exponential family), Z−1(⃗θ) = ( R ef(⃗x,⃗θ)d⃗x)−1, l.c. in ⃗θ whenever f(⃗x, ⃗θ) is convex in ⃗θ for all ⃗x. Finally, log-concavity is preserved under taking products (as noted above), affine translations of the domain, and/or pointwise limits, since concavity is preserved under addition, affine translations, and pointwise limits, respectively. Sums of l.c. functions are not necessarily l.c., as is easily shown (e.g., a mixture of Gaussians with widely-separated means, or the indicator of the union of disjoint convex sets). However, a key theorem (10,11) gives: Theorem (Integrating out preserves log-concavity). If f(⃗x, ⃗y) is jointly l.c. in (⃗x, ⃗y), for ⃗x and ⃗y finite dimensional, then f0(⃗x) ≡ Z f(⃗x, ⃗y)d⃗y is l.c. in ⃗x. Think of ⃗y as a latent variable or hyperparameter we want to marginalize over. This very useful fact has seen applications in various branches of statistics and operations research, but does not seem well-known in the machine learning community. The theorem implies, for example, that convolutions of l.c. functions are l.c.; thus the random vectors with l.c. densities form a vector space. Moreover, indefinite integrals of l.c. functions are l.c.; hence the error function, and more generally the cumulative distribution function of any l.c. density, is l.c., which is useful in the setting of generalized linear models (12) for classification. Finally, the mass under a l.c. probability measure of a convex set which is translated in a convex manner is itself a l.c. function of the convex translation parameter (11). Gaussian process methods background We now give a brief review of Gaussian process methods. Our goals are modest; we will do little more than define notation. See, e.g., (1) and references for further details. Gaussian process methods are based on a Bayesian “latent variable” approach: dependencies between the observed input and output data {⃗ti} and {⃗yi} are modeled as arising through a hidden (unobserved) Gaussian process G(⃗t). Recall that a Gaussian process is a stochastic process whose finite-dimensional projections are all multivariate Gaussian, with means and covariances defined consistently for all possible projections, and is therefore specified by its mean µ(⃗t) and covariance function C(⃗t1,⃗t2). The applications we will consider may be divided into two settings; “supervised” and “unsupervised” problems. We discuss the somewhat simpler unsupervised case first (however, it should be noted that the supervised cases have received significantly more attention in the machine learning literature to date, and might be considered of more importance to this community). Density estimation: We are given unordered data {⃗ti}; the setup is valid for any sample space, but assume ⃗ti ∈ℜd, d < ∞, for concreteness. We model the data as i.i.d. samples from an unknown distribution p. The prior over these unknown distributions, in turn, is modeled as a conditioned Gaussian process, p ∼G(⃗t): p is drawn from a Gaussian process G(⃗t) of mean µ(⃗t) and covariance C (to ensure that the resulting random measures are well-defined, we will assume throughout that G is moderately well-behaved; almost-sure local Lebesgue integrability is sufficient), conditioned to be nonnegative and to integrate to one over some arbitrarily large compact set (the latter by an obvious limiting argument, to prevent conditioning on a set of measure zero; the introduction of the compact set is to avoid problems of the sort encountered when trying to define uniform probability measures on unbounded spaces) with respect to some natural base measure on the sample space (e.g., Lebesgue measure in ℜd) (13). It is worth emphasizing that this setup differs somewhat from some earlier proposals (5,14,15), which postulated that nonnegativity be enforced by, e.g., modeling log p or √p as Gaussian, instead of the Gaussian p here; each approach has its own advantages, and it is unclear at the moment whether our results can be extended to this context (as will be clear below, the roadblock is in the normalization constraint, which is transformed nonlinearly along with the density in the nonlinear warping setup). Point process intensity estimation: A nearly identical setup can be used if we assume the data {⃗ti} represent a sample from a Poisson process with an unknown underlying intensity function (16–18); the random density above is simply replaced by the random intensity function here (this type of model is known as a Cox, or doubly-stochastic, process in the point-process literature). The only difference is that intensity functions are not required to be normalized, so we need only condition the Gaussian process G(⃗t) from which we draw the intensity functions to be nonnegative. It turns out we will be free to use any l.c. and convex warping of the range space of the Gaussian process G(⃗t) to enforce positivity; suitable warpings include exponentiation (corresponding to modeling the logarithm of the intensity as Gaussian (17)) or linear half-rectification. The supervised cases require a few extra ingredients. We are given paired data, inputs {⃗ti} with corresponding outputs {⃗yi}. We model the outputs as noise-corrupted observations from the Gaussian process ⃗G(⃗t) at the points {⃗ti}; denote the additional hidden “observation” noise process as {⃗n(⃗ti)}. This noise process is not always taken to be Gaussian; for computational reasons, {⃗n(⃗ti)} is typically assumed i.i.d., and also independent of ⃗G(⃗t), but both of these assumptions will be unnecessary for the results stated below. Regression: We assume ⃗y(⃗ti) = ⃗G(⃗ti) + σi⃗n(⃗ti); in words, draw ⃗G(⃗t) from a Gaussian process of mean ⃗µ(⃗t) and covariance C; the outputs are then obtained by sampling this function ⃗G(⃗t) at ⃗ti and adding noise ⃗n(⃗ti) of scale σi. Classification: y(⃗ti) = 1 G(⃗ti) + σin(⃗ti) > 0 , where 1(.) denotes the indicator function of an event. This case is as in the regression model, except we only observe a binarythresholded version of the real output. Results Our first result concerns the predictive densities associated with the above models: the posterior density of any continuous linear functional of G(⃗t), given observed data D = {⃗ti} and/or {yi}, under the Gaussian process prior for G(⃗t). The simplest and most important case of such a linear projection is the projection onto a finite collection of coordinates, {⃗tpred}, say; in this special case, the predictive density is the posterior density p({G(⃗tpred)}|D). It turns out that all we need to assume is the log-concavity of the distribution p(G,⃗n); this is clearly more general than what is needed for the strictly Gaussian cases considered above (for example, Laplacian priors on G are permitted, which could lead to more robust performance). Also note that dependence of (G,⃗n) is allowed; this permits, for example, coupling of the effective scales of the observation noise ⃗ni = ⃗n(⃗ti) for nearby points ⃗ti. Additonally, we allow nonstationarity and anisotropic correlations in G. The result applies for any of the applications discussed above. Proposition 1 (Predictive density). Given a l.c. prior on (G,⃗n), the predictive density is always l.c., for any data D. In other words, conditioning on data preserves these l.c. processes (where an l.c. process, like a Gaussian process, is defined by the log-concavity of its finite-dimensional projections). This represents a significant generalization of the obvious fact that in the regression setup under Gaussian noise, conditioning preserves Gaussian processes. Our second result applies to the likelihood of the hyperparameters corresponding to the above applications: the mean function µ, the covariance function C, and the observation noise scales {σi}. We first state the main result in some generality, then provide some useful examples and interpretation below. For each j > 0, let Aj,⃗θ denote a family of linear maps from some finite-dimensional vector space Gj to ℜNdG, where dG = dim(⃗G(⃗ti)), and N is the number of observed data points. Our main assumptions are as follows: first, assume A−1 j,⃗θ may be written A−1 j,⃗θ = P θkKj,k, where {Kj,k} is a fixed set of matrices and the inverse is defined as a map from range(Aj,⃗θ) to Gj/ker(Aj,⃗θ). Second, assume that dim(A−1 j,⃗θ(V )) is constant in ⃗θ for any set V . Finally, equip the (doubly) latent space Gj × ℜNdG = {(GL,⃗n)} with a translation family of l.c. measures pj,µL(GL,⃗n) indexed by the mean parameter µL, i.e., pj,µL(GL,⃗n) = pj((GL,⃗n)−µL), for some fixed measure pj(.). Then if the sequence pj(G,⃗n) induced by pj and Aj converges pointwise to the joint density p(G,⃗n), then: Proposition 2 (Likelihood). In the supervised cases, the likelihood is jointly l.c. in the latent mean function, covariance parameters, and inverse noise scales (µL, ⃗θ, {σ−1 i }), for all data D. In the unsupervised cases, the likelihood is l.c. in the mean function µ. Note that the mean function µ(⃗t) is induced in a natural way by µL and Ai,⃗θ, and that we allow the noise scale parameters {σi} to vary independently, increasing the robustness of the supervised methods (19) (since outliers can be “explained,” without large perturbations of the underlying predictive distributions of G(⃗t), by simply increasing the corresponding noise scale σi). Of course, in practice, it is likely that to avoid overfitting one would want to reduce the effective number of free parameters by representing µ(⃗t) and ⃗θ in finitedimensional spaces, and restricting the freedom of the inverse noise scales {σi}. The logconcavity in the mean function µ(⃗t) demonstrated here is perhaps most useful in the point process setting, where µ(⃗t) can model the effect of excitatory or inhibitory inputs on the intensity function, with spatially- or temporally-varying patterns of excitation, and/or selfexcitatory interactions between observation sites ⃗ti (by letting µ(⃗t) depend on the observed points ⃗ti (16,20)). In the special case that the l.c. prior measure pj is taken to be Gaussian with covariance C0, the main assumption here is effectively on the parameterization of the covariance C; ignoring the (technical) limiting operation in j for the moment, we are assuming roughly that there exists a single basis in which, for all allowed ⃗θ, the covariance may be written C = A⃗θC0At ⃗θ, where A⃗θ is of the special form described above. We may simplify further by assuming that C0 is white and stationary. One important example of a suitable two-parameter family of covariance kernels satisfying the conditions of Proposition 2 is provided by the Ornstein-Uhlenbeck kernels (which correspond to exponentially-filtered one-dimensional white noise): C(t1, t2) = σ2e−2|t1−t2|/τ For this kernel, one can parameterize C = A⃗θAt ⃗θ, with A−1 ⃗θ = θ1I −θ2D∗, where I and D denote the identity and differential operators, respectively, and θk > 0 to ensure that C is positive-definite. (To derive this reparameterization, note that C(|t1 −t2|) solves (I −aD2)C(|t1 −t2|) = bδ(t), for suitable constants a, b.) Thus Proposition 2 generalizes a recent neuroscientific result: the likelihood for a certain neural model (the leaky integrate-and-fire model driven by Gaussian noise, for which the corresponding covariance is Ornstein-Uhlenbeck) is l.c. (21,22) (of course, in this case the model was motivated by biophysical instead of learning-theoretic concerns). In addition, multidimensional generalizations of this family are straightforward: corresponding kernels solve the Helmholtz problem, (I −a∆)C(⃗t) = bδ(⃗t), with ∆denoting the Laplacian. Solutions to this problem are well-known: in the isotropic case, we obtain a family of radial Bessel functions, with a, b again setting the overall magnitude and correlation scale of C(⃗t1,⃗t2) = C(||⃗t1 −⃗t2||2). Generalizing in a different direction, we could let A⃗θ include higher-order differential terms, A−1 ⃗θ = P k=0 θkDk; the resulting covariance kernels correspond to higher-order autoregression process priors. An even broader class of kernel parameterizations may be developed in the spectral domain: still assuming stationary white noise inputs, we may diagonalize C in the Fourier basis, that is, C(⃗ω) = OtP(⃗ω)O, with O the (unitary) Fourier transform operator and P(⃗ω) the power spectral density. Thus, comparing to the conditions above, if the spectral density may be written as P(⃗ω)−1 = | P k θkhk(⃗ω)|2 (where |.| denotes complex magnitude), for θk > 0 and functions hk(⃗ω) such that sign(real(hk(⃗ω))) is constant in k for any ⃗ω, then the likelihood will be l.c. in ⃗θ; A⃗θ here may be taken as the multiplication operator Ot(P k θkhk(⃗ω))−1). Remember that the smoothness of the sample paths of G(⃗t) depends on the rate of decay of the spectral density (1,23); thus we may obtain smoother (or rougher) kernel families by choosing P k θkhk(⃗ω) as more rapidly- (or slowly-)increasing. Proofs Predictive density. This proof is a straightforward application of the Prekopa theorem (10). Write the predictive distributions as p({LkG}|D) = K(D) Z p({LkG}, {G(ti), n(ti)})p({yi, ti}|{LkG}, {G(ti), n(ti)}), where {Lk} is a finite set of continuous linear functionals of G, K(D) is a constant that depends only on the data, the integral is over all {G(ti), n(ti)}, and {ni, yi} is ignored in the unsupervised case. Now we need only prove that the multiplicands on the right hand side above are l.c. The log-concavity of the left term is assumed; the right term, in turn, can be rewritten as p({yi, ti}|{LkG}, {G(ti), n(ti)}) = p({yi, ti}|{G(ti), n(ti)}), by the Markovian nature of the models. We prove the log-concavity of the right individually for each of our applications. In the supervised cases, {ti} is given and so we only need to look at p({yi}|{G(ti), n(ti)}). In the classification case, this is simply an indicator of the set \ i G(ti) + σini ≤0, yi = 0 > 0, yi = 1 , which is jointly convex in {G(ti), n(ti)}, completing the proof in this case. The regression case is proven in a similar fashion: write p({yi}|{G(ti), n(ti)}) as the limit as ϵ →0 of the indicator of the convex set \ i (|G(ti) + σini −yi| < ϵ) , then use the fact that pointwise limits preserve log-concavity. (The predictive distributions of {y(t)} will also be l.c. here, by a nearly identical argument.) In the density estimation case, the term p({ti}|{G(ti)}) = Y i G(ti) is obviously l.c. in {G(ti)}. However, recall that we perturbed the distribution of G(t) in this case as well, by conditioning G(t) to be positive and normalized. The fact that p({LkG}, {G(ti)}) is l.c. follows upon writing this term as a marginalization of densities which are products of l.c. densities with indicators of convex sets (enforcing the linear normalization and positivity constraints). Finally, for the point process intensity case, write the likelihood term, as usual, p({ti}|{G(ti)}) = e− R f(G(⃗t))d⃗t Y i f(G(⃗ti)), where f is the scalar warping function that takes the original Gaussian function G(⃗t) into the space of intensity functions. This term is clearly l.c. whenever f(s) is both convex and l.c. in s; for more details on this class of functions, see e.g. (20). Likelihood. We begin with the unsupervised cases. In the density estimation case, write the likelihood as L(µ) = Z dpµ(G)1C({G(⃗t)}) Y i G(⃗ti), with pµ(G) the probability of G under µ. Here 1C is the (l.c.) indicator function of the convex set enforcing the linear constraints (positivity and normalization) on G. All three terms in the integrand on the right are clearly jointly l.c. in (G, µ). In the point process case, L(µ) = Z dpµ(G)e− R f(G(⃗t))d⃗t Y i f(G(⃗ti)); the joint log-concavity of the three multiplicands on the right is again easily demonstrated. The Prekopa theorem cannot be directly applied here, since the functions 1C(.) and e− R f(.) depend in an infinite-dimensional way on G and µ; however, we can apply the Prekopa theorem to any finite-dimensional approximation of these functions (e.g., by approximating the normalization condition and exponential integral by Riemann sums and the positivity condition at a finite number of points), then obtain the theorem in the limit as the approximation becomes infinitely fine, using the fact that pointwise limits preserve log-concavity. For the supervised cases, write L(µL, ⃗θ, {σ−1}) = lim j Z dpj(GL,⃗n)1 Aj,θ(GL + µG L) + ⃗σ.(⃗n + µn L) ∈V = lim j Z dpj(GL,⃗n)1 (GL,⃗n) ∈( X k θkKj,kV,⃗σ.−1.V ) + µL , with V an appropriate convex constraint set (or limit thereof) defined by the observed data {yi}, µG L and µn L the projection of µL into Gj or ℜNdG, respectively, and . denoting pointwise operations on vectors. The result now follows immediately from Rinott’s theorem on convex translations of sets under l.c. probability measures (11,22). Again, we have not assumed anything more about p(GL,⃗n) than log-concavity; as before, this allows dependence of G and ⃗n, anisotropic correlations, etc. It is worth noting, though, that the above result is somewhat stronger in the supervised case than the unsupervised; the proof of log-concavity in the covariance parameters ⃗θ does not seem to generalize easily to the unsupervised setup (briefly, because log(P k θkyk) is not jointly concave in (θk, yk) for all (θk, yk), θkyk > 0, precluding a direct application of the Prekopa or Rinott theorems in the unsupervised case). Extensions to ensure that the unsupervised likelihood is l.c. in ⃗θ are possible, but require further restrictions on the form of p(G|⃗θ) and will not be pursued here. Discussion We have provided some useful results on the log-concavity of the predictive densities and likelihoods associated with several common Gaussian process methods for machine learning. In particular, our results preclude the existence of non-global local maxima in these functions, for any observed data; moreover, Laplace approximations of these functions will not, in general, be disastrous, and efficient sampling methods are available. Perhaps the main practical implication of our results stems from our proposition on the likelihood; we recommend a certain simple way to obtain parameterized families of kernels which respect this log-concavity property. Kernel families which may be obtained in this manner can range from extremely smooth to singular, and may model anisotropies flexibly. Finally, these results indicate useful classes of constraints (or more generally, regularizing priors) on the hyperparameters; any prior which is l.c. (or any constraint set which is convex) in the parameterization discussed here will lead to l.c. a posteriori problems. More generally, we have introduced some straightforward applications of a useful and interesting theorem. We expect that further applications in machine learning (e.g., in latent variable models, marginalization of hyperparameters, etc.) will be easy to find. Acknowledgements: We thank Z. Ghahramani and C. Williams for many helpful conversations. LP is supported by an International Research Fellowship from the Royal Society. References 1. M. Seeger, International Journal of Neural Systems 14, 1 (2004). 2. P. Sollich, A. Halees, Neural Computation 14, 1393 (2002). 3. C. Williams, D. Barber, IEEE PAMI 20, 1342 (1998). 4. M. Gibbs, D. MacKay, IEEE Transactions on Neural Networks 11, 1458 (2000). 5. L. Csato, Gaussian processes - iterative sparse approximations, Ph.D. thesis, Aston U. (2002). 6. T. Minka, A family of algorithms for approximate bayesian inference, Ph.D. thesis, MIT (2001). 7. W. Gilks, P. Wild, Applied Statistics 41, 337 (1992). 8. R. Neal, Annals of Statistics 31, 705 (2003). 9. L. Lovasz, S. Vempala, The geometry of logconcave functions and an O∗(n3) sampling algorithm, Tech. Rep. 2003-04, Microsoft Research (2003). 10. A. Prekopa, Acad Sci. Math. 34, 335 (1973). 11. Y. Rinott, Annals of Probability 4, 1020 (1976). 12. P. McCullagh, J. Nelder, Generalized linear models (Chapman and Hall, London, 1989). 13. J. Oakley, A. O’Hagan, Biometrika under review (2003). 14. I. Good, R. Gaskins, Biometrika 58, 255 (1971). 15. W. Bialek, C. Callan, S. Strong, Physical Review Letters 77, 4693 (1996). 16. D. Snyder, M. Miller, Random Point Processes in Time and Space (Springer-Verlag, 1991). 17. J. Moller, A. Syversveen, R. Waagepetersen, Scandinavian Journal of Statistics 25, 451 (1998). 18. I. DiMatteo, C. Genovese, R. Kass, Biometrika 88, 1055 (2001). 19. R. Neal, Monte Carlo implementation of Gaussian process models for Bayesian regression and classification, Tech. Rep. 9702, University of Toronto (1997). 20. L. Paninski, Network: Computation in Neural Systems 15, 243 (2004). 21. J. Pillow, L. Paninski, E. Simoncelli, NIPS 17 (2003). 22. L. Paninski, J. Pillow, E. Simoncelli, Neural Computation 16, 2533 (2004). 23. H. Dym, H. McKean, Fourier Series and Integrals (Academic Press, New York, 1972).
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Incremental Algorithms for Hierarchical Classification∗ Nicol`o Cesa-Bianchi Universit`a di Milano Milano, Italy Claudio Gentile Universit`a dell’Insubria Varese, Italy Andrea Tironi Luca Zaniboni Universit`a di Milano Crema, Italy Abstract We study the problem of hierarchical classification when labels corresponding to partial and/or multiple paths in the underlying taxonomy are allowed. We introduce a new hierarchical loss function, the H-loss, implementing the simple intuition that additional mistakes in the subtree of a mistaken class should not be charged for. Based on a probabilistic data model introduced in earlier work, we derive the Bayes-optimal classifier for the H-loss. We then empirically compare two incremental approximations of the Bayes-optimal classifier with a flat SVM classifier and with classifiers obtained by using hierarchical versions of the Perceptron and SVM algorithms. The experiments show that our simplest incremental approximation of the Bayes-optimal classifier performs, after just one training epoch, nearly as well as the hierarchical SVM classifier (which performs best). For the same incremental algorithm we also derive an H-loss bound showing, when data are generated by our probabilistic data model, exponentially fast convergence to the H-loss of the hierarchical classifier based on the true model parameters. 1 Introduction and basic definitions We study the problem of classifying data in a given taxonomy of labels, where the taxonomy is specified as a tree forest. We assume that every data instance is labelled with a (possibly empty) set of class labels called multilabel, with the only requirement that multilabels including some node i in the taxonony must also include all ancestors of i. Thus, each multilabel corresponds to the union of one or more paths in the forest, where each path must start from a root but it can terminate on an internal node (rather than a leaf). Learning algorithms for hierarchical classification have been investigated in, e.g., [8, 9, 10, 11, 12, 14, 15, 17, 20]. However, the scenario where labelling includes multiple and partial paths has received very little attention. The analysis in [5], which is mainly theoretical, shows in the multiple and partial path case a 0/1-loss bound for a hierarchical learning algorithm based on regularized least-squares estimates. In this work we extend [5] in several ways. First, we introduce a new hierarchical loss function, the H-loss, which is better suited than the 0/1-loss to analyze hierarchical classification tasks, and we derive the corresponding Bayes-optimal classifier under the parametric data model introduced in [5]. Second, considering various loss functions, including the H-loss, we empirically compare the performance of the following three incremental kernel-based ∗This work was supported in part by the PASCAL Network of Excellence under EC grant no. 506778. This publication only reflects the authors’ views. algorithms: 1) a hierarchical version of the classical Perceptron algorithm [16]; 2) an approximation to the Bayes-optimal classifier; 3) a simplified variant of this approximation. Finally, we show that, assuming data are indeed generated according to the parametric model mentioned before, the H-loss of the algorithm in 3) converges to the H-loss of the classifier based on the true model parameters. Our incremental algorithms are based on training linear-threshold classifiers in each node of the taxonomy. A similar approach has been studied in [8], though their model does not consider multiple-path classifications as we do. Incremental algorithms are the main focus of this research, since we strongly believe that they are a key tool for coping with tasks where large quantities of data items are generated and the classification system needs to be frequently adjusted to keep up with new items. However, we found it useful to provide a reference point for our empirical results. Thus we have also included in our experiments the results achieved by nonincremental algorithms. In particular, we have chosen a flat and a hierarchical version of SVM [21, 7, 19], which are known to perform well on the textual datasets considered here. We assume data elements are encoded as real vectors x ∈Rd which we call instances. A multilabel for an instance x is any subset of the set {1, . . . , N} of all labels/classes, including the empty set. We denote the multilabel associated with x by a vector y = (y1, . . . , yN) ∈{0, 1}N, where i belongs to the multilabel of x if and only if yi = 1. A taxonomy G is a forest whose trees are defined over the set of labels. A multilabel y ∈{0, 1}N is said to respect a taxonomy G if and only if y is the union of one or more paths in G, where each path starts from a root but need not terminate on a leaf. See Figure 1. We assume the data-generating mechanism produces examples (x, y) such that y respects some fixed underlying taxonomy G with N nodes. The set of roots in G is denoted by root(G). We use par(i) to denote the unique parent of node i, anc(i) to denote the set of ancestors of i, and sub(i) to denote the set of nodes in the subtree rooted at i (including i). Finally, given a predicate φ over a set Ω, we will use {φ} to denote both the subset of Ω where φ is true and the indicator function of this subset. 2 The H-loss Though several hierarchical losses have been proposed in the literature (e.g., in [11, 20]), no one has emerged as a standard yet. Since hierarchical losses are defined over multilabels, we start by considering two very simple functions measuring the discrepancy between multilabels by = (by1, ..., byN) and y = (y1, ..., yN): the 0/1-loss ℓ0/1(by, y) = {∃i : byi ̸= yi} and the symmetric difference loss ℓ∆(by, y) = {by1 ̸= y1} + . . . + {byN ̸= yN}. There are several ways of making these losses depend on a given taxonomy G. In this work, we follow the intuition “if a mistake is made at node i, then further mistakes made in the subtree rooted at i are unimportant”. That is, we do not require the algorithm be able to make fine-grained distinctions on tasks when it is unable to make coarse-grained ones. For example, if an algorithm failed to label a document with the class SPORTS, then the algorithm should not be charged more loss because it also failed to label the same document with the subclass SOCCER and the sub-subclass CHAMPIONS LEAGUE. A function implementing this intuition is defined by ℓH(by, y) = PN i=1 ci {byi ̸= yi ∧byj = yj, j ∈anc(i)}, where c1, . . . , cN > 0 are fixed cost coefficients. This loss, which we call H-loss, can also be described as follows: all paths in G from a root down to a leaf are examined and, whenever we encounter a node i such that byi ̸= yi, we add ci to the loss, whereas all the loss contributions in the subtree rooted at i are discarded. Note that if c1 = . . . = cN = 1 then ℓ0/1 ≤ℓH ≤ℓ∆. Choices of ci depending on the structure of G are proposed in Section 4. Given a multilabel by ∈{0, 1}N define its G-truncation as the multilabel y′ = (y′ 1, ..., y′ N) ∈{0, 1}N where, for each i = 1, . . . , N, y′ i = 1 iff byi = 1 and byj = 1 for all j ∈anc(i). Note that the G-truncation of any multilabel always respects G. A graphical (a) (b) (c) (d) Figure 1: A one-tree forest (repeated four times). Each node corresponds to a class in the taxonomy G, hence in this case N = 12. Gray nodes are included in the multilabel under consideration, white nodes are not. (a) A generic multilabel which does not respect G; (b) its G-truncation. (c) A second multilabel that respects G. (d) Superposition of multilabel (b) on multilabel (c): Only the checked nodes contribute to the H-loss between (b) and (c). representation of the notions introduced so far is given in Figure 1. In the next lemma we show that whenever y respects G, then ℓH(by, y) cannot be smaller than ℓH(y′, y). In other words, when the multilabel y to be predicted respects a taxonomy G then there is no loss of generality in restricting to predictions which respect G. Lemma 1 Let G be a taxonomy, y, by ∈{0, 1}N be two multilabels such that y respects G, and y′ be the G-truncation of by. Then ℓH(y′, y) ≤ℓH(by, y) . Proof. For each i = 1, . . . , N we show that y′ i ̸= yi and y′ j = yj for all j ∈anc(i) implies byi ̸= yi and byj = yj for all j ∈anc(i). Pick some i and suppose y′ i ̸= yi and y′ j = yj for all j ∈anc(i). Now suppose y′ j = 0 (and thus yj = 0) for some j ∈anc(i). Then yi = 0 since y respects G. But this implies y′ i = 1, contradicting the fact that the G-truncation y′ respects G. Therefore, it must be the case that y′ j = yj = 1 for all j ∈anc(i). Hence the G-truncation of by left each node j ∈anc(i) unchanged, implying byj = yj for all j ∈anc(i). But, since the G-truncation of by does not change the value of a node i whose ancestors j are such that byj = 1, this also implies byi = y′ i. Therefore byi ̸= yi and the proof is concluded. □ 3 A probabilistic data model Our learning algorithms are based on the following statistical model for the data, originally introduced in [5]. The model defines a probability distribution fG over the set of multilabels respecting a given taxonomy G by associating with each node i of G a Bernoulli random variable Yi and defining fG(y | x) = QN i=1 P Yi = yi | Ypar(i) = ypar(i), X = x . To guarantee that fG(y | x) = 0 whenever y ∈{0, 1}N does not respect G, we set P Yi = 1 | Ypar(i) = 0, X = x = 0. Notice that this definition of fG makes the (rather simplistic) assumption that all Yk with the same parent node i (i.e., the children of i) are independent when conditioned on Yi and x. Through fG we specify an i.i.d. process {(X1, Y 1), (X2, Y 2), . . .}, where, for t = 1, 2, . . ., the multilabel Y t is distributed according to fG(· | Xt) and Xt is distributed according to a fixed and unknown distribution D. Each example (xt, yt) is thus a realization of the corresponding pair (Xt, Y t) of random variables. Our parametric model for fG is described as follows. First, we assume that the support of D is the surface of the d-dimensional unit sphere (i.e., instances x ∈Rd are such that ||x|| = 1). With each node i in the taxonomy, we associate a unit-norm weight vector ui ∈Rd. Then, we define the conditional probabilities for a nonroot node i with parent j by P (Yi = 1 | Yj = 1, X = x) = (1 + u⊤ i x)/2. If i is a root node, the previous equation simplifies to P (Yi = 1 | X = x) = (1 + u⊤ i x)/2. 3.1 The Bayes-optimal classifier for the H-loss We now describe a classifier, called H-BAYES, that is the Bayes-optimal classifier for the H-loss. In other words, H-BAYES classifies any instance x with the multilabel by = argmin¯y∈{0,1} E[ℓH(¯y, Y ) | x ]. Define pi(x) = P Yi = 1 | Ypar(i) = 1, X = x . When no ambiguity arises, we write pi instead of pi(x). Now, fix any unit-length instance x and let by be a multilabel that respects G. For each node i in G, recursively define Hi,x(by) = ci (pi(1 −byi) + (1 −pi)byi) + P k∈child(i) Hk,x(by) . The classifier H-BAYES operates as follows. It starts by putting all nodes of G in a set S; nodes are then removed from S one by one. A node i can be removed only if i is a leaf or if all nodes j in the subtree rooted at i have been already removed. When i is removed, its value byi is set to 1 if and only if pi 2 −P k∈child(i) Hk,x(by)/ci ≥1 . (1) (Note that if i is a leaf then (1) is equivalent to byi = {pi ≥1/2}.) If byi is set to zero, then all nodes in the subtree rooted at i are set to zero. Theorem 2 For any taxonomy G and all unit-length x ∈Rd, the multilabel generated by H-BAYES is the Bayes-optimal classification of x for the H-loss. Proof sketch. Let by be the multilabel assigned by H-BAYES and y∗be any multilabel minimizing the expected H-loss. Introducing the short-hand Ex[·] = E[· | x], we can write Ex ℓH(by, Y ) = PN i=1 ci (pi(1 −byi) + (1 −pi)byi) Q j∈anc(i) pj {byj = 1} . Note that we can recursively decompose the expected H-loss as Ex ℓH(by, Y ) = P i∈root(G) Ex Hi(by, Y ), where ExHi(by, Y ) = ci (pi(1 −byi) + (1 −pi)byi) Y j∈anc(i) pj {byj = 1} + X k∈child(i) ExHk(by, Y ) . (2) Pick a node i. If i is a leaf, then the sum in the RHS of (2) disappears and y∗ i = {pi ≥1/2}, which is also the minimizer of Hi,x(by) = ci (pi(1 −byi) + (1 −pi)byi), implying byi = y∗ i . Now let i be an internal node and inductively assume byj = by∗ j for all j ∈sub(i). Notice that the factors Q j∈anc(i) pj {byj = 1} occur in both terms in the RHS of (2). Hence y∗ i does not depend on these factors and we can equivalently minimize ci (pi(1 −byi) + (1 −pi)byi) + pi{byi = 1} P k∈child(i) Hk,x(by), (3) where we noted that, for each k ∈child(i), ExHk(by, Y ) = Q j∈anc(i) pj {byj = 1} pi{byi = 1}Hk,x(by) . Now observe that y∗ i minimizing (3) is equivalent to the assignment produced by H-BAYES. To conclude the proof, note that whenever y∗ i = 0, Lemma 1 requires that y∗ j = 0 for all nodes j ∈sub(i), which is exactly what H-BAYES does. □ 4 The algorithms We consider three incremental algorithms. Each one of these algorithms learns a hierarchical classifier by training a decision function gi : Rd →{0, 1} at each node i = 1, . . . , N. For a given set g1, . . . , gN of decision functions, the hierarchical classifier generated by these algorithms classifies an instance x through a multilabel by = (by1, ..., byN) defined as follows: byi = gi(x) if i ∈root(G) or byj = 1 for all j ∈anc(i) 0 otherwise. (4) Note that by computed this way respects G. The classifiers (4) are trained incrementally. Let gi,t be the decision function at node i after training on the first t −1 examples. When the next training example (xt, yt) is available, the algorithms compute the multilabel byt using classifier (4) based on g1,t(xt), . . . , gN,t(xt). Then, the algorithms consider for an update only those decision functions sitting at nodes i satisfying either i ∈root(G) or ypar(i),t = 1. We call such nodes eligible at time t. The decision functions of all other nodes are left unchanged. The first algorithm we consider is a simple hierarchical version of the Perceptron algorithm [16], which we call H-PERC. The decision functions at time t are defined by gi,t(xt) = {w⊤ i,txt ≥0}. In the update phase, the Perceptron rule wi,t+1 = wi,t +yi,txt is applied to every node i eligible at time t and such that byi,t ̸= yi,t. The second algorithm, called APPROX-H-BAYES, approximates the H-BAYES classifier of Section 3.1 by replacing the unknown quantities pi(xt) with estimates (1+w⊤ i,txt)/2. The weights wi,t are regularized least-squares estimates defined by wi,t = (I + Si,t−1 S⊤ i,t−1 + xt x⊤ t )−1Si,t−1y(i) t−1 . (5) The columns of the matrix Si,t−1 are all past instances xs that have been stored at node i; the s-th component of vector y(i) t−1 is the i-th component yi,s of the multilabel ys associated with instance xs. In the update phase, an instance xt is stored at node i, causing an update of wi,t, whenever i is eligible at time t and |w⊤ i,txt| ≤ p (5 ln t)/Ni,t, where Ni,t is the number of instances stored at node i up to time t −1. The corresponding decision functions gi,t are of the form gi,t(xt) = {w⊤ i,txt ≥τi,t}, where the threshold τi,t ≥0 at node i depends on the margin values w⊤ j,txt achieved by nodes j ∈sub(i) — recall (1). Note that gi,t is not a linear-threshold function, as xt appears in the definition of wi,t. The margin threshold p (5 ln t)/Ni,t, controlling the update of node i at time t, reduces the space requirements of the classifier by keeping matrices Si,t suitably small. This threshold is motivated by the work [4] on selective sampling. The third algorithm, which we call H-RLS (Hierarchical Regularized Least Squares), is a simplified variant of APPROX-H-BAYES in which the thresholds τi,t are set to zero. That is, we have gi,t(xt) = {w⊤ i,txt ≥0} where the weights wi,t are defined as in (5) and updated as in the APPROX-H-BAYES algorithm. Details on how to run APPROX-H-BAYES and H-RLS in dual variables and perform an update at node i in time O(N 2 i,t) are found in [3] (where a mistake-driven version of H-RLS is analyzed). 5 Experimental results The empirical evaluation of the algorithms was carried out on two well-known datasets of free-text documents. The first dataset consists of the first (in chronological order) 100,000 newswire stories from the Reuters Corpus Volume 1, RCV1 [2]. The associated taxonomy of labels, which are the topics of the documents, has 101 nodes organized in a forest of 4 trees. The forest is shallow: the longest path has length 3 and the the distribution of nodes, sorted by increasing path length, is {0.04, 0.53, 0.42, 0.01}. For this dataset, we used the bag-of-words vectorization performed by Xerox Research Center Europe within the EC project KerMIT (see [4] for details on preprocessing). The 100,000 documents were divided into 5 equally sized groups of chronologically consecutive documents. We then used each adjacent pair of groups as training and test set in an experiment (here the fifth and first group are considered adjacent), and then averaged the test set performance over the 5 experiments. The second dataset is a specific subtree of the OHSUMED corpus of medical abstracts [1]: the subtree rooted in “Quality of Health Care” (MeSH code N05.715). After removing overlapping classes (OHSUMED is not quite a tree but a DAG), we ended up with 94 Table 1: Experimental results on two hierarchical text classification tasks under various loss functions. We report average test errors along with standard deviations (in parenthesis). In bold are the best performance figures among the incremental algorithms. RCV1 0/1-loss unif. H-loss norm. H-loss ∆-loss PERC 0.702(±0.045) 1.196(±0.127) 0.100(±0.029) 1.695(±0.182) H-PERC 0.655(±0.040) 1.224(±0.114) 0.099(±0.028) 1.861(±0.172) H-RLS 0.456(±0.010) 0.743(±0.026) 0.057(±0.001) 1.086(±0.036) AH-BAY 0.550(±0.010) 0.815(±0.028) 0.090(±0.001) 1.465(±0.040) SVM 0.482(±0.009) 0.790(±0.023) 0.057(±0.001) 1.173(±0.051) H-SVM 0.440(±0.008) 0.712(±0.021) 0.055(±0.001) 1.050(±0.027) OHSU. 0/1-loss unif. H-loss norm. H-loss ∆-loss PERC 0.899(±0.024) 1.938(±0.219) 0.058(±0.005) 2.639(±0.226) H-PERC 0.846(±0.024) 1.560(±0.155) 0.057(±0.005) 2.528(±0.251) H-RLS 0.769(±0.004) 1.200(±0.007) 0.045(±0.000) 1.957(±0.011) AH-BAY 0.819(±0.004) 1.197(±0.006) 0.047(±0.000) 2.029(±0.009) SVM 0.784(±0.003) 1.206(±0.003) 0.044(±0.000) 1.872(±0.005) H-SVM 0.759(±0.002) 1.170(±0.005) 0.044(±0.000) 1.910(±0.007) classes and 55,503 documents. We made this choice based only on the structure of the subtree: the longest path has length 4, the distribution of nodes sorted by increasing path length is {0.26, 0.37, 0.22, 0.12, 0.03}, and there are a significant number of partial and multiple path multilabels. The vectorization of the subtree was carried out as follows: after tokenization, we removed all stopwords and also those words that did not occur at least 3 times in the corpus. Then, we vectorized the documents using the Bow library [13] with a log(1+ TF) log(IDF) encoding. We ran 5 experiments by randomly splitting the corpus in a training set of 40,000 documents and a test set of 15,503 documents. Test set performances are averages over these 5 experiments. In the training set we kept more documents than in the RCV1 splits since the OHSUMED corpus turned out to be a harder classification problem than RCV1. In both datasets instances have been normalized to unit length. We tested the hierarchical Perceptron algorithm (H-PERC), the hierarchical regularized leastsquares algorithm (H-RLS), and the approximated Bayes-optimal algorithm (APPROX-HBAYES), all described in Section 4. The results are summarized in Table 1. APPROX-HBAYES (AH-BAY in Table 1) was trained using cost coefficients ci chosen as follows: if i ∈root(G) then ci = |root(G)|−1. Otherwise, ci = cj/|child(j)|, where j is the parent of i. Note that this choice of coefficients amounts to splitting a unit cost equally among the roots and then splitting recursively each node’s cost equally among its children. Since, in this case, 0 ≤ℓH ≤1, we call the resulting loss normalized H-loss. We also tested a hierarchical version of SVM (denoted by H-SVM in Table 1) in which each node is an SVM classifier trained using a batch version of our hierarchical learning protocol. More precisely, each node i was trained only on those examples (xt, yt) such that ypar(i),t = 1 (note that, as no conditions are imposed on yi,t, node i is actually trained on both positive and negative examples). The resulting set of linear-threshold functions was then evaluated on the test set using the hierachical classification scheme (4). We tried both the C and ν parametrizations [18] for SVM and found the setting C = 1 to work best for our data.1 We finally tested the “flat” variants of Perceptron and SVM, denoted by PERC and SVM. In these variants, each node is trained and evaluated independently of the others, disregarding all taxonomical information. All SVM experiments were carried out using the libSVM implementation [6]. All the tested algorithms used a linear kernel. 1It should be emphasized that this tuning of C was actually chosen in hindsight, with no crossvalidation. As far as loss functions are concerned, we considered the 0/1-loss, the H-loss with cost coefficients set to 1 (denoted by uniform H-loss), the normalized H-loss, and the symmetric difference loss (denoted by ∆-loss). Note that H-SVM performs best, but our incremental algorithms were trained for a single epoch on the training set. The good performance of SVM (the flat variant of H-SVM) is surprising. However, with a single epoch of training H-RLS does not perform worse than SVM (except on OHSUMED under the normalized H-loss) and comes reasonably close to H-SVM. On the other hand, the performance of APPROX-H-BAYES is disappointing: on OHSUMED it is the best algorithm only for the uniform H-loss, though it was trained using the normalized H-loss; on RCV1 it never outperforms H-RLS, though it always does better than PERC and H-PERC. A possible explanation for this behavior is that APPROX-H-BAYES is very sensitive to errors in the estimates of pi(x) (recall Section 3.1). Indeed, the least-squares estimates (5), which we used to approximate H-BAYES, seem to work better in practice on simpler (and possibly more robust) algorithms, such as H-RLS. The lower values of normalized H-loss on OHSUMED (a harder corpus than RCV1) can be explained because a quarter of the 94 nodes in the OHSUMED taxonomy are roots, and thus each top-level mistake is only charged about 4/94. As a final remark, we observe that the normalized H-loss gave too small a range of values to afford fine comparisons among the best performing algorithms. 6 Regret bounds for the H-loss In this section we prove a theoretical bound on the H-loss of a slight variant of the algorithm H-RLS tested in Section 5. More precisely, we assume data are generated according to the probabilistic model introduced in Section 3 with unknown instance distribution D and unknown coefficients u1, . . . , uN. We define the regret of a classifier assigning label by to instance X as E ℓH(by, Y t) −E ℓH(y, Y ), where the expected value is with respect the random draw of (X, Y ) and y is the multilabel assigned by classifier (4) when the decision functions gi are zero-threshold functions of the form gi(x) = {u⊤ i x ≥0}. The theorem below shows that the regret of the classifier learned by a variant of H-RLS after t training examples, with t large enough, is exponentially small in t. In other words, H-RLS learns to classify as well as the algorithm that is given the true parameters u1, . . . , uN of the underlying data-generating process. We have been able to prove the theorem only for the variant of H-RLS storing all instances at each node. That is, every eligible node at time t is updated, irrespective of whether |w⊤ i,txt| ≤ p (5 ln t)/Ni,t. Given the i.i.d. data-generating process (X1, Y 1), (X2, Y 2), . . ., for each node k we define the derived process Xk1, Xk2, . . . including all and only the instances Xs of the original process that satisfy Ypar(k),s = 1. We call this derived process the process at node k. Note that, for each k, the process at node k is an i.i.d. process. However, its distribution might depend on k. The spectrum of the process at node k is the set of eigenvalues of the correlation matrix with entries E[Xk1,i Xk1,j] for i, j = 1, . . . , d. We have the following theorem, whose proof is omitted due to space limitations. Theorem 3 Let G be a taxonomy with N nodes and let fG be a joint density for G parametrized by N unit-norm vectors u1, . . . , uN ∈Rd. Assume the instance distribution is such that there exist γ1, . . . , γN > 0 satisfying P |u⊤ i Xt| ≥γi = 1 for i = 1, . . . , N. Then, for all t > max n maxi=1,...,N 16 µi λi γi , maxi=1,...,N 192d µi λ2 i o the regret E ℓH(byt, Y t) −E ℓH(yt, Y t) of the modified H-RLS algorithm is at most N X i=1 µi h t e−κ1 γ2 i λi t µi + t2 e−κ2λ2 i t µii X j∈sub(i) cj ! , where κ1, κ2 are constants, µi = E hQ j∈anc(i) (1 + u⊤ j X)/2 i and λi is the smallest eigenvalue in the spectrum of the process at node i. 7 Conclusions and open problems In this work we have studied the problem of hierarchical classification of data instances in the presence of partial and multiple path labellings. We have introduced a new hierarchical loss function, the H-loss, derived the corresponding Bayes-optimal classifier, and empirically compared an incremental approximation to this classifier with some other incremental and nonincremental algorithms. Finally, we have derived a theoretical guarantee on the H-loss of a simplified variant of the approximated Bayes-optimal algorithm. Our investigation leaves several open issues. The current approximation to the Bayesoptimal classifier is not satisfying, and this could be due to a bad choice of the model, of the estimators, of the datasets, or of a combination of them. Also, the normalized H-loss is not fully satisfying, since the resulting values are often too small. From the theoretical viewpoint, we would like to analyze the regret of our algorithms with respect to the Bayesoptimal classifier, rather than with respect to a classifier that makes a suboptimal use of the true model parameters. References [1] The OHSUMED test collection. URL: medir.ohsu.edu/pub/ohsumed/. [2] Reuters corpus volume 1. URL: about.reuters.com/researchandstandards/corpus/. [3] N. Cesa-Bianchi, A. Conconi, and C. Gentile. A second-order Perceptron algorithm. In Proc. 15th COLT, pages 121–137. Springer, 2002. [4] N. Cesa-Bianchi, A. Conconi, and C. Gentile. Learning probabilistic linear-threshold classifiers via selective sampling. In Proc. 16th COLT, pages 373–386. Springer, 2003. [5] N. Cesa-Bianchi, A. Conconi, and C. Gentile. Regret bounds for hierarchical classification with linear-threshold functions. In Proc. 17th COLT. Springer, 2004. To appear. [6] C.-C. Chang and C.-J. Lin. Libsvm — a library for support vector machines. URL: www.csie.ntu.edu.tw/∼cjlin/libsvm/. [7] N. Cristianini and J. Shawe-Taylor. An Introduction to Support Vector Machines. Cambridge University Press, 2001. [8] O. Dekel, J. Keshet, and Y. Singer. Large margin hierarchical classification. In Proc. 21st ICML. Omnipress, 2004. [9] S.T. Dumais and H. Chen. Hierarchical classification of web content. In Proc. 23rd ACM Int. Conf. RDIR, pages 256–263. ACM Press, 2000. [10] M. Granitzer. Hierarchical Text Classification using Methods from Machine Learning. PhD thesis, Graz University of Technology, 2003. [11] T. Hofmann, L. Cai, and M. Ciaramita. Learning with taxonomies: Classifying documents and words. In NIPS Workshop on Syntax, Semantics, and Statistics, 2003. [12] D. Koller and M. Sahami. Hierarchically classifying documents using very few words. In Proc. 14th ICML, Morgan Kaufmann, 1997. [13] A. McCallum. Bow: A toolkit for statistical language modeling, text retrieval, classification and clustering. URL: www-2.cs.cmu.edu/∼mccallum/bow/. [14] A.K. McCallum, R. Rosenfeld, T.M. Mitchell, and A.Y. Ng. Improving text classification by shrinkage in a hierarchy of classes. In Proc. 15th ICML. Morgan Kaufmann, 1998. [15] D. Mladenic. Turning yahoo into an automatic web-page classifier. In Proceedings of the 13th European Conference on Artificial Intelligence, pages 473–474, 1998. [16] F. Rosenblatt. The Perceptron: A probabilistic model for information storage and organization in the brain. Psychol. Review, 65:386–408, 1958. [17] M.E. Ruiz and P. Srinivasan. Hierarchical text categorization using neural networks. Information Retrieval, 5(1):87–118, 2002. [18] B. Sch¨olkopf, A. J. Smola, R. C. Williamson, and P. L. Bartlett. New support vector algorithms. Neural Computation, 12:1207–1245, 2000. [19] B. Sch¨olkopf and A. Smola. Learning with kernels. MIT Press, 2002. [20] A. Sun and E.-P. Lim. Hierarchical text classification and evaluation. In Proc. 2001 Int. Conf. Data Mining, pages 521–528. IEEE Press, 2001. [21] V.N. Vapnik. Statistical Learning Theory. Wiley, 1998.
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On Semi-Supervised Classification Balaji Krishnapuram, David Williams, Ya Xue, Alex Hartemink, Lawrence Carin Duke University, USA M´ario A. T. Figueiredo Instituto de Telecomunicac¸˜oes, Instituto Superior T´ecnico, Portugal Abstract A graph-based prior is proposed for parametric semi-supervised classification. The prior utilizes both labelled and unlabelled data; it also integrates features from multiple views of a given sample (e.g., multiple sensors), thus implementing a Bayesian form of co-training. An EM algorithm for training the classifier automatically adjusts the tradeoff between the contributions of: (a) the labelled data; (b) the unlabelled data; and (c) the co-training information. Active label query selection is performed using a mutual information based criterion that explicitly uses the unlabelled data and the co-training information. Encouraging results are presented on public benchmarks and on measured data from single and multiple sensors. 1 Introduction In many pattern classification problems, the acquisition of labelled training data is costly and/or time consuming, whereas unlabelled samples can be obtained easily. Semisupervised algorithms that learn from both labelled and unlabelled samples have been the focus of much research in the last few years; a comprehensive review up to 2001 can be found in [13], while more recent references include [1,2,6,7,16–18]. Most recent semi-supervised learning algorithms work by formulating the assumption that “nearby” points, and points in the same structure (e.g., cluster), should have similar labels [6,7,16]. This can be seen as a form of regularization, pushing the class boundaries toward regions of low data density. This regularization is often implemented by associating the vertices of a graph to all the (labelled and unlabelled) samples, and then formulating the problem on the vertices of the graph [6,16–18]. While current graph-based algorithms are inherently transductive — i.e., they cannot be used directly to classify samples not present when training — our classifier is parametric and the learned classifier can be used directly on new samples. Furthermore, our algorithm is trained discriminatively by maximizing a concave objective function; thus we avoid thorny local maxima issues that plague many earlier methods. Unlike existing methods, our algorithm automatically learns the relative importance of the labelled and unlabelled data. When multiple views of the same sample are provided (e.g. features from different sensors), we develop a new Bayesian form of co-training [4]. In addition, we also show how to exploit the unlabelled data and the redundant views of the sample (from co-training) in order to improve active label query selection [15]. The paper is organized as follows. Sec. 2 briefly reviews multinomial logistic regression. Sec. 3 describes the priors for semi-supervised learning and co-training. The EM algorithm derived to learn the classifiers is presented in Sec. 4. Active label selection is discussed in Sec. 5. Experimental results are shown in Sec. 6, followed by conclusions in Sec. 7. 2 Multinomial Logistic Regression In an m-class supervised learning problem, one is given a labelled training set DL = {(x1, y1), . . . , (xL, yL)}, where xi ∈Rd is a feature vector and yi the corresponding class label. In “1-of-m” encoding, yi = [y(1) i , . . . , y(m) i ] is a binary vector, such that y(c) i = 1 and y(j) i = 0, for j ̸= c, indicates that sample i belongs to class c. In multinomial logistic regression [5], the posterior class probabilities are modelled as log P(y(c) = 1|x) = xT w(c) −log Pm k=1 exp(xT w(k)), for c = 1, . . . , m, (1) where w(c) ∈Rd is the class-c weight vector. Notice that since Pm c=1 P(y(c)= 1|x) = 1, one of the weight vectors is redundant; we arbitrarily choose to set w(m) = 0, and consider the (d (m−1))-dimensional vector w = [(w(1))T , ..., (w(m−1))T ]T . Estimation of w may be achieved by maximizing the log-likelihood (with Y ≡{y1, ..., yL}) [5] ℓ(w) ≡log P(Y|w) = PL i=1 Pm c=1 y(c) i xT i w(c) −log Pm j=1 exp(xT i w(j)) . (2) In the presence of a prior p(w), we seek a maximum a posteriori (MAP) estimate, bw = arg maxw{ℓ(w) + log p(w)}. Actually, if the training data is separable, ℓ(w) is unbounded, and a prior is crucial. Although we focus on linear classifiers, we may see the d-dimensional feature vectors x as having resulted from some deterministic, maybe nonlinear, transformation of an input raw feature vector r; e.g., in a kernel classifier, xi = [1, K(ri, r1), ..., K(ri, rL)] (d = L+1). 3 Graph-Based Data-Dependent Priors 3.1 Graph Laplacians and Regularization for Semi-Supervised Learning Consider a scalar function f = [f1, ..., f|V |]T , defined on the set V = {1, 2, ..., |V |} of vertices of an undirected graph (V, E). Each edge of the graph, joining vertices i and j, is given a weight kij = kji ≥0, and we collect all the weights in a |V | × |V | matrix K. A natural way to measure how much f varies across the graph is by the quantity X i X j kij(fi −fj)2 = 2 f T ∆f, (3) where ∆= diag{P j k1j, ..., P j k|V |j}−K is the so-called graph Laplacian [2]. Notice that kij ≥0 (for all i, j) guarantees that ∆is positive semi-definite and also that ∆has (at least) one null eigenvalue (1T∆1 = 0, where 1 has all elements equal to one). In semi-supervised learning, in addition to DL, we are given U unlabelled samples DU = {xL+1, . . . , xL+U}. To use (3) for semi-supervised learning, the usual choice is to assign one vertex of the graph to each sample in X = [x1, . . . , xL+U]T (thus |V | = L + U), and to let kij represent some (non-negative) measure of “similarity” between xi and xj. A Gaussian random field (GRF) is defined on the vertices of V (with inverse variance λ) p(f) ∝exp{−λ f T∆f/2}, in which configurations that vary more (according to (3)) are less probable. Most graphbased approaches estimate the values of f, given the labels, using p(f) (or some modification thereof) as a prior. Accordingly, they work in a strictly transductive manner. 3.2 Non-Transductive Semi-Supervised Learning We first consider two-class problems (m = 2, thus w ∈Rd). In contrast to previous uses of graph-based priors, we define f as the real function f (defined over the entire observation space) evaluated at the graph nodes. Specifically, f is defined as a linear function of x, and at the graph node i, fi ≡f(xi) = wT xi. Then, f = [f1, ..., f|V |]T = Xw, and p(f) induces a Gaussian prior on w, with precision matrix A = XT ∆X, p(w) ∝exp{−(λ/2) wT XT∆Xw} = exp{−(λ/2) wT Aw}. (4) Notice that since ∆is singular, A may also be singular, and the corresponding prior may therefore be improper. This is no problem for MAP estimation of w because (as is well known) the normalization factor of the prior plays no role in this estimate. If we include extra regularization, by adding a non-negative diagonal matrix to A, the prior becomes p(w) ∝exp −(1/2) wT (λ0A + Λ) w , (5) where we may choose Λ = diag{λ1, ..., λd}, Λ = λ1I, or even Λ = 0. For m>2, we define (m−1) identical independent priors, one for each w(c), c = 1, ..., m. The joint prior on w = [(w(1))T , ..., (w(m−1))T ]T is then p(w|λ) ∝ m−1 Y c=1 exp{−1 2 (w(c))T λ(c) 0 A + Λ(c) w(c)} = exp{−1 2 wT Γ(λ)w}, (6) where λ is a vector containing all the λ(c) i parameters, Λ(c) = diag{λ(c) 1 , ..., λ(c) d }, and Γ(λ) = diag{λ(1) 0 , ..., λ(m−1) 0 } ⊗A + block-diag{Λ(1), ..., Λ(m−1)}. (7) Finally, since all the λ’s are inverses of variances, the conjugate priors are Gamma [3]: p(λ(c) 0 |α0, β0) = Ga(λ(c) 0 |α0, β0), and p(λ(c) i |α1, β1) = Ga(λ(c) i |α1, β1), for c = 1, ..., m −1 and i = 1, ..., d. Usually, α0, β0, α1, and β1 are given small values indicating diffuse priors. In the zero limit, we obtain scale-invariant (improper) Jeffreys hyper-priors. Summarizing, our model for semi-supervised learning includes the log-likelihood (2), a prior (6), and Gamma hyper-priors. In Section 4, we present a simple and computationally efficient expectation-maximization (EM) algorithm for obtaining the MAP estimate of w. 3.3 Exploiting Features from Multiple Sensors: The Co-Training Prior In some applications several sensors are available, each providing a different set of features. For simplicity, we assume two sensors s ∈{1, 2}, but everything discussed here is easily extended to any number of sensors. Denote the features from sensor s, for sample i, as x(s) i , and Ss as the set of sample indices for which we have features from sensor s (S1∪S2 = {1, ..., L + U}). Let O =S1 ∩S2 be the indices for which both sensors are available, and OU = O ∩{L + 1, ..., L + U} the unlabelled subset of O. By using the samples in S1 and S2 as two independent training sets, we may obtain two separate classifiers (denoted bw1 and bw2). However, we can coordinate the information from both sensors by using an idea known as co-training [4]: on the OU samples, classifiers bw1 and bw2 should agree as much as possible. Notice that, in a logistic regression framework, the disagreement between the two classifiers on the OU samples can be measured by P i∈OU [(w1)T x(1) i −(w2)T x(2) i ]2 = ωT C ω, (8) where ω = [(w1)T (w2)T ]T and C = P i∈OU [(x1 i )T (−x2 i )T ]T [(x1 i )T (−x2 i )T ]. This suggests the “co-training prior” (where λco is an inverse variance): p(w1, w2) = p(ω) ∝exp −(λco/2) ωTC ω . (9) This Gaussian prior can be combined with two smoothness Gaussian priors on w1 and w2 (obtained as described in Section 3.2); this leads to a prior which is still Gaussian, p(w1, w2) = p(ω) ∝exp −(1/2) ωT λcoC + block-diag{Γ1, Γ2} ω , (10) where Γ1 and Γ2 are the two graph-based precision matrices (see (7)) for w1 and w2. We can again adopt a Gamma hyper-prior for λco. Under this prior, and with a logistic regression likelihood as above, estimates of w1 and w2 can easily be found using minor modifications to the EM algorithm described in Section 4. Computationally, this is only slightly more expensive than separately training the two classifiers. 4 Learning Via EM To find the MAP estimate bw, we use the EM algorithm, with λ as missing data, which is equivalent to integrating out λ from the full posterior before maximization [8]. For simplicity, we will only describe the single sensor case (no co-training). E-step: We compute the expected value of the complete log-posterior, given Y and the current parameter estimate bw: Q(w| bw) ≡E[log p(w, λ|Y)| bw]. Since log p(w, λ|Y) = log p(Y|w) −(1/2)wT Γ(λ)w + K, (11) (where K collects all terms independent of w) is linear w.r.t. all the λ parameters (see (6) and (7)), we just have to plug their conditional expectations into (11): Q(w| bw) = log p(Y|w) −(1/2)wT E[Γ(λ)| bw] w = ℓ(w) −(1/2)wT Υ( bw) w. (12) We consider several different choices for the structure of the Γ matrix. The necessary expectations have well-known closed forms, due to the use of conjugate Gamma hyperpriors [3]. For example, if the λ(c) 0 are m −1 free non-negative parameters, we have γ(c) 0 ≡E[λ(c) 0 | bw] = (2 α0 + d) [2 β0 + ( bw(c))T A bw(c)]−1. for c = 1, ..., m −1. For λ(c) 0 = λ0, we still have a simple closed-form expression for E[λ0| bw], and the same is true for the λ(c) i parameters, for i > 0. Finally, Υ( bw) ≡E[Γ(λ)| bw] results from replacing the λ’s in (7) by the corresponding conditional expectations. M-step: Given matrix Υ( bw), the M-step reduces to a logistic regression problem with a quadratic regularizer, i.e., maximizing (12). To this end, we adopt the bound optimization approach (see details in [5,11]). Let B be a positive definite matrix such that −B bounds below (in the matrix sense) the Hessian of ℓ(w), which is negative definite, and g(w) is the gradient of ℓ(w). Then, we have the following lower bound on Q(w| bw): Q(w| bw) ≥l( bw) + (w −bw)T g( bw) −[(w −bw)T B(w −bw) + wT Υ( bw)w]/2. The maximizer of this lower bound, bwnew = (B + Υ( bw))−1 (B bw + g( bw)), is guaranteed to increase the Q-function, Q( bwnew| bw) ≥Q( bw| bw), and we thus obtain a monotonic generalized EM algorithm [5, 11]. This (maybe costly) matrix inversion can be avoided by a sequential approach where we only maximize w.r.t. one element of w at a time, preserving the monotonicity of the procedure. The sequential algorithm visits one particular element of w, say wu, and updates its estimate by maximizing the bound derived above, while keeping all other variables fixed at their previous values. This leads to bwnew u = bwu + [gu( bw) −(Υ( bw) bw)u] [(B + Υ( bw))uu]−1 , (13) and bwnew v = bwv, for v ̸= u. The total time required by a full sweep for all u = 1, ..., d is O(md(L + d)); this may be much better than the O((dm)3) of the matrix inversion. 5 Active Label Selection If we are allowed to obtain the label for one of the unlabelled samples, the following question arises: which sample, if labelled, would provide the most information? Consider the MAP estimate bw provided by EM. Our approach uses a Laplace approximation of the posterior p(w|Y) ≃N(w| bw, H−1), where H is the posterior precision matrix, i.e., the Hessian of minus the log-posterior H = ∇2(−log p(w|Y)). This approximation is known to be accurate for logistic regression under a Gaussian prior [14]. By treating Υ( bw) (the expectation of Γ(λ)) as deterministic, we obtain an evidence-type approximation [14] H=∇2[−log(p(Y|w)p(w|Υ( bw)))] = Υ( bw) +PL i=1(diag{pi} −pipT i ) ⊗xixT i , where pi is the (m −1)-dimensional vector computed from (1), the c-th element of which indicates the probability that sample xi belongs to class c. Now let x∗∈DU be an unlabelled sample and y∗its label. Assume that the MAP estimate bw remains unchanged after including y∗. In Sec. 7 we will discuss the merits and shortcomings of this assumption, which is only strictly valid when L →∞. Accepting it implies that after labeling x∗, and regardless of y∗, the posterior precision changes to H′ = H + (diag{p∗} −p∗pT ∗) ⊗x∗xT ∗. (14) Since the entropy of a Gaussian with precision H is (−1/2) log |H| (up to an additive constant), the mutual information (MI) between y∗and w (i.e., the expected decrease in entropy of w when y∗is observed) is I(w; y∗) = (1/2) log {|H′|/|H|}. Our criterion is then: the best sample to label is the one that maximizes I(w; y∗). Further insight into I(w; y∗) can be obtained in the binary case (where p is a scalar); here, the matrix identity |H + p∗(1 −p∗)x∗xT ∗| = |H|(1 + p∗(1 −p∗)xT ∗H−1x∗) yields I(w; y∗) = (1/2) log(1 + p∗(1 −p∗)xT ∗H−1x∗). (15) This MI is larger when p∗≈0.5, i.e., for samples with uncertain classifications. On the other hand, with p∗fixed, I(w; y∗) grows with xT ∗H−1x∗, i.e., it is large for samples with high variance of the corresponding class probability estimate. Summarizing, (15) favors samples with uncertain class labels and high uncertainty in the class probability estimate. 6 Experimental Results We begin by presenting two-dimensional synthetic examples to visually illustrate our semisupervised classifier. Fig. 1 shows the utility of using unlabelled data to improve the deciFigure 1: Synthetic two-dimensional examples. (a) Comparison of the supervised logistic linear classifier (boundary shown as dashed line) learned only from the labelled data (shown in color) with the proposed semi-supervised classifier (boundary shown as solid line) which also uses the unlabelled samples (shown as dots). (b) A RBF kernel classifier obtained by our algorithm, using two labelled samples (shaded circles) and many unlabelled samples. Figure 2: (a)-(c) Accuracy (on UCI datasets) of the proposed method, the supervised SVM, and the other semi-supervised classifiers mentioned in the text; a subset of samples is labelled and the others are treated as unlabelled samples. In (d), a separate holdout set is used to evaluate the accuracy of our method versus the amount of labelled and unlabelled data. sion boundary in linear and non-linear (kernel) classifiers (see figure caption for details). Next we show results with linear classifiers on three UCI benchmark datasets. Results with nonlinear kernels are similar, and therefore omitted to save space. We compare our method against state-of-the-art semi-supervised classifiers: the GRF method of [18], the SGT method of [10], and the transductive SVM (TSVM) of [9]. For reference, we also present results for a standard SVM. To avoid unduly helping our method, we always use a k=5 nearest neighbors graph, though our algorithm is not very sensitive to k. To avoid disadvantaging other methods that do depend on such parameters, we use their best settings. Since these adjustments cannot be made in practice, the difference between our algorithm and the others is under-represented. Each point on the plots in Fig. 2(a)-(c) is an average of 20 trials: we randomly select 20 labelled sets which are used by every method. All remaining samples are used as unlabelled by the semi-supervised algorithms. Figs. 2(a)-(c) are transductive, in the sense that the unlabelled and test data are the same. Our logistic GRF is non-transductive: after being trained, it may be applied to classify new data without re-training. In Fig. 2(d) we present non-transductive results for the Ionosphere data. Training took place using labelled and unlabelled data, and testing was performed on 200 new unseen samples. The results suggest that semi-supervised classifiers are most relevant when the labelled set is small relative to the unlabelled set (as is often the case). Our final set of results address co-training (Sec. 3.3) and active learning (Sec. 5), applied to airborne sensing data for the detection of surface and subsurface land mines. Two sensors were used: (1) a 70-band hyper-spectral electro-optic (EOIR) sensor; (2) an X-band synthetic aperture radar (SAR). A simple (energy) “prescreener” detected potential targets; for each of these, two feature vectors were extracted, of sizes 420 and 9, for the EOIR and SAR sensors, respectively. 123 samples have features from the EOIR sensor alone, 398 from the Figure 3: (a) Land mine detection ROC curves of classifiers designed using only hyperspectral (EOIR) features, only SAR features, and both. (b) Number of landmines detected during the active querying process (dotted lines), for active training and random selection (for the latter the bars reflect one standard deviation about the mean). ROC curves (solid) are for the learned classifier as applied to the remaining samples. SAR sensor alone, and 316 from both. This data will be made available upon request. We first consider supervised and semi-supervised classification. For the purely supervised case, a sparseness prior is used (as in [14]). In both cases a linear classifier is employed. For the data for which only one sensor is available, 20% of it is labelled (selected randomly). For the data for which both sensors are available, 80% is labelled (again selected randomly). The results presented in Fig. 3(a) show that, in general, the semi-supervised classifiers outperform the corresponding supervised ones, and the classifier learned from both sensors is markedly superior to classifiers learned from either sensor alone. In a second illustration, we use the active-learning algorithm (Sec. 5) to only acquire the 100 most informative labels. For comparison, we also show average results over 100 independent realizations for random label query selection (error bars indicate one standard deviation). The results in Fig. 3(b) are plotted in two stages: first, mines and clutter are selected during the labeling process (dashed curves); then, the 100 labelled examples are used to build the final semi-supervised classifier, for which the ROC curve is obtained using the remaining unlabelled data (solid curves). Interestingly, the active-learning algorithm finds almost half of the mines while querying for labels. Due to physical limitations of the sensors, the rate at which mines are detected drops precipitously after approximately 90 mines are detected — i.e., the remaining mines are poorly matched to the sensor physics. 7 Discussion 7.1 Principal Contributions Semi-supervised vs. Transductive: Unlike most earlier methods, after the training stage our algorithm can directly classify new samples without computationally expensive re-training. Tradeoff between labelled and unlabelled data: Automatically addressing the inherent tradeoff between their relative contributions, we have ensured that even a small amount of labelled data does not get overlooked because of an abundance of unlabelled samples. Bayesian co-training: Using the proposed prior, classifiers for all sensors are improved using: (a) the label information provided on the other types of data, and (b) samples drawn from the joint distribution of features from multiple sensors. Active label acquisition: We explicitly account for the knowledge of the unlabelled data and the co-training information while computing the well known mutual information criterion. 7.2 Quality of Assumptions and Empirically Observed Shortcomings The assumption that the mode of the posterior distribution of the classifier remains unchanged after seeing an additional label is clearly not true at the beginning of the active learning procedure. However, we have empirically found it a very good approximation after the active learning procedure has yielded as few as 15 labels. This assumption allows a tremendous saving in the computational cost, since it helps us avoid repeated re-training of classifiers in the active label acquisition process while evaluating candidate queries. A disturbing fact that has been reported in the literature (e.g., in [12]) and that we have confirmed (in unreported experiments) is that the error rate of the active query selection increases slightly when the number of labelled samples grows beyond an optimal number. We conjecture that this may be caused by keeping the hyper-prior parameters α0, β0, α1, β1 fixed at the same value; in all of our experiments we have set them to 10−4, corresponding to an almost uninformative hyper-prior. References [1] M. Belkin, I. Matveeva, and P. Niyogi. Regularization and regression on large graphs. In Proc. Computational Learning Theory – COLT’04, Banff, Canada, 2004. [2] M. Belkin and P. Niyogi. Using manifold structure for partially labelled classification. In NIPS 15, MIT Press, Cambridge, MA, 2003. [3] J. Bernardo and A. Smith. Bayesian Theory. J. Wiley & Sons, Chichester, UK, 1994. [4] A. Blum and T. Mitchell. Combining labelled and unlabelled data with co-training. In Proc. Computational Learning Theory – COLT’98, Madison, WI, 1998. [5] D. B¨ohning. Multinomial logistic regression algorithm. Annals Inst. Stat. Math., vol. 44, pp. 197–200, 1992. [6] O. Chapelle, J. Weston, and B. Sch¨olkopf. Cluster kernels for semi-supervised learning. In NIPS 15, MIT Press, Cambridge, MA, 2003. [7] A. Corduneanu and T. Jaakkola. On Information regularization. In Proc. Uncertainty in Artificial Intelligence – UAI’03, Acapulco, Mexico, 2003. [8] M. Figueiredo. Adaptive sparseness using Jeffreys’ prior. In NIPS 14, MIT Press, 2002. [9] T. Joachims. Transductive inference for text classification using support vector machines. In Int. Conf. Machine Learning – ICML’99, 1999. [10] T. Joachims. Transductive learning via spectral graph partitioning. In ICML’03, 2003. [11] K. Lange, D. Hunter, and I. Yang. Optimization transfer using surrogate objective functions. J. Computational and Graphical Statistics, vol. 9, pp. 1–59, 2000. [12] G. Schohn and D. Cohn. Less is more: Active learning with support vector machines. Intern. Conf. on Mach. Learn. – ICML’00. [13] M. Seeger. Learning with labelled and unlabelled data. Tech. Rep., Institute for Adaptive and Neural Computation, University of Edinburgh, UK, 2001. [14] M. Tipping. Sparse Bayesian learning and the relevance vector machine. J. Mach. Learn. Research, vol. 1, pp. 211–244, 2001. [15] S. Tong and D. Koller. Support vector machine active learning with applications to text classification. In J. Mach. Learn. Research, vol. 2, pp. 45–66, 2001. [16] D. Zhou, O. Bousquet, T. Lal, J. Weston, and B. Sch¨olkopf. Semi-supervised learning by maximizing smoothness. J. of Mach. Learn. Research, 2004 (submitted). [17] X. Zhu, J. Lafferty and Z. Ghahramani. Combining active learning and semi-supervised learning using Gaussian fields and harmonic functions. In ICML’03 Workshop on The Continuum from Labelled to Unlabelled Data in Mach. Learning, 2003. [18] X. Zhu, J. Lafferty and Z. Ghahramani. Semi-supervised learning: From Gaussian fields to Gaussian processes. Tech. Rep. CMU-CS-03-175, School of CS, CMU, 2003.
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Joint MRI Bias Removal Using Entropy Minimization Across Images Erik G. Learned-Miller Department of Computer Science University of Massachusetts, Amherst Amherst, MA 01003 Parvez Ahammad Division of Electrical Engineering University of California, Berkeley Berkeley, CA 94720 Abstract The correction of bias in magnetic resonance images is an important problem in medical image processing. Most previous approaches have used a maximum likelihood method to increase the likelihood of the pixels in a single image by adaptively estimating a correction to the unknown image bias field. The pixel likelihoods are defined either in terms of a pre-existing tissue model, or non-parametrically in terms of the image’s own pixel values. In both cases, the specific location of a pixel in the image is not used to calculate the likelihoods. We suggest a new approach in which we simultaneously eliminate the bias from a set of images of the same anatomy, but from different patients. We use the statistics from the same location across different images, rather than within an image, to eliminate bias fields from all of the images simultaneously. The method builds a “multi-resolution” non-parametric tissue model conditioned on image location while eliminating the bias fields associated with the original image set. We present experiments on both synthetic and real MR data sets, and present comparisons with other methods. 1 Introduction The problem of bias fields in magnetic resonance (MR) images is an important problem in medical imaging. This problem is illustrated in Figure 1. When a patient is imaged in the MR scanner, the goal is to obtain an image which is a function solely of the underlying tissue (left of Figure 1). However, typically the desired anatomical image is corrupted by a multiplicative bias field (2nd image of Figure 1) that is caused by engineering issues such as imperfections in the radio frequency coils used to record the MR signal. The result is a corrupted image (3rd image of Figure 1). (See [1] for background information.) The goal of MR bias correction is to estimate the uncorrupted image from the corrupted image. A variety of statistical methods have been proposed to address this problem. Wells et al. [7] developed a statistical model using a discrete set of tissues, with the brightness distribution for each tissue type (in a bias-free image) represented by a one-dimensional Guassian distribution. An expectation-maximization (EM) procedure was then used to simultaneouly estimate the bias field, the tissue type, and the residual noise. While this method works well in many cases, it has several drawbacks: (1) Models must be developed a priori for each type of acquistion (for each different setting of the MR scanner), for each Figure 1: On the left is an idealized mid-axial MR image of the human brain with little or no bias field. The second image is a simulated low-frequency bias field. It has been exaggerated for ease of viewing. The third image is the result of pixelwise multiplication of the image by the bias field. The goal of MR bias correction is to recover the low-bias image on the left from the biased image on the right. On the right is the sine/cosine basis, used to construct band-limited bias fiels (see text). new area of the body, and for different patient populations (like infants and adults). (2) Models must be developed from “bias-free” images, which may be difficult or impossible to obtain in many cases. (3) The model assumes a fixed number of tissues, which may be inaccurate. For example, during development of the human brain, there is continuous variability between gray matter and white matter. In addition, a discrete tissue model does not handle so-called partial volume effects in which a pixel represents a combination of several tissue types. This occurs frequently since many pixels occur at tissue boundaries. Non-parametric approaches have also been suggested, as for example by Viola [10]. In that work, a non-parametric model of the tissue was developed from a single image. Using the observation that the entropy of the pixel brightness distribution for a single image is likely to increase when a bias field is added, Viola’s method postulates a bias-correction field by minimizing the entropy of the resulting pixel brightness distribution. This approach addresses several of the problems of fixed-tissue parametric models, but has its own drawbacks: (1) The statistical model may be weak, since it is based on data from only a single image. (2) There is no mechanism for distinguishing between certain low-frequency image components and a bias field. That is, the method may mistake signal for noise in certain cases when removal of the true signal reduces the entropy of the brightness distriibution. We shall show that this is a problem in real medical images. The method we present overcomes or improves upon problems associated with both of these methods and their many variations (see, e.g., [1] for recent techniques). It models tissue brightness non-parametrically, but uses data from multiple images to provide improved distribution estimates and alleviate the need for bias-free images for making a model. It also conditions on spatial location, taking advantage of a rich information source ignored in other methods. Experimental results demonstrate the effectiveness of our method. 2 The Image Model and Problem Formulation We assume we are given a set I of observed images Ii with 1 ≤i ≤N, as shown on the left side of Figure 2. Each of these images is assumed to be the product of some bias-free image Li and a smooth bias field Bi ∈B. We shall refer to the bias-free images as latent images (also called intrinsic images by some authors). The set of all latent images shall be denoted L and the set of unknown bias fields B. Then each observed image can be written as the product Ii(x,y) = Li(x,y) ∗Bi(x,y), where (x,y) gives the pixel coordinates of each point, with P pixels per image. Consider again Figure 2. A pixel-stack through each image set is shown as the set of pixels corresponding to a particular location in each image (not necessarily the same tissue type). Our method relies on the principle that the pixel-stack values will have lower entropy when the bias fields have been removed. Figure 3 shows the simulated effect, on the distribution of values in a pixel-stack, of adding different bias fields to each image. The latent image generation model assumes that each pixel is drawn from a fixed distribution px,y(·) which gives the probability of each gray value at the the location (x,y) in the image. Furthermore, we assume that all pixels in the latent image are independent, given the distributions from which they are drawn. It is also assumed that the bias fields for each image are chosen independently from some fixed distribution over bias fields. Unlike most models for this problem which rely on statistical regularities within an image, we take a completely orthogonal approach by assuming that pixel values are independent given their image locations, but that pixel-stacks in general have low entropy when bias fields are removed. We formulate the problem as a maximum a posteriori (MAP) problem, searching for the most probable bias fields given the set of observed images. Letting B represent the 25dimensional product space of smooth bias fields (corresponding to the 25 basis images of Figure 1), we wish to find argmax B∈B P(B|I) (a) = argmax B∈B P(I|B)P(B) (1) (b) = argmax B∈B P(I|B) (2) (c) = argmax B∈B P(L(I,B)) (3) = argmax B∈B ∏ x,y N ∏ i=1 px,y(Li(x,y)) (4) = argmax B∈B ∑ x,y N ∑ i=1 log px,y(Li(x,y)) (5) (d) ≈ argmin B∈B ∑ x,y H(px,y) (6) (e) ≈ argmin B∈B ∑ x,y ˆHVasicek(L1(x,y),...,LN(x,y)) (7) = argmin B∈B ∑ x,y ˆHVasicek( I1(x,y) B1(x,y),..., IN(x,y) BN(x,y)). (8) Here H is the Shannon entropy (−E(logP(x))) and ˆHVasicek is a sample-based entropy estimator.1 (a) is just an application of Bayes rule. (b) assumes a uniform prior over the allowed bias fields. The method can easily be altered to incorporate a non-uniform prior. 1The entropy estimator used is similar to Vasicek’s estimator [6], given (up to minor details) by ˆHVasicek(Z1,...,ZN) = 1 N −m N−m ∑ i=1 log N m(Z(i+m) −Z(i)) , (9) where Zi’s represent the values in a pixel-stack, Z(i)’s represent those same values in rank order, N is the number of values in the pixel-stack and m is a function of N (like N0.5) such that m/N goes to 0 as m and N go to infinity. These entropy estimators are discussed at length elsewhere [3]. Figure 2: On the left are a set of mid-coronal brain images from eight different infants, showing clear signs of bias fields. A pixel-stack, a collection of pixels at the same point in each image, is represented by the small square near the top of each image. Although there are probably no more than two or three tissue types represented by the pixel-stack, the brightness distribution through the pixel-stack has high empirical entropy due to the presence of different bias fields in each image. On the right are a set of images that have been corrected using our bias field removal algorithm. While the images are still far from identical, the pixel-stack entropies have been reduced by mapping similar tissues to similar values in an “unsupervised” fashion, i.e. without knowing or estimating the tissue types. (c) expresses the fact that the probability of the observed image given a particular bias field is the same as the probability of the latent image associated with that observed image and bias field. The approximation (d) replaces the empirical mean of the log probability at each pixel with the negative entropy of the underlying distribution at that pixel. This entropy is in turn estimated (e) using the entropy estimator of Vasicek [6] directly from the samples in the pixel-stack, without ever estimating the distributions px,y explicitly. The inequality (d) becomes an equality as N grows large by the law of large numbers, while the consistency of Vasicek’s entropy estimator [2] implies that (e) also goes to equality with large N. (See [2] for a review of entropy estimators.) 3 The Algorithm Using these ideas, it is straightforward to construct algorithms for joint bias field removal. As mentioned above, we chose to optimize Equation (8) over the set of band-limited bias fields. To do this, we parameterize the set of bias fields using the sine/cosine basis images shown on the right of Figure 1: Bi = 25 ∑ j=1 αjφ j(x,y). We optimize Equation (8) by simultaneously updating the bias field estimates (taking a step along the numerical gradient) for each image to reduce the overall entropy. That is, at time step t, the coefficients αj for each bias field are updated using the latent image estimates and entropy estimates from time step t −1. After all α’s have been updated, a new set of latent images and pixel-stack entropies are calculated, and another gradient step is taken. Though it is possible to do a full gradient descent to convergence by optimizing one image at a time, the optimization landscape tends to have more local minima for the last few images in the process. The appeal of our joint gradient descent method, on the other hand, is that the ensemble of images provides a natural smoothing of the optimization landscape 0 50 100 150 200 250 0 1 2 3 4 5 6 7 8 0 50 100 150 200 250 0 1 2 3 4 5 6 7 8 Figure 3: On the left is a simulated distribution from a pixel-stack taken through a particular set of bias-free mid-axial MR images. The two sharp peaks in the brightness distribution represent two tissues which are commonly found at that particular pixel location. On the right is the result of adding an independent bias field to each image. In particular, the spread, or entropy, of the pixel distribution increases. In this work, we seek to remove bias fields by seeking to reduce the entropy of the pixel-stack distribution to its original state. in the joint process. It is in this sense that our method is “multi-resolution”, proceeding from a smooth optimization in the beginning to a sharper one near the end of the process. We now summarize the algorithm: 1. Initialize the bias field coefficients for each image to 0, with the exception of the coefficient for the DC-offset (the constant bias field component), which is initialized to 1. Initialize the gradient descent step size δ to some value. 2. Compute the summed pixelwise entropies for the set of images with initial “neutral” bias field corrections. (See below for method of computation.) 3. Iterate the following loop until no further changes occur in the images. (a) For each image: i. Calculate the numerical gradient ∇αHVasicek of (8) with respect to the bias field coefficients (αj’s) for the current image. ii. Set α = α+δ∇α ˆHVasicek. (b) Update δ (reduce its value according to some schedule). Upon convergence, it is assumed that the entropy has been reduced as much as possible by changing the bias fields, unless one or more of the gradient descents is stuck in a local minimum. Empirically, the likelihood of sticking in local minima is dramatically reduced by increasing the number of images (N) in the optimization. In our experiments described below with only 21 real infant brains, the algorithm appears to have found a global minimum of all bias fields, at least to the extent that this can be discerned visually. Note that for a set of identical images, the pixel-stack entropies are not increased by multiplying each image by the same bias field (since all images will still be the same). More generally, when images are approximately equivalent, their pixel-stack entropies are not signficantly affected by a “common” bias field, i.e. one that occurs in all of the images.2 This means that the algorithm cannot, in general, eliminate all bias fields from a set of images, but can only set all of the bias fields to be equivalent. We refer to any constant bias field remaining in all of the images after convergence as the residual bias field. 2Actually, multiplying each image by a bias field of small magnitude can artificially reduce the entropy of a pixel-stack, but this is only the result of the brightness values shrinking towards zero. Such artificial reductions in entropy can be avoided by normalizing a distribution to unit variance between iterations of computing its entropy, as is done in this work. Fortunately, there is an effect that tends to minimize the impact of the residual bias field in many test cases. In particular, the residual bias field tends to consist of components for each αj that approximate the mean of that component across images. For example, if half of the observed images have a positive value for a particular component’s coefficient, and half have a negative coefficient for that component, the residual bias field will tend to have a coefficient near zero for that component. Hence, the algorithm naturally eliminates bias field effects that are non-systematic, i.e. that are not shared across images. If the same type of bias field component occurs in a majority of the images, then the algorithm will not remove it, as the component is indistinguishable, under our model, from the underlying anatomy. In such a case, one could resort to within-image methods to further reduce the entropy. However, there is a risk that such methods will remove components that actually represent smooth gradations in the anatomy. This can be seen in the bottom third of Figure 4, and will be discussed in more detail below. 4 Experiments To test our algorithm, we ran two sets of experiments, the first on synthetic images for validation, and the second on real brain images. We obtained synthetic brain images from the BrainWeb project [8, 9] such as the one shown on the left of Figure 1. These images can be considered “idealized” MR images in the sense that the brightness values for each tissue are constant (up to a small amount of manually added isotropic noise). That is, they contain no bias fields. The initial goal was to ensure that our algorithm could remove synthetically added bias fields, in which the bias field coefficients were known. Using K copies of a single “latent” image, we added known but different bias fields to each one. For as few as five images, we could reliably recover the known bias field coefficients, up to a fixed offset for each image, to within 1% of the power of the original bias coefficients. More interesting are the results on real images, in which the latent images come from different patients. We obtained 21 pre-registered3 infant brain images (top of Figure 4) from Brigham and Women’s Hospital in Boston, Massachusetts. Large bias fields can be seen in many of the images. Probably the most striking is a “ramp-like” bias field in the sixth image of the second row. (The top of the brain is too bright, while the bottom is too dark.) Because the brain’s white matter is not fully developed in these infant scans, it is difficult to categorize tissues into a fixed number of classes as is typically done for adult brain images; hence, these images are not amenable to methods based on specific tissue models developed for adults (e.g. [7]). The middle third of Figure 4 shows the results of our algorithm on the infant brain images. (These results must be viewed in color on a good monitor to fully appreciate the results.) While a trained technician can see small imperfections in these images, the results are remarkably good. All major bias artifacts have been removed. It is interesting to compare these results to a method that reduces the entropy of each image individually, without using constraints between images. Using the results of our algorithm as a starting point, we continued to reduce the entropy of the pixels within each image (using a method akin to Viola’s [10]), rather than across images. These results are shown in the bottom third of Figure 4. Carefully comparing the central brain regions in the middle section of the figure and the bottom section of the figure, one can see that the butterfly shaped region in the middle of the brain, which represents developing white matter, has 3It is interesting to note that registration is not strictly necessary for this algorithm to work. The proposed MAP method works under very broad conditions, the main condition being that the bias fields do not span the same space as parts of the actual medical images. It is true, however, that as the latent images become less registered or differ in other ways, that a much larger number of images is needed to get good estimates of the pixel-stack distributions. been suppressed in the lower images. This is most likely because the entropy of the pixels within a particular image can be reduced by increasing the bias field “correction” in the central part of the image. In other words, the algorithm strives to make the image more uniform by removing the bright part in the middle of the image. However, our algorithm, which compares pixels across images, does not suppress these real structures, since they occur across images. Hence coupling across images can produce superior results. 5 Discussion The idea of minimizing pixelwise entropies to remove nuisance variables from a set of images is not new. In particular, Miller et al. [4, 5] presented an approach they call congealing in which the sum of pixelwise entropies is minimized by separate affine transforms applied to each image. Our method can thus be considered an extension of the congealing process to non-spatial transformations. Combining such approaches to do registration and bias removal simulataneously, or registration and lighting rectification of faces, for example, is an obvious direction for future work. This work uses information unused in other methods, i.e. information across images. This suggests an iterative scheme in which both types of information, both within and across images, are used. Local models could be based on weighted neighborhoods of pixels, pixel cylinders, rather than single pixel-stacks, in sparse data scenarios. For “easy” bias correction problems, such an approach may be overkill, but for difficult problems in bias correction, where the bias field is difficult to separate from the underlying tissue, as discussed in [1], such an approach could produce critical extra leverage. We would like to thank Dr. Terrie Inder and Dr. Simon Warfield for graciously providing the infant brain images for this work. The images were obtained under NIH grant P41 RR13218. Also, we thank Neil Weisenfeld and Sandy Wells for helpful discussions. This work was partially supported by Army Research Office grant DAAD 19-02-1-0383. References [1] Fan, A., Wells, W., Fisher, J., Cetin, M., Haker, S., Mulkern, C., Tempany, C., Willsky, A. A unified variational approach to denoising and bias correction in MR. Proceedings of IPMI, 2003. [2] Beirlant, J., Dudewicz, E., Gyorfi, L. and van der Meulen, E. Nonparametric entropy estimation: An overview. International Journal of Mathematical and Statistical Sciences, 6. pp.17-39. 1997. [3] Learned-Miller, E. G. and Fisher, J. ICA using spacings estimates of entropy. Journal of Machine Learning Research, Volume 4, pp. 1271-1295, 2003. [4] Miller, E. G., Matsakis, N., Viola, P. A. Learning from one example through shared densities on transforms. IEEE Conference on Computer Vision and Pattern Recognition. 2000. [5] Miller, E. G. Learning from one example in machine vision by sharing probability densities. Ph.D. thesis. Massachusetts Institute of Technology. 2002. [6] Vasicek, O. A test for normality based on sample entropy. Journal of the Royal Statistical Society Series B, 31. pp. 632-636, 1976. [7] Wells, W. M., Grimson, W. E. L., Kikinis, R., Jolesz, F. Adaptive segmentation of MRI data. IEEE Transactions on Medical Imaging, 15. pp. 429-442, 1996. [8] Collins, D.L., Zijdenbos, A.P., Kollokian, J.G., Sled, N.J., Kabani, C.J., Holmes, C.J., Evans, A.C. Design and Construction of a realistic digital brain phantom. IEEE Transactions on Medical Imaging, 17. pp. 463-468, 1998. [9] http://www.bic.mni.mcgill.ca/brainweb/ [10] Viola, P.A. Alignment by maximization of mutual information. Ph.D. Thesis. Massachusetts Institute of Technology. 1995. Figure 4: NOTE: This image must be viewed in color (preferably on a bright display) for full effect. Top. Original infant brain images. Middle. The same images after bias removal with our algorithm. Note that developing white matter (butterfly-like structures in middle brain) is well-preserved. Bottom. Bias removal using a single image based algorithm. Notice that white matter structures are repressed.
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Chemosensory processing in a spiking model of the olfactory bulb: chemotopic convergence and center surround inhibition Baranidharan Raman and Ricardo Gutierrez-Osuna Department of Computer Science Texas A&M University College Station, TX 77840 {barani,rgutier}@cs.tamu.edu Abstract This paper presents a neuromorphic model of two olfactory signalprocessing primitives: chemotopic convergence of olfactory receptor neurons, and center on-off surround lateral inhibition in the olfactory bulb. A self-organizing model of receptor convergence onto glomeruli is used to generate a spatially organized map, an olfactory image. This map serves as input to a lattice of spiking neurons with lateral connections. The dynamics of this recurrent network transforms the initial olfactory image into a spatio-temporal pattern that evolves and stabilizes into odor- and intensity-coding attractors. The model is validated using experimental data from an array of temperature-modulated gas sensors. Our results are consistent with recent neurobiological findings on the antennal lobe of the honeybee and the locust. 1 Introduction An artificial olfactory system comprises of an array of cross-selective chemical sensors followed by a pattern recognition engine. An elegant alternative for the processing of sensor-array signals, normally performed with statistical pattern recognition techniques [1], involves adopting solutions from the biological olfactory system. The use of neuromorphic approaches provides an opportunity for formulating new computational problems in machine olfaction, including mixture segmentation, background suppression, olfactory habituation, and odor-memory associations. A biologically inspired approach to machine olfaction involves (1) identifying key signal processing primitives in the olfactory pathway, (2) adapting these primitives to account for the unique properties of chemical sensor signals, and (3) applying the models to solving specific computational problems. The biological olfactory pathway can be divided into three general stages: (i) olfactory epithelium, where primary reception takes place, (ii) olfactory bulb (OB), where the bulk of signal processing is performed and, (iii) olfactory cortex, where odor associations are stored. A review of literature on olfactory signal processing reveals six key primitives in the olfactory pathway that can be adapted for use in machine olfaction. These primitives are: (a) chemical transduction into a combinatorial code by a large population of olfactory receptor neurons (ORN), (b) chemotopic convergence of ORN axons onto glomeruli (GL), (c) logarithmic compression through lateral inhibition at the GL level by periglomerular interneurons, (d) contrast enhancement through lateral inhibition of mitral (M) projection neurons by granule interneurons, (e) storage and association of odor memories in the piriform cortex, and (f) bulbar modulation through cortical feedback [2, 3]. This article presents a model that captures the first three abovementioned primitives: population coding, chemotopic convergence and contrast enhancement. The model operates as follows. First, a large population of cross-selective pseudosensors is generated from an array of metal-oxide (MOS) gas sensors by means of temperature modulation. Next, a self-organizing model of convergence is used to cluster these pseudo-sensors according to their relative selectivity. This clustering generates an initial spatial odor map at the GL layer. Finally, a lattice of spiking neurons with center on-off surround lateral connections is used to transform the GL map into identity- and intensity-specific attractors. The model is validated using a database of temperature-modulated sensor patterns from three analytes at three concentration levels. The model is shown to address the first problem in biologically-inspired machine olfaction: intensity and identity coding of a chemical stimulus in a manner consistent with neurobiology [4, 5]. 2 Modeling chemotopic convergence The projection of sensory signals onto the olfactory bulb is organized such that ORNs expressing the same receptor gene converge onto one or a few GLs [3]. This convergence transforms the initial combinatorial code into an organized spatial pattern (i.e., an olfactory image). In addition, massive convergence improves the signal to noise ratio by integrating signals from multiple receptor neurons [6]. When incorporating this principle into machine olfaction, a fundamental difference between the artificial and biological counterparts must be overcome: the input dimensionality at the receptor/sensor level. The biological olfactory system employs a large population of ORNs (over 100 million in humans, replicated from 1,000 primary receptor types), whereas its artificial analogue uses a few chemical sensors (commonly one replica of up to 32 different sensor types). To bridge this gap, we employ a sensor excitation technique known as temperature modulation [7]. MOS sensors are conventionally driven in an isothermal fashion by maintaining a constant temperature. However, the selectivity of these devices is a function of the operating temperature. Thus, capturing the sensor response at multiple temperatures generates a wealth of additional information as compared to the isothermal mode of operation. If the temperature is modulated slow enough (e.g., mHz), the behavior of the sensor at each point in the temperature cycle can then be treated as a pseudo-sensor, and thus used to simulate a large population of cross-selective ORNs (refer to Figure 1(a)). To model chemotopic convergence, these temperature-modulated pseudo-sensors (referred to as ORNs in what follows) must be clustered according to their selectivity [8]. As a first approximation, each ORN can be modeled by an affinity vector [9] consisting of the responses across a set of C analytes: [ ] C i i i i K K K K ,..., , 2 1 = r (1) where a i K is the response of the ith ORN to analyte a. The selectivity of this ORN is then defined by the orientation of the affinity vector i Κ r . A close look at the OB also shows that neighboring GLs respond to similar odors [10]. Therefore, we model the ORN-GL projection with a Kohonen self-organizing map (SOM) [11]. In our model, the SOM is trained to model the distribution of ORNs in chemical sensitivity space, defined by the affinity vector i Κ r . Once the training of the SOM is completed, each ORN is assigned to the closest SOM node (a simulated GL) in affinity space, thereby forming a convergence map. The response of each GL can then be computed as ( ) ∑= ⋅ = N i a i ij a j ORN W G 1 σ (2) where a i ORN is the response of pseudo-sensor i to analyte a, Wij=1 if pseudo-sensor i converges to GL j and zero otherwise, and ( )⋅ σ is a squashing sigmoidal function that models saturation. This convergence model works well under the assumption that the different sensory inputs are reasonably uncorrelated. Unfortunately, most gas sensors are extremely collinear. As a result, this convergence model degenerates into a few dominant GLs that capture most of the sensory activity, and a large number of dormant GLs that do not receive any projections. To address this issue, we employ a form of competition known as conscience learning [12], which incorporates a habituation mechanism to prevent certain SOM nodes from dominating the competition. In this scheme, the fraction of times that a particular SOM node wins the competition is used as a bias to favor non-winning nodes. This results in a spreading of the ORN projections to neighboring units and, therefore, significantly reduces the number of dormant units. We measure the performance of the convergence mapping with the entropy across the lattice, ∑ − = i i P P H log , where Pi is the fraction of ORNs that project to SOM node i [13]. To compare Kohonen and conscience learning, we built convergence mappings with 3,000 pseudo-sensors and 400 GL units (refer to section 4 for details). The theoretical maximum of the entropy for this network, which corresponds to a uniform distribution, is 8.6439. When trained with Kohonen’s algorithm, the entropy of the SOM is 7.3555. With conscience learning, the entropy increases to 8.2280. Thus, conscience is an effective mechanism to improve the spreading of ORN projections across the GL lattice. 3 Modeling the olfactory bulb network Mitral cells, which synapse ORNs at the GL level, transform the initial olfactory image into a spatio-temporal code by means of lateral inhibition. Two roles have been suggested for this lateral inhibition: (a) sharpening of the molecular tuning range of individual M cells with respect to that of their corresponding ORNs [10], and (b) global redistribution of activity, such that the bulb-wide representation of an odorant, rather than the individual tuning ranges, becomes specific and concise over time [3]. More recently, center on-off surround inhibitory connections have been found in the OB [14]. These circuits have been suggested to perform pattern normalization, noise reduction and contrast enhancement of the spatial patterns. We model each M cell using a leaky integrate-and-fire spiking neuron [15]. The input current I(t) and change in membrane potential u(t) of a neuron are given by: ] [ ) ( ) ( ) ( ) ( RC t I R t u dt du dt du C R t u t I = ⋅ + − = + = τ τ (3) Each M cell receives current Iinput from ORNs and current Ilateral from lateral connections with other M cells: )1 , ( ) , ( ) ( − ⋅ = ⋅ = ∑ ∑ t k L t j I ORN W j I k kj lateral i i ij input α (4) where Wij indicates the presence/absence of a synapse between ORNi and Mj, as determined by the chemotopic mapping, Lkj is the efficacy of the lateral connection between Mk and Mj, and α(k,t-1) is the post-synaptic current generated by a spike at Mk: ] )1 , ( [ )1 , ( )1 , ( syn E t j u t k g t k − − ⋅ − − = − + α (5) g(k,t-1) is the conductance of the synapse between Mk and Mj at time t-1, u(j,t-1) is the membrane potential of Mj at time t-1 and the + subscript indicates this value becomes zero if negative, and Esyn is the reverse synaptic potential. The change in conductance of post-synaptic membrane is: ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( t k spk g t k z dt t k dz t k z t k z t k g dt t k dg t k g norm syn syn ⋅ + − = = + − = = τ τ & & (6) where z(.) and g(.) are low pass filters of the form exp(-t/τsyn) and ) / exp( syn t t τ − ⋅ , respectively, τsyn is the synaptic time constant, gnorm is a normalization constant, and spk(j,t) marks the occurrence of a spike in neuron i at time t: ≠ = = spike spike V t j u V t j u t j spk ) , ( 0 ) , ( 1 ) , ( (7) Combining equations (3) and (4), the membrane potential can be expressed as: ≥ < ⋅ − + − = + + − = = threshold spike threshold input lateral V t j u V V t j u dt t j u t j u t j u C j I C t j I RC t j u dt t j du t j u ) , ( ) , ( )1 , ( )1 , ( ) , ( ) ( ) , ( ) , ( ) , ( ) , ( & & (8) When the membrane potential reaches Vthreshold, a spike is generated, and the membrane potential is reset to Vrest. Any further inputs to the neuron are ignored during the subsequent refractory period. Following [14], lateral interactions are modeled with a center on-off surround matrix Lij. Each M cell makes excitatory synapses to nearby M cells (d<de), where d is the Manhattan distance measured in the lattice, and inhibitory synapses with distant M cells (de<d<di) through granule cells (implicit in our model). Excitatory synapses are assigned uniform random weights between [0, 0.1]. Inhibitory synapses are assigned negative weights in the same interval. Model parameters are summarized in Table 1. Table 1. Parameters of the OB spiking neuron lattice Parameter Value Parameter Value Peak synaptic conductance (Gpeak) 0.01 Synaptic time constants (τsyn) 10 ms Capacitance (C) 1 nF Total simulation time (ttot) 500 ms Resistance (R) 10 MOhm Integration time step (dt) 1 ms Spike voltage (Vspike) 70 mV Refractory period (tref) 3 ms Threshold voltage (Vthreshold) 5 mV Number of mitral cells (N) 400 Synapse Reverse potential (Esyn) 70 mV Normalization constant (gnorm) 0.0027 Excitatory distance (de) N d 6 1 < Inhibitory distance (di) N d N 6 2 6 1 < < 4 Results The proposed model is validated on an experimental dataset containing gas sensor signals for three analytes: acetone (A), isopropyl alcohol (B) and ammonia (C), at three different concentration levels per analyte. Two Figaro MOS sensors (TGS 2600, TGS 2620) were temperature modulated using a sinusoidal heater voltage (0-7 V; 2.5min period; 10Hz sampling frequency). The response of the two sensors to the three analytes at the three concentration levels is shown in Figure 1(a). This response was used to generate a population of 3,000 ORNs, which were then mapped onto a GL layer with 400 units arranged as a 20×20 lattice. Pseudo-Sensors 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 2 4 6 8 10 12 14 16 18 20 5 10 15 20 2 4 6 8 10 12 14 16 18 20 5 10 15 20 2 4 6 8 10 12 14 16 18 20 5 10 15 20 2 4 6 8 10 12 14 16 18 20 5 10 15 20 2 4 6 8 10 12 14 16 18 20 5 10 15 20 2 4 6 8 10 12 14 16 18 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 A1 A2 A3 B1 B3 B2 C1 C2 C3 Concentration 500 1000 1500 2000 2500 3000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Sensor conductance (Acetone) A3 A2 A1 500 1000 1500 2000 2500 3000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Sensor Conductance (Iso-propyl alcohol B3 B2 B1 500 1000 1500 2000 2500 3000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Sensor Conductance (Ammonia) C3 C2 C1 Sensor 1 Sensor 2 (a) (b) Figure 1. (a) Temperature modulated response to the three analytes (A,B,C) at three concentrations (A3: highest concentration of A), and (b) initial GL maps. The sensor response to the highest concentration of each analyte was used to generate the SOM convergence map. Figure 1(b) shows the initial odor map of the three analytes following conscience learning of the SOM. These olfactory images show that the identity of the stimulus is encoded by the spatial pattern across the lattice, whereas the intensity is encoded by the overall amplitude of this pattern. Analytes A and B, which induce similar responses on the MOS sensors, also lead to very similar GL maps. The GL maps are input to the lattice of spiking neurons for further processing. As a result of the dynamics induced by the recurrent connections, these initial maps are transformed into a spatio-temporal pattern. Figure 2 shows the projection of membrane potential of the 400 M cells along their first three principal components. Three trajectories are shown per analyte, which correspond to the sensor response to the highest analyte concentration on three separate days of data collection. These results show that the spatio-temporal pattern is robust to the inherent drift of chemical sensors. The trajectories originate close to each other, but slowly migrate and converge into unique odor-specific attractors. It is important to note that these trajectories do not diverge indefinitely, but in fact settle into an attractor, as illustrated by the insets in Figure 2. -200 -150 -100 -50 0 50 100 150 -250 -200 -150 -100 -50 0 50 100 -15 -10 -5 0 5 10 15 20 Odor A Odor B Odor C Figure 2. Odor-specific attractors from experimental sensor data. Three trajectories are shown per analyte, corresponding to the sensor response on three separate days. These results show that the attractors are repeatable and robust to sensor drift. To illustrate the coding of identity and intensity performed by the model, Figure 3 shows the trajectories of the three analytes at three concentrations. The OB network activity evolves to settle into an attractor, where the identity of the stimulus is encoded by the direction of the trajectory relative to the initial position, and the intensity is encoded by the length along the trajectory. This emerging code is also consistent with recent findings in neurobiology, as discussed next. 5 Discussion A recent study of spatio-temporal activity in projection neurons (PN) of the honeybee antennal lobe (analogous to M cells in mammalian OB) reveals evolution and convergence of the network activity into odor-specific attractors [4]. Figure 4(a) shows the projection of the spatio-temporal response of the PNs along their first three principal components. These trajectories begin close to each other, and evolve over time to converge into odor specific regions. These experimental results are consistent with the attractor patterns emerging from our model. Furthermore, an experimental study of odor identity and intensity coding in the locust show hierarchical groupings of spatio-temporal PN activity according to odor identity, followed by odor intensity [5]. Figure 4(b) illustrates this grouping in the activity of 14 PNs when exposed to three odors at five concentrations. Again, these results closely resemble the grouping of attractors in our model, shown in Figure 3. -300 -250 -200 -150 -100 -50 0 50 -50 0 50 100 150 200 250 300 350 -50 0 50 100 150 200 PC1 PC2 PC3 C3 C2 C1 A3 A1 A2 B2 B3 B1 -300 -250 -200 -150 -100 -50 0 50 -50 0 50 100 150 200 250 300 350 -50 0 50 100 150 200 PC1 PC2 PC3 C3 C2 C1 A3 A1 A2 B2 B3 B1 Figure 3. Identity and intensity coding using dynamic attractors. Previous studies by Pearce et al. [6] using a large population of optical micro-bead chemical sensors have shown that massive convergence of sensory inputs can be used to provide sensory hyperacuity by averaging out uncorrelated noise. In contrast, the focus of our work is on the coding properties induced by chemotopic convergence. Our model produces an initial spatial pattern or olfactory image, whereby odor identity is coded by the spatial activity across the GL lattice, and odor intensity is encoded by the amplitude of this pattern. Hence, the bulk of the identity/intensity coding is performed by this initial convergence primitive. Subsequent processing by a lattice of spiking neurons introduces time as an additional coding dimension. The initial spatial maps are transformed into a spatiotemporal pattern by means of center on-off surround lateral connections. Excitatory lateral connections allow the model to spread M cell activity, and are responsible for moving the attractors away from their initial coordinates. In contrast, inhibitory connections ensure that these trajectories eventually converge onto an attractor, rather than diverge indefinitely. It is the interplay between excitatory and inhibitory connections that allows the model to enhance the initial coding produced by the chemotopic convergence mapping. (a) (b) hexanol octanol nonanol isoamylacetate (a) (b) hexanol octanol nonanol isoamylacetate Figure 4. (a) Odor trajectories formed by spatio-temporal activity in the honeybee AL (adapted from [4]). (b) Identity and intensity clustering of spatio-temporal activity in the locust AL (adapted from [5]; arrows indicate the direction of increasing concentration). At present, our model employs a center on-off surround kernel that is constant throughout the lattice. Further improvements can be achieved through adaptation of these lateral connections by means of Hebbian and anti-Hebbian learning. These extensions will allow us to investigate additional computational functions (e.g., pattern completion, orthogonalization, coding of mixtures) in the processing of information from chemosensor arrays. Acknowledgments This material is based upon work supported by the National Science Foundation under CAREER award 9984426/0229598. Takao Yamanaka, Alexandre PereraLluna and Agustin Gutierrez-Galvez are gratefully acknowledged for valuable suggestions during the preparation of this manuscript. References [1] Gutierrez-Osuna, R. (2002) Pattern Analysis for Machine Olfaction: A Review. IEEE Sensors Journal 2(3): 189-202. [2] Pearce, T. C. (1999) Computational parallels between the biological olfactory pathway and its analogue ‘The Electronic Nose’: Part I. Biologiacal olfaction. BioSystems 41: 43-67. [3] Laurent, G. (1999) A Systems Perspective on Early Olfactory Coding. Science 286(22): 723-728. [4] Galán, R. F.,Sachse, S., Galizia, C.G., & Herz, A.V. (2003) Odor-driven attractor dynamics in the antennal lobe allow for simple and rapid olfactory pattern classification. Neural Computation 16(5): 999-1012. [5] Stopfer, M., Jayaraman, V., & Laurent, G. (2003) Intensity versus Identity Coding in an Olfactory System. Neuron 39: 991-1004. [6] Pearce, T.C., Verschure, P.F.M.J., White, J., & Kauer, J. S. (2001) Robust Stimulus Encoding in Olfactory Processing: Hyperacuity and Efficient Signal Transmission. In S. Wermter, J. Austin and D. Willshaw (Eds.), Emergent Neural Computation Architectures Based on Neuroscience. pp. 461-479. Springer-Verlag. [7] Lee. A. P., & Reedy, B. J. (1999) Temperature modulation in semiconductor gas sensing. Sensors and Actuators B 60: 35-42. [8] Vassar, R., Chao, S.K., Sitcheran, R., Nunez, J. M., Vosshall, L.B., & Axel, A. (1994) Topographic Organization of Sensory Projections to the Olfactory Bulb. Cell 79(6): 981991. [9] Gutierrez-Osuna, R. (2002) A Self-organizing Model of Chemotopic Convergence for Olfactory Coding. In Proceedings of the 2nd EMBS-BMES Conference, pp. 23-26. Texas. [10] Mori, K., Nagao, H., & Yoshihara, Y. (1999) The Olfactory Bulb: Coding and Processing of Odor molecule information. Science 286: 711-715. [11] Kohonen, T. (1982) Self-organized formation of topologically correct feature maps. Biological Cybernetics 43: 59-69. [12] DeSieno, D. (1988) Adding conscience to competitive learning. In Proceedings of International Conference on Neural Networks (ICNN), pp. 117-124. Piscataway, NJ. [13] Laaksonen, J., Koskela, M., & Oja, E. (2003) Probability interpretation of distributions on SOM surfaces. In Proceedings of Workshop on Self-Organizing Maps. Hibikino, Japan. [14] Aungst et al. (2003) Center-surround inhibition among olfactory bulb glomeruli. Nature 26: 623- 629. [15] Gerstner, W., & Kistler, W. (2002) Spiking Neuron Models: Single Neurons, Populations, Plasticity. Cambridge, University Press.
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Using Random Forests in the Structured Language Model Peng Xu and Frederick Jelinek Center for Language and Speech Processing Department of Electrical and Computer Engineering The Johns Hopkins University {xp,jelinek}@jhu.edu Abstract In this paper, we explore the use of Random Forests (RFs) in the structured language model (SLM), which uses rich syntactic information in predicting the next word based on words already seen. The goal in this work is to construct RFs by randomly growing Decision Trees (DTs) using syntactic information and investigate the performance of the SLM modeled by the RFs in automatic speech recognition. RFs, which were originally developed as classifiers, are a combination of decision tree classifiers. Each tree is grown based on random training data sampled independently and with the same distribution for all trees in the forest, and a random selection of possible questions at each node of the decision tree. Our approach extends the original idea of RFs to deal with the data sparseness problem encountered in language modeling. RFs have been studied in the context of n-gram language modeling and have been shown to generalize well to unseen data. We show in this paper that RFs using syntactic information can also achieve better performance in both perplexity (PPL) and word error rate (WER) in a large vocabulary speech recognition system, compared to a baseline that uses Kneser-Ney smoothing. 1 Introduction In many systems dealing with speech or natural language, such as Automatic Speech Recognition and Statistical Machine Translation, a language model is a crucial component for searching in the often prohibitively large hypothesis space. Most state-of-the-art systems use n-gram language models, which are simple and effective most of the time. Many smoothing techniques that improve language model probability estimation have been proposed and studied in the n-gram literature [1]. There has so far been work in exploring Decision Tree (DT) language models [2, 3], which attempt to cluster similar histories together to achieve better probability estimation. However, the results were negative [3]: decision tree language models failed to improve upon the baseline n-gram models with the same order n. Random Forest (RF) language models, which are generalizations of DT language models, have been recently applied to word n-grams [4]. DT growing is randomized to construct RFs efficiently. Once constructed, the RFs function as a randomized history clustering, which helps in dealing with the data sparseness problem. In general, the weakness of some trees can be compensated for by other trees. The collective contribution of all DTs in an RF n-gram model results in significant improvements in both perplexity (PPL) and word error rate (WER) in a large vocabulary speech recognition system. Language models can also be improved with better representations of the history. Recent efforts have studied various ways of using information from a longer history span than that usually captured by normal n-gram language models, as well as ways of using syntactic information that is not available to the word-based n-gram models [5, 6, 7]. All these language models are based on stochastic parsing techniques that build up parse trees for the input word sequence and condition the generation of words on syntactic and lexical information available in the parse trees. Since these language models capture useful hierarchical characteristics of language, they can improve PPL and WER significantly for various tasks. However, due to the n-gram nature of the components of the syntactic language models, the data sparseness problem can be severe. In order to reduce the data sparseness problem for using rich syntactic information in the context, we study the use of RFs in the structured language model (SLM) [5]. Our results show that although the components of the SLM have high order n-grams, our RF approach can still achieve better performance, reducing both the perplexity (PPL) and word error rate (WER) in a large vocabulary speech recognition system compared to a Kneser-Ney smoothing baseline. 2 Basic Language Modeling The purpose of a language model is to estimate the probability of a word string. Let W denote a string of N words, that is, W = w1, w2, . . . , wN. Then, by the chain rule of probability, we have P (W )=P (w1)×Q N i=2 P (wi|w1,...,wi−1). (1) In order to estimate the probabilities P(wi|w1, . . . , wi−1), we need a training corpus consisting of a large number of words. However, in any practical natural language system of even moderate vocabulary size, it is clear that the number of probabilities to be estimated and stored is prohibitively large. Therefore, histories w1, . . . , wi−1 for a word wi are usually grouped into equivalence classes. The most widely used language models, n-gram language models, use the identities of the last n −1 words as equivalence classes. In an n-gram model, we then have P (W )=P (w1)×Q N i=2 P (wi|wi−1 i−n+1), (2) where we have used wi−1 i−n+1 to denote the word sequence wi−n+1, . . . , wi−1. If we could handle unlimited amounts of training data, the maximum likelihood (ML) estimate of P(wi|wi−1 i−n+1) would be the best: P (wi|wi−1 i−n+1)= C(wi i−n+1) C(wi−1 i−n+1) , (3) where C(wi i−n+1) is the number of times the string wi−n+1, . . . , wi is seen in the training data. 2.1 Language Model Smoothing An n-gram model when n = 3 is called a trigram model. For a vocabulary of size |V | = 104, there are |V |3 = 1012 trigram probabilities to be estimated. For any training data of a manageable size, many of the probabilities will be zero if the ML estimate is used. In order to solve this problem, many smoothing techniques have been studied (see [1] and the references therein). Smoothing adjusts the ML estimates to produce more accurate probabilities and to assign nonzero probabilities to any word string. Details about various smoothing techniques will not be presented in this paper, but we will outline a particular way of smoothing, namely interpolated Kneser-Ney smoothing [8], for later reference. Interpolated Kneser-Ney smoothing assumes the following form: PKN(wi|wi−1 i−n+1) = max(C(wi i−n+1)−D,0) C(wi−1 i−n+1) +λ(wi−1 i−n+1)PKN(wi|wi−1 i−n+2), (4) where D is a discounting constant and λ(wi−1 i−n+1) is the interpolation weight for the lower order probabilities ((n −1)-gram). The discount constant is often estimated using the leave-one-out method, leading to the approximation D = n1 n1+2n2 , where n1 is the number of n-grams with count one and n2 is the number of n-grams with count two. To ensure that the probabilities sum to one, we have λ(wi−1 i−n+1)= D P wi:C(wi i−n+1)>0 1 C(wi−1 i−n+1) . The lower order probabilities in interpolated Kneser-Ney smoothing can be estimated as (assuming ML estimation): PKN(wi|wi−1 i−n+2)= P wi−n+1:C(wi i−n+1)>0 1 P wi−n+1,wi:C(wi i−n+1)>0 1 . (5) Note that the lower order probabilities are usually recursively smoothed using Equation 4. 2.2 Language Model Evalution A commonly used task-independent quality measure for a given language model is related to the cross-entropy of the underlying model and is referred to as perplexity (PPL): P P L=exp(−1/N P N i=1 log [P (wi|wi−1 1 )]), (6) where w1, . . . , wN is the test text that consists of N words. For different tasks, there are different task-dependent quality measures of language models. For example, in an automatic speech recognition system, the performance is usually measured by word error rate (WER). 3 The Structured Language Model (SLM) The SLM uses rich syntactic information beyond regular word n-grams to improve language model quality. An extensive presentation of the SLM can be found in Chelba and Jelinek, 2000 [5]. The model assigns a probability P(W, T) to every sentence W and every possible binary parse T. The terminals of T are the words of W with POS tags, and the nodes of T are annotated with phrase headwords and non-terminal labels. Let W be (<s>, SB) ....... (w_p, t_p) (w_{p+1}, t_{p+1}) ........ (w_k, t_k) w_{k+1}.... </s> h_0 = (h_0.word, h_0.tag) h_{-1} h_{-m} = (<s>, SB) Figure 1: A word-parse k-prefix a sentence of length n words to which we have prepended the sentence beginning marker <s> and appended the sentence end marker </s> so that w0 =<s> and wn+1 =</s>. Let Wk = w0 . . . wk be the word k-prefix of the sentence — the words from the beginning of the sentence up to the current position k — and WkTk the word-parse k-prefix. Figure 1 shows a word-parse k-prefix; h_0, .., h_{-m} are the exposed heads, each head being a pair (headword, non-terminal label), or (word, POS tag) in the case of a root-only tree. The exposed heads at a given position k in the input sentence are a function of the word-parse k-prefix [5]. The joint probability P(W, T) of a word sequence W and a complete parse T comes from contributions of three components: WORD-PREDICTOR, TAGGER and CONSTRUCTOR. The SLM works in the following way: first, the WORD-PREDICTOR predicts a word based on the word-parse prefix; the TAGGER then assigns a POS tag to the predicted word based on the word itself and the word-parse prefix; the CONSTRUCTOR takes a series of actions each of which turns a parse prefix into a new parse prefix (the series of actions ends with a NULL action which tells the WORD-PREDICTOR to predict the next word). Details about the three components can be found in [5]. Each of the three components can be seen as an n-gram model and can be estimated independently because of the product form of the joint probability. They are parameterized (approximated) as follows: P (wk|Wk−1Tk−1) = P (wk|h0.tag,h0.word,h−1.tag,h−1.word), (7) P (tk|wk,Wk−1Tk−1) = P (tk|wk,h0.tag,h−1.tag), (8) P (pk i |Wk−1Tk−1,wk,tk,pk 1...pk i−1) = P (pk i |h0.tag,h−1.tag,h−2.tag,h0.word,h−1.word), (9) where pk i is the ith CONSTRUCTOR action after the kth word and POS tag have been predicted. Since the number of parses for a given word prefix Wk grows exponentially with k, |{Tk}| ∼O(2k), the state space of our model is huge even for relatively short sentences. Thus we must use a search strategy that prunes the allowable parse set. One choice is a synchronous multi-stack search algorithm [5] which is very similar to a beam search. The language model probability assignment for the word at position k + 1 in the input sentence is made using: PSLM(wk+1|Wk) = P Tk∈Sk P (wk+1|WkTk)·ρ(Wk,Tk), ρ(Wk,Tk) = P (WkTk)/ P Tk∈Sk P (WkTk), (10) which ensures a proper probability normalization over strings of words, where Sk is the set of all parses present in the stacks at the current stage k and P(WkTk) is the joint probability of word-parse prefix WkTk. Each model component —WORD-PREDICTOR, TAGGER, CONSTRUCTOR— is estimated independently from a set of parsed sentences after undergoing headword percolation and binarization (see details in [5]). 4 Using Random Forests in the Structured Language Model 4.1 Random Forest n-gram Modeling A Random Forest (RF) n-gram model is a collection of randomly constructed decision tree (DT) n-gram models. Unlike RFs in classification and regression tasks [9, 10, 11], RFs are used in language modeling to deal with the data sparsenes problem [4]. Therefore, the training data is not randomly sampled for each DT. Figure 2 shows the algorithm DT-Grow and Node-Split used for generating random DT language models. We define a position in the history as the distance between a word in the history and the predicted word. The randomization is carried out in two places: a random selection of Algorithm DT-Grow Input: counts for training and heldout data Initialize: Create a root node containing all histories in the training data and put it in set Φ While Φ is not empty 1. Get a node p from Φ 2. If Node-Split(p) is successful, eliminate p from Φ and put the two children of p in Φ Foreach internal node p in the tree 1. LH p ←normalized likelihood of heldout data associated with p, using training data statistics in p 2. Get the set of leaves P rooted in p 3. LH P ←normalized likelihood of heldout data associated with all leaves in P, using training data statistics in the corresponding leaves 4. if LH P −LH p < 0, prune the subtree rooted in p Output: a Decistion Tree language model Algorithm Node-Split(p) Input: node p and training data associated Initialize: Randomly select a subset of positions I in the history Foreach position i in I 1. Group all histories into basic elements β(v) 2. Randomly split the elements β(v) into sets L and R 3. While there are elements moved, Do (a) Move each element from L to R if the move results in positive gain in training data likelihood (b) Move each element from R to L if the move results in positive gain in training data likelihood Select the position from I that results in the largest gain Output: a split L and R, or failure if the largest gain is not positive Figure 2: The algorithm DT-Grow and Node-Split positions in the history and an initial random split of basic elements. Since our splitting criterion is to maximize the log-likelihood of the training data, each split uses only statistics (from training data) associated with the node under consideration. Smoothing is not needed in the splitting and we can use a fast exchange algorithm [12] in Node-Split. Given a position i in the history, β(v) is defined to be the set of histories belonging to the node p, such that they all have word v at position i. It is clear that for every position i in the history, the union ∪vβ(v) is all histories in the node p. In DT-Grow, after a DT is fully grown, we use some heldout data to prune it. Pruning is done in such a way that we maximize the likelihood of the heldout data, where smoothing is applied according to Equation 4: PDT (wi|ΦDT (wi−1 i−n+1)) = max(C(wi,ΦDT (wi−1 i−n+1))−D,0) C(ΦDT (wi−1 i−n+1)) +λ(ΦDT (wi−1 i−n+1))PKN(wi|wi−1 i−n+2) (11) where ΦDT (·) is one of the DT nodes the history can be mapped to and PKN(wi|wi−1 i−n+2) is as defined in Equation 5. This pruning is similar to the pruning strategy used in CART [13]. Once we get the DTs, we only use the leaf nodes as equivalence classes of histories. If a new history is encountered, it is very likely that we will not be able to place it at a leaf node in the DT. In this case, λ(ΦDT (wi−1 i−n+1)) = 1 in Equation 11 and we simply use PKN(wi|wi−1 i−n+2) to get the probabilities. The randomized version of the DT growing algorithm can be run many times and finally we will get a collection of randomly grown DTs: a Random Forest (RF). Since each DT is a smoothed language model, we simply aggregate all DTs in our RF to get the RF language model. Suppose we have M randomly grown DTs, DT1, . . . , DTM. In the n-gram case, the RF language model probabilities can be computed as: PRF (wi|wi−1 i−n+1)= 1 M P M j=1 PDTj (wi|ΦDTj (wi−1 i−n+1)) (12) where ΦDTj(wi−1 i−n+1) maps the history wi−1 i−n+1 to a leaf node in DTj. If wi−1 i−n+1 can not be mapped to a leaf node in some DT, we back-off to the lower order KN probability as mentioned earlier. It can be shown by the Law of Large Numbers that the probability in Equation 12 converges as the number of DTs grows. It converges to ET PT (wi|ΦT (wi−1 i−n+1) where T is a random variable representing the random DTs. The advantage of the RF approach over the KN smoothing lies in the fact that different DTs have different weaknesses and strengths for word prediction. As the number of trees grows, the weakness of some trees can be compensated for by some other trees. This advantage and the convergence have been shown experimentally in [4]. 4.2 Using RFs in the SLM Since the three model components in the SLM as in Equation 7-9 can be estimated independently, we can construct an RF for each component using the algorithm DT-Grow in the previous section. The only difference is that we will have different n-gram orders and different items in the history for each model. Ideally, we would like to use RFs for each component in the SLM. However, due to the nature of the SLM, there are difficulties. The SLM uses a synchronous multi-stack search algorithm to dynamically construct stacks and compute the language model probabilities as in Equation 10. If we use RFs for all components, we need to load all DTs in the RFs into memory at runtime. This is impractical for RFs of any reasonable size. There is a different approach that can take advantage of the randomness in the RFs. Suppose we have M randomly grown DTs, DT a 1 , . . . , DT a M for each component a of the SLM, where a ∈{P, T, C} for WORD-PREDICTOR, TAGGER and CONSTRUCTOR, respectively. The DTs are grouped into M triples {DT P j , DT T j , DT C j } j = 1, . . . , M. We calculate the joint probability P(W, T) for the jth DT triple according to: Pj(W,T )= Q n+1 k=1[PDT P j (wk|Wk−1Tk−1)·PDT T j (tk|Wk−1Tk−1,wk)· Q Nk i=1 PDT C j (pk i |Wk−1Tk−1,wk,tk,pk 1...pk i−1)]. (13) Then, the language model probability assignment for the jth DT triple is made using: Pj(wk+1|Wk) = P T j k ∈Sj k PDT P j (wk+1|WkT j k)·ρj(Wk,T j k), ρj(Wk,T j k) = Pj(WkT j k)/ P T j k ∈Sj k Pj(WkT j k), (14) which is achieved by running the synchronous multi-stack algorithm using the jth DT triple as a model. Finally, after the SLM is run M times, the RF language model probability is an average of the probabilities above: PRF (wk+1|Wk)= 1 M P M j=1 Pj(wk+1|Wk). (15) The triple {DT P j , DT T j , DT C j } can be considered as a single DT in which the root node has three children corresponding to the three root nodes of DT P j , DT T j and DT C j . The root node of this DT asks the question: Which model component does the history belong to? According to the answer, we can proceed to one of the three children nodes (one of the three components, in fact). Since the multi-stack search algorithm is deterministic given the DT, the probability in Equation 15 can be shown to converge. 5 Experiments 5.1 Perplexity (PPL) We have used the UPenn Treebank portion of the WSJ corpus to carry out our experiments. The UPenn Treebank contains 24 sections of hand-parsed sentences, for a total of about one million words. We used section 00-20 for training our models, section 21-22 as heldout data for pruning the DTs, and section 23-24 to test our models. Before carrying out our experiments, we normalized the text in the following ways: numbers in arabic form were replaced by a single token “N”, punctuation was removed, all words were mapped to lower case, extra information in the parse trees was ignored, and, finally, traces were ignored. The word vocabulary contains 10k words including a special token for unknown words. There are 40 items in the part-of-speech set and 54 items in the non-terminal set, respectively. The three components in the SLM were treated independently during training. We trained an RF for each component and each RF contained 100 randomly grown DTs. The baseline SLM used KN smoothing (KN-SLM). The 100 probability sequences from the 100 triples were aggregated to get the final PPL. The results are shown in Table 1. We also interpolated the SLM with the KN-trigram to get further improvements. The interpolation weight α in Table 1 is on KN-trigram. The RF-SLM achieved a 10.9% and a 7.5% improvement over the KN-SLM, before and after interpolation with KN-trigram, respectively. Compared to the improvements reported in [4] (10.5% from RF-trigram to KN-trigram), the RF-SLM achieved greater improvement by using syntactic information. Figure 3 shows the convergence of the PPL as the number of DTs grows from 1 to 100. 20 40 60 80 100 120 125 130 135 140 145 150 155 160 Number of DTs PPL Figure 3: PPL convergence Model α=0.0 α=0.4 α=1.0 KN-SLM 137.9 127.2 145.0 RF-SLM 122.8 117.6 145.0 Gain 10.9% 7.5% Table 1: PPL comparison between KNSLM and RF-SLM, interpolated with KN-trigram 5.2 Word Error Rate by N-best Re-scoring To test our RF modeling approach in the context of speech recognition, we evaluated the models in the WSJ DARPA’93 HUB1 test setup. The size of the test set is 213 utterances, 3446 words. The 20k word open vocabulary and baseline 3-gram model are the standard ones provided by NIST and LDC — see [5] for details. The N-best lists were generated using the standard 3-gram model trained on 40M words of WSJ. The N-best size was at most 50 for each utterance, and the average size was about 23. For the KN-SLM and RFSLM, we used 20M words automatically parsed, binarized and enriched with headwords and NT/POS tag information. As the size of RF-SLM becomes very large, we only used RF for the WORD-PREDICTOR component (RF-SLM-P). The other two components used KN smoothing. The results are reported in Table 2. Model α=0.0 α=0.2 α=0.4 α=0.6 α=0.8 KN-SLM 12.8 12.5 12.6 12.7 12.7 RF-SLM-P 11.9 12.2 12.3 12.3 12.6 Table 2: N-best rescoring WER results For purpose of comparison, we interpolated all models with the KN-trigram built from 40M words at different level of interpolation weights α (on KN-trigram). However, it is the α = 0.0 column that is the most interesting. We can see that the RF approach improved over the regular KN approach with an absolute WER reduction of 0.9%. 6 Conclusions Based on the idea of Random Forests in classification and regression, we developed algorithms for constructing and using Random Forests in language modeling. In particular, we applied this new probability estimation technique to the Structured Language Model, in which there are three model components that can be estimated independently. The independently constructed Random Forests can be considered as a more general single Random Forest, which ensures the convergence of the probabilities as the number of Decision Trees grows. The results on a large vocabulary speech recognition system show that we can achieve significant reduction in both perplexity and word error rate, compared to a baseline using Kneser-Ney smoothing. References [1] Stanley F. Chen and Joshua Goodman, “An empirical study of smoothing techniques for language modeling,” Tech. Rep. TR-10-98, Computer Science Group, Harvard University, Cambridge, Massachusetts, 1998. [2] L. Bahl, P. Brown, P. de Souza, and R. Mercer, “A tree-based statistical language model for natural language speech recognition,” in IEEE Transactions on Acoustics, Speech and Signal Processing, July 1989, vol. 37, pp. 1001–1008. [3] Gerasimos Potamianos and Frederick Jelinek, “A study of n-gram and decision tree letter language modeling methods,” Speech Communication, vol. 24(3), pp. 171–192, 1998. [4] Peng Xu and Frederick Jelinek, “Random forests in language modeling,” in Proceedings of the 2004 Conference on Empirical Methods in Natural Language Processing, Barcelona, Spain, July 2004. [5] Ciprian Chelba and Frederick Jelinek, “Structured language modeling,” Computer Speech and Language, vol. 14, no. 4, pp. 283–332, October 2000. [6] Eugene Charniak, “Immediate-head parsing for language models,” in Proceedings of the 39th Annual Meeting and 10th Conference of the European Chapter of ACL, Toulouse, France, July 2001, pp. 116–123. [7] Brian Roark, Robust Probabilistic Predictive Syntactic Processing: Motivations, Models and Applications, Ph.D. thesis, Brown University, Providence, RI, 2001. [8] Reinhard Kneser and Hermann Ney, “Improved backing-off for m-gram language modeling,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, 1995, vol. 1, pp. 181–184. [9] Y. Amit and D. Geman, “Shape quantization and recognition with randomized trees,” Neural Computation, , no. 9, pp. 1545–1588, 1997. [10] Leo Breiman, “Random forests,” Tech. Rep., Statistics Department, University of California, Berkeley, Berkeley, CA, 2001. [11] T.K. Ho, “The random subspace method for constructing decision forests,” IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 20, no. 8, pp. 832–844, 1998. [12] S. Martin, J. Liermann, and H. Ney, “Algorithms for bigram and trigram word clustering,” Speech Communication, vol. 24(3), pp. 171–192, 1998. [13] L. Breiman, J.H. Friedman, R.A. Olshen, and C.J. Stone, Classification and Regression Trees, Chapman and Hall, New York, 1984.
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Exploration-Exploitation Tradeoffs for Experts Algorithms in Reactive Environments Daniela Pucci de Farias Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, MA 02139 pucci@mit.edu Nimrod Megiddo IBM Almaden Research Center 650 Harry Road, K53-B2 San Jose, CA 95120 megiddo@almaden.ibm.com Abstract A reactive environment is one that responds to the actions of an agent rather than evolving obliviously. In reactive environments, experts algorithms must balance exploration and exploitation of experts more carefully than in oblivious ones. In addition, a more subtle definition of a learnable value of an expert is required. A general exploration-exploitation experts method is presented along with a proper definition of value. The method is shown to asymptotically perform as well as the best available expert. Several variants are analyzed from the viewpoint of the exploration-exploitation tradeoff, including explore-then-exploit, polynomially vanishing exploration, constant-frequency exploration, and constant-size exploration phases. Complexity and performance bounds are proven. 1 Introduction Real-world environments require agents to choose actions sequentially. For example, a driver has to choose everyday a route from one point to another, based on past experience and perhaps some current information. In another example, an airline company has to set prices dynamically, also based on past experience and current information. One important difference between these two examples is that the effect of the driver’s decision on the future traffic patterns is negligible, whereas prices set by one airline can affect future market prices significantly. In this sense the decisions of the airlines are made in a reactive environment, whereas the driver performs in a non-reactive one. For this reason, the driver’s problem is essentially a problem of prediction while the airline’s problem has an additional element of control. In the decision problems we consider, an agent has to repeatedly choose currently feasible actions. The agent then observes a reward, which depends both on the chosen action and the current state of the environment. The state of the environment may depend both on the agent’s past choices and on choices made by the environment independent of the agent’s current choice. There are various known approaches to sequential decision making under uncertainty. In this paper we focus on the so-called experts algorithm approach. An “expert” (or “oracle”) is simply a particular strategy recommending actions based on the past history of the process. An experts algorithm is a method that combines the recommendations of several given “experts” (or “oracles”) into another strategy of choosing actions (e.g., [4, 1, 3]). Many learning algorithms can be interpreted as “exploration-exploitation” methods. Roughly speaking, such algorithms blend choices of exploration, aimed at acquiring knowledge, and exploitation that capitalizes on gained knowledge to accumulate rewards. In particular, some experts algorithms can be interpreted as blending the testing of all experts and following those experts that observed to be more rewarding. Our previous paper [2] presented a specific exploration-exploitation experts algorithm. The reader is referred to [2] for more definitions, examples and discussion. That algorithm was designed especially for learning in reactive environments. The difference between our algorithm and previous experts algorithms is that our algorithm tests each expert for multiple consecutive stages of the decision process, in order to acquire knowledge about how the environment reacts to the expert. We pointed out that the “Minimum Regret” criterion often used for evaluating experts algorithms was not suitable for reactive environments, since it ignored the possibility that different experts may induce different states of the environment. The previous paper, however, did not attempt to optimize the exploration-exploitation tradeoff. It rather focused on one particular possibility, which was shown to perform in the long-run as well as the best expert. In this paper, we present a more general exploration-exploitation experts method and provide results about the convergence of several of its variants. We develop performance guarantees showing that the method achieves average payoff comparable to that achieved by the best expert. We characterize convergence rates that hold both in expected value and with high probability. We also introduce a definition for the long-term value of an expert, which captures the reactions of the environment to the expert’s actions, as well as the fact that any learning algorithm commits mistakes. Finally, we characterize how fast the method learns the value of each expert. An important aspect of our results is that they provide an explicit characterization of the tradeoff between exploration and exploitation. The paper is organized as follows. The method is described in section 2. Convergence rates based on actual expert performance are presented in section 3. In section 4, we define the experts’ long-rum values, whereas in section 5 we address the question of how fast the method learns the values of the experts. Finally, in section 6 we analyze various explorations schemes. These results assume that the number of stages. 2 The Exploration-Exploitation Method The problem we consider in this paper can be described as follows. At times t = 1, 2, . . ., an agent has to choose actions at ∈A. At the same times the environment also “chooses” bt ∈B, and then the agent receives a reward R(at, bt). The choices of the environment may depend on various factors, including the past choices of the agent. As in the particular algorithm of [2], the general method follows chosen experts for multiple stages rather than picking a different expert each time. A maximal set of consecutive stages during which the same expert is followed is called a phase. Phase numbers are denoted by i, The number of phases during which expert e has been followed is denoted by Ne, the total number of stages during which expert e has been followed is denoted by Se, and the average payoff from phases in which expert e has been followed is denoted by Me. The general method is stated as follows. • Exploration. An exploration phase consists of picking a random expert e (i.e., from the uniform distribution over {1, . . . , r}), and following e’s recommendations for a certain number of stages depending on the variant of the method. • Exploitation. An exploitation phase consists of picking an expert e with maximum Me, breaking ties at random, and following e’s recommendations for a certain number of stages depending on the variant of the method. A general Exploration-Exploitation Experts Method: 1. Initialize Me = Ne = Se = 0 (e = 1, . . . , r) and i = 1. 2. With probability pi, perform an exploration phase, and with probability 1 −pi perform an exploitation phase; denote by e the expert chosen to be followed and by n the number of stages chosen for the current phase. 3. Follow expert e’s instructions for the next n stages. Increment Ne = Ne + 1 and update Se = Se + n. Denote by ˜R the average payoff accumulated during the current phase of n stages and update Me = Me + n Se ( ˜R −Me) . 4. Increment i = i + 1 and go to step 2. We denote stage numbers by s and phase numbers by i. We denote by M1(i), . . . , Mr(i) the values of the registers M1, . . . , Mr, respectively, at the end of phase i. Similarly, we denote by N1(i), . . . , Nr(i) the values of the registers N1, . . . , Nr, respectively, and by S1(i), . . . , Sr(i) the values of the registers S1, . . . , Sr, respectively, at the end of phase i. In sections 3 and 5, we present performance bounds for the EEE method when the length of the phase is n = Ne. In section 6.4 we consider the case where n = L for a fixed L. Due to space limitations, proofs are omitted and can be found in the online appendix CITE. 3 Bounds Based on Actual Expert Performance The original variant of the EEE method [2] used pi = 1/i and n = Ne. The following was proven: Pr ³ lim inf s→∞M(s) ≥max e lim inf i→∞Me(i) ´ = 1 . (1) In words, the algorithm achieves asymptotically an average reward that is as large as that of the best expert. In this section we generalize this result. We present several bounds characterizing the relationship between M(i) and Me(i). These bounds are valuable in several ways. First, they provide worst-case guarantees about the performance of the EEE method. Second, they provide a starting point for analyzing the behavior of the method under various assumptions about the environment. Third, they quantify the relationship between amount of exploration, represented by the exploration probabilities pi, and the loss of performance. Together with the analysis of Section 5, which characterizes how fast the EEE method learns the value of each expert, the bounds derived here describe explicitly the tradeoff between exploration and exploitation. We denote by Zej the event “phase j performs exploration with expert e,” and let Zj = P e Zej and ¯Zi0i = E h i X j=i0+1 Zj i = i X j=i0+1 pi . Note that ¯Zi0i denotes the expected number of exploration phases between phases i0 + 1 and i. The first theorem establishes that, with high probability, after a finite number of iterations, the EEE method performs comparably to the best expert. The performance of each expert is defined as the smallest average reward achieved by that expert in the interval between an (arbitrary) phase i0 and the current phase i. It can be shown via a counterexample that this bound cannot be extended into a (somewhat more natural) comparison between the average reward of the EEE method and the average reward of each expert at iteration i. Theorem 3.1. For all i0, i and ϵ such that ¯Zi0i ≤iϵ2/(4√ru2) −i0ϵ/(4u), Pr ³ M(i) ≤max e min i0+1≤j≤i Me(j) −2ϵ ´ ≤exp ( −1 2i µ iϵ2 4√ru2 −i0ϵ 4u −¯Zi0i ¶2) . The following theorem characterizes the expected difference between the average reward of EEE method and that of the best expert. Theorem 3.2. For all i0 ≤i and ϵ > 0, E h M(i) −max e min i0+1≤j≤i Me(i) i ≥−ϵ −u i0(i0 + 1) i (i/r + 1) −2u µ3u + 2ϵ ϵ ¶2 ¯Zi0i i . It follows from Theorem 3.1 that, under certain assumptions on the exploration probabilities, the EEE method performs asymptotically at least as well as the expert that did best. Corollary 3.1 generalizes the asymptotic result established in [2]. Corollary 3.1. If limi→∞¯Z0i/i = 0, then Pr ³ lim inf s→∞M(s) ≥max e lim inf i→∞Me(i) ´ = 1 . (2) Note that here the average reward obtained by the EEE method is compared with the reward actually achieved by each expert during the same run of the method. It does not have any implication on the behavior of Me(i), which is analyzed in the next section. 4 The Value of an Expert In this section we analyze the behavior of the average reward Me(i) that is computed by the EEE method for each expert e. This average reward is also used by the method to intuitively estimate the value of expert e. So, the question is whether the EEE method is indeed capable of learning the value of the best experts. Thus, we first discuss what is a “learnable value” of an expert. This concept is not trivial especially when the environment is reactive. The obvious definition of a value as the expected average reward the expert could achieve, if followed exclusively, does not work. The previous paper presented an example (see Section 4 in [2]) of a repeated Matching Pennies game, which proved this impossibility. That example shows that an algorithm that attempts to learn what an expert would achieve, if played exclusively, cannot avoid committing fatal “mistakes.” In certain environments, every non-trivial learning algorithm must commit such fatal mistakes. Hence, such mistakes cannot, in general, be considered necessarily a weakness of the algorithm. A more realistic concept of value, relative to a certain environment policy π, is defined as follows, using a real parameter τ. Definition 4.1. (i) Achievable τ-Value. A real µ is called an achievable τ-value for expert e against an environment policy π, if there exists a constant cτ ≥0 such that, for every stage s0, every possible history hs0 at stage s0 and any number of stages s, E h 1 s Ps0+s s=s0+1R(ae(s), b(s)) : ae(s) ∼σe(hs), b(s) ∼π(hs) i ≥µ −cτ sτ . (ii) τ-Value. The τ-value µτ e of expert e with respect to π is the largest achievable τ-value of e: µτ e = sup{ µ : µ is an achievable τ-value} . (3) In words, a value µ is achievable by expert e if the expert can secure an expected average reward during the s stages, between stage s0 and stage s0 + s, which is asymptotically at least as much as µ, regardless of the history of the play prior to stage s0. In [2], we introduced the notion of flexibility as a way of reasoning about the value of an expert and when it can be learned. The τ-value can be viewed as a relaxation of the previous assumptions and hence the results here strengthen those of [2]. We note, however, that flexibility does hold when the environment reacts with bounded memory or as a finite automaton. 5 Bounds Based on Expected Expert Performance In this section we characterize how fast the EEE method learns the τ-value of each expert. We can derive the rate at which the average reward achieved by the EEE method approaches the τ-value of the best expert. Theorem 5.1. Denote ¯τ = min(τ, 1). For all ϵ > 0 and i, if 4r 3 µ 4cτ ϵ(2 −¯τ) ¶1/¯τ ≤¯Z0i , then Pr ³ inf j≥i Me(j) < µτ e −ϵ ´ ≤33u2 ϵ2 exp µ −ϵ2 ¯Z0i 43u2r ¶ . Note from the definition of τ-values that we can only expect the average reward of expert e to be close to µτ e if the phase lengths when the expert is chosen are sufficiently large. This is necessary to ensure that the bias term cτ/sτ, present in the definition of the τ-value, is small. The condition on ¯Z0i reflects this observation. It ensures that each expert is chosen sufficiently many phases; since phase lengths grow proportionally to the number of phases an expert is chosen, this implies that phase lengths are large enough. We can combine Theorems 3.1 and 5.1 to provide an overall bound on the difference of the average reward achieved by the EEE method and the τ-value of the best expert. Corollary 5.1. For all ϵ > 0, i0 and i, if (i) 4r 3 µ 4cτ ϵ(2 −¯τ) ¶1/¯τ ≤¯Z0i0 , and (ii) ¯Zi0i ≤ iϵ2 4√ru2 −i0ϵ 4u , then Pr ³ M(i) ≤max e µτ e −3ϵ ´ ≤33u2 ϵ2 exp µ −ϵ2 ¯Z0i0 43u2r ¶ + exp ( −1 2i µ iϵ2 4√ru2 −i0ϵ 4u −¯Zi0i ¶2) . (4) Corollary 5.1 explicitly quantifies the tradeoff between exploration and exploitation. In particular, one would like to choose pj such that ¯Z0i0 is large enough to make the first term in the bound small, and ¯Zi0i as small as possible. In Section 6, we analyze several exploration schemes and their effect on the convergence rate of the EEE method. Here we can also derive from Theorems 3.1 and 5.1 asymptotic guarantees for the EEE method. Corollary 5.2. If limi→∞¯Z0i = ∞, then Pr (lim infi→∞Me(i) ≥µτ e) = 1. The following is an immediate result from Corollaries 3.1 and 5.2: Corollary 5.3. If limi→∞¯Z0i = ∞and limi→∞¯Z0i/i = 0, then Pr ³ lim inf i→∞M(i) ≥max e µτ e ´ = 1 . 6 Exploration Schemes The results of the previous sections hold under generic choices of the probabilities pi. Here, we discuss how various particular choices affect the speed of exploiting accumulated information, gathering new information and adapting to changes in the environment. 6.1 Explore-then-Exploit One approach to determining exploration schemes is to minimize the upper bound provided in Corollary 5.1. This gives rise to a scheme where the whole exploration takes place before any exploitation. Indeed, according to expression (4), for any fixed number of iterations i, it is optimal to let ¯Z0i0 = i0 (i.e., pj = 1 for all j ≤i0) and ¯Zi0i = 0 (i.e., pj = 0 for all j > i0). Let U denote the upper bound given by (4). It can be shown that the smallest number of phases i, such that U ≤β, is bounded between two polynomials in 1/ϵ, u, and r. Moreover, its dependence on the the total number of experts r is asymptotically O(r1.5). The main drawback of explore-then-exploit is its inability to adapt to changes in the policy of the environment — since the whole exploration occurs first, any change that occurs after exploration has ended cannot be learned. Moreover, the choice of the last exploration phase i0 depends on parameters of the problem that may not be observable. Finally, it requires fixing β and ϵ a priori, and can only achieve optimality within these tolerance parameters. 6.2 Polynomially Decreasing Exploration In [2] asymptotic results were described that were equivalent to Corollaries 3.1 and 5.3 when pj = 1/j. This choice of exploration probabilities satisfies lim i→∞ ¯Z0i = ∞and lim i→∞ ¯Z0i/i = 0 , so the corollaries apply. We have, however, ¯Z0i0 ≤log(i0) + 1 . It follows that the total number of phases required for U to hold grows exponentially in 1/ϵ, u and r. An alternative scheme, leading to polynomial complexity, can be developed by choosing pj = j−α, for some α ∈(0, 1). In this case, ¯Z0i0 ≥(i0 + 1)1−α 1 −α −1 and ¯Z0i ≤i1−α 1 −α . It follows that the smallest number of phases that guarantees that U ≤β is on the order of i = O à max " u 3−α 1−α r 3−α 2(1−α) ϵ 3−α 1−α µ log u2 ϵ2β ¶ 1 1−α , u 2 α r 1 2α ϵ 2 α #! . 6.3 Constant-Rate Exploration The previous exploration schemes have the property that the frequency of exploration vanishes as the number of phases grows. This property is required in order to achieve the asymptotic optimality results described in Corollaries 3.1 and 5.3. However, it also makes the EEE method increasingly slower in tracking changes in the policy of the environment. An alternative approach is to use a constant frequency pj = η ∈(0, 1) of exploration. Constant-rate exploration does not satisfy the conditions of Corollaries 3.1 and 5.3. However, for any given tolerance level ϵ, the value of η can be chosen so that Pr ³ lim inf i→∞M(i) ≥max e µτ e −ϵ ´ = 1 . Moreover, constant-rate exploration yields complexity results similar to those of the explore-then-exploit scheme. For example, given any tolerance level ϵ, if pj = ηϵ2 8√ru2 (j = 1, 2, . . .) ; then it follows that U ≤β if the number of phases i is on the order of i = O µr2u5 ϵ5 log u2 ϵ2β ¶ . 6.4 Constant Phase Lengths In all the variants of the EEE method considered so far, the number of stages per phase increases linearly as a function of the number of phases during which the same expert has been followed previously. This growth is used to ensure that, as long as the policy of the environment exhibits some regularity, that regularity is captured by the algorithm. For instance, if that policy is cyclic, then the EEE method correctly learns the long-term value of each expert, regardless of the lengths of the cycles. For practical purposes, it may be necessary to slow down the growth of phase lengths in order to get some meaningful results in reasonable time. In this section, we consider the possibility of a constant number L of stages in each phase. Following the same steps that we took to prove Theorems 3.1, 3.2 and 5.1, we can derive the following results: Theorem 6.1. If the EEE method is implemented with phases of fixed length L, then for all i0, i, and ϵ, such that ¯Zi0i ≤ iϵ2 2u2 −i0ϵ 2u , the following bound holds: Pr ³ M(i) ≤max e min i0+1≤j≤i Me(j) −2ϵ ´ ≤exp ( −1 2i µ iϵ2 2u2 −i0ϵ 2u −¯Zi0i ¶2) . We can also characterize the expected difference between the average reward of EEE method and that of the best expert. Theorem 6.2. If the EEE method is implemented with phases of fixed length L, then for all i0 ≤i and ϵ > 0, E h M(i) −max e min i0+1≤j≤i Me(i) i ≥−ϵ −ui0 i −2u2 ϵ ¯Zi0i i . Theorem 6.3. If the EEE method is implemented with phases of fixed length L ≥2, then for all ϵ > 0, Pr ³ inf j≥i Me(j) < µτ e −cτ Lτ −ϵ ´ ≤2L2u2 ϵ2 · exp µ −ϵ2 ¯Z0i 4L2u2r ¶ . An important qualitative difference between fixed-length phases and increasing-length ones is the absence of the number of experts r in the bound given in Theorem 6.2. This implies that, in the explore-then-exploit or constant-rate exploration schemes, the algorithm requires a number of phases which grows only linearly with r to ensure that Pr(M(i) ≤max e M τ e −c/Lτ −ϵ) ≤β . Note, however, that we cannot ensure performance better than maxe µτ e −cτ/Lτ. References [1] Auer, P., Cesa-Bianchi, N., Freund, Y. and Schapire, R.E. (1995) Gambling in a rigged casino: The adversarial multi-armed bandit problem. In Proc. 36th Annual IEEE Symp. on Foundations of Computer Science, pp. 322–331, Los Alamitos, CA: IEEE Computer Society Press. [2] de Farias, D. P. and Megiddo, N. (2004) How to Combine Expert (and Novice) Advice when Actions Impact the Environment. In Advances in Neural Information Processing Systems 16, S. Thrun, L. Saul and B. Sch¨olkopf, Eds., Cambridge, MA:MIT Press. http://books.nips.cc/papers/files/nips16/NIPS2003 CN09.pdf [3] Freund, Y. and Schapire, R.E. (1999) Adaptive game playing using multiplicative weights. Games and Economic Behavior 29:79–103. [4] Littlestone, N. and Warmuth, M.K. (1994) The weighted majority algorithm. Information and Computation 108 (2):212–261.
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Unsupervised Variational Bayesian Learning of Nonlinear Models Antti Honkela and Harri Valpola Neural Networks Research Centre, Helsinki University of Technology P.O. Box 5400, FI-02015 HUT, Finland {Antti.Honkela, Harri.Valpola}@hut.fi http://www.cis.hut.fi/projects/bayes/ Abstract In this paper we present a framework for using multi-layer perceptron (MLP) networks in nonlinear generative models trained by variational Bayesian learning. The nonlinearity is handled by linearizing it using a Gauss–Hermite quadrature at the hidden neurons. This yields an accurate approximation for cases of large posterior variance. The method can be used to derive nonlinear counterparts for linear algorithms such as factor analysis, independent component/factor analysis and state-space models. This is demonstrated with a nonlinear factor analysis experiment in which even 20 sources can be estimated from a real world speech data set. 1 Introduction Linear latent variable models such as factor analysis, principal component analysis (PCA) and independent component analysis (ICA) [1] are used in many applications ranging from engineering to social sciences and psychology. In many of these cases, the effect of the desired factors or sources to the observed data is, however, not linear. A nonlinear model could therefore produce better results. The method presented in this paper can be used as a basis for many nonlinear latent variable models, such as nonlinear generalizations of the above models. It is based on the variational Bayesian framework, which provides a solid foundation for nonlinear modeling that would otherwise be prone to overfitting [2]. It also allows for easy comparison of different model structures, which is even more important for flexible nonlinear models than for simpler linear models. General nonlinear generative models for data x(t) of the type x(t) = f(s(t), θf) + n(t) = Bφ(As(t) + a) + b + n(t) (1) often employ a multi-layer perceptron (MLP) (as in the equation) or a radial basis function (RBF) network to model the nonlinearity. Here s(t) are the latent variables of the model, n(t) is noise and θf are the parameters of the nonlinearity, in case of MLP the weight matrices A, B and bias vectors a, b. In context of variational Bayesian methods, RBF networks seem more popular of the two because it is easier to evaluate analytic expressions and bounds for certain key quantities [3]. With MLP networks such values are not as easily available and one usually has to resort to numeric approximations. Nevertheless, MLP networks can often, especially for nearly linear models and in high dimensional spaces, provide an equally good model with fewer parameters [4]. This is important with generative models whose latent variables are independent or at least uncorrelated and the intrinsic dimensionality of the input is large. A reasonable approximate bound for a good model is also often better than a strict bound for a bad model. Most existing applications of variational Bayesian methods for nonlinear models are concerned with the supervised case where the inputs of the network are known and only the weights have to be learned [3, 5]. This is easier as there are fewer parameters with related posterior variance above the nonlinear hidden layer and the distributions thus tend to be easier to handle. In this paper we present a novel method for evaluating the statistics of the outputs of an MLP network in context of unsupervised variational Bayesian learning of its weights and inputs. The method is demonstrated with a nonlinear factor analysis problem. The new method allows for reliable estimation of a larger number of factors than before [6,7]. 2 Variational learning of unsupervised MLPs Let us denote the observed data by X = {x(t)|t}, the latent variables of the model by S = {s(t)|t} and the model parameters by θ = (θi). The nonlinearity (1) can be used as a building block of many different models depending on the model assumed for the sources S. Simple Gaussian prior on S leads to a nonlinear factor analysis (NFA) model [6,7] that is studied here because of its simplicity. The method could easily be extended with a mixture-of-Gaussians prior on S [8] to get a nonlinear independent factor analysis model, but this is omitted here. In many nonlinear blind source separation (BSS) problems it is enough to apply simple NFA followed by linear ICA postprocessing to achieve nonlinear BSS [6, 7]. Another possible extension would be to include dynamics for S as in [9]. In order to deal with the flexible nonlinear models, a powerful learning paradigm resistant to overfitting is needed. The variational Bayesian method of ensemble learning [2] has proven useful here. Ensemble learning is based on approximating the true posterior p(S, θ|X) with a tractable approximation q(S, θ), typically a multivariate Gaussian with a diagonal covariance. The approximation is fitted to minimize the cost C = log q(S, θ) p(S, θ, X) = D(q(S, θ)||p(S, θ|X)) −log p(X) (2) where ⟨·⟩denotes expectation over q(S, θ) and D(q||p) is the Kullback-Leibler divergence between q and p. As the Kullback-Leibler divergence is always non-negative, C yields an upper bound for −log p(X) and thus a lower bound for the evidence p(X). The cost can be evaluated analytically for a large class of mainly linear models [10,11] leading to simple and efficient learning algorithms. 2.1 Evaluating the cost Unfortunately, the cost (2) cannot be evaluated analytically for the nonlinear model (1). Assuming a Gaussian noise model, the likelihood term of C becomes Cx = ⟨−log p(X|S, θ)⟩= X t ⟨−log N(x(t); f(s(t), θf), Σx)⟩. (3) The term Cx depends on the first and second moments of f(s(t), θf) over the posterior approximation q(S, θ), and they cannot easily be evaluated analytically. Assuming the noise covariance is diagonal, the cross terms of the covariance of the output are not needed, only the scalar variances of the different components. If the activation functions of the MLP network were linear, the output mean and variance could be evaluated exactly using only the mean and variance of the inputs s(t) and θf. Thus a natural first approximation would be to linearize the network about the input mean using derivatives [6]. Taking the derivative with respect to s(t), for instance, yields ∂f(s(t), θf) ∂s(t) = B diag(φ′(y(t))) A, (4) where diag(v) denotes a diagonal matrix with elements of vector v on the main diagonal and y(t) = As(t) + a. Due to the local nature of the approximation, this can lead to severe underestimation of the variance, especially when the hidden neurons of the MLP network operate in the saturated region. This makes the nonlinear factor analysis algorithm using this approach unstable with large number of factors because the posterior variance corresponding to the last factors is typically large. To avoid this problem, we propose using a Gauss–Hermite quadrature to evaluate an effective linearization of the nonlinear activation functions φ(yi(t)). The Gauss– Hermite quadrature is a method for approximating weighted integrals Z ∞ −∞ f(x) exp(−x2) dx ≈ X k wk f(tk), (5) where the weights wk and abscissas tk are selected by requiring exact result for suitable number of low-order polynomials. This allows evaluating the mean and variance of φ(yi(t)) by quadratures φ(yi(t))GH = X k w′ kφ yi(t) + t′ k p eyi(t) (6) eφ(yi(t))GH = X k w′ k h φ yi(t) + t′ k p eyi(t) −φ(yi(t))GH i2 , (7) respectively. Here the weights and abscissas have been scaled to take into account the Gaussian pdf weight instead of exp(−x2), and yi(t) and eyi(t) are the mean and variance of yi(t), respectively. We used a three point quadrature that yields accurate enough results but can be evaluated quickly. Using e.g. five points improves the accuracy slightly, but slows the computation down significantly. As both of the quadratures depend on φ at the same points, they can be evaluated together easily. Using the approximation formula eφ(yi(t)) = φ′(yi(t))2eyi(t), the resulting mean and variance can be interpreted to yield an effective linearization of φ(yi(t)) through ⟨φ(yi(t))⟩:= φ(yi(t))GH ⟨φ′(yi(t))⟩:= s eφ(yi(t))GH eyi(t) . (8) The positive square root is used here because the derivative of the logistic sigmoid used as activation function is always positive. Using these to linearize the MLP as in Eq. (4), the exact mean and variance of the linearized model can be evaluated in a relatively straightforward manner. Evaluation of the variance due to the sources requires propagating matrices through the network to track the correlations between the hidden units. Hence the computational complexity depends quadratically on the number of sources. The same problem does not affect the network weights as each parameter only affects the value of one hidden neuron. 2.2 Details of the approximation The mean and variance of φ(yi(t)) depend on the distribution of yi(t). The Gauss– Hermite quadrature assumes that yi(t) is Gaussian. This is not true in our case, as the product of two independent normally distributed variables aij and sj(t) is super-Gaussian, although rather close to Gaussian if the mean of one of the variables is significantly larger in absolute value than the standard deviation. In case of N sources, the actual input yi(t) is a sum of N of these and a Gaussian variable and therefore rather close to a Gaussian, at least for larger values of N. Ignoring the non-Gaussianity, the quadrature depends on the mean and variance of yi(t). These can be evaluated exactly because of the linearity of the mapping as eyi,tot(t) = X j eAij(sj(t)2 + esj(t)) + A 2 ijesj(t) + eai, (9) where θ denotes the mean and eθ the variance of θ. Here it is assumed that the posterior approximations q(S) and q(θf) have diagonal covariances. Full covariances can be used instead without too much difficulty, if necessary. In an experiment investigating the approximation accuracy with a random MLP [12], the Taylor approximation was found to underestimate the output variance by a factor of 400, at worst. The worst case result of the above approximation was underestimation by a factor of 40, which is a great improvement over the Taylor approximation, but still far from perfect. The worst case behavior could be improved to underestimation by a factor of 5 by introducing another quadrature evaluated with a different variance for yi(t). This change cannot be easily justified except by the fact that it produces better results. The difference in behavior of the two methods in more realistic cases is less drastic, but the version with two quadratures seems to provide more accurate approximations. The more accurate approximation is implemented by evaluating another quadrature using the variance of yi(t) originating mainly from θf, eyi,weight(t) = X j eAij(sj(t)2 + esj(t)) + eai, (10) and using the implied ⟨φ′(yi(t))⟩in the evaluation of the effects of these variances. The total variance (9) is still used in evaluation of the means and the evaluation of the effects of the variance of s(t). 2.3 Learning algorithm for nonlinear factor analysis The nonlinear factor analysis (NFA) model [6] is learned by numerically minimizing the cost C evaluated above. The minimization algorithm is a combination of conjugate gradient for the means of S and θf, fixed point iteration for the variances of S and θf, and EM like updates for other parameters and hyperparameters. The fixed point update algorithm for the variances follows from writing the cost function as a sum C = Cq + Cp = ⟨log q(S, θ)⟩+ ⟨−log p(S, θ, X)⟩. (11) A parameter θi that is assumed independent of others under q and has a Gaussian posterior approximation q(θi) = N(θi; θi, eθi), only affects the corresponding negentropy term −1/2 log(2πeeθi) in Cq. Differentiating this with respect to eθi and setting the result to zero leads to a fixed point update rule eθi = 2∂Cp/∂eθi −1 . In order to get a stable update algorithm for the variances, dampening by halving the step on log scale until the cost function does not increase must be added to the fixed point updates. The variance is increased at most by 10 % on one iteration and not set to a negative value even if the gradient is negative. The required partial derivatives can be evaluated analytically with simple backpropagation like computations with the MLP network. The quadratures used at hidden nodes lead to analytical expressions for the means and variances of the hidden nodes and the corresponding feedback gradients are easy to derive. Along with the derivatives with respect to variances, it is easy to evaluate them with respect to means of the same parameters. These derivatives can then be used in a conjugate gradient algorithm to update the means of S and θf. Due to the flexibility of the MLP network and the gradient based learning algorithm, the nonlinear factor analysis method is sensitive to the initialization. We have used linear PCA for initialization of the means of the sources S. The means of the weights θf are initialized randomly while all the variances are initialized to small constant values. After this, the sources are kept fixed for 20 iterations while only the network weights are updated. The hyperparameters governing noise and parameter distributions are only updated after 80 more iterations to update the sources and the MLP. By that time, a reasonable model of the data has been learned and the method is not likely to prune away all the sources and other parameters as unnecessary. 2.4 Other approximation methods Another way to get a more robust approximation for the statistics of f would be to use the deterministic sampling approach used in unscented transform [13] and consecutively in different unscented algorithms. Unfortunately this approach does not work very well in high dimensional cases. The unscented transform also ignores all the prior information on the form of the nonlinearity. In case of the MLP network, everything except the scalar activation functions is known to be linear. All information on the correlations of variables is also ignored, which leads to loss of accuracy when the output depends on products of input variables like in our case. In an experiment of mean and log-variance approximation accuracy with a relatively large random MLP [12], the unscented transform needed over 100 % more time to achieve results with 10 times the mean squared error of the proposed approach. Part of our problem was also faced by Barber and Bishop in their work on ensemble learning for supervised learning of MLP networks [5]. In their work the inputs s(t) of the network are part of the data and thus have no associated variance. This makes the problem easier as the inputs y(t) of the hidden neurons are Gaussian. By using the cumulative Gaussian distribution or the error function erf as the activation function, the mean of the outputs of the hidden neurons and thus of the outputs of the whole network can be evaluated analytically. The covariances still need to be evaluated numerically, and that is done by evaluating all the correlations of the hidden neurons separately. In a network with H hidden neurons, this requires O(H2) quadrature evaluations. In our case the inputs of the hidden neurons are not Gaussian and hence even the error function as the activation function would not allow for exact evaluation of the means. This is why we have decided to use the standard logistic sigmoid activation function in form of tanh which is more common and faster to evaluate numerically. In our approach all the required means and variances can be evaluated with O(H) quadratures. 3 Experiments The proposed nonlinear factor analysis method was tested on natural speech data set consisting of spectrograms of 24 individual words of Finnish speech, spoken by 20 different speakers. The spectra were modified to mimic the reception abilities of the human ear. This is a standard preprocessing procedure for speech recognition. No speaker or word information was used in learning, the spectrograms of different words were simply blindly concatenated. The preprocessed data consisted of 2547 30-dimensional spectrogram vectors. The data set was tested with two different learning algorithms for the NFA model, one based on the Taylor approximation introduced in [6] and another based on the proposed approximation. Contrary to [6], the algorithm based on Taylor approximation used the same conjugate gradient based optimization algorithm as the new approximation. This helped greatly in stabilizing the algorithm that used to be rather unstable with high source dimensionalities due to sensitivity of the Taylor approximation in regions where it is not really valid. Both algorithms were tested using 1 to 20 sources, each number with four different random initializations for the MLP network weights. The number of hidden neurons in the MLP network was 40. The learning algorithm was run for 2000 iterations.1 45 50 55 45 50 55 Reference cost (nats / sample) Proposed cost (nats / sample) 45 50 55 45 50 55 Reference cost (nats / sample) Taylor cost (nats / sample) Figure 1: The attained values of C in different simulations as evaluated by the different approximations plotted against reference values evaluated by sampling. The left subfigure shows the values from experiments using the proposed approximation and the right subfigure from experiments using the Taylor approximation. Fig. 1 shows a comparison of the cost function values evaluated by the different approximations and a reference value evaluated by sampling. The reference cost values were evaluated by sampling 400 points from the distribution q(S, θf), evaluating f(s, θf) at those points, and using the mean and variance of the output points in the cost function evaluation. The accuracy of the procedure was checked by performing the evaluation 100 times for one of the simulations. The standard deviation of the values was 5·10−3 nats per sample which should not show at all in the figures. The unit nat here signifies the use of natural logarithm in Eq. (2). The results in Fig. 1 show that the proposed approximation yields consistently very 1The Matlab code used in the experiments is available at http://www.cis.hut.fi/ projects/bayes/software/. 5 10 15 20 44 46 48 50 52 54 56 # of sources Cost function value (nats / sample) Proposed approximation Reference value 5 10 15 20 44 46 48 50 52 54 56 # of sources Cost function value (nats / sample) Taylor approximation Reference value Figure 2: The attained value of C in simulations with different numbers of sources. The values shown are the means of 4 simulations with different random initializations. The left subfigure shows the values from experiments using the proposed approximation and the right subfigure from experiments using the Taylor approximation. Both values are compared to reference values evaluated by sampling. reliable estimates of the true cost, although it has a slight tendency to underestimate it. The older Taylor approximation [6] breaks down completely in some cases and reports very small costs even though the true value can be significantly larger. The situations where the Taylor approximation fails are illustrated in Fig. 2, which shows the attained cost as a function of number of sources used. The Taylor approximation shows a decrease in cost as the number of the sources increases even though the true cost is increasing rapidly. The behavior of the proposed approximation is much more consistent and qualitatively correct. 4 Discussion The problem of estimating the statistics of a nonlinear transform of a probability distribution is also encountered in nonlinear extensions of Kalman filtering. The Taylor approximation corresponds to extended Kalman filter and the new approximation can be seen as a modification of it with a more accurate linearization. This opens up many new potential applications in time series analysis and elsewhere. The proposed method is somewhat similar to unscented Kalman filtering based on the unscented transform [13], but much better suited for high dimensional MLP-like nonlinearities. This is not very surprising, as worst case complexity of general Gaussian integration is exponential with respect to the dimensionality of the input [14] and unscented transform as a general method with linear complexity is bound to be less accurate in high dimensional problems. In case of the MLP, the complexity of the unscented transform depends on the number of all weights, which in our case with 20 sources can be more than 2000. 5 Conclusions In this paper we have proposed a novel approximation method for unsupervised MLP networks in variational Bayesian learning. The approximation is based on using numerical Gauss–Hermite quadratures to evaluate the global effect of the nonlinear activation function of the network to produce an effective linearization of the MLP. The statistics of the outputs of the linearized network can be evaluated exactly to get accurate and reliable estimates of the statistics of the MLP outputs. These can be used to evaluate the standard variational Bayesian ensemble learning cost function C and numerically minimize it using a hybrid fixed point / conjugate gradient algorithm. We have demonstrated the method with a nonlinear factor analysis model and a real world speech data set. It was able to reliably estimate all the 20 factors we attempted from the 30-dimensional data set. The presented method can be used together with linear ICA for nonlinear BSS [7], and the approximation can be easily applied to more complex models such as nonlinear independent factor analysis [6] and nonlinear state-space models [9]. Acknowledgments The authors wish to thank David Barber, Markus Harva, Bert Kappen, Juha Karhunen, Uri Lerner and Tapani Raiko for useful comments and discussions. This work was supported in part by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778. This publication only reflects the authors’ views. References [1] A. Hyv¨arinen, J. Karhunen, and E. Oja. Independent Component Analysis. J. Wiley, 2001. [2] G. E. Hinton and D. van Camp. Keeping neural networks simple by minimizing the description length of the weights. In Proc. of the 6th Ann. ACM Conf. on Computational Learning Theory, pp. 5–13, Santa Cruz, CA, USA, 1993. [3] P. Sykacek and S. Roberts. Adaptive classification by variational Kalman filtering. In Advances in Neural Information Processing Systems 15, pp. 753–760. MIT Press, 2003. [4] S. Haykin. Neural Networks – A Comprehensive Foundation, 2nd ed. Prentice-Hall, 1999. [5] D. Barber and C. Bishop. Ensemble learning for multi-layer networks. In Advances in Neural Information Processing Systems 10, pp. 395–401. MIT Press, 1998. [6] H. Lappalainen and A. Honkela. Bayesian nonlinear independent component analysis by multi-layer perceptrons. In M. Girolami, ed., Advances in Independent Component Analysis, pp. 93–121. Springer-Verlag, Berlin, 2000. [7] H. Valpola, E. Oja, A. Ilin, A. Honkela, and J. Karhunen. Nonlinear blind source separation by variational Bayesian learning. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E86-A(3):532–541, 2003. [8] H. Attias. Independent factor analysis. Neural Computation, 11(4):803–851, 1999. [9] H. Valpola and J. Karhunen. An unsupervised ensemble learning method for nonlinear dynamic state-space models. Neural Computation, 14(11):2647–2692, 2002. [10] H. Attias. A variational Bayesian framework for graphical models. In Advances in Neural Information Processing Systems 12, pp. 209–215. MIT Press, 2000. [11] Z. Ghahramani and M. Beal. Propagation algorithms for variational Bayesian learning. In Advances in Neural Information Processing Systems 13, pp. 507–513. MIT Press, 2001. [12] A. Honkela. Approximating nonlinear transformations of probability distributions for nonlinear independent component analysis. In Proc. 2004 IEEE Int. Joint Conf. on Neural Networks (IJCNN 2004), pp. 2169–2174, Budapest, Hungary, 2004. [13] S. Julier and J. K. Uhlmann. A general method for approximating nonlinear transformations of probability distributions. Technical report, Robotics Research Group, Department of Engineering Science, University of Oxford, 1996. [14] F. Curbera. Delayed curse of dimension for Gaussian integration. Journal of Complexity, 16(2):474–506, 2000.
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Bayesian Surprise Attracts Human Attention Laurent Itti Department of Computer Science University of Southern California Los Angeles, California 90089-2520, USA itti@usc.edu Pierre Baldi Department of Computer Science University of California, Irvine Irvine, California 92697-3425, USA pfbaldi@ics.uci.edu Abstract The concept of surprise is central to sensory processing, adaptation, learning, and attention. Yet, no widely-accepted mathematical theory currently exists to quantitatively characterize surprise elicited by a stimulus or event, for observers that range from single neurons to complex natural or engineered systems. We describe a formal Bayesian definition of surprise that is the only consistent formulation under minimal axiomatic assumptions. Surprise quantifies how data affects a natural or artificial observer, by measuring the difference between posterior and prior beliefs of the observer. Using this framework we measure the extent to which humans direct their gaze towards surprising items while watching television and video games. We find that subjects are strongly attracted towards surprising locations, with 72% of all human gaze shifts directed towards locations more surprising than the average, a figure which rises to 84% when considering only gaze targets simultaneously selected by all subjects. The resulting theory of surprise is applicable across different spatio-temporal scales, modalities, and levels of abstraction. Life is full of surprises, ranging from a great christmas gift or a new magic trick, to wardrobe malfunctions, reckless drivers, terrorist attacks, and tsunami waves. Key to survival is our ability to rapidly attend to, identify, and learn from surprising events, to decide on present and future courses of action [1]. Yet, little theoretical and computational understanding exists of the very essence of surprise, as evidenced by the absence from our everyday vocabulary of a quantitative unit of surprise: Qualities such as the “wow factor” have remained vague and elusive to mathematical analysis. Informal correlates of surprise exist at nearly all stages of neural processing. In sensory neuroscience, it has been suggested that only the unexpected at one stage is transmitted to the next stage [2]. Hence, sensory cortex may have evolved to adapt to, to predict, and to quiet down the expected statistical regularities of the world [3, 4, 5, 6], focusing instead on events that are unpredictable or surprising. Electrophysiological evidence for this early sensory emphasis onto surprising stimuli exists from studies of adaptation in visual [7, 8, 4, 9], olfactory [10, 11], and auditory cortices [12], subcortical structures like the LGN [13], and even retinal ganglion cells [14, 15] and cochlear hair cells [16]: neural response greatly attenuates with repeated or prolonged exposure to an initially novel stimulus. Surprise and novelty are also central to learning and memory formation [1], to the point that surprise is believed to be a necessary trigger for associative learning [17, 18], as supported by mounting evidence for a role of the hippocampus as a novelty detector [19, 20, 21]. Finally, seeking novelty is a well-identified human character trait, with possible association with the dopamine D4 receptor gene [22, 23, 24]. In the Bayesian framework, we develop the only consistent theory of surprise, in terms of the difference between the posterior and prior distributions of beliefs of an observer over the available class of models or hypotheses about the world. We show that this definition derived from first principles presents key advantages over more ad-hoc formulations, typically relying on detecting outlier stimuli. Armed with this new framework, we provide direct experimental evidence that surprise best characterizes what attracts human gaze in large amounts of natural video stimuli. We here extend a recent pilot study [25], adding more comprehensive theory, large-scale human data collection, and additional analysis. 1 Theory Bayesian Definition of Surprise. We propose that surprise is a general concept, which can be derived from first principles and formalized across spatio-temporal scales, sensory modalities, and, more generally, data types and data sources. Two elements are essential for a principled definition of surprise. First, surprise can exist only in the presence of uncertainty, which can arise from intrinsic stochasticity, missing information, or limited computing resources. A world that is purely deterministic and predictable in real-time for a given observer contains no surprises. Second, surprise can only be defined in a relative, subjective, manner and is related to the expectations of the observer, be it a single synapse, neuronal circuit, organism, or computer device. The same data may carry different amount of surprise for different observers, or even for the same observer taken at different times. In probability and decision theory it can be shown that the only consistent and optimal way for modeling and reasoning about uncertainty is provided by the Bayesian theory of probability [26, 27, 28]. Furthermore, in the Bayesian framework, probabilities correspond to subjective degrees of beliefs in hypotheses or models which are updated, as data is acquired, using Bayes’ theorem as the fundamental tool for transforming prior belief distributions into posterior belief distributions. Therefore, within the same optimal framework, the only consistent definition of surprise must involve: (1) probabilistic concepts to cope with uncertainty; and (2) prior and posterior distributions to capture subjective expectations. Consistently with this Bayesian approach, the background information of an observer is captured by his/her/its prior probability distribution {P(M)}M∈M over the hypotheses or models M in a model space M. Given this prior distribution of beliefs, the fundamental effect of a new data observation D on the observer is to change the prior distribution {P(M)}M∈M into the posterior distribution {P(M|D)}M∈M via Bayes theorem, whereby ∀M ∈M, P(M|D) = P(D|M) P(D) P(M). (1) In this framework, the new data observation D carries no surprise if it leaves the observer beliefs unaffected, that is, if the posterior is identical to the prior; conversely, D is surprising if the posterior distribution resulting from observing D significantly differs from the prior distribution. Therefore we formally measure surprise elicited by data as some distance measure between the posterior and prior distributions. This is best done using the relative entropy or Kullback-Leibler (KL) divergence [29]. Thus, surprise is defined by the average of the log-odd ratio: S(D, M) = KL(P(M|D), P(M)) = Z M P(M|D) log P(M|D) P(M) dM (2) taken with respect to the posterior distribution over the model class M. Note that KL is not symmetric but has well-known theoretical advantages, including invariance with respect to Figure 1: Computing surprise in early sensory neurons. (a) Prior data observations, tuning preferences, and top-down influences contribute to shaping a set of “prior beliefs” a neuron may have over a class of internal models or hypotheses about the world. For instance, M may be a set of Poisson processes parameterized by the rate λ, with {P(M)}M∈M = {P(λ)}λ∈IR+∗the prior distribution of beliefs about which Poisson models well describe the world as sensed by the neuron. New data D updates the prior into the posterior using Bayes’ theorem. Surprise quantifies the difference between the posterior and prior distributions over the model class M. The remaining panels detail how surprise differs from conventional model fitting and outlier-based novelty. (b) In standard iterative Bayesian model fitting, at every iteration N, incoming data DN is used to update the prior {P(M|D1, D2, ..., DN−1)}M∈M into the posterior {P(M|D1, D2, ..., DN)}M∈M. Freezing this learning at a given iteration, one then picks the currently best model, usually using either a maximum likelihood criterion, or a maximum a posteriori one (yielding MMAP shown). (c) This best model is used for a number of tasks at the current iteration, including outlier-based novelty detection. New data is then considered novel at that instant if it has low likelihood for the best model (e.g., Db N is more novel than Da N). This focus onto the single best model presents obvious limitations, especially in situations where other models are nearly as good (e.g., M∗in panel (b) is entirely ignored during standard novelty computation). One palliative solution is to consider mixture models, or simply P(D), but this just amounts to shifting the problem into a different model class. (d) Surprise directly addresses this problem by simultaneously considering all models and by measuring how data changes the observer’s distribution of beliefs from {P(M|D1, D2, ..., DN−1)}M∈M to {P(M|D1, D2, ..., DN)}M∈M over the entire model class M (orange shaded area). reparameterizations. A unit of surprise — a “wow” — may then be defined for a single model M as the amount of surprise correspondingto a two-fold variation between P(M|D) and P(M), i.e., as log P(M|D)/P(M) (with log taken in base 2), with the total number of wows experienced for all models obtained through the integration in eq. 2. Surprise and outlier detection. Outlier detection based on the likelihood P(D|Mbest) of D given a single best model Mbest is at best an approximation to surprise and, in some cases, is misleading. Consider, for instance, a case where D has very small probability both for a model or hypothesis M and for a single alternative hypothesis M. Although D is a strong outlier, it carries very little information regarding whether M or M is the better model, and therefore very little surprise. Thus an outlier detection method would strongly focus attentional resources onto D, although D is a false positive, in the sense that it carries no useful information for discriminating between the two alternative hypotheses M and M. Figure 1 further illustrates this disconnect between outlier detection and surprise. 2 Human experiments To test the surprise hypothesis — that surprise attracts human attention and gaze in natural scenes — we recorded eye movements from eight na¨ıve observers (three females and five males, ages 23-32, normal or corrected-to-normal vision). Each watched a subset from 50 videoclips totaling over 25 minutes of playtime (46,489 video frames, 640 × 480, 60.27 Hz, mean screen luminance 30 cd/m2, room 4 cd/m2, viewing distance 80cm, field of view 28◦× 21◦). Clips comprised outdoors daytime and nighttime scenes of crowded environments, video games, and television broadcast including news, sports, and commercials. Right-eye position was tracked with a 240 Hz video-based device (ISCAN RK-464), with methods as previously [30]. Two hundred calibrated eye movement traces (10,192 saccades) were analyzed, corresponding to four distinct observers for each of the 50 clips. Figure 2 shows sample scanpaths for one videoclip. To characterize image regions selected by participants, we process videoclips through computational metrics that output a topographic dynamic master response map, assigning in real-time a response value to every input location. A good master map would highlight, more than expected by chance, locations gazed to by observers. To score each metric we hence sample, at onset of every human saccade, master map activity around the saccade’s future endpoint, and around a uniformly random endpoint (random sampling was repeated 100 times to evaluate variability). We quantify differences between histograms of master Figure 2: (a) Sample eye movement traces from four observers (squares denote saccade endpoints). (b) Our data exhibits high inter-individual overlap, shown here with the locations where one human saccade endpoint was nearby (≈5◦) one (white squares), two (cyan squares), or all three (black squares) other humans. (c) A metric where the master map was created from the three eye movement traces other than that being tested yields an upper-bound KL score, computed by comparing the histograms of metric values at human (narrow blue bars) and random (wider green bars) saccade targets. Indeed, this metric’s map was very sparse (many random saccades landing on locations with nearzero response), yet humans preferentially saccaded towards the three active hotspots corresponding to the eye positions of three other humans (many human saccades landing on locations with near-unity responses). map samples collected from human and random saccades using again the Kullback-Leibler (KL) distance: metrics which better predict human scanpaths exhibit higher distances from random as, typically, observers non-uniformly gaze towards a minority of regions with highest metric responses while avoiding a majority of regions with low metric responses. This approach presents several advantages over simpler scoring schemes [31, 32], including agnosticity to putative mechanisms for generating saccades and the fact that applying any continuous nonlinearity to master map values would not affect scoring. Experimental results. We test six computational metrics, encompassing and extending the state-of-the-art found in previous studies. The first three quantify static image properties (local intensity variance in 16 × 16 image patches [31]; local oriented edge density as measured with Gabor filters [33]; and local Shannon entropy in 16 × 16 image patches [34]). The remaining three metrics are more sensitive to dynamic events (local motion [33]; outlier-based saliency [33]; and surprise [25]). For all metrics, we find that humans are significantly attracted by image regions with higher metric responses. However, the static metrics typically respond vigorously at numerous visual locations (Figure 3), hence they are poorly specific and yield relatively low KL scores between humans and random. The metrics sensitive to motion, outliers, and surprising events, in comparison, yield sparser maps and higher KL scores. The surprise metric of interest here quantifies low-level surprise in image patches over space and time, and at this point does not account for high-level or cognitive beliefs of our human observers. Rather, it assumes a family of simple models for image patches, each processed through 72 early feature detectors sensitive to color, orientation, motion, etc., and computes surprise from shifts in the distribution of beliefs about which models better describe the patches (see [25] and [35] for details). We find that the surprise metric significantly outperforms all other computational metrics (p < 10−100 or better on t-tests for equality of KL scores), scoring nearly 20% better than the second-best metric (saliency) and 60% better than the best static metric (entropy). Surprising stimuli often substantially differ from simple feature outliers; for example, a continually blinking light on a static background elicits sustained flicker due to its locally outlier temporal dynamics but is only surprising for a moment. Similarly, a shower of randomly-colored pixels continually excites all low-level feature detectors but rapidly becomes unsurprising. Strongest attractors of human attention. Clearly, in our and previous eye-tracking experiments, in some situations potentially interesting targets were more numerous than in others. With many possible targets, different observers may orient towards different locations, making it more difficult for a single metric to accurately predict all observers. Hence we consider (Figure 4) subsets of human saccades where at least two, three, or all four observers simultaneously agreed on a gaze target. Observers could have agreed based on bottom-up factors (e.g., only one location had interesting visual appearance at that time), top-down factors (e.g., only one object was of current cognitive interest), or both (e.g., a single cognitively interesting object was present which also had distinctive appearance). Irrespectively of the cause for agreement, it indicates consolidated belief that a location was attractive. While the KL scores of all metrics improved when progressively focusing onto only those locations, dynamic metrics improved more steeply, indicating that stimuli which more reliably attracted all observers carried more motion, saliency, and surprise. Surprise remained significantly the best metric to characterize these agreed-upon attractors of human gaze (p < 10−100 or better on t-tests for equality of KL scores). Overall, surprise explained the greatest fraction of human saccades, indicating that humans are significantly attracted towards surprising locations in video displays. Over 72% of all human saccades were targeted to locations predicted to be more surprising than on average. When only considering saccades where two, three, or four observers agreed on a common gaze target, this figure rose to 76%, 80%, and 84%, respectively. Figure 3: (a) Sample video frames, with corresponding human saccades and predictions from the entropy, surprise, and human-derived metrics. Entropy maps, like intensity variance and orientation maps, exhibited many locations with high responses, hence had low specificity and were poorly discriminative. In contrast, motion, saliency, and surprise maps were much sparser and more specific, with surprise significantly more often on target. For three example frames (first column), saccades from one subject are shown (arrows) with corresponding apertures over which master map activity at the saccade endpoint was sampled (circles). (b) KL scores for these metrics indicate significantly different performance levels, and a strict ranking of variance < orientation < entropy < motion < saliency < surprise < human-derived. KL scores were computed by comparing the number of human saccades landing onto each given range of master map values (narrow blue bars) to the number of random saccades hitting the same range (wider green bars). A score of zero would indicate equality between the human and random histograms, i.e., humans did not tend to hit various master map values any differently from expected by chance, or, the master map could not predict human saccades better than random saccades. Among the six computational metrics tested in total, surprise performed best, in that surprising locations were relatively few yet reliably gazed to by humans. Figure 4: KL scores when considering only saccades where at least one (all 10,192 saccades), two (7,948 saccades), three (5,565 saccades), or all four (2,951 saccades) humans agreed on a common gaze location, for the static (a) and dynamic metrics (b). Static metrics improved substantially when progressively focusing onto saccades with stronger inter-observer agreement (average slope 0.56 ± 0.37 percent KL score units per 1,000 pruned saccades). Hence, when humans agreed on a location, they also tended to be more reliably predicted by the metrics. Furthermore, dynamic metrics improved 4.5 times more steeply (slope 2.44 ± 0.37), suggesting a stronger role of dynamic events in attracting human attention. Surprising events were significantly the strongest (t-tests for equality of KL scores between surprise and other metrics, p < 10−100). 3 Discussion While previous research has shown with either static scenes or dynamic synthetic stimuli that humans preferentially fixate regions of high entropy [34], contrast [31], saliency [32], flicker [36], or motion [37], our data provides direct experimental evidence that humans fixate surprising locations even more reliably. These conclusions were made possible by developing new tools to quantify what attracts human gaze over space and time in dynamic natural scenes. Surprise explained best where humans look when considering all saccades, and even more so when restricting the analysis to only those saccades for which human observers tended to agree. Surprise hence represents an inexpensive, easily computable approximation to human attentional allocation. In the absence of quantitative tools to measure surprise, most experimental and modeling work to date has adopted the approximation that novel events are surprising, and has focused on experimental scenarios which are simple enough to ensure an overlap between informal notions of novelty and surprise: for example, a stimulus is novel during testing if it has not been seen during training [9]. Our definition opens new avenues for more sophisticated experiments, where surprise elicited by different stimuli can be precisely compared and calibrated, yielding predictions at the single-unit as well as behavioral levels. The definition of surprise — as the distance between the posterior and prior distributions of beliefs over models — is entirely general and readily applicable to the analysis of auditory, olfactory, gustatory, or somatosensory data. While here we have focused on behavior rather than detailed biophysical implementation, it is worth noting that detecting surprise in neural spike trains does not require semantic understanding of the data carried by the spike trains, and thus could provide guiding signals during self-organization and development of sensory areas. At higher processing levels, top-down cues and task demands are known to combine with stimulus novelty in capturing attention and triggering learning [1, 38], ideas which may now be formalized and quantified in terms of priors, posteriors, and surprise. Surprise, indeed, inherently depends on uncertainty and on prior beliefs. Hence surprise theory can further be tested and utilized in experiments where the prior is biased, for example by top-down instructions or prior exposures to stimuli [38]. In addition, simple surprise-based behavioral measures such as the eye-tracking one used here may prove useful for early diagnostic of human conditions including autism and attention-deficit hyperactive disorder, as well as for quantitative comparison between humans and animals which may have lower or different priors, including monkeys, frogs, and flies. Beyond sensory biology, computable surprise could guide the development of data mining and compression systems (giving more bits to surprising regions of interest), to find surprising agents in crowds, surprising sentences in books or speeches, surprising sequences in genomes, surprising medical symptoms, surprising odors in airport luggage racks, surprising documents on the world-wide-web, or to design surprising advertisements. Acknowledgments: Supported by HFSP, NSF and NGA (L.I.), NIH and NSF (P.B.). 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Convex Neural Networks Yoshua Bengio, Nicolas Le Roux, Pascal Vincent, Olivier Delalleau, Patrice Marcotte Dept. IRO, Universit´e de Montr´eal P.O. Box 6128, Downtown Branch, Montreal, H3C 3J7, Qc, Canada {bengioy,lerouxni,vincentp,delallea,marcotte}@iro.umontreal.ca Abstract Convexity has recently received a lot of attention in the machine learning community, and the lack of convexity has been seen as a major disadvantage of many learning algorithms, such as multi-layer artificial neural networks. We show that training multi-layer neural networks in which the number of hidden units is learned can be viewed as a convex optimization problem. This problem involves an infinite number of variables, but can be solved by incrementally inserting a hidden unit at a time, each time finding a linear classifier that minimizes a weighted sum of errors. 1 Introduction The objective of this paper is not to present yet another learning algorithm, but rather to point to a previously unnoticed relation between multi-layer neural networks (NNs),Boosting (Freund and Schapire, 1997) and convex optimization. Its main contributions concern the mathematical analysis of an algorithm that is similar to previously proposed incremental NNs, with L1 regularization on the output weights. This analysis helps to understand the underlying convex optimization problem that one is trying to solve. This paper was motivated by the unproven conjecture (based on anecdotal experience) that when the number of hidden units is “large”, the resulting average error is rather insensitive to the random initialization of the NN parameters. One way to justify this assertion is that to really stay stuck in a local minimum, one must have second derivatives positive simultaneously in all directions. When the number of hidden units is large, it seems implausible for none of them to offer a descent direction. Although this paper does not prove or disprove the above conjecture, in trying to do so we found an interesting characterization of the optimization problem for NNs as a convex program if the output loss function is convex in the NN output and if the output layer weights are regularized by a convex penalty. More specifically, if the regularization is the L1 norm of the output layer weights, then we show that a “reasonable” solution exists, involving a finite number of hidden units (no more than the number of examples, and in practice typically much less). We present a theoretical algorithm that is reminiscent of Column Generation (Chv´atal, 1983), in which hidden neurons are inserted one at a time. Each insertion requires solving a weighted classification problem, very much like in Boosting (Freund and Schapire, 1997) and in particular Gradient Boosting (Mason et al., 2000; Friedman, 2001). Neural Networks, Gradient Boosting, and Column Generation Denote ˜x ∈Rd+1 the extension of vector x ∈Rd with one element with value 1. What we call “Neural Network” (NN) here is a predictor for supervised learning of the form ˆy(x) = Pm i=1 wihi(x) where x is an input vector, hi(x) is obtained from a linear discriminant function hi(x) = s(vi · ˜x) with e.g. s(a) = sign(a), or s(a) = tanh(a) or s(a) = 1 1+e−a . A learning algorithm must specify how to select m, the wi’s and the vi’s. The classical solution (Rumelhart, Hinton and Williams, 1986) involves (a) selecting a loss function Q(ˆy, y) that specifies how to penalize for mismatches between ˆy(x) and the observed y’s (target output or target class), (b) optionally selecting a regularization penalty that favors “small” parameters, and (c) choosing a method to approximately minimize the sum of the losses on the training data D = {(x1, y1), . . . , (xn, yn)} plus the regularization penalty. Note that in this formulation, an output non-linearity can still be used, by inserting it in the loss function Q. Examples of such loss functions are the quadratic loss ||ˆy −y||2, the hinge loss max(0, 1 −yˆy) (used in SVMs), the cross-entropy loss −y log ˆy −(1 −y) log(1 −ˆy) (used in logistic regression), and the exponential loss e−yˆy (used in Boosting). Gradient Boosting has been introduced in (Friedman, 2001) and (Mason et al., 2000) as a non-parametric greedy-stagewise supervised learning algorithm in which one adds a function at a time to the current solution ˆy(x), in a steepest-descent fashion, to form an additive model as above but with the functions hi typically taken in other kinds of sets of functions, such as those obtained with decision trees. In a stagewise approach, when the (m+1)-th basis hm+1 is added, only wm+1 is optimized (by a line search), like in matching pursuit algorithms.Such a greedy-stagewise approach is also at the basis of Boosting algorithms (Freund and Schapire, 1997), which is usually applied using decision trees as bases and Q the exponential loss. It may be difficult to minimize exactly for wm+1 and hm+1 when the previous bases and weights are fixed, so (Friedman, 2001) proposes to “follow the gradient” in function space, i.e., look for a base learner hm+1 that is best correlated with the gradient of the average loss on the ˆy(xi) (that would be the residue ˆy(xi) −yi in the case of the square loss). The algorithm analyzed here also involves maximizing the correlation between Q′ (the derivative of Q with respect to its first argument, evaluated on the training predictions) and the next basis hm+1. However, we follow a “stepwise”, less greedy, approach, in which all the output weights are optimized at each step, in order to obtain convergence guarantees. Our approach adapts the Column Generation principle (Chv´atal, 1983), a decomposition technique initially proposed for solving linear programs with many variables and few constraints. In this framework, active variables, or “columns”, are only generated as they are required to decrease the objective. In several implementations, the column-generation subproblem is frequently a combinatorial problem for which efficient algorithms are available. In our case, the subproblem corresponds to determining an “optimal” linear classifier. 2 Core Ideas Informally, consider the set H of all possible hidden unit functions (i.e., of all possible hidden unit weight vectors vi). Imagine a NN that has all the elements in this set as hidden units. We might want to impose precision limitations on those weights to obtain either a countable or even a finite set. For such a NN, we only need to learn the output weights. If we end up with a finite number of non-zero output weights, we will have at the end an ordinary feedforward NN. This can be achieved by using a regularization penalty on the output weights that yields sparse solutions, such as the L1 penalty. If in addition the loss function is convex in the output layer weights (which is the case of squared error, hinge loss, ϵ-tube regression loss, and logistic or softmax cross-entropy), then it is easy to show that the overall training criterion is convex in the parameters (which are now only the output weights). The only problem is that there are as many variables in this convex program as there are elements in the set H, which may be very large (possibly infinite). However, we find that with L1 regularization, a finite solution is obtained, and that such a solution can be obtained by greedily inserting one hidden unit at a time. Furthermore, it is theoretically possible to check that the global optimum has been reached. Definition 2.1. Let H be a set of functions from an input space X to R. Elements of H can be understood as “hidden units” in a NN. Let W be the Hilbert space of functions from H to R, with an inner product denoted by a · b for a, b ∈W. An element of W can be understood as the output weights vector in a neural network. Let h(x) : H →R the function that maps any element hi of H to hi(x). h(x) can be understood as the vector of activations of hidden units when input x is observed. Let w ∈W represent a parameter (the output weights). The NN prediction is denoted ˆy(x) = w · h(x). Let Q : R × R →R be a cost function convex in its first argument that takes a scalar prediction ˆy(x) and a scalar target value y and returns a scalar cost. This is the cost to be minimized on example pair (x, y). Let D = {(xi, yi) : 1 ≤i ≤n} a training set. Let Ω: W →R be a convex regularization functional that penalizes for the choice of more “complex” parameters (e.g., Ω(w) = λ||w||1 according to a 1-norm in W, if H is countable). We define the convex NN criterion C(H, Q, Ω, D, w) with parameter w as follows: C(H, Q, Ω, D, w) = Ω(w) + n X t=1 Q(w · h(xt), yt). (1) The following is a trivial lemma, but it is conceptually very important as it is the basis for the rest of the analysis in this paper. Lemma 2.2. The convex NN cost C(H, Q, Ω, D, w) is a convex function of w. Proof. Q(w · h(xt), yt) is convex in w and Ωis convex in w, by the above construction. C is additive in Q(w · h(xt), yt) and additive in Ω. Hence C is convex in w. Note that there are no constraints in this convex optimization program, so that at the global minimum all the partial derivatives of C with respect to elements of w cancel. Let |H| be the cardinality of the set H. If it is not finite, it is not obvious that an optimal solution can be achieved in finitely many iterations. Lemma 2.2 says that training NNs from a very large class (with one or more hidden layer) can be seen as convex optimization problems, usually in a very high dimensional space, as long as we allow the number of hidden units to be selected by the learning algorithm. By choosing a regularizer that promotes sparse solutions, we obtain a solution that has a finite number of “active” hidden units (non-zero entries in the output weights vector w). This assertion is proven below, in theorem 3.1, for the case of the hinge loss. However, even if the solution involves a finite number of active hidden units, the convex optimization problem could still be computationally intractable because of the large number of variables involved. One approach to this problem is to apply the principles already successfully embedded in Gradient Boosting, but more specifically in Column Generation (an optimization technique for very large scale linear programs), i.e., add one hidden unit at a time in an incremental fashion. The important ingredient here is a way to know that we have reached the global optimum, thus not requiring to actually visit all the possible hidden units. We show that this can be achieved as long as we can solve the sub-problem of finding a linear classifier that minimizes the weighted sum of classification errors. This can be done exactly only on low dimensional data sets but can be well approached using weighted linear SVMs, weighted logistic regression, or Perceptron-type algorithms. Another idea (not followed up here) would be to consider first a smaller set H1, for which the convex problem can be solved in polynomial time, and whose solution can theoretically be selected as initialization for minimizing the criterion C(H2, Q, Ω, D, w), with H1 ⊂H2, and where H2 may have infinite cardinality (countable or not). In this way we could show that we can find a solution whose cost satisfies C(H2, Q, Ω, D, w) ≤C(H1, Q, Ω, D, w), i.e., is at least as good as the solution of a more restricted convex optimization problem. The second minimization can be performed with a local descent algorithm, without the necessity to guarantee that the global optimum will be found. 3 Finite Number of Hidden Neurons In this section we consider the special case with Q(ˆy, y) = max(0, 1 −yˆy) the hinge loss, and L1 regularization, and we show that the global optimum of the convex cost involves at most n + 1 hidden neurons, using an approach already exploited in (R¨atsch, Demiriz and Bennett, 2002) for L1-loss regression Boosting with L1 regularization of output weights. The training criterion is C(w) = K∥w∥1 + n X t=1 max (0, 1 −ytw · h(xt)). Let us rewrite this cost function as the constrained optimization problem: min w,ξ L(w, ξ) = K∥w∥1 + n X t=1 ξt s.t. yt [w · h(xt)] ≥1 −ξt (C1) and ξt ≥0, t = 1, . . . , n (C2) Using a standard technique, the above program can be recast as a linear program. Defining λ = (λ1, . . . , λn) the vector of Lagrangian multipliers for the constraints C1, its dual problem (P) takes the form (in the case of a finite number J of base learners): (P) : max λ n X t=1 λt s.t. λ · Zi −K ≤0, i ∈I and λt ≤1, t = 1, . . . , n with (Zi)t = ythi(xt). In the case of a finite number J of base learners, I = {1, . . . , J}. If the number of hidden units is uncountable, then I is a closed bounded interval of R. Such an optimization problem satisfies all the conditions needed for using Theorem 4.2 from (Hettich and Kortanek, 1993). Indeed: • I is compact (as a closed bounded interval of R); • F : λ 7→Pn t=1 λt is a concave function (it is even a linear function); • g : (λ, i) 7→λ · Zi −K is convex in λ (it is actually linear in λ); • ν(P) ≤n (therefore finite) (ν(P) is the largest value of F satisfying the constraints); • for every set of n + 1 points i0, . . . , in ∈I, there exists ˜λ such that g(˜λ, ij) < 0 for j = 0, . . . , n (one can take ˜λ = 0 since K > 0). Then, from Theorem 4.2 from (Hettich and Kortanek, 1993), the following theorem holds: Theorem 3.1. The solution of (P) can be attained with constraints C′ 2 and only n + 1 constraints C′ 1 (i.e., there exists a subset of n+1 constraints C′ 1 giving rise to the same maximum as when using the whole set of constraints). Therefore, the primal problem associated is the minimization of the cost function of a NN with n + 1 hidden neurons. 4 Incremental Convex NN Algorithm In this section we present a stepwise algorithm to optimize a NN, and show that there is a criterion that allows to verify whether the global optimum has been reached. This is a specialization of minimizing C(H, Q, Ω, D, w), with Ω(w) = λ||w||1 and H = {h : h(x) = s(v·˜x)} is the set of soft or hard linear classifiers (depending on choice of s(·)). Algorithm ConvexNN(D,Q,λ,s) Input: training set D = {(x1, y1), . . . , (xn, yn)}, convex loss function Q, and scalar regularization penalty λ. s is either the sign function or the tanh function. (1) Set v1 = (0, 0, . . . , 1) and select w1 = argminw1 P t Q(w1s(1), yt) + λ|w1|. (2) Set i = 2. (3) Repeat (4) Let qt = Q′(Pi−1 j=1 wjhj(xt), yt) (5) If s = sign (5a) train linear classifier hi(x) = sign(vi · ˜x) with examples {(xt, sign(qt))} and errors weighted by |qt|, t = 1 . . . n (i.e., maximize P t qthi(xt)) (5b) else (s = tanh) (5c) train linear classifier hi(x) = tanh(vi · ˜x) to maximize P t qthi(xt). (6) If P t qthi(xt) < λ, stop. (7) Select w1, . . . , wi (and optionally v2, . . . , vi) minimizing (exactly or approximately) C = P t Q(Pi j=1 wjhj(xt), yt) + λ P j=1 |wj| such that ∂C ∂wj = 0 for j = 1 . . . i. (8) Return the predictor ˆy(x) = Pi j=1 wjhj(x). A key property of the above algorithm is that, at termination, the global optimum is reached, i.e., no hidden unit (linear classifier) can improve the objective. In the case where s = sign, we obtain a Boosting-like algorithm, i.e., it involves finding a classifier which minimizes the weighted cost P t qtsign(v · ˜xt). Theorem 4.1. Algorithm ConvexNN stops when it reaches the global optimum of C(w) = P t Q(w · h(xt), yt) + λ||w||1. Proof. Let w be the output weights vector when the algorithm stops. Because the set of hidden units H we consider is such that when h is in H, −h is also in H, we can assume all weights to be non-negative. By contradiction, if w′ ̸= w is the global optimum, with C(w′) < C(w), then, since C is convex in the output weights, for any ϵ ∈(0, 1), we have C(ϵw′ + (1 −ϵ)w) ≤ϵC(w′) + (1 −ϵ)C(w) < C(w). Let wϵ = ϵw′ + (1 −ϵ)w. For ϵ small enough, we can assume all weights in w that are strictly positive to be also strictly positive in wϵ. Let us denote by Ip the set of strictly positive weights in w (and wϵ), by Iz the set of weights set to zero in w but to a non-zero value in wϵ, and by δϵk the difference wϵ,k −wk in the weight of hidden unit hk between w and wϵ. We can assume δϵj < 0 for j ∈Iz, because instead of setting a small positive weight to hj, one can decrease the weight of −hj by the same amount, which will give either the same cost, or possibly a lower one when the weight of −hj is positive. With o(ϵ) denoting a quantity such that ϵ−1o(ϵ) →0 when ϵ →0, the difference ∆ϵ(w) = C(wϵ) −C(w) can now be written: ∆ϵ(w) = λ (∥wϵ∥1 −∥w∥1) + X t (Q(wϵ · h(xt), yt) −Q(w · h(xt), yt)) = λ X i∈Ip δϵi + X j∈Iz −δϵj + X t X k (Q′(w · h(xt), yt)δϵkhk(xt)) + o(ϵ) = X i∈Ip λδϵi + X t qtδϵihi(xt) ! + X j∈Iz −λδϵj + X t qtδϵjhj(xt) ! + o(ϵ) = X i∈Ip δϵi ∂C ∂wi (w) + X j∈Iz −λδϵj + X t qtδϵjhj(xt) ! + o(ϵ) = 0 + X j∈Iz −λδϵj + X t qtδϵjhj(xt) ! + o(ϵ) since for i ∈Ip, thanks to step (7) of the algorithm, we have ∂C ∂wi (w) = 0. Thus the inequality ϵ−1∆ϵ(w) < 0 rewrites into X j∈Iz ϵ−1δϵj −λ + X t qthj(xt) ! + ϵ−1o(ϵ) < 0 which, when ϵ →0, yields (note that ϵ−1δϵj does not depend on ϵ since δϵj is linear in ϵ): X j∈Iz ϵ−1δϵj −λ + X t qthj(xt) ! ≤0 (2) But, hi being the optimal classifier chosen in step (5a) or (5c), all hidden units hj verify P t qthj(xt) ≤P t qthi(xt) < λ and ∀j ∈Iz, ϵ−1δϵj (−λ + P t qthj(xt)) > 0 (since δϵj < 0), contradicting eq. 2. (Mason et al., 2000) prove a related global convergence result for the AnyBoost algorithm, a non-parametric Boosting algorithm that is also similar to Gradient Boosting (Friedman, 2001). Again, this requires solving as a sub-problem an exact minimization to find a function hi ∈H that is maximally correlated with the gradient Q′ on the output. We now show a simple procedure to select a hyperplane with the best weighted classification error. Exact Minimization In step (5a) we are required to find a linear classifier that minimizes the weighted sum of classification errors. Unfortunately, this is an NP-hard problem (w.r.t. d, see theorem 4 in (Marcotte and Savard, 1992)). However, an exact solution can be easily found in O(n3) computations for d = 2 inputs. Proposition 4.2. Finding a linear classifier that minimizes the weighted sum of classification error can be achieved in O(n3) steps when the input dimension is d = 2. Proof. We want to maximize P i cisign(u · xi + b) with respect to u and b, the ci’s being in R. Consider u fixed and sort the xi’s according to their dot product with u and denote r the function which maps i to r(i) such that xr(i) is in i-th position in the sort. Depending on the value of b, we will have n+1 possible sums, respectively −Pk i=1 cr(i) +Pn i=k+1 cr(i), k = 0, . . . , n. It is obvious that those sums only depend on the order of the products u · xi, i = 1, . . . , n. When u varies smoothly on the unit circle, as the dot product is a continuous function of its arguments, the changes in the order of the dot products will occur only when there is a pair (i, j) such that u · xi = u · xj. Therefore, there are at most as many order changes as there are pairs of different points, i.e., n(n −1)/2. In the case of d = 2, we can enumerate all the different angles for which there is a change, namely a1, . . . , az with z ≤n(n−1) 2 . We then need to test at least one u = [cos(θ), sin(θ)] for each interval ai < θ < ai+1, and also one u for θ < a1, which makes a total of n(n−1) 2 possibilities. It is possible to generalize this result in higher dimensions, and as shown in (Marcotte and Savard, 1992), one can achieve O(log(n)nd) time. Algorithm 1 Optimal linear classifier search Maximizing Pn i=1 ciδ(sign(w · xi), yi) in dimension 2 (1) for i = 1, . . . , n for j = i + 1, . . . , n (3) Θi,j = θ(xi, xj) + π 2 where θ(xi, xj) is the angle between xi and xj (6) sort the Θi,j in increasing order (7) w0 = (1, 0) (8) for k = 1, . . . , n(n−1) 2 (9) wk = (cos Θi,j, sin Θi,j), uk = wk+wk−1 2 (10) sort the xi according to the value of uk · xi (11) compute S(uk) = Pn i=1 ciδ(uk · xi), yi) (12) output: argmaxukS Approximate Minimization For data in higher dimensions, the exact minimization scheme to find the optimal linear classifier is not practical. Therefore it is interesting to consider approximate schemes for obtaining a linear classifier with weighted costs. Popular schemes for doing so are the linear SVM (i.e., linear classifier with hinge loss), the logistic regression classifier, and variants of the Perceptron algorithm. In that case, step (5c) of the algorithm is not an exact minimization, and one cannot guarantee that the global optimum will be reached. However, it might be reasonable to believe that finding a linear classifier by minimizing a weighted hinge loss should yield solutions close to the exact minimization. Unfortunately, this is not generally true, as we have found out on a simple toy data set described below. On the other hand, if in step (7) one performs an optimization not only of the output weights wj (j ≤i) but also of the corresponding weight vectors vj, then the algorithm finds a solution close to the global optimum (we could only verify this on 2-D data sets, where the exact solution can be computed easily). It means that at the end of each stage, one first performs a few training iterations of the whole NN (for the hidden units j ≤i) with an ordinary gradient descent mechanism (we used conjugate gradients but stochastic gradient descent would work too), optimizing the wj’s and the vj’s, and then one fixes the vj’s and obtains the optimal wj’s for these vj’s (using a convex optimization procedure). In our experiments we used a quadratic Q, for which the optimization of the output weights can be done with a neural network, using the outputs of the hidden layer as inputs. Let us consider now a bit more carefully what it means to tune the vj’s in step (7). Indeed, changing the weight vector vj of a selected hidden neuron to decrease the cost is equivalent to a change in the output weights w’s. More precisely, consider the step in which the value of vj becomes v′ j. This is equivalent to the following operation on the w’s, when wj is the corresponding output weight value: the output weight associated with the value vj of a hidden neuron is set to 0, and the output weight associated with the value v′ j of a hidden neuron is set to wj. This corresponds to an exchange between two variables in the convex program. We are justified to take any such step as long as it allows us to decrease the cost C(w). The fact that we are simultaneously making such exchanges on all the hidden units when we tune the vj’s allows us to move faster towards the global optimum. Extension to multiple outputs The multiple outputs case is more involved than the single-output case because it is not enough to check the condition P t htqt > λ. Consider a new hidden neuron whose output is hi when the input is xi. Let us also denote α = [α1, . . . , αno]′ the vector of output weights between the new hidden neuron and the no output neurons. The gradient with respect to αj is gj = ∂C ∂αj = P t htqtj −λsign(αj) with qtj the value of the j-th output neuron with input xt. This means that if, for a given j, we have | P t htqtj| < λ, moving αj away from 0 can only increase the cost. Therefore, the right quantity to consider is (| P t htqtj| −λ)+. We must therefore find argmaxv P j (| P t htqtj| −λ)2 +. As before, this sub-problem is not convex, but it is not as obvious how to approximate it by a convex problem. The stopping criterion becomes: if there is no j such that | P t htqtj| > λ, then all weights must remain equal to 0 and a global minimum is reached. Experimental Results We performed experiments on the 2-D double moon toy dataset (as used in (Delalleau, Bengio and Le Roux, 2005)), to be able to compare with the exact version of the algorithm. In these experiments, Q(w · h(xt), yt) = [w · h(xt) −yt]2. The set-up is the following: • Select a new linear classifier, either (a) the optimal one or (b) an approximate using logistic regression. • Optimize the output weights using a convex optimizer. • In case (b), tune both input and output weights by conjugate gradient descent on C and finally re-optimize the output weights using LASSO regression. • Optionally, remove neurons whose output weight has been set to 0. Using the approximate algorithm yielded for 100 training examples an average penalized (λ = 1) squared error of 17.11 (over 10 runs), an average test classification error of 3.68% and an average number of neurons of 5.5 . The exact algorithm yielded a penalized squared error of 8.09, an average test classification error of 5.3%, and required 3 hidden neurons. A penalty of λ = 1 was nearly optimal for the exact algorithm whereas a smaller penalty further improved the test classification error of the approximate algorithm. Besides, when running the approximate algorithm for a long time, it converges to a solution whose quadratic error is extremely close to the one of the exact algorithm. 5 Conclusion We have shown that training a NN can be seen as a convex optimization problem, and have analyzed an algorithm that can exactly or approximately solve this problem. We have shown that the solution with the hinge loss involved a number of non-zero weights bounded by the number of examples, and much smaller in practice. We have shown that there exists a stopping criterion to verify if the global optimum has been reached, but it involves solving a sub-learning problem involving a linear classifier with weighted errors, which can be computationally hard if the exact solution is sought, but can be easily implemented for toy data sets (in low dimension), for comparing exact and approximate solutions. The above experimental results are in agreement with our initial conjecture: when there are many hidden units we are much less likely to stall in the optimization procedure, because there are many more ways to descend on the convex cost C(w). They also suggest, based on experiments in which we can compare with the exact sub-problem minimization, that applying Algorithm ConvexNN with an approximate minimization for adding each hidden unit while continuing to tune the previous hidden units tends to lead to fast convergence to the global minimum. What can get us stuck in a “local minimum” (in the traditional sense, i.e., of optimizing w’s and v’s together) is simply the inability to find a new hidden unit weight vector that can improve the total cost (fit and regularization term) even if there exists one. Note that as a side-effect of the results presented here, we have a simple way to train neural networks with hard-threshold hidden units, since increasing P t Q′(ˆy(xt), yt)sign(vixt) can be either achieved exactly (at great price) or approximately (e.g. by using a cross-entropy or hinge loss on the corresponding linear classifier). Acknowledgments The authors thank the following for support: NSERC, MITACS, and the Canada Research Chairs. They are also grateful for the feedback and stimulating exchanges with Sam Roweis, Nathan Srebro, and Aaron Courville. References Chv´atal, V. (1983). Linear Programming. W.H. Freeman. Delalleau, O., Bengio, Y., and Le Roux, N. (2005). Efficient non-parametric function induction in semi-supervised learning. In Cowell, R. and Ghahramani, Z., editors, Proceedings of AISTATS’2005, pages 96–103. Freund, Y. and Schapire, R. E. (1997). A decision theoretic generalization of on-line learning and an application to boosting. Journal of Computer and System Science, 55(1):119–139. Friedman, J. (2001). Greedy function approximation: a gradient boosting machine. Annals of Statistics, 29:1180. Hettich, R. and Kortanek, K. (1993). Semi-infinite programming: theory, methods, and applications. SIAM Review, 35(3):380–429. Marcotte, P. and Savard, G. (1992). Novel approaches to the discrimination problem. Zeitschrift fr Operations Research (Theory), 36:517–545. Mason, L., Baxter, J., Bartlett, P. L., and Frean, M. (2000). Boosting algorithms as gradient descent. In Advances in Neural Information Processing Systems 12, pages 512–518. R¨atsch, G., Demiriz, A., and Bennett, K. P. (2002). Sparse regression ensembles in infinite and finite hypothesis spaces. Machine Learning. Rumelhart, D., Hinton, G., and Williams, R. (1986). Learning representations by back-propagating errors. Nature, 323:533–536.
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Efficient Estimation of OOMs Herbert Jaeger, Mingjie Zhao, Andreas Kolling International University Bremen Bremen, Germany h.jaeger|m.zhao|a.kolling@iu-bremen.de Abstract A standard method to obtain stochastic models for symbolic time series is to train state-emitting hidden Markov models (SE-HMMs) with the Baum-Welch algorithm. Based on observable operator models (OOMs), in the last few months a number of novel learning algorithms for similar purposes have been developed: (1,2) two versions of an ”efficiency sharpening” (ES) algorithm, which iteratively improves the statistical efficiency of a sequence of OOM estimators, (3) a constrained gradient descent ML estimator for transition-emitting HMMs (TE-HMMs). We give an overview on these algorithms and compare them with SE-HMM/EM learning on synthetic and real-life data. 1 Introduction Stochastic symbol sequences with memory effects are frequently modelled by training hidden Markov models with the Baum-Welch variant of the EM algorithm. More specifically, state-emitting HMMs (SE-HMMs) are standardly employed, which emit observable events from hidden states. Known weaknesses of HMM training with Baum-Welch are long runtimes and proneness to getting trapped in local maxima. Over the last few years, an alternative to HMMs has been developed, observable operator models (OOMs). The class of processes that can be described by (finite-dimensional) OOMs properly includes the processes that can be described by (finite-dimensional) HMMs. OOMs identify the observable events a of a process with linear observable operators τa acting on a real vector space of predictive states w [1]. A basic learning algorithm for OOMs [2] estimates the observable operators τa by solving a linear system of learning equations. The learning algorithm is constructive, fast and yields asymptotically correct estimates. Two problems that so far prevented OOMs from practical use were (i) poor statistical efficiency, (ii) the possibility that the obtained models might predict negative “probabilities” for some sequences. Since a few months the first problem has been very satisfactorily solved [2]. In this novel approach to learning OOMs from data we iteratively construct a sequence of estimators whose statistical efficiency increases, which led us to call the method efficiency sharpening (ES). Another, somewhat neglected class of stochastic models is transition-emitting HMMs (TEHMMs). TE-HMMs fall in between SE-HMMs and OOMs w.r.t. expressiveness. TEHMMs are equivalent to OOMs whose operator matrices are non-negative. Because TEHMMs are frequently referred to as Mealy machines (actually a misnomer because originally Mealy machines are not probabilistic but only non-deterministic), we have started to call non-negative OOMs “Mealy OOMs” (MOOMs). We use either name according to the way the models are represented. A variant of Baum-Welch has recently been described for TE-HMMs [3]. We have derived an alternative learning constrained log gradient (CLG) algorithm for MOOMs which performs a constrained gradient descent on the log likelihood surface in the log model parameter space of MOOMs. In this article we give a compact introduction to the basics of OOMs (Section 2), outline the new ES and CLG algorithms (Sections 3 and 4), and compare their performance on a variety of datasets (Section 5). In the conclusion (Section 6) we also provide a pointer to a Matlab toolbox. 2 Basics of OOMs Let (Ω, A, P, (Xn)n≥0) or (Xn) for short be a discrete-time stochastic process with values in a finite symbol set O = {a1, . . . , aM}. We will consider only stationary processes here for notational simplicity; OOMs can equally model nonstationary processes. An mdimensional OOM for (Xn) is a structure A = (Rm, (τa)a∈O, w0), where each observable operator τa is a real-valued m×m matrix and w0 ∈Rm is the starting state, provided that for any finite sequence ai0 . . . ain it holds that P(X0 = ai0, . . . Xn = ain) = 1mτain · · · τai0 w0, (1) where 1m always denotes a row vector of units of length m (we drop the subscript if it is clear from the context). We will use the shorthand notation ¯a to denote a generic sequence and τ¯a to denote a concatenation of the corresponding operators in reverse order, which would condense (1) into P(¯a) = 1τ¯aw0. Conversely, if a structure A = (Rm, (τa)a∈O, w0) satisfies (i) 1w0 = 1, (ii) 1( a∈O τa) = 1, (iii) ∀¯a ∈O∗: 1τ¯aw0 ≥0, (2) (where O∗denotes the set of all finite sequences over O), then there exists a process whose distribution is described by A via (1). The process is stationary iff ( a∈O τa)w0 = w0. Conditions (i) and (ii) are easy to check, but no efficient criterium is known to decide whether the non-negativity criterium (iii) holds for a structure A (for recent progress in this problem, which is equivalent to a problem of general interest in linear algebra, see [4]). Models A learnt from data tend to marginally violate (iii) – this is the unresolved non-negativity problem in the theory of OOMs. The state w¯a of an OOM after an initial history ¯a is obtained by normalizing τ¯aw0 to unit component sum via w¯a = τ¯aw0/1τ¯aw0. A fundamental (and nontrivial) theorem for OOMs characterizes equivalence of two OOMs. Two m-dimensional OOms A = (Rm, (τa)a∈O, w0) and ˜ A = (Rm, (˜τa)a∈O, ˜w0) are defined to be equivalent if they generate the same probability distribution according to (1). By the equivalence theorem, A is equivalent to ˜ A if and only if there exists a transformation matrix ϱ of size m × m, satisfying 1ϱ = 1, such that ˜τa = ϱτaϱ−1 for all symbols a. We mentioned in the Introduction that OOM states represent the future probability distribution of the process. This can be algebraically captured in the notion of characterizers. Let A = (Rm, (τa)a∈O, w0) be an OOM for (Xn) and choose k such that κ = |O|k ≥m. Let ¯b1, . . . ,¯bκ be the alphabetical enumeration of Ok. Then a m×κ matrix C is a characterizer of length k for A iff 1C = 1 (that is, C has unit column sums) and ∀¯a ∈O∗: w¯a = C(P(¯b1|¯a) · · · P(¯bκ|¯a))′, (3) where ′ denotes the transpose and P(¯b|¯a) is the conditional probability that the process continues with ¯b after an initial history ¯a. It can be shown [2] that every OOM has characterizers of length k for suitably large k. Intuitively, a characterizer “bundles” the length k future distribution into the state vector by projection. If two equivalent OOMs A, ˜ A are related by ˜τa = ϱτaϱ−1, and C is a characterizer for A, it is easy to check that ϱC is a characterizer for ˜ A. We conclude this section by explaining the basic learning equations. An analysis of (1) reveals that for any state w¯a and operator τb from an OOM it holds that τaw¯a = P(a|¯a)w¯aa, (4) where ¯aa is the concatenation of ¯a with a. The vectors w¯a and P(a|¯a)w¯aa thus form an argument-value pair for τa. Let ¯a1, . . . , ¯al be a finite sequence of finite sequences over O, and let V = (w¯a1 · · · w¯al) be the matrix containing the corresponding state vectors. Let again C be a m × κ sized characterizer of length k and ¯b1, . . . ,¯bκ be the alphabetical enumeration of Ok. Let V = (P(¯bi|¯aj)) be the κ × l matrix containing the conditional continuation probabilities of the initial sequences ¯aj by the sequences ¯bi. It is easy to see that V = CV . Likewise, let Wa = (P(a|¯a1)w¯a1a · · · P(a|¯al)w¯ala) contain the vectors corresponding to the rhs of (4), and let W a = (P(a¯bi|¯aj)) be the analog of V . It is easily verified that Wa = CW a. Furthermore, by construction it holds that τaV = Wa. A linear operator on Rm is uniquely determined by l ≥m argument-value pairs provided there are at least m linearly independent argument vectors in these pairs. Thus, if a characterizer C is found such that V = CV has rank m, the operators τa of an OOM characterized by C are uniquely determined by V and the matrices W a via τa = WaV † = CW a(CV )†, where † denotes the pseudo-inverse. Now, given a training sequence S, the conditional continuation probabilities P(¯bi|¯aj), P(a¯bi|¯aj) that make up V , W a can be estimated from S by an obvious counting scheme, yielding estimates ˆP(¯bi|¯aj), ˆP(a¯bi|¯aj) for making up ˆV and ˆW a, respectively. This leads to the general form of OOM learning equations: ˆτa = C ˆW a(C ˆV )†. (5) In words, to learn an OOM from S, first fix a model dimension m, a characterizer C, indicative sequences ¯a1, . . . , ¯al, then construct estimates ˆV and ˆW a by frequency counting, and finally use (5) to obtain estimates of the operators. This estimation procedure is asymptotically correct in the sense that, if the training data were generated by an m-dimensional OOM in the first place, this generator will almost surely be perfectly recovered as the size of training data goes to infinity. The reason for this is that the estimates ˆV and ˆW a converge almost surely to V and W a. The starting state can be recovered from the estimated operators by exploiting ( a∈O τa)w0 = w0 or directly from C and ˆV (see [2] for details). 3 The ES Family of Learning Algorithms All learning algorithms based on (5) are asymptotically correct (which EM algorithms are not, by the way), but their statistical efficiency (model variance) depends crucially on (i) the choice of indicative sequences ¯a1, . . . , ¯al and (ii) the characterizer C (assuming that the model dimension m is determined by other means, e.g. by cross-validation). We will first address (ii) and describe an iterative scheme to obtain characterizers that lead to a low model variance. The choice of C has a twofold impact on model variance. First, the pseudoinverse operation in (5) blows up variation in C ˆV depending on the matrix condition number of this matrix. Thus, C should be chosen such that the condition of C ˆV gets close to 1. This strategy was pioneered in [5], who obtained the first halfway statistically satisfactory learning procedures. In contrast, here we set out from the second mechanism by which C influences model variance, namely, choose C such that the variance of C ˆV itself is minimized. We need a few algebraic preparations. First, observe that if some characterizer C is used with (5), obtaining a model ˆ A, and ϱ is an OOM equivalence transformation, then if ˜C = ϱC is used with (5), the obtained model ˆ˜ A is an equivalent version of ˆ A via ϱ. Furthermore, it is easy to see [2] that two characterizers C1, C2 characterize the same OOM iff C1V = C2V . We call two characterizers similar if this holds, and write C1 ∼C2. Clearly C1 ∼C2 iff C2 = C1+G for some G satisfying GV = 0 and 1G = 0. That is, the similarity equivalence class of some characterizer C is the set {C + G|GV = 0, 1G = 0}. Together with the first observation this implies that we may confine our search for “good” characterizers to a single (and arbitrary) such equivalence class of characterizers. Let C0 in the remainder be a representative of an arbitrarily chosen similarity class whose members all characterize A. In [2] it is explained that the variance of C ˆV is monotonically tied to i=1,...,κ;j=1,...,l P(¯aj¯bi)∥w¯aj −C(:, i)∥2, where C(:, i) is the i-th column of C. This observation allows us to determine an optimal (minimal variance of C ˆV within the equivalence class of C0) characterizer Copt as the solution to the following minimization problem: Copt = C0 + Gopt, where Gopt = arg min G i=1,...,κ;j=1,...,l P(¯aj¯bi)∥w¯aj −(C0 + G)(:, i)∥2 (6) under the constraints GV = 0 and 1G = 0. This problem can be analytically solved [2] and has a surprising and beautiful solution, which we now explain. In a nutshell, Copt is composed column-wise by certain states of a time-reversed version of A. We describe in more detail time-reversal of OOMs. Given an OOM A = (Rm, (τa)a∈O, w0) with an induced probability distribution PA, its reverse OOM Ar = (Rm, (τ r a)a∈O, wr 0) is characterized by a probability distribution PAr satisfying ∀a0 · · · an ∈O∗: PA(a0 · · · an) = PAr(an · · · a0). (7) A reverse OOM can be easily computed from the “forward” OOM as follows. If A = (Rm, (τa)a∈O, w0) is an OOM for a stationary process, and w0 has no zero entry, then Ar = (Rm, (Dτ ′ aD−1)a∈O, w0) (8) is a reverse OOM to A, where D = diag(w0) is a diagonal matrix with w0 on its diagonal. Now let ¯b1, . . . ,¯bκ again be the sequences employed in V . Let Ar = (Rm, (τ r a)a∈O, w0) be the reverse OOM to A, which was characterized by C0. Furthermore, for ¯bi = b1 . . . bk let wr ¯bi = τ r b1 · · · τ r bkw0/1τ r b1 · · · τ r bkw0. Then it holds that C = (wr ¯b1 · · · wr ¯bκ) is a characterizer for an OOM equivalent to A. C can effectively be transformed into a characterizer Cr for A by Cr = ϱrC, where ϱr = (C 1τ¯b1 ... 1τ¯bκ )−1. (9) We call Cr the reverse characterizer of A, because it is composed from the states of a reverse OOM to A. The analytical solution to (6) turns out to be [2] Copt = Cr. (10) To summarize, within a similarity class of characterizers, the one which minimizes model variance is the (unique) reverse characterizer in this class. It can be cheaply computed from the “forward” OOM via (8) and (9). This analytical finding suggests the following generic, iterative procedure to obtain characterizers that minimize model variance: 1. Setup. Choose a model dimension m and a characterizer length k. Compute V , W a from the training string S. 2. Initialization. Estimate an initial model ˆ A(0) with some “classical” OOM estimation method (a refined such method is detailed out in [2]). 3. Efficiency sharpening iteration. Assume that ˆ A(n) is given. Compute its reverse characterizer ˆCr(n+1). Use this in (5) to obtain a new model estimate ˆ A(n+1). 4. Termination. Terminate when the training log-likelihood of models ˆ A(n) appear to settle on a plateau. The rationale behind this scheme is that the initial model ˆ A(0) is obtained essentially from an uninformed, ad hoc characterizer, for which one has to expect a large model variation and thus (on the average) a poor ˆ A(0). However, the characterizer ˆCr(1) obtained from the reversed ˆ A(0) is not uninformed any longer but shaped by a reasonable reverse model. Thus the estimator producing ˆ A(1) can be expected to produce a model closer to the correct one due to its improved efficiency, etc. Notice that this does not guarantee a convergence of models, nor any monotonic development of any performance parameter in the obtained model sequence. In fact, the training log likelihood of the model sequence typically shoots to a plateau level in about 2 to 5 iterations, after which it starts to jitter about this level, only slowly coming to rest – or even not stabilizing at all; it is sometimes observed that the log likelihood enters a small-amplitude oscillation around the plateau level. An analytical understanding of the asymptotic learning dynamics cannot currently be offered. We have developed two specific instantiations of the general ES learning scheme, differentiated by the set of indicative sequences used. The first simply uses l = κ, ¯a1, . . . , ¯al = ¯b1, . . . ,¯bκ, which leads to a computationally very cheap iterated recomputation of (5) with updated reverse characterizers. We call this the “poor man’s” ES algorithm. The statistical efficiency of the poor man’s ES algorithm is impaired by the fact that only the counting statistics of subsequences of length 2k are exploited. The other ES instantiation exploits the statistics of all subsequences in the original training string. It is technically rather involved and rests on a suffix tree (ST) representation of S. We can only give a coarse sketch here (details in [2]). In each iteration, the current reverse model is run backwards through S and the obtained reverse states are additively collected bottom-up in the nodes of the ST. From the ST nodes the collected states are then harvested into matrices corresponding directly to C ˆV and C ˆW a, that is, an explicit computation of the reverse characterizer is not required. This method incurs a computational load per iteration which is somewhat lower than Baum-Welch for SE-HMMs (because only a backward pass of the current model has to be computed), plus the required initial ST construction which is linear in the size of S. 4 The CLG Algorithm We must be very brief here due to space limitations. The CLG algorithm will be detailed out in a future paper. It is an iterative update scheme for the matrix parameters [ˆτa]ij of a MOOM. This scheme is analytically derived as gradient descent in the model log likelihood surface over the log space of these matrix parameters, observing constraints of non-negativity of these parameters and the general OOM constraints (i) and (ii) from Eqn. (2). Note that the constraint (iii) from (2) is automatically satisfied in MOOMs. We skip the derivation of the CLG scheme and describe only its “mechanics”. Let S = s1 . . . sN be the training string and for 1 ≤k ≤N define ¯ak = s1 . . . sk,¯bk = sk+1 . . . sN. Define for some m-dimensional OOM and a ∈O σk = 1τ¯bk 1τ¯bkw¯ak , ya = sk=a σ′ kw′ ¯ak−1 1τskw¯ak−1 , y0 = max i,j,a {[ya]ij}, [ya0]i,j = [ya]i,j/y0. (11) Then the update equation is [ˆτ + a ]ij = ηj · [ˆτa]ij · [ya0]λ ij, (12) where ˆτ + a is the new estimate of τa, ηj’s are normalization parameters determined by the constraint (ii) from Eqn. (2), and λ is a learning rate which here unconventionally appears in the exponent because the gradient descent is carried out in the log parameter space. Note that by (12) [ˆτ + a ]ij remains non-negative if [ˆτa]ij is. This update scheme is derived in a way that is unrelated to the derivation of the EM algorithm; to our surprise we found that for λ = 1 (12) is equivalent to the Baum-Welch algorithm for TE-HMMs. However, significantly faster convergence is achieved with non-unit λ; in the experiments carried out so far a value close to 2 was heuristically found to work best. 5 Numerical Comparisons We compared the poor man’s ES algorithm, the suffix-tree based algorithm, the CLG algorithm and the standard SE-HMM/Baum-Welch method on four different types of data, which were generated by (a) randomly constructed, 10-dimensional, 5-symbol SE-HMMs, (b) randomly constructed, 10-dimensional, 5-symbol MOOMs, (c) a 3-dimensional, 2symbol OOM which is not equivalent to any HMM nor MOOM (the “probability clock” process [2]), (d) a belletristic text (Mark Twain’s short story “The 1,000,000 Pound Note”). For each of (a) and (b), 40 experiments were carried out with freshly constructed generators per experiment; a training string of length 1000 and a test string of length 10000 was produced from each generator. For (c), likewise 40 experiments were carried out with freshly generated training/testing sequences of same lengthes as before; here however the generator was identical for all experiments. For (a) – (c), the results reported below are averaged numbers over the 40 experiments. For the (d) dataset, after preprocessing which shrunk the number of different symbols to 27, the original string was sorted sentence-wise into a training and a testing string, each of length ∼21000 (details in [2]). The following settings were used with the various training methods. (i) The poor man’s ES algorithm was used with a length k = 2 of indicative sequences on all datasets. Two ES iterations were carried out and the model of the last iteration was used to compute the reported log likelihoods. (ii) For the suffix-tree based ES algorithm, on datasets (a) – (c), likewise two ES iterations were done and the model from the iteration with the lowest (reverse) training LL was used for reporting. On dataset (d), 4 ES iterations were called and similarly the model with the best reverse training LL was chosen. (iii) In the MOOM studies, a learning rate of λ = 1.85 was used. Iterations were stopped when two consecutive training LL’s differed by less than 5e-5% or after 100 iterations. (iv) For HMM/BaumWelch training, the public-domain implementation provided by Kevin Murphy was used. Iterations were stopped after 100 steps or if LL’s differed by less than 1e-5%. All computations were done in Matlab on 2 GHz PCs except the HMM training on dataset (d) which was done on a 330 MHz machine (the reported CPU times were scaled by 330/2000 to make them comparable with the other studies). Figure 1 shows the training and testing loglikelihoods as well as the CPU times for all methods and datasets. 2 4 6 8 10 12 −1500 −1450 −1400 −1350 −1300 −1250 −1200 2 4 6 8 10 12 −2 −1 0 1 2 (a) 2 4 6 8 10 12 −1700 −1650 −1600 −1550 −1500 −1450 −1400 2 4 6 8 10 12 −2 −1 0 1 2 (b) 2 3 4 6 8 10 12 −690 −685 −680 −675 −670 2 3 4 6 8 10 12 −2 −1 0 1 2 (c) 51020 40 60 100 150 −5.5 −5 −4.5 −4 −3.5 x 10 4 51020 40 60 100 150 2 2.5 3 3.5 4 (d) Figure 1: Findings for datasets (a)–(d). In each panel, the left y-axis shows log likelihoods for training and testing (testing LL normalized to training stringlength) and the right y-axis measures the log 10 of CPU times. HMM models are documented in solid/black lines, poor man’s ES models in dotted/magenta lines, suffix-tree ES models in broken/blue, and MOOMs in dash-dotted/red lines. The thickest lines in each panel show training LL, the thinnest CPU time, and intermediate testing LL. The x-axes indicate model dimension. On dataset (c), no results of the poor man’s algorithm are given because the learning equations became ill-conditioned for all but the lowest dimensions. Some comments on Fig. 1. (1) The CPU times roughly exhibit an even log spread over almost 2 orders of magnitude, in the order poor man’s (fastest) – suffix-tree ES – CLG – Baum-Welch. (2) CLG has the lowest training LL throughout, which needs an explanation because the proper OOMs trained by ES are more expressive. Apparently the ES algorithm does not lead to local ML optima; otherwise suffix-tree ES models should show the lowest training LL. (3) On HMM-generated data (a), Baum-Welch HMMs can play out their natural bias for this sort of data and achieve a lower test error than the other methods. (4) On the MOOM data (b), the test LL of MOOM/CLG and OOM/poor man models of dimension 2 equals the best HMM/Baum-Welch test LL which is attained at a dimension of 4; the OOM/suffix-tree test LL at dimension 2 is superior to the best HMM test LL. (5) On the “probability clock” data (c), the suffix-tree ES trained OOMs surpassed the non-OOM models in test LL, with the optimal value obtained at the (correct) model dimension 3. This comes as no surprise because these data come from a generator that is incommensurable with either HMMs or MOOMs. (6) On the large empirical dataset (d) the CLG/MOOMs have by a fair margin the highest training LL, but the test LL quickly drops to unacceptable lows. It is hard to explain this by overfitting, considering the complexity and the size of the training string. The other three types of models are evenly ordered in both training and testing error from HMMs (poorest) to suffix-tree ES trained OOMs. Overfitting does not occur up to the maximal dimension investigated. Depending on whether one wants a very fast algorithm with good, or a fast algorithm with very good train/test LL, one here would choose the poor man’s or the suffix-tree ES algorithm as the winner. (7) One detail in panel (d) needs an explanation. The CPU time for the suffix-tree ES has an isolated peak for the smallest dimension. This is earned by the construction of the suffix tree, which was built only for the smallest dimension and re-used later. 6 Conclusion We presented, in a sadly condensed fashion, three novel learning algorithms for symbol dynamics. A detailed treatment of the Efficiency Sharpening algorithm is given in [2], and a Matlab toolbox for it can be fetched from http://www.faculty.iubremen.de/hjaeger/OOM/OOMTool.zip. The numerical investigations reported here were done using this toolbox. Our numerical simulations demonstrate that there is an altogether new world of faster and often statistically more efficient algorithms for sequence modelling than Baum-Welch/SE-HMMs. The topics that we will address next in our research group are (i) a mathematical analysis of the asymptotic behaviour of the ES algorithms, (ii) online adaptive versions of these algorithms, and (iii) versions of the ES algorithms for nonstationary time series. References [1] M. L. Littman, R. S. Sutton, and S. Singh. Predictive representation of state. In Advances in Neural Information Processing Systems 14 (Proc. NIPS 01), pages 1555–1561, 2001. http://www.eecs.umich.edu/∼baveja/Papers/psr.pdf. [2] H. Jaeger, M. Zhao, K. Kretzschmar, T. Oberstein, D. Popovici, and A. Kolling. Learning observable operator models via the es algorithm. In S. Haykin, J. Principe, T. Sejnowski, and J. McWhirter, editors, New Directions in Statistical Signal Processing: from Systems to Brains, chapter 20. MIT Press, to appear in 2005. [3] H. Xue and V. Govindaraju. Stochastic models combining discrete symbols and continuous attributes in handwriting recognition. In Proc. DAS 2002, 2002. [4] R. Edwards, J.J. McDonald, and M.J. Tsatsomeros. On matrices with common invariant cones with applications in neural and gene networks. Linear Algebra and its Applications, in press, 2004 (online version). http://www.math.wsu.edu/math/faculty/tsat/files/emt.pdf. [5] K. Kretzschmar. Learning symbol sequences with Observable Operator Models. GMD Report 161, Fraunhofer Institute AIS, 2003. http://omk.sourceforge.net/files/OomLearn.pdf.
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AER Building Blocks for Multi-Layer Multi-Chip Neuromorphic Vision Systems R. Serrano-Gotarredona1, M. Oster2, P. Lichtsteiner2, A. Linares-Barranco4, R. PazVicente4, F. Gómez-Rodríguez4, H. Kolle Riis3, T. Delbrück2, S. C. Liu2, S. Zahnd2, A. M. Whatley2, R. Douglas2, P. Häfliger3, G. Jimenez-Moreno4, A. Civit4, T. Serrano-Gotarredona1, A. Acosta-Jiménez1, B. Linares-Barranco1 1Instituto de Microelectrónica de Sevilla (IMSE-CNM-CSIC) Sevilla Spain, 2Institute of Neuroinformatics (INI-ETHZ) Zurich Switzerland, 3University of Oslo Norway (UIO), 4University of Sevilla Spain (USE). Abstract A 5-layer neuromorphic vision processor whose components communicate spike events asychronously using the address-eventrepresentation (AER) is demonstrated. The system includes a retina chip, two convolution chips, a 2D winner-take-all chip, a delay line chip, a learning classifier chip, and a set of PCBs for computer interfacing and address space remappings. The components use a mixture of analog and digital computation and will learn to classify trajectories of a moving object. A complete experimental setup and measurements results are shown. 1 Introduction The Address-Event-Representation (AER) is an event-driven asynchronous inter-chip communication technology for neuromorphic systems [1][2]. Senders (e.g. pixels or neurons) asynchronously generate events that are represented on the AER bus by the source addresses. AER systems can be easily expanded. The events can be merged with events from other senders and broadcast to multiple receivers [3]. Arbitrary connections, remappings and transformations can be easily performed on these digital addresses. A potentially huge advantage of AER systems is that computation is event driven and thus can be very fast and efficient. Here we describe a set of AER building blocks and how we assembled them into a prototype vision system that learns to classify trajectories of a moving object. All modules communicate asynchronously using AER. The building blocks and demonstration system have been developed in the EU funded research project CAVIAR (Convolution AER VIsion Architecture for Real-time). The building blocks (Fig. 1) consist of: (1) a retina loosely modeled on the magnocellular pathway that responds to brightness changes, (2) a convolution chip with programmable convolution kernel of arbitrary shape and size, (3) a multi-neuron 2D competition chip, (4) a spatiotemporal pattern classification learning module, and (5) a set of FPGA-based PCBs for address remapping and computer interfaces. Using these AER building blocks and tools we built the demonstration vision system shown schematically in Fig. 1, that detects a moving object and learns to classify its trajectories. It has a front end retina, followed by an array of convolution chips, each programmed to detect a specific feature with a given spatial scale. The competition or ‘object’ chip selects the most salient feature and scale. A spatio-temporal pattern classification module categorizes trajectories of the object chip outputs. 2 Retina Biological vision uses asynchronous events (spikes) delivered from the retina. The stream of events encodes dynamic scene contrast. Retinas are optimized to deliver relevant information and to discard redundancy. CAVIAR’s input is a dynamic visual scene. We developed an AER silicon retina chip ‘TMPDIFF’ that generates events corresponding to relative changes in image intensity [8]. These address-events are broadcast asynchronously on a shared digital bus to the convolution chips. Static scenes produce no output. The events generated by TMPDIFF represent relative changes in intensity that exceed a user-defined threshold and are ON or OFF type depending on the sign of the change since the last event. This silicon retina loosely models the magnocellular retinal pathway. The front-end of the pixel core (see Fig. 2a) is an active unity-gain logarithmic photoreceptor that can be self-biased by the average photocurrent [7]. The active feedback speeds up the response compared to a passive log photoreceptor and greatly increases bandwidth at low illumination. The photoreceptor output is buffered to a voltage-mode capacitive-feedback amplifier with closed-loop gain set by a well-matched capacitor ratio. The amplifier is balanced after transmission of each event by the AER handshake. ON and OFF events are detected by the comparators that follow. Mismatch of the event threshold is determined by only 5 transistors and is effectively further reduced by the gain of the amplifier. Much higher contrast resolution than in previous work [6] is obtained by using the excellent matching between capacitors to form a self-clocked switched-capacitor change amplifier, allowing for operation with scene contrast down to about 20%. A chip photo is shown in Fig. 2b. Fig. 1: Demonstration AER vision system Fig. 2. Retina. a) core of pixel circuit, b) chip photograph. (a) (b) TMPDIFF has 64x64 pixels, each with 2 outputs (ON and OFF), which are communicated off-chip on a 16-bit AER bus. It is fabricated in a 0.35µm process. Each pixel is 40x40 µm2 and has 28 transistors and 3 capacitors. The operating range is at least 5 decades and minimum scene illumination with f/1.4 lens is less than 10 lux. 3 Convolution Chip The convolution chip is an AER transceiver with an array of event integrators. Foreach incoming event, integrators within a projection field around the addressed pixel compute a weighted event integration. The weight of this integration is defined by the convolution kernel [4]. This event-driven computation puts the kernel onto the integrators. Fig. 3a shows the block diagram of the convolution chip. The main parts of the chip are: (1) An array of 32x32 pixels. Each pixel contains a binary weighted signed current source and an integrate-and-fire signed integrator [5]. The current source is controlled by the kernel weight read from the RAM and stored in a dynamic register. (2) A 32x32 kernel RAM. Each kernel weight value is stored with signed 4-bit resolution. (3) A digital controller handles all sequence of operations. (4) A monostable. For each incoming event, it generates a pulse of fixed duration that enables the integration simultaneously in all the pixels. (5) X-Neighborhood Block. This block performs a displacement of the kernel in the x direction. (6) Arbitration and decoding circuitry that generate the output address events. It uses Boahen’s burst mode fully parallel AER [2]. The chip operation sequence is as follows: (1) Each time an input address event is received, the digital control block stores the (x,y) address and acknowledges reception of the event. (2) The control block computes the x-displacement that has to be applied to the kernel and the limits in the y addresses where the kernel has to be copied. (3) The Afterwards, the control block generates signals that control on a row-by-row basis the copy of the kernel to the corresponding rows in the pixel array. (4) Once the kernel copy is finished, the control block activates the generation of a monostable pulse. This way, in each pixel a current weighted by the corresponding kernel weight is integrated during a fixed time interval. Afterwards, kernel weights in the pixels are erased. (5) When the integrator voltage in a pixel reaches a threshold, that pixel asynchronously sends an event, which is arbitrated and decoded in the periphery of the array. The pixel voltage is reset upon reception of the acknowledge from the periphery. 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 −5 0 5 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 −4 −2 0 2 High Monostable Ack Rqst Address Clock Control Block I/O Speed x−neighbourhood y−Decoder Kernel−RAM Pixel Array Row−Arbiter Line Buffer & Column Arbiter out AER (x,y) Fig. 3. Convolution chip (a) architecture of the convolution chip. (b) microphotograph of fabricated chip. (c) kernel for detecting circumferences of radius close to 4 pixels and (d) close to 9 pixels. (a) (b) Control kernel-RAM X-neighb. pixel array (c) (d) A prototype convolution chip has been fabricated in a CMOS 0.35µm process. Both the size of the pixel array and the size of the kernel storage RAM are 32x32. The input address space can be up to 128x128. In the experimental setup of Section 7, the 64x64 retina output is fed to the convolution chip, whose pixel array addresses are centered on that of the retina. The pixel size is 92.5µm x 95µm. The total chip area is 5.4x4.2 mm2. Fig. 3b shows the microphotograph of the fabricated chip. AER events can be fed-in up to a peak rate of 50 Mevent/s. Output event rate depends on kernel lines nk. The measured output AER peak delay is (40 + 20 x nk) ns/event. 4 Competition ‘Object’ Chip This AER transceiver chip consists of a group of VLSI integrate-and-fire neurons with various types of synapses [9]. It reduces the dimensionality of the input space by preserving the strongest input and suppressing all other inputs. The strongest input is determined by configuring the architecture on the ’Object’ chip as a spiking winner-takeall network. Each convolution chip convolves the output spikes of the retina with its preprogrammed feature kernel (in our example, this kernel consists of a ring filter of a particular resolution). The ’Object’ chip receives the outputs of several convolution chips and computes the winner (strongest input) in two dimensions. First, it determines the strongest input in each feature map and in addition, it determines the strongest feature. The computation to determine the strongest input in each feature map is carried out using a two-dimensional winner-take-all circuit as shown in Fig. 4. The network is configured so that it implements a hard winner-take-all, that is, only one neuron is active at a time. The activity of the winner is proportional to the winner’s input activity. The winner-take-all circuit can reliably select the winner given a difference of input firing rate of only 10% assuming that it receives input spike trains having a regular firing rate [10]. Each excitatory input spike charges the membrane of the post-synaptic neuron until one neuron in the array--the winner--reaches threshold and is reset. All other neurons are then inhibited via a global inhibitory neuron which is driven by all the excitatory neurons. Self-excitation provides hysteresis for the winning neuron by facilitating the selection of this neuron as the next winner. Because of the moving stimulus, the network has to determine the winner using an estimate of the instantaneous input firing rates. The number of spikes that the neuron must integrate before eliciting an output spike can be adjusted by varying the efficacies of the input synapses. To determine the winning feature, we use the activity of the global inhibitory neuron (which reflects the activity of the strongest input within a feature map) of each feature map in a second layer of competition. By adding a second global inhibitory neuron to each feature map and by driving this neuron through the outputs of the first global inhibitory neurons of all feature maps, only the strongest feature map will survive. The output of the object chip will be spikes encoding the spatial location of the stimulus and the identity of the winning feature. (In the characterization shown in Section 7, the global competition was disabled, so both objects could be simultaneously localized by the object chip). We integrated the winner-take-all circuits for four feature maps on a single chip with a total of 16x16 neurons; each feature uses an 8x8 array. The chip was fabricated in a 0.35 µm CMOS process with an area of 8.5 mm2. 5 Learning Spatio-Temporal Pattern Classification The last step of data reduction in the CAVIAR demonstrator is a subsystem that learns to classify the spatio-temporal patterns provided by the object chip. It consists of three components: a delay line chip, a competitive Hebbian learning chip [11], and an AER mapper that connects the two. The task of the delay line chip is to project the temporal dimension into a spatial dimension. The competitive Hebbian learning chip will then learn to classify the resulting patterns. The delay line chip consists of one cascade of 880 delay elements. 16 monostables in series form one delay element. The output of every delay element produces an output address event. A pulse can be inserted at every delay-element by an input address event. The cascade can be programmed to be interrupted or connected between any two subsequent delay-elements. The associative Hebbian learning chip consists of 32 neurons with 64 learning synapses each. Each synapse includes learning circuitry with a weak multi-level memory cell for spike-based learning [11]. A simple example of how this system may be configured is depicted in Fig. 5: the mapper between the object chip and the delay line chip is programmed to project all activity from the left half of the field of vision onto the input of one delay line, and from the right half of vision onto another. The mapper between the delay line chip and the competitive Hebbian learning chip taps these two delay lines at three different delays and maps these 6 outputs onto 6 synapses of each of the 32 neurons in the competitive Hebbian learning chip. This configuration lets the system learn the direction of motion. right half visual field left half visual field ∆t ∆t ∆t ∆t ..... delay line chip competitive Hebbian learning chip AER mapper 0 1 2 3 4 5 Fig. 4: Architecture of ’Object’ chip configured for competition within two feature maps and competition across the two feature maps. Fig. 5: System setup for learning direction of motion Fig. 6: Developed AER interfacing PCBs. (a) PCI-AER, (b) USB-AER, (c) AER-switch, (d) mini-USB (a) (b) (c) (d) 6 Computer Interfaces When developing and tuning complex hierarchical multi-chips AER systems it is crucial to have available proper computer interfaces for (a) reading AER traffic and visualizing it, and (b) for injecting synthesized or recorded AER traffic into AER buses. We developed several solutions. Fig. 6(a) shows a PCI-AER interfacing PCB capable of transmitting AER streams from within the computer or, vice versa, capturing them from an AER bus and into computer memory. It uses a Spartan-II FPGA, and can achieve a peak rate of 15 Mevent/s using PCI mastering. Fig. 5(b) shows a USB-AER board that does not require a PCI slot and can be controlled through a USB port. It uses a Spartan II 200 FPGA with a Silicon Labs C8051F320 microcontroller. Depending on the FPGA firmware, it can be used to perform five different functions: (a) transform sequence of frames into AER in real time [13], (b) histogram AER events into sequences of frames in real time, (c) do remappings of addresses based on look-up-tables, (d) capture timestamped events for offline analysis, (e) reproduce time-stamped sequences of events in real time. This board can also work without a USB connection (stand-alone mode) by loading the firmware through MMC/SD cards, used in commercial digital cameras. This PCB can handle AER traffic of up to 25 Mevent/s. It also includes a VGA output for visualizing histogrammed frames. The third PCB, based on a simple CPLD, is shown in Fig. 6(c). It splits one AER bus into 2, 3 or 4 buses, and vice versa, merges 2, 3 or 4 buses into a single bus, with proper handling of handshaking signals. The last board in Fig. 6(d) is a lower performance but more compact single-chip bus-powered USB interface based on a C8051F320 microcontroller. It captures timestamped events to a computer at rates of up to 100 kevent/ s and is particularly useful for demonstrations and field capture of retina output. 7 Demonstration Vision System To test CAVIAR’s capabilities, we built a demonstration system that could simultaneously track two objects of different size. A block diagram of the complete system is shown in Fig. 7(a), and a photograph of the complete experimental setup is given in Fig. 7(b). The USB AER Merger AER monitor AER USB monitor Splitter AER USB Monitor PCI−AER AER mapper AER mapper 17 16 15 14 13 12 11 10 9 8 6 5 7 4 3 2 mapper AER USB mapper AER USB monitor AER USB learning chip AER chip line AER delay chip object AER motion retina AER AER chip Convolution USB AER chip Convolution Fig. 7: Experimental setup of multi-layered AER vision system for ball tracking (white boxes include custom designed chips, blue boxes are interfacing PCBs). (a) block diagram, (b) photograph of setup. 1 2 3 5 6 4 8 9 10 11 12 13 14 15 16 17 (b) 7 (a) complete chain consisted of 17 pieces (chips and PCBs), all numbered in Fig. 7: (1) The rotating wheel stimulus. (2) The retina. The retina looked at a rotating disc with two solid circles on it of two different radii. (3) A USB-AER board as mapper to reassign addresses and eliminate the polarity of brightness change. (4) A 1-to-3 splitter (one output for the PCI-AER board (7) to visualize the retina output, as shown in Fig. 8(a), and two outputs for two convolution chips). (5-6) Two convolution chips programmed with the kernels in Fig. 3c-d, to detect circumferences of radius 4 pixels and 9 pixels, respectively. They see the complete 64x64 retina image (with rectified activity; polarity is ignored) but provide a 32x32 output for only the central part of the retina image. This eliminates convolution edge effects. The output of each convolution chip is fed to a USB-AER board working as a monitor (8-9) to visualize their outputs (Fig. 8b). The left half is for the 4-radius kernel and the right half for the 9-radius kernel. The outputs of the convolution chips provide the center of the circumferences only if they have radius close to 4 pixels or 9 pixels, respectively. As can be seen, each convolution chip detects correctly the center of its corresponding circumference, but not the other. Both chips are tuned for the same feature but with different spatial scale. Both convolution chips outputs are merged onto a single AER bus using a merger (10) and then fed to a mapper (11) to properly reassign the address and bit signs for the winner-take-all ‘object’ chip (12), which correctly decides the centers of the convolution chip outputs. The object chip output is fed to a monitor (13) for visualization purposes. This output is shown in Fig. 8(c). The output of this chip is transformed using a mapper (14) and fed to the delay line chip (15), the outputs of which are fed through a mapper (16) to the learning (17) chip. The system as characterized can simultaneously trach two objects of different shape; we have connected but not yet studied trajectory learning and classification. 8 Conclusions In terms of the number of independent components, CAVIAR demonstrates the largest AER system yet assembled. It consists of 5 custom neuromorphic AER chips and at least 6 custom AER digital boards. Its functioning shows that AER can be used for assembling complex real time sensory processing systems and that relevant information about object size and location can be extracted and restored through a chain of feedforward stages. The CAVIAR system is a useful environment to develop reusable AER infrastructure and is capable of fast visual computation that is not limited by normal imager frame rate. Its continued development will result in insights about spike coding and representation. Acknowledgements This work was sponsored by EU grant IST-2001-34124 (CAVIAR), and Spanish grant TIC-2003-08164-C03 (SAMANTA). We thank K. Boahen for sharing AER interface Fig. 8: Captured AER outputs at different stages of processing chain. (a) at the retina output, (b) at the output of the 2 convolution chips, (c) at the output of the object chip. ‘I’ labels the activity of the inhibitory neurons. (a) (b) (b) (c) technology and the EU project ALAVLSI for sharing chip development and other AER computer interfaces [14]. References [1] M. Sivilotti, Wiring Considerations in Analog VLSI Systems with Application to FieldProgrammable Networks, Ph.D. Thesis, California Institute of Technology, Pasadena CA, 1991. [2] K. Boahen, “Point-to-Point Connectivity Between Neuromorphic Chips Using Address Events,” IEEE Trans. on Circuits and Systems Part-II, vol. 47, No. 5, pp. 416-434, May 2000. [3] J. P. Lazzaro and J. Wawrzynek, “A Multi-Sender Asynchronous Extension to the Address-Event Protocol,” 16th Conference on Advanced Research in VLSI, W. J. Dally, J. W. Poulton, and A. T. Ishii (Eds.), pp. 158-169, 1995. [4] T. Serrano-Gotarredona, A. G. Andreou, and B. Linares-Barranco, "AER Image Filtering Architecture for Vision Processing Systems," IEEE Trans. Circuits and Systems (Part II): Analog and Digital Signal Processing, vol. 46, No. 9, pp. 1064-1071, September 1999. [5] R. Serrano-Gotarredona, B. Linares-Barranco, and T. Serrano-Gotarredona, “A New Charge-Packet Driven Mismatch-Calibrated Integrate-and-Fire Neuron for Processing Positive and Negative Signals in AER-based Systems,” In Proc. of the IEEE Int. Symp. Circ. Syst., (ISCAS04), vol. 5, pp. 744-747 ,Vancouver, Canada, May 2004. [6] P. Lichtsteiner, T. Delbrück, and J. Kramer, "Improved ON/OFF temporally differentiating address-event imager," in 11th IEEE International Conference on Electronics, Circuits and Systems (ICECS2004), Tel Aviv, Israel, 2004, pp. 211-214. [7] T. Delbrück and D. Oberhoff, "Self-biasing low-power adaptive photoreceptor," in Proc. of the IEEE Int. Symp. Circ. Syst. (ISCAS04), pp. IV-844-847, 2004. [8] P. Lichtsteiner and T. Delbrück “64x64 AER Logarithmic Temporal Derivative Silicon Retina,” Research in Microelectronics and Electronics, Vol. 2, pp. 202-205, July 2005. [9] Liu, S.-C. and Kramer, J. and Indiveri, G. and Delbrück, T. and Burg, T. and Douglas, R. “Orientation-selective aVLSI spiking neurons”, Neural Networks, 14:(6/7) 629-643, Jul, 2001 [10]Oster, M. and Liu, S.-C. “A Winner-take-all Spiking Network with Spiking Inputs”, in 11th IEEE International Conference on Electronics, Circuits and Systems (ICECS 2004), Tel Aviv, pp. 203-206, 2004 [11]H. Kolle Riis and P. Haefliger, “Spike based learning with weak multi-level static memory,” In Proc. of the IEEE Int. Symp. Circ. Syst. (ISCAS04), vol. 5, pp. 393-395, Vancouver, Canada, May 2004. [12]P. Häfliger and H. Kolle Riis, “A Multi-Level Static Memory Cell,” In Proc. of the IEEE Int. Symp. Circ. Syst. (ISCAS04), vol. 1, pp. 22-25, Bangkok, Thailand, May 2003. [13]A. Linares-Barranco, G. Jiménez-Moreno, B. Linares-Barranco, and A. Civit-Ballcels, “On Algorithmic Rate-Coded AER Generation,” accepted for publication in IEEE Trans. Neural Networks, May 2006 (tentatively). [14]V. Dante, P. Del Giudice, and A. M. Whatley, “PCI-AER Hardware and Software for Interfacing to Address-Event Based Neuromorphic Systems”, The Neuromorphic Engineer, 2:(1) 5-6, 2005.
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A Hierarchical Compositional System for Rapid Object Detection Long Zhu and Alan Yuille Department of Statistics University of California at Los Angeles Los Angeles, CA 90095 {lzhu,yuille}@stat.ucla.edu Abstract We describe a hierarchical compositional system for detecting deformable objects in images. Objects are represented by graphical models. The algorithm uses a hierarchical tree where the root of the tree corresponds to the full object and lower-level elements of the tree correspond to simpler features. The algorithm proceeds by passing simple messages up and down the tree. The method works rapidly, in under a second, on 320 × 240 images. We demonstrate the approach on detecting cats, horses, and hands. The method works in the presence of background clutter and occlusions. Our approach is contrasted with more traditional methods such as dynamic programming and belief propagation. 1 Introduction Detecting objects rapidly in images is very important. There has recently been great progress in detecting objects with limited appearance variability, such as faces and text [1,2,3]. The use of the SIFT operator also enables rapid detection of rigid objects [4]. The detection of such objects can be performed in under a second even in very large images which makes real time applications practical, see [3]. There has been less progress for the rapid detection of deformable objects, such as hands, horses, and cats. Such objects can be represented compactly by graphical models, see [5,6,7,8], but their variations in shape and appearance makes searching for them considerably harder. Recent work has included the use of dynamic programming [5,6] and belief propagation [7,8] to perform inference on these graphical models by searching over different spatial configurations. These algorithms are successful at detecting objects but pruning was required to obtain reasonable convergence rates [5,7,8]. Even so, algorithms can take minutes to converge on images of size 320 × 240. In this paper, we propose an alternative methods for performing inference on graphical models of deformable objects. Our approach is based on representing objects in a probabilistic compositional hierarchical tree structure. This structure enables rapid detection of objects by passing messages up and down the tree structure. Our approach is fast with a typical speed of 0.6 seconds on a 320 × 240 image (without optimized code). Our approach can be applied to detect any object that can be represented by a graphical model. This includes the models mentioned above [5,6,7,8], compositional models [9], constellation models [10], models using chamfer matching [11] and models using deformable blur filters [12]. 2 Background Graphical models give an attractive framework for modeling object detection problems in computer vision. We use the models and notation described in [8]. The positions of feature points on the object are represented by {xi : i ∈Λ}. We augment this representation to include attributes of the points and obtain a representation {qi : i ∈ Λ}. These attributes can be used to model the appearance of the features in the image. For example, a feature point can be associated with an oriented intensity edge and qi can represent the orientation [8]. Alternatively, the attribute could represent the output of a blurred edge filter [12], or the appearance properties of a constellation model part [10]. There is a prior probability distribution on the configuration of the model P({qi}) and a likelihood function for generating the image data P(D|{qi}). We use the same likelihood model as [8]. Our priors are similar to [5,8,12], being based on deformations away from a prototype template. Inference consists of maximizing the posterior P({qi}|D) = P(D|{qi})P({qi})/P(D). As described in [8], this corresponds to a maximizing a posterior of form: P({qi}|D) = 1 Z Y i ψi(qi) Y i,j ψij(qi, qj), (1) where {ψi(qi)} and {ψij(qi, qj)} are the unary and pairwise potentials of the graph. The unary potentials model how well the individual features match to positions in the image. The binary potentials impose (probabilistic) constraints about the spatial relationships between feature points. Algorithms such as dynamic programming [5,6] and belief propagation [7,8] have been used to search for optima of P({qi}|D). But the algorithms are time consuming because each state variable qi can take a large number of values (each feature point on the template can, in principle, match any point in the 240 × 320 image). Pruning and other ingenious techniques are used to speed up the search [5,7,8]. But performance remains at speeds of seconds to minutes. 3 The Hierarchical Compositional System We define a compositional hierarchy by breaking down the representation {qi : i ∈Λ} into substructures which have their own probability models. At the first level, we group elements into K1 subsets {qi : i ∈S1 a} where Λ = ∪K1 a=1S1 a, S1 a ∩S1 b = ∅, a ̸= b. These subsets correspond to meaningful parts of the object, such as ears and other features. See figure (1) for the basic structure. Specific examples for cats and horses will be given later. For each of these subsets we define a generative model Pa(D|{qi : i ∈S1 a}) and a prior Pa({qi : i ∈S1 a}). These generative and prior models are inherited from the full model, see equation (1), by simply cutting the connections between the subset S1 a and the Λ/S1 a (the remaining features on the object). Hence Pa1(D|{qi : i ∈S1 a}) = 1 Za1 Y i∈S1 a ψi(qi) Figure 1: The Hierarchical Compositional structure. The full model contains all the nodes S3 1. This is decomposed into subsets S2 1, S2 2, S2 3 corresponding to sub-features. These, in turn, can be decomposed into subsets corresponding to more elementary features. Pa1({qi : i ∈S1 a}) = 1 ˆZa1 Y i,j∈S1a ψij(qi, qj). (2) We repeat the same process at the second and higher levels. The subsets {S1 a : a = 1, ..., K1} are composed to form a smaller selection of subsets {S2 b : b = 1, ..., K2}, so that Λ = ∪K2 a=1S2 a, S2 a ∩S2 b = ∅, a ̸= b and each S1 a is contained entirely inside one S2 b . Again the S2 b are selected to correspond to meaningful parts of the object. Their generative models and prior distributions are again obtained from the full model, see equation (1). by cutting them off the links to the remaining nodes Λ/S2 b . The algorithm is run using two thresholds T1, T2. For each subset, say S1 a, we define the evidence to be Pa1(D|{zµ i : i ∈S1 a})Pa1({zµ i : i ∈S1 a}). We determine all possible configurations {zµ i : i ∈S1 a} such that evidence of each configuration is above T1. This gives a (possibly large) set of positions for the {qi : i ∈S1 a}. We apply non-maximum suppression to reduce many similar configurations in same local area to the one with maximum evidence (measured locally). We observe that a little displacement of position does not change optimality much for upper level matching. Typically, non-maximum suppression keeps around 30 ∼500 candidate configurations for each node. These remaining configurations can be considered as proposals [13] and are passed up the tree to the subset S2 b which contains S1 a. Node S2 b evaluates the proposals to determine which ones are consistent, thus detecting composites of the subfeatures. There is also top-down message passing which occurs when one part of a node S2 b contains high evidence – e.g. Pa1(D|{zµ i : i ∈S1 a})Pa1({zµ i : i ∈S1 a}) > T2 – but the other child nodes have no consistent values. In this case, we allow the matching to proceed if the combined matching strength is above threshold T1. This mechanism enables the high-level models and, in particular, the priors for the relative positions of the sub-nodes to overcome weak local evidence. This performs a similar function to Coughlan and Shen’s dynamic quantization scheme [8]. More sophisticated versions of this approach can be considered. For example, we could use the proposals to activate a data driven Monte Carlo Markov Chain (DDMCMC) algorithm [13]. To our knowledge, the use of hierarchical proposals of this type is unknown in the Monte Carlo sampling literature. 4 Experimental Results We illustrate our hierarchical compositional system on examples of cats, horses, and hands. The images include background clutter and the objects can be partially occluded. Figure 2: The prototype cat (top left panel), edges after grouping (top right panel), prototype template for ears and top of head (bottom left panel), and prototype for ears and eyes (bottom right panel). 15 points are used for the ears and 24 for the head. First we preprocess the image using a Canny edge detector followed by simple edge grouping which eliminates isolated edges. Edge detection and edge grouping is illustrated in the top panels of figure (2). This figure is used to construct a prototype template for the ears, eyes, and head – see bottom panels of figure (2). We construct a graphical model for the cat as described in section (2). Then we define a hierarchical structure, see figure (3). Figure 3: Hierarchy Structure for Cat Template. Next we illustrate the results on several cat images, see figure (4). Several of these images were used in [8] and we thank Coughlan and Shen for supplying them. In all examples, our algorithm detects the cat correctly despite the deformations of the cat from the prototype, see figure (2). The detection was performed in less than 0.6 seconds (with unoptimized code). The images are 320 × 240 and the preprocessing time is included. The algorithm is efficient since the subfeatures give bottom-up proposals which constraint the positions of the full model. For example, figure (5) shows the proposals for ears for the cluttered cat image (center panel of figure (4). Figure 4: Cat with Occlusion (top panels). Cat with clutter (centre panel). Cat with eyes (bottom panel). We next illustrate our approach on the tasks of detecting horses. This requires a more complicated hierarchy, see figure (6). The algorithm succeeds in detecting the horse, see right panels of figure (7), using the prototype template shown in the left panel of figure (7). Finally, we illustrate this approach for the much studied task of detecting hands, see [5,11]. Our approach detects hand from the Cambridge dataset in under a second, see figure (8). (We are grateful to Thayananthan, Stenger, Torr, and Cipolla for supplying these images). Figure 5: Cat Proposals: Left ears (left three panels). Right ears (right three panels). Figure 6: Horse Hierarchy. This is more complicated than the cat. Figure 7: The left panels show the prototype horse (top left panel) and its feature points (bottom left panel). The right panel shows the input image (top right panel) and the position of the horse as detected by the algorithm (bottom right panel). Figure 8: Prototype hand (top left panel), edge map of prototype hand (bottom left panel), Test hand (top right panel), Test hand edges (bottom right panel). 40 points are used. 5 Comparison with alternative methods We ran the algorithm on image of typical size 320×240. There were usually 4000 segments after edge grouping. The templates had between 15 and 24 points. The average speed was 0.6 seconds on a laptop with 1.6 G Intel Pentium CPU (including all processing: edge detector, edge grouping, and object detection. Other papers report times of seconds to minutes for detecting deformable objects from similar images [5,6,7,8]. So our approach is up to 100 times faster. The Soft-Assign method in [15] has the ability to deal with objects with around 200 key points, but requires the initialization of the template to be close to the target object. This requirement is not practical in many applications. In our proposed method, there is no need to initialize the template near to the target. Our hierarchical compositional tree structure is similar to the standard divide and conquer strategy used in some computer science algorithms. This may roughly be expected to scale as log N where N is the number of points on the deformable template. But precise complexity convergence results are difficult to obtain because they depend on the topology of the template, the amount of clutter in the background, and other factors. This approach can be applied to any graphical model such as [10,12]. It is straightforward to design hierarchial compositional structures for objects based on their natural decompositions into parts. There are alternative, and more sophisticated ways, to perform inference on graphical models by decomposing them into sub-graphs, see for example [14]. But these are typically far more computationally demanding. 6 Conclusion We have presented a hierarchical compositional system for rapidly detecting deformable objects in images by performing inference on graphical models. Computation is performed by passing messages up and down the tree. The systems detects objects in under a second on images of size 320×240. This makes the approach practical for real world applications. Our approach is similar in spirit to DDMCMC [13] in that we use proposals to guide the search for objects. In this paper, the proposals are based on a hierarchy of features which enables efficient computation. The low-level features propose more complex features which are validated by the probability models of the complex features. We have not found it necessary to perform stochastic sampling, though it is straightforward to do so in this framework. Acknowledgments This research was supported by NSF grant 0413214. References [1] Viola, P. and Jones, M. (2001). ”Fast and Robust Classification using Asymmetric AdaBoost and a Detector Cascade”. In Proceedings NIPS01. [2] Schniederman, H. and Kanade, T. (2000). ”A Statistical method for 3D object detection applied to faces and cars”. In Computer Vision and Pattern Recognition. [3] Chen, X. and Yuille, A.L. (2004). AdaBoost Learning for Detecting and Reading Text in City Scenes. Proceedings CVPR. [4] Lowe, D.G. (1999). “Object recognition from local scale-invariant features.” In Proc. International Conference on Computer Vision ICCV. Corfu, pages 1150-1157. [5] Coughlan, J.M., Snow, D., English, C. and Yuille, A.L. (2000). “Efficient Deformable Template Detection and Localization without User Initialization”. Computer Vision and Image Understanding. 78, pp 303-319. [6] Felzenswalb, P. (2005). “Representation and Detection of Deformable Shapes”. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 27, No. 2. [7] Coughlan, J.M., and Ferreira, S. (2002). “Finding Deformable Shapes using Loopy Belief Propoagation”. In Proceedings European Conference of Computer Vision.. 2002. [8] Coughlan, J.M., and Shen, H. (2004). “Shape Matching with Belief Propagation: Using Dynamic Quantization to Accomodate Occlusion and Clutter”. In GMBV . [9] Geman, S. Potter, D. and Chi, Z. (2002). “ Composition systems”. Quarterly of Applied Mathematics, LX, pp 707-736. [10] Fergus, R., Perona, P. and Zisserman, A. (2003) “Object Class Recognition by Unsupervised Scale-Invariant Learning”. Proceeding CVPR. (2), pp 264-271. [11] Thayananthan, A. Stenger, B., Torr, P. and Cipolla, R. (2003). ”Shape context and chamfer matching in cluttered scenes,” In Proc. Conf. Comp. Vision Pattern Rec., pp. 127–133. [12] Berg, A.C., Berg, T.L., and Malik, J. (2005). “Shape Matching and Object Recognition using Low Distortion Correspondence”. Proceedings CVPR. [13] Tu, Z., Chen, X., Yuille, A.L., and Zhu, S.C. (2003). “Image Parsing: Unifying Segmentation, Detection, and Recognition”. In Proceedings ICCV. [14] Wainwright, M.J., Jaakkola, T.S., and Willsky., A.S. “Tree-Based Reparamterization Framework for Analysis of Sum-Product and Related Algorithms”. IEEE Transactions on Information Theory. Vol. 49, pp 1120-1146. No. 5. 2003. [15] Chui,H. and Rangarajan, A., A New Algorithm for Non-Rigid Point Matching. In Proceedings CVPR 2000.
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Conditional Visual Tracking in Kernel Space Cristian Sminchisescu1,2,3 Atul Kanujia3 Zhiguo Li3 Dimitris Metaxas3 1TTI-C, 1497 East 50th Street, Chicago, IL, 60637, USA 2University of Toronto, Department of Computer Science, Canada 3Rutgers University, Department of Computer Science, USA crismin@cs.toronto.edu, {kanaujia,zhli,dnm}@cs.rutgers.edu Abstract We present a conditional temporal probabilistic framework for reconstructing 3D human motion in monocular video based on descriptors encoding image silhouette observations. For computational efficiency we restrict visual inference to low-dimensional kernel induced non-linear state spaces. Our methodology (kBME) combines kernel PCA-based non-linear dimensionality reduction (kPCA) and Conditional Bayesian Mixture of Experts (BME) in order to learn complex multivalued predictors between observations and model hidden states. This is necessary for accurate, inverse, visual perception inferences, where several probable, distant 3D solutions exist due to noise or the uncertainty of monocular perspective projection. Low-dimensional models are appropriate because many visual processes exhibit strong non-linear correlations in both the image observations and the target, hidden state variables. The learned predictors are temporally combined within a conditional graphical model in order to allow a principled propagation of uncertainty. We study several predictors and empirically show that the proposed algorithm positively compares with techniques based on regression, Kernel Dependency Estimation (KDE) or PCA alone, and gives results competitive to those of high-dimensional mixture predictors at a fraction of their computational cost. We show that the method successfully reconstructs the complex 3D motion of humans in real monocular video sequences. 1 Introduction and Related Work We consider the problem of inferring 3D articulated human motion from monocular video. This research topic has applications for scene understanding including human-computer interfaces, markerless human motion capture, entertainment and surveillance. A monocular approach is relevant because in real-world settings the human body parts are rarely completely observed even when using multiple cameras. This is due to occlusions form other people or objects in the scene. A robust system has to necessarily deal with incomplete, ambiguous and uncertain measurements. Methods for 3D human motion reconstruction can be classified as generative and discriminative. They both require a state representation, namely a 3D human model with kinematics (joint angles) or shape (surfaces or joint positions) and they both use a set of image features as observations for state inference. The computational goal in both cases is the conditional distribution for the model state given image observations. Generative model-based approaches [6, 16, 14, 13] have been demonstrated to flexibly reconstruct complex unknown human motions and to naturally handle problem constraints. However it is difficult to construct reliable observation likelihoods due to the complexity of modeling human appearance. This varies widely due to different clothing and deformation, body proportions or lighting conditions. Besides being somewhat indirect, the generative approach further imposes strict conditional independence assumptions on the temporal observations given the states in order to ensure computational tractability. Due to these factors inference is expensive and produces highly multimodal state distributions [6, 16, 13]. Generative inference algorithms require complex annealing schedules [6, 13] or systematic non-linear search for local optima [16] in order to ensure continuing tracking. These difficulties motivate the advent of a complementary class of discriminative algorithms [10, 12, 18, 2], that approximate the state conditional directly, in order to simplify inference. However, inverse, observation-to-state multivalued mappings are difficult to learn (see e.g. fig. 1a) and a probabilistic temporal setting is necessary. In an earlier paper [15] we introduced a probabilistic discriminative framework for human motion reconstruction. Because the method operates in the originally selected state and observation spaces that can be task generic, therefore redundant and often high-dimensional, inference is more expensive and can be less robust. To summarize, reconstructing 3D human motion in a Figure 1: (a, Left) Example of 180o ambiguity in predicting 3D human poses from silhouette image features (center). It is essential that multiple plausible solutions (e.g. F1 and F2) are correctly represented and tracked over time. A single state predictor will either average the distant solutions or zig-zag between them, see also tables 1 and 2. (b, Right) A conditional chain model. The local distributions p(yt|yt−1, zt) or p(yt|zt) are learned as in fig. 2. For inference, the predicted local state conditional is recursively combined with the filtered prior c.f. (1). conditional temporal framework poses the following difficulties: (i) The mapping between temporal observations and states is multivalued (i.e. the local conditional distributions to be learned are multimodal), therefore it cannot be accurately represented using global function approximations. (ii) Human models have multivariate, high-dimensional continuous states of 50 or more human joint angles. The temporal state conditionals are multimodal which makes efficient Kalman filtering algorithms inapplicable. General inference methods (particle filters, mixtures) have to be used instead, but these are expensive for high-dimensional models (e.g. when reconstructing the motion of several people that operate in a joint state space). (iii) The components of the human state and of the silhouette observation vector exhibit strong correlations, because many repetitive human activities like walking or running have low intrinsic dimensionality. It appears wasteful to work with high-dimensional states of 50+ joint angles. Even if the space were truly high-dimensional, predicting correlated state dimensions independently may still be suboptimal. In this paper we present a conditional temporal estimation algorithm that restricts visual inference to low-dimensional, kernel induced state spaces. To exploit correlations among observations and among state variables, we model the local, temporal conditional distributions using ideas from Kernel PCA [11, 19] and conditional mixture modeling [7, 5], here adapted to produce multiple probabilistic predictions. The corresponding predictor is referred to as a Conditional Bayesian Mixture of Low-dimensional Kernel-Induced Experts (kBME). By integrating it within a conditional graphical model framework (fig. 1b), we can exploit temporal constraints probabilistically. We demonstrate that this methodology is effective for reconstructing the 3D motion of multiple people in monocular video. Our contribution w.r.t. [15] is a probabilistic conditional inference framework that operates over a non-linear, kernel-induced low-dimensional state spaces, and a set of experiments (on both real and artificial image sequences) that show how the proposed framework positively compares with powerful predictors based on KDE, PCA, or with the high-dimensional models of [15] at a fraction of their cost. 2 Probabilistic Inference in a Kernel Induced State Space We work with conditional graphical models with a chain structure [9], as shown in fig. 1b, These have continuous temporal states yt, t = 1 . . . T, observations zt. For compactness, we denote joint states Yt = (y1, y2, . . . , yt) or joint observations Zt = (z1, . . . , zt). Learning and inference are based on local conditionals: p(yt|zt) and p(yt|yt−1, zt), with yt and zt being low-dimensional, kernel induced representations of some initial model having state xt and observation rt. We obtain zt, yt from rt, xt using kernel PCA [11, 19]. Inference is performed in a low-dimensional, non-linear, kernel induced latent state space (see fig. 1b and fig. 2 and (1)). For display or error reporting, we compute the original conditional p(x|r), or a temporally filtered version p(xt|Rt), Rt = (r1, r2, . . . , rt), using a learned pre-image state map [3]. 2.1 Density Propagation for Continuous Conditional Chains For online filtering, we compute the optimal distribution p(yt|Zt) for the state yt, conditioned by observations Zt up to time t. The filtered density can be recursively derived as: p(yt|Zt) = Z yt−1 p(yt|yt−1, zt)p(yt−1|Zt−1) (1) We compute using a conditional mixture for p(yt|yt−1, zt) (a Bayesian mixture of experts c.f. §2.2) and the prior p(yt−1|Zt−1), each having, say M components. We integrate M 2 pairwise products of Gaussians analytically. The means of the expanded posterior are clustered and the centers are used to initialize a reduced M-component Kullback-Leibler approximation that is refined using gradient descent [15]. The propagation rule (1) is similar to the one used for discrete state labels [9], but here we work with multivariate continuous state spaces and represent the local multimodal state conditionals using kBME (fig. 2), and not log-linear models [9] (these would require intractable normalization). This complex continuous model rules out inference based on Kalman filtering or dynamic programming [9]. 2.2 Learning Bayesian Mixtures over Kernel Induced State Spaces (kBME) In order to model conditional mappings between low-dimensional non-linear spaces we rely on kernel dimensionality reduction and conditional mixture predictors. The authors of KDE [19] propose a powerful structured unimodal predictor. This works by decorrelating the output using kernel PCA and learning a ridge regressor between the input and each decorrelated output dimension. Our procedure is also based on kernel PCA but takes into account the structure of the studied visual problem where both inputs and outputs are likely to be low-dimensional and the mapping between them multivalued. The output variables xi are projected onto the column vectors of the principal space in order to obtain their principal coordinates yi. A z ∈P(Fr) p(y|z) / y ∈P(Fx) (Q Q Q Q Q Q Q Q Q Q Q Q Q Φr(r) ⊂Fr kP CA O Φx(x) ⊂Fx kP CA O x ≈PreImage(y) r ∈R ⊂Rr Φr O x ∈X ⊂Rx Φx O p(x|r) ≈p(x|y) Figure 2: The learned low-dimensional predictor, kBME, for computing p(x|r) ≡ p(xt|rt), ∀t. (We similarly learn p(xt|xt−1, rt), with input (x, r) instead of r – here we illustrate only p(x|r) for clarity.) The input r and the output x are decorrelated using Kernel PCA to obtain z and y respectively. The kernels used for the input and output are Φr and Φx, with induced feature spaces Fr and Fx, respectively. Their principal subspaces obtained by kernel PCA are denoted by P(Fr) and P(Fx), respectively. A conditional Bayesian mixture of experts p(y|z) is learned using the low-dimensional representation (z, y). Using learned local conditionals of the form p(yt|zt) or p(yt|yt−1, zt), temporal inference can be efficiently performed in a low-dimensional kernel induced state space (see e.g. (1) and fig. 1b). For visualization and error measurement, the filtered density, e.g. p(yt|Zt), can be mapped back to p(xt|Rt) using the pre-image c.f. (3). similar procedure is performed on the inputs ri to obtain zi. In order to relate the reduced feature spaces of z and y (P(Fr) and P(Fx)), we estimate a probability distribution over mappings from training pairs (zi, yi). We use a conditional Bayesian mixture of experts (BME) [7, 5] in order to account for ambiguity when mapping similar, possibly identical reduced feature inputs to very different feature outputs, as common in our problem (fig. 1a). This gives a model that is a conditional mixture of low-dimensional kernel-induced experts (kBME): p(y|z) = M X j=1 g(z|δj)N(y|Wjz, Σj) (2) where g(z|δj) is a softmax function parameterized by δj and (Wj, Σj) are the parameters and the output covariance of expert j, here a linear regressor. As in many Bayesian settings [17, 5], the weights of the experts and of the gates, Wj and δj, are controlled by hierarchical priors, typically Gaussians with 0 mean, and having inverse variance hyperparameters controlled by a second level of Gamma distributions. We learn this model using a double-loop EM and employ ML-II type approximations [8, 17] with greedy (weight) subset selection [17, 15]. Finally, the kBME algorithm requires the computation of pre-images in order to recover the state distribution x from it’s image y ∈P(Fx). This is a closed form computation for polynomial kernels of odd degree. For more general kernels optimization or learning (regression based) methods are necessary [3]. Following [3, 19], we use a sparse Bayesian kernel regressor to learn the pre-image. This is based on training data (xi, yi): p(x|y) = N(x|AΦy(y), Ω) (3) with parameters and covariances (A, Ω). Since temporal inference is performed in the low-dimensional kernel induced state space, the pre-image function needs to be calculated only for visualizing results or for the purpose of error reporting. Propagating the result from the reduced feature space P(Fx) to the output space X produces a Gaussian mixture with M elements, having coefficients g(z|δj) and components N(x|AΦy(Wjz), AJΦyΣjJ⊤ ΦyA⊤+Ω), where JΦy is the Jacobian of the mapping Φy. 3 Experiments We run experiments on both real image sequences (fig. 5 and fig. 6) and on sequences where silhouettes were artificially rendered. The prediction error is reported in degrees (for mixture of experts, this is w.r.t. the most probable one, but see also fig. 4a), and normalized per joint angle, per frame. The models are learned using standard cross-validation. Pre-images are learned using kernel regressors and have average error 1.7o. Training Set and Model State Representation: For training we gather pairs of 3D human poses together with their image projections, here silhouettes, using the graphics package Maya. We use realistically rendered computer graphics human surface models which we animate using human motion capture [1]. Our original human representation (x) is based on articulated skeletons with spherical joints and has 56 skeletal d.o.f. including global translation. The database consists of 8000 samples of human activities including walking, running, turns, jumps, gestures in conversations, quarreling and pantomime. Image Descriptors: We work with image silhouettes obtained using statistical background subtraction (with foreground and background models). Silhouettes are informative for pose estimation although prone to ambiguities (e.g. the left / right limb assignment in side views) or occasional lack of observability of some of the d.o.f. (e.g. 180o ambiguities in the global azimuthal orientation for frontal views, e.g. fig. 1a). These are multiplied by intrinsic forward / backward monocular ambiguities [16]. As observations r, we use shape contexts extracted on the silhouette [4] (5 radial, 12 angular bins, size range 1/8 to 3 on log scale). The features are computed at different scales and sizes for points sampled on the silhouette. To work in a common coordinate system, we cluster all features in the training set into K = 50 clusters. To compute the representation of a new shape feature (a point on the silhouette), we ‘project’ onto the common basis by (inverse distance) weighted voting into the cluster centers. To obtain the representation (r) for a new silhouette we regularly sample 200 points on it and add all their feature vectors into a feature histogram. Because the representation uses overlapping features of the observation the elements of the descriptor are not independent. However, a conditional temporal framework (fig. 1b) flexibly accommodates this. For experiments, we use Gaussian kernels for the joint angle feature space and dot product kernels for the observation feature space. We learn state conditionals for p(yt|zt) and p(yt|yt−1, zt) using 6 dimensions for the joint angle kernel induced state space and 25 dimensions for the observation induced feature space, respectively. In fig. 3b) we show an evaluation of the efficacy of our kBME predictor for different dimensions in the joint angle kernel induced state space (the observation feature space dimension is here 50). On the analyzed dancing sequence, that involves complex motions of the arms and the legs, the non-linear model significantly outperforms alternative PCA methods and gives good predictions for compact, low-dimensional models.1 In tables 1 and 2, as well as fig. 4, we perform quantitative experiments on artificially rendered silhouettes. 3D ground truth joint angles are available and this allows a more 1Running times: On a Pentium 4 PC (3 GHz, 2 GB RAM), a full dimensional BME model with 5 experts takes 802s to train p(xt|xt−1, rt), whereas a kBME (including the pre-image) takes 95s to train p(yt|yt−1, zt). The prediction time is 13.7s for BME and 8.7s (including the pre-image cost 1.04s) for kBME. The integration in (1) takes 2.67s for BME and 0.31s for kBME. The speed-up for kBME is significant and likely to increase with original models having higher dimensionality. 1 2 3 4 5 6 7 8 1 10 100 1000 Degree of Multimodality Number of Clusters 0 20 40 60 1 10 100 Number of Dimensions Prediction Error kBME KDE_RVM PCA_BME PCA_RVM Figure 3: (a, Left) Analysis of ‘multimodality’ for a training set. The input zt dimension is 25, the output yt dimension is 6, both reduced using kPCA. We cluster independently in (yt−1, zt) and yt using many clusters (2100) to simulate small input perturbations and we histogram the yt clusters falling within each cluster in (yt−1, zt). This gives intuition on the degree of ambiguity in modeling p(yt|yt−1, zt), for small perturbations in the input. (b, Right) Evaluation of dimensionality reduction methods for an artificial dancing sequence (models trained on 300 samples). The kBME is our model §2.2, whereas the KDE-RVM is a KDE model learned with a Relevance Vector Machine (RVM) [17] feature space map. PCA-BME and PCA-RVM are models where the mappings between feature spaces (obtained using PCA) is learned using a BME and a RVM. The non-linearity is significant. Kernel-based methods outperform PCA and give low prediction error for 5-6d models. systematic evaluation. Notice that the kernelized low-dimensional models generally outperform the PCA ones. At the same time, they give results competitive to the ones of high-dimensional BME predictors, while being lower-dimensional and therefore significantly less expensive for inference, e.g. the integral in (1). In fig. 5 and fig. 6 we show human motion reconstruction results for two real image sequences. Fig. 5 shows the good quality reconstruction of a person performing an agile jump. (Given the missing observations in a side view, 3D inference for the occluded body parts would not be possible without using prior knowledge!) For this sequence we do inference using conditionals having 5 modes and reduced 6d states. We initialize tracking using p(yt|zt), whereas for inference we use p(yt|yt−1, zt) within (1). In the second sequence in fig. 6, we simultaneously reconstruct the motion of two people mimicking domestic activities, namely washing a window and picking an object. Here we do inference over a product, 12-dimensional state space consisting of the joint 6d state of each person. We obtain good 3D reconstruction results, using only 5 hypotheses. Notice however, that the results are not perfect, there are small errors in the elbow and the bending of the knee for the subject at the l.h.s., and in the different wrist orientations for the subject at the r.h.s. This reflects the bias of our training set. KDE-RR RVM KDE-RVM BME kBME Walk and turn 10.46 4.95 7.57 4.27 4.69 Conversation 7.95 4.96 6.31 4.15 4.79 Run and turn left 5.22 5.02 6.25 5.01 4.92 Table 1: Comparison of average joint angle prediction error for different models. All kPCA-based models use 6 output dimensions. Testing is done on 100 video frames for each sequence, the inputs are artificially generated silhouettes, not in the training set. 3D joint angle ground truth is used for evaluation. KDE-RR is a KDE model with ridge regression (RR) for the feature space mapping, KDE-RVM uses an RVM. BME uses a Bayesian mixture of experts with no dimensionality reduction. kBME is our proposed model. kPCAbased methods use kernel regressors to compute pre-images. 1 2 3 4 5 0 5 10 15 20 25 30 Expert Prediction Expert Number Frequency − Close to ground truth 1 2 3 4 5 0 2 4 6 8 10 12 14 Current Expert Frequency − Closest to Ground truth 1st Probable Prev Output 2nd Probable Prev Output 3rd Probable Prev Output 4th Probable Prev Output 5th Probable Prev Output Figure 4: (a, Left) Histogram showing the accuracy of various expert predictors: how many times the expert ranked as the k-th most probable by the model (horizontal axis) is closest to the ground truth. The model is consistent (the most probable expert indeed is the most accurate most frequently), but occasionally less probable experts are better. (b, Right) Histograms show the dynamics of p(yt|yt−1, zt), i.e. how the probability mass is redistributed among experts between two successive time steps, in a conversation sequence. KDE-RR RVM KDE-RVM BME kBME Walk and turn back 7.59 6.9 7.15 3.6 3.72 Run and turn 17.7 16.8 16.08 8.2 8.01 Table 2: Joint angle prediction error computed for two complex sequences with walks, runs and turns, thus more ambiguity (100 frames). Models have 6 state dimensions. Unimodal predictors average competing solutions. kBME has significantly lower error. Figure 5: Reconstruction of a jump (selected frames). Top: original image sequence. Middle: extracted silhouettes. Bottom: 3D reconstruction seen from a synthetic viewpoint. 4 Conclusion We have presented a probabilistic framework for conditional inference in latent kernelinduced low-dimensional state spaces. Our approach has the following properties: (a) Figure 6: Reconstructing the activities of 2 people operating in an 12-d state space (each person has its own 6d state). Top: original image sequence. Bottom: 3D reconstruction seen from a synthetic viewpoint. Accounts for non-linear correlations among input or output variables, by using kernel nonlinear dimensionality reduction (kPCA); (b) Learns probability distributions over mappings between low-dimensional state spaces using conditional Bayesian mixture of experts, as required for accurate prediction. In the resulting low-dimensional kBME predictor ambiguities and multiple solutions common in visual, inverse perception problems are accurately represented. (c) Works in a continuous, conditional temporal probabilistic setting and offers a formal management of uncertainty. We show comparisons that demonstrate how the proposed approach outperforms regression, PCA or KDE alone for reconstructing the 3D human motion in monocular video. Future work we will investigate scaling aspects for large training sets and alternative structured prediction methods. References [1] CMU Human Motion DataBase. Online at http://mocap.cs.cmu.edu/search.html, 2003. [2] A. Agarwal and B. Triggs. 3d human pose from silhouettes by Relevance Vector Regression. In CVPR, 2004. [3] G. Bakir, J. Weston, and B. Scholkopf. Learning to find pre-images. In NIPS, 2004. [4] S. Belongie, J. Malik, and J. Puzicha. Shape matching and object recognition using shape contexts. PAMI, 24, 2002. [5] C. Bishop and M. Svensen. Bayesian mixtures of experts. In UAI, 2003. [6] J. Deutscher, A. Blake, and I. Reid. Articulated Body Motion Capture by Annealed Particle Filtering. In CVPR, 2000. [7] M. Jordan and R. Jacobs. Hierarchical mixtures of experts and the EM algorithm. Neural Computation, (6):181–214, 1994. [8] D. Mackay. Bayesian interpolation. Neural Computation, 4(5):720–736, 1992. [9] A. McCallum, D. Freitag, and F. Pereira. Maximum entropy Markov models for information extraction and segmentation. In ICML, 2000. [10] R. Rosales and S. Sclaroff. Learning Body Pose Via Specialized Maps. In NIPS, 2002. [11] B. Sch¨olkopf, A. Smola, and K. M¨uller. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10:1299–1319, 1998. [12] G. Shakhnarovich, P. Viola, and T. Darrell. Fast Pose Estimation with Parameter Sensitive Hashing. In ICCV, 2003. [13] L. Sigal, S. Bhatia, S. Roth, M. Black, and M. Isard. Tracking Loose-limbed People. In CVPR, 2004. [14] C. Sminchisescu and A. Jepson. Generative Modeling for Continuous Non-Linearly Embedded Visual Inference. In ICML, pages 759–766, Banff, 2004. [15] C. Sminchisescu, A. Kanaujia, Z. Li, and D. Metaxas. Discriminative Density Propagation for 3D Human Motion Estimation. In CVPR, 2005. [16] C. Sminchisescu and B. Triggs. Kinematic Jump Processes for Monocular 3D Human Tracking. In CVPR, volume 1, pages 69–76, Madison, 2003. [17] M. Tipping. Sparse Bayesian learning and the Relevance Vector Machine. JMLR, 2001. [18] C. Tomasi, S. Petrov, and A. Sastry. 3d tracking = classification + interpolation. In ICCV, 2003. [19] J. Weston, O. Chapelle, A. Elisseeff, B. Scholkopf, and V. Vapnik. Kernel dependency estimation. In NIPS, 2002.
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Fast Krylov Methods for N-Body Learning Nando de Freitas Department of Computer Science University of British Columbia nando@cs.ubc.ca Yang Wang School of Computing Science Simon Fraser University ywang12@cs.sfu.ca Maryam Mahdaviani Department of Computer Science University of British Columbia maryam@cs.ubc.ca Dustin Lang Department of Computer Science University of Toronto dalang@cs.ubc.ca Abstract This paper addresses the issue of numerical computation in machine learning domains based on similarity metrics, such as kernel methods, spectral techniques and Gaussian processes. It presents a general solution strategy based on Krylov subspace iteration and fast N-body learning methods. The experiments show significant gains in computation and storage on datasets arising in image segmentation, object detection and dimensionality reduction. The paper also presents theoretical bounds on the stability of these methods. 1 Introduction Machine learning techniques based on similarity metrics have gained wide acceptance over the last few years. Spectral clustering [1] is a typical example. Here one forms a Laplacian matrix L = D−1/2WD−1/2, where the entries of W measure the similarity between data points xi ∈X, i = 1, . . . , N. For example, a popular choice is to set the entries of W to wij = e−1 σ ∥xi−xj∥2 where σ is a user-specified parameter. D is a normalizing diagonal matrix with entries di = P j wij. The clusters can be found by running, say, K-means on the eigenvectors of L. K-means generates better clusters on this nonlinear embedding of the data provided one adopts a suitable similarity metric. The list of machine learning domains where one forms a covariance or similarity matrix (be it W, D−1W or D −W) is vast and includes ranking on nonlinear manifolds [2], semi-supervised and active learning [3], Gaussian processes [4], Laplacian eigen-maps [5], stochastic neighbor embedding [6], multi-dimensional scaling, kernels on graphs [7] and many other kernel methods for dimensionality reduction, feature extraction, regression and classification. In these settings, one is interested in either inverting the similarity matrix or finding some of its eigenvectors. The computational cost of both of these operations is O(N 3) while the storage requirement is O(N 2). These costs are prohibitively large in applications where one encounters massive quantities of data points or where one is interested in real-time solutions such as spectral image segmentation for mobile robots [8]. In this paper, we present general numerical techniques for reducing the computational cost to O(N log N), or even O(N) in specific cases, and the storage cost to O(N). These reductions are achieved by combining Krylov subspace iterative solvers (such as Arnoldi, Lanczos, GMRES and conjugate gradients) with fast kernel density estimation (KDE) techniques (such as fast multipole expansions, the fast Gauss transform and dual tree recursions [9, 10, 11]). Specific Krylov methods have been applied to kernel problems. For example, [12] uses Lanczos for spectral clustering and [4] uses conjugate gradients for Gaussian processes. However, the use of fast KDE methods, in particular fast multipole methods, to further accelerate these techniques has only appeared in the context of interpolation [13] and our paper on semi-supervised learning [8]. Here, we go for a more general exposition and present several new examples, such as fast nonlinear embeddings and fast Gaussian processes. More importantly, we attack the issue of stability of these methods. Fast KDE techniques have guaranteed error bounds. However, if these techiques are used inside iterative schemes based on orthogonalization of the Krylov subspace, there is a danger that the errors might grow over iterations. In practice, good behaviour has been observed. In Section 4, we present theoretical results that explain these observations and shed light on the behaviour of these algorithms. Before doing so, we begin with a very brief review of Krylov solvers and fast KDE methods. 2 Krylov subspace iteration This section is a compressed overview of Krylov subspace iteration. The main message is that Krylov methods are very efficient algorithms for solving linear systems and eigenvalue problems, but they require a matrix vector multiplication at each iteration. In the next section, we replace this expensive matrix-vector multiplication with a call to fast KDE routines. Readers happy with this message and familiar with Krylov methods, such as conjugate gradients and Lanczos, can skip the rest of this section. For ease of presentation, let the similarity matrix be simply A = W ∈RN×N, with entries aij = a(xi, xj). (One can easily handle other cases, such as A = D−1W and A = D−W.) Typical measures of similarity include polynomial a(xi, xj) = (xixT j +b)p, Gaussian a(xi, xj) = e−1 σ (xi−xj)(xi−xj)T and sigmoid a(xi, xj) = tanh(αxixT j −β) kernels, where xixT j denotes a scalar inner product. Our goal is to solve linear systems Ax = b and (possibly generalized) eigenvalue problems Ax = λx. The former arise, for example, in semi-supervised learning and Gaussian processes, while the latter arise in spectral clustering and dimensionality reduction. One could attack these problems with naive iterative methods such as the power method, Jacobi and Gauss-Seidel [14]. The problem with these strategies is that the estimate x(t), at iteration t, only depends on the previous estimate x(t−1). Hence, these methods do typically take too many iterations to converge. It is well accepted in the numerical computation field that Krylov methods [14, 15], which make use of the entire history of solutions {x(1), . . . , x(t−1)}, converge at a faster rate. The intuition behind Krylov subspace methods is to use the history of the solutions we have already computed. We formulate this intuition in terms of projecting an N-dimensional problem onto a lower dimensional subspace. Given a matrix A and a vector b, the associated Krylov matrix is: K = [b Ab A2b . . . ]. The Krylov subspaces are the spaces spanned by the column vectors of this matrix. In order to find a new estimate of x(t) we could project onto the Krylov subspace. However, K is a poorly conditioned matrix. (As in the power method, Atb is converging to the eigenvector corresponding to the largest eigenvalue of A.) We therefore need to construct a well-conditioned orthogonal matrix Q(t) = [q(1) · · · q(t)], with q(i) ∈RN, that spans the Krylov space. That is, the leading t columns of K and Q span the same space. This is easily done using the QR-decomposition of K [14], yielding the following Arnoldi relation (augmented Schuur factorization): AQ(t) = Q(t+1) eH(t), where eH(t) is the augmented Hessenberg matrix: eH(t) = h1,1 h1,2 h1,3 · · · h1, t h2,1 h2,2 h2,3 · · · h2,t ... ... ... ... ... 0 · · · 0 ht,t−1 ht,t 0 · · · 0 0 ht+1,t . The eigenvalues of the smaller (t + 1) × t Hessenberg matrix approximate the eigenvalues of A as t increases. These eigenvalues can be computed efficiently by applying the Arnoldi relation recursively as shown in Figure 1. (If A is symmetric, then eH is tridiagonal and we obtain the Lanczos algorithm.) Notice that the matrix vector multiplication v = Aq is the expensive step in the Arnoldi algorithm. Most Krylov algorithms resemble the Arnoldi algorithm in this. To solve systems of equations, we can minimize either the residual Initialization: b = arbitrary, q(1) = b/∥b∥ FOR t = 1, 2, 3, . . . • v = Aq(t) • FOR j = 1, . . . , N – hj,t = q(t)T v – v = v −hj,tq(j) • ht+1,t = ∥v∥ • q(t+1) = v/ht+1,t Initialization: q(1) = b/∥b∥ FOR t = 1, 2, 3, . . . • Perform step t of the Arnoldi algorithm • miny ‚‚‚ eH(t)y −∥b∥i ‚‚‚ • Set x(t) = Q(t)y(t) Figure 1: The Arnoldi (left) and GMRES (right) algorithms. r(t) ≜b−Ax(t), leading to the GMRES and MINRES algorithms, or the A-norm, leading to conjugate gradients (CG) [14]. GMRES, MINRES and CG apply to general, symmetric, and spd matrices respectively. For ease of presentation, we focus on the GMRES algorithm. At step t of GMRES, we approximate the solution by the vector in the Krylov subspace x(t) ∈K(t) that minimizes the norm of the residual. Since x(t) is in the Krylov subspace, it can be written as a linear combination of the columns of the Krylov matrix K(t). Our problem therefore reduces to finding the vector y ∈Rt that minimizes ∥AK(t)y −b∥. As before, stability considerations force us to use the QR decomposition of K(t). That is, instead of using a linear combination of the columns of K(t), we use a linear combination of the columns of Q(t). So our least squares problem becomes y(t) = miny ∥AQ(t)y−b∥. Since AQ(t) = Q(t+1) eH(t), we only need to solve a problem of dimension (t + 1) × t: y(t) = miny ∥Q(t+1) eH(t)y −b∥. Keeping in mind that the columns of the projection matrix Q are orthonormal, we can rewrite this least squares problem as miny ∥eH(t)y − Q(t+1)T b∥. We start the iterations with q(1) = b/∥b∥and hence Q(t+1)T b = ∥b∥i, where i is the unit vector with a 1 in the first entry. The final form of our least squares problem at iteration t is: y(t) = min y
eH(t)y −∥b∥i
, with solution x(t) = Q(t)y(t). The algorithm is shown in Figure 1. The least squares problem of size (t + 1) × t to compute y(t) can be solved in O(t) steps using Givens rotations [14]. Notice again that the expensive step in each iteration is the matrix-vector product v = Aq. This is true also of CG and other Krylov methods. One important property of the Arnoldi relation is that the residuals are orthogonal to the space spanned by the columns of V = Q(t+1) eH(t). That is, VT r(t) = eH(t)T Q(t+1)T (b −Q(t+1) eH(t)y(t)) = eH(t)T ∥b∥i −eH(t)T eH(t)y(t) = 0 In the following section, we introduce methods to speed up the matrix-vector product v = Aq. These methods will incur, at most, a pre-specified (tolerance) error e(t) at iteration t. Later, we present theoretical bounds on how these errors affect the residuals and the orthogonality of the Krylov subspace. 3 Fast KDE The expensive step in Krylov methods is the operation v = Aq(t). This step requires that we solve two O(N 2) kernel estimates: vi = N X j=1 q(t) j a(xi, xj) i = 1, 2, . . ., M. It is possible to reduce the storage and computational cost to O(N) at the expense of a small specified error tolerance ϵ, say 10−6, using the fast Gauss transform (FGT) algorithm [16, 17]. This algorithm is an instance of more general fast multipole methods for solving N-body interactions [9]. The FGT applies when the problem is low dimensional, say xk ∈R3. However, to attack larger dimensions one can adopt clustering-based partitions as in the improved fast Gauss transform (IFGT) [10]. Fast multipole methods tend to work only in low dimensions and are specific to the choice of similarity metric. Dual tree recursions based on KD-trees and ball trees [11, 18] overcome these difficulties, but on average cost O(N log N). Due to space constraints, we can only mention these techniques here, but refer the reader to [18] for a thorough comparison. 4 Stability results The problem with replacing the matrix-vector multiplication at each iteration of the Krylov methods is that we do not know how the errors accumulate over successive iterations. In this section, we will derive bounds that describe what factors influence these errors. In particular, the bounds will state what properties of the similarity metric and measurable quantities affect the residuals and the orthogonality of the Krylov subspaces. Several papers have addressed the issue of Krylov subspace stability [19, 20, 21]. Our approach follows from [21]. For presentation purposes, we focus on the GMRES algorithm. Let e(t) denote the errors introduced in the approximate matrix-vector multiplication at each iteration of Arnoldi. For the purposes of upper-bounding, this is the tolerance of the fast KDE methods. Then, the fast KDE methods change the Arnoldi relation to: AQ(t) + E(t) = h Aq(1) + e(1), . . . , Aq(t) + e(t)i = Q(t+1) eH(t), where E(t) = e(1), . . . , e(t) . The new true residuals are therefore: r(t) = b −Ax(t) = b −AQ(t)y(t) = b −Q(t+1) eH(t)y(t) + E(t)y(t) and er(t) = b −Q(t+1) eH(t)y(t) are the measured residuals. We need to ensure two bounds when using fast KDE methods in Krylov iterations. First, the measured residuals er(t) should not deviate too far from the true residuals r(t). Second, deviations from orthogonality should be upper-bounded. Let us address the first question. The deviation in residuals is given by ∥er(t) −r(t)∥= ∥E(t)y(t)∥. Let y(t) = [y1, . . . , yt]T . Then, this deviation satisfies: ∥er(t) −r(t)∥=
t X k=1 yke(k)
≤ t X k=1 |yk|∥e(k)∥. (1) The deviation from orthogonality can be upper-bounded in a similar fashion: ∥VT r(t)∥=∥eH(t)TQ(t+1)T (er(t) + E(t)y(t))∥=
eH(t)T E(t)y(t)
≤∥eH(t)∥ t X k=1 |yk|∥e(k)∥ (2) The following lemma provides a relation between the yk and the measured residuals er(k−1). Lemma 1. [21, Lemma 5.1] Assume that t iterations of the inexact Arnoldi method have been carried out. Then, for any k = 1, . . . , t, |yk| ≤ 1 σt( eH(t)) ∥er(k−1)∥ (3) where σt( eH(t)) denotes the t-th singular value of eH(t). The proof of the lemma follows from the QR decomposition of eH(t), see [15, 21]. This lemma, in conjunction with equations (1) and (2), allows us to establish the main theoretical result of this section: Proposition 1. Let ϵ > 0. If for every k ≤t we have ∥e(k)∥< σt( eH(t)) t 1 ∥er(k−1)∥ϵ, then ∥er(t) −r(t)∥< ϵ. Moreover, if ∥e(k)∥< σt( eH(t)) t∥eH(t)∥ 1 ∥er(k−1)∥ϵ, then ∥VT r(t)∥< ϵ. Proof: First, we have ∥er(t) −r(t)∥≤ t X k=1 |yk|∥e(k)∥< t X k=1 σt( eH(t)) t 1 ∥er(k−1)∥ϵ 1 σt( eH(t)) ∥er(k−1)∥= ϵ. and similarly, ∥VT r(t)∥≤∥eH(t)∥Pt k=1 |yk|∥e(k)∥< ϵ □ Proposition 1 tells us that in order to keep the residuals bounded while ensuring bounded deviations from orthogonality at iteration k, we need to monitor the eigenvalues of eH(t) and the measured residuals er(k−1). Of course, we have no access to eH(t). However, monitoring the residuals is of practical value. If the residuals decrease, we can increase the tolerance of the fast KDE algorithms and viceversa. The bounds do lead to a natural way of constructing adaptive algorithms for setting the tolerance of the fast KDE algorithms. (d) (c) (b) (a) 1000 2000 3000 4000 5000 6000 7000 200 400 600 800 1000 1200 Data Set Size (Number of Features) Time (Seconds) Time Comparison NAIVE CG CG−DT Figure 2: Figure (a) shows a test image from the PASCAL database. Figure (b) shows the SIFT features extracted from the image. Figure (c) shows the positive feature predictions for the label ”car”. Figure (d) shows the centroid of the positive features as a black dot. The plot on the right shows the computational gains obtained by using fast Krylov methods. 5 Experimental results The results of this section demonstrate that significant computational gains may be obtained by combining fast KDE methods with Krylov iterations. We present results in three domains: spectral clustering and image segmentation [1, 12], Gaussian process regression [4] and stochastic neighbor embedding [6]. 5.1 Gaussian processes with large dimensional features In this experiment we use Gaussian processes to predict the labels of 128-dimensional SIFT features [22] for the purposes of object detection and localization as shown in Figure 2. There are typically thousands of features per image, so it is of paramount importance to generate fast predictions. The hard computational task here involves inverting the covariance matrix of the Gaussian process. The figure shows that it is possible to do this efficiently, under the same ROC error, by combining conjugate gradients [4] with dual trees. 5.2 Spectral clustering and image segmentation We applied spectral clustering to color image segmentation; a generalized eigenvalue problem. The types of segmentations obtained are shown in Figure 3. There are no perceptible differences between them. We observed that fast Krylov methods run approximately twice as fast as the Nystrom method. One should note that the result of Nystrom depends on the quality of sampling, while fast N-body methods enable us to work directly with the full matrix, so the solution is less sensitive. Once again, fast KDE methods lead to significant computational improvements over Krylov algorithms (Lanczos in this case). 5.3 Stochastic neighbor embedding Our final example is again a generalized eigenvalue problem arising in dimensionality reduction. We use the stochastic neighbor embedding algorithm of [6] to project two 3-D structures to 2-D, as shown in Figure 4. Again, we observe significant computational improvements. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 N Running time(seconds) Lanczos IFGT Dual Tree Figure 3: (left) Segmentation results (order: original image, IFGT, dual trees and Nystrom) and (right) computational improvements obtained in spectral clustering. 6 Conclusions We presented a general approach for combining Krylov solvers and fast KDE methods to accelerate machine learning techniques based on similarity metrics. We demonstrated some of the methods on several datasets and presented results that shed light on the stability and convergence properties of these methods. One important point to make is that these methods work better when there is structure in the data. There is no computational gain if there is not statistical information in the data. This is a fascinating relation between computation and statistical information, which we believe deserves further research and understanding. One question is how can we design pre-conditioners in order to improve the convergence behavior of these algorithms. Another important avenue for further research is the application of the bounds presented in this paper in the design of adaptive algorithms. Acknowledgments We would like to thank Arnaud Doucet, Firas Hamze, Greg Mori and Changjiang Yang. References [1] A Y Ng, M I Jordan, and Y Weiss. On spectral clustering: Analysis and algorithm. In Advances in Neural Information Processing Systems, pages 849–856, 2001. [2] D Zhou, J Weston, A Gretton, O Bousquet, and B Scholkopf. Ranking on data manifolds. In Advances on Neural Information Processing Systems, 2004. [3] X Zhu, J Lafferty, and Z Ghahramani. Semi-supervised learning using Gaussian fields and harmonic functions. In International Conference on Machine Learning, pages 912–919, 2003. [4] M N Gibbs. Bayesian Gaussian processes for regression and classification. In PhD Thesis, University of Cambridge, 1997. [5] M Belkin and P Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15(6):1373–1396, 2003. [6] G Hinton and S Roweis. Stochastic neighbor embedding. In Advances in Neural Information Processing Systems, pages 833–840, 2002. [7] A Smola and R Kondor. Kernels and regularization of graphs. In Computational Learning Theory, pages 144–158, 2003. True manifold Sampled data Embedding of SNE Embedding of SNE withIFGT 0 1000 2000 3000 4000 5000 10 1 10 2 10 3 10 4 N Running time(seconds) S−curve SNE SNE with IFGT 0 1000 2000 3000 4000 5000 10 1 10 2 10 3 10 4 N Running time(seconds) Swissroll SNE SNE with IFGT Figure 4: Examples of embedding on S-curve and Swiss-roll datasets. [8] M Mahdaviani, N de Freitas, B Fraser, and F Hamze. Fast computational methods for visually guided robots. In IEEE International Conference on Robotics and Automation, 2004. [9] L Greengard and V Rokhlin. A fast algorithm for particle simulations. Journal of Computational Physics, 73:325–348, 1987. [10] C Yang, R Duraiswami, N A Gumerov, and L S Davis. Improved fast Gauss transform and efficient kernel density estimation. In International Conference on Computer Vision, Nice, 2003. [11] A Gray and A Moore. Rapid evaluation of multiple density models. In Artificial Iintelligence and Statistics, 2003. [12] J Shi and J Malik. Normalized cuts and image segmentation. In IEEE Conference on Computer Vision and Pattern Recognition, pages 731–737, 1997. [13] R K Beatson, J B Cherrie, and C T Mouat. Fast fitting of radial basis functions: Methods based on preconditioned GMRES iteration. Advances in Computational Mathematics, 11:253–270, 1999. [14] J W Demmel. Applied Numerical Linear Algebra. SIAM, 1997. [15] Y Saad. Iterative Methods for Sparse Linear Systems. The PWS Publishing Company, 1996. [16] L Greengard and J Strain. The fast Gauss transform. SIAM Journal of Scientific Statistical Computing, 12(1):79–94, 1991. [17] B J C Baxter and G Roussos. A new error estimate of the fast Gauss transform. SIAM Journal of Scientific Computing, 24(1):257–259, 2002. [18] D Lang, M Klaas, and N de Freitas. Empirical testing of fast kernel density estimation algorithms. Technical Report TR-2005-03, Department of Computer Science, UBC, 2005. [19] G H Golub and Q Ye. Inexact preconditioned conjugate gradient method with inner-outer iteration. SIAM Journal of Scientific Computing, 21:1305–1320, 1999. [20] G W Stewart. Backward error bounds for approximate Krylov subspaces. Linear Algebra and Applications, 340:81–86, 2002. [21] V Simoncini and D B Szyld. Theory of inexact Krylov subspace methods and applications to scientific computing. SIAM Journal on Scientific Computing, 25:454–477, 2003. [22] D G Lowe. Object recognition from local scale-invariant features. In ICCV, 1999.
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A Connectionist Model for Constructive Modal Reasoning Artur S. d’Avila Garcez Department of Computing, City University London London EC1V 0HB, UK aag@soi.city.ac.uk Lu´ıs C. Lamb Institute of Informatics, Federal University of Rio Grande do Sul Porto Alegre RS, 91501-970, Brazil LuisLamb@acm.org Dov M. Gabbay Department of Computer Science, King’s College London Strand, London, WC2R 2LS, UK dg@dcs.kcl.ac.uk Abstract We present a new connectionist model for constructive, intuitionistic modal reasoning. We use ensembles of neural networks to represent intuitionistic modal theories, and show that for each intuitionistic modal program there exists a corresponding neural network ensemble that computes the program. This provides a massively parallel model for intuitionistic modal reasoning, and sets the scene for integrated reasoning, knowledge representation, and learning of intuitionistic theories in neural networks, since the networks in the ensemble can be trained by examples using standard neural learning algorithms. 1 Introduction Automated reasoning and learning theory have been the subject of intensive investigation since the early developments in computer science [14]. However, while (machine) learning has focused mainly on quantitative and connectionist approaches [16], the reasoning component of intelligent systems has been developed mainly by formalisms of classical and non-classical logics [7, 9]. More recently, the recognition of the need for systems that integrate reasoning and learning into the same foundation, and the evolution of the fields of cognitive and neural computation, has led to a number of proposals that attempt to integrate reasoning and learning [1, 3, 12, 13, 15]. We claim that an effective integration of reasoning and learning can be obtained by neuralsymbolic learning systems [3, 4]. Such systems concern the application of problem-specific symbolic knowledge within the neurocomputing paradigm. By integrating logic and neural networks, they may provide (i) a sound logical characterisation of a connectionist system, (ii) a connectionist (parallel) implementation of a logic, or (iii) a hybrid learning system bringing together advantages from connectionism and symbolic reasoning. Intuitionistic logical systems have been advocated by many as providing adequate logical foundations for computation (see [2] for a survey). We argue, therefore, that intuitionism could also play an important part in neural computation. In this paper, we follow the research path outlined in [4, 5], and develop a computational model for integrated reasoning, representation, and learning of intuitionistic modal knowledge. We concentrate on reasoning and knowledge representation issues, which set the scene for connectionist intuitionistic learning, since effective knowledge representation should precede learning [15]. Still, we base the representation on standard, simple neural network architectures, aiming at future work on experimental learning within the model proposed here. A key contribution of this paper is the proposal to shift the notion of logical implication (and negation) in neural networks from the standard notion of implication as a partial function from input to output (and of negation as failure to activate a neuron), to an intuitionistic notion which we will see can be implemented in neural networks if we make use of network ensembles. We claim that the intuitionistic interpretation introduced here will make sense for a number of problems in neural computation in the same way that intuitionistic logic is more appropriate than classical logic in a number of computational settings. We will start by illustrating the proposed computational model in an appropriate constructive reasoning, distributed knowledge representation scenario, namely, the wise men puzzle [7]. Then, we will show how ensembles of Connectionist Inductive Learning and Logic Programming (C-ILP) networks [3] can compute intuitionistic modal knowledge. The networks are set up by an Intuitionistic Modal Algorithm introduced in this paper. A proof that the algorithm produces a neural network ensemble that computes a semantics of its associated intuitionistic modal theory is then given. Furthermore, the networks in the ensemble are kept simple and in a modular structure, and may be trained from examples with the use of standard learning algorithms such as backpropagation [11]. In Section 2, we present the basic concepts of intuitionistic reasoning used in the paper. In Section 3, we motivate the proposed model using the wise men puzzle. In Section 4, we introduce the Intuitionistic Modal Algorithm, which translates intuitionistic modal theories into neural network ensembles, and prove that the ensemble computes a semantics of the theory. Section 5 concludes the paper and discusses directions for future work. 2 Background In this section, we present some basic concepts of artificial neural networks and intuitionistic programs used throughout the paper. We concentrate on ensembles of single hidden layer feedforward networks, and on recurrent networks typically with feedback only from the output to the input layer. Feedback is used with the sole purpose of denoting that the output of a neuron should serve as the input of another neuron when we run the network, i.e. the weight of any feedback connection is fixed at 1. We use bipolar semi-linear activation functions h(x) = 2 1+e−βx −1 with inputs in {−1, 1}. Throughout, we will use 1 to denote truth-value true, and −1 to denote truth-value false. Intuitionistic logic was originally developed by Brouwer, and later by Heyting and Kolmogorov [2]. In intuitionistic logics, a statement that there exists a proof of a proposition x is only made if there is a constructive method of the proof of x. One of the consequences of Brouwer’s ideas is the rejection of the law of the excluded middle, namely α∨¬α, since one cannot always state that there is a proof of α or of its negation, as accepted in classical logic and in (classical) mathematics. The development of these ideas and applications in mathematics has led to developments in constructive mathematics and has influenced several lines of research on logic and computing science [2]. An intuitionistic modal language L includes propositional letters (atoms) p, q, r..., the connectives ¬, ∧, an intuitionistic implication ⇒, the necessity (□) and possibility (♦) modal operators, where an atom will be necessarily true in a possible world if it is true in every world that is related to this possible world, while it will be possibly true if it is true in some world related to this world. Formally, we interpret the language as follows, where formulas are denoted by α, β, γ... Definition 1 (Kripke Models for Intuitionistic Modal Logic) Let L be an intuitionistic language. A model for L is a tuple M = ⟨Ω, R, v⟩where Ωis a set of worlds, v is a mapping that assigns to each ω ∈Ωa subset of the atoms of L, and R is a reflexive, transitive, binary relation over Ω, such that: (a) (M, ω) |= p iff p ∈v(ω) (for atom p); (b) (M, ω) |= ¬α iff for all ω′ such that R(ω, ω′), (M, ω′) ̸⊨α; (c) (M, ω) |= α ∧β iff (M, ω) |= α and (M, ω) |= β; (d) (M, ω) |= α ⇒β iff for all ω′ with R(ω, ω′) we have (M, ω′) |= β whenever we have (M, ω′) |= α; (e) (M, ω) |= □α iff for all ω′ ∈Ωif R(ω, ω′) then (M, ω′) |= α; (f) (M, ω) |= ♦α iff there exists ω′ ∈Ωsuch that R(ω, ω′) and (M, ω′) |= α. We now define labelled intuitionistic programs as sets of intuitionistic rules, where each rule is labelled by the world at which it holds, similarly to Gabbay’s Labelled Deductive Systems [8]. Definition 2 (Labelled Intuitionistic Program) A Labelled Intuitionistic Program is a finite set of rules C of the form ωi : A1, ..., An ⇒A0 (where “,” abbreviates “∧”, as usual), and a finite set of relations R between worlds ωi (1 ≤i ≤m) in C, where Ak (0 ≤k ≤n) are atoms and ωi is a label representing a world in which the associated rule holds. To deal with intuitionistic negation, we adopt the approach of [10], as follows. We rename any negative literal ¬A as an atom A′ not present originally in the language. This form of renaming allows our definition of labelled intuitionistic programs above to consider atoms only. For example, given A1, ..., A′ k, ..., An ⇒A0, where A′ k is a renaming of ¬Ak, an interpretation that assigns true to A′ k represents that ¬Ak is true; it does not represent that Ak is false. Following Definition 1 (intuitionistic negation), A′ will be true in a world ωi if and only if A does not hold in every world ωj such that R(ωi, ωj). Finally, we extend labelled intuitionistic programs to include modalities. Definition 3 (Labelled Intuitionistic Modal Program) A modal atom is of the form MA where M ∈{□, ♦} and A is an atom. A Labelled Intuitionistic Modal Program is a finite set of rules C of the form ωi : MA1, ..., MAn ⇒MA0, where MAk (0 ≤k ≤n) are modal atoms and ωi is a label representing a world in which the associated rule holds, and a finite set of (accessibility) relations R between worlds ωi (1 ≤i ≤m) in C. 3 Motivating Scenario In this section, we consider an archetypal testbed for distributed knowledge representation, namely, the wise men puzzle [7], and model it intuitionistically in a neural network ensemble. Our aim is to illustrate the combination of neural networks and intuitionistic modal reasoning. The formalisation of our computational model will be given in Section 4. A certain king wishes to test his three wise men. He arranges them in a circle so that they can see and hear each other. They are all perceptive, truthful and intelligent, and this is common knowledge in the group. It is also common knowledge among them that there are three red hats and two white hats, and five hats in total. The king places a hat on the head of each wise man in a way that they are not able to see the colour of their own hats, and then asks each one whether they know the colour of the hats on their heads. The puzzle illustrates a situation in which intuitionistic implication and intuitionistic negation occur. Knowledge evolves in time, with the current knowledge persisting in time. For example, at the first round it is known that there are at most two white hats on the wise men’s heads. Then, if the wise men get to a second round, it becomes known that there is at most one white hat on their heads.1 This new knowledge subsumes the previous knowledge, which in turn persists. This means that if A ⇒B is true at a world t1 then A ⇒B will be true at a world t2 that is related to t1 (intuitionistic implication). Now, in any situation in which a wise man knows that his hat is red, this knowledge - constructed with the use of sound reasoning processes - cannot be refuted. In other words, in this puzzle, if ¬A is true at world t1 then A cannot be true at a world t2 that is related to t1 (intuitionistic negation). We model the wise men puzzle by constructing the relative knowledge of each wise man along time points. This allows us to explicitly represent the relativistic notion of knowledge, which is a principle of intuitionistic reasoning. For simplicity, we refer to wise man 1 (respectively, 2 and 3) as agent 1 (respectively, 2 and 3). The resulting model is a twodimensional network ensemble (agents × time), containing three networks in each dimension. In addition to pi - denoting the fact that wise man i wears a red hat - to model each agent’s individual knowledge, we need to use a modality Kj, j ∈{1, 2, 3}, which represents the relative notion of knowledge at each time point t1, t2, t3. Thus, Kjpi denotes the fact that agent j knows that agent i wears a red hat. The K modality above corresponds to the □modality in intuitionistic modal reasoning, as customary in the logics of knowledge [7], and as exemplified below. First, we model the fact that each agent knows the colour of the others’ hats. For example, if wise man 3 wears a red hat (neuron p3 is active) then wise man 1 knows that wise man 3 wears a red hat (neuron Kp3 is active for wise man 1). We then need to model the reasoning process of each wise man. In this example, let us consider the case in which neurons p1 and p3 are active. For agent 1, we have the rule t1 : K1¬p2 ∧K1¬p3 ⇒K1p1, which states that agent 1 can deduce that he is wearing a red hat if he knows that the other agents are both wearing white hats. Analogous rules exist for agents 2 and 3. As before, the implication is intuitionistic, so that it persists at t2 and t3 as depicted in Figure 1 for wise man 1 (represented via hidden neuron h1 in each network). In addition, according to the philosophy of intuitionistic negation, we may only conclude that agent 1 knows ¬p2, if in every world envisaged by agent 1, p2 is not derived. This is illustrated with the use of dotted lines in Figure 1, in which, e.g., if neuron Kp2 is not active at t3 then neuron K¬p2 will be active at t2. As a result, the network ensemble will never derive p2 (as one should expect), and thus it will derive K1¬p2 and K3¬p2.2 4 Connectionist Intuitionistic Modal Reasoning The wise men puzzle example of Section 3 shows that simple, single-hidden layer neural networks can be combined in a modular structure where each network represents a possible world in the Kripke structure of Definition 1. The way that the networks should then be inter-connected can be defined by following a semantics for ⇒and ¬, and for □and ♦from intuitionistic logic. In this section, we see how exactly we construct a network ensemble 1This is because if there were two white hats on their heads, one of them would have known (and have said), in the first round, that his hat was red, for he would have been seeing the other two with white hats. 2To complete the formalisation of the problem, the following rules should also hold at t2 (and at t3): K1¬p2 ⇒K1p1 and K1¬p3 ⇒K1p1. Analogous rules exist for agents 2 and 3. wise man 1 at point t2 wise man 1 at point t3 K¬¬¬¬p3 K¬¬¬¬p2 K¬¬¬¬p2 Kp1 K¬¬¬¬p3 Kp2 Kp3 h1 h2 h3 h4 h5 K¬¬¬¬p2 K¬¬¬¬p2 K¬¬¬¬p3 K¬¬¬¬p3 Kp1 Kp2 Kp3 h1 h2 h3 h4 h5 wise man 1 at point t1 K¬¬¬¬p3 K¬¬¬¬p2 K¬¬¬¬p2 Kp1 K¬¬¬¬p3 Kp2 Kp3 h1 h2 h3 h4 h5 −−−−1 −−−−1 −−−−1 −−−−1 Figure 1: Wise men puzzle: Intuitionistic negation and implication. given an intuitionistic modal program. We introduce a translation algorithm, which takes the program as input and produces the ensemble as output by setting the initial architecture, set of weights, and thresholds of the networks according to a Kripke semantics for the program. We then prove that the translation is correct, and thus that the network ensemble can be used to compute the logical consequences of the program in parallel. Before we present the algorithm, let us illustrate informally how ⇒, ¬, □, and ♦are represented in the ensemble. We follow the key idea behind Connectionist Modal Logics (CML) to represent Kripke models in neural networks [6]. Each possible world is represented by a single hidden layer neural network. In each network, input and output neurons represent atoms or modal atoms of the form A, ¬A, □A, or ♦A, while each hidden neuron encodes a rule. For example, in Figure 1, hidden neuron h1 encodes a rule of the form A ∧B ⇒C. Thresholds and weights must be such that the hidden layer computes a logical and of the input layer, while the output layer computes a logical or of the hidden layer.3 Furthermore, in each network, each output neuron is connected to its corresponding input neuron with a weight fixed at 1.0 (as depicted in Figure 1 for K¬p2 and K¬p3), so that chains of the form A ⇒B and B ⇒C can be represented and computed. This basically characterises C-ILP networks [3]. Now, in CML, we allow for an ensemble of C-ILP networks, each network representing knowledge in a (learnable) possible world. In addition, we allow for a number of fixed feedforward and feedback connections to occur among different networks in the ensemble, as shown in Figure 1. These are defined as follows: in the case of □, if neuron □A is activated (true) in network (world) ωi then A must be activated in every network ωj that is related to ωi (this is analogous to the situation in which we activate K1p3 and K2p3 whenever p3 is active). Dually, if A is active in every ωj then □A must be activated 3For example, if A ∧B ⇒D and C ⇒D then a hidden neuron h1 is used to connect A and B to D, and a hidden neuron h2 is used to connect C to D such that if h1 or h2 is activated then D is activated. in ωi (this is done with the use of feedback connections and a hidden neuron that computes a logical and, as detailed in the algorithm below). In the case of ♦, if ♦A is activated in network ωi then A must be activated in at least one network ωj that is related to ωi (we do this by choosing an arbitrary ωj to make A active). Dually, if A is activated in any ωj that is related to ωi then ♦A must be activated in ωi (this is done with the use of a hidden neuron that computes a logical or, also as detailed in the algorithm below). Now, in the case of ⇒, according to the semantics of intuitionistic implication, ωi : A ⇒B and R(ωi, ωj) imply ωj : A ⇒B. We implement this by copying the neural representation of A ⇒B from ωi to ωj, as done via h1 in Figure 1. Finally, in the case of ¬, we need to make sure that ¬A is activated in ωi if, for every ωj such that R(ωi, ωj), A is not active in ωj. This is implemented with the use of negative weights (to account for the fact that the non-activation of a neuron needs to activate another neuron), as depicted in Figure 1 (dashed arrows), and detailed in the algorithm below. We are now in a position to introduce the Intuitionistic Modal Algorithm. Let P = {P1, ..., Pn} be a labelled intuitionistic modal program with rules of the form ωi : MA1, ..., MAk →MA0, where each Aj (0 ≤j ≤k) is an atom and M ∈{□, ♦}, 1 ≤i ≤n. Let N = {N1, ..., Nn} be a neural network ensemble with each network Ni corresponding to program Pi. Let q denote the number of rules occurring in P. Consider that the atoms of Pi are numbered from 1 to ηi such that the input and output layers of Ni are vectors of length ηi, where the j-th neuron represents the j-th atom of Pi. In addition, let Amin denote the minimum activation for a neuron to be considered active (or true), Amin ∈(0, 1); for each rule rl in each program Pi, let kl denote the number of atoms in the body of rule rl, and let µl denote the number of rules in Pi with the same consequent as rl (including rl). Let MAXrl(kl, µl) denote the greater of kl and µl for rule rl, and let MAXP(k1, ..., kq, µ1, ..., µq) denote the greatest of k1, ..., kq, µ1, ..., µq for program P. Below, we use k as a shorthand for k1, ..., kq, and µ as a shorthand for µ1, ..., µq. The equations in the algorithm come from the proof of Theorem 1, given in the sequel. Intuitionistic Modal Algorithm 1. Rename each modal atom MAj by a new atom not occurring in P of the form A□ j if M = □, or A♦ j if M = ♦; 2. For each rule rl of the form A1, ..., Ak ⇒A0 in Pi (1 ≤i ≤n) such that R(ωi, ωj), do: add a rule A1, ..., Ak ⇒A0 to Pj (1 ≤j ≤n). 3. Calculate Amin > (MAXP(k,µ, n) −1)⧸(MAXP(k,µ, n) + 1); 4. Calculate W ≥(2⧸β)·(ln (1 + Amin)−ln (1 −Amin))⧸(MAXP(k,µ)·(Amin −1)+Amin+ 1); 5. For each rule rl of the form A1, ..., Ak ⇒A0 (k ≥0) in Pi (1 ≤i ≤n), do: (a) Add a neuron Nl to the hidden layer of neural network Ni associated with Pi; (b) Connect each neuron Ai (1 ≤i ≤k) in the input layer of Ni to Nl and set the connection weight to W; (c) Connect Nl to neuron A0 in the output layer of Ni and set the connection weight to W; (d) Set the threshold θl of Nl to θl = ((1 + Amin) · (kl −1) ⧸2)W; (e) Set the threshold θA0 of A0 in the output layer of Ni to θA0 = ((1 + Amin) · (1 −µl)⧸2)W. (f) For each atom of the form A′ in rl, do: (i) Add a hidden neuron NA′ to Ni; (ii) Set the step function s(x) as the activation function of NA′;4 (iii) Set the threshold θA′ of NA′ such that n −(1 + Amin) < θA′ < nAmin; (iv) For each 4Any hidden neuron created to encode negation (such as h4 in Figure 1) shall have a non-linear activation function s(x) = y, where y = 1 if x > 0, and y = 0 otherwise. Such neurons encode (meta-level) knowledge about negation, while the other hidden neurons encode (object-level) knowledge about the problem domain. The former are not expected to be trained by examples and, as a result, the use of the step function will simplify the algorithm. The latter are to be trained, and therefore require a differentiable, semi-linear activation function. network Nj corresponding to program Pj (1 ≤j ≤n) in P such that R(ωi, ωj), do: Connect the output neuron A of Nj to the hidden neuron NA′ of Ni and set the connection weight to −1; and Connect the hidden neuron NA′ of Ni to the output neuron A′ of Ni and set the connection weight to W I such that W I > h−1(Amin) +µA′.W + θA′. 6. For each output neuron A♦ j in network Ni, do: (a) Add a hidden neuron AM j and an output neuron Aj to an arbitrary network Nz such that R(ωi, ωz); (b) Set the step function s(x) as the activation function of AM j , and set the semi-linear function h(x) as the activation function of Aj; (c) Connect A♦ j in Ni to AM j and set the connection weight to 1; (d) Set the threshold θM of AM j such that −1 < θM < Amin; (e) Set the threshold θAj of Aj in Nz such that θAj = ((1 + Amin) · (1 −µAj)⧸2)W; (f) Connect AM j to Aj in Nz and set the connection weight to W M > h−1(Amin) + µAjW + θAj. 7. For each output neuron A□ j in network Ni, do: (a) Add a hidden neuron AM j to each Nu (1 ≤u ≤n) such that R(ωi, ωu), and add an output neuron Aj to Nu if Aj /∈Nu; (b) Set the step function s(x) as the activation function of AM j , and set the semi-linear function h(x) as the activation function of Aj; (c) Connect A□ j in Ni to AM j and set the connection weight to 1; (d) Set the threshold θM of AM j such that −1 < θM < Amin; (e) Set the threshold θAj of Aj in each Nu such that θAj = ((1 + Amin) · (1 −µAj)⧸2)W; (f) Connect AM j to Aj in Nu and set the connection weight to W M > h−1(Amin) + µAjW + θAj. 8. For each output neuron Aj in network Nu such that R(ωi, ωu), do: (a) Add a hidden neuron A∨ j to Ni; (b) Set the step function s(x) as the activation function of A∨ j ; (c) For each output neuron A♦ j in Ni, do: (i) Connect Aj in Nu to A∨ j and set the connection weight to 1; (ii) Set the threshold θ∨of A∨ j such that −nAmin < θ∨< Amin −(n −1); (iii) Connect A∨ j to A♦ j in Ni and set the connection weight to W M > h−1(Amin) + µAjW + θAj. 9. For each output neuron Aj in network Nu such that R(ωi, ωu), do: (a) Add a hidden neuron A∧ j to Ni; (b) Set the step function s(x) as the activation function of A∧ j ; (c) For each output neuron A□ j in Ni, do: (i) Connect Aj in Nu to A∧ j and set the connection weight to 1; (ii) Set the threshold θ∧of A∧ j such that n −(1 + Amin) < θ∧< nAmin; (iii) Connect A∧ j to A□ j in Ni and set the connection weight to W M > h−1(Amin) + µAjW + θAj. Finally, we prove that N is equivalent to P. Theorem 1 (Correctness of Intuitionistic Modal Algorithm) For any intuitionistic modal program P there exists an ensemble of neural networks N such that N computes the intuitionistic modal semantics of P. Proof The algorithm to build each individual network in the ensemble is that of C-ILP, which we know is provably correct [3]. The algorithm to include modalities is that of CML, which is also provably correct [6]. We need to consider when modalities and intuitionistic negation are to be encoded together. Consider an output neuron A0 with neurons M (encoding modalities) and neurons n (encoding negation) among its predecessors in a network’s hidden layer. There are four cases to consider. (i) Both neurons M and neurons n are not activated: since the activation function of neurons M and n is the step function, their activation is zero, and thus this case reduces to C-ILP. (ii) Only neurons M are activated: from the algorithm above, A0 will also be activated (with minimum input potential W M + ς, where ς ∈R). (iii) Only neurons n are activated: as before, A0 will also be activated (now with minimum input potential W I + ς). (iv) Both neurons M and neurons n are activated: the input potential of A0 is at least W M + W I + ς. Since W M > 0 and W I > 0, and since the activation function of A0, h(x), is monotonically increasing, A0 will be activated whenever both M and n neurons are activated. This completes the proof. 5 Concluding Remarks In this paper, we have presented a new model of computation that integrates neural networks and constructive, intuitionistic modal reasoning. We have defined labelled intuitionistic modal programs, and have presented an algorithm to translate the intuitionistic theories into ensembles of C-ILP neural networks, and showed that the ensembles compute a semantics of the corresponding theories. As a result, each ensemble can be seen as a new massively parallel model for the computation of intuitionistic modal logic. In addition, since each network can be trained efficiently using, e.g., backpropagation, one can adapt the network ensemble by training possible world representations from examples. Work along these lines has been done in [4, 5], where learning experiments in possible worlds settings were investigated. As future work, we shall consider learning experiments based on the constructive model introduced in this paper. Extensions of this work also include the study of how to represent other non-classical logics such as branching time temporal logics, and conditional logics of normality, which are relevant for cognitive and neural computation. Acknowledgments Artur Garcez is partly supported by the Nuffield Foundation and The Royal Society. Luis Lamb is partly supported by the Brazilian Research Council CNPq and by the CAPES and FAPERGS foundations. References [1] A. Browne and R. Sun. Connectionist inference models. Neural Networks, 14(10):1331–1355, 2001. [2] D. Van Dalen. Intuitionistic logic. In D. M. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 5. Kluwer, 2nd edition, 2002. [3] A. S. d’Avila Garcez, K. Broda, and D. M. Gabbay. Neural-Symbolic Learning Systems: Foundations and Applications. Perspectives in Neural Computing. Springer-Verlag, 2002. [4] A. S. d’Avila Garcez and L. C. Lamb. Reasoning about time and knowledge in neural-symbolic learning systems. In Advances in Neural Information Processing Systems 16, Proceedings of NIPS 2003, pages 921–928, Vancouver, Canada, 2004. MIT Press. [5] A. S. d’Avila Garcez, L. C. Lamb, K. Broda, and D. M. Gabbay. Applying connectionist modal logics to distributed knowledge representation problems. International Journal on Artificial Intelligence Tools, 13(1):115–139, 2004. [6] A. S. d’Avila Garcez, L. C. Lamb, and D. M. Gabbay. Connectionist modal logics. Theoretical Computer Science. Forthcoming. [7] R. Fagin, J. Halpern, Y. Moses, and M. Vardi. Reasoning about Knowledge. MIT Press, 1995. [8] D. M. Gabbay. Labelled Deductive Systems. Clarendom Press, Oxford, 1996. [9] D. M. Gabbay, C. Hogger, and J. A. Robinson, editors. Handbook of Logic in Artificial Intelligence and Logic Programming, volume 1-5, Oxford, 1994-1999. Clarendom Press. [10] M. Gelfond and V. Lifschitz. Classical negation in logic programs and disjunctive databases. New Generation Computing, 9:365–385, 1991. [11] D. E. Rumelhart, G. E. Hinton, and R. J. Williams. Learning representations by backpropagating errors. Nature, 323:533–536, 1986. [12] L. Shastri. Advances in SHRUTI: a neurally motivated model of relational knowledge representation and rapid inference using temporal synchrony. Applied Intelligence, 11:79–108, 1999. [13] G. G. Towell and J. W. Shavlik. Knowledge-based artificial neural networks. Artificial Intelligence, 70(1):119–165, 1994. [14] A. M. Turing. 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Spiking Inputs to a Winner-take-all Network Matthias Oster and Shih-Chii Liu Institute of Neuroinformatics University of Zurich and ETH Zurich Winterthurerstrasse 190 CH-8057 Zurich, Switzerland {mao,shih}@ini.phys.ethz.ch Abstract Recurrent networks that perform a winner-take-all computation have been studied extensively. Although some of these studies include spiking networks, they consider only analog input rates. We present results of this winner-take-all computation on a network of integrate-and-fire neurons which receives spike trains as inputs. We show how we can configure the connectivity in the network so that the winner is selected after a pre-determined number of input spikes. We discuss spiking inputs with both regular frequencies and Poisson-distributed rates. The robustness of the computation was tested by implementing the winner-take-all network on an analog VLSI array of 64 integrate-and-fire neurons which have an innate variance in their operating parameters. 1 Introduction Recurrent networks that perform a winner-take-all computation are of great interest because of the computational power they offer. They have been used in modelling attention and recognition processes in cortex [Itti et al., 1998,Lee et al., 1999] and are thought to be a basic building block of the cortical microcircuit [Douglas and Martin, 2004]. Descriptions of theoretical spike-based models [Jin and Seung, 2002] and analog VLSI (aVLSI) implementations of both spike and non-spike models [Lazzaro et al., 1989,Indiveri, 2000,Hahnloser et al., 2000] can be found in the literature. Although the competition mechanism in these models uses spike signals, they usually consider the external input to the network to be either an analog input current or an analog value that represents the spike rate. We describe the operation and connectivity of a winner-take-all network that receives input spikes. We consider the case of the hard winner-take-all mode, where only the winning neuron is active and all other neurons are suppressed. We discuss a scheme for setting the excitatory and inhibitory weights of the network so that the winner which receives input with the shortest inter-spike interval is selected after a pre-determined number of input spikes. The winner can be selected with as few as two input spikes, making the selection process fast [Jin and Seung, 2002]. We tested this computation on an aVLSI chip with 64 integrate-and-fire neurons and various dynamic excitatory and inhibitory synapses. The distribution of mismatch (or variance) in the operating parameters of the neurons and synapses has been reduced using a spike coding (a) (b) VSelf VI Figure 1: Connectivity of the winner-take-all network: (a) in biological networks, inhibition is mediated by populations of global inhibitory interneurons (filled circle). To perform a winner-take-all operation, they are driven by excitatory neurons (unfilled circles) and in return, they inhibit all excitatory neurons (black arrows: excitatory connections; dark arrows: inhibitory). (b) Network model in which the global inhibitory interneuron is replaced by full inhibitory connectivity of efficacy VI. Self excitation of synaptic efficacy Vself stabilizes the selection of the winning neuron. mismatch compensation procedure described in [Oster and Liu, 2004]. The results shown in Section 3 of this paper were obtained with a network that has been calibrated so that the neurons have about 10% variance in their firing rates in response to a common input current. 1.1 Connectivity We assume a network of integrate-and-fire neurons that receive external excitatory or inhibitory spiking input. In biological networks, inhibition between these array neurons is mediated by populations of global inhibitory interneurons (Fig. 1a). They are driven by the excitatory neurons and inhibit them in return. In our model, we assume the forward connections between the excitatory and the inhibitory neurons to be strong, so that each spike of an excitatory neuron triggers a spike in the global inhibitory neurons. The strength of the total inhibition between the array neurons is adjusted by tuning the backward connections from the global inhibitory neurons to the array neurons. This configuration allows the fastest spreading of inhibition through the network and is consistent with findings that inhibitory interneurons tend to fire at high frequencies. With this configuration, we can simplify the network by replacing the global inhibitory interneurons with full inhibitory connectivity between the array neurons (Fig. 1b). In addition, each neuron has a self-excitatory connection that facilitates the selection of this neuron as winner for repeated input. 2 Network Connectivity Constraints for a Winner-Take-All Mode We first discuss the conditions for the connectivity under which the network operates in a hard winner-take-all mode. For this analysis, we assume that the neurons receive spike trains of regular frequency. We also assume the neurons to be non-leaky. The membrane potentials Vi, i = 1 . . . N then satisfy the equation of a non-leaky integrateVself VE VE Vth Vth VE VE VI VI (a) (b) Figure 2: Membrane potential of the winning neuron k (a) and another neuron in the array (b). Black bars show the times of input spikes. Traces show the changes in the membrane membrane potential caused by the various synaptic inputs. Black dots show the times of output spikes of neuron k. and-fire neuron model with non-conductance-based synapses: dVi dt = VE X n δ(t −t(n) i ) −VI N X j=1 j̸=i X m δ(t −s(m) j ) (1) The membrane resting potential is set to 0. Each neuron receives external excitatory input and inhibitory connections from all other neurons. All inputs to a neuron are spikes and its output is also transmitted as spikes to other neurons. We neglect the dynamics of the synaptic currents and the delay in the transmission of the spikes. Each input spike causes a fixed discontinuous jump in the membrane potential (VE for the excitatory synapse and VI for the inhibitory). Each neuron i spikes when Vi ≥Vth and is reset to Vi =0. Immediately afterwards, it receives a self-excitation of weight Vself. All potentials satisfy 0 ≤Vi ≤Vth, that is, an inhibitory spike can not drive the membrane potential below ground. All neurons i ∈1 . . . N, i̸=k receive excitatory input spike trains of constant frequency ri. Neuron k receives the highest input frequency (rk >ri ∀i̸=k). As soon as neuron k spikes once, it has won the computation. Depending on the initial conditions, other neurons can at most have transient spikes before the first spike of neuron k. For this hard winner-take-all mode, the network has to fulfill the following constraints (Fig. 2): (a) Neuron k (the winning neuron) spikes after receiving nk = n input spikes that cause its membrane potential to exceed threshold. After every spike, the neuron is reset to Vself: Vself + nkVE ≥Vth (2) (b) As soon as neuron k spikes once, no other neuron i ̸= k can spike because it receives an inhibitory spike from neuron k. Another neuron can receive up to n spikes even if its input spike frequency is lower than that of neuron k because the neuron is reset to Vself after a spike, as illustrated in Figure 2. The resulting membrane voltage has to be smaller than before: ni · VE ≤nk · VE ≤VI (3) (c) If a neuron j other than neuron k spikes in the beginning, there will be some time in the future when neuron k spikes and becomes the winning neuron. From then on, the conditions (a) and (b) hold, so a neuron j ̸=k can at most have a few transient spikes. Let us assume that neurons j and k spike with almost the same frequency (but rk > rj). For the inter-spike intervals ∆i=1/ri this means ∆j>∆k. Since the spike trains are not synchronized, an input spike to neuron k has a changing phase offset φ from an input spike of neuron j. At every output spike of neuron j, this phase decreases by ∆φ = nk(∆j−∆k) until φ < nk(∆j−∆k). When this happens, neuron k receives (nk+1) input spikes before neuron j spikes again and crosses threshold: (nk + 1) · VE ≥Vth (4) We can choose Vself =VE and VI =Vth to fulfill the inequalities (2)-(4). VE is adjusted to achieve the desired nk. Case (c) happens only under certain initial conditions, for example when Vk ≪Vj or when neuron j initially received a spike train of higher frequency than neuron k. A leaky integrate-and-fire model will ensure that all membrane potentials are discharged (Vi =0) at the onset of a stimulus. The network will then select the winning neuron after receiving a pre-determined number of input spikes and this winner will have the first output spike. 2.1 Poisson-Distributed Inputs In the case of Poisson-distributed spiking inputs, there is a probability associated with the correct winner being selected. This probability depends on the Poisson rate ν and the number of spikes needed for the neuron to reach threshold n. The probability that m input spikes arrive at a neuron in the period T is given by the Poisson distribution P(m, νT) = e−νT (νT)m m! (5) We assume that all neurons i receive an input rate νi, except the winning neuron which receives a higher rate νk. All neurons are completely discharged at t = 0. The network will make a correct decision at time T, if the winner crosses threshold exactly then with its nth input spike, while all other neuron received less than n spikes until then. The winner receives the nth input spike at T, if it received n−1 input spikes in [0; T[ and one at time T. This results in the probability density function pk(T) = νkP(n−1, νkT) (6) The probability that the other N−1 neurons receive less or equal than n−1 spikes in [0; T[ is P0(T) = N Y i=1 i̸=k n−1 X j=0 P(j, νiT) (7) For a correct decision, the output spike of the winner can happen at any time T >0, so we integrate over all times T: P = ∞ Z 0 pk(T) · P0(T) dT = ∞ Z 0 νkP(n−1, νkT) · N Y j=1 i̸=k n−1 X i=0 P(j, νiT) ! dT (8) We did not find a closed solution for this integral, but we can discuss its properties n is varied by changing the synaptic efficacies. For n = 1 every input spike elicits an output spike. The probability of a having an output spike from neuron k is then directly dependent on the input rates, since no computation in the network takes place. For n →∞, the integration times to determine the rates of the Poisson-distributed input spike trains are large, and the neurons perform a good estimation of the input rate. The network can then discriminate small changes in the input frequencies. This gain in precision leads a slow response time of the network, since a large number of input spike is integrated before an output spike of the network. The winner-take-all architecture can also be used with a latency spike code. In this case, the delay of the input spikes after a global reset determines the strength of the signal. The winner is selected after the first input spike to the network (nk = 1). If all neurons are discharged at the onset of the stimulus, the network does not require the global reset. In general, the computation is finished at a time nk·∆k after the stimulus onset. 3 Results We implemented this architecture on a chip with 64 integrate-and-fire neurons implemented in analog VLSI technology. These neurons follow the model equation 1, except that they also show a small linear leakage. Spikes from the neurons are communicated off-chip using an asynchronous event representation transmission protocol (AER). When a neuron spikes, the chip outputs the address of this neuron (or spike) onto a common digital bus (see Figure 3). An external spike interface module (consisting of a custom computer board that can be programmed through the PCI bus) receives the incoming spikes from the chip, and retransmits spikes back to the chip using information stored in a routing table. This module can also monitor spike trains from the chip and send spikes from a stored list. Through this module and the AER protocol, we implement the connectivity needed for the winnertake-all network in Figure 1. All components have been used and described in previous work [Boahen, 2000,Liu et al., 2001]. neuron array spike interface module monitor sequence reroute Figure 3: The connections are implemented by transmitting spikes over a common bus (grey arrows). Spikes from aVLSI neurons in the network are recorded by the digital interface and can be monitored and rerouted to any neuron in the array. Additionally, externally generated spike trains can be transmitted to the array through the sequencer. We configure this network according to the constraints which are described above. Figure 4 illustrates the network behaviour with a spike raster plot. At time t = 0, the neurons receive inputs with the same regular firing frequency of 100Hz except for one neuron which received a higher input frequency of 120Hz. The synaptic efficacies were tuned so that threshold is reached with 6 input spikes, after which the network does select the neuron with the strongest input as the winner. We characterized the discrimination capability of the winner-take-all implementation by 1 32 64 1 32 64 −50 0 50 100 150 1 32 64 Time [ms] (a) (b) (c) Figure 4: Example raster plot of the spike trains to and from the neurons: (a) Input: starting from 0 ms, the neurons are stimulated with spike trains of a regular frequency of 100Hz, but randomized phase. Neuron number 42 receives an input spike train with an increased frequency of 120Hz. (b) Output without WTA connectivity: after an adjustable number of input spikes, the neurons start to fire with a regular output frequency. The output frequencies of the neurons are slightly different due to mismatch in the synaptic efficacies. Neuron 42 has the highest output frequency since it receives the strongest input. (c) Output with WTA connectivity: only neuron 42 with the strongest input fires, all other neurons are suppressed. measuring to which minimal frequency, compared to the other input, the input rate to this neuron has to be raised to select it as the winner. The neuron being tested receives an input of regular frequency of f ·100Hz, while all other neuron receive 100Hz. The histogram of the minimum factors f for all neurons is shown in Figure 5. On average, the network can discriminate a difference in the input frequency of 10%. This value is identical with the variation in the synaptic efficacies of the neurons, which had been compensated to a mismatch of 10%. We can therefore conclude that the implemented winner-take-all network functions according to the above discussion of the constraints. Since only the timing information of the spike trains is used, the results can be extended to a wide range of input frequencies different from 100Hz. To test the performance of the network with Poisson inputs, we stimulated all neurons with Poisson-distibuted spike rates of rate ν, except neuron k which received the rate νk = fν. Eqn. 8 then simplifies to P = ∞ Z 0 fν P(n−1, fν T) · n−1 X i=0 P(i, νT) !N−1 dT (9) We show measured data and theoretical predictions for a winner-take-all network of 2 and 8 neurons (Fig. 6). Obviously, the discrimation performance of the network is substantially limited by the Poisson nature of the spike trains compared to spike trains of regular frequency. 1 1.05 1.1 1.15 1.2 0 2 4 6 8 10 12 Increase factor f # neurons Figure 5: Discrimination capability of the winner-take-all network: X-axis: factor f to which the input frequency of a neuron has to be increased, compared to the input rate of the other neurons, in order for that neuron to be selected as the winner. Y-axis: histogram of all 64 neurons. 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 fincrease (n=8) Pcorrect 2 4 6 8 0 0.2 0.4 0.6 0.8 1 n (fincrease=1.5) Pcorrect Figure 6: Probability of a correct decision of the winner-take-all network, versus difference in frequencies (left), and number of input spikes n for a neuron to reach threshold (right). The measured data (crosses/circles) is shown with the prediction of the model (continuous lines), for a winner-take-all network of 2 neurons (red,circles) and 8 neurons (blue, crosses). 4 Conclusion We analysed the performance and behavior of a winner-take-all spiking network that receives input spike trains. The neuron that receives spikes with the highest rate is selected as the winner after a pre-determined number of input spikes. Assuming a non-leaky integrate-and-fire model neuron with constant synaptic weights, we derived constraints for the strength of the inhibitory connections and the self-excitatory connection of the neuron. A large inhibitory synaptic weight is in agreement with previous analysis for analog inputs [Jin and Seung, 2002]. The ability of a single spike from the inhibitory neuron to inhibit all neurons removes constraints on the matching of the time constants and efficacy of the connections from the excitatory neurons to the inhibitory neuron and vice versa. This feature makes the computation tolerant to variance in the synaptic parameters as demonstrated by the results of our experiment. We also studied whether the network is able to select the winner in the case of input spike trains which have a Poisson distribution. Because of the Poisson distributed inputs, the network does not always chose the right winner (that is, the neuron with the highest input frequency) but there is a certain probability that the network does select the right winner. Results from the network show that the measured probabilities match that of the theoretical results. We are currently extending our analysis to a leaky integrate-and-fire neuron model and conductance-based synapses, which results in a more complex description of the network. Acknowledgments This work was supported in part by the IST grant IST-2001-34124. We acknowledge Sebastian Seung for discussions on the winner-take-all mechanism. References [Boahen, 2000] Boahen, K. A. (2000). Point-to-point connectivity between neuromorphic chips using address-events. IEEE Transactions on Circuits & Systems II, 47(5):416–434. [Douglas and Martin, 2004] Douglas, R. and Martin, K. (2004). Cortical microcircuits. Annual Review of Neuroscience, 27(1f). [Hahnloser et al., 2000] Hahnloser, R., Sarpeshkar, R., Mahowald, M. A., Douglas, R. J., and Seung, S. (2000). Digital selection and analogue amplification coexist in a cortexinspired silicon circuit. Nature, 405:947–951. [Indiveri, 2000] Indiveri, G. (2000). Modeling selective attention using a neuromorphic analog VLSI device. Neural Computation, 12(12):2857–2880. [Itti et al., 1998] Itti, C., Niebur, E., and Koch, C. (1998). A model of saliency-based fast visual attention for rapid scene analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, 20(11):1254–1259. [Jin and Seung, 2002] Jin, D. Z. and Seung, H. S. (2002). Fast computation with spikes in a recurrent neural network. Physical Review E, 65:051922. [Lazzaro et al., 1989] Lazzaro, J., Ryckebusch, S., Mahowald, M. A., and Mead, C. A. (1989). Winner-take-all networks of O(n) complexity. In Touretzky, D., editor, Advances in Neural Information Processing Systems, volume 1, pages 703–711. Morgan Kaufmann, San Mateo, CA. [Lee et al., 1999] Lee, D., Itti, C., Koch, C., and Braun, J. (1999). Attention activates winner-take-all competition among visual filters. Nature Neuroscience, 2:375–381. [Liu et al., 2001] Liu, S.-C., Kramer, J., Indiveri, G., Delbr¨uck, T., Burg, T., and Douglas, R. (2001). Orientation-selective aVLSI spiking neurons. Neural Networks: Special Issue on Spiking Neurons in Neuroscience and Technology, 14(6/7):629–643. [Oster and Liu, 2004] Oster, M. and Liu, S.-C. (2004). A winner-take-all spiking network with spiking inputs. In 11th IEEE International Conference on Electronics, Circuits and Systems. ICECS ’04: Tel Aviv, Israel, 13–15 December.
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Estimation of Intrinsic Dimensionality Using High-Rate Vector Quantization Maxim Raginsky and Svetlana Lazebnik Beckman Institute, University of Illinois 405 N Mathews Ave, Urbana, IL 61801 {maxim,slazebni}@uiuc.edu Abstract We introduce a technique for dimensionality estimation based on the notion of quantization dimension, which connects the asymptotic optimal quantization error for a probability distribution on a manifold to its intrinsic dimension. The definition of quantization dimension yields a family of estimation algorithms, whose limiting case is equivalent to a recent method based on packing numbers. Using the formalism of high-rate vector quantization, we address issues of statistical consistency and analyze the behavior of our scheme in the presence of noise. 1. Introduction The goal of nonlinear dimensionality reduction (NLDR) [1, 2, 3] is to find low-dimensional manifold descriptions of high-dimensional data. Most NLDR schemes require a good estimate of the intrinsic dimensionality of the data to be available in advance. A number of existing methods for estimating the intrinsic dimension (e.g., [3, 4, 5]) rely on the fact that, for data uniformly distributed on a d-dimensional compact smooth submanifold of IRD, the probability of a small ball of radius ǫ around any point on the manifold is Θ(ǫd). In this paper, we connect this argument with the notion of quantization dimension [6, 7], which relates the intrinsic dimension of a manifold (a topological property) to the asymptotic optimal quantization error for distributions on the manifold (an operational property). Quantization dimension was originally introduced as a theoretical tool for studying “nonstandard” signals, such as singular distributions [6] or fractals [7]. However, to the best of our knowledge, it has not been previously used for dimension estimation in manifold learning. The definition of quantization dimension leads to a family of dimensionality estimation algorithms, parametrized by the distortion exponent r ∈[1, ∞), yielding in the limit of r = ∞a scheme equivalent to K´egl’s recent technique based on packing numbers [4]. To date, many theoretical aspects of intrinsic dimensionality estimation remain poorly understood. For instance, while the estimator bias and variance are assessed either heuristically [4] or exactly [5], scant attention is paid to robustness of each particular scheme against noise. Moreover, existing schemes do not fully utilize the potential for statistical consistency afforded by ergodicity of i.i.d. data: they compute the dimensionality estimate from a fixed training sequence (typically, the entire dataset of interest), whereas we show that an independent test sequence is necessary to avoid overfitting. In addition, using the framework of high-rate vector quantization allows us to analyze the performance of our scheme in the presence of noise. 2. Quantization-based estimation of intrinsic dimension Let us begin by introducing the definitions and notation used in the rest of the paper. A D-dimensional k-point vector quantizer [6] is a measurable map Qk : IRD →C, where C = {y1, . . . , yk} ⊂IRD is called the codebook and the yi’s are called the codevectors. The number log2 k is called the rate of the quantizer, in bits per vector. The sets Ri △= {x ∈IRD : Qk(x) = yi}, 1 ≤i ≤k, are called the quantizer cells (or partition regions). The quantizer performance on a random vector X distributed according to a probability distribution µ (denoted X ∼µ) is measured by the average rth-power distortion δr(Qk|µ) △= Eµ ∥X −Qk(X)∥r, r ∈[1, ∞), where ∥· ∥is the Euclidean norm on IRD. In the sequel, we will often find it more convenient to work with the quantizer error er(Qk|µ) △= δr(Qk|µ)1/r. Let Qk denote the set of all D-dimensional k-point quantizers. Then the performance achieved by an optimal k-point quantizer on X is δ∗ r(k|µ) △= infQk∈Qk δr(Qk|µ) or equivalently, e∗ r(k|µ) △= δ∗ r(k|µ)1/r. 2.1. Quantization dimension The dimensionality estimation method presented in this paper exploits the connection between the intrinsic dimension d of a smooth compact manifold M ⊂IRD (from now on, simply referred to as “manifold”) and the asymptotic optimal quantization error for a regular probability distribution1 on M. When the quantizer rate is high, the partition cells can be well approximated by D-dimensional balls around the codevectors. Then the regularity of µ ensures that the probability of such a ball of radius ǫ is Θ(ǫd), and it can be shown [7, 6] that e∗ r(k|µ) = Θ(k−1/d). This is referred to as the high-rate (or high-resolution) approximation, and motivates the definition of quantization dimension of order r: dr(µ) △= −lim k→∞ log k log e∗r(k|µ). The theory of high-rate quantization confirms that, for a regular µ supported on the manifold M, dr(µ) exists for all 1 ≤r ≤∞and equals the intrinsic dimension of M [7, 6]. (The r = ∞limit will be treated in Sec. 2.2.) This definition immediately suggests an empirical procedure for estimating the intrinsic dimension of a manifold from a set of samples. Let Xn = (X1, . . . , Xn) be n i.i.d. samples from an unknown regular distribution µ on the manifold. We also fix some r ∈[1, ∞). Briefly, we select a range k1 ≤k ≤k2 of codebook sizes for which the high-rate approximation holds (see Sec. 3 for implementation details), and design a sequence of quantizers { ˆQk}k2 k=k1 that give us good approximations ˆer(k|µ) to the optimal error e∗ r(k|µ) over the chosen range of k. Then an estimate of the intrinsic dimension is obtained by plotting log k vs. −log ˆer(k|µ) and measuring the slope of the plot over the chosen range of k (because the high-rate approximation holds, the plot is linear). This method hinges on estimating reliably the optimal errors e∗ r(k|µ). Let us explain how this can be achieved. The ideal quantizer for each k should minimize the training error er(Qk|µtrain) = 1 n n X i=1 ∥Xi −Qk(Xi)∥r !1/r , 1A probability distribution µ on IRD is regular of dimension d [6] if it has compact support and if there exist constants c, ǫ0 > 0, such that c−1ǫd ≤µ(B(a, ǫ)) ≤cǫd for all a ∈supp(µ) and all ǫ ∈(0, ǫ0), where B(a, ǫ) is the open ball of radius ǫ centered at a. If M ⊂IRD is a d-dimensional smooth compact manifold, then any µ with M = supp(µ) that possesses a smooth, strictly positive density w.r.t. the normalized surface measure on M is regular of dimension d. where µtrain is the corresponding empirical distribution. However, finding this empirically optimal quantizer is, in general, an intractable problem, so in practice we merely strive to produce a quantizer ˆQk whose error er( ˆQk|µtrain) is a good approximation to the minimal empirical error e∗ r(k|µtrain) △= infQk∈Qk er(Qk|µtrain) (the issue of quantizer design is discussed in Sec. 3). However, while minimizing the training error is necessary for obtaining a statistically consistent approximation to an optimal quantizer for µ, the training error itself is an optimistically biased estimate of e∗ r(k|µ) [8]: intuitively, this is due to the fact that an empirically designed quantizer overfits the training set. A less biased estimate is given by the performance of ˆQk on a test sequence independent from the training set. Let Zm = (Z1, . . . , Zm) be m i.i.d. samples from µ, independent from Xn. Provided m is sufficiently large, the law of large numbers guarantees that the empirical average er( ˆQk|µtest) = 1 m m X i=1 ∥Zi −ˆQk(Zi)∥r !1/r will be a good estimate of the test error er( ˆQk|µ). Using learning-theoretic formalism [8], one can show that the test error of an empirically optimal quantizer is a strongly consistent estimate of e∗ r(k|µ), i.e., it converges almost surely to e∗ r(k|µ) as n →∞. Thus, we take ˆer(k|µ) = er( ˆQk|µtest). In practice, therefore, the proposed scheme is statistically consistent to the extent that ˆQk is close to the optimum. 2.2. The r = ∞limit and packing numbers If the support of µ is compact (which is the case with all probability distributions considered in this paper), then the limit e∞(Qk|µ) = limr→∞er(Qk|µ) exists and gives the “worstcase” quantization error of X by Qk: e∞(Qk|µ) = max x∈supp(µ) ∥x −Qk(x)∥. The optimum e∗ ∞(k|µ) = infQk∈Qk e∞(Qk|µ) has an interesting interpretation as the smallest covering radius of the most parsimonious covering of supp(µ) by k or fewer balls of equal radii [6]. Let us describe how the r = ∞case is equivalent to dimensionality estimation using packing numbers [4]. The covering number NM(ǫ) of a manifold M ⊂IRD is defined as the size of the smallest covering of M by balls of radius ǫ > 0, while the packing number PM(ǫ) is the cardinality of the maximal set S ⊂M with ∥x −y∥≥ǫ for all distinct x, y ∈S. If d is the dimension of M, then NM(ǫ) = Θ(ǫ−d) for small enough ǫ, leading to the definition of the capacity dimension: dcap(M) △= −limǫ→0 log NM(ǫ) log ǫ . If this limit exists, then it equals the intrinsic dimension of M. Alternatively, K´egl [4] suggests using the easily proved inequality NM(ǫ) ≤PM(ǫ) ≤NM(ǫ/2) to express the capacity dimension in terms of packing numbers as dcap(M) = −limǫ→0 log PM(ǫ) log ǫ . Now, a simple geometric argument shows that, for any µ supported on M, PM(e∗ ∞(k|µ)) > k [6]. On the other hand, NM(e∗ ∞(k|µ)) ≤k, which implies that PM(2e∗ ∞(k|µ)) ≤k. Let {ǫk} be a sequence of positive reals converging to zero, such that ǫk = e∗ ∞(k|µ). Let k0 be such that log ǫk < 0 for all k ≥k0. Then it is not hard to show that −log PM(2ǫk) log 2ǫk −1 ≤− log k log e∗∞(k|µ) < −log PM(ǫk) log ǫk , k ≥k0. In other words, there exists a decreasing sequence {ǫk}, such that for sufficiently large values of k (i.e., in the high-rate regime) the ratio −log k/ log e∗ ∞(k|µ) can be approximated increasingly finely both from below and from above by quantities involving the packing numbers PM(ǫk) and PM(2ǫk) and converging to the common value dcap(M). This demonstrates that the r = ∞case of our scheme is numerically equivalent to K´egl’s method based on packing numbers. For a finite training set, the r = ∞case requires us to find an empirically optimal kpoint quantizer w.r.t. the worst-case ℓ2 error — a task that is much more computationally complex than for the r = 2 case (see Sec. 3 for details). In addition to computational efficiency, other important practical considerations include sensitivity to sampling density and noise. In theory, this worst-case quantizer is completely insensitive to variations in sampling density, since the optimal error e∗ ∞(k|µ) is the same for all µ with the same support. However, this advantage is offset in practice by the increased sensitivity of the r = ∞scheme to noise, as explained next. 2.3. Estimation with noisy data Random noise transforms “clean” data distributed according to µ into “noisy” data distributed according to some other distribution ν. This will cause the empirically designed quantizer to be matched to the noisy distribution ν, whereas our aim is to estimate optimal quantizer performance on the original clean data. To do this, we make use of the rth-order Wasserstein distance [6] between µ and ν, defined as ¯ρr(µ, ν) △= infX∼µ,Y ∼ν(E ∥X −Y ∥r)1/r, r ∈[1, ∞), where the infimum is taken over all pairs (X, Y ) of jointly distributed random variables with the respective marginals µ and ν. It is a natural measure of quantizer mismatch, i.e., the difference in performance that results from using a quantizer matched to ν on data distributed according to µ [9]. Let νn denote the empirical distribution of n i.i.d. samples of ν. It is possible to show (details omitted for lack of space) that for an empirically optimal k-point quantizer Q∗ k,r trained on n samples of ν, |er(Q∗ k,r|ν) −e∗ r(k|µ)| ≤2¯ρr(νn, ν) + ¯ρr(µ, ν). Moreover, νn converges to ν in the Wasserstein sense [6]: limn→∞¯ρr(νn, ν) = 0. Thus, provided the training set is sufficiently large, the distortion estimation error is controlled by ¯ρr(µ, ν). Consider the case of isotropic additive Gaussian noise. Let W be a D-dimensional zeromean Gaussian with covariance matrix K = σ2ID, where ID is the D ×D identity matrix. The noisy data are described by the random variable X + W = Y ∼ν, and ¯ρr(µ, ν) ≤ √ 2σ Γ((r + D)/2) Γ(D/2) 1/r , where Γ is the gamma function. In particular, ¯ρ2(µ, ν) ≤σ √ D. The magnitude of the bound, and hence the worst-case sensitivity of the estimation procedure to noise, is controlled by the noise variance, by the extrinsic dimension, and by the distortion exponent. The factor involving the gamma functions grows without bound both as D →∞and as r →∞, which suggests that the susceptibility of our algorithm to noise increases with the extrinsic dimension of the data and with the distortion exponent. 3. Experimental results We have evaluated our quantization-based scheme for two choices of the distortion exponent, r = 2 and r = ∞. For r = 2, we used the k-means algorithm to design the quantizers. For r = ∞, we have implemented a Lloyd-type algorithm, which alternates two steps: (1) the minimum-distortion encoder, where each sample Xi is mapped to its nearest neighbor in the current codebook, and (2) the centroid decoder, where the center of each region is recomputed as the center of the minimum enclosing ball of the samples assigned to that region. It is clear that the decoder step locally minimizes the worst-case error (the largest distance of any sample from the center). Using a simple randomized algorithm, the minimum enclosing ball can be found in O((D + 1)!(D + 1)N) time, where N is the number of samples in the region [10]. Because of this dependence on D, the running time of the Lloyd algorithm becomes prohibitive in high dimensions, and even for D < 10 it is an 500 1000 1500 2000 0 2 4 6 8 10 Codebook size (k) Training error Training error r=2 r=∞ Lloyd r=∞ greedy 500 1000 1500 2000 0 2 4 6 8 10 Codebook size (k) Test error Test error r=2 r=∞ Lloyd r=∞ greedy Training error Test error Figure 1: Training and test error vs. codebook size on the swiss roll (Figure 2 (a)). Dashed line: r = 2 (k-means), dash-dot: r = ∞(Lloyd-type), solid: r = ∞(greedy). −10 −5 0 5 10 −10 −5 0 5 10 20 40 −2 −1 0 1 5 6 7 8 9 10 11 −log(Error) Rate (log k) Training error Test error Training fit Test fit 6 7 8 9 10 1 1.25 1.5 1.75 2 2.25 2.5 Rate Dim. Estimate Training estimate Test estimate (a) (b) (c) −4 −2 0 2 4 −4 −2 0 2 4 −0.5 0 0.5 1 2 3 4 5 6 7 8 9 10 11 −log(Error) Rate (log k) Training error Test error 6 7 8 9 10 0.75 1 1.25 1.5 1.75 2 2.25 Rate Dim. Estimate Training estimate Test estimate (d) (e) (f) Figure 2: (a) The swiss roll (20,000 samples). (b) Plot of rate vs. negative log of the quantizer error (log-log curves), together with parametric curves fitted using linear least squares (see text). (c) Slope (dimension) estimates: 1.88 (training) and 2.04 (test). (d) Toroidal spiral (20,000 samples). (e) Log-log curves, exhibiting two distinct linear parts. (f) Dimension estimates: 1.04 (training), 2.02 (test) in the low-rate region, 0.79 (training), 1.11 (test) in the high-rate region. order of magnitude slower than k-means. Thus, we were compelled to also implement a greedy algorithm reminiscent of K´egl’s algorithm for estimating the packing number [4]: supposing that k−1 codevectors have already been selected, the kth one is chosen to be the sample point with the largest distance from the nearest codevector. Because this is the point that gives the worst-case error for codebook size k−1, adding it to the codebook lowers the error. We generate several codebooks, initialized with different random samples, and then choose the one with the smallest error. For the experiment shown in Figure 3, the training error curves produced by this greedy algorithm were on average 21% higher than those of the Lloyd algorithm, but the test curves were only 8% higher. In many cases, the two test curves are visually almost coincident (Figure 1). Therefore, in the sequel, we report only the results for the greedy algorithm for the r = ∞case. Our first synthetic dataset (Fig. 2 (a)) is the 2D “swiss roll” embedded in IR3 [2]. We split the samples into 4 equal parts and use each part in turn for training and the rest for testing. This cross-validation setup produces four sets of error curves, which we average to obtain an improved estimate. We sample quantizer rates in increments of 0.1 bits. The lowest rate is 5 bits, and the highest rate is chosen as log(n/2), where n is the size of the training set. The high-rate approximation suggests the asymptotic form Θ(k−1/d) for the quantizer error as a function of codebook size k. To validate this approximation, we use linear least squares to fit curves of the form a + b k−1/2 to the r = 2 training and test distortion curves for the the swiss roll. The fitting procedure yields estimates of −0.22 + 29.70k−1/2 and 0.10 + 28.41k−1/2 for the training and test curves, respectively. These estimates fit the observed data well, as shown in Fig. 2(b), a plot of rate vs. the negative logarithm of the training and test error (“log-log curves” in the following). Note that the additive constant for the training error is negative, reflecting the fact that the training error of the empirical quantizer is identically zero when n = k (each sample becomes a codevector). On the other hand, the test error has a positive additive constant as a consequence of quantizer suboptimality. Significantly, the fit deteriorates as n/k →1, as the average number of training samples per quantizer cell becomes too small to sustain the exponentially slow decay required for the high-rate approximation. Fig. 2(c) shows the slopes of the training and test log-log curves, obtained by fitting a line to each successive set of 10 points. These slopes are, in effect, rate-dependent dimensionality estimates for the dataset. Note that the training slope is always below the test slope; this is a consequence of the “optimism” of the training error and the “pessimism” of the test error (as reflected in the additive constants of the parametric fits). The shapes of the two slope curves are typical of many “well-behaved” datasets. At low rates, both the training and the test slopes are close to the extrinsic dimension, reflecting the global geometry of the dataset. As rate increases, the local manifold structure is revealed, and the slope yields its intrinsic dimension. However, as n/k →1, the quantizer begins to “see” isolated samples instead of the manifold structure. Thus, the training slope begins to fall to zero, and the test slope rises, reflecting the failure of the quantizer to generalize to the test set. For most datasets in our experiments, a good intrinsic dimensionality estimate is given by the first minimum of the test slope where the line-fitting residual is sufficiently low (marked by a diamond in Fig. 2(c)). For completeness, we also report the slope of the training curve at the same rate (note that the training curve may not have local minima because of its tendency to fall as the rate increases). Interestingly, some datasets yield several well-defined dimensionality estimates at different rates. Fig. 2(d) shows a toroidal spiral embedded in IR3, which at larger scales “looks” like a torus, while at smaller scales the 1D curve structure becomes more apparent. Accordingly, the log-log plot of the test error (Fig. 2(e)) has two distinct linear parts, yielding dimension estimates of 2.02 and 1.11, respectively (Fig. 2(f)). Recall from Sec. 2.1 that the high-rate approximation for regular probability distributions is based on the assumption that the intersection of each quantizer cell with the manifold is a d-dimensional neighborhood of that manifold. Because we compute our dimensionality estimate at a rate for which this approximation is valid, we know that the empirically optimal quantizer at this rate partitions the data into clusters that are locally d-dimensional. Thus, our dimensionality estimation procedure is also useful for finding a clustering of the data that respects the intrinsic neighborhood structure of the manifold from which it is sampled. As an expample, for the toroidal spiral of Fig. 2(c), we obtain two distinct dimensionality estimates of 2 and 1 at rates 6.6 and 9.4, respectively (Fig. 2(f)). Accordingly, quantizing the spiral at the lower (resp. higher) rate yields clusters that are locally two-dimensional (resp. one-dimensional). To ascertain the effect of noise and extrinsic dimension on our method, we have embedded the swiss roll in dimensions 4 to 8 by zero-padding the coordinates and applying a random orthogonal matrix, and added isotropic zero-mean Gaussian noise in the high-dimensional space, with σ = 0.2, 0.4, . . . , 1. First, we have verified that the r = 2 estimator behaves in agreement with the Wasserstein bound from Sec. 2.3. The top part of Fig. 3(a) shows the maximum differences between the noisy and the noiseless test error curves for each combination of D and σ, and the bottom part shows the corresponding values of the Wasserstein bound σ √ D for comparison. For each value of σ, the test error of the empirically designed quantizer differs from the noiseless case by O( √ D), while, for a fixed D, the difference 3 4 5 6 7 8 0 0.5 1 1.5 2 2.5 3 D empirical difference σ = 0.0 σ = 0.2 σ = 0.4 σ = 0.6 σ = 0.8 σ = 1.0 0 0.2 0.4 0.6 0.8 1 3 4 5 6 7 8 2 2.5 3 3.5 σ r = 2 training D d estimate 0 0.2 0.4 0.6 0.8 1 3 4 5 6 7 8 2 2.5 3 3.5 σ r = ∞ training D d estimate 3 4 5 6 7 8 0 0.5 1 1.5 2 2.5 3 D bound σ = 0.0 σ = 0.2 σ = 0.4 σ = 0.6 σ = 0.8 σ = 1.0 0 0.2 0.4 0.6 0.8 1 3 4 5 6 7 8 2 2.5 3 3.5 σ r = 2 test D d estimate 0 0.2 0.4 0.6 0.8 1 3 4 5 6 7 8 2 2.5 3 3.5 σ r = ∞ test D d estimate (a) Noise bounds (b) r = 2 (c) r = ∞ Figure 3: (a) Top: empirically observed differences between noisy and noiseless test curves; bottom: theoretically derived bound ` σ √ D ´ . (b) Height plot of dimension estimates for the r = 2 algorithm as a function of D and σ. Top: training estimates, bottom: test estimates. (c) Dimension estimates for r = ∞. Top: training, bottom: test. Note that the training estimates are consistently lower than the test estimates: the average difference is 0.17 (resp. 0.28) for the r = 2 (resp. r = ∞) case. of the noisy and noiseless test errors grows as O(σ). As predicted by the bound, the additive constant in the parametric form of the test error increases with σ, resulting in larger slopes of the log-log curve and therefore higher dimension estimates. This is reflected in Figs. 3(b) and (c), which show training and test dimensionality estimates for r = 2 and r = ∞, respectively. The r = ∞estimates are much less stable than those for r = 2 because the r = ∞(worst-case) error is controlled by outliers and often stays constant over a range of rates. The piecewise-constant shape of the test error curves (see Fig. 1) results in log-log plots with unstable slopes. Table 1 shows a comparative evaluation on the MNIST handwritten digits database2 and a face video.3 The MNIST database contains 70,000 images at resolution 28×28 (D = 784), and the face video has 1965 frames at resolution 28 × 20 (D = 560). For each of the resulting 11 datasets (taking each digit separately), we used half the samples for training and half for testing. The first row of the table shows dimension estimates obtained using a baseline regression method [3]: for each sample point, a local estimate is given by the first local minimum of the curve d log ℓ d log ǫ(ℓ), where ǫ(ℓ) is the distance from the point to its ℓth nearest neighbor, and a global estimate is then obtained by averaging the local estimates. The rest of the table shows the estimates obtained from the training and test curves of the r = 2 quantizer and the (greedy) r = ∞quantizer. Comparative examination of the results shows that the r = ∞estimates tend to be fairly low, which is consistent with the experimental findings of K´egl [4]. By contrast, the r = 2 estimates seem to be most resistant to negative bias. The relatively high values of the dimension estimates reflect the many degrees of freedom found in handwritten digits, including different scale, slant and thickness of the strokes, as well as the presence of topological features (i.e., loops in 2’s or extra horizontal bars in 7’s). The lowest dimensionality is found for 1’s, while the highest is found for 8’s, reflecting the relative complexities of different digits. For the face dataset, the different dimensionality estimates range from 4.25 to 8.30. This dataset certainly contains enough degrees of freedom to justify such high estimates, including changes in pose 2http://yann.lecun.com/exdb/mnist/ 3http://www.cs.toronto.edu/˜roweis/data.html, B. Frey and S. Roweis. Table 1: Performance on the MNIST dataset and on the Frey faces dataset. Handwritten digits (MNIST data set) Faces # samples 6903 7877 6990 7141 6824 6313 6876 7293 6825 6958 1965 Regression 11.14 7.86 12.79 13.39 11.98 13.05 11.19 10.42 13.79 11.26 5.63 r = 2 train 12.39 6.51 16.04 15.38 13.22 14.63 12.05 12.32 19.80 13.44 5.70 r = 2 test 15.47 7.11 20.89 19.78 16.79 19.80 16.02 16.02 20.07 17.46 8.30 r = ∞train 10.33 8.19 10.15 12.63 9.87 8.49 9.85 8.10 10.88 7.40 4.25 r = ∞test 9.02 6.61 13.98 12.21 7.26 10.46 9.08 9.92 14.03 9.59 6.39 and facial expression, as well as camera jitter.4 Finally, for both the digits and the faces, significant noise in the dataset additionally inflated the estimates. 4. Discussion We have demonstrated an approach to intrinsic dimensionality estimation based on highrate vector quantization. A crucial distinguishing feature of our method is the use of an independent test sequence to ensure statistical consistency and avoid underestimating the dimension. Many existing methods are well-known to exhibit a negative bias in high dimensions [4, 5]. This can have serious implications in practice, as it may result in lowdimensional representations that lose essential features of the data. Our results raise the possibility that this negative bias may be indicative of overfitting. In the future we plan to integrate our proposed method into a unified package of quantization-based algorithms for estimating the intrinsic dimension of the data, obtaining its dimension-reduced manifold representation, and compressing the low-dimensional data [11]. Acknowledgments Maxim Raginsky was supported by the Beckman Institute Postdoctoral Fellowship. Svetlana Lazebnik was partially supported by the National Science Foundation grants IIS0308087 and IIS-0535152. References [1] S.T. Roweis and L.K. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290:2323–2326, December 2000. [2] J.B. Tenenbaum, V. de Silva, and J.C. Langford. A global geometric framework for nonlinear dimensionality reduction. Science, 290:2319–2323, December 2000. [3] M. Brand. Charting a manifold. In NIPS 15, pages 977–984, Cambridge, MA, 2003. MIT Press. [4] B. K´egl. Intrinsic dimension estimation using packing numbers. In NIPS 15, volume 15, Cambridge, MA, 2003. MIT Press. [5] E. Levina and P.J. Bickel. Maximum likelihood estimation of intrinsic dimension. In NIPS 17, Cambridge, MA, 2005. MIT Press. [6] S. Graf and H. Luschgy. Foundations of Quantization for Probability Distributions. SpringerVerlag, Berlin, 2000. [7] P.L. Zador. Asymptotic quantization error of continuous signals and the quantization dimension. IEEE Trans. Inform. Theory, IT-28:139–149, March 1982. [8] T. Linder. Learning-theoretic methods in vector quantization. In L. Gy¨orfi, editor, Principles of Nonparametric Learning. Springer-Verlag, New York, 2001. [9] R.M. Gray and L.D. Davisson. Quantizer mismatch. IEEE Trans. Commun., 23:439–443, 1975. [10] E. Welzl. Smallest enclosing disks (balls and ellipsoids). In New Results and New Trends in Computer Science, volume 555 of LNCS, pages 359–370. Springer, 1991. [11] M. Raginsky. A complexity-regularized quantization approach to nonlinear dimensionality reduction. Proc. 2005 IEEE Int. Symp. Inform. Theory, pages 352–356. 4Interestingly, Brand [3] reports an intrinsic dimension estimate of 3 for this data set. However, he used only a 500-frame subsequence and introduced additional mirror symmetry.
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Generalization error bounds for classifiers trained with interdependent data Nicolas Usunier, Massih-Reza Amini, Patrick Gallinari Department of Computer Science, University of Paris VI 8, rue du Capitaine Scott, 75015 Paris France {usunier, amini, gallinari}@poleia.lip6.fr Abstract In this paper we propose a general framework to study the generalization properties of binary classifiers trained with data which may be dependent, but are deterministically generated upon a sample of independent examples. It provides generalization bounds for binary classification and some cases of ranking problems, and clarifies the relationship between these learning tasks. 1 Introduction Many machine learning (ML) applications deal with the problem of bipartite ranking where the goal is to find a function which orders relevant elements over irrelevant ones. Such problems appear for example in Information Retrieval, where the system returns a list of documents, ordered by relevancy to the user’s demand. The criterion widely used to measure the ranking quality is the Area Under the ROC Curve (AUC) [6]. Given a training set S = ((xp, yp))n p=1 with yp ∈{±1}, its optimization over a class of real valued functions G can be carried out by finding a classifier of the form cg(x, x′) = sign(g(x)−g(x′)), g ∈G which minimizes the error rate over pairs of examples (x, 1) and (x′, −1) in S [6]. More generally, it is well-known that the learning of scoring functions can be expressed as a classification task over pairs of examples [7, 5]. The study of the generalization properties of ranking problems is a challenging task, since the pairs of examples violate the central i.i.d. assumption of binary classification. Using task-specific studies, this issue has recently been the focus of a large amount of work. [2] showed that SVM-like algorithms optimizing the AUC have good generalization guarantees, and [11] showed that maximizing the margin of the pairs, defined by the quantity g(x) −g(x′), leads to the minimization of the generalization error. While these results suggest some similarity between the classification of the pairs of examples and the classification of independent data, no common framework has been established. As a major drawback, it is not possible to directly deduce results for ranking from those obtained in classification. In this paper, we present a new framework to study the generalization properties of classifiers over data which can exhibit a suitable dependency structure. Among others, the problems of binary classification, bipartite ranking, and the ranking risk defined in [5] are special cases of our study. It shows that it is possible to infer generalization bounds for classifiers trained over interdependent examples using generalization results known for binary classification. We illustrate this property by proving a new margin-based, data-dependent bound for SVM-like algorithms optimizing the AUC. This bound derives straightforwardly from the same kind of bounds for SVMs for classification given in [12]. Since learning algorithms aim at minimizing the generalization error of their chosen hypothesis, our results suggest that the design of bipartite ranking algorithms can follow the design of standard classification learning systems. The remainder of this paper is as follows. In section 2, we give the formal definition of our framework and detail the progression of our analysis over the paper. In section 3, we present a new concentration inequality which allows to extend the notion of Rademacher complexity (section 4), and, in section 5, we prove generalization bounds for binary classification and bipartite ranking tasks under our framework. Finally, the missing proofs are given in a longer version of the paper [13]. 2 Formal framework We distinguish between the input and the training data. The input data S = (sp)n p=1 is a set of n independent examples, while the training data Z = (zi)N i=1 is composed of N binary classified elements where each zi is in Xtr × {−1, +1}, with Xtr the space of characteristics. For example, in the general case of bipartite ranking, the input data is the set of elements to be ordered, while the training data is constituted by the pairs of examples to be classified. The purpose of this work is the study of generalization properties of classifiers trained using a possibly dependent training data, but in the special case where the latter is deterministically generated from the input data. The aim here is to select a hypothesis h ∈H = {hθ : Xtr →{−1, 1}|θ ∈Θ} which optimizes the empirical risk L(h, Z) = 1 N N i=1 ℓ(h, zi), ℓbeing the instantaneous loss of h, over the training set Z. Definition 1 (Classifiers trained with interdependent data). A classification algorithm over interdependent training data takes as input data a set S = (sp)n p=1 supposed to be drawn according to an unknown product distribution ⊗n p=1Dp over a product sample space Sn 1, outputs a binary classifier chosen in a hypothesis space H : {h : Xtr →{+1, −1}}, and has a two-step learning process. In a first step, the learner applies to its input data S a fixed function ϕ : Sn →(Xtr × {−1, 1})N to generate a vector Z = (zi)N i=1 = ϕ(S) of N training examples zi ∈Xtr × {−1, 1}, i = 1, ..., N. In the second step, the learner runs a classification algorithm in order to obtain h which minimizes the empirical classification loss L(h, Z), over its training data Z = ϕ(S). Examples Using the notations above, when S = Xtr × {±1}, n = N, ϕ is the identity function and S is drawn i.i.d. according to an unknown distribution D, we recover the classical definition of a binary classification algorithm. Another example is the ranking task described in [5] where S = X ×R, Xtr = X 2, N = n(n−1) and, given S = ((xp, yp))n p=1 drawn i.i.d. according to a fixed D, ϕ generates all the pairs ((xk, xl), sign( yk−yl 2 )), k ̸= l. In the remaining of the paper, we will prove generalization error bounds of the selected hypothesis by upper bounding sup h∈H L(h) −L(h, ϕ(S)) (1) with high confidence over S, where L(h) = ESL(h, ϕ(S)). To this end we decompose Z = ϕ(S) using the dependency graph of the random variables composing Z with a technique similar to the one proposed by [8]. We go towards this result by first bounding 1It is equivalent to say that the input data is a vector of independent, but not necessarilly identically distributed random variables. supq∈Q E ˜S 1 N N i=1 q(ϕ( ˜S)i) −1 N N i=1 q(ϕ(S)i) with high confidence over samples S, where ˜S is also drawn according to ⊗n p=1Dp, Q is a class of functions taking values in [0, 1], and ϕ(S)i denotes the i-th training example (Theorem 4). This bound uses an extension of the Rademacher complexity [3], the fractional Rademacher complexity (FRC) (definition 3), which is a weighted sum of Rademacher complexities over independent subsets of the training data. We show that the FRC of an arbitrary class of real-valued functions can be trivially computed given the Rademacher complexity of this class of functions and ϕ (theorem 6). This theorem shows that generalization error bounds for classes of classifiers over interdependent data (in the sense of definition 1) trivially follows from the same kind of bounds for the same class of classifiers trained over i.i.d. data. Finally, we show an example of the derivation of a margin-based, data-dependent generalization error bound (i.e. a bound on equation (1) which can be computed on the training data) for the bipartite ranking case when H = {(x, x′) →sign(K(θ, x) −K(θ, x′))|K(θ, θ) ≤B2}, assuming that the input examples are drawn i.i.d. according to a distribution D over X × {±1}, X ⊂Rd and K is a kernel over X 2. Notations Throughout the paper, we will use the notations of the preceding subsection, except for Z = (zi)N i=1, which will denote an arbitrary element of (Xtr × {−1, 1})N. In order to obtain the dependency graph of the random variables ϕ(S)i, we will consider, for each 1 ≤i ≤N, a set [i] ⊂{1, ..., n} such that ϕ(S)i depends only on the variables sp ∈S for which p ∈[i]. Using these notations, if we consider two indices k, l in {1, ..., N}, we can notice that the two random variables ϕ(S)k and ϕ(S)l are independent if and only if [k]∩[l] = ∅. The dependency graph of the ϕ(S)is follows, by constructing the graph Γ(ϕ), with the set of vertices V = {1, ..., N}, and with an edge between k and l if and only if [k] ∩[l] ̸= ∅. The following definitions, taken from [8], will enable us to separate the set of partly dependent variables into sets of independent variables: • A subset A of V is independent if all the elements in A are independent. • A sequence C = (Cj)m j=1 of subsets of V is a proper cover of V if, for all j, Cj is independent, and j Cj = V • A sequence C = (Cj, wj)m j=1 is a proper, exact fractional cover of Γ if wj > 0 for all j, and, for each i ∈V , m j=1 wjICj(i) = 1, where ICj is the indicator function of Cj. • The fractional chromatic number of Γ, noted χ(Γ), is equal to the minimum of j wj over all proper, exact fractional cover. It is to be noted that from lemma 3.2 of [8], the existence of proper, exact fractional covers is ensured. Since Γ is fully determined by the function ϕ, we will note χ(Γ) = χ(ϕ). Moreover, we will denote by C(ϕ) = (Cj, wj)κ j=1 a proper, exact fractional cover of Γ such that j wj = χ(ϕ). Finally, for a given C(ϕ), we denote by κj the number of elements in Cj, and we fix the notations: Cj = {Cj1, ..., Cjκj}. It is to be noted that if (ti)N i=1 ∈RN, and C(ϕ) = (Cj, wj)κ j=1, lemma 3.1 of [8] states that: N i=1 ti = κ j=1 wjTj, where Tj = κj k=1 tCjk (2) 3 A new concentration inequality Concentration inequalities bound the probability that a random variable deviates too much from its expectation (see [4] for a survey). They play a major role in learning theory as they can be used for example to bound the probability of deviation of the expected loss of a function from its empirical value estimated over a sample set. A well-known inequality is McDiarmid’s theorem [9] for independent random variables, which bounds the probability of deviation from its expectation of an arbitrary function with bounded variations over each one of its parameters. While this theorem is very general, [8] proved a large deviation bound for sums of partly random variables where the dependency structure of the variables is known, which can be tighter in some cases. Since we also consider variables with known dependency structure, using such results may lead to tighter bounds. However, we will bound functions like in equation (1), which do not write as a sum of partly dependent variables. Thus, we need a result on more general functions than sums of random variables, but which also takes into account the known dependency structure of the variables. Theorem 2. Let ϕ : X n →X ′N. Using the notations defined above, let C(ϕ) = (Cj, wj)κ j=1. Let f : X ′N →R such that: 1. There exist κ functions fj : X ′κj →R which satisfy ∀Z = (z1, ..., zN) ∈X ′N, f(Z) = j wjfj(zCj1, ..., zCjκj ). 2. There exist β1, ..., βN ∈R+ such that ∀j, ∀Zj, Zk j ∈X ′κj such that Zj and Zk j differ only in the k-th dimension, |fj(Zj) −fj(Zk j )| ≤βCjk. Let finally D1, ..., Dn be n probability distributions over X. Then, we have: PX∼⊗n i=1Di(f ◦ϕ(X) −Ef ◦ϕ > ϵ) ≤exp(− 2ϵ2 χ(ϕ) N i=1 β2 i ) (3) and the same holds for P(Ef ◦ϕ −f ◦ϕ > ϵ). The proof of this theorem (given in [13]) is a variation of the demonstrations in [8] and McDiarmid’s theorem. The main idea of this theorem is to allow the decomposition of f, which will take as input partly dependent random variables when applied to ϕ(S), into a sum of functions which, when considering f ◦ϕ(S), will be functions of independent variables. As we will see, this theorem will be the major tool in our analysis. It is to be noted that when X = X ′, N = n and ϕ is the identity function of X n, the theorem 2 is exactly McDiarmid’s theorem. On the other hand, when f takes the form N i=1 qi(zi) with for all z ∈X ′, a ≤qi(z) ≤a + βi with a ∈R, then theorem 2 reduces to a particular case of the large deviation bound of [8]. 4 The fractional Rademacher complexity Let Z = (zi)N i=1 ∈ZN. If Z is supposed to be drawn i.i.d. according to a distribution DZ over Z, for a class F of functions from Z to R, the Rademacher complexity of F is defined by [10] RN(F) = EZ∼DZRN(F, Z), where RN(F, Z) = Eσ supf∈F N i=1 σif(zi) is the empirical Rademacher complexity of F on Z, and σ = (σi)n i=1 is a sequence of independent Rademacher variables, i.e. ∀i, P(σi = 1) = P(σi = −1) = 1 2. This quantity has been extensively used to measure the complexity of function classes in previous bounds for binary classification [3, 10]. In particular, if we consider a class of functions Q = {q : Z →[0, 1]}, it can be shown (theorem 4.9 in [12]) that with probability at least 1−δ over Z, all q ∈Q verify the following inequality, which serves as a preliminary result to show data-dependent bounds for SVMs in [12]: EZ∼DZq(z) ≤1 N N i=1 q(zi) + RN(Q) + ln(1/δ) 2N (4) In this section, we generalize equation (4) to our case with theorem 4, using the following definition2 (we denote λ(q, ϕ(S)) = 1 N N i=1 q(ϕ(S)i) and λ(q) = ESλ(q, ϕ(S))): Definition 3. Let Q, be class of functions from a set Z to R, Let ϕ : X n →ZN and S a sample of size n drawn according to a product distribution ⊗n p=1Dp over X n. Then, we define the empirical fractional Rademacher complexity 3 of Q given ϕ as: R∗ n(Q, S, ϕ) = 2 N Eσ j wj sup q∈Q i∈Cj σiq(ϕ(S)i) As well as the fractional Rademacher complexity of Q as R∗ n(Q, ϕ) = ESR∗ n(Q, S, ϕ) Theorem 4. Let Q be a class of functions from Z to [0, 1]. Then, with probability at least 1 −δ over the samples S drawn according to n p=1 Dp, for all q ∈Q: λ(q) −1 N N i=1 q(ϕi(S)) ≤R∗ n(Q, ϕ) + χ(ϕ) ln(1/δ) 2N And: λ(q) −1 N N i=1 q(ϕi(S)) ≤R∗ n(Q, S, ϕ) + 3 χ(ϕ) ln(2/δ) 2N In the definition of the fractional Rademacher complexity (FRC), if ϕ is the identity function, we recover the standard Rademacher complexity, and theorem 4 reduces to equation (4). These results are therefore extensions of equation (4), and show that the generalization error bounds for the tasks falling in our framework will follow from a unique approach. Proof. In order to find a bound for all q in Q of λ(q) −λ(q, ϕ(S)), we write: λ(q) −λ(q, ϕ(S)) ≤sup q∈Q E ˜S 1 N N i=1 q(ϕ( ˜S)i) −1 N N i=1 q(ϕ(S)i) ≤1 N j wj sup q∈Q ⎡ ⎣E ˜S i∈Cj q(ϕ( ˜S)i) − i∈Cj q(ϕ(S)i) ⎤ ⎦ (5) Where we have used equation (2). Now, consider, for each j, fj : Zκj →R such that, for all z(j) ∈Zκj, fj(z(j)) = 1 N supq∈Q E ˜S κj k=1 q(ϕ( ˜S)Cjk) −κj k=1 q(z(j) k ). It is clear that if f : ZN →R is defined by: for all Z ∈ZN, f(Z) = N j=1 wjfj(zCj1, ..., zCjκj), then the right side of equation (5) is equal to f ◦ϕ(S), and that f satisfies all the conditions of theorem 2 with, for all i ∈{1, ..., N}, βi = 1 N . Therefore, with a direct application of theorem 2, we can claim that, with probability at least 1 −δ over samples S drawn according to n p=1 Dp (we denote λj(q, ϕ(S)) = 1 N i∈Cj q(ϕ(S)i)): λ(q) −λ(q, ϕ(S)) ≤ES j wj sup q∈Q E ˜Sλj(q, ϕ( ˜S)) −λj(q, ϕ(S)) + χ(ϕ) ln(1/δ) 2N ≤ES, ˜S j wj N sup q∈Q i∈Cj [q(ϕ( ˜S)i) −q(ϕ(S)i)] + χ(ϕ) ln(1/δ) 2N (6) 2The fractional Rademacher complexity depends on the cover C(ϕ) chosen, since it is not unique. However in practice, our bounds only depend on χ(ϕ) (see section 4.1). 3this denomination stands as it is a sum of Rademacher averages over independent parts of ϕ(S). Now fix j, and consider σ = (σi)N i=1, a sequence of N independent Rademacher variables. For a given realization of σ, we have that ES, ˜S sup q∈Q i∈Cj [q(ϕ( ˜S)i) −q(ϕ(S)i)] = ES, ˜S sup q∈Q i∈Cj σi[q(ϕ( ˜S)i) −q(ϕ(S)i)] (7) because, for each σi considered, σi = −1 simply corresponds to permutating, in S, ˜S, the two sequences S[i] and ˜S[i] (where S[i] denotes the subset of S ϕ(S)i really depends on) which have the same distribution (even though the sp’s are not identically distributed), and are independent from the other S[k] and ˜S[k] since we are considering i, k ∈Cj. Therefore, taking the expection over S, ˜S is the same with the elements permuted this way as if they were not permuted. Then, from equation (6), the first inequality of the theorem follow. The second inequality is due to an application of theorem 2 to R∗ n(Q, S, ϕ). Remark 5. The symmetrization performed in equation (7) requires the variables ϕ(S)i appearing in the same sum to be independent. Thus, the generalization of Rademacher complexities could only be performed using a decomposition in independent sets, and the cover C assures some optimality of the decomposition. Moreover, even though McDiarmid’s theorem could be applied each time we used theorem 2, the derivation of the real numbers bounding the differences is not straightforward, and may not lead to the same result. The creation of the dependency graph of ϕ and theorem 2 are therefore necessary tools for obtaining theorem 4. Properties of the fractional Rademacher complexity Theorem 6. Let Q be a class of functions from a set Z to R, and ϕ : X n →ZN. For S ∈X n, the following results are true. 1. Let φ : R →R, an L-Lipschitz function. Then R∗ n(φ ◦Q, S, ϕ) ≤LR∗ n(Q, S, ϕ) 2. If there exist M > 0 such that for every k, and samples Sk of size k Rk(Q, Sk) ≤ M √ k, then R∗ n(Q, S, ϕ) ≤M χ(ϕ) N 3. Let K be a kernel over Z, B > 0, denote ||x||K = K(x, x) and define HK,B = {hθ : Z →R, hθ(x) = K(θ, x)|||θ||K ≤B}. Then: R∗ n(HK,B, S, ϕ) ≤2B χ(ϕ) N N i=1 ||ϕ(S)i||2 K The first point of this theorem is a direct consequence of a Rademacher process comparison theorem, namely theorem 7 of [10], and will enable the obtention of margin-based bounds. The second and third points show that the results regarding the Rademacher complexity can be used to immediately deduce bounds on the FRC. This result, as well as theorem 4 show that binary classifiers of i.i.d. data and classifiers of interdependent data will have generalization bounds of the same form, but with different convergence rate depending on the dependency structure imposed by ϕ. elements of proof. The second point results from Jensen’s inequality, using the facts that j wj = χ(ϕ) and, from equation (2), j wj|Cj| = N. The third point is based on the same calculations by noting that (see e.g. [3]), if Sk = ((xp, yp))k p=1, then Rk(HK,B, Sk) ≤2B k k p=1 ||xp||2 K. 5 Data-dependent bounds The fact that classifiers trained on interdependent data will ”inherit” the generalization bound of the same classifier trained on i.i.d. data suggests simple ways of obtaining bipartite ranking algorithms. Indeed, suppose we want to learn a linear ranking function, for example a function h ∈HK,B as defined in theorem 6, where K is a linear kernel, and consider a sample S ∈(X × {−1, 1})n with X ⊂Rd, drawn i.i.d. according to some D. Then we have, for input examples (x, 1) and (x′, −1) in S, h(x) −h(x′) = h(x −x′). Therefore, we can learn a bipartite ranking function by applying an SVM algorithm to the pairs ((x, 1), (x′, −1)) in S, each pair being represented by x −x′, and our framework allows to immediately obtain generalization bounds for this learning process based on the generalization bounds for SVMs. We show these bounds in theorem 7. To derive the bounds, we consider φ, the 1-Lipschitz function defined by φ(x) = min(1, max(1 −x, 0)) ≥[[x ≤0]]4. Given a training example z, we denote by zl its label and zf its feature representation. With an abuse of notation, we denote φ(h, Z) = 1 N N i=1 φ(zl ih(zf i )). For a sample S drawn according to n p=1 Dp, we have, for all h in some function class H: ES 1 N i ℓ(h, zi) ≤ES 1 N i φ(zl ih(zf i )) ≤φ(h, Z) + Eσ j 2wj N sup h∈H i∈Cj σiφ(zl ih(zf i )) + 3 χ(ϕ) ln(2/δ) 2N where ℓ(h, zi) = [[zl ih(zf i ) ≤0]], and the last inequality holds with probability at least 1 −δ over samples S from theorem 4. Notice that when σCjk is a Rademacher variable, it has the same distribution as zl CjkσCjk since zl Cjk ∈{−1, 1}. Thus, using the first result of theorem 6 we have that with probability 1 −δ over the samples S, all h in H satisfy: ES 1 N i ℓ(h, zi) ≤1 N i φ(zlh(zf)) + R∗ n(H, S, ϕ) + 3 χ(ϕ) ln(2/δ) 2N (8) Now putting in equation (8) the third point of theorem 6, with H = HK,B as defined in theorem 6 with Z = X, we obtain the following theorem: Theorem 7. Let S ∈(X × {−1, 1})n be a sample of size n drawn i.i.d. according to an unknown distribution D. Then, with probability at least 1 −δ, all h ∈HK,B verify: ES[[yih(xi) ≤0]] ≤1 n n i=1 φ(yih(xi)) + 2B n n i=1 ||xi||2 K + 3 ln(2/δ) 2n And E{[[h(x) ≤h(x′)]]|y = 1, y′ = −1} ≤ 1 nS +nS − nS + i=1 nS − j=1 φ(h(xσ(i)) −h(xν(j))) + 2B max(nS +, nS −) nS +nS − nS + i=1 nS − j=1 ||xσ(i) −xν(j)||2 K + 3 ln(2/δ) 2 min(nS +, nS −) Where nS +, nS −are the number of positive and negative instances in S, and σ and ν also depend on S, and are such that xσ(i) is the i-th positive instance in S and ν(j) the j-th negative instance. 4remark that φ is upper bounded by the slack variables of the SVM optimization problem (see e.g. [12]). It is to be noted that when h ∈HK,B with a non linear kernel, the same bounds apply, with, for the case of bipartite ranking, ||xσ(i) −xν(j)||2 K replaced by ||xσ(i)||2 K + ||xν(j)||2 K − 2K(xσ(i), xν(j)). For binary classification, we recover the bounds of [12], since our framework is a generalization of their approach. As expected, the bounds suggest that kernel machines will generalize well for bipartite ranking. Thus, we recover the results of [2] obtained in a specific framework of algorithmic stability. However, our bound suggests that the convergence rate is controlled by 1/ min(nS +, nS −), while their results suggested 1/nS + + 1/nS −. The full proof, in which we follow the approach of [1], is given in [13]. 6 Conclusion We have shown a general framework for classifiers trained with interdependent data, and provided the necessary tools to study their generalization properties. It gives a new insight on the close relationship between the binary classification task and the bipartite ranking, and allows to prove the first data-dependent bounds for this latter case. Moreover, the framework could also yield comparable bounds on other learning tasks. Acknowledgments This work was supported in part by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778. This publication only reflects the authors views. References [1] Agarwal S., Graepel T., Herbrich R., Har-Peled S., Roth D. (2005) Generalization Error Bounds for the Area Under the ROC curve, Journal of Machine Learning Research. [2] Agarwal S., Niyogi P. (2005) Stability and generalization of bipartite ranking algorithms, Conference on Learning Theory 18. [3] Bartlett P., Mendelson S. (2002) Rademacher and Gaussian Complexities: Risk Bounds and Structural Results, Journal of Machine Learning Research 3, pp. 463-482. [4] Boucheron S., Bousquet O., Lugosi G. (2004) Concentration inequalities, in O. Bousquet, U.v. Luxburg, and G. Rtsch (editors), Advanced Lectures in Machine Learning, Springer, pp. 208-240. [5] Clemenc¸on S., Lugosi G., Vayatis N. (2005) Ranking and scoring using empirical risk minimization, Conference on Learning Theory 18. [6] Cortes C., Mohri M. (2004) AUC optimization vs error rate miniminzation NIPS 2003, [7] Freund Y., Iyer R.D., Schapire R.E., Singer Y. (2003) An Efficient Boosting Algorithm for Combining Preferences, Journal of Machine Learning Research 4, pp. 933-969. [8] Janson S. (2004) Large deviations for sums of partly dependent random variables, Random Structures and Algorithms 24, pp. 234-248. [9] McDiarmid C. (1989) On the method of bounded differences, Surveys in Combinatorics. [10] Meir R., Zhang T. (2003) Generalization Error Bounds for Bayesian Mixture Algorithms, Journal of Machine Learning Research 4, pp. 839-860. [11] Rudin C., Cortes C., Mohri M., Schapire R.E. (2005) Margin-Based Ranking meets Boosting in the middle, Conference on Learning Theory 18. [12] Shawe-Taylor J., Cristianini N. (2004) Kernel Methods for Pattern Analysis, Cambridge U. Prs. [13] Long version of this paper, Available at http://www-connex.lip6.fr/˜usunier/nips05-lv.pdf
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Bayesian model learning in human visual perception Gerg˝o Orb´an Collegium Budapest Institute for Advanced Study 2 Szenth´aroms´ag utca, Budapest, 1014 Hungary ogergo@colbud.hu J´ozsef Fiser Department of Psychology and Volen Center for Complex Systems Brandeis University Waltham, Massachusetts 02454, USA fiser@brandeis.edu Richard N. Aslin Department of Brain and Cognitive Sciences, Center for Visual Science University of Rochester Rochester, New York 14627, USA aslin@cvs.rochester.edu M´at´e Lengyel Gatsby Computational Neuroscience Unit University College London 17 Queen Square, London WC1N 3AR United Kingdom lmate@gatsby.ucl.ac.uk Abstract Humans make optimal perceptual decisions in noisy and ambiguous conditions. Computations underlying such optimal behavior have been shown to rely on probabilistic inference according to generative models whose structure is usually taken to be known a priori. We argue that Bayesian model selection is ideal for inferring similar and even more complex model structures from experience. We find in experiments that humans learn subtle statistical properties of visual scenes in a completely unsupervised manner. We show that these findings are well captured by Bayesian model learning within a class of models that seek to explain observed variables by independent hidden causes. 1 Introduction There is a growing number of studies supporting the classical view of perception as probabilistic inference [1, 2]. These studies demonstrated that human observers parse sensory scenes by performing optimal estimation of the parameters of the objects involved [3, 4, 5]. Even single neurons in primary sensory cortices have receptive field properties that seem to support such a computation [6]. A core element of this Bayesian probabilistic framework is an internal model of the world, the generative model, that serves as a basis for inference. In principle, inference can be performed on several levels: the generative model can be used for inferring the values of hidden variables from observed information, but also the model itself may be inferred from previous experience [7]. Most previous studies testing the Bayesian framework in human psychophysical experiments used highly restricted generative models of perception, usually consisting of a few observed and latent variables, of which only a limited number of parameters needed to be adjusted by experience. More importantly, the generative models considered in these studies were tailor-made to the specific pscychophysical task presented in the experiment. Thus, it remains to be shown whether more flexible, ‘open-ended’ generative models are used and learned by humans during perception. Here, we use an unsupervised visual learning task to show that a general class of generative models, sigmoid belief networks (SBNs), perform similarly to humans (also reproducing paradoxical aspects of human behavior), when not only the parameters of these models but also their structure is subject to learning. Crucially, the applied Bayesian model learning embodies the Automatic Occam’s Razor (AOR) effect that selects the models that are ‘as simple as possible, but no simpler’. This process leads to the extraction of independent causes that efficiently and sufficiently account for sensory experience, without a pre-specification of the number or complexity of potential causes. In section 2, we describe the experimental protocol we used in detail. Next, the mathematical framework is presented that is used to study model learning in SBNs (Section 3). In Section 4, experimental results on human performance are compared to the prediction of our Bayes-optimal model learning in the SBN framework. All the presented human experimental results were reproduced and had identical roots in our simulations: the modal model developed latent variables corresponding to the unknown underlying causes that generated the training scenes. In Section 5, we discuss the implications of our findings. Although structure and parameter learning are not fundamentally different computations in Bayesian inference, we argue that the natural integration of these two kinds of learning lead to a behavior that accounts for human data which cannot be reproduced in some simpler alternative learning models with parameter but without structure learning. Given the recent surge of biologically plausible neural network models performing inference in belief networks we also point out challenges that our findings present for future models of probabilistic neural computations. 2 Experimental paradigm Human adult subjects were trained and then tested in an unsupervised learning paradigm with a set of complex visual scenes consisting of 6 of 12 abstract unfamiliar black shapes arranged on a 3x3 (Exp 1) or 5x5 (Exps 2-4) white grid (Fig. 1, left panel). Unbeknownst to subjects, various subsets of the shapes were arranged into fixed spatial combinations (combos) (doublets, triplets, quadruplets, depending on the experiment). Whenever a combo appeared on a training scene, its constituent shapes were presented in an invariant spatial arrangement, and in no scenes elements of a combo could appear without all the other elements of the same combo also appearing. Subjects were presented with 100–200 training scenes, each scene was presented for 2 seconds with a 1-second pause between scenes. No specific instructions were given to subjects prior to training, they were only asked to pay attention to the continuous sequence of scenes. The test phase consisted of 2AFC trials, in which two arrangements of shapes were shown sequentially in the same grid that was used in the training, and subjects were asked which of the two scenes was more familiar based on the training. One of the presented scenes was either a combo that was actually used for constructing the training set (true combo), or a part of it (embedded combo) (e.g., a pair of adjacent shapes from a triplet or quadruplet combo). The other scene consisted of the same number of shapes as the first scene in an arrangement that might or might not have occurred during training, but was in fact a mixture of shapes from different true combos (mixture combo). Here four experiments are considered that assess various aspects of human observational 11 w12 w22 w24 wx2 wx1 y1 x1 y3 y4 y2 wy1 wy4 wy2 wy3 x2 w23 w Figure 1: Experimental design (left panel) and explanation of graphical model parameters (right panel). learning, the full set of experiments are presented elsewhere [8, 9]. Each experiment was run with 20 na¨ıve subjects. 1. Our first goal was to establish that humans are sensitive to the statistical structure of visual experience, and use this experience for judging familiarity. In the baseline experiment 6 doublet combos were defined, three of which were presented simultaneously in any given training scene, allowing 144 possible scenes [8]. Because the doublets were not marked in any way, subjects saw only a group of random shapes arranged on a grid. The occurrence frequency of doublets and individual elements was equal across the set of scenes, allowing no obvious bias to remember any element more than others. In the test phase a true and a mixture doublet were presented sequentially in each 2AFC trial. The mixture combo was presented in a spatial position that had never appeared before. 2. In the previous experiment the elements of mixture doublets occurred together fewer times than elements of real doublets, thus a simple strategy based on tracking co-occurrence frequencies of shape-pairs would be sufficient to distinguish between them. The second, frequency-balanced experiment tested whether humans are sensitive to higher-order statistics (at least cross-correlations, which are co-occurence frequencies normalized by respective invidual occurence frequencies). The structure of Experiment 1 was changed so that while the 6 doublet combo architecture remained, their appearance frequency became non-uniform introducing frequent and rare combos. Frequent doublets were presented twice as often as rare ones, so that certain mixture doublets consisting of shapes from frequent doublets appeared just as often as rare doublets. Note, that the frequency of the constituent shapes of these mixture doublets was higher than that of rare doublets. The training session consisted of 212 scenes, each scene being presented twice. In the test phase, the familiarity of both single shapes and doublet combos was tested. In the doublet trials, rare combos with low appearance frequency but high correlations between elements were compared to mixed combos with higher element and equal pair appearance frequency, but lower correlations between elements. 3. The third experiment tested whether human performance in this paradigm can be fully accounted for by learning cross-correlations. Here, four triplet combos were formed and presented with equal occurrence frequencies. 112 scenes were presented twice to subjects. In the test phase two types of tests were performed. In the first type, the familiarity of a true triplet and a mixture triplet was compared, while in the second type doublets consisting of adjacent shapes embedded in a triplet combo (embedded doublet) were tested against mixture doublets. 4. The fourth experiment compared directly how humans treat embedded and independent (non-embedded) combos of the same spatial dimensions. Here two quadruplet combos and two doublet combos were defined and presented with equal frequency. Each training scene consisted of six shapes, one quadruplet and one doublet. 120 such scenes were constructed. In the test phase three types of tests were performed. First, true quadruplets were compared to mixture quadruplets; next, embedded doublets were compared to mixture doublets, finally true doublets were compared to mixture doublets. 3 Modeling framework The goal of Bayesian learning is to ‘reverse-engineer’ the generative model that could have generated the training data. Because of inherent ambiguity and stochasticity assumed by the generative model itself, the objective is to establish a probability distribution over possible models. Importantly, because models with parameter spaces of different dimensionality are compared, the likelihood term (Eq. 3) will prefer the simplest model (in our case, the one with fewest parameters) that can effectively account for (generate) the training data due to the AOR effect in Bayesian model comparison [7]. Sigmoid belief networks The class of generative models we consider is that of two-layer sigmoid belief networks (SBNs, Fig. 1). The same modelling framework has been successfully aplied to animal learning in classical conditioning [10, 11]. The SBN architecture assumes that the state of observed binary variables (yj, in our case: shapes being present or absent in a training scene) depends through a sigmoidal activation function on the state of a set of hidden binary variables (x), which are not directly observable: P (yj = 1|x, wm, m) = 1 + exp − X i wijxi −wyj !!−1 (1) where wij describes the (real-valued) influence of hidden variable xi on observed variable yj, wyj determines the spontaneous activation bias of yj, and m indicates the model structure, including the number of latent variables and identity of the observeds they can influence (the wij weights that are allowed to have non-zero value). Observed variables are independent conditioned on the latents (i.e. any correlation between them is assumed to be due to shared causes), and latent variables are marginally independent and have Bernoulli distributions parametrised by wx: P (y|x, wm, m) = Y j P (yj|x, wm, m) , P (x|wm, m) = Y i (1 + exp (−1xiwxi))−1 (2) Finally, scenes (y(t)) are assumed to be iid samples from the same generative distribution, and so the probability of the training data (D) given a specific model is: P (D|wm, m) = Y t P y(t)|wm, m = Y t X x Y j P y(t) j , x|wm, m (3) The ‘true’ generative model that was actually used for generating training data in the experiments (Section 2) is closely related to this model, with the combos corresponding to latent variables. The main difference is that here we ignore the spatial aspects of the task, i.e. only the occurrence of a shape matters but not where it appears on the grid. Although in general, space is certainly not a negligible factor in vision, human behavior in the present experiments depended on the fact of shape-appearances sufficiently strongly so that this simplification did not cause major confounds in our results. A second difference between the model and the human experiments was that in the experiments, combos were not presented completely randomly, because the number of combos per scene was fixed (and not binomially distributed as implied by the model, Eq. 2). Nevertheless, our goal was to demonstrate the use of a general-purpose class of generative models, and although truly independent causes are rare in natural circumstances, always a fixed number of them being present is even more so. Clearly, humans are able to capture dependences between latent variables, and these should be modeled as well ([12]). Similarly, for simplicity we also ignored that subsequent scenes are rarely independent (Eq. 3) in natural vision. Training Establishing the posterior probability of any given model is straightforward using Bayes’ rule: P (wm, m|D) ∝P (D|wm, m) P (wm, m) (4) where the first term is the likelihood of the model (Eq. 3), and the second term is the prior distribution of models. Prior distributions for the weights were: P (wij) = Laplace (12, 2), P (wxi) = Laplace (0, 2), P wxj = δ (−6). The prior over model structure preferred simple models and was such that the distributions of the number of latents and of the number of links conditioned on the number of latents were both Geometric (0.1). The effect of this preference is ‘washed out’ with increasing training length as the likelihood term (Eq. 3) sharpens. Testing When asked to compare the familiarity of two scenes (yA and yB) in the testing phase, the optimal strategy for subjects would be to compute the posterior probability of both scenes based on the training data P yZ|D = X m Z dwm X x P yZ, x|wm, m P (wm, m|D) (5) and always (ie, with probability one) choose the one with the higher probability. However, as a phenomenological model of all kinds of possible sources of noise (sensory noise, model noise, etc) we chose a soft threshold function for computing choice probability: P (choose A) = 1 + exp −β log P yA|D P (yB|D) !!−1 (6) and used β = 1 (β = ∞corresponds to the optimal strategy). Note that when computing the probability of a test scene, we seek the probability that exactly the given scene was generated by the learned model. This means that we require not only that all the shapes that are present in the test scene are present in the generated data, but also that all the shapes that are absent from the test scene are absent from the generated data. A different scheme, in which only the presence but not the absence of the shapes need to be matched (i.e. absent observeds are marginalized out just as latents are in Eq. 5) could also be pursued, but the results of the embedding experiments (Exp. 3 and 4, see below) discourage it. The model posterior in Eq. 4 is analytically intractable, therefore an exchange reversiblejump Markov chain Monte Carlo sampling method [10, 13, 14] was applied, that ensured fair sampling from a model space containing subspaces of differring dimensionality, and integration over this posterior in Eq. 5 was approximated by a sum over samples. 4 Results Pilot studies were performed with reduced training datasets in order to test the performance of the model learning framework. First, we trained the model on data consisting of 8 observed variables (‘shapes’). The 8 ‘shapes’ were partitioned into three ‘combos’ of different . −6 −6 −6 −6 13 12 12 −1 −6 −6 −6 −6 12 13 12 13 12 −1 −1 . 0 10 20 30 0 1 2 3 4 Training length Avarage latent # Figure 2: Bayesian learning in sigmoid belief networks. Left panel: MAP model of a 30trial-long training with 8 observed variables and 3 combos. Latent variables of the MAP model reflect the relationships defined by the combos. Right panel: Increasing model complexity with increasing training experience. Average number of latent variables (±SD) in the model posterior distribution as a function of the length of training data was obtained by marginalizing Eq. 4 over weights w. sizes (5, 2, 1), two of which were presented simultaneously in each training trial. The AOR effect in Bayesian model learning should select the model structure that is of just the right complexity for describing the data. Accordingly, after 30 trials, the maximum a posteriori (MAP) model had three latents corresponding to the underlying ‘combos’ (Fig. 2, left panel). Early on in training simpler model structures dominated because of the prior preference for low latent and link numbers, but due to the simple structure of the training data the likelihood term won over in as few as 10 trials, and the model posterior converged to the true generative model (Fig. 2, right panel, gray line). Importantly, presenting more data with the same statistics did not encourage the fitting of over-complicated model structures. On the other hand, if data was generated by using more ‘combos’ (4 ‘doublets’), model learning converged to a model with a correspondingly higher number of latents (Fig. 2, right panel, black line). In the baseline experiment (Experiment 1) human subjects were trained with six equalsized doublet combos and were shown to recognize true doublets over mixture doublets (Fig. 3, first column). When the same training data was used to compute the choice probability in 2AFC tests with model learning, true doublets were reliably preferred over mixture doublets. Also, the MAP model showed that the discovered latent variables corresponded to the combos generating the training data (data not shown). In Experiment 2, we sought to answer the question whether the statistical learning demonstrated in Experiment 1 was solely relying on co-occurrence frequencies, or was using something more sophisticated, such as at least cross-correlations between shapes. Bayesian model learning, as well as humans, could distinguish between rare doublet combos and mixtures from frequent doublets (Fig. 3, second column) despite their balanced co-occurrence frequencies. Furthermore, although in this comparison rare doublet combos were preferred, both humans and the model learned about the frequencies of their constituent shapes and preferred constituent single shapes of frequent doublets over those of rare doublets. Nevertheless, it should be noted that while humans showed greater preference for frequent singlets than for rare doublets our simulations predicted an opposite trend1. We were interested whether the performance of humans could be fully accounted for by the learning of cross-correlations, or they demonstrated more sophisticated computations. 1This discrepancy between theory and experiments may be explained by Gestalt effects in human vision that would strongly prefer the independent processing of constituent shapes due to their clear spatial separation in the training scenes. The reconciliation of such Gestalt effects with pure statistical learning is the target of further investigations. Experiment 1 Experiment 2 Experiment 3 Experiment 4 EXPERIMENT dbls 0 20 40 60 80 100 Percent correct dbls sngls 0 20 40 60 80 100 trpls e’d dbls 0 20 40 60 80 100 qpls idbls e’d dbls 0 20 40 60 80 100 MODEL dbls 0 20 40 60 80 100 Percent correct dbls sngls 0 20 40 60 80 100 trpls e’d dbls 0 20 40 60 80 100 qpls idbls e’d dbls 0 20 40 60 80 100 Figure 3: Comparison of human and model performance in four experiments. Bars show percent ‘correct’ values (choosing a true or embedded combo over a mixture combo, or a frequent singlet over a rare singlet) for human experiments (average over subjects ±SEM), and ‘correct’ choice probabilities (Eq. 6) for computer simulations. Sngls: Single shapes; dbls: Doublet combos; trpls: triplet combos; e’d dbls: embedded doublet combos; qpls: quadruple combos; idbls: independent doublet combos. In Experiment 3, training data was composed of triplet combos, and beside testing true triplets against mixture triplets, we also tested embedded doublets (pairs of shapes from the same triplet) against mixture doublets (pairs of shapes from different triplets). If learning only depends on cross-correlations, we expect to see similar performance on these two types of tests. In contrast, human performace was significantly different for triplets (true triplets were preferred) and doublets (embedded and mixture doublets were not distinguished) (Fig. 3, third column). This may be seen as Gestalt effects being at work: once the ‘whole’ triplet is learned, its constituent parts (the embedded doublets) loose their significance. Our model reproduced this behavior and provided a straightforward explanation: latent-to-observed weights (wij) in the MAP model were so strong that whenever a latent was switched on it could almost only produce triplets, therefore doublets were created by spontaneous independent activation of observeds which thus produced embedded and mixture doublets with equal chance. In other words, doublets were seen as mere noise under the MAP model. The fourth experiment tested explicitly whether embedded combos and equal-sized independent real combos are distinguished and not only size effects prevented the recognition of embedded small structures in the previous experiment. Both human experiments and Bayesian model selection demonstrated that quadruple combos as well as stand-alone doublets were reliably recognized (Fig. 3, fourth column), while embedded doublets were not. 5 Discussion We demonstrated that humans flexibly yet automatically learn complex generative models in visual perception. Bayesian model learning has been implicated in several domains of high level human cognition, from causal reasoning [15] to concept learning [16]. Here we showed it being at work already at a pre-verbal stage. We emphasized the importance of learning the structure of the generative model, not only its parameters, even though it is quite clear that the two cannot be formally distinguished. Nevertheless we have two good reasons to believe that structure learning is indeed important in our case. (1) Sigmoid belief networks identical to ours but without structure learning have been shown to perform poorly on a task closely related to ours [17], F¨oldi´ak’s bar test [18]. More complicated models will of course be able to produce identical results, but we think our model framework has the advantage of being intuitively simple: it seeks to find the simplest possible explanation for the data assuming that it was generated by independent causes. (2) Structure learning allows Occam’s automatic razor to come to play. This is computationally expensive, but together with the generative model class we use provides a neat and highly efficient way to discover ‘independent components’ in the data. We experienced difficulties with other models [17] developed for similar purposes when trying to reproduce our experimental findings. Our approach is very much in the tradition that sees the finding of independent causes behind sensory data as one of the major goals of perception [2]. Although neural network models that can produce such computations exist [6, 19], none of these does model selection. Very recently, several models have been proposed for doing inference in belief networks [20, 21] but parameter learning let alone structure learning proved to be non-trivial in them. Our results highlight the importance of considering model structure learning in neural models of Bayesian inference. Acknowledgements We were greatly motivated by the earlier work of Aaron Courville and Nathaniel Daw [10, 11], and hugely benefited from several useful discussions with them. We would also like to thank the insightful comments of Peter Dayan, Maneesh Sahani, Sam Roweis, and Zolt´an Szatm´ary on an earlier version of this work. This work was supported by IST-FET1940 program (GO), NIH research grant HD-37082 (RNA, JF), and the Gatsby Charitable Foundation (ML). References [1] Helmholtz HLF. Treatise on Physiological Optics. New York: Dover, 1962. [2] Barlow HB. Vision Res 30:1561, 1990. [3] Ernst MO, Banks MS. Nature 415:429, 2002. [4] K¨ording KP, Wolpert DM. Nature 427:244, 2004. [5] Kersten D, et al. Annu Rev Psychol 55, 2004. [6] Olshausen BA, Field DJ. Nature 381:607, 1996. [7] MacKay DJC. Network: Comput Neural Syst 6:469, 1995. [8] Fiser J, Aslin RN. Psych Sci 12:499, 2001. [9] Fiser J, Aslin RN. J Exp Psychol Gen , in press. [10] Courville AC, et al. In NIPS 16 , Cambridge, MA, 2004. MIT Press. [11] Courville AC, et al. In NIPS 17 , Cambridge, MA, 2005. MIT Press. [12] Hinton GE, et al. In Artificial Intelligence and Statistics , Barbados, 2005. [13] Green PJ. Biometrika 82:711, 1995. [14] Iba Y. Int J Mod Phys C 12:623, 2001. [15] Tenenbaum JB, Griffiths TL. In NIPS 15 , 35, Cambridge, MA, 2003. MIT Press. [16] Tenenbaum JB. In NIPS 11 , 59, Cambridge, MA, 1999. MIT Press. [17] Dayan P, Zemel R. Neural Comput 7:565, 1995. [18] F¨oldiak P. Biol Cybern 64:165, 1990. [19] Dayan P, et al. Neural Comput 7:1022, 1995. [20] Rao RP. Neural Comput 16:1, 2004. [21] Deneve S. In NIPS 17 , Cambridge, MA, 2005. MIT Press.
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Temporal Abstraction in Temporal-difference Networks Richard S. Sutton, Eddie J. Rafols, Anna Koop Department of Computing Science University of Alberta Edmonton, AB, Canada T6G 2E8 {sutton,erafols,anna}@cs.ualberta.ca Abstract We present a generalization of temporal-difference networks to include temporally abstract options on the links of the question network. Temporal-difference (TD) networks have been proposed as a way of representing and learning a wide variety of predictions about the interaction between an agent and its environment. These predictions are compositional in that their targets are defined in terms of other predictions, and subjunctive in that that they are about what would happen if an action or sequence of actions were taken. In conventional TD networks, the inter-related predictions are at successive time steps and contingent on a single action; here we generalize them to accommodate extended time intervals and contingency on whole ways of behaving. Our generalization is based on the options framework for temporal abstraction. The primary contribution of this paper is to introduce a new algorithm for intra-option learning in TD networks with function approximation and eligibility traces. We present empirical examples of our algorithm’s effectiveness and of the greater representational expressiveness of temporallyabstract TD networks. The primary distinguishing feature of temporal-difference (TD) networks (Sutton & Tanner, 2005) is that they permit a general compositional specification of the goals of learning. The goals of learning are thought of as predictive questions being asked by the agent in the learning problem, such as “What will I see if I step forward and look right?” or “If I open the fridge, will I see a bottle of beer?” Seeing a bottle of beer is of course a complicated perceptual act. It might be thought of as obtaining a set of predictions about what would happen if certain reaching and grasping actions were taken, about what would happen if the bottle were opened and turned upside down, and of what the bottle would look like if viewed from various angles. To predict seeing a bottle of beer is thus to make a prediction about a set of other predictions. The target for the overall prediction is a composition in the mathematical sense of the first prediction with each of the other predictions. TD networks are the first framework for representing the goals of predictive learning in a compositional, machine-accessible form. Each node of a TD network represents an individual question—something to be predicted—and has associated with it a value representing an answer to the question—a prediction of that something. The questions are represented by a set of directed links between nodes. If node 1 is linked to node 2, then node 1 represents a question incorporating node 2’s question; its value is a prediction about node 2’s prediction. Higher-level predictions can be composed in several ways from lower ones, producing a powerful, structured representation language for the targets of learning. The compositional structure is not just in a human designer’s head; it is expressed in the links and thus is accessible to the agent and its learning algorithm. The network of these links is referred to as the question network. An entirely separate set of directed links between the nodes is used to compute the values (predictions, answers) associated with each node. These links collectively are referred to as the answer network. The computation in the answer network is compositional in a conventional way—node values are computed from other node values. The essential insight of TD networks is that the notion of compositionality should apply to questions as well as to answers. A secondary distinguishing feature of TD networks is that the predictions (node values) at each moment in time can be used as a representation of the state of the world at that time. In this way they are an instance of the idea of predictive state representations (PSRs) introduced by Littman, Sutton and Singh (2002), Jaeger (2000), and Rivest and Schapire (1987). Representing a state by its predictions is a potentially powerful strategy for state abstraction (Rafols et al., 2005). We note that the questions used in all previous work with PSRs are defined in terms of concrete actions and observations, not other predictions. They are not compositional in the sense that TD-network questions are. The questions we have discussed so far are subjunctive, meaning that they are conditional on a certain way of behaving. We predict what we would see if we were to step forward and look right, or if we were to open the fridge. The questions in conventional TD networks are subjunctive, but they are conditional only on primitive actions or open-loop sequences of primitive actions (as are conventional PSRs). It is natural to generalize this, as we have in the informal examples above, to questions that are conditional on closed-loop temporally extended ways of behaving. For example, opening the fridge is a complex, high-level action. The arm must be lifted to the door, the hand shaped for grasping the handle, etc. To ask questions like “if I were to go to the coffee room, would I see John?” would require substantial temporal abstraction in addition to state abstraction. The options framework (Sutton, Precup & Singh, 1999) is a straightforward way of talking about temporally extended ways of behaving and about predictions of their outcomes. In this paper we extend the options framework so that it can be applied to TD networks. Significant extensions of the original options framework are needed. Novel features of our option-extended TD networks are that they 1) predict components of option outcomes rather than full outcome probability distributions, 2) learn according to the first intra-option method to use eligibility traces (see Sutton & Barto, 1998), and 3) include the possibility of options whose ‘policies’ are indifferent to which of several actions are selected. 1 The options framework In this section we present the essential elements of the options framework (Sutton, Precup & Singh, 1999) that we will need for our extension of TD networks. In this framework, an agent and an environment interact at discrete time steps t = 1, 2, 3.... In each state st ∈S, the agent selects an action at ∈A, determining the next state st+1.1 An action is a way of behaving for one time step; the options framework lets us talk about temporally extended ways of behaving. An individual option consists of three parts. The first is the initiation set, I ⊂S, the subset of states in which the option can be started. The second component of an option is its policy, π : S × A ⇒[0, 1], specifying how the agent behaves when 1Although the options framework includes rewards, we omit them here because we are concerned only with prediction, not control. following the option. Finally, a termination function, β : S × A ⇒[0, 1], specifies how the option ends: β(s) denotes the probability of terminating when in state s. The option is thus completely and formally defined by the 3-tuple (I, π, β). 2 Conventional TD networks In this section we briefly present the details of the structure and the learning algorithm comprising TD networks as introduced by Sutton and Tanner (2005). TD networks address a prediction problem in which the agent may not have direct access to the state of the environment. Instead, at each time step the agent receives an observation ot ∈O dependent on the state. The experience stream thus consists of a sequence of alternating actions and observations, o1, a1, o2, a2, o3 · · ·. The TD network consists of a set of nodes, each representing a single scalar prediction, interlinked by the question and answer networks as suggested previously. For a network of n nodes, the vector of all predictions at time step t is denoted yt = (y1 t , . . . , yn t )T . The predictions are estimates of the expected value of some scalar quantity, typically of a bit, in which case they can be interpreted as estimates of probabilities. The predictions are updated at each time step according to a vector-valued function u with modifiable parameter W, which is often taken to be of a linear form: yt = u(yt−1, at−1, ot, Wt) = σ(Wtxt), (1) where xt ∈ℜm is an m-vector of features created from (yt−1, at−1, ot), Wt is an n × m matrix (whose elements are sometimes referred to as weights), and σ is the n-vector form of either the identity function or the S-shaped logistic function σ(s) = 1 1+e−s . The feature vector is an arbitrary vector-valued function of yt−1, at−1, and ot. For example, in the simplest case the feature vector is a unit basis vector with the location of the one communicating the current state. In a partially observable environment, the feature vector may be a combination of the agent’s action, observations, and predictions from the previous time step. The overall update u defines the answer network. The question network consists of a set of target functions, zi : O ×ℜn →ℜ, and condition functions, ci : A×ℜn →[0, 1]n. We define zi t = zi(ot+1, ˜yt+1) as the target for prediction yi t.2 Similarly, we define ci t = ci(at, yt) as the condition at time t. The learning algorithm for each component wij t of Wt can then be written wij t+1 = wij t + α zi t −yi t ci t ∂yi t ∂wij t , (2) where α is a positive step-size parameter. Note that the targets here are functions of the observation and predictions exactly one time step later, and that the conditions are functions of a single primitive action. This is what makes this algorithm suitable only for learning about one-step TD relationships. By chaining together multiple nodes, Sutton and Tanner (2005) used it to predict k steps ahead, for various particular values of k, and to predict the outcome of specific action sequences (as in PSRs, e.g., Littman et al., 2002; Singh et al., 2004). Now we consider the extension to temporally abstract actions. 3 Option-extended TD networks In this section we present our intra-option learning algorithm for TD networks with options and eligibility traces. As suggested earlier, each node’s outgoing link in the question 2The quantity ˜y is almost the same as y, and we encourage the reader to think of them as identical here. The difference is that ˜y is calculated by weights that are one step out of date as compared to y, i.e., ˜yt = u(yt−1, at−1, ot, Wt−1) (cf. equation 1). network will now correspond to an option applying over possibly many steps. The policy of the ith node’s option corresponds to the condition function ci, which we think of as a recognizer for the option. It inspects each action taken to assess whether the option is being followed: ci t = 1 if the agent is acting consistently with the option policy and ci t = 0 otherwise (intermediate values are also possible). When an agent ceases to act consistently with the option policy, we say that the option has diverged. The possibility of recognizing more than one action as consistent with the option is a significant generalization of the original idea of options. If no actions are recognized as acceptable in a state, then the option cannot be followed and thus cannot be initiated. Here we take the set of states with at least one recognized action to be the initiation set of the option. The option-termination function β generalizes naturally to TD networks. Each node i is given a corresponding termination function, βi : O×ℜn →[0, 1], where βi t = βi(ot+1, yt) is the probability of terminating at time t.3 βi t = 1 indicates that the option has terminated at time t; βi t = 0 indicates that it has not, and intermediate values of β correspond to soft or stochastic termination conditions. If an option terminates, then zi t acts as the target, but if the option is ongoing without termination, then the node’s own next value, ˜yi t+1, should be the target. The termination function specifies which of the two targets (or mixture of the two targets) is used to produce a form of TD error for each node i: δi t = βi tzi t + (1 −βi t)˜yi t+1 −yi t. (3) Our option-extended algorithm incorporates eligibility traces (see Sutton & Barto, 1998) as short-term memory variables organized in an n × m matrix E, paralleling the weight matrix. The traces are a record of the effect that each weight could have had on each node’s prediction during the time the agent has been acting consistently with the node’s option. The components eij of the eligibility matrix are updated by eij t = ci t λeij t−1(1 −βi t) + ∂yi t ∂wij t , (4) where 0 ≤λ ≤1 is the trace-decay parameter familiar from the TD(λ) learning algorithm. Because of the ci t factor, all of a node’s traces will be immediately reset to zero whenever the agent deviates from the node’s option’s policy. If the agent follows the policy and the option does not terminate, then the trace decays by λ and increments by the gradient in the way typical of eligibility traces. If the policy is followed and the option does terminate, then the trace will be reset to zero on the immediately following time step, and a new trace will start building. Finally, our algorithm updates the weights on each time step by wij t+1 = wij t + α δi t eij t . (5) 4 Fully observable experiment This experiment was designed to test the correctness of the algorithm in a simple gridworld where the environmental state is observable. We applied an options-extended TD network to the problem of learning to predict observations from interaction with the gridworld environment shown on the left in Figure 1. Empty squares indicate spaces where the agent can move freely, and colored squares (shown shaded in the figure) indicate walls. The agent is egocentric. At each time step the agent receives from the environment six bits representing the color it is facing (red, green, blue, orange, yellow, or white). In this first experiment we also provided 6 × 6 × 4 = 144 other bits directly indicating the complete state of the environment (square and orientation). 3The fact that the option depends only on the current predictions, action, and observation means that we are considering only Markov options. Figure 1: The test world (left) and the question network (right) used in the experiments. The triangle in the world indicates the location and orientation of the agent. The walls are labeled R, O, Y, G, and B representing the colors red, orange, yellow, green and blue. Note that the left wall is mostly blue but partly green. The right diagram shows in full the portion of the question network corresponding to the red bit. This structure is repeated, but not shown, for the other four (non-white) colors. L, R, and F are primitive actions, and Forward and Wander are options. There are three possible actions: A ={F, R, L}. Actions were selected according to a fixed stochastic policy independent of the state. The probability of the F, L, and R actions were 0.5, 0.25, and 0.25 respectively. L and R cause the agent to rotate 90 degrees to the left or right. F causes the agent to move ahead one square with probability 1 −p and to stay in the same square with probability p. The probability p is called the slipping probability. If the forward movement would cause the agent to move into a wall, then the agent does not move. In this experiment, we used p = 0, p = 0.1, and p = 0.5. In addition to these primitive actions, we provided two temporally abstract options, Forward and Wander. The Forward option takes the action F in every state and terminates when the agent senses a wall (color) in front of it. The policy of the Wander option is the same as that actually followed by the agent. Wander terminates with probability 1 when a wall is sensed, and spontaneously with probability 0.5 otherwise. We used the question network shown on the right in Figure 1. The predictions of nodes 1, 2, and 3 are estimates of the probability that the red bit would be observed if the corresponding primitive action were taken. Node 4 is a prediction of whether the agent will see the red bit upon termination of the Wander option if it were taken. Node 5 predicts the probability of observing the red bit given that the Forward option is followed until termination. Nodes 6 and 7 represent predictions of the outcome of a primitive action followed by the Forward option. Nodes 8 and 9 take this one step further: they represent predictions of the red bit if the Forward option were followed to termination, then a primitive action were taken, and then the Forward option were followed again to termination. We applied our algorithm to learn the parameter W of the answer network for this question network. The step-size parameter α was 1.0, and the trace-decay parameter λ was 0.9. The initial W0, E0, and y0 were all 0. Each run began with the agent in the state indicated in Figure 1 (left). In this experiment σ(·) was the identity function. For each value of p, we ran 50 runs of 20,000 time steps. On each time step, the root-meansquared (RMS) error in each node’s prediction was computed and then averaged over all the nodes. The nodes corresponding to the Wander option were not included in the average because of the difficulty of calculating their correct predictions. This average was then 0 0.4 0 5000 10000 15000 20000 p = 0.5 p = 0.1 p = 0 RMS Error Steps Fully Observable 0 0.4 0 100000 200000 300000 RMS Error Steps Partially Observable Figure 2: Learning curves in the fully-observable experiment for each slippage probability (left) and in the partially-observable experiment (right). itself averaged over the 50 runs and bins of 1,000 time steps to produce the learning curves shown on the left in Figure 2. For all slippage probabilities, the error in all predictions fell almost to zero. After approximately 12,000 trials, the agent made almost perfect predictions in all cases. Not surprisingly, learning was slower at the higher slippage probabilities. These results show that our augmented TD network is able to make a complete temporally-abstract model of this world. 5 Partially observable experiment In our second experiment, only the six color observation bits were available to the agent. This experiment provides a more challenging test of our algorithm. To model the environment well, the TD network must construct a representation of state from very sparse information. In fact, completely accurate prediction is not possible in this problem with our question network. In this experiment the input vector consisted of three groups of 46 components each, 138 in total. If the action was R, the first 46 components were set to the 40 node values and the six observation bits, and the other components were 0. If the action was L, the next group of 46 components was filled in in the same way, and the first and third groups were zero. If the action was F, the third group was filled. This technique enables the answer network as function approximator to represent a wider class of functions in a linear form than would otherwise be possible. In this experiment, σ(·) was the S-shaped logistic function. The slippage probability was p = 0.1. As our performance measure we used the RMS error, as in the first experiment, except that the predictions for the primitive actions (nodes 1-3) were not included. These predictions can never become completely accurate because the agent can’t tell in detail where it is located in the open space. As before, we averaged RMS error over 50 runs and 1,000 time step bins, to produce the learning curve shown on the right in Figure 2. As before, the RMS error approached zero. Node 5 in Figure 1 holds the prediction of red if the agent were to march forward to the wall ahead of it. Corresponding nodes in the other subnetworks hold the predictions of the other colors upon Forward. To make these predictions accurately, the agent must keep track of which wall it is facing, even if it is many steps away from it. It has to learn a sort of compass that it can keep updated as it turns in the middle of the space. Figure 3 is a demonstration of the compass learned after a representative run of 200,000 time steps. At the end of the run, the agent was driven manually to the state shown in the first row (relative 1 O Y R B G O Y R B G O Y R B G O Y R B G O Y R B G O Y R B G O Y R B G O Y R B G O Y R B G O Y R B G O Y R B G O Y R B G O Y R B G O Y R B G st yt 5 yt 8 t 1 2 3 4 5 25 29 Figure 3: An illustration of part of what the agent learns in the partially observable environment. The second column is a sequence of states with (relative) time index as given by the first column. The sequence was generated by controlling the agent manually. On steps 1-25 the agent was spun clockwise in place, and the trajectory after that is shown by the line in the last state diagram. The third and fourth columns show the values of the nodes corresponding to 5 and 8 in Figure 1, one for each color-observation bit. time index t = 1). On steps 1-25 the agent was spun clockwise in place. The third column shows the prediction for node 5 in each portion of the question network. That is, the predictions shown are for each color-observation bit at termination of the Forward option. At t = 1, the agent is facing the orange wall and it predicts that the Forward option would result in seeing the orange bit and none other. Over steps 2-5 we see that the predictions are maintained accurately as the agent spins despite the fact that its observation bits remain the same. Even after spinning for 25 steps the agent knows exactly which way it is facing. While spinning, the agent correctly never predicts seeing the green bit (after Forward), but if it is driven up and turned, as in the last row of the figure, the green bit is accurately predicted. The fourth column shows the prediction for node 8 in each portion of the question network. Recall that these nodes correspond to the sequence Forward, L, Forward. At time t = 1, the agent accurately predicts that Forward will bring it to orange (third column) and also predicts that Forward, L, Forward will bring it to green. The predictions made for node 8 at each subsequent step of the sequence are also correct. These results show that the agent is able to accurately maintain its long term predictions without directly encountering sensory verification. How much larger would the TD network have to be to handle a 100x100 gridworld? The answer is not at all. The same question network applies to any size problem. If the layout of the colored walls remain the same, then even the answer network transfers across worlds of widely varying sizes. In other experiments, training on successively larger problems, we have shown that the same TD network as used here can learn to make all the long-term predictions correctly on a 100x100 version of the 6x6 gridworld used here. 6 Conclusion Our experiments show that option-extended TD networks can learn effectively. They can learn facts about their environments that are not representable in conventional TD networks or in any other method for learning models of the world. One concern is that our intra-option learning algorithm is an off-policy learning method incorporating function approximation and bootstrapping (learning from predictions). The combination of these three is known to produce convergence problems for some methods (see Sutton & Barto, 1998), and they may arise here. A sound solution may require modifications to incorporate importance sampling (see Precup, Sutton & Dasgupta, 2001). In this paper we have considered only intra-option eligibility traces—traces extending over the time span within an option but not persisting across options. Tanner and Sutton (2005) have proposed a method for inter-option traces that could perhaps be combined with our intra-option traces. The primary contribution of this paper is the introduction of a new learning algorithm for TD networks that incorporates options and eligibility traces. Our experiments are small and do little more than exercise the learning algorithm, showing that it does not break immediately. More significant is the greater representational power of option-extended TD networks. Options are a general framework for temporal abstraction, predictive state representations are a promising strategy for state abstraction, and TD networks are able to represent compositional questions. The combination of these three is potentially very powerful and worthy of further study. Acknowledgments The authors gratefully acknowledge the ideas and encouragement they have received in this work from Mark Ring, Brian Tanner, Satinder Singh, Doina Precup, and all the members of the rlai.net group. References Jaeger, H. (2000). Observable operator models for discrete stochastic time series. Neural Computation, 12(6):1371-1398. MIT Press. Littman, M., Sutton, R. S., & Singh, S. (2002). Predictive representations of state. In T. G. Dietterich, S. Becker and Z. Ghahramani (eds.), Advances In Neural Information Processing Systems 14, pp. 1555-1561. MIT Press. Precup, D., Sutton, R. S., & Dasgupta, S. (2001). Off-policy temporal-difference learning with function approximation. In C. E. Brodley, A. P. Danyluk (eds.), Proceedings of the Eighteenth International Conference on Machine Learning, pp. 417-424. San Francisco, CA: Morgan Kaufmann. Rafols, E. J., Ring, M., Sutton, R.S., & Tanner, B. (2005). Using predictive representations to improve generalization in reinforcement learning. To appear in Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence. Rivest, R. L., & Schapire, R. E. (1987). Diversity-based inference of finite automata. In Proceedings of the Twenty Eighth Annual Symposium on Foundations of Computer Science, (pp. 78–87). IEEE Computer Society. Singh, S., James, M. R., & Rudary, M. R. (2004). Predictive state representations: A new theory for modeling dynamical systems. In Uncertainty in Artificial Intelligence: Proceedings of the Twentieth Conference in Uncertainty in Artificial Intelligence, (pp. 512–519). AUAI Press. Sutton, R. S., & Barto, A. G. (1998). Reinforcement learning: An introduction. Cambridge, MA: MIT Press. Sutton, R. S., Precup, D., Singh, S. (1999). Between MDPs and semi-MDPs: A framework for temporal abstraction in reinforcement learning. Artificial Intelligence, 112, pp. 181-211. Sutton, R. S., & Tanner, B. (2005). Temporal-difference networks. To appear in Neural Information Processing Systems Conference 17. Tanner, B., Sutton, R. S. (2005) Temporal-difference networks with history. To appear in Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence.
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Active Learning For Identifying Function Threshold Boundaries Brent Bryan Center for Automated Learning and Discovery Carnegie Mellon University Pittsburgh, PA 15213 bryanba@cs.cmu.edu Jeff Schneider Robotics Institute Carnegie Mellon University Pittsburgh, PA 15213 schneide@cs.cmu.edu Robert C. Nichol Institute of Cosmology and Gravitation University of Portsmouth Portsmouth, PO1 2EG, UK bob.nichol@port.ac.uk Christopher J. Miller Observatorio Cerro Tololo Observatorio de AURA en Chile La Serena, Chile cmiller@noao.edu Christopher R. Genovese Department of Statistics Carnegie Mellon University Pittsburgh, PA 15213 genovese@stat.cmu.edu Larry Wasserman Department of Statistics Carnegie Mellon University Pittsburgh, PA 15213 larry@stat.cmu.edu Abstract We present an efficient algorithm to actively select queries for learning the boundaries separating a function domain into regions where the function is above and below a given threshold. We develop experiment selection methods based on entropy, misclassification rate, variance, and their combinations, and show how they perform on a number of data sets. We then show how these algorithms are used to determine simultaneously valid 1 −α confidence intervals for seven cosmological parameters. Experimentation shows that the algorithm reduces the computation necessary for the parameter estimation problem by an order of magnitude. 1 Introduction In many scientific and engineering problems where one is modeling some function over an experimental space, one is not necessarily interested in the precise value of the function over an entire region. Rather, one is curious about determining the set of points for which the function exceeds some particular value. Applications include determining the functional range of wireless networks [1], factory optimization analysis, and gaging the extent of environmental regions in geostatistics. In this paper, we use this idea to compute confidence intervals for a set of cosmological parameters that affect the shape of the temperature power spectrum of the Cosmic Microwave Background (CMB). In one dimension, the threshold discovery problem is a root-finding problem where no hints as to the location or number of solutions are given; several methods exist which can be used to solve this problem (e.g. bisection, Newton-Raphson). However, one dimensional algorithms cannot be easily extended to the multivariate case. In particular, the ideas of root bracketing and function transversal are not well defined [2]; given a particular bracket of a continuous surface, there will be an infinite number of solutions to the equation f(⃗x)−t = 0, since the solution in multiple dimensions is a set of surfaces, rather than a set of points. Numerous active learning papers deal with similar problems in multiple dimensions. For instance, [1] presents a method for picking experiments to determine the localities of local extrema when the input space is discrete. Others have used a variety of techniques to reduce the uncertainty over the problem’s entire domain to map out the function (e.g. [3], and [4]), or locate the optimal value (e.g. [5]). We are interested in locating the subset of the input space wherein the function is above a given threshold. Algorithms that merely find a local optimum and search around it will not work in general, as there may be multiple disjoint regions above the threshold. While techniques that map out the entire surface of the underlying function will correctly identify those regions which are above a given threshold, we assert that methods can be developed that are more efficient at localizing a particular contour of the function. Intuitively, points on the function that are located far from the boundary are less interesting, regardless of their variance. In this paper, we make the following contributions to the literature: • We present a method for choosing experiments that is more efficient than global variance minimization, as well as other heuristics, when one is solely interested in localizing a function contour. • We show that this heuristic can be used in continuous valued input spaces, without defining a priori a set of possible experiments (e.g. imposing a grid). • We use our function threshold detection method to determine 1−α simultaneously valid confidence intervals of CMB parameters, making no assumptions about the model being fit and few assumptions about the data in general. 2 Algorithm We begin by formalizing the problem. Assume that we are given a bounded sample space S ⊂Rn and a scoring function: f : S →R, but possibly no data points ({s, f(s)}, s ∈S). Given a threshold t, we want to find the set of points S′ where f is equal to or above the threshold: {s ∈S′|s ∈S, f(s) ≥t}. If f is invertible, then the solution is trivial. However, it is often the case that f is not trivially invertible, such as the CMB model mentioned in §1. In these cases, we can discover S′ by modeling S given some experiments. Thus, we wish to know how to choose experiments that help us determine S′ efficiently. We assume that the cost to compute f(s) given s is significant. Thus, care should be taken when choosing the next experiment, as picking optimum points may reduce the runtime of the algorithm by orders of magnitude. Therefore, it is preferable to analyze current knowledge about the underlying function and select experiments which quickly refine the estimate of the function around the threshold of interest. There are several methods one could use to create a model of the data, notably some form of parametric regression. However, we chose to approximate the unknown boundary as a Gaussian Process (GP), as many forms of regression (e.g. linear) necessarily smooths the data, ignoring subtle features of the function that may become pronounced with more data. In particular, we use ordinary kriging, a form of GPs, which assumes that the semivariogram (K(·, ·) is a linear function of the distance between samples [6]; this estimation procedure assumes the the sampled data are normal with mean equal to the true function and variance given by the sampling noise . The expected value of K(si, sj) for si, sj ∈S, is can be written as E[K(si, sj)] = k 2 h n X l=1 αl(sil −sjl)2i1/2 + c where k is a constant — known as the kriging parameter — which is an estimated limit on the first derivate of the function, αl is a scaling factor for each dimension, and c is the variance (e.g. experimental noise) of the sampled points. Since, the joint distribution of a finite set of sampled points for GPs is Gaussian, the predicted distribution of a query point sq given a known set A is normal with mean and variance given by µsq = µA + Σ′ AqΣ−1 AA(yA −µA) (1) σ2 sq = Σ′ AqΣ−1 AAΣAq (2) where ΣAq denotes the column vector with the ith entry equal to K(si, sq), ΣAA denotes the semivariance matrix between the elements of A (the ij element of ΣAA is K(si, sj)), yA denotes the column vector with the ith entry equal to f(si), the true value of the function for each point in A, and µA is the mean of the yA’s. As given, prediction with GP requires O(n3) time, as an n × n linear system of equations must be solved. However, for many GPs — and ordinary kriging in particular — the correlation between two points decreases as a function of distance. Thus, the full GP model can be approximated well by a local GP, where only the k nearest neighbors of the query point are used to compute the prediction value; this reduces the computation time to O(k3 log(n)) per prediction, since O(log(n)) time is required to find the k-nearest neighbors using spatial indexing structures such as balanced kd-trees. Since we have assumed that experimentation is expensive, it would be ideal to iteratively analyze the entire input space and pick the next experiment in such a manner that minimized the total number of experiments necessary. If the size of the parameter space (|S|) is finite, such an approach may be feasible. However, if |S| is large or infinite, testing all points may be impractical. Instead of imposing some arbitrary structure on the possible experimental points (such as using a grid), our algorithm chooses candidate points uniformly at random from the input space, and then selects the candidate point with the highest score according to the metrics given in §2.1. This allows the input space to be fully explored (in expectation), and ensures that interesting regions of space that would have fallen between successive grid points are not missed; in §4 we show how imposing a grid upon the input space results in just such a situation. While the algorithm is unable to consider the entire space for each sampling iteration, over multiple iterations it does consider most of the space, resulting in the function boundaries being quickly localized, as can be seen in §3. 2.1 Choosing experiments from among candidates Given a set of random input points, the algorithm evaluates each one and chooses the point with the highest score as the location for the next experiment. Below is the list of evaluation methods we considered. Random: One of the candidate points is chosen uniformly at random. This method serves as a baseline for comparison, Probability of incorrect classification: Since we are trying to map the boundary between points above and below a threshold, we consider choosing the point from our random sample which has the largest probability of being misclassified by our model. Using the distribution defined by Equations 1 and 2, the probability, p, that the point is above the given threshold can be computed. The point is predicted to be above the threshold if p > 0.5 and thus the expected misclassification probability is min(p, 1 −p). Entropy: Instead of misclassification probability we can consider entropy: −p log2(p) − (1 −p) log2(1 −p). Entropy is a monotonic function of the misclassification rate so these two will not choose different experiments. They are listed separately because they have different effects when mixed with other evaluations. Both entropy and misclassification will choose points near the boundary. Unfortunately, they have the drawback that once they find a point near the boundary they continue to choose points near that location and will not explore the rest of the parameter space. Variance: Both entropy and probability of incorrect classification suffer from a lack of incentive to explore the space. To rectify this problem, we consider the variance of each query point (given by Equation 2) as an evaluation metric. This metric is common in active learning methods whose goal is to map out an entire function. Since variance is related to the distance to nearest neighbors, this strategy chooses points that are far from areas currently searched, and hence will not get stuck at one boundary point. However, it is well known that such approaches tend to spend a large portion of their time on the edges of the parameter space and ultimately cover the space exhaustively [7]. Information gain: Information gain is a common myopic metric used in active learning. Information gain at the query point is the same as entropy in our case because all run experiments are assumed to have the same variance. Computing a full measure of information gain over the whole state space would provide an optimal 1-step experiment choice. In some discrete or linear problems this can be done, but it is intractable for continuous non-linear spaces. We believe the good performance of the evaluation metrics proposed below stems from their being heuristic proxies for global information gain or reduction in misclassification error. Products of metrics: One way to rectify the problems of point policies that focus solely on points near the boundary or points with large variance regardless of their relevance to refining the predictive model, is to combine the two measures. Intuitively, doing this can mimic the idea of information gain; the entropy of a query point measures the classification uncertainty, while the variance is a good estimator of how much impact a new observation would have in this region, and thus what fraction the uncertainty would be reduced. [1] proposed scoring points based upon the product of their entropy and variance to identify the presence of local maxima and minima, a problem closely related to boundary detection. We shall also consider scoring points based upon the product of their probability of incorrect classification and variance. Note that while entropy and probability of incorrect classification are monotonically related, entropy times variance and probability of incorrect classification times variance are not. Straddle: Using the same intuition as for products of heuristics, we define straddle heuristic, as straddle(sq) = 1.96ˆσq − ˆf(sq) −t , The straddle algorithm scores points highest that are both unknown and near the boundary. As such, the straddle algorithm prefers points near the threshold, but far from previous examples. The straddle score for a point may be negative, which indicates that the model currently estimates the probability that the point is on a boundary is less than five percent. Since the straddle heuristic relies on the variance estimate, it is also subject to oversampling edge positions. 3 Experiments We now assess the accuracy with which our model reproduces a known function for the point policies just described. This is done by computing the fraction of test points in which the predictive model agrees with the true function about which side of the threshold the test points are on after some fixed number of experiments. This process is repeated several times to account for variations due to the random sampling of the input space. The first model we consider is a 2D sinusoidal function given by f(x, y) = sin(10x) + cos(4y) −cos(3xy) x ∈[0, 1], y ∈[0, 2], with a boundary threshold of t = 0. This function and threshold were examined for the following reasons: 1) the target threshold winds through the plot giving ample length to 0 1 2 0 0.5 1 A 0 0.5 1 B 0 0.5 1 0 1 2 C Figure 1: Predicted function boundary (solid), true function boundary (dashed), and experiments (dots) for the 2D sinusoid function after A) 50 experiments and B) 100 experiments using the straddle heuristic and C) 100 experiments using the variance heuristic. Table 1: Number of experiments required to obtain 99% classification accuracy for the 2D models and 95% classification accuracy for the 4D model for various heuristics. Heuristics requiring more than 10,000 experiments to converge are labeled “did not converge”. 2D Sin.(1K Cand.) 2D Sin.(31 Cand.) 2D DeBoor 4D Sinusoid Random 617 ± 158 617 ± 158 7727 ± 987 6254 ± 364 Entropy did not converge did not converge did not converge 6121 ± 1740 Variance 207 ± 7 229 ± 9 4306 ± 573 2320 ± 57 Entropy×Var 117 ± 5 138 ± 6 1621 ± 201 1210 ± 43 Prob. Incor.×Std 113 ± 11 129 ± 14 740 ± 117 1362 ± 89 Straddle 106 ± 5 123 ± 6 963 ± 136 1265 ± 94 test the accuracy of the approximating model, 2) the boundary is discontinuous with several small pieces, 3) there is an ambiguous region (around (0.9, 1), where the true function is approximately equal to the threshold, and the gradient is small and 4) there are areas in the domain where the function is far from the threshold and hence we can ensure that the algorithm is not oversampling in these regions. Table 1 shows the number of experiments necessary to reach a 99% and 95% accuracy for the 2D and 4D models, respectively. Note that picking points solely on entropy does not converge in many cases, while both the straddle algorithm and probability incorrect times standard deviation heuristic result in approximations that are significantly better than random and variance heuristics. Figures 1A-C confirm that the straddle heuristic is aiding in boundary prediction. Note that most of the 50 experiments sampled between Figures 1A and 1B are chosen near the boundary. The 100 experiments chosen to minimize the variance result in an even distribution over the input space and a worse boundary approximation, as seen in Figure 1C. These results indicate that the algorithm is correctly modeling the test function and choosing experiments that pinpoint the location of the boundary. From the Equations 1 and 2, it is clear that the algorithm does not depend on data dimensionality directly. To ensure that heuristics are not exploiting some feature of the 2D input space, we consider the 4D sinusoidal function f(⃗x) = sin(10x1) + cos(4x2) −cos(3x1x2) + cos(2x3) + cos(3x4) −sin(5x3x4) where ⃗x ∈[(0, 0, 1, 0), (1, 2, 2, 2)] and t = 0. Comparison of the 2D and 4D results in Table 1 reveals that the relative performance of the heuristics remains unchanged, indicating that the best heuristic for picking experiments is independent of the problem dimension. To show that the decrease in the number candidate points relative to the input parameter space that occurs with higher dimensional problems is not an issue, we reconsider the 2D sinusoidal problem. Now, we use only 31 candidate points instead of 1000 to simulate the point density difference between 4D and 2D. Results shown in Table 1, indicate that reducing the number of candidate points does not drastically alter the realized performance. Additional experiments were performed on a discontinuous 2D function (the DeBoor function given in [1]) with similar results, as can be seen in Table 1. 4 Statistical analysis of cosmological parameters Let us now look at a concrete application of this work: a statistical analysis of cosmological parameters that affect formation and evolution of our universe. One key prediction of the Big Bang model for the origin of our universe is the presence of a 2.73K cosmic microwave background radiation (CMB). Recently, the Wilkinson Microwave Anisotropy Project (WMAP) has completed a detailed survey of the this radiation exhibiting small CMB temperature fluctuations over the sky [8]. It is believed that the size and spatial proximity of these temperature fluctuations depict the types and rates of particle interactions in the early universe and consequently characterize the formation of large scale structure (galaxies, clusters, walls and voids) in the current observable universe. It is conjectured that this radiation permeated through the universe unchanged since its formation 15 billion years ago. Therefore, the sizes and angular separations of these CMB fluctuations give an unique picture of the universe immediately after the Big Bang and have a large implication on our understanding of primordial cosmology. An important summary of the temperature fluctuations is the CMB power spectrum shown in Figure 2, which gives the temperature variance of the CMB as a function of spatial frequency (or multi-pole moment). It is well known that the shape of this curve is affected by at least seven cosmological parameters: optical depth (τ), dark energy mass fraction (ΩΛ), total mass fraction (Ωm), baryon density (ωb), dark matter density (ωdm), neutrino fraction (fn), and spectral index (ns). For instance, the height of first peak is determined by the total energy density of the universe, while the third peak is related to the amount of dark matter. Thus, by fitting models of the CMB power spectrum for given values of the seven parameters, we can determine how the parameters influence the shape of the model spectrum. By examining those models that fit the data, we can then establish the ranges of the parameters that result in models which fit the data. Previous work characterizing confidence intervals for cosmological parameters either used marginalization over the other parameters, or made assumptions about the values of the parameters and/or the shape of the CMB power spectrum. However, [9] notes that “CMB data have now become so sensitive that the key issue in cosmological parameter determination is not always the accuracy with which the CMB power spectrum features can be measured, but often what prior information is used or assumed.” In this analysis, we make no assumptions about the ranges or values of the parameters, and assume only that the data are normally distributed around the unknown CMB spectrum with covariance known up to a constant multiple. Using the method of [10], we create a non-parametric confidence ball (under a weighted squared-error loss) for the unknown spectrum that is centered on a nonparametric estimate with a radius for each specified confidence level derived from the asymptotic distribution of a pivot statistic1. For any candidate spectrum, membership in the confidence ball can be determined by comparing the ball’s radius to the variance weighted sum of squares deviation between the candidate function and the center of the ball. One advantage of this method is that it gives us simultaneously valid confidence intervals on all seven of our input parameters; this is not true for 1 −α confidence intervals derived from a collection of χ2 distributions where the confidence intervals often have substantially lower coverage [11]. However, there is no way to invert the modeling process to determine parameter ranges given a fixed sum of squared error. Thus, we use the algorithm detailed 1See Appendix 3 in [10] for the derivation of this radius 0 2000 4000 6000 0 200 400 600 800 Temperature Variance Multipole Moment Figure 2: WMAP data, overlaid with regressed model (solid) and an example of a model CMB spectrum that barely fits at the 95% confidence level (dashed; parameter values are ωDM = 0.1 and ωB = 0.028). 0 0.05 0.1 0 0.2 0.4 0.6 0.8 ωB ωDM Figure 3: 95% confidence bounds for ωB as a function of ωDM. Gray dots denote models which are rejected at a 95% confidence level, while the black dots denote those that are not. in §2 to map out the confidence surface as a function of the input parameters; that is, we use the algorithm to pick a location in the seven dimensional parameter space to perform an experiment, and then run CMBFast [12] to create simulated power spectrum given this set of input parameters. We can then compute the sum of squares of error for this spectrum (relative to the regressed model) and easily tell if the 7D input point is inside the confidence ball. In practice, we model the sum of squared error, not the confidence level of the model. This creates a more linear output space, as the confidence level for most of the models is zero, and thus it is impossible to distinguish between poor and terrible model fits. Due to previous efforts on this project, we were able to estimate the semivariogram of the GP from several hundred thousand random points already run through CMBFast. For this work, we chose the αl’s such that the partials in each dimension where approximately unity, resulting in k ≃1; c was set to a small constant to account for instabilities in the simulator. These points also gave a starting point for our algorithm2. Subsequently, we have run several hundred thousand more CMBFast models. We find that it takes 20 seconds to pick an experiment from among a set of 2,000 random candidates. CMBFast then takes roughly 3 minutes to compute the CMB spectrum given our chosen point in parameter space. In Figure 3, we show a plot of baryon density (ωB) versus the dark matter density (ωDM) of the universe over all values of the other five parameters (τ, ΩDE, ΩM, fn, ns). Experiments that are within a 95% confidence ball given the CMB data are plotted in black, while those that are rejected at the 95% level are gray. Note how there are areas that remain unsampled, while the boundary regions (transitions between gray and black points) are heavily sampled, indicating that our algorithm is choosing reasonable points. Moreover, the results of Figure 3 agree well with results in the literature (derived using parametric models and Bayesian analysis), as well as with predictions favored by nucleosynthesis [9]. While hard to distinguish in Figure 3, the bottom left group of points above the 95% confidence boundary splits into two separate peaks in parameter space. The one to the left is the concordance model, while the second peak (the one to the right) is not believed to represent the correct values of the parameters (due to constraints from other data). The existence of high probability points in this region of the parameter space has been suggested before, but computational limitations have prevented much characterization of it. Moreover, the third peak, near the top right corner of Figure 3 was basically ignored by previous grid based approaches. Comparison of the number of experiments performed by our straddle 2While initial values are not required (as we have seen in §3), it is possible to incorporate this background knowledge into the model to help the algorithm converge more quickly. Table 2: Number of points found in the three peaks for the grid based approach of [9] and our straddle algorithm. Peak Center # Points in Effective Radius ωDM ωB Grid Straddle Concordance Model 0.116 0.024 2118 16055 Peak 2 0.165 0.023 2825 9634 Peak 3 0.665 0.122 0 5488 Total Points 5613300 603384 algorithm with the grid based approach used by [9] is shown in Table 2. Even with only 10% of the experiments used in the grid approach, we sampled the concordance peak 8 times more frequently, and the second peak 3.4 times more frequently than the grid based approach. Moreover, it appears that the grid completely missed the third peak, while our method sampled it over 5000 times. These results dramatically illustrate the power of our adaptive method, and show how it does not suffer from assumptions made by a grid-based approaches. We are following up on the scientific ramifications of these results in a separate astrophysics paper. 5 Conclusions We have developed an algorithm for locating a specified contour of a function while minimizing the number queries necessary. We described and showed how several different methods for picking the next experimental point from a group of candidates perform on synthetic test functions. Our experiments indicate that the straddle algorithm outperforms previously published methods, and even handles functions with large discontinuities. Moreover, the algorithm is shown to work on multi-dimensional data, correctly classifying the boundary at a 99% level with half the points required for variance minimizing methods. We have then applied this algorithm to a seven dimensional statistical analysis of cosmological parameters affecting the Cosmic Microwave Background. With only a few hundred thousand simulations we are able to accurately describe the interdependence of the cosmological parameters, leading to a better understanding of fundamental physical properties. References [1] N. Ramakrishnan, C. Bailey-Kellogg, S. Tadepalli, and V. N. Pandey. Gaussian processes for active data mining of spatial aggregates. In Proceedings of the SIAM International Conference on Data Mining, 2005. [2] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in C. Cambridge University Press, 2nd edition, 1992. [3] D. A. Cohn, Z. Ghahramani, and M. I. Jordan. Active learning with statistical models. In G. Tesauro, D. Touretzky, and T. Leen, editors, Advances in Neural Information Processing Systems, volume 7, pages 705–712. The MIT Press, 1995. [4] Simon Tong and Daphne Koller. Active learning for parameter estimation in bayesian networks. In NIPS, pages 647–653, 2000. [5] A. Moore and J. Schneider. Memory-based stochastic optimization. In D. Touretzky, M. Mozer, and M. Hasselm, editors, Neural Information Processing Systems 8, volume 8, pages 1066–1072. MIT Press, 1996. [6] Noel A. C. Cressie. Statistics for Spatial Data. Wiley, New York, 1991. [7] D. MacKay. Information-based objective functions for active data selection. Neural Computation, 4(4):590–604, 1992. [8] C. L. Bennett et al. First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results. Astrophysical Journal Supplement Series, 148:1–27, September 2003. [9] M. Tegmark, M. Zaldarriaga, and A. J. Hamilton. Towards a refined cosmic concordance model: Joint 11-parameter constraints from the cosmic microwave background and large-scale structure. Physical Review D, 63(4), February 2001. [10] C. Genovese, C. J. Miller, R. C. Nichol, M. Arjunwadkar, and L. Wasserman. Nonparametric inference for the cosmic microwave background. Statistic Science, 19(2):308–321, 2004. [11] C. J. Miller, R. C. Nichol, C. Genovese, and L. Wasserman. A non-parametric analysis of the cmb power spectrum. Bulletin of the American Astronomical Society, 33:1358, December 2001. [12] U. Seljak and M. Zaldarriaga. A Line-of-Sight Integration Approach to Cosmic Microwave Background Anisotropies. Astrophyical Journal, 469:437–+, October 1996.
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Cue Integration for Figure/Ground Labeling Xiaofeng Ren, Charless C. Fowlkes and Jitendra Malik Computer Science Division, University of California, Berkeley, CA 94720 {xren,fowlkes,malik}@cs.berkeley.edu Abstract We present a model of edge and region grouping using a conditional random field built over a scale-invariant representation of images to integrate multiple cues. Our model includes potentials that capture low-level similarity, mid-level curvilinear continuity and high-level object shape. Maximum likelihood parameters for the model are learned from human labeled groundtruth on a large collection of horse images using belief propagation. Using held out test data, we quantify the information gained by incorporating generic mid-level cues and high-level shape. 1 Introduction Figure/ground organization, the binding of contours to surfaces, is a classical problem in vision. In the 1920s, Edgar Rubin pointed to several generic properties, such as closure, which governed the perception of figure/ground. However, it is clear that in the context of natural scenes, such processing must be closely intertwined with many low- and mid-level grouping cues as well as a priori object knowledge [10]. In this paper, we study a simplified task of figure/ground labeling in which the goal is to label every pixel as belonging to either a figural object or background. Our goal is to understand the role of different cues in this process, including low-level cues, such as edge contrast and texture similarity; mid-level cues, such as curvilinear continuity; and highlevel cues, such as characteristic shape or texture of the object. We develop a conditional random field model [7] over edges, regions and objects to integrate these cues. We train the model from human-marked groundtruth labels and quantify the relative contributions of each cue on a large collection of horse images[2]. In computer vision, the work of Geman and Geman [3] inspired a whole subfield of work on Markov Random Fields in relation to segmentation and denoising. More recently, Conditional Random Fields (CRF) have been applied to low-level segmentation [6, 12, 4] and have shown performance superior to traditional MRFs. However, most of the existing MRF/CRF models focus on pixel-level labeling, requiring inferences over millions of pixels. Being tied to the pixel resolution, they are also unable to deal with scale change or explicitly capture mid-level cues such as junctions. Our approach overcomes these difficulties by utilizing a scale-invariant representation of image contours and regions where each variable in our model can correspond to hundreds of pixels. It is also quite straightforward to design potentials which capture complicated relationships between these mid-level tokens in a transparent way. Interest in combining object knowledge with segmentation has grown quickly over the Yt Xe Z Ys (1) (2) (3) (4) Figure 1: A scale-invariant representation of images: Given the input (1), we estimate the local probability of boundary Pb based on gradients (2). We then build a piecewise linear approximation of the edge map and complete it with Constrained Delaunay Triangulation (CDT). The black edges in (3) are gradient edges detected in (2); the green edges are potential completions generated by CDT. (4) We perform inference in a probabilistic model built on top of this representation and extract marginal distributions on edges X, triangular regions Y and object pose Z. last few years [2, 16, 14]. Our probabilistic approach is similar in spirit to [14] however we focus on learning parameters of a discriminative model and quantify our performance on test data. Compared to previous techniques which rely heavily on top-down template matching [2, 5], our approach has three major advantages: (1) We are able to use midlevel grouping cues including junctions and continuity. Our results show these cues make quantitatively significant contributions. (2) We combine cues in a probabilistic framework where the relative weighting of cues is learned from training data resulting in weights that are easy to interpret. (3) The role of different cues can be easily studied by ”surgically removing” them refitting the remaining parameters. 2 A conditional random field for figure/ground labeling Figure 1 provides an overview of our technique for building a discrete, scale-independent representation of image boundaries from a low-level detector. First we compute an edge map using the boundary detector of [9] which utilizes both brightness and texture contrast to estimate the probability of boundary, Pb at each pixel. Next we use Canny’s hysteresis thresholding to trace the Pb boundaries and then recursively split the boundaries using angles, a scale-invariant measure, until each segment is approximately linear. Finally we utilize the Constrained Delaunay Triangulation [13] to complete the piecewise linear approximations. CDT often completes gaps in object boundaries where local gradient information is absent. More details about this construction can be found in [11]. Let G be the resulting CDT graph. The edges and triangles in G are natural entities for figure/ground labeling. We introduce the following random variables: • Edges: Xe is 1 if edge e in the CDT is a true boundary and 0 otherwise. • Regions: Yt is 1 if triangle t corresponds to figure and 0 otherwise. • Pose: Z encodes the figural object’s pose in the scene. We use a very simple Z which considers a discrete configuration space given by a grid of 25 possible image locations. Z is easily augmented to include an indicator of object category or aspect as well as location. We now describe a conditional random field model on {X, Y, Z} used to integrate multiple grouping cues. The model takes the form of a log-linear combination of features which are functions of variables and image measurements. We consider Z a latent variable which is marginalized out by assuming a uniform distribution over aspects and locations. P(X, Y |Z, I, Θ) = 1 Z(I, Θ)e−E(X,Y |Z,I,Θ) where the energy E of a configuration is linear in the parameters Θ = {α, ⃗β,⃗δ, γ, ⃗η, κ,⃗ν} and given by E = −α X e L1(Xe|I) −⃗β · X ⟨s,t⟩ ⃗L2(Ys, Yt|I) −⃗δ · X V ⃗M1(XV |I) −γ X ⟨s,t⟩ M2(Ys, Yt, Xe) −⃗η · X t ⃗H1(Yt|I) −κ X t H2(Yt|Z, I) −⃗ν · X e ⃗H3(Xe|Z, I) The table below gives a summary of each potential. The next section fills in details. Similarity Edge energy along e L1(Xe|I) Brightness/Texture similarity between s and t L2(Ys, Yt|I) Continuity Collinearity and junction frequency at vertex V M1(XV |I) Closure Consistency of edge and adjoining regions M2(Ys, Yt, Xe) Familiarity Similarity of region t to exemplar texture H1(Yt|I) Compatibility of region shape with pose H2(Yt|Z, I) Compatibility of local edge shape with pose H3(Xe|Z, I) 3 Cues for figure/ground labeling 3.1 Low-level Cues: Similarity of Brightness and Texture Yt Xe L1 L2 Ys To capture the locally measured edge contrast, we assign a singleton edge potential whose energy is L1(Xe|I) = log(Pbe)Xe where Pbe is the average Pb recorded over the pixels corresponding to edge e. Since the triangular regions have larger support than the local edge detector, we also include a pairwise, region-based similarity cue, computed as ⃗β · ⃗L2(Ys, Yt|I) = (βB log(f(|Is −It|)) + βT log(g(χ2(hs, ht))))1{Ys=Yt} where f predicts the likelihood of s and t belonging to the same group given the difference of average image brightness and g makes a similar prediction based on the χ2 difference between histograms of vector quantized filter responses (referred to as textons [8]) which describe the texture in the two regions. 3.2 Mid-level Cues: Curvilinear Continuity and Closure Yt Xe M2 Ys M1 There are two types of edges in the CDT graph, gradient-edges (detected by Pb) and completed-edges (filled in by the triangulation). Since true boundaries are more commonly marked by a gradient, we keep track of these two types of edges separately when modeling junctions. To capture continuity and the frequency of different junction types, we assign energy: ⃗δ · ⃗M1(XV |I) = X i,j δi,j1{degg(V )=i,degc(V )=j} + δC1{degg(V )+degc(V )=2} log(h(θ)) where XV = {Xe1, Xe2, . . .} is the set of edge variables incident on V , degg(V ) is the number of gradient-edges at vertex V for which Xe = 1. Similarly degc(V ) is the number of completed-edges that are “turned on”. When the total degree of a vertex is 2, δC weights the continuity of the two edges. h is the output of a logistic function fit to |θ| and the probability of continuation. It is smooth and symmetric around θ = 0 and falls of as θ →π. If the angle between the two edges is close to 0, they form a good continuation, f(θ) is large, and they are more likely to both be turned on. In order to assert the duality between segments and boundaries, we use a compatibility term M2(Ys, Yt, Xe) = 1{Ys=Yt,Xe=0} + 1{Ys̸=Yt,Xe=1} which simply counts when the label of s and t is consistent with that of e. 3.3 High-level Cues: Familiarity of Shape and Texture We are interested in encoding high-level knowledge about object categories. In this paper we experiment with a single object category, horses, but we believe our high-level cues will scale to multiple objects in a natural way. Yt Xe Ys H2 H3 Z H1 We compute texton histograms ht for each triangular region (as in L1). From the set of training images, we use k-medoids to find 10 representative histograms {hF 1 , . . . , hF 10} for the collection of segments labeled as figure and 10 histograms {hG l , . . . , hG l0} for the set of background segments. Each segment in a test image is compared to the set of exemplar histograms using the χ2 histogram difference. We use the energy term H1(Yt|I) = log µmini χ2(ht, hF i ) mini χ2(ht, hG i ) ¶ Yt to capture the cue of texture familiarity. We describe the global shape of the object using a template T(x, y) generated by averaging the groundtruth object segmentation masks. This yields a silhouette with quite fuzzy boundaries due to articulations and scale variation. Figure 3.3(a) shows the template extracted from our training data. Let O(Z, t) be the normalized overlap between template centered at Z = (x0, y0) with the triangular region corresponding to Yt. This is computed as the integral of T(x, y) over the triangle t divided by the area of t. We then use energy ⃗η · ⃗H2(Yt|Z) = ηF log(O(Z, t))Yt + ηG log(1 −O(Z, t))(1 −Yt) In the case of multiple objects or aspects of a single object, we use multiple templates and augment Z with an indicator of the aspect Z = (x, y, a). In our experiments on the dataset considered here, we found that the variability is too small (all horses facing left) to see a significant impact on performance from adding multiple aspects. Lastly, we would like to capture the spatial layout of articulated structures such as the horses legs and head. To describe characteristic configuration of edges, we utilize the geometric blur[1] descriptor applied to the output of the Pb boundary detector. The geometric blur centered at location x, GBx(y), is a linear operator applied to Pb(x, y) whose value is another image given by the “convolution” of Pb(x, y) with a spatially varying Gaussian. Geometric blur is motivated by the search for a linear operator which will respond strongly to a particular object feature and is invariant to some set of transformations of the image. We use the geometric blur computed at the set of image edges (Pb > 0.05) to build a library of 64 prototypical ”shapemes” from the training data by vector quantization. For each edge Xe which expresses a particular shapeme we would like to know whether Xe should be (a) (b) (c) (d) (e) Figure 2: Using a priori shape knowledge: (a) average horse template. (b) one shapeme, capturing long horizontal curves. Shown here is the average shape in this shapeme cluster. (c) on a horse, this shapeme occurs at horse back and stomach. Shown here is the density of the shapeme M ON overlayed with a contour plot of the average mask. (d) another shapeme, capturing parallel vertical lines. (e) on a horse, this shapeme occurs at legs. “turned on”. This is estimated from training data by building spatial maps M ON i (x, y) and M OF F i (x, y) for each shapeme relative to the object center which record the frequency of a true/false boundary expressing shapeme i. Figure 3.3(b-e) shows two example shapemes and their corresponding M ON map. Let Se,i(x, y) be the indicator of the set of pixels on edge e which express shapeme i. For an object in pose Z = (x0, y0) we use the energy ⃗ν · X e ⃗H3(Xe|Z, I) = X e 1 |e|(νON X i,x,y log(M ON i (x −x0, y −y0))Se,i(x, y)Xe+ νOF F X i,x,y log(M OF F i (x −x0, y −y0))Se,i(x, y)(1 −Xe)) 4 Learning cue integration We carry out approximate inference using loopy belief propagation [15] which appears to converge quickly to a reasonable solution for the graphs and potentials in question. To fit parameters of the model, we maximize the joint likelihood over X, Y, Z taking each image as an iid sample. Since our model is log-linear in the parameters Θ, partial derivatives always yield the difference between the empirical expectation of a feature given by the training data and the expected value given the model parameters. For example, the derivative with respect to the continuation parameter δ0 for a single training image/ground truth labeling, (I, X, Y, Z) is: ∂ ∂δ0 −log P(X, Y |Z, I, Θ) = ∂ ∂δ0 log Z(In, Θ) − X V ∂ ∂δ0 {δ01{degg(V )+degc(V )=2}log(f(θ))} = *X V 1{degg(V )+degc(V )=2}log(f(θ)) + − X V 1{degg(V )+degc(V )=2}log(f(θ)) where the expectation is taken with respect to P(X, Y |Z, I, Θ). Given this estimate, we optimize the parameters by gradient descent. We have also used the difference of the energy and the Bethe free energy given by the beliefs as an estimate of the log likelihood in order to support line-search in conjugate gradient or quasi-newton routines. For our model, we find that gradient descent with momentum is efficient enough. deg=0 deg=1 deg=2 deg=3 weight=2.4607 weight=0.8742 weight=1.1458 weight=0.0133 Figure 3: Learning about junctions: (a) deg=0, no boundary detected; the most common case. (b) line endings. (c) continuations of contours, more common than line endings. (d) T-junctions, very rare for the horse dataset. Compare with hand set potentials of Geman and Geman [3]. 5 Experiments In our experiments we use 344 grayscale images of the horse dataset of Borenstein et al [2]. Half of the images are used for training and half for testing. Human-marked segmentations are used1 for both training and evaluation. Training: loopy belief propagation on a typical CDT graph converges in about 1 second. The gradient descent learning described above converges within 1000 iterations. To understand the weights given by the learning procedure, Figure 3 shows some of the junction types in M1 and their associated weights δ. Testing: we evaluate the performance of our model on both edge and region labels. We present the results using a precision-recall curve which shows the trade-off between false positives and missed detections. For each edge e, we assign the marginal probability E[Xe] to all pixels (x, y) belonging to e. Then for each threshold r, pixels above r are matched to human-marked boundaries H. The precision P = P(H(x, y) = 1|PE(x, y) > r) and recall R = P(PE(x, y) > r|H(x, y) = 1) are recorded. Similarly, each pixel in a triangle t is assigned the marginal probability E[Yt] and the precision and recall of the ground-truth figural pixels computed. The evaluations are shown in Figure 4 for various combinations of cues. Figure 5 shows our results on some of the test images. 6 Conclusion We have introduced a conditional random field model on a triangulated representation of images for figure/ground labeling. We have measured the contributions of mid- and highlevel cues by quantitative evaluations on held out test data. Our findings suggest that midlevel cues provide useful information, even in the presence of high-level shape cues. In future work we plan to extend this model to multiple object categories. References [1] A. Berg and J. Malik. Geometric blur for template matching. In CVPR, 2001. [2] E. Borenstein and S. Ullman. Class-specific, top-down segmentation. In Proc. 7th Europ. Conf. Comput. Vision, volume 2, pages 109–124, 2002. [3] S. Geman and D. Geman. Stochastic relaxation, gibbs distribution, and the bayesian retoration of images. IEEE Trans. Pattern Analysis and Machine Intelligence, 6:721–41, Nov. 1984. 1From the human segmentations on pixel-grid, we use two simple techniques to establish groundtruth labels on the CDT edges Xe and triangles Yt. For Xe, we run a maximum-cardinality bipartite matching between the human marked boundaries and the CDT edges. We label Xe = 1 if 75% of the pixels lying under the edge e are matched to human boundaries. For Yt, we label Yt = 1 if at least half of the pixels within the triangle are figural pixels in the human segmentation. 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 Recall Precision Boundaries Pb [F=0.54] Pb + M [F=0.56] Pb + H [F=0.62] Pb + M + H [F=0.66] Ground Truth [F=0.80] 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 Recall Precision Regions L+M [F=0.66] L+H [F=0.82] L+M+H [F=0.83] Ground Truth [F=0.95] Figure 4: Performance evaluation: (a) precision-recall curves for horse boundaries, models with low-level cues only (Pb), low- plus mid-level cues (Pb+M), low- plus high-level cues (Pb + H), and all three classes of cues combined (Pb + M + H). The F-measure recorded in the legend is the maximal harmonic mean of precision and recall and provides an overall ranking. Using high-level cues greatly improves the boundary detection performance. Midlevel continuity cues are useful with or without high-level cues. (b) precision-recall for regions. The poor performance of the baseline L + M model indicates the ambiguity of figure/ground labeling at low-level despite successful boundary detection. High-level shape knowledge is the key, consistent with evidence from psychophysics [10]. In both boundary and region cases, the groundtruth labels on CDTs are nearly perfect, indicating that the CDT graphs preserve most of the image structure. [4] X. He, R. Zemel, and M. Carreira-Perpinan. Multiscale conditional random fields for image labelling. In IEEE Conference on Computer Vision and Pattern Recognition, 2004. [5] M. P. Kumar, P. H. S. Torr, and A. Zisserman. OBJ CUT. In CVPR, 2005. [6] S. Kumar and M. Hebert. Discriminative random fields: A discriminative framework for contextual interaction in classification. In ICCV, 2003. [7] John Lafferty, Andrew McCallum, and Fernando Pereira. Conditional random fields: Probabilistic models for segmenting and labeling sequence data. In Proc. 18th International Conf. on Machine Learning, 2001. [8] J. Malik, S. Belongie, J. Shi, and T. Leung. Textons, contours and regions: Cue integration in image segmentation. In Proc. 7th Int’l. Conf. Computer Vision, pages 918–925, 1999. [9] D. Martin, C. Fowlkes, and J. Malik. Learning to detect natural image boundaries using brightness and texture. In Advances in Neural Information Processing Systems 15, 2002. [10] M. A. Peterson and B. S. Gibson. Object recognition contributions to figure-ground organization. Perception and Psychophysics, 56:551–564, 1994. [11] X. Ren, C. Fowlkes, and J. Malik. Mid-level cues improve boundary detection. Technical Report UCB//CSD-05-1382, UC Berkeley, January 2005. [12] N. Shental, A. Zomet, T. Hertz, and Y. Weiss. Pairwise clustering and graphical models. In NIPS 2003, 2003. [13] J. Shewchuk. Triangle: Engineering a 2d quality mesh generator and delaunay triangulator. In First Workshop on Applied Computational Geometry, pages 124–133, 1996. [14] Z.W. Tu, X.R. Chen, A.L Yuille, and S.C. Zhu. Image parsing: segmentation, detection, and recognition. In ICCV, 2003. [15] Y. Weiss. Correctness of local probability propagation in graphical models with loops. Neural Computation, 2000. [16] S. Yu, R. Gross, and J. Shi. Concurrent object segmentation and recognition with graph partitioning. In Advances in Neural Information Processing Systems 15, 2002. (a) (b) (c) (d) Figure 5: Sample results. (a) the input grayscale images. (b) the low-level boundary map output by Pb. (c) the edge marginals under our full model and (d) the image masked by the output region marginals. A red cross in (d) indicates the most probably object center. By combining relatively simple low-/mid-/high-level cues in a learning framework, We are able to find and segment horses under varying conditions with only a simple object mode. The boundary maps show the model is capable of suppressing strong gradients in the scene background while boosting low-contrast edges between figure and ground. (Row 3) shows an example of an unusual pose. In (Row 5) we predict a correct off-center object location and (Row 8) demonstrates grouping together figure with non-homogeneous appearance.
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Inferring Motor Programs from Images of Handwritten Digits Geoffrey Hinton and Vinod Nair Department of Computer Science, University of Toronto 10 King’s College Road, Toronto, M5S 3G5 Canada {hinton,vnair}@cs.toronto.edu Abstract We describe a generative model for handwritten digits that uses two pairs of opposing springs whose stiffnesses are controlled by a motor program. We show how neural networks can be trained to infer the motor programs required to accurately reconstruct the MNIST digits. The inferred motor programs can be used directly for digit classification, but they can also be used in other ways. By adding noise to the motor program inferred from an MNIST image we can generate a large set of very different images of the same class, thus enlarging the training set available to other methods. We can also use the motor programs as additional, highly informative outputs which reduce overfitting when training a feed-forward classifier. 1 Overview The idea that patterns can be recognized by figuring out how they were generated has been around for at least half a century [1, 2] and one of the first proposed applications was the recognition of handwriting using a generative model that involved pairs of opposing springs [3, 4]. The “analysis-by-synthesis” approach is attractive because the true generative model should provide the most natural way to characterize a class of patterns. The handwritten 2’s in figure 1, for example, are very variable when viewed as pixels but they have very similar motor programs. Despite its obvious merits, analysis-by-synthesis has had few successes, partly because it is computationally expensive to invert non-linear generative models and partly because the underlying parameters of the generative model are unknown for most large data sets. For example, the only source of information about how the MNIST digits were drawn is the images themselves. We describe a simple generative model in which a pen is controlled by two pairs of opposing springs whose stiffnesses are specified by a motor program. If the sequence of stiffnesses is specified correctly, the model can produce images which look very like the MNIST digits. Using a separate network for each digit class, we show that backpropagation can be used to learn a “recognition” network that maps images to the motor programs required to produce them. An interesting aspect of this learning is that the network creates its own training data, so it does not require the training images to be labelled with motor programs. Each recognition network starts with a single example of a motor program and grows an “island of competence” around this example, progressively extending the region over which it can map small changes in the image to the corresponding small changes in the motor program (see figure 2). Figure 1: An MNIST image of a 2 and the additional images that can be generated by inferring the motor program and then adding random noise to it. The pixels are very different, but they are all clearly twos. Fairly good digit recognition can be achieved by using the 10 recognition networks to find 10 motor programs for a test image and then scoring each motor program by its squared error in reconstructing the image. The 10 scores are then fed into a softmax classifier. Recognition can be improved by using PCA to model the distribution of motor trajectories for each class and using the distance of a motor trajectory from the relevant PCA hyperplane as an additional score. Each recognition network is solving a difficult global search problem in which the correct motor program must be found by a single, “open-loop” pass through the network. More accurate recognition can be achieved by using this open-loop global search to initialize an iterative, closed-loop local search which uses the error in the reconstructed image to revise the motor program. This requires reconstruction errors in pixel space to be mapped to corrections in the space of spring stiffnesses. We cannot backpropagate errors through the generative model because it is just a hand-coded computer program. So we learn “generative” networks, one per digit class, that emulate the generator. After learning, backpropagation through these generative networks is used to convert pixel reconstruction errors into stiffness corrections. Our final system gives 1.82% error on the MNIST test set which is similar to the 1.7% achieved by a very different generative approach [5] but worse than the 1.53% produced by the best backpropagation networks or the 1.4% produced by support vector machines [6]. It is much worse than the 0.4% produced by convolutional neural networks that use cleverly enhanced training sets [7]. Recognition of test images is quite slow because it uses ten different recognition networks followed by iterative local search. There is, however, a much more efficient way to make use of our ability to extract motor programs. They can be treated as additional output labels when using backpropagation to train a single, multilayer, discriminative neural network. These additional labels act as a very informative regularizer that reduces the error rate from 1.53% to 1.27% in a network with two hidden layers of 500 units each. This is a new method of improving performance that can be used in conjunction with other tricks such as preprocessing the images, enhancing the training set or using convolutional neural nets [8, 7]. 2 A simple generative model for drawing digits The generative model uses two pairs of opposing springs at right angles. One end of each spring is attached to a frictionless horizontal or vertical rail that is 39 pixels from the center of the image. The other end is attached to a “pen” that has significant mass. The springs themselves are weightless and have zero rest length. The pen starts at the equilibrium position defined by the initial stiffnesses of the four springs. It then follows a trajectory that is determined by the stiffness of each spring at each of the 16 subsequent time steps in the motor program. The mass is large compared with the rate at which the stiffnesses change, so the system is typically far from equilibrium as it follows the smooth trajectory. On each time step, the momentum is multiplied by 0.9 to simulate viscosity. A coarse-grain trajectory is computed by using one step of forward integration for each time step in the motor program, so it contains 17 points. The code is at www.cs.toronto.edu/∼hinton/code. Figure 2: The training data for each class-specific recognition network is produced by adding noise to motor programs that are inferred from MNIST images using the current parameters of the recognition network. To initiate this process, the biases of the output units are set by hand so that they represent a prototypical motor program for the class. Given a coarse-grain trajectory, we need a way of assigning an intensity to each pixel. We tried various methods until we hand-evolved one that was able to reproduce the MNIST images fairly accurately, but we suspect that many other methods would be just as good. For each point on the coarse trajectory, we share two units of ink between the the four closest pixels using bilinear interpolation. We also use linear interpolation to add three fine-grain trajectory points between every pair of coarse-grain points. These fine-grain points also contribute ink to the pixels using bilinear interpolation, but the amount of ink they contribute is zero if they are less than one pixel apart and rises linearly to the same amount as the coarse-grain points if they are more than two pixels apart. This generates a thin skeleton with a fairly uniform ink density. To flesh-out the skeleton, we use two “ink parameters”, a, b, to specify a 3×3 kernel of the form b(1+a)[ a 12, a 6, a 12; a 6, 1−a, a 6; a 12, a 6, a 12] which is convolved with the image four times. Finally, the pixel intensities are clipped to lie in the interval [0,1]. The matlab code is at www.cs.toronto.edu/∼hinton/code. The values of 2a and b/1.5 are additional, logistic outputs of the recognition networks1. 3 Training the recognition networks The obvious way to learn a recognition network is to use a training set in which the inputs are images and the target outputs are the motor programs that were used to generate those images. If we knew the distribution over motor programs for a given digit class, we could easily produce such a set by running the generator. Unfortunately, the distribution over motor programs is exactly what we want to learn from the data, so we need a way to train 1We can add all sorts of parameters to the hand-coded generative model and then get the recognition networks to learn to extract the appropriate values for each image. The global mass and viscosity as well as the spacing of the rails that hold the springs can be learned. We can even implement affinelike transformations by attaching the four springs to endpoints whose eight coordinates are given by the recognition networks. These extra parameters make the learning slower and, for the normalized digits, they do not improve discrimination, probably because they help the wrong digit models as much as the right one. the recognition network without knowing this distribution in advance. Generating scribbles from random motor programs will not work because the capacity of the network will be wasted on irrelevant images that are far from the real data. Figure 2 shows how a single, prototype motor program can be used to initialize a learning process that creates its own training data. The prototype consists of a sequence of 4 × 17 spring stiffnesses that are used to set the biases on 68 of the 70 logistic output units of the recognition net. If the weights coming from the 400 hidden units are initially very small, the recognition net will then output a motor program that is a close approximation to the prototype, whatever the input image. Some random noise is then added to this motor program and it is used to generate a training image. So initially, all of the generated training images are very similar to the one produced by the prototype. The recognition net will therefore devote its capacity to modeling the way in which small changes in these images map to small changes in the motor program. Images in the MNIST training set that are close to the prototype will then be given their correct motor programs. This will tend to stretch the distribution of motor programs produced by the network along the directions that correspond to the manifold on which the digits lie. As time goes by, the generated training set will expand along the manifold for that digit class until all of the MNIST training images of that class are well modelled by the recognition network. It takes about 10 hours in matlab on a 3 GHz Xeon to train each recognition network. We use minibatches of size 100, momentum of 0.9, and adaptive learning rates on each connection that increase additively when the sign of the gradient agrees with the sign of the previous weight change and decrease multiplicatively when the signs disagree [9]. The net is generating its own training data, so the objective function is always changing which makes it inadvisable to use optimization methods that go as far as they can in a carefully chosen direction. Figures 3 and 4 show some examples of how well the recognition nets perform after training. Nearly all models achieve an average squared pixel error of less than 15 per image on their validation set (pixel intensities are between 0 and 1 with a preponderance of extreme values). The inferred motor programs are clearly good enough to capture the diverse handwriting styles in the data. They are not good enough, however, to give classification performance comparable to the state-of-the-art on the MNIST database. So we added a series of enhancements to the basic system to improve the classification accuracy. 4 Enhancements to the basic system Extra strokes in ones and sevens. One limitation of the basic system is that it draws digits using only a single stroke (i.e. the trajectory is a single, unbroken curve). But when people draw digits, they often add extra strokes to them. Two of the most common examples are the dash at the bottom of ones, and the dash through the middle of sevens (see examples in figure 5). About 2.2% of ones and 13% of sevens in the MNIST training set are dashed and not modelling the dashes reduces classification accuracy significantly. We model dashed ones and sevens by augmenting their basic motor programs with another motor program to draw the dash. For example, a dashed seven is generated by first drawing an ordinary seven using the motor program computed by the seven model, and then drawing the dash with a motor program computed by a separate neural network that models only dashes. Dashes in ones and sevens are modeled with two different networks. Their training proceeds the same way as with the other models, except now there are only 50 hidden units and the training set contains only the dashed cases of the digit. (Separating these cases from the rest of the MNIST training set is easy because they can be quickly spotted by looking at the difference between the images and their reconstructions by the dashless digit model.) The net takes the entire image of a digit as input, and computes the motor program for just the dash. When reconstructing an unlabelled image as say, a seven, we compute both Figure 3: Examples of validation set images reconstructed by their corresponding model. In each case the original image is on the left and the reconstruction is on the right. Superimposed on the original image is the pen trajectory. the dashed and dashless versions of seven and pick the one with the lower squared pixel error to be that image’s reconstruction as a seven. Figure 5 shows examples of images reconstructed using the extra stroke. Local search. When reconstructing an image in its own class, a digit model often produces a sensible, overall approximation of the image. However, some of the finer details of the reconstruction may be slightly wrong and need to be fixed up by an iterative local search that adjusts the motor program to reduce the reconstruction error. We first approximate the graphics model with a neural network that contains a single hidden layer of 500 logistic units. We train one such generative network for each of the ten digits and for the dashed version of ones and sevens (for a total of 12 nets). The motor programs used for training are obtained by adding noise to the motor programs inferred from the training data by the relevant, fully trained recognition network. The images produced from these motor programs by the graphics model are used as the targets for the supervised learning of each generative network. Given these targets, the weight updates are computed in the same way as for the recognition networks. Figure 4: To model 4’s we use a single smooth trajectory, but turn off the ink for timesteps 9 and 10. For images in which the pen does not need to leave the paper, the recognition net finds a trajectory in which points 8 and 11 are close together so that points 9 and 10 are not needed. For 5’s we leave the top until last and turn off the ink for timesteps 13 and 14. Figure 5: Examples of dashed ones and sevens reconstructed using a second stroke. The pen trajectory for the dash is shown in blue, superimposed on the original image. Initial squared pixel error = 33.8 10 iterations, error = 15.2 20 iterations, error = 10.5 30 iterations, error = 9.3 Figure 6: An example of how local search improves the detailed registration of the trajectory found by the correct model. After 30 iterations, the squared pixel error is less than a third of its initial value. Once the generative network is trained, we can use it to iteratively improve the initial motor program computed by the recognition network for an image. The main steps in one iteration are: 1) compute the error between the image and the reconstruction generated from the current motor program by the graphics model; 2) backpropagate the reconstruction error through the generative network to calculate its gradient with respect to the motor program; 3) compute a new motor program by taking a step along the direction of steepest descent plus 0.5 times the previous step. Figure 6 shows an example of how local search improves the reconstruction by the correct model. Local search is usually less effective at improving the fits of the wrong models, so it eliminates about 20% of the classification errors on the validation set. PCA model of the image residuals. The sum of squared pixel errors is not the best way of comparing an image with its reconstruction, because it treats the residual pixel errors as independent and zero-mean Gaussian distributed, which they are not. By modelling the structure in the residual vectors, we can get a better estimate of the conditional probability of the image given the motor program. For each digit class, we construct a PCA model of the image residual vectors for the training images. Then, given a test image, we project the image residual vector produced by each inferred motor program onto the relevant PCA hyperplane and compute the squared distance between the residual and its projection. This gives ten scores for the image that measure the quality of its reconstructions by the digit models. We don’t discard the old sum of squared pixel errors as they are still useful for classifying most images correctly. Instead, all twenty scores are used as inputs to the classifier, which decides how to combine both types of scores to achieve high classification accuracy. PCA model of trajectories. Classifying an image by comparing its reconstruction errors for the different digit models tacitly relies on the assumption that the incorrect models will reconstruct the image poorly. Since the models have only been trained on images in their Squared error = 24.9, Shape prior score = 31.5 Squared error = 15.0, Shape prior score = 104.2 Figure 7: Reconstruction of a two image by the two model (left box) and by the three model (right box), with the pen trajectory superimposed on the original image. The three model sharply bends the bottom of its trajectory to better explain the ink, but the trajectory prior for three penalizes it with a high score. The two model has a higher squared error, but a much lower prior score, which allows the classifier to correctly label the image. own class, they often do reconstruct images from other classes poorly, but occasionally they fit an image from another class well. For example, figure 7 shows how the three model reconstructs a two image better than the two model by generating a highly contorted three. This problem becomes even more pronounced with local search which sometimes contorts the wrong model to fit the image really well. The solution is to learn a PCA model of the trajectories that a digit model infers from images in its own class. Given a test image, the trajectory computed by each digit model is scored by its squared distance from the relevant PCA hyperplane. These 10 “prior” scores are then given to the classifier along with the 20 “likelihood” scores described above. The prior scores eliminate many classification mistakes such as the one in figure 7. 5 Classification results To classify a test image, we apply multinomial logistic regression to the 30 scores – i.e. we use a neural network with no hidden units, 10 softmax output units and a cross-entropy error. The net is trained by gradient descent using the scores for the validation set images. To illustrate the gain in classification accuracy achieved by the enhancements explained above, table 1 gives the percent error on the validation set as each enhancement is added to the system. Together, the enhancements almost halve the number of mistakes. Enhancements Validation set Test set % error % error None 4.43 1 3.84 1, 2 3.01 1, 2, 3 2.67 1, 2, 3, 4 2.28 1.82 Table 1: The gain in classification accuracy on the validation set as the following enhancements are added: 1) extra stroke for dashed ones and sevens, 2) local search, 3) PCA model of image residual, and 4) PCA trajectory prior. To avoid using the test set for model selection, the performance on the official test set was only measured for the final system. 6 Discussion After training a single neural network to output both the class label and the motor program for all classes (as described in section 1) we tried ignoring the label output and classifying the test images by using the cost, under 10 different PCA models, of the trajectory defined by the inferred motor program. Each PCA model was fitted to the trajectories extracted from the training images for a given class. This gave 1.80% errors which is as good as the 1.82% we got using the 10 separate recognition networks and local search. This is quite surprising because the motor programs produced by the single network were simplified to make them all have the same dimensionality and they produced significantly poorer reconstructions. By only using the 10 digit-specific recognition nets to create the motor programs for the training data, we get much faster recognition of test data because at test time we can use a single recognition network for all classes. It also means we do not need to trade-off prior scores against image residual scores because there is only one image residual. The ability to extract motor programs could also be used to enhance the training set. [7] shows that error rates can be halved by using smooth vector distortion fields to create extra training data. They argue that these fields simulate “uncontrolled oscillations of the hand muscles dampened by inertia”. Motor noise may be better modelled by adding noise to an actual motor program as shown in figure 1. Notice that this produces a wide variety of non-blurry images and it can also change the topology. The techniques we have used for extracting motor programs from digit images may be applicable to speech. There are excellent generative models that can produce almost perfect speech if they are given the right formant parameters [10]. Using one of these generative models we may be able to train a large number of specialized recognition networks to extract formant parameters from speech without requiring labeled training data. Once this has been done, labeled data would be available for training a single feed-forward network that could recover accurate formant parameters which could be used for real-time recognition. Acknowledgements We thank Steve Isard, David MacKay and Allan Jepson for helpful discussions. This research was funded by NSERC, CFI and OIT. GEH is a fellow of the Canadian Institute for Advanced Research and holds a Canada Research Chair in machine learning. References [1] D. M. MacKay. Mindlike behaviour in artefacts. British Journal for Philosophy of Science, 2:105–121, 1951. [2] M. Halle and K. Stevens. Speech recognition: A model and a program for research. IRE Transactions on Information Theory, IT-8 (2):155–159, 1962. [3] Murray Eden. Handwriting and pattern recognition. IRE Transactions on Information Theory, IT-8 (2):160–166, 1962. [4] J.M. Hollerbach. An oscillation theory of handwriting. Biological Cybernetics, 39:139–156, 1981. [5] G. Mayraz and G. E. Hinton. Recognizing hand-written digits using hierarchical products of experts. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24:189–197, 2001. [6] D. Decoste and B. Schoelkopf. Training invariant support vector machines. Machine Learning, 46:161–190, 2002. [7] Patrice Y. Simard, Dave Steinkraus, and John Platt. Best practice for convolutional neural networks applied to visual document analysis. In International Conference on Document Analysis and Recogntion (ICDAR), IEEE Computer Society, Los Alamitos, pages 958–962, 2003. [8] 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. [9] A. Jacobs R. Increased Rates of Convergence Through Learning Rate Adaptation. Technical Report: UM-CS-1987-117. University of Massachusetts, Amherst, MA, 1987. [10] W. Holmes, J. Holmes, and M. Judd. Extension of the bandwith of the jsru parallel-formant synthesizer for high quality synthesis of male and female speech. In Proceedings of ICASSP 90 (1), pages 313–316, 1990.
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Dual-Tree Fast Gauss Transforms Dongryeol Lee Computer Science Carnegie Mellon Univ. dongryel@cmu.edu Alexander Gray Computer Science Carnegie Mellon Univ. agray@cs.cmu.edu Andrew Moore Computer Science Carnegie Mellon Univ. awm@cs.cmu.edu Abstract In previous work we presented an efficient approach to computing kernel summations which arise in many machine learning methods such as kernel density estimation. This approach, dual-tree recursion with finitedifference approximation, generalized existing methods for similar problems arising in computational physics in two ways appropriate for statistical problems: toward distribution sensitivity and general dimension, partly by avoiding series expansions. While this proved to be the fastest practical method for multivariate kernel density estimation at the optimal bandwidth, it is much less efficient at larger-than-optimal bandwidths. In this work, we explore the extent to which the dual-tree approach can be integrated with multipole-like Hermite expansions in order to achieve reasonable efficiency across all bandwidth scales, though only for low dimensionalities. In the process, we derive and demonstrate the first truly hierarchical fast Gauss transforms, effectively combining the best tools from discrete algorithms and continuous approximation theory. 1 Fast Gaussian Summation Kernel summations are fundamental in both statistics/learning and computational physics. This paper will focus on the common form G(xq) = NR P r=1 e −||xq−xr||2 2h2 i.e. where the kernel is the Gaussian kernel with scaling parameter, or bandwidth h, there are NR reference points xr, and we desire the sum for NQ different query points xq. Such kernel summations appear in a wide array of statistical/learning methods [5], perhaps most obviously in kernel density estimation [11], the most widely used distribution-free method for the fundamental task of density estimation, which will be our main example. Understanding kernel summation algorithms from a recently developed unified perspective [5] begins with the picture of Figure 1, then separately considers the discrete and continuous aspects. Discrete/geometric aspect. In terms of discrete algorithmic structure, the dual-tree framework of [5], in the context of kernel summation, generalizes all of the well-known algorithms. 1 It was applied to the problem of kernel density estimation in [7] using a simple 1These include the Barnes-Hut algorithm [2], the Fast Multipole Method [8], Appel’s algorithm [1], and the WSPD [4]: the dual-tree method is a node-node algorithm (considers query regions rather than points), is fully recursive, can use distribution-sensitive data structures such as kd-trees, and is bichromatic (can specialize for differing query and reference sets). Figure 1: The basic idea is to approximate the kernel sum contribution of some subset of the reference points XR, lying in some compact region of space R with centroid xR, to a query point. In more efficient schemes a query region is considered, i.e. the approximate contribution is made to an entire subset of the query points XQ lying in some region of space Q, with centroid xQ. finite-difference approximation, which is tantamount to a centroid approximation. Partially by avoiding series expansions, which depend explicitly on the dimension, the result was the fastest such algorithm for general dimension, when operating at the optimal bandwidth. Unfortunately, when performing cross-validation to determine the (initially unknown) optimal bandwidth, both suboptimally small and large bandwidths must be evaluated. The finite-difference-based dual-tree method tends to be efficient at or below the optimal bandwidth, and at very large bandwidths, but for intermediately-large bandwidths it suffers. Continuous/approximation aspect. This motivates investigating a multipole-like series approximation which is appropriate for the Gaussian kernel, as introduced by [9], which can be shown the generalize the centroid approximation. We define the Hermite functions hn(t) by hn(t) = e−t2Hn(t), where the Hermite polynomials Hn(t) are defined by the Rodrigues formula: Hn(t) = (−1)net2Dne−t2, t ∈R1. After scaling and shifting the argument t appropriately, then taking the product of univariate functions for each dimension, we obtain the multivariate Hermite expansion G(xq) = NR X r=1 e −||xq−xr||2 2h2 = NR X r=1 X α≥0 1 α! xr −xR √ 2h2 α hα xq −xR √ 2h2 (1) where we’ve adopted the usual multi-index notation as in [9]. This can be re-written as G(xq) = NR X r=1 e −||xq−xr||2 2h2 = NR X r=1 X α≥0 1 α!hα xr −xQ √ 2h2 xq −xQ √ 2h2 α (2) to express the sum as a Taylor (local) expansion about a nearby representative centroid xQ in the query region. We will be using both types of expansions simultaneously. Since series approximations only hold locally, Greengard and Rokhlin [8] showed that it is useful to think in terms of a set of three ‘translation operators’ for converting between expansions centered at different points, in order to create their celebrated hierarchical algorithm. This was done in the context of the Coulombic kernel, but the Gaussian kernel has importantly different mathematical properties. The original Fast Gauss Transform (FGT) [9] was based on a flat grid, and thus provided only one operator (“H2L” of the next section), with an associated error bound (which was unfortunately incorrect). The Improved Fast Gauss Transform (IFGT) [14] was based on a flat set of clusters and provided no operators with a rearranged series approximation, which intended to be more favorable in higher dimensions but had an incorrect error bound. We will show the derivations of all the translation operators and associated error bounds needed to obtain, for the first time, a hierarchical algorithm for the Gaussian kernel. 2 Translation Operators and Error Bounds The first operator converts a multipole expansion of a reference node to form a local expansion centered at the centroid of the query node, and is our main approximation workhorse. Lemma 2.1. Hermite-to-local (H2L) translation operator for Gaussian kernel (as presented in Lemma 2.2 in [9, 10]): Given a reference node XR, a query node XQ, and the Hermite expansion centered at a centroid xR of XR: G(xq) = P α≥0 Aαhα xq−xR √ 2h2 , the Taylor expansion of the Hermite expansion at the centroid xQ of the query node XQ is given by G(xq) = P β≥0 Bβ xq−xQ √ 2h2 β where Bβ = (−1)|β| β! P α≥0 Aαhα+β xQ−xR √ 2h2 . Proof. (sketch) The proof consists of replacing the Hermite function portion of the expansion with its Taylor series. Note that we can rewrite G(xq) = P α≥0 NR P r=1 1 α! xr−xR √ 2h2 α hα xq−xR √ 2h2 by interchanging the summation order, such that the term in the brackets depends only on the reference points, and can thus be computed indepedent of any query location – we will call such terms Hermite moments. The next operator allows the efficient pre-computation of the Hermite moments in the reference tree in a bottom-up fashion from its children. Lemma 2.2. Hermite-to-Hermite (H2H) translation operator for Gaussian kernel: Given the Hermite expansion centered at a centroid xR′ in a reference node XR′: G(xq) = P α≥0 A′ αhα xq−xR′ √ 2h2 , this same Hermite expansion shifted to a new location xR of the parent node of XR is given by G(xq) = P γ≥0 Aγhγ xq−xR √ 2h2 where Aγ = P 0≤α≤γ 1 (γ−α)!A′ α xR′−xR √ 2h2 γ−α . Proof. We simply replace the Hermite function part of the expansion by a new Taylor series, as follows: G(xq) = X α≥0 A′ αhα „xq −xR′ √ 2h2 « = X α≥0 A′ α X β≥0 1 β! „xR −xR′ √ 2h2 «β (−1)|β|hα+β „xq −xR √ 2h2 « = X α≥0 X β≥0 A′ α 1 β! „xR −xR′ √ 2h2 «β (−1)|β|hα+β „xq −xR √ 2h2 « = X α≥0 X β≥0 A′ α 1 β! „xR′ −xR √ 2h2 «β hα+β „xq −xR √ 2h2 « = X γ≥0 2 4 X 0≤α≤γ 1 (γ −α)!A′ α „xR′ −xR √ 2h2 «γ−α 3 5 hγ „xq −xR √ 2h2 « where γ = α + β. The next operator acts as a “clean-up” routine in a hierarchical algorithm. Since we can approximate at different scales in the query tree, we must somehow combine all the approximations at the end of the computation. By performing a breadth-first traversal of the query tree, the L2L operator shifts a node’s local expansion to the centroid of each child. Lemma 2.3. Local-to-local (L2L) translation operator for Gaussian kernel: Given a Taylor expansion centered at a centroid xQ′ of a query node XQ′: G(xq) = P β≥0 Bβ xq−xQ′ √ 2h2 β , the Taylor expansion obtained by shifting this expansion to the new centroid xQ of the child node XQ is G(xq) = P α≥0 " P β≥α β! α!(β−α)!Bβ xQ−xQ′ √ 2h2 β−α # xq−xQ √ 2h2 α . Proof. Applying the multinomial theorem to to expand about the new center xQ yields: G(xq) = X β≥0 Bβ „xq −xQ′ √ 2h2 «β = X β≥0 X α≤β Bβ β! α!(β −α)! „xQ −xQ′ √ 2h2 «β−α „xq −xQ √ 2h2 «α . whose summation order can be interchanged to achieve the result. Because the Hermite and the Taylor expansion are truncated after taking pD terms, we incur an error in approximation. The original error bounds for the Gaussian kernel in [9, 10] were wrong and corrections were shown in [3]. Here, we will present all necessary three error bounds incurred in performing translation operators. We note that these error bounds place limits on the size of the query node and the reference node. 2 Lemma 2.4. Error Bound for Truncating an Hermite Expansion (as presented in [3]): Suppose we are given an Hermite expansion of a reference node XR about its centroid xR: G(xq) = P α≥0 Aαhα xq−xR √ 2h2 where Aα = NR P r=1 1 α! xr−xR √ 2h2 α . For any query point xq, the error due to truncating the series after the first pD term is |ǫM(p)| ≤ NR (1−r)D D−1 P k=0 D k (1 − rp)k rp √p! D−k where ∀xr ∈XR satisfies ||xr −xR||∞< rh for r < 1. Proof. (sketch) We expand the Hermite expansion as a product of one-dimensional Hermite functions, and utilize a bound on one-dimensional Hermite functions due to [13]: 1 n!|hn(x)| ≤2 n 2 √ n!e −x2 2 , n ≥0, x ∈R1. Lemma 2.5. Error Bound for Truncating a Taylor Expansion Converted from an Hermite Expansion of Infinite Order: Suppose we are given the following Taylor expansion about the centroid xQ of a query node G(xq) = P β≥0 Bβ xq−xQ √ 2h2 β where 2Strain [12] proposed the interesting idea of using Stirling’s formula (for any non-negative integer n: ` n+1 e ´n ≤n!) to lift the node size constraint; one might imagine that this could allow approximation of larger regions in a tree-based algorithm. Unfortunately, the error bounds developed in [12] were also incorrect. We have derived the three necessary corrected error bounds based on the techniques in [3]. However, due to space, and because using these bounds actually degraded performance slightly, we do not include those lemmas here. Bβ = (−1)|β| β! P α≥0 Aαhα+β xQ−xR √ 2h2 and Aα’s are the coefficients of the Hermite expansion centered at the reference node centroid xR. Then, truncating the series after pD terms satisfies the error bound |ǫL(p)| ≤ NR (1−r)D D−1 P k=0 D k (1 −rp)k rp √p! D−k where ||xq −xQ||∞< rh for r < 1, ∀xq ∈XQ. Proof. Taylor expansion of the Hermite function yields e −||xq−xr||2 2h2 = X β≥0 (−1)|β| β! X α≥0 1 α! „xr −xR √ 2h2 «α hα+β „xQ −xR √ 2h2 « „xq −xQ √ 2h2 «β = X β≥0 (−1)|β| β! X α≥0 1 α! „xR −xr √ 2h2 «α (−1)|α|hα+β „xQ −xR √ 2h2 « „xq −xQ √ 2h2 «β = X β≥0 (−1)|β| β! hβ „xQ −xr √ 2h2 « „xq −xQ √ 2h2 «β Use e −||xq−xr||2 2h2 = DQ i=1 (up(xqi, xri, xQi) + vp(xqi, xri, xQi)) for 1 ≤i ≤D, where up(xqi, xri, xQi) = p−1 X ni=0 (−1)ni ni! hni „xQi −xri √ 2h2 « „xqi −xQi √ 2h2 «ni vp(xqi, xri, xQi) = ∞ X ni=p (−1)ni ni! hni „xQi −xri √ 2h2 « „xqi −xQi √ 2h2 «ni . These univariate functions respectively satisfy up(xqi, xri, xQi) ≤ 1−rp 1−r and vp(xqi, xri, xQi) ≤ 1 √p! rp 1−r, for 1 ≤i ≤D, achieving the multivariate bound. Lemma 2.6. Error Bound for Truncating a Taylor Expansion Converted from an Already Truncated Hermite Expansion: A truncated Hermite expansion centered about the centroid xR of a reference node G(xq) = P α<p Aαhα xq−xR √ 2h2 has the following Taylor expansion about the centroid xQ of a query node: G(xq) = P β≥0 Cβ xq−xQ √ 2h2 β where the coefficients Cβ are given by Cβ = (−1)|β| β! P α<p Aαhα+β xQ−xR √ 2h2 . Truncating the series after pD terms satisfies the error bound |ǫL(p)| ≤ NR (1−2r)2D D−1 P k=0 D k ((1 − (2r)p)2)k ((2r)p)(2−(2r)p) √p! D−k for a query node XQ for which ||xq −xQ||∞< rh, and a reference node XR for which ||xr −xR||∞< rh for r < 1 2, ∀xq ∈XQ, ∀xr ∈XR. Proof. We define upi = up(xqi, xri, xQi, xRi), vpi = vp(xqi, xri, xQi, xRi), wpi = wp(xqi, xri, xQi, xRi) for 1 ≤i ≤D: upi = p−1 X ni=0 (−1)ni ni! p−1 X nj=0 1 nj! „xRi −xri √ 2h2 «nj (−1)njhni+nj „xQi −xRi √ 2h2 « „xqi −xQi √ 2h2 «ni vpi = p−1 X ni=0 (−1)ni ni! ∞ X nj=p 1 nj! „xRi −xri √ 2h2 «nj (−1)nj hni+nj „xQi −xRi √ 2h2 « „xqi −xQi √ 2h2 «ni wpi = ∞ X ni=p (−1)ni ni! ∞ X nj=0 1 nj! „xRi −xri √ 2h2 «nj (−1)njhni+nj „xQi −xRi √ 2h2 « „xqi −xQi √ 2h2 «ni Note that e −||xq−xr||2 2h2 = DQ i=1 (upi + vpi + wpi) for 1 ≤i ≤D. Using the bound for Hermite functions and the property of geometric series, we obtain the following upper bounds: upi ≤ p−1 X ni=0 p−1 X nj=0 (2r)ni(2r)nj = „1 −(2r)p) 1 −2r «2 vpi ≤ 1 √p! p−1 X ni=0 ∞ X nj =p (2r)ni(2r)nj = 1 √p! „1 −(2r)p 1 −2r « „ (2r)p 1 −2r « wpi ≤ 1 √p! ∞ X ni=p ∞ X nj=0 (2r)ni(2r)nj = 1 √p! „ 1 1 −2r « „ (2r)p 1 −2r « Therefore, ˛˛˛˛˛e −||xq−xr||2 2h2 − D Y i=1 upi ˛˛˛˛˛ ≤(1 −2r)−2D D−1 X k=0 D k ! ((1 −(2r)p)2)k „((2r)p)(2 −(2r)p) √p! «D−k ˛˛˛˛˛˛ G(xq) − X β<p Cβ „ xq −xQ √ 2h2 «β ˛˛˛˛˛˛ ≤ NR (1 −2r)2D D−1 X k=0 “D k ” ((1 −(2r)p)2)k „ ((2r)p)(2 −(2r)p) √p! «D−k 3 Algorithm and Results Algorithm. The algorithm mainly consists of making the function call DFGT(Q.root,R.root), i.e. calling the recursive function DFGT() with the root nodes of the query tree and reference tree. After the DFGT() routine is completed, the pre-order traversal of the query tree implied by the L2L operator is performed. Before the DFGT() routine is called, the reference tree could be initialized with Hermite coefficients stored in each node using the H2H translation operator, but instead we will compute them as needed on the fly. It adaptively chooses among three possible methods for approximating the summation contribution of the points in node R to the queries in node Q, which are self-explanatory, based on crude operation count estimates. Gmin Q , a running lower bound on the kernel sum G(xq) for any xq ∈XQ, is used to ensure locally that the global relative error is ǫ or less. This automatic mechanism allows the user to specify only an error tolerance ǫ rather than other tweak parameters. Upon approximation, the upper and lower bounds on G for Q and all its children are updated; the latter can be done in an O(1) delayed fashion as in [7]. The remainder of the routine implements the characteristic four-way dual-tree recursion. We also tested a hybrid method (DFGTH) which approximates if either of the DFD or DFGT approximation criteria are met. Experimental results. We empirically studied the runtime 3 performance of five algorithms on five real-world datasets for kernel density estimation at every query point with a range of bandwidths, from 3 orders of magnitude smaller than optimal to three orders larger than optimal, according to the standard least-squares cross-validation score [11]. The naive 3All times include all preprocessing costs including any data structure construction. Times are measured in CPU seconds on a dual-processor AMD Opteron 242 machine with 8 Gb of main memory and 1 Mb of CPU cache. All the codes that we have written and obtained are written in C and C++, and was compiled under -O6 -funroll-loops flags on Linux kernel 2.4.26. algorithm computes the sum explicitly and thus exactly. We have limited all datasets to 50K points so that true relative error, i.e. | bG(xq) −Gtrue(xq)| /Gtrue(xq), can be evaluated, and set the tolerance at 1% relative error for all query points. When any method fails to achieve the error tolerance in less time than twice that of the naive method, we give up. Codes for the FGT [9] and for the IFGT [14] were obtained from the authors’ websites. Note that both of these methods require the user to tweak parameters, while the others are automatic. 4 DFD refers to the depth-first dual-tree finite-difference method [7]. DFGT(Q, R) pDH = pDL = pH2L = ∞ if R.maxside < 2h, pDH = the smallest p ≥1 such that NR (1−r)D D−1 P k=0 D k (1 −rp)k rp √p! D−k < ǫGmin Q . if Q.maxside < 2h, pDL = the smallest p ≥1 such that NR (1−r)D D−1 P k=0 D k (1 −rp)k rp √p! D−k < ǫGmin Q . if max(Q.maxside,R.maxside) < h, pH2L = the smallest p ≥1 such that NR (1−2r)2D D−1 P k=0 D k ((1 −(2r)p)2)k ((2r)p)(2−(2r)p) √p! D−k < ǫGmin Q . cDH = pD DHNQ. cDL = pD DLNR. cH2L = DpD+1 H2L. cDirect = DNQNR. if no Hermite coefficient of order pDH exists for XR, Compute it. cDH = cDH + pD DHNR. if no Hermite coefficient of order pH2L exists for XR, Compute it. cH2L = cH2L + pD H2LNR. c = min(cDH, cDL, cH2L, cDirect). if c = cDH < ∞, (Direct Hermite) Evaluate each xq at the Hermite series of order pDH centered about xR of XR using Equation 1. if c = cDL < ∞, (Direct Local) Accumulate each xr ∈XR as the Taylor series of order pDL about the center xQ of XQ using Equation 2. if c = cH2L < ∞, (Hermite-to-Local) Convert the Hermite series of order pH2L centered about xR of XR to the Taylor series of the same order centered about xQ of XQ using Lemma 2.1. if c ̸= cDirect, Update Gmin and Gmax in Q and all its children. return. if leaf(Q) and leaf(R), Perform the naive algorithm on every pair of points in Q and R. else DFGT(Q.left, R.left). DFGT(Q.left, R.right). DFGT(Q.right, R.left). DFGT(Q.right, R.right). 4For the FGT, note that the algorithm only ensures: ˛˛˛ bG(xq) −Gtrue(xq) ˛˛˛ ≤τ. Therefore, we first set τ = ǫ, halving τ until the error tolerance ǫ was met. For the IFGT, which has multiple parameters that must be tweaked simultaneously, an automatic scheme was created, based on the recommendations given in the paper and software documentation: For D = 2, use p = 8; for D = 3, use p = 6; set ρx = 2.5; start with K = √ N and double K until the error tolerance is met. When this failed to meet the tolerance, we resorted to additional trial and error by hand. The costs of parameter selection for these methods in both computer and human time is not included in the table. Algorithm \ scale 0.001 0.01 0.1 1 10 100 1000 sj2-50000-2 (astronomy: positions), D = 2, N = 50000, h∗= 0.00139506 Naive 301.696 301.696 301.696 301.696 301.696 301.696 301.696 FGT out of RAM out of RAM out of RAM 3.892312 2.01846 0.319538 0.183616 IFGT > 2×Naive > 2×Naive > 2×Naive > 2×Naive > 2×Naive > 2×Naive 7.576783 DFD 0.837724 1.087066 1.658592 6.018158 62.077669 151.590062 1.551019 DFGT 0.849935 1.11567 4.599235 72.435177 18.450387 2.777454 2.532401 DFGTH 0.846294 1.10654 1.683913 6.265131 5.063365 1.036626 0.68471 colors50k (astronomy: colors), D = 2, N = 50000, h∗= 0.0016911 Naive 301.696 301.696 301.696 301.696 301.696 301.696 301.696 FGT out of RAM out of RAM out of RAM > 2×Naive > 2×Naive 0.475281 0.114430 IFGT > 2×Naive > 2×Naive > 2×Naive > 2×Naive > 2×Naive > 2×Naive 7.55986 DFD 1.095838 1.469454 2.802112 30.294007 280.633106 81.373053 3.604753 DFGT 1.099828 1.983888 29.231309 285.719266 12.886239 5.336602 3.5638 DFGTH 1.081216 1.47692 2.855083 24.598749 7.142465 1.78648 0.627554 edsgc-radec-rnd (astronomy: angles), D = 2, N = 50000, h∗= 0.00466204 Naive 301.696 301.696 301.696 301.696 301.696 301.696 301.696 FGT out of RAM out of RAM out of RAM 2.859245 1.768738 0.210799 0.059664 IFGT > 2×Naive > 2×Naive > 2×Naive > 2×Naive > 2×Naive > 2×Naive 7.585585 DFD 0.812462 1.083528 1.682261 5.860172 63.849361 357.099354 0.743045 DFGT 0.84023 1.120015 4.346061 73.036687 21.652047 3.424304 1.977302 DFGTH 0.821672 1.104545 1.737799 6.037217 5.7398 1.883216 0.436596 mockgalaxy-D-1M-rnd (cosmology: positions), D = 3, N = 50000, h∗= 0.000768201 Naive 354.868751 354.868751 354.868751 354.868751 354.868751 354.868751 354.868751 FGT out of RAM out of RAM out of RAM out of RAM > 2×Naive > 2×Naive > 2×Naive IFGT > 2×Naive > 2×Naive > 2×Naive > 2×Naive > 2×Naive > 2×Naive > 2×Naive DFD 0.70054 0.701547 0.761524 0.843451 1.086608 42.022605 383.12048 DFGT 0.73007 0.733638 0.799711 0.999316 50.619588 125.059911 109.353701 DFGTH 0.724004 0.719951 0.789002 0.877564 1.265064 22.6106 87.488392 bio5-rnd (biology: drug activity), D = 5, N = 50000, h∗= 0.000567161 Naive 364.439228 364.439228 364.439228 364.439228 364.439228 364.439228 364.439228 FGT out of RAM out of RAM out of RAM out of RAM out of RAM out of RAM out of RAM IFGT > 2×Naive > 2×Naive > 2×Naive > 2×Naive > 2×Naive > 2×Naive > 2×Naive DFD 2.249868 2.4958865 4.70948 12.065697 94.345003 412.39142 107.675935 DFGT > 2×Naive > 2×Naive > 2×Naive > 2×Naive > 2×Naive > 2×Naive > 2×Naive DFGTH > 2×Naive > 2×Naive > 2×Naive > 2×Naive > 2×Naive > 2×Naive > 2×Naive Discussion. The experiments indicate that the DFGTH method is able to achieve reasonable performance across all bandwidth scales. Unfortunately none of the series approximation-based methods do well on the 5-dimensional data, as expected, highlighting the main weakness of the approach presented. Pursuing corrections to the error bounds necessary to use the intriguing series form of [14] may allow an increase in dimensionality. References [1] A. W. Appel. An Efficient Program for Many-Body Simulations. SIAM Journal on Scientific and Statistical Computing, 6(1):85–103, 1985. [2] J. Barnes and P. Hut. A Hierarchical O(NlogN) Force-Calculation Algorithm. Nature, 324, 1986. [3] B. Baxter and G. Roussos. A new error estimate of the fast gauss transform. SIAM Journal on Scientific Computing, 24(1):257–259, 2002. [4] P. Callahan and S. Kosaraju. A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. Journal of the ACM, 62(1):67–90, January 1995. [5] A. Gray and A. W. Moore. N-Body Problems in Statistical Learning. In T. K. Leen, T. G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems 13 (December 2000). MIT Press, 2001. [6] A. G. Gray. Bringing Tractability to Generalized N-Body Problems in Statistical and Scientific Computation. PhD thesis, Carnegie Mellon University, 2003. [7] A. G. Gray and A. W. Moore. Rapid Evaluation of Multiple Density Models. In Artificial Intelligence and Statistics 2003, 2003. [8] L. Greengard and V. Rokhlin. A Fast Algorithm for Particle Simulations. Journal of Computational Physics, 73, 1987. [9] L. Greengard and J. Strain. The fast gauss transform. SIAM Journal on Scientific and Statistical Computing, 12(1):79–94, 1991. [10] L. Greengard and X. Sun. A new version of the fast gauss transform. Documenta Mathematica, Extra Volume ICM(III):575– 584, 1998. [11] B. W. Silverman. Density Estimation for Statistics and Data Analysis. Chapman and Hall, 1986. [12] J. Strain. The fast gauss transform with variable scales. SIAM Journal on Scientific and Statistical Computing, 12:1131– 1139, 1991. [13] O. Sz´asz. On the relative extrema of the hermite orthogonal functions. J. Indian Math. Soc., 15:129–134, 1951. [14] C. Yang, R. Duraiswami, N. A. Gumerov, and L. Davis. Improved fast gauss transform and efficient kernel density estimation. International Conference on Computer Vision, 2003.
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Describing Visual Scenes using Transformed Dirichlet Processes Erik B. Sudderth, Antonio Torralba, William T. Freeman, and Alan S. Willsky Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology esuddert@mit.edu, torralba@csail.mit.edu, billf@mit.edu, willsky@mit.edu Abstract Motivated by the problem of learning to detect and recognize objects with minimal supervision, we develop a hierarchical probabilistic model for the spatial structure of visual scenes. In contrast with most existing models, our approach explicitly captures uncertainty in the number of object instances depicted in a given image. Our scene model is based on the transformed Dirichlet process (TDP), a novel extension of the hierarchical DP in which a set of stochastically transformed mixture components are shared between multiple groups of data. For visual scenes, mixture components describe the spatial structure of visual features in an object–centered coordinate frame, while transformations model the object positions in a particular image. Learning and inference in the TDP, which has many potential applications beyond computer vision, is based on an empirically effective Gibbs sampler. Applied to a dataset of partially labeled street scenes, we show that the TDP’s inclusion of spatial structure improves detection performance, flexibly exploiting partially labeled training images. 1 Introduction In this paper, we develop methods for analyzing the features composing a visual scene, thereby localizing and categorizing the objects in an image. We would like to design learning algorithms that exploit relationships among multiple, partially labeled object categories during training. Working towards this goal, we propose a hierarchical probabilistic model for the expected spatial locations of objects, and the appearance of visual features corresponding to each object. Given a new image, our model provides a globally coherent explanation for the observed scene, including estimates of the location and category of an a priori unknown number of objects. This generative approach is motivated by the pragmatic need for learning algorithms which require little manual supervision and labeling. While discriminative models may produce accurate classifiers, they typically require very large training sets even for relatively simple categories [1]. In contrast, generative approaches can discover large, visually salient categories (such as foliage and buildings [2]) without supervision. Partial segmentations can then be used to learn semantically interesting categories (such as cars and pedestrians) which are less visually distinctive, or present in fewer training images. Moreover, generative models provide a natural framework for learning contextual relationships between objects, and transferring knowledge between related, but distinct, visual scenes. Constellation LDA Transformed DP Figure 1: A scene with faces as described by three generative models. Constellation: Fixed parts of a single face in unlocalized clutter. LDA: Bag of unlocalized face and background features. TDP: Spatially localized clusters of background clutter, and one or more faces (in this case, the sample contains one face and two background clusters). Note: The LDA and TDP images are sampled from models learned from training images, while the Constellation image is a hand-constructed illustration. The principal challenge in developing hierarchical models for scenes is specifying tractable, scalable methods for handling uncertainty in the number of objects. This issue is entirely ignored by most existing models. We address this problem using Dirichlet processes [3], a tool from nonparametric Bayesian analysis for learning mixture models whose number of components is not fixed, but instead estimated from data. In particular, we extend the recently proposed hierarchical Dirichlet process (HDP) [4, 5] framework to allow more flexible sharing of mixture components between images. The resulting transformed Dirichlet process (TDP) is naturally suited to our scene understanding application, as well as many other domains where “style and content” are combined to produce the observed data [6]. We begin in Sec. 2 by reviewing several related generative models for objects and scenes. Sec. 3 then introduces Dirichlet processes and develops the TDP model, including MCMC methods for learning and inference. We specialize the TDP to visual scenes in Sec. 4, and conclude in Sec. 5 by demonstrating object recognition and segmentation in street scenes. 2 Generative Models for Objects and Scenes Constellation models [7] describe single objects via the appearance of a fixed, and typically small, set of spatially constrained parts (see Fig. 1). Although they can successfully recognize objects in cluttered backgrounds, they do not directly provide a mechanism for detecting multiple object instances. In addition, it seems difficult to generalize the fixed set of constellation parts to problems where the number of objects is uncertain. Grammars, and related rule–based systems, were one of the earliest approaches to scene understanding [8]. More recently, distributions over hierarchical tree–structured partitions of image pixels have been used to segment simple scenes [9, 10]. In addition, an image parsing [11] framework has been proposed which explains an image using a set of regions generated by generic or object–specific processes. While this model allows uncertainty in the number of regions, and hence the number of objects, the high dimensionality of the model state space requires good, discriminatively trained bottom–up proposal distributions for acceptable MCMC performance. We also note that the BLOG language [12] provides a promising framework for reasoning about unknown objects. As of yet, however, the computational tools needed to apply BLOG to large–scale applications are unavailable. Inspired by techniques from the text analysis literature, several recent papers analyze scenes using a spatially unstructured bag of features extracted from local image patches (see Fig. 1). In particular, latent Dirichlet allocation (LDA) [13] describes the features xji in image j using a K component mixture model with parameters θk. Each image reuses these same mixture parameters in different proportions πj (see the graphical model of Fig. 2). By appropriately defining these shared mixtures, LDA may be used to discover object categories from images of single objects [2], categorize natural scenes [14], and (with a slight extension) parse presegmented captioned images [15]. While these LDA models are sometimes effective, their neglect of spatial structure ignores valuable information which is critical in challenging object detection tasks. We recently proposed a hierarchical extension of LDA which learns shared parts describing the internal structure of objects, and contextual relationships among known groups of objects [16]. The transformed Dirichlet process (TDP) addresses a key limitation of this model by allowing uncertainty in the number and identity of the objects depicted in each image. As detailed in Sec. 4 and illustrated in Fig. 1, the TDP effectively provides a textural model in which locally unstructured clumps of features are given global spatial structure by the inferred set of objects underlying each scene. 3 Hierarchical Modeling using Dirichlet Processes In this section, we review Dirichlet process mixture models (Sec. 3.1) and previously proposed hierarchical extensions (Sec. 3.2). We then introduce the transformed Dirichlet process (TDP) (Sec. 3.3), and discuss Monte Carlo methods for learning TDPs (Sec. 3.4). 3.1 Dirichlet Process Mixture Models Let θ denote a parameter taking values in some space Θ, and H be a measure on Θ. A Dirichlet process (DP), denoted by DP(γ, H), is then a distribution over measures on Θ, where the concentration parameter γ controls the similarity of samples G ∼DP(γ, H) to the base measure H. Samples from DPs are discrete with probability one, a property highlighted by the following stick–breaking construction [4]: G(θ) = ∞ X k=1 βkδ(θ, θk) β′ k ∼Beta(1, γ) βk = β′ k k−1 Y ℓ=1 (1 −β′ ℓ) (1) Each parameter θk ∼H is independently sampled, while the weights β = (β1, β2, . . .) use Beta random variables to partition a unit–length “stick” of probability mass. In nonparametric Bayesian statistics, DPs are commonly used as prior distributions for mixture models with an unknown number of components [3]. Let F(θ) denote a family of distributions parameterized by θ. Given G ∼DP(γ, H), each observation xi from an exchangeable data set x is generated by first choosing a parameter ¯θi ∼G, and then sampling xi ∼F(¯θi). Computationally, this process is conveniently described by a set z of independently sampled variables zi ∼Mult(β) indicating the component of the mixture G(θ) (see eq. (1)) associated with each data point xi ∼F(θzi). Integrating over G, the indicator variables z demonstrate an important clustering property. Letting nk denote the number of times component θk is chosen by the first (i−1) samples, p (zi | z1, . . . , zi−1, γ) = 1 γ + i −1 "X k nkδ(zi, k) + γδ(zi, ¯k) # (2) Here, ¯k indicates a previously unused mixture component (a priori, all unused components are equivalent). This process is sometimes described by analogy to a Chinese restaurant in which the (infinite collection of) tables correspond to the mixture components θk, and customers to observations xi [4]. Customers are social, tending to sit at tables with many other customers (observations), and each table shares a single dish (parameter). 3.2 Hierarchical Dirichlet Processes In many domains, there are several groups of data produced by related, but distinct, generative processes. For example, in this paper’s applications each group is an image, and the data are visual features composing a scene. Given J groups of data, let xj = (xj1, . . . , xjnj) denote the nj exchangeable data points in group j. Hierarchical Dirichlet processes (HDPs) [4, 5] describe grouped data with a coupled set of xji θ zji πj α H k J nj K xji θ tji πj kjt β α H γ k J nj ∞ ∞ xji θ tji πj kjt jt β α φ H R γ k k J nj ∞ ∞ ρ θ tji πj β α φ H R γ k k J nj ∞ wji λ η O o kjt jt ∞ yji oji ρ LDA Hierarchical DP Transformed DP Visual Scene TDP Figure 2: Graphical representations of the LDA, HDP, and TDP models for sharing mixture components θk, with proportions πj, among J groups of exchangeable data xj = (xj1, . . . , xjnj). LDA directly assigns observations xji to clusters via indicators zji. HDP and TDP models use “table” indicators tji as an intermediary between observations and assignments kjt to an infinite global mixture with weights β. TDPs augment each table t with a transformation ρjt sampled from a distribution parameterized by φkjt. Specializing the TDP to visual scenes (right), we model the position yji and appearance wji of features using distributions ηo indexed by unobserved object categories oji. mixture models. To construct an HDP, a global probability measure G0 ∼DP(γ, H) is first chosen to define a set of shared mixture components. A measure Gj ∼DP(α, G0) is then independently sampled for each group. Because G0 is discrete (as in eq. (1)), groups Gj will reuse the same mixture components θk in different proportions: Gj(θ) = ∞ X k=1 πjkδ(θ, θk) πj ∼DP(α, β) (3) In this construction, shared components improve generalization when learning from few examples, while distinct mixture weights capture differences between groups. The generative process underlying HDPs may be understood in terms of an extension of the DP analogy known as the Chinese restaurant franchise [4]. Each group defines a separate restaurant in which customers (observations) xji sit at tables tji. Each table shares a single dish (parameter) θ, which is ordered from a menu G0 shared among restaurants (groups). Letting kjt indicate the parameter θkjt assigned to table t in group j, we may integrate over G0 and Gj (as in eq. (2)) to find the conditional distributions of these indicator variables: p (tji | tj1, . . . , tji−1, α) ∝ X t njtδ(tji, t) + αδ(tji, ¯t) (4) p (kjt | k1, . . . , kj−1, kj1, . . . , kjt−1, γ) ∝ X k mkδ(kjt, k) + γδ(kjt, ¯k) (5) Here, mk is the number of tables previously assigned to θk. As before, customers prefer tables t at which many customers njt are already seated (eq. (4)), but sometimes choose a new table ¯t. Each new table is assigned a dish kj¯t according to eq. (5). Popular dishes are more likely to be ordered, but a new dish θ¯k ∼H may also be selected. The HDP generative process is summarized in the graphical model of Fig. 2. Given the assignments tj and kj for group j, observations are sampled as xji ∼F(θzji), where zji = kjtji indexes the shared parameters assigned to the table associated with xji. 3.3 Transformed Dirichlet Processes In the HDP model of Fig. 2, the group distributions Gj are derived from the global distribution G0 by resampling the mixture weights from a Dirichlet process (see eq. (3)), leaving the component parameters θk unchanged. In many applications, however, it is difficult to define θ so that parameters may be exactly reused between groups. Consider, for example, a Gaussian distribution describing the location at which object features are detected in an image. While the covariance of that distribution may stay relatively constant across object instances, the mean will change dramatically from image to image (group to group), depending on the objects’ position relative to the camera. Motivated by these difficulties, we propose the Transformed Dirichlet Process (TDP), an extension of the HDP in which global mixture components undergo a set of random transformations before being reused in each group. Let ρ denote a transformation of the parameter vector θ ∈Θ, φ ∈Φ the parameters of a distribution Q over transformations, and R a measure on Φ. We begin by augmenting the DP stick–breaking construction of eq. (1) to create a global measure describing both parameters and transformations: G0(θ, ρ) = ∞ X k=1 βkδ(θ, θk)q(ρ | φk) θk ∼H φk ∼R (6) As before, β is sampled from a stick–breaking process with parameter γ. For each group, we then sample a measure Gj ∼DP(α, G0). Marginalizing over transformations ρ, Gj(θ) reuses parameters from G0(θ) exactly as in eq. (3). Because samples from DPs are discrete, the joint measure for group j then has the following form: Gj(θ, ρ) = ∞ X k=1 πjkδ(θ, θk) " ∞ X ℓ=1 ωjkℓδ(ρ, ρjkℓ) # ∞ X ℓ=1 ωjkℓ= 1 (7) Note that within the jth group, each shared parameter vector θk may potentially be reused multiple times with different transformations ρjkℓ. Conditioning on θk, it can be shown that Gj(ρ | θk) ∼DP(αβk, Q(φk)), so that the proportions ωjk of features associated with each transformation of θk follow a stick–breaking process with parameter αβk. Each observation xji is now generated by sampling (¯θji, ¯ρji) ∼Gj, and then choosing xji ∼F(¯θji, ¯ρji) from a distribution which transforms ¯θji by ¯ρji. Although the global family of transformation distributions Q(φ) is typically non–atomic, the discreteness of Gj ensures that transformations are shared between observations within group j. Computationally, the TDP is more conveniently described via an extension of the Chinese restaurant franchise analogy (see Fig. 2). As before, customers (observations) xji sit at tables tji according to the clustering bias of eq. (4), and new tables choose dishes according to their popularity across the franchise (eq. (5)). Now, however, the dish (parameter) θkjt at table t is seasoned (transformed) according to ρjt ∼q(ρjt | φkjt). Each time a dish is ordered, the recipe is seasoned differently. 3.4 Learning via Gibbs Sampling To learn the parameters of a TDP, we extend the HDP Gibbs sampler detailed in [4]. The simplest implementation samples table assignments t, cluster assignments k, transformations ρ, and parameters θ, φ. Let t−ji denote all table assignments excluding tji, and define k−jt, ρ−jt similarly. Using the Markov properties of the TDP (see Fig. 2), we have p ¡ tji = t | t−ji, k, ρ, θ, x ¢ ∝p ¡ t | t−ji¢ f ¡ xji | θkjt, ρjt ¢ (8) The first term is given by eq. (4). For a fixed set of transformations ρ, the second term is a simple likelihood evaluation for existing tables, while new tables may be evaluated by marginalizing over possible cluster assignments (eq. (5)). Because cluster assignments kjt and transformations ρjt are strongly coupled in the posterior, a blocked Gibbs sampler which jointly resamples them converges much more rapidly: p ¡ kjt = k, ρjt | k−jt, ρ−jt, t, θ, φ, x ¢ ∝p ¡ k | k−jt¢ q (ρjt | φk) Y tji=t f (xji | θk, ρjt) For the models considered in this paper, F is conjugate to Q for any fixed observation value. We may thus analytically integrate over ρjt and, combined with eq. (5), sample a Training Data TDP HDP Figure 3: Comparison of hierarchical models learned via Gibbs sampling from synthetic 2D data. Left: Four of 50 “images” used for training. Center: Global distribution G0(θ) for the HDP, where ellipses are covariance estimates and intensity is proportional to prior probability. Right: Global TDP distribution G0(θ, ρ) over both clusters θ (solid) and translations ρ of those clusters (dashed). new cluster assignment ¯kjt. Conditioned on ¯kjt, we again use conjugacy to sample ¯ρjt. We also choose the parameter priors H and R to be conjugate to Q and F, respectively, so that standard formulas may be used to resample θ, φ. 4 Transformed Dirichlet Processes for Visual Scenes 4.1 Context–Free Modeling of Multiple Object Categories In this section, we adapt the TDP model of Sec. 3.3 to describe the spatial structure of visual scenes. Groups j now correspond to training, or test, images. For the moment, we assume that the observed data xji = (oji, yji), where yji is the position of a feature corresponding to object category oji, and the number of object categories O is known (see Fig. 2). We then choose cluster parameters θk = (¯ok, µk, Λk) to describe the mean µk and covariance Λk of a Gaussian distribution over feature positions, as well as the single object category ¯ok assigned to all observations sampled from that cluster. Although this cluster parameterization does not capture contextual relationships between object categories, the results of Sec. 5 demonstrate that it nevertheless provides an effective model of the spatial variability of individual categories across many different scenes. To model the variability in object location from image to image, transformation parameters ρjt are defined to translate feature position relative to that cluster’s “canonical” mean µk: p ¡ oji, yji | tji = t, kj, ρj, θ ¢ = δ(oji, ¯okjt) × N ¡ yji; µkjt + ρjt, Λkjt ¢ (9) We note that there is a different translation ρjt associated with each table t, allowing the same object cluster to be reused at multiple locations within a single image. This flexibility, which is not possible with HDPs, is critical to accurately modeling visual scenes. Density models for spatial transformations have been previously used to recognize isolated objects [17], and estimate layered decompositions of video sequences [18]. In contrast, the proposed TDP models the variability of object positions across scenes, and couples this with a nonparametric prior allowing uncertainty in the number of objects. To ensure that the TDP scene model is identifiable, we define p (ρjt | kj, φ) to be a zero– mean Gaussian with covariance φkjt. The parameter prior R is uniform across object categories, while R and H both use inverse–Wishart position distributions, weakly biased towards moderate covariances. Fig. 3 shows a 2D synthetic example based on a single object category (O = 1). Following 100 Gibbs sampling iterations, the TDP correctly discovers that the data is composed of elongated “bars” in the upper right, and round “blobs” in the lower left. In contrast, the learned HDP uses a large set of global clusters to discretize the transformations underlying the data, and thus generalizes poorly to new translations. 4.2 Detecting Objects from Image Features To apply the TDP model of Sec. 4.1 to images, we must learn the relationship between object categories and visual features. As in [2, 16], we obtain discrete features by vector quantizing SIFT descriptors [19] computed over locally adapted elliptical regions. To improve discriminative power, we divide these elliptical regions into three groups (roughly circular, and horizontally or vertically elongated) prior to quantizing SIFT values, producing a discrete vocabulary with 1800 appearance “words”. Given the density of feature detection, these descriptors essentially provide a multiscale over–segmentation of the image. We assume that the appearance wji of each detected feature is independently sampled conditioned on the underlying object category oji (see Fig. 2). Placing a symmetric Dirichlet prior, with parameter λ, on each category’s multinomial appearance distribution ηo, p ¡ wji = b | oji = o, w−ji, t, k, θ ¢ ∝cbo + λ (10) where cbo is the number of times feature b is currently assigned to object o. Because a single object category is associated with each cluster, the Gibbs sampler of Sec. 3.4 may be easily adapted to this case by incorporating eq. (10) into the assignment likelihoods. 5 Analyzing Street Scenes To demonstrate the potential of our TDP scene model, we consider a set of street scene images (250 training, 75 test) from the MIT-CSAIL database. These images contain three “objects”: buildings, cars (side views), and roads. All categories were labeled in 112 images, while in the remainder only cars were segmented. Training from semi–supervised data is accomplished by restricting object category assignments for segmented features. Fig. 4 shows the four global object clusters learned following 100 Gibbs sampling iterations. There is one elongated car cluster, one large building cluster, and two road clusters with differing shapes. Interestingly, the model has automatically determined that building features occur in large homogeneous patches, while road features are sparse and better described by many smaller transformed clusters. To segment test images, we run the Gibbs sampler for 50 iterations from each of 10 random initializations. Fig. 4 shows segmentations produced by averaging these samples, as well as transformed clusters from the final iteration. Qualitatively, results are typically good, although foliage is often mislabeled as road due to the textural similarities with features detected in shadows across roads. For comparison, we also trained an LDA model based solely on feature appearance, allowing three topics per object category and again using object labels to restrict the Gibbs sampler’s assignments [16]. As shown by the ROC curves of Fig. 4, our TDP model of spatial scene structure significantly improves segmentation performance. In addition, through the set of transformed car clusters generated by the Gibbs sampler, the TDP explicitly estimates the number of object instances underlying each image. These detections, which are not possible using LDA, are based on a single global parsing of the scene which automatically estimates object locations without a “sliding window” [1]. 6 Discussion We have developed the transformed Dirichlet process, a hierarchical model which shares a set of stochastically transformed clusters among groups of data. Applied to visual scenes, TDPs provide a model of spatial structure which allows the number of objects generating an image to be automatically inferred, and lead to improved detection performance. We are currently investigating extensions of the basic TDP scene model presented in this paper which describe the internal structure of objects, and also incorporate richer contextual cues. Acknowledgments Funding provided by the National Geospatial-Intelligence Agency NEGI-1582-04-0004, the National Science Foundation NSF-IIS-0413232, the ARDA VACE program, and a grant from BAE Systems. References [1] P. Viola and M. J. Jones. Robust real–time face detection. IJCV, 57(2):137–154, 2004. [2] J. Sivic, B. C. Russell, A. A. Efros, A. Zisserman, and W. T. Freeman. Discovering objects and their location in images. In ICCV, 2005. 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 Alarm Rate Detection Rate Car (TDP) Building (TDP) Road (TDP) Car (LDA) Building (LDA) Road (LDA) Road Road Car Building Figure 4: TDP analysis of street scenes containing cars (red), buildings (green), and roads (blue). Top right: Global model G0 describing object shape (solid) and expected transformations (dashed). Bottom right: ROC curves comparing TDP feature segmentation performance to an LDA model of feature appearance. Left: Four test images (first row), estimated segmentations of features into object categories (second row), transformed global clusters associated with each image interpretation (third row), and features assigned to different instances of the transformed car cluster (fourth row). [3] M. D. Escobar and M. West. Bayesian density estimation and inference using mixtures. J. Amer. Stat. Assoc., 90(430):577–588, June 1995. [4] Y. W. Teh, M. I. Jordan, M. J. Beal, and D. M. Blei. Hierarchical Dirichlet processes. Technical Report 653, U.C. Berkeley Statistics, October 2004. [5] Y. W. Teh, M. I. Jordan, M. J. Beal, and D. M. Blei. Hierarchical Dirichlet processes. In NIPS 17, pages 1385–1392. MIT Press, 2005. [6] J. B. Tenenbaum and W. T. Freeman. 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Latent Dirichlet allocation. JMLR, 3:993–1022, 2003. [14] L. Fei-Fei and P. Perona. A Bayesian hierarchical model for learning natural scene categories. In CVPR, volume 2, pages 524–531, 2005. [15] K. Barnard et al. Matching words and pictures. JMLR, 3:1107–1135, 2003. [16] E. B. Sudderth, A. Torralba, W. T. Freeman, and A. S. Willsky. Learning hierarchical models of scenes, objects, and parts. In ICCV, 2005. [17] E. G. Miller, N. E. Matsakis, and P. A. Viola. Learning from one example through shared densities on transforms. In CVPR, volume 1, pages 464–471, 2000. [18] N. Jojic and B. J. Frey. Learning flexible sprites in video layers. In CVPR, volume 1, pages 199–206, 2001. [19] D. G. Lowe. Distinctive image features from scale–invariant keypoints. IJCV, 60(2):91–110, 2004.
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Worst-Case Bounds for Gaussian Process Models Sham M. Kakade University of Pennsylvania Matthias W. Seeger UC Berkeley Dean P. Foster University of Pennsylvania Abstract We present a competitive analysis of some non-parametric Bayesian algorithms in a worst-case online learning setting, where no probabilistic assumptions about the generation of the data are made. We consider models which use a Gaussian process prior (over the space of all functions) and provide bounds on the regret (under the log loss) for commonly used non-parametric Bayesian algorithms — including Gaussian regression and logistic regression — which show how these algorithms can perform favorably under rather general conditions. These bounds explicitly handle the infinite dimensionality of these non-parametric classes in a natural way. We also make formal connections to the minimax and minimum description length (MDL) framework. Here, we show precisely how Bayesian Gaussian regression is a minimax strategy. 1 Introduction We study an online (sequential) prediction setting in which, at each timestep, the learner is given some input from the set X, and the learner must predict the output variable from the set Y. The sequence {(xt, yt)| t = 1, . . . , T} is chosen by Nature (or by an adversary), and importantly, we do not make any statistical assumptions about its source: our statements hold for all sequences. Our goal is to sequentially code the next label yt, given that we have observed x≤t and y<t (where x≤t and y<t denote the sequences {x1, . . . xt} and {y1, . . . yt−1}). At each time t, we have a conditional distribution P(·|x≤t, y<t) over Y, which is our prediction strategy that is used to predict the next variable yt. We then incur the instantaneous loss −log P(yt|x≤t, y<t) (referred to as log loss), and the cumulative loss is the sum of these instantaneous losses over t = 1, . . . , T. Let Θ be a parameter space indexing elementary prediction rules in some model class, where P(y|x, θ) for θ ∈Θ is a conditional distribution over Y called the likelihood. An expert is a single atom θ ∈Θ, or, more precisely, the algorithm which outputs the predictive distribution P(·|xt, θ) for every t. We are interested in bounds on the regret — the difference in the cumulative loss of a given adaptive prediction strategy and the the cumulative loss of the best possible expert chosen in hindsight from a subset of Θ. Kakade and Ng [2004] considered a parametric setting where Θ = Rd, X = Rd, and the prediction rules were generalized linear models, in which P(y|x, θ) = P(y|θ · x). They derived regret bounds for the Bayesian strategy (assuming a Gaussian prior over Θ), which showed that many simple Bayesian algorithms (such as Gaussian linear regression and logistic regression) perform favorably when compared, in retrospect, to the best θ ∈Θ. Importantly, these regret bounds have a time and dimensionality dependence of the form d 2 log T — a dependence common in in most MDL procedures (see Grunwald [2005]). For Gaussian linear regression, the bounds of Kakade and Ng [2004] are comparable to the best bounds in the literature, such as those of Foster [1991], Vovk [2001], Azoury and Warmuth [2001] (though these latter bounds are stated in terms of the closely related square loss). In this paper, we provide worst-case regret bounds on Bayesian non-parametric methods, which show how these algorithms can have low regret. In particular, we examine the case where the prior (over functions) is a Gaussian process — thereby extending the work of Kakade and Ng [2004] to infinite-dimensional spaces of experts. There are a number of important differences between this and the parametric setting. First, it turns out that the natural competitor class is the reproducing kernel Hilbert space (RKHS) H. Furthermore, the notion of dimensionality is more subtle, since the space H may be infinite dimensional. In general, there is no apriori reason that any strategy (including the Bayesian one) should be able to compete favorably with the complex class H. However, for some input sequences x≤T and kernels, we show that it is possible to compete favorably. Furthermore, the relation of our results to Kakade and Ng [2004] is made explicit in Section 3.2. Our second contribution is in making formal connections to minimax theory, where we show precisely how Bayesian Gaussian regression is a minimax algorithm. In a general setting, Shtarkov [1987] showed that a certain normalized maximum likelihood (NML) distribution minimizes the regret in the worst case. Unfortunately, for some “complex” model classes, there may exist no strategy which achieves finite regret, and so the NML distribution may not exist.1 Gaussian density estimation (formally described in Example 4.2) is one such case where this NML distribution does not exist. If one makes further restrictions (on Y), then minimax results can be derived, such as in Takimoto and Warmuth [2000], Barron et al. [1998], Foster and Stine [2001]. Instead of making further restrictions, we propose minimizing a form of a penalized regret, where one penalizes more “complex” experts as measured by their cost under a prior q(θ). This penalized regret essentially compares our cumulative loss to the loss of a two part code (common in MDL, see Grunwald [2005]), where one first codes the model θ under a prior q and then codes the data using this θ. Here, we show that a certain normalized maximum a posteriori distribution is the corresponding minimax strategy, in general. Our main result here is in showing that for Gaussian regression, the Bayesian strategy is precisely this minimax strategy. The differences between this result and that of Takimoto and Warmuth [2000] are notable. In the later, they assume Y ⊂R is bounded and derive (near) minimax algorithms which hold the variance of their predictions constant at each timestep (so they effectively deal with the square loss). Under Bayes rule, the variance of the predictions adapts, which allows the minimax property to hold with Y = R being unbounded. Other minimax results have been considered in the non-parametric setting. The work of Opper and Haussler [1998] and Cesa-Bianchi and Lugosi [2001] provide minimax bounds in some non-parametric cases (in terms of a covering number of the comparator class), though they do not consider input sequences. The rest of the paper is organized as follows: Section 2 summarizes our model, Section 3 presents and discusses our bounds, and Section 4 draws out the connections to the minimax and MDL framework. All proofs are available in a forthcoming longer version of this paper. 2 Bayesian Methods with Gaussian Process Priors With a Bayesian prior distribution Pbayes(θ) over Θ, the Bayesian predicts yt using the rule Pbayes(yt|x≤t, y<t) = Z P(yt|xt, θ)Pbayes(θ|x<t, y<t) dθ where the posterior is given by Pbayes(θ|x<t, y<t) ∝P(y<t|x<t, θ)Pbayes(θ). 1For these cases, the normalization constant of the NML distribution is not finite. Assuming the Bayesian learner models the data to be independent given θ, then P(y<t|x<t, θ) = t−1 Y t′=1 P(yt′|xt′, θ) . It is important to stress that these are “working assumptions” in the sense that they lead to a prediction strategy (the Bayesian one), but the analysis does not make any probabilistic assumptions about the generation of the data. The cumulative loss of the Bayesian strategy is then − T X t=1 log Pbayes(yt|x≤t, y<t) = −log Pbayes(y≤T |x≤T ). which follows form the chain rule of conditional probabilities. In this paper, we are interested in non-parametric prediction, which can be viewed as working with an infinite-dimensional function space Θ — assume Θ consists of real-valued functions u(x). The likelihood P(y|x, u(·)) is thus a distribution over y given x and the function u(·). Similar to Kakade and Ng [2004] (where they considered generalized linear models), we make the natural restriction that P(y|x, u(·)) = P(y|u(x)). We can think of u as a latent function and of P(y|u(x)) as a noise distribution. Two particularly important cases are that of Gaussian regression and logistic regression. In Gaussian regression, we have that Y = R and that P(y|u(x)) = N(y|u(x), σ2) (so y is distributed as a Gaussian with mean u(x) and fixed variance σ2). In logistic regression, Y = {−1, 1} and P(y|u(x)) = (1 + e−yu(x))−1. In this paper, we consider the case in which the prior dPbayes(u(·)) is a zero-mean Gaussian process (GP) with covariance function K, i.e. a real-valued random process which has the property that for every finite set x1, . . . , xn the random vector (u(x1), . . . , u(xn))T is multivariate Gaussian, distributed as N(0, K), where K ∈Rn,n is the covariance (or kernel) matrix with Ki,j = K(xi, xj). Note that K has to be a positive semidefinite function in that for all finite sets x1, . . . , xn the corresponding kernel matrices K are positive semidefinite. Finally, we specify the subset of experts we would like the Bayesian prediction strategy to compete against. Every positive semidefinite kernel K is associated with a unique reproducing kernel Hilbert space (RKHS) H, defined as follows: consider the linear space of all finite kernel expansions (over any x1, . . . , xn) of the form f(x) = Pn i=1 αiK(x, xi) with the inner product X i αiK(·, xi), X j βjK(·, yj) K = X i,j αiβjK(xi, yj). and define the RKHS H as the completion of this space. By construction, H contains all finite kernel expansions f(x) = Pn i=1 αiK(x, xi) with ∥f∥2 K = αT Kα, Ki,j = K(xi, xj) . (1) The characteristic property of H is that all (Dirac) evaluation functionals are represented in H itself by the functions K(·, xi), meaning (f, K(·, xi))K = f(xi). The RKHS H turns out to be the largest subspace of experts for which our results are meaningful. 3 Worst-Case Bounds In this section, we present our worst-case bounds, give an interpretation, and relate the results to the parametric case of Kakade and Ng [2004]. The proofs are available in a forthcoming longer version. Theorem 3.1: Let (x≤T , y≤T ) be a sequence from (X × Y)T . For all functions f in the RKHS H associated with the prior covariance function K, we have −log Pbayes(y≤T |x≤T ) ≤−log P(y≤T |x≤T , f(·)) + 1 2∥f∥2 K + 1 2 log |I + cK| , where ∥f∥K is the RKHS norm of f, K = (K(xt, xt′)) ∈RT,T is the kernel matrix over the input sequence x≤T , and c > 0 is a constant such that for all yt ∈y≤T , −d2 du2 log P(yt|u) ≤c for all u ∈R. The proof of this theorem parallels that provided by Kakade and Ng [2004], with a number of added complexities for handling GP priors. For the special case of Gaussian regression where c = σ−2, the following theorem shows the stronger result that the bound is satisfied with an equality for all sequences. Theorem 3.2: Assume P(yt|u(xt)) = N(yt|u(xt), σ2) and that Y = R. Let (x≤T , y≤T ) be a sequence from (X × Y)T . Then, −log Pbayes(y≤T |x≤T ) = min f∈H −log P(y≤T |x≤T , f(·)) + 1 2∥f∥2 K + 1 2 log I + σ−2K (2) and the minimum is attained for a kernel expansion over x≤T . This equality has important implications in our minimax theory (in Corollary 4.4, we make this precise). It is not hard to see that the equality does not hold for other likelihoods. 3.1 Interpretation The regret bound depends on two terms, ∥f∥2 K and log |I + cK|. We discuss each in turn. The dependence on ∥f∥2 K states the intuitive fact that a meaningful bound can only be obtained under smoothness assumptions on the set of experts. The more complicated f is (as measured by ∥· ∥K), the higher the regret may be. The equality shows in Theorem 3.2 shows this dependence is unavoidable. We come back to this dependence in Section 4. Let us now interpret the log |I + cK| term, which we refer to as the regret term. The constant c, which bounds the curvature of the likelihood, exists for most commonly used exponential family likelihoods. For logistic regression, we have c = 1/4, and for the Gaussian regression, we have c = σ−2. Also, interestingly, while f is an arbitrary function in H, this regret term depends on K only at the sequence points x≤T . For most infinite-dimensional kernels and without strong restrictions on the inputs, the regret term can be as large as Ω(T) — the sequence can be chosen s.t. K ≈c′I, which implies that log |I + cK| ≈T log(1 + cc′). For example, for an isotropic kernel (which is a function of the norm ∥x −x′∥2) we can choose the xt to be mutually far from each other. For kernels which barely enforce smoothness — e.g. the Ornstein-Uhlenbeck kernel exp(−b∥x −x′∥1) — the regret term can easily Ω(T). The cases we are interested in are those where the regret term is o(T), in which case the average regret tends to 0 with time. A spectral interpretation of this term helps us understand the behavior. If we let the λ1, λ2, . . . λT be the eigenvalues of K, then log |I + cK| = T X t=1 log(1 + cλt) ≤c tr K where tr K is the trace of K. This last quantity is closely related to the “degrees of freedom” in a system (see Hastie et al. [2001]). Clearly, if the sum of the eigenvalues has a sublinear growth rate of o(T), then the average regret tends to 0. Also, if one assumes that the input sequence, x≤T , is i.i.d. then the above eigenvalues are essentially the process eigenvalues. In a forthcoming longer version, we explore this spectral interpretation in more detail and provide a case using the exponential kernel in which the regret grows as O(poly(log T)). We now review the parametric case. 3.2 The Parametric Case Here we obtain a slight generalization of the result in Kakade and Ng [2004] as a special case. Namely, the familiar linear model — with u(x) = θ · x, θ, x ∈Rd and Gaussian prior θ ∼N(0, I) — can be seen as a GP model with the linear kernel: K(x, x′) = x · x′. With X = (x1, . . . xT )T we have that a kernel expansion f(x) = P i αixi ·x = θ ·x with θ = XTα, and ∥f∥2 K = αTXXTα = ∥θ∥2 2, so that H = {θ · x | θ ∈Rd}, and so log |I + cK| = log I + cXTX Therefore, our result gives an input-dependent version of the result of Kakade and Ng [2004]. If we make the further assumption that ∥x∥2 ≤1 (as done in Kakade and Ng [2004]), then we can obtain exactly their regret term: log |I + cK| ≤d log 1 + cT d which can seen by rotating K into an diagonal matrix and maximizing the expression subject to the constraint that ∥x∥2 ≤1 (i.e. that the eigenvalues must sum to 1). In general, this example shows that if K is a finite-dimension kernel such as the linear or the polynomial kernel, then the regret term is only O(log T). 4 Relationships to Minimax Procedures and MDL This section builds the framework for understanding the minimax property of Gaussian regression. We start by reviewing Shtarkov’s theorem, which shows that a certain normalized maximum likelihood density is the minimax strategy (when using the log loss). In many cases, this minimax strategy does not exist — in those cases where the minimax regret is infinite. We then propose a different, penalized notion of regret, and show that a certain normalized maximum a posteriori density is the minimax strategy here. Our main result (Corollary 4.4) shows that for Gaussian regression the Bayesian strategy is precisely this minimax strategy 4.1 Normalized Maximum Likelihood Here, let us assume that there are no inputs — sequences consist of only yt ∈Y. Given a measurable space with base measure µ, we employ a countable number of random variables yt in Y. Fix the sequence length T and define the model class F = {Q(·|θ) | θ ∈Θ)}, where Q(·|θ) denotes a joint probability density over YT with respect to µ. We assume that for our model class there exists a parameter, θml(y≤T ), maximizing the likelihood Q(y≤T |θ) over Θ for all y≤T ∈YT . We make this assumption to make the connections to maximum likelihood (and, later, MAP) estimation clear. Define the regret of a joint density P on y≤T as: R(y≤T , P, Θ) = −log P(y≤T ) −inf θ∈Θ{−log Q(y≤T |θ)} (3) = −log P(y≤T ) + log Q(y≤T |θml(y≤T )). (4) where the latter step uses our assumption on the existence of θml(y≤T ). Define the minimax regret with respect to Θ as: R(Θ) = inf P sup y≤T ∈YT R(y≤T , P, Θ) where the inf is over all probability densities on YT . The following theorem due to Shtarkov [1987] characterizes the minimax strategy. Theorem 4.1: [Shtarkov, 1987]If the following density exists (i.e. if it has a finite normalization constant), then define it to be the normalized maximum likelihood (NML) density. Pml(y≤T ) = Q(y≤T |θml(y≤T )) R Q(y≤T |θml(y≤T ))dµ(y≤T ) (5) If Pml exists, it is a minimax strategy, i.e. for all y≤T , the regret R(y≤T , Pml, Θ) does not exceed R(Θ). Note that this density exists only if the normalizing constant is finite, which is not the case in general. The proof is straightforward using the fact that the NML density is an equalizer — meaning that it has constant regret on all sequences. Proof: First note that the regret R(y≤T , Pml, Θ) is the constant log R Q(y≤T |θml(y≤T ))dµ(y≤T ). To see this, simply substitute Eq. 5 into Eq. 4 and simplify. For convenience, define the regret of any P as R(P, Θ) = supy≤T ∈YT R(y≤T , P, Θ). For any P ̸= Pml (differing on a set with positive measure), there exists some y≤T such that P(y≤T ) < Pml(y≤T ), since the densities are normalized. This implies that R(P, Θ) ≥R(y≤T , P, Θ) > R(y≤T , Pml, Θ) = R(Pml, Θ) where the first step follows from the definition of R(P, Θ), the second from −log P(y≤T ) > −log Pml(y≤T ), and the last from the fact that Pml is an equalizer (its regret is constant on all sequences). Hence, P has a strictly larger regret, implying that Pml is the unique minimax strategy. □ Unfortunately, in many important model classes, the minimax regret R(Θ) is not finite, and the NML density does not exist. We now provide one example (see Grunwald [2005] for further discussion). Example 4.2: Consider a model which assumes the sequence is generated i.i.d. from a Gaussian with unknown mean and unit variance. Specifically, let Θ = R, Y = R, and P(y≤T |θ) be the product ΠT t=1N(yt; θ, 1). It is easy to see that for this class the minimax regret is infinite and Pml does not exist (see Grunwald [2005]). This example can be generalized to the Gaussian regression model (if we know the sequence x≤T in advance). For this problem, if one modifies the space of allowable sequences (i.e. YT is modified), then one can obtain finite regret, such as those in Barron et al. [1998], Foster and Stine [2001]. This technique may not be appropriate in general. 4.2 Normalized Maximum a Posteriori To remedy this problem, consider placing some structure on the model class F = {Q(·|θ)|θ ∈Θ}. The idea is to penalize Q(·|θ) ∈F based on this structure. The motivation is similar to that of structural risk minimization [Vapnik, 1998]. Assume that Θ is a measurable space and place a prior distribution with density function q on Θ. Define the penalized regret of P on y≤T as: Rq(y≤T , P, Θ) = −log P(y≤T ) −inf θ∈Θ{−log Q(y≤T |θ) −log q(θ)} . Note that −log Q(y≤T |θ) −log q(θ) can be viewed as a “two part” code, in which we first code θ under the prior q and then code y≤T under the likelihood Q(·|θ). Unlike the standard regret, the penalized regret can be viewed as a comparison to an actual code. These two part codes are common in the MDL literature (see Grunwald [2005]). However, in MDL, they consider using minimax schemes (via Pml) for the likelihood part of the code, while we consider minimax schemes for this penalized regret. Again, for clarity, assume there exists a parameter, θmap(y≤T ) maximizing log Q(y≤T |θ)+ log q(θ). Notice that this is just the maximum aposteriori (MAP) parameter, if one were to use a Bayesian strategy with the prior q (since the posterior density would be proportional to Q(y≤T |θ)q(θ)). Here, Rq(y≤T , P, Θ) = −log P(y≤T ) + log Q(y≤T |θmap(y≤T )) + log q(θmap(y≤T )) Similarly, with respect to Θ, define the minimax penalized regret as: Rq(Θ) = inf P sup y≤T ∈YT Rq(y≤T P, Θ) where again the inf is over all densities on YT . If Θ is finite or countable and Q(·|θ) > 0 for all θ, then the Bayes procedure has the desirable property of having penalized regret which is non-positive.2 However, in general, the Bayes procedure does not achieve the minimax penalized regret, Rq(Θ), which is what we desire — though, for one case, we show that it does (in the next section). We now characterize this minimax strategy in general. Theorem 4.3: Define the normalized maximum a posteriori (NMAP) density, if it exists, as: Pmap(y≤T ) = Q(y≤T |θmap(y≤T ))q(θmap(y≤T )) R Q(y≤T |θmap(y≤T ))q(θmap(y≤T )) dµ(y≤T ) . (6) If Pmap exists, it is a minimax strategy for the penalized regret, i.e. for all y≤T , the penalized regret Rq(y≤T , Pmap, Θ) does not exceed Rq(Θ). The proof relies on Pmap being an equalizer for the penalized regret and is identical to that of Theorem 4.1 — just replace all quantities with their penalized equivalents. 4.3 Bayesian Gaussian Regression as a Minimax Procedure We now return to the setting with inputs and show how the Bayesian strategy for the Gaussian regression model is a minimax strategy for all input sequences x≤T . If we fix the input sequence x≤T , we can consider the competitor class to be F = {P(y≤T |x≤T , θ) | θ ∈ Θ)}. In other words, we make the more stringent comparison against a model class which has full knowledge of the input sequence in advance. Importantly, note that the learner only observes the past inputs x<t at time t. Consider the Gaussian regression model, with likelihood P(y≤T |x≤T , u(·)) = N(y≤T |u(x≤T ), σ2I), where u(·) is some function and I is the T × T identity. For 2To see this, simply observe that Pbayes(y≤T ) = P θ Q(y≤T |θ)q(θ) ≥ Q(y≤T |θmap(y≤T ))q(θmap(y≤T )) and take the −log of both sides. technical reasons, we do not define the class of competitor functions Θ to be the RKHS H, but instead define Θ = {u(·)| u(x) = PT t=1 αtK(x, xt), α ∈RT } — the set of kernel expansions over x≤T . The model class is then F = {P(·|x≤T , u(·)) | u ∈Θ}. The representer theorem implies that competing against Θ is equivalent to competing against the RKHS. It is easy to see that for this case, the NML density does not exist (recall Example 4.2) — the comparator class Θ contains very complex functions. However, the case is quite different for the penalized regret. Now let us consider using a GP prior. We choose q to be the corresponding density over Θ, which means that q(u) is proportional to exp(−∥u∥2 K/2), where ∥u∥2 K = αT Kα with Ki,j = K(xi, xj) (recall Eq. 1). Now note that the penalty −log q(u) is just the RKHS norm ∥u∥2 K/2, up to an additive constant. Using Theorem 4.3 and the equality in Theorem 3.2, we have the following corollary, which shows that the Bayesian strategy is precisely the NMAP distribution (for Gaussian regression). Corollary 4.4: For any x≤T , in the Gaussian regression setting described above — where F and Θ are defined with respect to x≤T and where q is the GP prior over Θ — we have that Pbayes is a minimax strategy for the penalized regret, i.e. for all y≤T , the regret Rq(y≤T , Pbayes, Θ) does not exceed Rq(Θ). Furthermore, Pbayes and Pmap are densities of the same distribution. Importantly, note that, while the competitor class F is constructed with full knowledge of x≤T in advance, the Bayesian strategy, Pbayes, can be implemented in an online manner in that it only needs to know x<t for prediction at time t. Acknowledgments We thank Manfred Opper and Manfred Warmuth for helpful discussions. References K. S. Azoury and M. Warmuth. Relative loss bounds for on-line density estimation with the exponential family of distributions. Machine Learning, 43(3), 2001. A. Barron, J. Rissanen, and B. Yu. The minimum description length principle in coding and modeling. IEEE Trans. Information Theory, 44, 1998. Nicolo Cesa-Bianchi and Gabor Lugosi. Worst-case bounds for the logarithmic loss of predictors. Machine Learning, 43, 2001. D. P. Foster. Prediction in the worst case. Annals of Statistics, 19, 1991. D. P. Foster and R. A. Stine. The competitive complexity ratio. Proceedings of 2001 Conf on Info Sci and Sys, WP8, 2001. P.D. Grunwald. A tutorial introduction to the minimum description length principle. Advances in MDL: Theory and Applications, 2005. T. Hastie, R. Tibshirani, , and J. Friedman. The Elements of Statistical Learning. Springer, 2001. S. M. Kakade and A. Y. Ng. Online bounds for bayesian algorithms. Proceedings of Neural Information Processing Systems, 2004. M. Opper and D. Haussler. Worst case prediction over sequences under log loss. The Mathematics of Information Coding, Extraction and Distribution, 1998. Y. Shtarkov. Universal sequential coding of single messages. Problems of Information Transmission, 23, 1987. E. Takimoto and M. Warmuth. The minimax strategy for Gaussian density estimation. Proc. 13th Annu. Conference on Comput. Learning Theory, 2000. Vladimir N. Vapnik. Statistical Learning Theory. Wiley, 1st edition, 1998. V. Vovk. Competitive on-line statistics. International Statistical Review, 69, 2001.
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Analyzing Auditory Neurons by Learning Distance Functions Inna Weiner1 Tomer Hertz1,2 Israel Nelken2,3 Daphna Weinshall1,2 1School of Computer Science and Engineering, 2The Center for Neural Computation, 3Department of Neurobiology, The Hebrew University of Jerusalem, Jerusalem, Israel, 91904 weinerin,tomboy,daphna@cs.huji.ac.il,israel@md.huji.ac.il Abstract We present a novel approach to the characterization of complex sensory neurons. One of the main goals of characterizing sensory neurons is to characterize dimensions in stimulus space to which the neurons are highly sensitive (causing large gradients in the neural responses) or alternatively dimensions in stimulus space to which the neuronal response are invariant (defining iso-response manifolds). We formulate this problem as that of learning a geometry on stimulus space that is compatible with the neural responses: the distance between stimuli should be large when the responses they evoke are very different, and small when the responses they evoke are similar. Here we show how to successfully train such distance functions using rather limited amount of information. The data consisted of the responses of neurons in primary auditory cortex (A1) of anesthetized cats to 32 stimuli derived from natural sounds. For each neuron, a subset of all pairs of stimuli was selected such that the responses of the two stimuli in a pair were either very similar or very dissimilar. The distance function was trained to fit these constraints. The resulting distance functions generalized to predict the distances between the responses of a test stimulus and the trained stimuli. 1 Introduction A major challenge in auditory neuroscience is to understand how cortical neurons represent the acoustic environment. Neural responses to complex sounds are idiosyncratic, and small perturbations in the stimuli may give rise to large changes in the responses. Furthermore, different neurons, even with similar frequency response areas, may respond very differently to the same set of stimuli. The dominant approach to the functional characterization of sensory neurons attempts to predict the response of the cortical neuron to a novel stimulus. Prediction is usually estimated from a set of known responses of a given neuron to a set of stimuli (sounds). The most popular approach computes the spectrotemporal receptive field (STRF) of each neuron, and uses this linear model to predict neuronal responses. However, STRFs have been recently shown to have low predictive power [10, 14]. In this paper we take a different approach to the characterization of auditory cortical neurons. Our approach attempts to learn the non-linear warping of stimulus space that is induced by the neuronal responses. This approach is motivated by our previous observations [3] that different neurons impose different partitions of the stimulus space, which are not necessarily simply related to the spectro-temporal structure of the stimuli. More specifically, we characterize a neuron by learning a pairwise distance function over the stimulus domain that will be consistent with the similarities between the responses to different stimuli, see Section 2. Intuitively a good distance function would assign small values to pairs of stimuli that elicit a similar neuronal response, and large values to pairs of stimuli that elicit different neuronal responses. This approach has a number of potential advantages: First, it allows us to aggregate information from a number of neurons, in order to learn a good distance function even when the number of known stimuli responses per neuron is small, which is a typical concern in the domain of neuronal characterization. Second, unlike most functional characterizations that are limited to linear or weakly non-linear models, distance learning can approximate functions that are highly non-linear. Finally, we explicitly learn a distance function on stimulus space; by examining the properties of such a function, it may be possible to determine the stimulus features that most strongly influence the responses of a cortical neuron. While this information is also implicitly incorporated into functional characterizations such as the STRF, it is much more explicit in our new formulation. In this paper we therefore focus on two questions: (1) Can we learn distance functions over the stimulus domain for single cells using information extracted from their neuronal responses?? and (2) What is the predictive power of these cell specific distance functions when presented with novel stimuli? In order to address these questions we used extracellular recordings from 22 cells in the auditory cortex of cats in response to natural bird chirps and some modified versions of these chirps [1]. To estimate the distance between responses, we used a normalized distance measure between the peri-stimulus time histograms of the responses to the different stimuli. Our results, described in Section 4, show that we can learn compatible distance functions on the stimulus domain with relatively low training errors. This result is interesting by itself as a possible characterization of cortical auditory neurons, a goal which eluded many previous studies [3]. Using cross validation, we measure the test error (or predictive power) of our method, and report generalization power which is significantly higher than previously reported for natural stimuli [10]. We then show that performance can be further improved by learning a distance function using information from pairs of related neurons. Finally, we show better generalization performance for wide-band stimuli as compared to narrow-band stimuli. These latter two contributions may have some interesting biological implications regarding the nature of the computations done by auditory cortical neurons. Related work Recently, considerable attention has been focused on spectrotemporal receptive fields (STRFs) as characterizations of the function of auditory cortical neurons [8, 4, 2, 11, 16]. The STRF model is appealing in several respects: it is a conceptually simple model that provides a linear description of the neuron’s behavior. It can be interpreted both as providing the neuron’s most efficient stimulus (in the time-frequency domain), and also as the spectro-temporal impulse response of the neuron [10, 12]. Finally, STRFs can be efficiently estimated using simple algebraic techniques. However, while there were initial hopes that STRFs would uncover relatively complex response properties of cortical neurons, several recent reports of large sets of STRFs of cortical neurons concluded that most STRFs are somewhat too simple [5], and that STRFs are typically rather sluggish in time, therefore missing the highly precise synchronization of some cortical neurons [11]. Furthermore, when STRFs are used to predict neuronal responses to natural stimuli they often fail to predict the correct responses [10, 6]. For example, in Machens et al. only 11% of the response power could be predicted by STRFs on average [10]. Similar results were also reported in [14], who found that STRF models account for only 18 −40% (on average) of the stimulus related power in auditory cortical neural responses to dynamic random chord stimuli. Various other studies have shown that there are significant and relevant non-linearities in auditory cortical responses to natural stimuli [13, 1, 9, 10]. Using natural sounds, Bar-Yosef et. al [1] have shown that auditory neurons are extremely sensitive to small perturbations in the (natural) acoustic context. Clearly, these non-linearities cannot be sufficiently explained using linear models such as the STRF. 2 Formalizing the problem as a distance learning problem Our approach is based on the idea of learning a cell-specific distance function over the space of all possible stimuli, relying on partial information extracted from the neuronal responses of the cell. The initial data consists of stimuli and the resulting neural responses. We use the neuronal responses to identify pairs of stimuli to which the neuron responded similarly and pairs to which the neuron responded very differently. These pairs can be formally described by equivalence constraints. Equivalence constraints are relations between pairs of datapoints, which indicate whether the points in the pair belong to the same category or not. We term a constraint positive when they points are known to originate from the same class, and negative belong to different classes. In this setting the goal of the algorithm is to learn a distance function that attempts to comply with the equivalence constraints. This formalism allows us to combine information from a number of cells to improve the resulting characterization. Specifically, we combine equivalence constraints gathered from pairs of cells which have similar responses, and train a single distance function for both cells. Our results demonstrate that this approach improves prediction results of the “weaker” cell, and almost always improves the result of the “stronger” cell in each pair. Another interesting result of this formalism is the ability to classify stimuli based on the responses of the total recorded cortical cell ensemble. For some stimuli, the predictive performance based on the learned inter-stimuli distance was very good, whereas for other stimuli it was rather poor. These differences were correlated with the acoustic structure of the stimuli, partitioning them into narrowband and wideband stimuli. 3 Methods Experimental setup Extracellular recordings were made in primary auditory cortex of nine halothane-anesthetized cats. Anesthesia was induced by ketamine and xylazine and maintained with halothane (0.25-1.5%) in 70% N2O using standard protocols authorized by the committee for animal care and ethics of the Hebrew University - Haddasah Medical School. Single neurons were recorded using metal microelectrodes and an online spike sorter (MSD, alpha-omega). All neurons were well separated. Penetrations were performed over the whole dorso-ventral extent of the appropriate frequency slab (between about 2 and 8 kHz). Stimuli were presented 20 times using sealed, calibrated earphones at 60-80 dB SPL, at the preferred aurality of the neurons as determined using broad-band noise bursts. Sounds were taken from the Cornell Laboratory of Ornithology and have been selected as in [1]. Four stimuli, each of length 60-100 ms, consisted of a main tonal component with frequency and amplitude modulation and of a background noise consisting of echoes and unrelated components. Each of these stimuli was further modified by separating the main tonal component from the noise, and by further separating the noise into echoes and background. All possible combinations of these components were used here, in addition to a stylized artificial version that lacked the amplitude modulation of the natural sound. In total, 8 versions of each stimulus were used, and therefore each neuron had a dataset consisting of 32 datapoints. For more detailed methods, see Bar-Yosef et al. [1]. Data representation We used the first 60 ms of each stimulus. Each stimulus was represented using the first d real Cepstral coefficients. The real Cepstrum of a signal x was calculated by taking the natural logarithm of magnitude of the Fourier transform of x and then computing the inverse Fourier transform of the resulting sequence. In our experiments we used the first 21-30 coefficients. Neuronal responses were represented by creating PeriStimulus Time Histograms (PSTHs) using 20 repetitions recorded for each stimuli. Response duration was 100 ms. Obtaining equivalence constraints over stimuli pairs The distances between responses were measured using a normalized χ2 distance measure. All responses to both stimuli (40 responses in total) were superimposed to generate a single high-resolution PSTH. Then, this PSTH was non-uniformly binned so that each bin contained at least 10 spikes. The same bins were then used to generate the PSTHs of the responses to the two stimuli separately. For similar responses, we would expect that on average each bin in these histograms would contain 5 spikes. Formally, let N denote the number of bins in each histogram, and let ri 1,ri 2 denote the number of spikes in the i’th bin in each of the two histograms respectively. The distance between pairs of histograms is given by: χ2(ri 1, ri 2) = PN i=1 (ri 1−ri 2)2 (ri 1+ri 2)/2/(N −1). In order to identify pairs (or small groups) of similar responses, we computed the normalized χ2 distance matrix over all pairs of responses, and used the complete-linkage algorithm to cluster the responses into 8 −12 clusters. All of the points in each cluster were marked as similar to one another, thus providing positive equivalence constraints. In order to obtain negative equivalence constraints, for each cluster ci we used the 2−3 furthest clusters from it to define negative constraints. All pairs, composed of a point from cluster ci and another point from these distant clusters, were used as negative constraints. Distance learning method In this paper, we use the DistBoost algorithm [7], which is a semi-supervised boosting learning algorithm that learns a distance function using unlabeled datapoints and equivalence constraints. The algorithm boosts weak learners which are soft partitions of the input space, that are computed using the constrained ExpectationMaximization (cEM) algorithm [15]. The DistBoost algorithm, which is briefly summarized in 1, has been previously used in several different applications and has been shown to perform well [7, 17]. Evaluation methods In order to evaluate the quality of the learned distance function, we measured the correlation between the distances computed by our distance learning algorithm to those induced by the χ2 distance over the responses. For each stimulus we measured the distances to all other stimuli using the learnt distance function. We then computed the rank-order (Spearman) correlation coefficient between these learnt distances in the stimulus domain and the χ2 distances between the appropriate responses. This procedure produced a single correlation coefficient for each of the 32 stimuli, and the average correlation coefficient across all stimuli was used as the overall performance measure. Parameter selection The following parameters of the DistBoost algorithm can be finetuned: (1) the input dimensionality d = 21-30, (2) the number of Gaussian models in each weak learner M = 2-4, (3) the number of clusters used to extract equivalence constraints C = 8-12, and (4) the number of distant clusters used to define negative constraints numAnti = 2-3. Optimal parameters were determined separately for each of the 22 cells, based solely on the training data. Specifically, in the cross-validation testing we used a validation paradigm: Using the 31 training stimuli, we removed an additional datapoint and trained our algorithm on the remaining 30 points. We then validated its performance using the left out datapoint. The optimal cell specific parameters were determined using this approach. Algorithm 1 The DistBoost Algorithm Input: Data points: (x1, ..., xn), xk ∈X A set of equivalence constraints: (xi1, xi2, yi), where yi ∈{−1, 1} Unlabeled pairs of points: (xi1, xi2, yi = ∗), implicitly defined by all unconstrained pairs of points • Initialize W 1 i1i2 = 1/(n2) i1, i2 = 1, . . . , n (weights over pairs of points) wk = 1/n k = 1, . . . , n (weights over data points) • For t = 1, .., T 1. Fit a constrained GMM (weak learner) on weighted data points in X using the equivalence constraints. 2. Generate a weak hypothesis ˜ht : X × X → [−1, 1] and define a weak distance function as ht(xi, xj) = 1 2 “ 1 −˜ht(xi, xj) ” ∈[0, 1] 3. Compute rt = P (xi1 ,xi2 ,yi=±1) W t i1i2yiht(xi1, xi2), only over labeled pairs. Accept the current hypothesis only if rt > 0. 4. Choose the hypothesis weight αt = 1 2 ln( 1+rt 1−rt ) 5. Update the weights of all points in X × X as follows: W t+1 i1i2 = ( W t i1i2 exp(−αtyi˜ht(xi1, xi2)) yi ∈{−1, 1} W t i1i2 exp(−αt) yi = ∗ 6. Normalize: W t+1 i1i2 = W t+1 i1i2 n P i1,i2=1 W t+1 i1i2 7. Translate the weights from X × X to X: wt+1 k = P j W t+1 kj Output: A final distance function D(xi, xj) = PT t=1 αtht(xi, xj) 4 Results Cell-specific distance functions We begin our analysis with an evaluation of the fitting power of the method, by training with the entire set of 32 stimuli (see Fig. 1). In general almost all of the correlation values are positive and they are quite high. The average correlation over all cells is 0.58 with ste = 0.023. In order to evaluate the generalization potential of our approach, we used a Leave-OneOut (LOU) cross-validation paradigm. In each run, we removed a single stimulus from the dataset, trained our algorithm on the remaining 31 stimuli, and then tested its performance on the datapoint that was left out (see Fig. 3). In each histogram we plot the test correlations of a single cell, obtained when using the LOU paradigm over all of the 32 stimuli. As can be seen, on some cells our algorithm obtains correlations that are as high as 0.41, while for other cells the average test correlation is less then 0.1. The average correlation over all cells is 0.26 with ste = 0.019. Not surprisingly, the train results (Fig. 1) are better than the test results (Fig. 3). Interestingly, however, we found that there was a significant correlation between the training performance and the test performance C = 0.57, p < 0.05 (see Fig. 2, left). Boosting the performance of weak cells In order to boost the performance of cells with low average correlations, we constructed the following experiment: We clustered the responses of each cell, using the complete-linkage algorithm over the χ2 distances with 4 clusters. We then used the F 1 2 score that evaluates how well two clustering partitions are in agreement with one another (F 1 2 = 2∗P ∗R P +R , where P denotes precision and R denotes recall.). This measure was used to identify pairs of cells whose partition of the stimuli was most similar to each other. In our experiment we took the four cells with the lowest −1 −0.5 0 0.5 1 All cells −1 −0.5 0 0.5 1 0 5 10 15 20 25 30 Cell 13 −1 −0.5 0 0.5 1 0 5 10 15 20 25 30 Cell 18 Figure 1: Left: Histogram of train rank-order correlations on the entire ensemble of cells. The rank-order correlations were computed between the learnt distances and the distances between the recorded responses for each single stimulus (N = 22 ∗32). Center: train correlations for a “strong” cell. Right: train correlations for a “weak” cell. Dotted lines represent average values. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 Train correlation Test correlation 16 20 18 14 25 38 37 19 0 0.1 0.2 0.3 0.4 0.5 Cell number Mean Test Rank−Order Correlation Original constraints Intersection of const. Figure 2: Left: Train vs. test cell specific correlations. Each point marks the average correlation of a single cell. The correlation between train and test is 0.57 with p = 0.05. The distribution of train and test correlations is displayed as histograms on the top and on the right respectively. Right: Test rankorder correlations when training using constraints extracted from each cell separately, and when using the intersection of the constraints extracted from a pair of cells. This procedure always improves the performance of the weaker cell, and usually also improves the performance of the stronger cell performance (right column of Fig 3), and for each of them used the F 1 2 score to retrieve the most similar cell. For each of these pairs, we trained our algorithm once more, using the constraints obtained by intersecting the constraints derived from the two cells in the pair, in the LOU paradigm. The results can be seen on the right plot in Fig 2. On all four cells, this procedure improved LOUT test results. Interestingly and counter-intuitively, when training the better performing cell in each pair using the intersection of its constraints with those from the poorly performing cell, results deteriorated only for one of the four better performing cells. Stimulus classification The cross-validation results induced a partition of the stimulus space into narrowband and wideband stimuli. We measured the predictability of each stimulus by averaging the LOU test results obtained for the stimulus across all cells (see Fig. 4). Our analysis shows that wideband stimuli are more predictable than narrowband stimuli, despite the fact that the neuronal responses to these two groups are not different as a whole. Whereas the non-linearity in the interactions between narrowband and wideband stimuli has already been noted before [9], here we further refine this observation by demonstrating a significant difference between the behavior of narrow and wideband stimuli with respect to the predictability of the similarity between their responses. 5 Discussion In the standard approach to auditory modeling, a linear or weakly non-linear model is fitted to the data, and neuronal properties are read from the resulting model. The usefulness of this approach is limited however by the weak predictability of A1 responses when using such models. In order to overcome this limitation, we reformulated the problem of char−1 −0.5 0 0.5 1 0 5 10 15 Cell 49 −1 −0.5 0 0.5 1 0 5 10 15 Cell 15 −1 −0.5 0 0.5 1 0 5 10 15 Cell 52 −1 −0.5 0 0.5 1 0 5 10 15 Cell 2 −1 −0.5 0 0.5 1 0 5 10 15 Cell 25 −1 −0.5 0 0.5 1 0 5 10 15 Cell 37 −1 −0.5 0 0.5 1 0 5 10 15 Cell 38 −1 −0.5 0 0.5 1 0 5 10 15 Cell 24 −1 −0.5 0 0.5 1 0 5 10 15 Cell 19 −1 −0.5 0 0.5 1 0 5 10 15 Cell 3 −1 −0.5 0 0.5 1 0 5 10 15 Cell 1 −1 −0.5 0 0.5 1 0 5 10 15 Cell 16 −1 −0.5 0 0.5 1 0 5 10 15 Cell 54 −1 −0.5 0 0.5 1 0 5 10 15 Cell 20 −1 −0.5 0 0.5 1 0 5 10 15 Cell 13 −1 −0.5 0 0.5 1 0 5 10 15 Cell 17 −1 −0.5 0 0.5 1 0 5 10 15 Cell 51 −1 −0.5 0 0.5 1 0 5 10 15 Cell 18 −1 −0.5 0 0.5 1 0 5 10 15 Cell 14 −1 −0.5 0 0.5 1 0 5 10 15 Cell 36 −1 −0.5 0 0.5 1 0 5 10 15 Cell 21 −1 −0.5 0 0.5 1 0 5 10 15 Cell 48 −1 −0.5 0 0.5 1 All cells Figure 3: Histograms of cell specific test rank-order correlations for the 22 cells in the dataset. The rank-order correlations compare the predicted distances to the distances between the recorded responses, measured on a single stimulus which was left out during the training stage. For visualization purposes, cells are ordered (columns) by their average test correlation per stimulus in descending order. Negative correlations are in yellow, positive in blue. acterizing neuronal responses of highly non-linear neurons. We use the neural data as a guide for training a highly non-linear distance function on stimulus space, which is compatible with the neural responses. The main result of this paper is the demonstration of the feasibility of this approach. Two further results underscore the usefulness of the new formulation. First, we demonstrated that we can improve the test performance of a distance function by using constraints on the similarity or dissimilarity between stimuli derived from the responses of multiple neurons. Whereas we expected this manipulation to improve the test performance of the algorithm on the responses of neurons that were initially poorly predicted, we found that it actually improved the performance of the algorithm also on neurons that were rather well predicted, although we paired them with neurons that were poorly predicted. Thus, it is possible that intersecting constraints derived from multiple neurons uncover regularities that are hard to extract from individual neurons. Second, it turned out that some stimuli consistently behaved better than others across the neuronal population. This difference was correlated with the acoustic structure of the stimuli: those stimuli that contained the weak background component (wideband stimuli) were generally predicted better. This result is surprising both because background component is substantially weaker than the other acoustic components in the stimuli (by as much as 35-40 dB). It may mean that the relationships between physical structure (as characterized by the Cepstral parameters) and the neuronal responses becomes simpler in the presence of the background component, but is much more idiosyncratic when this component is absent. This result underscores the importance of interactions between narrow and wideband stimuli for understanding the complexity of cortical processing. The algorithm is fast enough to be used in near real-time. It can therefore be used to guide real experiments. One major problem during an experiment is that of stimulus selection: choosing the best set of stimuli for characterizing the responses of a neuron. The distance functions trained here can be used to direct this process. For example, they can be used to Time (ms) Frequency (kHz) Main 50 100 0 10 20 Time (ms) Frequency (kHz) Natural 50 100 0 10 20 Time (ms) Frequency (kHz) Echo 50 100 0 10 20 Time (ms) Frequency (kHz) Background 50 100 0 10 20 Narrowband Wideband 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 A1 Stimuli Mean Test Rank−Order Correlation Narrow band Wide band Figure 4: Left: spectrograms of input stimuli, which are four different versions of a single natural bird chirp. Right: Stimuli specific correlation values averaged over the entire ensemble of cells. The predictability of wideband stimuli is clearly better than that of the narrowband stimuli. find surprising stimuli: either stimuli that are very different in terms of physical structure but that would result in responses that are similar to those already measured, or stimuli that are very similar to already tested stimuli but that are predicted to give rise to very different responses. References [1] O. Bar-Yosef, Y. Rotman, and I. Nelken. Responses of Neurons in Cat Primary Auditory Cortex to Bird Chirps: Effects of Temporal and Spectral Context. J. Neurosci., 22(19):8619–8632, 2002. [2] D. T. Blake and M. M. Merzenich. Changes of AI Receptive Fields With Sound Density. J Neurophysiol, 88(6):3409–3420, 2002. [3] G. Chechik, A. Globerson, M.J. Anderson, E.D. Young, I. Nelken, and N. Tishby. Group redundancy measures reveal redundancy reduction in the auditory pathway. In NIPS, 2002. [4] R. C. deCharms, D. T. Blake, and M. M. Merzenich. Optimizing Sound Features for Cortical Neurons. Science, 280(5368):1439–1444, 1998. [5] D. A. Depireux, J. Z. Simon, D. J. Klein, and S. A. Shamma. Spectro-Temporal Response Field Characterization With Dynamic Ripples in Ferret Primary Auditory Cortex. J Neurophysiol, 85(3):1220–1234, 2001. [6] J. J. Eggermont, P. M. Johannesma, and A. M. Aertsen. Reverse-correlation methods in auditory research. Q Rev Biophys., 16(3):341–414, 1983. [7] T. Hertz, A. Bar-Hillel, and D. Weinshall. Boosting margin based distance functions for clustering. In ICML, 2004. [8] N. Kowalski, D. A. Depireux, and S. A. Shamma. Analysis of dynamic spectra in ferret primary auditory cortex. I. Characteristics of single-unit responses to moving ripple spectra. J Neurophysiol, 76(5):3503–3523, 1996. [9] L. Las, E. A. Stern, and I. Nelken. Representation of Tone in Fluctuating Maskers in the Ascending Auditory System. J. Neurosci., 25(6):1503–1513, 2005. [10] C. K. Machens, M. S. Wehr, and A. M. Zador. Linearity of Cortical Receptive Fields Measured with Natural Sounds. J. Neurosci., 24(5):1089–1100, 2004. [11] L. M. Miller, M. A. Escabi, H. L. Read, and C. E. Schreiner. Spectrotemporal Receptive Fields in the Lemniscal Auditory Thalamus and Cortex. J Neurophysiol, 87(1):516–527, 2002. [12] I. Nelken. Processing of complex stimuli and natural scenes in the auditory cortex. Current Opinion in Neurobiology, 14(4):474–480, 2004. [13] Y. Rotman, O. Bar-Yosef, and I. Nelken. Relating cluster and population responses to natural sounds and tonal stimuli in cat primary auditory cortex. Hearing Research, 152(1-2):110–127, 2001. [14] M. Sahani and J. F. Linden. How linear are auditory cortical responses? In NIPS, 2003. [15] N. Shental, A. Bar-Hilel, T. Hertz, and D. Weinshall. Computing Gaussian mixture models with EM using equivalence constraints. In NIPS, 2003. [16] F. E. Theunissen, K. Sen, and A. J. Doupe. Spectral-Temporal Receptive Fields of Nonlinear Auditory Neurons Obtained Using Natural Sounds. J. Neurosci., 20(6):2315–2331, 2000. [17] C. Yanover and T. Hertz. Predicting protein-peptide binding affinity by learning peptide-peptide distance functions. In RECOMB, 2005.
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Cyclic Equilibria in Markov Games Martin Zinkevich and Amy Greenwald Department of Computer Science Brown University Providence, RI 02912 {maz,amy}@cs.brown.edu Michael L. Littman Department of Computer Science Rutgers, The State University of NJ Piscataway, NJ 08854–8019 mlittman@cs.rutgers.edu Abstract Although variants of value iteration have been proposed for finding Nash or correlated equilibria in general-sum Markov games, these variants have not been shown to be effective in general. In this paper, we demonstrate by construction that existing variants of value iteration cannot find stationary equilibrium policies in arbitrary general-sum Markov games. Instead, we propose an alternative interpretation of the output of value iteration based on a new (non-stationary) equilibrium concept that we call “cyclic equilibria.” We prove that value iteration identifies cyclic equilibria in a class of games in which it fails to find stationary equilibria. We also demonstrate empirically that value iteration finds cyclic equilibria in nearly all examples drawn from a random distribution of Markov games. 1 Introduction Value iteration (Bellman, 1957) has proven its worth in a variety of sequential-decisionmaking settings, most significantly single-agent environments (Puterman, 1994), team games, and two-player zero-sum games (Shapley, 1953). In value iteration for Markov decision processes and team Markov games, the value of a state is defined to be the maximum over all actions of the value of the combination of the state and action (or Q value). In zero-sum environments, the max operator becomes a minimax over joint actions of the two players. Learning algorithms based on this update have been shown to compute equilibria in both model-based scenarios (Brafman & Tennenholtz, 2002) and Q-learning-like model-free scenarios (Littman & Szepesv´ari, 1996). The theoretical and empirical success of such algorithms has led researchers to apply the same approach in general-sum games, in spite of exceedingly weak guarantees of convergence (Hu & Wellman, 1998; Greenwald & Hall, 2003). Here, value-update rules based on select Nash or correlated equilibria have been evaluated empirically and have been shown to perform reasonably in some settings. None has been identified that computes equilibria in general, however, leaving open the question of whether such an update rule is even possible. Our main negative theoretical result is that an entire class of value-iteration update rules, including all those mentioned above, can be excluded from consideration for computing stationary equilibria in general-sum Markov games. Briefly, existing value-iteration algorithms compute Q values as an intermediate result, then derive policies from these Q values. We demonstrate a class of games in which Q values, even those corresponding to an equilibrium policy, contain insufficient information for reconstructing an equilibrium policy. Faced with the impossibility of developing algorithms along the lines of traditional value iteration that find stationary equilibria, we suggest an alternative equilibrium concept— cyclic equilibria. A cyclic equilibrium is a kind of non-stationary joint policy that satisfies the standard conditions for equilibria (no incentive to deviate unilaterally). However, unlike conditional non-stationary policies such as tit-for-tat and finite-state strategies based on the “folk theorem” (Osborne & Rubinstein, 1994), cyclic equilibria cycle rigidly through a set of stationary policies. We present two positive results concerning cyclic equilibria. First, we consider the class of two-player two-state two-action games used to show that Q values cannot reconstruct all stationary equilibrium. Section 4.1 shows that value iteration finds cyclic equilibria for all games in this class. Second, Section 5 describes empirical results on a more general set of games. We find that on a significant fraction of these games, value iteration updates fail to converge. In contrast, value iteration finds cyclic equilibria for nearly all the games. The success of value iteration in finding cyclic equilibria suggests this generalized solution concept could be useful for constructing robust multiagent learning algorithms. 2 An Impossibility Result for Q Values In this section, we consider a subclass of Markov games in which transitions are deterministic and are controlled by one player at a time. We show that this class includes games that have no deterministic equilibrium policies. For this class of games, we present (proofs available in an extended technical report) two theorems. The first, a negative result, states that the Q values used in existing value-iteration algorithms are insufficient for deriving equilibrium policies. The second, presented in Section 4.1, is a positive result that states that value iteration does converge to cyclic equilibrium policies in this class of games. 2.1 Preliminary Definitions Given a finite set X, define ∆(X) to be the set of all probability distributions over X. Definition 1 A Markov game Γ = [S, N, A, T, R, γ] is a set of states S, a set of players N = {1, . . . , n}, a set of actions for each player in each state {Ai,s}s∈S,i∈N (where we represent the set of all state-action pairs as A ≡S s∈S {s} × Q i∈N Ai,s , a transition function T : A →∆(S), a reward function R : A →Rn, and a discount factor γ. Given a Markov game Γ, let As = Q i∈N Ai,s. A stationary policy is a set of distributions {π(s) : s ∈S}, where for all s ∈S, π(s) ∈∆(As). Given a stationary policy π, define V π,Γ : S →Rn and Qπ,Γ : A →Rn to be the unique pair of functions satisfying the following system of equations: for all i ∈N, for all (s, a) ∈A, V π,Γ i (s) = X a∈As π(s)(a)Qπ,Γ i (s, a), (1) Qπ,Γ i (s, a) = Ri(s, a) + γ X s′∈S T(s, a)(s′)V π,Γ i (s′). (2) A deterministic Markov game is a Markov game Γ where the transition function is deterministic: T : A →S. A turn-taking game is a Markov game Γ where in every state, only one player has a choice of action. Formally, for all s ∈S, there exists a player i ∈N such that for all other players j ∈N\{i}, |Aj,s| = 1. 2.2 A Negative Result for Stationary Equilibria A NoSDE (pronounced “nasty”) game is a deterministic turn-taking Markov game Γ with two players, two states, no more than two actions for either player in either state, and no deterministic stationary equilibrium policy. That the set of NoSDE games is non-empty is demonstrated by the game depicted in Figure 1. This game has no deterministic stationary equilibrium policy: If Player 1 sends, Player 2 prefers to send; but, if Player 2 sends, Player 1 prefers to keep; and, if Player 1 keeps, Player 2 prefers to keep; but, if Player 2 keeps, Player 1 prefers to send. No deterministic policy is an equilibrium because one player will always have an incentive to change policies. 1 2 R1(1, send, noop) = 0 R1(2, noop, send) = 0 R1(1, keep, noop) = 1 R1(2, noop, keep) = 3 1 2 R2(1, send, noop) = 3 R2(2, noop, send) = 0 R2(1, keep, noop) = 0 R2(2, noop, keep) = 1 Figure 1: An example of a NoSDE game. Here, S = {1, 2}, A1,1 = A2,2 = {keep, send}, A1,2 = A2,1 = {noop}, T(1, keep, noop) = 1, T(1, send, noop) = 2, T(2, noop, keep) = 2, T(2, noop, send) = 1, and γ = 3/4. In the unique stationary equilibrium, Player 1 sends with probability 2/3 and Player 2 sends with probability 5/12. Lemma 1 Every NoSDE game has a unique stationary equilibrium policy.1 It is well known that, in general Markov games, random policies are sometimes needed to achieve an equilibrium. This fact can be demonstrated simply by a game with one state where the utilities correspond to a bimatrix game with no deterministic equilibria (penny matching, say). Random actions in these games are sometimes linked with strategies that use “faking” or “bluffing” to avoid being exploited. That NoSDE games exist is surprising, in that randomness is needed even though actions are always taken with complete information about the other player’s choice and the state of the game. However, the next result is even more startling. Current value-iteration algorithms attempt to find the Q values of a game with the goal of using these values to find a stationary equilibrium of the game. The main theorem of this paper states that it is not possible to derive a policy from the Q values for NoSDE games, and therefore in general Markov games. Theorem 1 For any NoSDE game Γ = [S, N, A, T, R] with a unique equilibrium policy π, there exists another NoSDE game Γ′ = [S, N, A, T, R′] with its own unique equilibrium policy π′ such that Qπ,Γ = Qπ′,Γ′ but π ̸= π′ and V π,Γ ̸= V π′,Γ′. This result establishes that computing or learning Q values is insufficient to compute a stationary equilibrium of a game.2 In this paper we suggest an alternative, where we still 1The policy is both a correlated equilibrium and a Nash equilibrium. 2Although maintaining Q values and state values and deriving policies from both sets of functions might circumvent this problem, we are not aware of existing value-iteration algorithms or learning algorithms that do so. This observation presents a possible avenue of research not followed in this paper. do value iteration in the same way, but we extract a cyclic equilibrium from the sequence of values instead of a stationary one. 3 A New Goal: Cyclic Equilibria A cyclic policy is a finite sequence of stationary policies π = {π1, . . . , πk}. Associated with π is a sequence of value functions {V π,Γ,j} and Q-value functions {Qπ,Γ,j} such that V π,Γ,j i (s) = X a∈As πj(s)(a) Qπ,Γ,j i (s, a) and (3) Qπ,Γ,j i (s, a) = Ri(s, a) + γ X s′∈S T(s, a)(s′) V π,Γ,inck(j) i (s′) (4) where for all j ∈{1, . . . , k}, inck(k) = 1 and inck(j) = j + 1 if j < k. Definition 2 Given a Markov game Γ, a cyclic correlated equilibrium is a cyclic policy π, where for all j ∈{1, . . . , k}, for all i ∈N, for all s ∈S, for all ai, a′ i ∈Ai,s: X a−i∈A−i,s πj(s)(ai, a−i) Qπ,Γ,j i (s, ai, a−i) ≥ X (ai,a−i)∈As πj(s)(ai, a−i) Qπ,Γ,j(s, a′ i, a−i). (5) Here, a−i denotes a joint action for all players except i. A similar definition can be constructed for Nash equilibria by insisting that all policies πj(s) are product distributions. In Definition 2, we imagine that action choices are moderated by a referee with a clock that indicates the current stage j of the cycle. At each stage, a typical correlated equilibrium is executed, meaning that the referee chooses a joint action a from πj(s), tells each agent its part of that joint action, and no agent can improve its value by eschewing the referee’s advice. If no agent can improve its value by more than ǫ at any stage, we say π is an ǫ-correlated cyclic equilibrium. A stationary correlated equilibrium is a cyclic correlated equilibrium with k = 1. In the next section, we show how value iteration can be used to derive cyclic correlated equilibria. 4 Value Iteration in General-Sum Markov Games For a game Γ, define QΓ = (Rn)A to be the set of all state-action (Q) value functions, VΓ = (Rn)S to be the set of all value functions, and ΠΓ to be the set of all stationary policies. Traditionally, value iteration can be broken down into estimating a Q value based upon a value function, selecting a policy π given the Q values, and deriving a value function based upon π and the Q value functions. Whereas the first and the last step are fairly straightforward, the step in the middle is quite tricky. A pair (π, Q) ∈ΠΓ × QΓ agree (see Equation 5) if, for all s ∈S, i ∈N, ai, a′ i ∈Ai,s: X a−i∈A−i,s π(s)(ai, a−i) Qi(s, ai, a−i) ≥ X (ai,a−i)∈As π(s)(ai, a−i) Q(s, a′ i, a−i). (6) Essentially, Q and π agree if π is a best response for each player given payoffs Q. An equilibrium-selection rule is a function f : QΓ →ΠΓ such that for all Q ∈QΓ, (f(Q), Q) agree. The set of all such rules is FΓ. In essence, these rules update values assuming an equilibrium policy for a one-stage game with Q(s, a) providing the terminal rewards. Examples of equilibrium-selection rules are best-Nash, utilitarian-CE, dictatorialCE, plutocratic-CE, and egalitarian-CE (Greenwald & Hall, 2003). (Utilitarian-CE, which we return to later, selects the correlated equilibrium in which total of the payoffs is maximized.) Foe-VI and Friend-VI (Littman, 2001) do not fit into our formalism, but it can be proven that in NoSDE games they converge to deterministic policies that are neither stationary nor cyclic equilibria. Define dΓ : VΓ × VΓ →R to be a distance metric over value functions, such that dΓ(V, V ′) = max s∈S,i∈N |Vi(s) −V ′ i (s)|. (7) Using our notation, the value-iteration algorithm for general-sum Markov games can be described as follows. Algorithm 1: ValueIteration(game Γ, V 0 ∈VΓ, f ∈FΓ, Integer T) For t := 1 to T: 1. ∀s ∈S, a ∈A, Qt(s, a) := R(s, a) + γ P s′∈S T(s, a)(s′) V t−1(s′). 2. πt = f(Qt). 3. ∀s ∈S, V t(s) = P a∈As πt(s)(a) Qt(s, a). Return {Q1, . . . , QT }, {π1, . . . , πT }, {V 1, . . . , V T }. If a stationary equilibrium is sought, the final policy is returned. Algorithm 2: GetStrategy(game Γ, V 0 ∈VΓ, f ∈FΓ, Integer T) 1. Run (Q1 . . . QT , π1 . . . πT , V 1 . . . V T ) = ValueIteration(Γ, V 0, f, T). 2. Return πT . For cyclic equilibria, we have a variety of options for how many past stationary policies we want to consider for forming a cycle. Our approach searches for a recent value function that matches the final value function (an exact match would imply a true cycle). Ties are broken in favor of the shortest cycle length. Observe that the order of the policies returned by value iteration is reversed to form a cyclic equilibrium. Algorithm 3: GetCycle(game Γ, V 0 ∈VΓ, f ∈FΓ, Integer T, Integer maxCycle) 1. If maxCycle ≥T, maxCycle := T −1. 2. Run (Q1 . . . QT , π1 . . . πT , V 1 . . . V T ) = ValueIteration(Γ, V 0, f, T). 3. Define k := argmint∈{1,...,maxCycle} d(V T , V T −t). 4. For each t ∈{1, . . . , k} set πt := πT +1−t. 4.1 Convergence Conditions Fact 1 If d(V T , V T −1) = ǫ in GetStrategy, then GetStrategy returns an ǫγ 1−γ -correlated equilibrium. Fact 2 If GetCycle returns a cyclic policy of length k and d(V T , V T −k) = ǫ, then GetCycle returns an ǫγ 1−γk -correlated cyclic equilibrium. Since, given V 0 and Γ, the space of value functions is bounded, eventually there will be two value functions in {V 1, . . . , V T } that are close according to dΓ. Therefore, the two practical (and open) questions are (1) how many iterations does it take to find an ǫ-correlated cyclic equilibrium? and (2) How large is the cyclic equilibrium that is found? In addition to approximate convergence described above, in two-player turn-taking games, one can prove exact convergence. In fact, all the members of FΓ described above can be construed as generalizations of utilitarian-CE in turn-taking games, and utilitarian-CE is proven to converge. Theorem 2 Given the utilitarian-CE equilibrium-selection rule f, for every NoSDE game Γ, for every V 0 ∈VΓ, there exists some finite T such that GetCycle(Γ, V 0, f, T, ⌈T/2⌉) returns a cyclic correlated equilibrium. Theoretically, we can imagine passing infinity as a parameter to value iteration. Doing so shows the limitation of value-iteration in Markov games. Theorem 3 Given the utilitarian-CE equilibrium-selection rule f, for any NoSDE game Γ with unique equilibrium π, for every V 0 ∈VΓ, the value-function sequence {V 1, V 2, . . .} returned from ValueIteration(Γ, V 0, f, ∞) does not converge to V π. Since all of the other rules specified above (except friend-VI and foe-VI) can be implemented with the utilitarian-CE equilibrium-selection rule, none of these rules will be guaranteed to converge, even in such a simple class as turn-taking games! Theorem 4 Given the game Γ in Figure 1 and its stationary equilibrium π, given V 0 i (s) = 0 for all i ∈N, s ∈S, then for any update rule f ∈FΓ, the value-function sequence {V 1, V 2, . . .} returned from ValueIteration(Γ, V 0, f, ∞) does not converge to V π. 5 Empirical Results To complement the formal results of the previous sections, we ran two batteries of tests on value iteration in randomly generated games. We assessed the convergence behavior of value iteration to stationary and cyclic equilibria. 5.1 Experimental Details Our game generator took as input the set of players N, the set of states S, and for each player i and state s, the actions Ai,s. To construct a game, for each state-joint action pair (s, a) ∈A, for each agent i ∈N, the generator sets Ri(s, a) to be an integer between 0 and 99, chosen uniformly at random. Then, it selects T(s, a) to be deterministic, with the resulting state chosen uniformly at random. We used a consistent discount factor of γ = 0.75 to decrease experimental variance. The primary dependent variable in our results was the frequency with which value iteration converged to a stationary Nash equilibrium or a cyclic Nash equilibrium (of length less than 100). To determine convergence, we first ran value iteration for 1000 steps. If dΓ(V 1000, V 999) ≤0.0001, then we considered value iteration to have converged to a stationary policy. If for some k ≤100 max t∈{1,...,k} dΓ(V 1001−t, V 1001−(t+k)) ≤0.0001, (8) then we considered value iteration to have converged to a cycle.3 To determine if a game has a deterministic equilibrium, for every deterministic policy π, we ran policy evaluation (for 1000 iterations) to estimate V π,Γ and Qπ,Γ, and then checked if π was an ǫ-correlated equilibrium for ǫ=0.0001. 5.2 Turn-taking Games In the first battery of tests, we considered sets of turn-taking games with x states and y actions: formally, there were x states {1, . . . , x}. In odd-numbered states, Player 1 had y 3In contrast to the GetCycle algorithm, we are here concerned with finding a cyclic equilibrium so we check an entire cycle for convergence. 0 20 40 60 80 100 120 2 3 4 5 6 7 8 9 10 Games Without Convergence (out of 1000) Actions 5 States 4 States 3 States 2 States 40 50 60 70 80 90 100 2 3 4 5 6 7 8 9 10 Percent Converged Games States Cyclic uCE OTComb OTBest uCE Figure 2: (Left) For each combination of states and actions, 1000 deterministic turn-taking games were generated. The graph plots the number of games where value iteration did not converge to a stationary equilibrium. (Right) Frequency of convergence on 100 randomly generated games with simultaneous actions. Cyclic uCE is the number of times utilitarian-CE converged to a cyclic equilibrium. OTComb is the number of games where any one of Friend-VI, Foe-VI, utilitarian-NE-VI, and 5 variants of correlated equilibriumVI: dictatorial-CE-VI (First Player), dictatorial-CE-VI (Second Player), utilitarian-CE-VI, plutocratic-CE-VI, and egalitarian-VI converged to a stationary equilibrium. OTBest is the maximum number of games where the best fixed choice of the equilibrium-selection rule converged. uCE is the number of games in which utilitarian-CE-VI converged to a stationary equilibrium. actions and Player 2 had one action: in even-numbered states, Player 1 had one action and Player 2 had y actions. We varied x from 2 to 5 and y from 2 to 10. For each setting of x and y, we generated and tested one thousand games. Figure 2 (left) shows the number of generated games for which value iteration did not converge to a stationary equilibrium. We found that nearly half (48%, as many as 5% of the total set) of these non-converged games had no stationary, deterministic equilibria (they were NoSDE games). The remainder of the stationary, deterministic equilibria were simply not discovered by value iteration. We also found that value iteration converged to cycles of length 100 or less in 99.99% of the games. 5.3 Simultaneous Games In a second set of experiments, we generated two-player Markov games where both agents have at least two actions in every state. We varied the number of states between 2 and 9, and had either 2 or 3 actions for every agent in every state. Figure 2 (right) summarizes results for 3-action games (2-actions games were qualitatively similar, but converged more often). Note that the fraction of random games on which the algorithms converged to stationary equilibria decreases as the number of states increases. This result holds because the larger the game, the larger the chance that value iteration will fall into a cycle on some subset of the states. Once again, we see that the cyclic equilibria are found much more reliably than stationary equilibria by value-iteration algorithms. For example, utilitarian-CE converges to a cyclic correlated equilibrium about 99% of the time, whereas with 10 states and 3 actions, on 26% of the games none of the techniques converge. 6 Conclusion In this paper, we showed that value iteration, the algorithmic core of many multiagent planning reinforcement-learning algorithms, is not well behaved in Markov games. Among other impossibility results, we demonstrated that the Q-value function retains too little information for constructing optimal policies, even in 2-state, 2-action, deterministic turntaking Markov games. In fact, there are an infinite number of such games with different Nash equilibrium value functions that have identical Q-value functions. This result holds for proposed variants of value iteration from the literature such as updating via a correlated equilibrium or a Nash equilibrium, since, in turn-taking Markov games, both rules reduce to updating via the action with the maximum value for the controlling player. Our results paint a bleak picture for the use of value-iteration-based algorithms for computing stationary equilibria. However, in a class of games we called NoSDE games, a natural extension of value iteration converges to a limit cycle, which is in fact a cyclic (nonstationary) Nash equilibrium policy. Such cyclic equilibria can also be found reliably for randomly generated games and there is evidence that they appear in some naturally occurring problems (Tesauro & Kephart, 1999). One take-away message of our work is that nonstationary policies may hold the key to improving the robustness of computational approaches to planning and learning in general-sum games. Acknowledgements This research was supported by NSF Grant #IIS-0325281, NSF Career Grant #IIS0133689, and the Alberta Ingenuity Foundation through the Alberta Ingenuity Centre for Machine Learning. References Bellman, R. (1957). Dynamic programming. Princeton, NJ: Princeton University Press. Brafman, R. I., & Tennenholtz, M. (2002). R-MAX—a general polynomial time algorithm for near-optimal reinforcement learning. Journal of Machine Learning Research, 3, 213–231. Greenwald, A., & Hall, K. (2003). Correlated Q-learning. Proceedings of the Twentieth International Conference on Machine Learning (pp. 242–249). Hu, J., & Wellman, M. (1998). Multiagent reinforcement learning:theoretical framework and an algorithm. Proceedings of the Fifteenth International Conference on Machine Learning (pp. 242–250). Morgan Kaufman. Littman, M. (2001). Friend-or-foe Q-learning in general-sum games. Proceedings of the Eighteenth International Conference on Machine Learning (pp. 322–328). Morgan Kaufmann. Littman, M. L., & Szepesv´ari, C. (1996). A generalized reinforcement-learning model: Convergence and applications. Proceedings of the Thirteenth International Conference on Machine Learning (pp. 310–318). Osborne, M. J., & Rubinstein, A. (1994). A Course in Game Theory. The MIT Press. Puterman, M. (1994). Markov decision processes: Discrete stochastic dynamic programming. Wiley-Interscience. Shapley, L. (1953). Stochastic games. Proceedings of the National Academy of Sciences of the United States of America, 39, 1095–1100. Tesauro, G., & Kephart, J. (1999). Pricing in agent economies using multi-agent Qlearning. Proceedings of Fifth European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (pp. 71–86).
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The Information-Form Data Association Filter Brad Schumitsch, Sebastian Thrun, Gary Bradski, and Kunle Olukotun Stanford AI Lab Stanford University, Stanford, CA 94305 Abstract This paper presents a new filter for online data association problems in high-dimensional spaces. The key innovation is a representation of the data association posterior in information form, in which the “proximity” of objects and tracks are expressed by numerical links. Updating these links requires linear time, compared to exponential time required for computing the exact posterior probabilities. The paper derives the algorithm formally and provides comparative results using data obtained by a real-world camera array and by a large-scale sensor network simulation. 1 Introduction This paper addresses the problem of data association in online object tracking [6]. The data association problem arises in a large number of application domains, including computer vision, robotics, and sensor networks. Our setup assumes an online tracking system that receives two types of data: sensor data, conveying information about the identity or type of objects that are being tracked; and transition data, characterizing the uncertainty introduced through the tracker’s inability to reliably track individual objects over time. The setup is motivated by a camera network which we recently deployed in our lab. Here sensor data relates to the color of clothing of individual people, which enables us to identify them. Tracks are lost when people walk too closely together, or when they occlude each other. We show that the standard probabilistic solution to the discrete data association problem requires exponential update time and exponential memory. This is because each data association hypothesis is expressed by a permutation matrix that assigns computer-internal tracks to objects in the physical world. An optimal filter would therefore need to maintain a probability distribution over the space of all permutation matrices, which grows exponentially with N, the number of objects in the world. The common remedy involves the selection of a small number K of likely hypotheses. This is the core of numerous widelyused multi-hypothesis tracking algorithms [9, 1]. More recent solutions involve particle filters [3], which maintain stochastic samples of hypotheses. Both of these techniques are very effective for small N, but the number of hypothesis they require grows exponentially with N. This paper provides a filter algorithm that scales to much larger problems. This filter maintains an information matrix Ωof size N × N, which relates tracks to physical objects in the world. The rows of Ωcorrespond to object identities, the columns to the tracks of the tracker. Ωis a matrix in information form, that is, it can be thought of as a non-normalized log-probability. Fig. 1a shows an example. The highlighted first column corresponds to track 1 in the tracker. The numerical values in this column suggest that this track is most strongly (a) Example: Information matrix Ω= 2 12 4 4 1 2 11 0 10 4 4 15 5 2 1 2 (b) Most likely data association ˆA = argmax A tr AT Ω= 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 (c) Update: Associating track 2 with object 4 2 12 4 4 1 2 11 0 10 4 4 15 5 2 1 2 −→ 2 12 4 4 1 2 11 0 10 4 4 15 5 3 1 2 (d) Update: Tracks 2 and 3 merge 2 12 4 4 1 2 11 0 10 4 4 15 5 3 1 2 −→ 2 11.31 11.31 4 1 10.31 10.31 0 10 4 4 15 5 2.43 2.43 2 (e) Graphical network interpretation of the information form Figure 1: Illustration of the information form filter for data association in object tracking associated with object 3, since the value 10 dominates all other values in this column. Thus, looking at column 1 of Ωin isolation would have us conclude that the most likely association of track 1 is object 3. However, the most likely permutation matrix is shown in Fig. 1b; from all possible data association assignments, this matrix receives the highest score. Its score is tr ˆAT Ω= 5 + 12 + 11 + 15 = 43 (here “tr” denotes the trace of a matrix). This permutation matrix associates object 3 with track 4, while associating track 1 with object 4. The key question now pertains to the construction of Ω. As we shall see, the update operations for Ωare simple and parallelizable. Suppose we receive a measurement that associates track 2 with object 4 (e.g., track 2’s hair color appears to be the same as person 4’s hair color in our camera array). As a result, our approach adds a value to the element in Ωthat links object 4 and track 2, as illustrated in Fig. 1c (the exact magnitude of this value will be discussed below). Similarly, suppose our tracker is unable to distinguish between objects 2 and 3, perhaps because these objects are so close together in a camera image that they cannot be tracked individually. Such a situation leads to a new information matrix, in which both columns assume the same values, as illustrated in Fig. 1d. The exact values in this new information matrix are the result of an exponentiated averaging explained below. All of these updates are easily parallelized, and hence are applicable to a decentralized network of cameras. The exact update and inference rules are based on a probabilistic model that is also discussed below. Given the importance of data association, it comes as no surprise that our algorithm is related to a rich body of prior work. The data association problem has been studied as an offline problem, in which all data is memorized and inference takes place after data collection. There exists a wealth of powerful methods, such as RANSAC [4] and MCMC [6, 2], but those are inherently offline and their memory requirements increase over time. The dominant online, or filter, paradigm involves the selection of K representative samples of the data association matrix, but such algorithms tend to work only for small N [11]. Relatively little work has focused on the development of compact sufficient statistics for data association. One alternative O(N 2) technique to the one proposed here was explored in [8]. This technique uses doubly stochastic matrices, which are computationally hard to maintain. The first mention of information filters is in [8], but the update rules there were computationally less efficient (in O(N 4)) and required central optimization. The work in this paper does not address the continuous-valued aspects of object tracking. Those are very well understood, and information representations have been successfully applied [5, 10]. Information representations are popular in the field of graphical networks. Our approach can be viewed as a learning algorithm for a Markov network [7] of a special topology, where any track and any object are connected by an edge. Such a network is shown in Fig. 1e. The filter update equations manipulate the strength of the edges based on data. 2 Problem Setup and Bayes Filter Solution We begin with a formal definition of the data association problem and derive the obvious but inefficient Bayes filter solution. Throughout this paper, we make the closed world assumption, that is, there are always the same N known objects in the world. 2.1 Data Association We assume that we are given a tracking algorithm that maintains N internal tracks of the moving objects. Due to insufficient information, this assumed tracking algorithm does not always know the exact mapping of identities to internal tracks. Hence, the same internal track may correspond to different identities at different times. The data association problem is the problem of assigning these N tracks to N objects. Each data association hypothesis is characterized by a permutation matrix of the type shown in Fig. 1b. The columns of this matrix correspond to the internal tracks, and the rows to the objects. We will denote the data association matrix by A (not to be confused with the information matrix Ω). In our closed world, A is always a permutation matrix; hence all elements are 0 or 1. There are exponentially many permutation matrices, which is a reason why data association is considered a hard problem. 2.2 Identity Measurement The correct data association matrix A is unobservable. Instead, the sensors produce local information about the relation of individual tracks to individual objects. We will denote sensor measurements by zj, where j is the index of the corresponding track. Each zj = {zij} specifies a local probability distribution in the corresponding object space: p(xi = yj | zj) = zij with X i zij = 1 (1) Here xi is the i-th object in the world, and yj is the j-th track. The measurement in our introductory example (see Fig. 1c) was of a special form, in that it elevated one specific correspondence over the others. This occurs when zij = α for some α ≈1, and zkj = 1−α N−1 for all k ̸= i. Such a measurement arises when the tracker receives evidence that a specific track yj corresponds with high likelihood to a specific object xi. Specifically, the measurement likelihood of this correspondence is α, and the error probability is 1 −α. 2.3 State Transitions As time passes by, our tracker may confuse tracks, which is a loss of information with respect to the data association. The tracker confusing two objects amounts to a random flip of two columns in the data association matrix A. The model adopted in this paper generalizes this example to arbitrary distributions over permutations of the columns in A. Let {B1, . . . , BM} be a set of permutation matrices, and {β1, . . . , βM} with P m βm = 1 be a set of associated probabilities. The “true” permutation matrix undergoes a random transition from A to A Bm with probability βm: A prob=βm −→ A Bm (2) The sets {B1, . . . , BM} and {β1, . . . , βM} are given to us by the tracker. For the example in Fig. 1d, in which tracks 2 and 3 merge, the following two permutation matrices will implement such a merge: B1 = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ; β1 = 0.5 B2 = 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 ; β2 = 0.5 (3) The first such matrix leaves the association unchanged, whereas the second swaps columns 2 and 3. Since β1 = β2 = 0.5, such a swap happens exactly with probability 0.5. 2.4 Inefficient Bayesian Solution For small N, the data association problem now has an obvious Bayes filter solution. Specifically, let A be the space of all permutation matrices. The Bayesian filter solves the identity tracking problem by maintaining a probabilistic belief over the space of all permutation matrices A ∈A. For each A, it maintains a posterior probability denoted p(A). This probability is updated in two different ways, reminiscent of the measurement and state transition updates in DBNs and EKFs. The measurement step updates the belief in response to a measurement zj. This update is an application of Bayes rule: p(A) ←− 1 L p(A) X i aij zij (4) with L = X ¯ A p( ¯A) X i ¯aij zij (5) Here aij denotes the ij-th element of the matrix A. Because A is a permutation matrix, only one element in the sum over i is non-zero (hence there is not really a summation here). The state transition updates the belief in accordance with the permutation matrices Bm and associated probabilities βm (see Eq. 2): p(A) ←− X m βm p(A BT m) (6) We use here that the inverse of a permutation matrix is its transpose. This Bayesian filter is an exact solution to our identity tracking problem. Its problem is complexity: there are N! permutation matrices A, and we have to compute probabilities for all of them. Thus, the exact filter is only applicable to problems with small N. Even if we want to keep track of K ≪N likely permutations—as attempted by filters like the multihypothesis EKF or the particle filter—the required number of tracks K will generally have to scale exponentially with N (albeit at a slower rate). This exponential scaling renders the Bayesian filter ultimately inapplicable to the identity tracking problem with large N. 3 The Information-Form Solution Our data association filter represents the posterior in condensed form, using an N × N information matrix. As a result, it requires linear update time and quadratic memory, instead of the exponential time and memory requirements of the Bayes filter. However, we give two caveats regarding our method: it is approximate, and it does not maintain probabilities. The approximation is the result of a Jensen approximation, which we will show is empirically accurate. The calculation of probabilities from an information matrix requires inference, and we will provide several options for performing this inference. 3.1 The Information Matrix The information matrix, denoted Ω, is a matrix of size N × N whose elements are nonnegative. Ωinduces a probability distribution over the space of all data association matrices A, through the following definition: p(A) = 1 Z exp tr A Ω with Z = X A exp tr A Ω (7) Here tr is the trace of a matrix, and Z is the partition function. Computing the posterior probability p(A) from Ωis hard, due to the difficulty of computing the partition function Z. However, as we shall see, maintaining Ωis surprisingly easy, and it is also computationally efficient. 3.2 Measurement Update in Information Form In information form, the measurement update is a local addition of the form: Ω ←− Ω+ 0 · · · 0 log z1j 0 · · · 0 ... ... ... ... ... ... ... 0 · · · 0 log z1N 0 · · · 0 (8) This follows directly from Eq. 4. The complexity of this update is O(N). Of particular interest is the case where one specific association was affirmed with probability zij = α, while all others were true with the error probability zkj = 1−α N−1. Then the update is of the form Ω ←− Ω+ 0 · · · 0 c 0 · · · 0 ... ... ... ... ... ... ... 0 · · · 0 c 0 · · · 0 ... ... ... log α ... ... ... 0 · · · 0 c 0 · · · 0 ... ... ... ... ... ... ... 0 · · · 0 c 0 · · · 0 with c = log 1 −α N −1 (9) However, since Ωis a non-normalized matrix (it is normalized via the partition function Z in Eq. 7), we can modify Ωas long as exp tr A Ωis changed by the same factor for any A. In particular, we can subtract c from an entire column in Ω; this will affect the result of exp tr A Ωby a factor of exp c, which is independent of A and hence will be subsumed by the normalizer Z. This allows us to perform a more efficient update ωij ←− ωij + log α −log 1 −α N −1 (10) where ωij is the ij-th element of Ω. This update is indeed of the form shown in Fig. 1c. It requires O(1) time, is entirely local, and is an exact realization of Bayes rule in information form. 3.3 State Transition Update in Information Form The state transition update is also simple, but it is approximate. We show that using a Jensen bound, we obtain the following update for the information matrix: Ω ←− log X m βm BT m exp Ω (11) Here the expression “exp Ω” denotes a component-wise exponentiation of the matrix Ω; the result is also a matrix. This update implements a “dual” of a geometric mean; here the exponentiation is applied to the individual elements of this mean, and the logarithm is applied to the result. It is important to notice that this update only affects elements in Ω that might be affected by a permutation Bm; all others remain the same. A numerical example of this update was given in Fig. 1d, assuming the permutation matrices in Eq. 3. The values there are the result of applying this update formula. For example, for the first row we get log 1 2(exp 12 + exp 4) = 11.3072. The derivation of this update formula is straightforward. We begin with Eq. 6, written in logarithmic form. The transformations rely heavily on the fact that A and Bm are permutation matrices. We use the symbol “tr∗” for a multiplicative version of the matrix trace, in which all elements on the diagonal are multiplied. log p(A) ←− log X m βm p(A BT m) = const. + log X m βm exp tr A BT m Ω = const. + log X m βm tr∗exp A BT m Ω = const. + log X m βm tr∗A BT m exp Ω ≤ const. + log tr∗A X m βm BT m exp Ω = const. + tr A " log X m βm BT m exp Ω # (12) The result is of the form of (the logarithm of) Eq. 7. The expression in brackets is equivalent to the right-hand side of the update Eq. 11. A benefit of this update rule is that it only affects columns in Ωthat are affected by a permutation Bm; all other columns are unchanged. We note that the approximation in this derivation is the result of applying a Jensen bound. As a result, we gain a compact closed-form solution to the update problem, but the state transition step may sacrifice information in doing so (as indicated by the “≤” sign). In our experimental results section, however, we find that this approximation is extremely accurate in practice. 4 Computing the Data Association The previous section formally derived our update rules, which are simple and local. We now address the problem of recovering actual data association hypotheses from the information matrix, along with the associated probabilities. We consider three cases: the computation of the most likely data association matrix as illustrated in Fig. 1b; the computation of a relative probability of the form p(A)/p(A′); and the computation of an absolute probability or expectation. To recover argmaxA p(A), we need only solve a linear program. Relative probabilities are also easy to recover. Consider, for example, the quotient of the probability p(A)/p(A′) for two identity matrices A and A′. When calculating this quotient from Eq. 7, the normalizer Z cancels out: p(A) p(A′) = exp tr(A −A′) Ω (13) Absolute probabilities and expectations are generally the most difficult to compute. This is because of the partition function Z in Eq. 7, whose exact calculation requires considering N! permutation matrices. Our approximate method for recovering probabilities/expectations is based on the Metropolis algorithm. Specifically, consider the expectation of a function f: E[f(A)] = X A f(A) p(A) (14) Our method approximates this expression through a finite sample of matrices A[1], A[2], . . ., using Metropolis and the proposal distribution defined in Eq. 13. This proposal generates excellent results for simple functions f (e.g., the marginal of a single identity). For more (a) camera (b) array of 16 ceiling-mounted cameras (c) camera images (d) 2 of the tracks Figure 2: The camera array, part of the common area in the Stanford AI Lab. Panel (d) compares our esitmate with ground truth for two of the tracks. The data association is essentially correct at all times. (a) Comparison K-hypothesis vs. information-theoretic tracker our approach K-hypotheses (b) Comparison using a DARPA challenge data set produced by Northrop Grumman our approach @ @ I Figure 3: Results for our approach information-form filter the common multi-hypothesis approach for (a) synthetic data and (b) a DARPA challenge data set. The comparison (b) involves additional algorithms, including one published in [8]. complex functions f, we refer the reader to improved proposal distributions that have been found to be highly efficient in related problems [6, 2]. 5 Experimental Results To evaluate this algorithm, we deployed a network of ceiling-mounted cameras in our lab, shown in Fig. 2. We used 16 cameras to track individuals walking through the lab. The tracker uses background subtraction to find blobs and uses a color histogram to classify these blobs. Only when two or more people come very close to each other might the tracker lose track of individual people. We find that for N = 5 our method tracks people nearly perfectly, but so does the full-blown Bayesian solution, as well as the K-best multihypothesis method that is popular in the tracking literature. To investigate scaling to larger N, we compared our approach on two data sets: a synthetic one with up to N = 1, 600 objects, and a dataset using an sensor network simulation provided to us by Northrop Grumman through an ongoing DARPA program. The latter set is thought to be realistic. It was chosen because it involves a large number (N = 200) of moving objects, whose motion patterns come from a behavioral model. In all cases, we measured the number of objects mislabeled in the maximum likelihood hypothesis (as found by solving the LP). All results are averaged over 50 runs. The comparison in Fig. 3a shows that our approach outperforms the traditional K-best hypothesis approach (with K = N) by a large margin. Furthermore, our approach seems to be unaffected by N, the number of entities in the environment, whereas the traditional approach deteriorates. This comes as no surprise, since the traditional approach requires increasing numbers of samples to cover the space of all data associations. The results in Fig. 3b compare (from left to right), the most likely hypothesis, the most recent sensor measurement, the K-best approach with K = 200, an approach proposed in [8], and our approach. Notice that this plot is in log-form. No comparisons were attempted with offline techniques, such as the ones in [4, 6], because the data sets used here are quite large and our interest is online filtering. 6 Conclusion We have provided an information form algorithm for the data association problem in object tracking. The key idea of this approach is to maintain a cumulative matrix of information associating computer-internal tracks with physical objects. Updating this matrix is easy; furthermore, efficient methods were proposed for extracting concrete data association hypotheses from this representation. Empirical work using physical networks of camera arrays illustrated that our approach outperforms alternative paradigms that are commonly used throughout all of science. Despite these advances, the work possesses a number of limitations. Specifically, our closed world assumption is problematic, although we believe the extension to open worlds is relatively straightforward. Also missing is a tight integration of our discrete formulation into continuous-valued traditional tracking algorithms such as EKFs. Such extensions warrant further research. We believe the key innovation here is best understood from a graphical model perspective. Sampling K good data associations cannot exploit conditional independence in the data association posterior, hence will always require that K is an exponential function of N. The information form and the equivalent graphical network in Fig. 1e exploits conditional independences. This subtle difference makes it possible to get away with O(N 2) memory and O(N) computation without a loss of accuracy when N increases, as shown in Fig. 3a. The information form discussed here—and the associated graphical networks— promise to overcome a key brittleness associated with the current state-of-the-art in online data association. Acknowledgements We gratefully thank Jaewon Shin and Leo Guibas for helpful discussions. This research was sponsored by the Defense Advanced Research Projects Agency (DARPA) under the ACIP program and grant number NBCH104009. References [1] Y. Bar-Shalom and X.-R. Li. Estimation and Tracking: Principles, Techniques, and Software. YBS, Danvers, MA, 1998. [2] F. Dellaert, S.M. Seitz, C. Thorpe, and S. Thrun. EM, MCMC, and chain flipping for structure from motion with unknown correspondence. Machine Learning, 50(1-2):45–71, 2003. [3] A. Doucet, J.F.G. de Freitas, and N.J. Gordon, editors. Sequential Monte Carlo Methods in Practice. Springer, 2001. [4] M. A. Fischler and R. C. Bolles. Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography. Communications of the ACM, 24:381–395, 1981. [5] P. Maybeck. Stochastic Models, Estimation, and Control, Volume 1. Academic Press, 1979. [6] H. Pasula, S. Russell, M. Ostland, and Y. Ritov. Tracking many objects with many sensors. IJCAI-99. [7] J. Pearl. Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kaufmann, 1988. [8] J. Shin, N. Lee, S. Thrun, and L. Guibas. Lazy inference on object identities in wireless sensor networks. IPSN-05. [9] D.B. Reid. An algorithm for tracking multiple targets. IEEE Transactions on Aerospace and Electronic Systems, AC-24:843–854, 1979. [10] S. Thrun, Y. Liu, D. Koller, A.Y. Ng, Z. Ghahramani, and H. Durrant-Whyte. Simultaneous localization and mapping with sparse extended information filters. IJRR, 23(7/8), 2004. [11] D. Fox, J. Hightower, L. Lioa, D. Schulz, and G. Borriello. Bayesian Filtering for Location Estimation. IEEE Pervasive Computing, 2003.
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Consistency of one-class SVM and related algorithms R´egis Vert Laboratoire de Recherche en Informatique Universit´e Paris-Sud 91405, Orsay Cedex, France Masagroup 24 Bd de l’Hˆopital 75005, Paris, France Regis.Vert@lri.fr Jean-Philippe Vert Geostatistics Center Ecole des Mines de Paris - ParisTech 77300 Fontainebleau, France Jean-Philippe.Vert@ensmp.fr Abstract We determine the asymptotic limit of the function computed by support vector machines (SVM) and related algorithms that minimize a regularized empirical convex loss function in the reproducing kernel Hilbert space of the Gaussian RBF kernel, in the situation where the number of examples tends to infinity, the bandwidth of the Gaussian kernel tends to 0, and the regularization parameter is held fixed. Non-asymptotic convergence bounds to this limit in the L2 sense are provided, together with upper bounds on the classification error that is shown to converge to the Bayes risk, therefore proving the Bayes-consistency of a variety of methods although the regularization term does not vanish. These results are particularly relevant to the one-class SVM, for which the regularization can not vanish by construction, and which is shown for the first time to be a consistent density level set estimator. 1 Introduction Given n i.i.d. copies (X1, Y1), . . . , (Xn, Yn) of a random variable (X, Y ) ∈Rd×{−1, 1}, we study in this paper the limit and consistency of learning algorithms that solve the following problem: arg min f∈Hσ ( 1 n n X i=1 φ (Yif(Xi)) + λ∥f ∥2 Hσ ) , (1) where φ : R →R is a convex loss function and Hσ is the reproducing kernel Hilbert space (RKHS) of the normalized Gaussian radial basis function kernel (denoted simply Gaussian kernel below): kσ(x, x′) = 1 √ 2πσ d exp −∥x −x′ ∥2 2σ2 , σ > 0 . (2) This framework encompasses in particular the classical support vector machine (SVM) [1] when φ(u) = max(1 −u, 0). Recent years have witnessed important theoretical advances aimed at understanding the behavior of such regularized algorithms when n tends to infinity and λ decreases to 0. In particular the consistency and convergence rates of the two-class SVM (see, e.g., [2, 3, 4] and references therein) have been studied in detail, as well as the shape of the asymptotic decision function [5, 6]. All results published so far study the case where λ decreases as the number of points tends to infinity (or, equivalently, where λσ−d converges to 0 if one uses the classical non-normalized version of the Gaussian kernel instead of (2)). Although it seems natural to reduce regularization as more and more training data are available –even more than natural, it is the spirit of regularization [7, 8]–, there is at least one important situation where λ is typically held fixed: the one-class SVM [9]. In that case, the goal is to estimate an α-quantile, that is, a subset of the input space X of given probability α with minimum volume. The estimation is performed by thresholding the function output by the one-class SVM, that is, the SVM (1) with only positive examples; in that case λ is supposed to determine the quantile level1. Although it is known that the fraction of examples in the selected region converges to the desired quantile level α [9], it is still an open question whether the region converges towards a quantile, that is, a region of minimum volume. Besides, most theoretical results about the consistency and convergence rates of two-class SVM with vanishing regularization constant do not translate to the one-class case, as we are precisely in the seldom situation where the SVM is used with a regularization term that does not vanish as the sample size increases. The main contribution of this paper is to show that Bayes consistency can be obtained for algorithms that solve (1) without decreasing λ, if instead the bandwidth σ of the Gaussian kernel decreases at a suitable rate. We prove upper bounds on the convergence rate of the classification error towards the Bayes risk for a variety of functions φ and of distributions P, in particular for SVM (Theorem 6). Moreover, we provide an explicit description of the function asymptotically output by the algorithms, and establish converge rates towards this limit for the L2 norm (Theorem 7). In particular, we show that the decision function output by the one-class SVM converges towards the density to be estimated, truncated at the level 2λ (Theorem 8); we finally show that this implies the consistency of one-class SVM as a density level estimator for the excess-mass functional [10] (Theorem 9). Due to lack of space we limit ourselves in this extended abstract to the statement of the main results (Section 2) and sketch the proof of the main theorem (Theorem 3) that underlies all other results in Section 3. All detailed proofs are available in the companion paper [11]. 2 Notations and main results Let (X, Y ) be a pair of random variables taking values in Rd × {−1, 1}, with distribution P. We assume throughout this paper that the marginal distribution of X is absolutely continuous with respect to Lebesgue measure with density ρ : Rd →R, and that is has a support included in a compact set X ⊂Rd. We denote η : Rd →[0, 1] a measurable version of the conditional distribution of Y = 1 given X. The normalized Gaussian radial basis function (RBF) kernel kσ with bandwidth parameter σ > 0 is defined for any (x, x′) ∈Rd × Rd by: kσ(x, x′) = 1 √ 2πσ d exp −∥x −x′ ∥2 2σ2 , and the corresponding reproducing kernel Hilbert space (RKHS) is denoted by Hσ. We note κσ = √ 2πσ −d the normalizing constant that ensures that the kernel integrates to 1. 1While the original formulation of the one-class SVM involves a parameter ν, there is asymptotically a one-to-one correspondance between λ and ν Denoting by M the set of measurable real-valued functions on Rd, we define several risks for functions f ∈M: • The classification error rate, usually refered to as (true) risk of f, when Y is predicted by the sign of f(X), is denoted by R (f) = P (sign (f(X)) ̸= Y ) . • For a scalar λ > 0 fixed throughout this paper and a convex function φ : R →R, the φ-risk regularized by the RKHS norm is defined, for any σ > 0 and f ∈Hσ, by Rφ,σ (f) = EP [φ (Y f (X))] + λ∥f ∥2 Hσ Furthermore, for any real r ≥0, we denote by L (r) the Lipschitz constant of the restriction of φ to the interval [−r, r]. For example, for the hinge loss φ(u) = max(0, 1 −u) one can take L(r) = 1, and for the squared hinge loss φ(u) = max(0, 1 −u)2 one can take L(r) = 2(r + 1). • Finally, the L2-norm regularized φ-risk is, for any f ∈M: Rφ,0 (f) = EP [φ (Y f (X))] + λ∥f ∥2 L2 where, ∥f ∥2 L2 = Z Rd f(x)2dx ∈[0, +∞]. The minima of the three risk functionals defined above over their respective domains are denoted by R∗, R∗ φ,σ and R∗ φ,0 respectively. Each of these risks has an empirical counterpart where the expectation with respect to P is replaced by an average over an i.i.d. sample T = {(X1, Y1) , . . . , (Xn, Yn)}. In particular, the following empirical version of Rφ,σ will be used ∀σ > 0, f ∈Hσ, bRφ,σ (f) = 1 n n X i=1 φ (Yif (Xi)) + λ∥f ∥2 Hσ . The main focus of this paper is the analysis of learning algorithms that minimize the empirical φ-risk regularized by the RKHS norm bRφ,σ, and their limit as the number of points tends to infinity and the kernel width σ decreases to 0 at a suitable rate when n tends to ∞, λ being kept fixed. Roughly speaking, our main result shows that in this situation, if φ is a convex loss function, the minimization of bRφ,σ asymptotically amounts to minimizing Rφ,0. This stems from the fact that the empirical average term in the definition of bRφ,σ converges to its corresponding expectation, while the norm in Hσ of a function f decreases to its L2 norm when σ decreases to zero. To turn this intuition into a rigorous statement, we need a few more assumptions about the minimizer of Rφ,0 and about P. First, we observe that the minimizer of Rφ,0 is indeed well-defined and can often be explicitly computed: Lemma 1 For any x ∈Rd, let fφ,0(x) = arg min α∈R ρ(x) [η(x)φ(α) + (1 −η)φ(−α)] + λα2 . Then fφ,0 is measurable and satisfies: Rφ,0 (fφ,0) = inf f∈M Rφ,0 (f) Second, we provide below a general result that shows how to control the excess Rφ,0-risk of the empirical minimizer of the Rφ,σ-risk, for which we need to recall the notion of modulus of continuity [12]. Definition 2 (Modulus of continuity) Let f be a Lebesgue measurable function from Rd to R. Then its modulus of continuity in the L1-norm is defined for any δ ≥0 as follows ω(f, δ) = sup 0≤∥t ∥≤δ ∥f(. + t) −f(.) ∥L1 , (3) where ∥t ∥is the Euclidian norm of t ∈Rd. Our main result can now be stated as follows: Theorem 3 (Main Result) Let σ1 > σ > 0, 0 < p < 2, δ > 0, and let ˆfφ,σ denote a minimizer of the bRφ,σ risk over Hσ. Assume that the marginal density ρ is bounded, and let M = supx∈Rd ρ(x). Then there exist constants (Ki)i=1...4 (depending only on p, δ, λ, d, and M) such that, for any x > 0, the following holds with probability greater than 1−e−x over the draw of the training data: Rφ,0( ˆfφ,σ) −R∗ φ,0 ≤K1L r κσφ (0) λ ! 4 2+p 1 σ [2+(2−p)(1+δ)]d 2+p 1 n 2 2+p + K2L r κσφ (0) λ !2 1 σ d x n + K3 σ2 σ2 1 + K4ω(fφ,0, σ1) . (4) The first two terms in the r.h.s. of (4) bound the estimation error associated with the gaussian RKHS, which naturally tends to be small when the number of training data increases and when the RKHS is ’small’, i.e., when σ is large. As is usually the case in such variance/bias splitings, the variance term here depends on the dimension d of the input space. Note that it is also parametrized by both p and δ. The third term measures the error due to penalizing the L2-norm of a fixed function in Hσ1 by its ∥. ∥Hσ-norm, with 0 < σ < σ1. This is a price to pay to get a small estimation error. As for the fourth term, it is a bound on the approximation error of the Gaussian RKHS. Note that, once λ and σ have been fixed, σ1 remains a free variable parameterizing the bound itself. In order to highlight the type of convergence rates one can obtain from Theorem 3, let us assume that the φ loss function is Lipschitz on R (e.g., take the hinge loss), and suppose that for some 0 ≤β ≤1, c1 > 0, and for any h ≥0, the function fφ,0 satisfies the following inequality ω(fφ,0, h) ≤c1hβ . (5) Then we can optimize the right hand side of (4) w.r.t. σ1, σ, p and δ by balancing the four terms. This eventually leads to: Rφ,0 ˆfφ,σ −R∗ φ,0 = OP 1 n 2β 4β+(2+β)d −ǫ! , (6) for any ǫ > 0. This rate is achieved by choosing σ1 = 1 n 2 4β+(2+β)d −ǫ β , (7) σ = σ 2+β 2 1 = 1 n 2+β 4β+(2+β)d −ǫ(2+β) 2β , (8) p = 2 and δ as small as possible (that is why an arbitray small quantity ǫ appears in the rate). Theorem 3 shows that minimizing the bRφ,σ risk for well-chosen width σ is a an algorithm consistant for the Rφ,0-risk. In order to relate this consistency with more traditional measures of performance of learning algorithms, the next theorem shows that under a simple additionnal condition on φ, Rφ,0-risk-consistency implies Bayes consistency: Theorem 4 If φ is convex, differentiable at 0, with φ′(0) < 0, then for every sequence of functions (fi)i≥1 ∈M, lim i→+∞Rφ,0 (fi) = R∗ φ,0 =⇒ lim i→+∞R (fi) = R∗ This theorem results from a more general quantitative analysis of the relationship between the excess Rφ,0-risk and the excess R-risk, in the spirit of [13]. In order to state a refined version in the particular case of the support vector machine algorithm, we first need the following definition: Definition 5 We say that a distribution P with ρ as marginal density of X w.r.t. Lebesgue measure has a low density exponent γ ≥0 if there exists (c2, ǫ0) ∈(0, +∞)2 such that ∀ǫ ∈[0, ǫ0], P x ∈Rd : ρ(x) ≤ǫ ≤c2ǫγ. We are now in position to state a quantitative relationship between the excess Rφ,0-risk and the excess R-risk in the case of support vector machines: Theorem 6 Let φ1(α) = max (1 −α, 0) be the hinge loss function, and φ2(α) = max (1 −α, 0)2, be the squared hinge loss function. Then for any distribution P with low density exponent γ, there exist constant (K1, K2, r1, r2) ∈(0, +∞)4 such that for any f ∈M with an excess Rφ1,0-risk upper bounded by r1 the following holds: R(f) −R∗≤K1 Rφ1,0(f) −R∗ φ1,0 γ 2γ+1 , and if the excess regularized Rφ2,0-risk upper bounded by r2 the following holds: R(f) −R∗≤K2 Rφ2,0(f) −R∗ φ2,0 γ 2γ+1 , This result can be extended to any loss function through the introduction of variational arguments, in the spirit of [13]; we do not further explore this direction, but the reader is invited to consult [11] for more details. Hence we have proved the consistency of SVM, together with upper bounds on the convergence rates, in a situation where the effect of regularization does not vanish asymptotically. Another consequence of the Rφ,0-consistency of an algorithm is the L2-convergence of the function output by the algorithm to the minimizer of the Rφ,0-risk: Lemma 7 For any f ∈M, the following holds: ∥f −fφ,0 ∥2 L2 ≤1 λ Rφ,0(f) −R∗ φ,0 . This result is particularly relevant to study algorithms whose objective are not binary classification. Consider for example the one-class SVM algorithm, which served as the initial motivation for this paper. Then we claim the following: Theorem 8 Let ρλ denote the density truncated as follows: ρλ(x) = ( ρ(x) 2λ if ρ(x) ≤2λ, 1 otherwise. (9) Let ˆfσ denote the function output by the one-class SVM, that is the function that solves (1) in the case φ is the hinge-loss function and Yi = 1 for all i ∈{1, . . . , n}. Then, under the general conditions of Theorem 3, for σ choosen as in Equation (8), lim n→+∞∥ˆfσ −ρλ ∥L2 = 0 . An interesting by-product of this theorem is the consistency of the one-class SVM algorithm for density level set estimation: Theorem 9 Let 0 < µ < 2λ < M, let Cµ be the level set of the density function ρ at level µ, and bCµ be the level set of 2λ ˆfσ at level µ, where ˆfσ is still the function outptut by the one-class SVM. For any distribution Q, for any subset C of Rd, define the excess-mass of C with respect to Q as follows: HQ (C) = Q (C) −µLeb (C) , (10) where Leb is the Lebesgue measure. Then, under the general assumptions of Theorem 3, we have lim n→+∞HP (Cµ) −HP bCµ = 0 , (11) for σ choosen as in Equation (8). The excess-mass functional was first introduced in [10] to assess the quality of density level set estimators. It is maximized by the true density level set Cµ and acts as a risk functional in the one-class framework. The proof ef Theorem 9 is based on the following result: if ˆρ is a density estimator converging to the true density ρ in the L2 sense, then for any fixed 0 < µ < sup {ρ}, the excess mass of the level set of ˆρ at level µ converges to the excess mass of Cµ. In other words, as is the case in the classification framework, plug-in estimators built on L2-consistent density estimators are consistent with respect to the excess mass. 3 Proof of Theorem 3 (sketch) In this section we sketch the proof of the main learning theorem of this contribution, which underlies most other results stated in Section 2 The proof of Theorem 3 is based on the following decomposition of the excess Rφ,0-risk for the minimizer ˆfφ,σ of bRφ,σ, valid for any 0 < σ < √ 2σ1 and any sample (xi, yi)i=1,...,n: Rφ,0( ˆfφ,σ) −R∗ φ,0 = h Rφ,0 ˆfφ,σ −Rφ,σ ˆfφ,σ i + h Rφ,σ( ˆfφ,σ) −R∗ φ,σ i + R∗ φ,σ −Rφ,σ(kσ1 ∗fφ,0) (12) + [Rφ,σ(kσ1 ∗fφ,0) −Rφ,0(kσ1 ∗fφ,0)] + Rφ,0(kσ1 ∗fφ,0) −R∗ φ,0 . It can be shown that kσ1 ∗fφ,0 ∈H√ 2σ1 ⊂Hσ ⊂L2(Rd) which justifies the introduction of Rφ,σ(kσ1∗fφ,0) and Rφ,0(kσ1∗fφ,0). By studying the relationship between the Gaussian RKHS norm and the L2 norm, it can be shown that Rφ,0 ˆfφ,σ −Rφ,σ ˆfφ,σ = λ ∥ˆfφ,σ ∥2 L2 −∥ˆfφ,σ ∥2 Hσ ≤0, while the following stems from the definition of R∗ φ,σ: R∗ φ,σ −Rφ,σ(kσ1 ∗fφ,0) ≤0. Hence, controlling Rφ,0( ˆfφ,σ)−R∗ φ,0 boils down to controlling each of the remaining three terms in (12). • The second term in (12) is usually referred to as the sample error or estimation error. The control of such quantities has been the topic of much research recently, including for example [14, 15, 16, 17, 18, 4]. Using estimates of local Rademacher complexities through covering numbers for the Gaussian RKHS due to [4], the following result can be shown: Lemma 10 For any σ > 0 small enough, let ˆfφ,σ be the minimizer of the bRφ,σrisk on a sample of size n, where φ is a convex loss function. For any 0 < p < 2, δ > 0, and x ≥1, the following holds with probability at least 1 −ex over the draw of the sample: Rφ,σ( ˆfφ,σ) −Rφ,σ(fφ,σ) ≤K1L r κσφ (0) λ ! 4 2+p 1 σ [2+(2−p)(1+δ)]d 2+p 1 n 2 2+p + K2L r κσφ (0) λ !2 1 σ d x n , where K1 and K2 are positive constants depending neither on σ, nor on n. • In order to upper bound the fourth term in (12), the analysis of the convergence of the Gaussian RKHS norm towards the L2 norm when the bandwidth of the kernel tends to 0 leads to: Rφ,σ(kσ1 ∗fφ,0) −Rφ,0(kσ1 ∗fφ,0) = ∥kσ1 ∗fφ,0 ∥2 Hσ −∥kσ1 ∗fφ,0 ∥2 L2 ≤σ2 2σ2 1 ∥fφ,0 ∥2 L2 ≤φ (0) σ2 2λσ2 1 . • The fifth term in (12) corresponds to the approximation error. It can be shown that for any bounded function in L1(Rd) and all σ > 0, the following holds: ∥kσ ∗f −f ∥L1 ≤(1 + √ d)ω(f, σ) , (13) where ω(f, .) denotes the modulus of continuity of f in the L1 norm. From this the following inequality can be derived: Rφ,0(kσ1 ∗fφ,0) −Rφ,0(fφ,0) ≤(2λ∥fφ,0 ∥L∞+ L (∥fφ,0 ∥L∞) M) 1 + √ d ω (fφ,0, σ1) . 4 Conclusion We have shown that consistency of learning algorithms that minimize a regularized empirical risk can be obtained even when the so-called regularization term does not asymptotically vanish, and derived the consistency of one-class SVM as a density level set estimator. Our method of proof is based on an unusual decomposition of the excess risk due to the presence of the regularization term, which plays an important role in the determination of the asymptotic limit of the function that minimizes the empirical risk. Although the upper bounds on the convergence rates we obtain are not optimal, they provide a first step toward the analysis of learning algorithms in this context. Acknowledgments The authors are grateful to St´ephane Boucheron, Pascal Massart and Ingo Steinwart for fruitful discussions. This work was supported by the ACI “Nouvelles interfaces des Math´ematiques” of the French Ministry for Research, and by the IST Program of the European Community, under the Pascal Network of Excellence, IST-2002-506778. References [1] B. E. Boser, I. M. Guyon, and V. N. Vapnik. A training algorithm for optimal margin classifiers. In Proceedings of the 5th annual ACM workshop on Computational Learning Theory, pages 144–152. ACM Press, 1992. [2] I. Steinwart. Support vector machines are universally consistent. J. Complexity, 18:768–791, 2002. [3] T. Zhang. Statistical behavior and consistency of classification methods based on convex risk minimization. Ann. Stat., 32:56–134, 2004. [4] I. Steinwart and C. Scovel. Fast rates for support vector machines using gaussian kernels. Technical report, Los Alamos National Laboratory, 2004. submitted to Annals of Statistics. [5] I. Steinwart. Sparseness of support vector machines. J. Mach. Learn. Res., 4:1071–1105, 2003. [6] P. L. Bartlett and A. Tewari. Sparseness vs estimating conditional probabilities: Some asymptotic results. In Lecture Notes in Computer Science, volume 3120, pages 564–578. Springer, 2004. [7] A.N. Tikhonov and V.Y. Arsenin. Solutions of ill-posed problems. W.H. Winston, Washington, D.C., 1977. [8] B. W. Silverman. On the estimation of a probability density function by the maximum penalized likelihood method. Ann. Stat., 10:795–810, 1982. [9] B. Sch¨olkopf, J. C. Platt, J. Shawe-Taylor, A. J. Smola, and R. C. Williamson. Estimating the support of a high-dimensional distribution. Neural Comput., 13:1443–1471, 2001. [10] J. A. Hartigan. Estimation of a convex density contour in two dimensions. J. Amer. Statist. Assoc., 82(397):267–270, 1987. [11] R. Vert and J.-P. Vert. Consistency and convergence rates of one-class svm and related algorithms. J. Mach. Learn. Res., 2006. To appear. [12] R. A. DeVore and G. G. Lorentz. Constructive Approximation. Springer Grundlehren der Mathematischen Wissenschaften. Springer Verlag, 1993. [13] P.I. Bartlett, M.I. Jordan, and J.D. McAuliffe. Convexity, classification and risk bounds. Technical Report 638, UC Berkeley Statistics, 2003. [14] A. B. Tsybakov. On nonparametric estimation of density level sets. Ann. Stat., 25:948–969, June 1997. [15] E. Mammen and A. Tsybakov. Smooth discrimination analysis. Ann. Stat., 27(6):1808–1829, 1999. [16] P. Massart. Some applications of concentration inequalities to statistics. Ann. Fac. Sc. Toulouse, IX(2):245–303, 2000. [17] P. L. Bartlett, O. Bousquet, and S. Mendelson. Local rademacher complexities. Annals of Statistics, 2005. To appear. [18] V. Koltchinskii. Localized rademacher complexities. Manuscript, september 2003.
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Robust Fisher Discriminant Analysis Seung-Jean Kim Alessandro Magnani Stephen P. Boyd Information Systems Laboratory Electrical Engineering Department, Stanford University Stanford, CA 94305-9510 sjkim@stanford.edu alem@stanford.edu boyd@stanford.edu Abstract Fisher linear discriminant analysis (LDA) can be sensitive to the problem data. Robust Fisher LDA can systematically alleviate the sensitivity problem by explicitly incorporating a model of data uncertainty in a classification problem and optimizing for the worst-case scenario under this model. The main contribution of this paper is show that with general convex uncertainty models on the problem data, robust Fisher LDA can be carried out using convex optimization. For a certain type of product form uncertainty model, robust Fisher LDA can be carried out at a cost comparable to standard Fisher LDA. The method is demonstrated with some numerical examples. Finally, we show how to extend these results to robust kernel Fisher discriminant analysis, i.e., robust Fisher LDA in a high dimensional feature space. 1 Introduction Fisher linear discriminant analysis (LDA), a widely-used technique for pattern classification, finds a linear discriminant that yields optimal discrimination between two classes which can be identified with two random variables, say X and Y in Rn. For a (linear) discriminant characterized by w ∈Rn, the degree of discrimination is measured by the Fisher discriminant ratio f(w, µx, µy, Σx, Σy) = wT (µx −µy)(µx −µy)T w wT (Σx + Σy)w = (wT (µx −µy))2 wT (Σx + Σy)w , where µx and Σx (µy and Σy) denote the mean and covariance of X (Y). A discriminant that maximizes the Fisher discriminant ratio is given by wnom = (Σx + Σy)−1(µx −µy), which gives the maximum Fisher discriminant ratio (µx −µy)T (Σx + Σy)−1(µx −µy) = max w̸=0 f(w, µx, µy, Σx, Σy). In applications, the problem data µx, µy, Σx, and Σy are not known but are estimated from sample data. Fisher LDA can be sensitive to the problem data: the discriminant wnom computed from an estimate of the parameters µx, µy, Σx, and Σy can give very poor discrimination for another set of problem data that is also a reasonable estimate of the parameters. In this paper, we attempt to systematically alleviate this sensitivity problem by explicitly incorporating a model of data uncertainty in the classification problem and optimizing for the worst-case scenario under this model. We assume that the problem data µx, µy, Σx, and Σy are uncertain, but known to belong to a convex compact subset U of Rn × Rn × Sn ++ × Sn ++. Here we use Sn ++ (Sn +) to denote the set of all n × n symmetric positive definite (semidefinite) matrices. We make one technical assumption: for each (µx, µy, Σx, Σy) ∈U, we have µx ̸= µy. This assumption simply means that for each possible value of the means and covariances, the classes are distinguishable via Fisher LDA. The worst-case analysis problem of finding the worst-case means and covariances for a given discriminant w can be written as minimize f(w, µx, µy, Σx, Σy) subject to (µx, µy, Σx, Σy) ∈U, (1) with variables µx, µy, Σx, and Σy. The optimal value of this problem is the worst-case Fisher discriminant ratio (over the class U of possible means and covariances), and any optimal points for this problem are called worst-case means and covariances. These depend on w. We will show in §2 that (1) is a convex optimization problem, since the Fisher discriminant ratio is a convex function of µx, µy, Σx, Σy for a given discriminant w. As a result, it is computationally tractable to find the worst-case performance of a discriminant w over the set of possible means and covariances. The robust Fisher LDA problem is to find a discriminant that maximizes the worst-case Fisher discriminant ratio. This can be cast as the optimization problem maximize min (µx,µy,Σx,Σy)∈U f(w, µx, µy, Σx, Σy) subject to w ̸= 0, (2) with variable w. We denote any optimal w for this problem as w⋆. Here we choose a linear discriminant that maximizes the Fisher discrimination ratio, with the worst possible means and covariances that are consistent with our data uncertainty model. The main result of this paper is to give an effective method for solving the robust Fisher LDA problem (2). We will show in §2 that the robust optimal Fisher discriminant w⋆can be found as follows. First, we solve the (convex) optimization problem minimize max w̸=0 f(w, µx, µy, Σx, Σy) = (µx −µy)T (Σx + Σy)−1(µx −µy) subject to (µx, µy, Σx, Σy) ∈U, (3) with variables (µx, µy, Σx, Σy). Let (µ⋆ x, µ⋆ y, Σ⋆ x, Σ⋆ y) denote any optimal point. Then the discriminant w⋆= Σ⋆ x + Σ⋆ y −1 (µ⋆ x −µ⋆ y) (4) is a robust optimal Fisher discriminant, i.e., it is optimal for (2). Moreover, we will see that µ⋆ x, µ⋆ y and Σ⋆ x, Σ⋆ y are worst-case means and covariances for the robust optimal Fisher discriminant w⋆. Since convex optimization problems are tractable, this means that we have a tractable general method for computing a robust optimal Fisher discriminant. A robust Fisher discriminant problem of modest size can be solved by standard convex optimization methods, e.g., interior-point methods [3]. For some special forms of the uncertainty model, the robust optimal Fisher discriminant can be solved more efficiently than by a general convex optimization formulation. In §3, we consider an important special form for U for which a more efficient formulation can be given. In comparison with the ‘nominal’ Fisher LDA, which is based on the means and covariances estimated from the sample data set without considering the estimation error, the robust Fisher LDA performs well even when the sample size used to estimate the means and covariances is small, resulting in estimates which are not accurate. This will be demonstrated with some numerical examples in §4. Recently, there has been a growing interest in kernel Fisher discriminant analysis i.e., Fisher LDA in a higher dimensional feature space, e.g., [7]. Our results can be extended to robust kernel Fisher discriminant analysis under certain uncertainty models. This will be briefly discussed in §5. Various types of robust classification problems have been considered in the prior literature, e.g., [2, 5, 6]. Most of the research has focused on formulating robust classification problems that can be efficiently solved via convex optimization. In particular, the robust classification method developed in [6] is based on the criterion g(w, µx, µy, Σx, Σy) = |wT (µx −µy)| (wT Σxw)1/2 + (wT Σyw)1/2 , which is similar to the Fisher discriminant ratio f. With a specific uncertainty model on the means and covariances, the robust classification problem with discrimination criterion g can be cast as a second-order cone program, a special type of convex optimization problem [5]. With general uncertainty models, however, it is not clear whether robust discriminant analysis with g can be performed via convex optimization. 2 Robust Fisher LDA We first consider the worst-case analysis problem (1). Here we consider the discriminant w as fixed, and the parameters µx, µy, Σx, and Σy are variables, constrained to lie in the convex uncertainty set U. To show that (1) is a convex optimization problem, we must show that the Fisher discriminant ratio is a convex function of µx, µy, Σx, and Σy. To show this, we express the Fisher discriminant ratio f as the composition f(w, µx, µy, Σx, Σy) = g(H(µx, µy, Σx, Σy)), where g(u, t) = u2/t and H is the function H(µx, µy, Σx, Σy) = (wT (µx −µy), wT (Σx + Σy)w). The function H is linear (as a mapping from µx, µy, Σx, and Σy into R2), and the function g is convex (provided t > 0, which holds here). Thus, the composition f is a convex function of µx, µy, Σx, and Σy. (See [3].) Now we turn to the main result of this paper. Consider a function of the form R(w, a, B) = (wT a)2 wT Bw , (5) which is the Rayleigh quotient for the matrix pair aaT ∈Sn + and B ∈Sn ++, evaluated at w. The robust Fisher LDA problem (2) is equivalent to a problem of the form maximize min (a,B)∈V R(w, a, B) subject to w ̸= 0, (6) where a = µx−µy, B = Σx+Σy, V = {(µx−µy, Σx+Σy) | (µx, µy, Σx, Σy) ∈U}. (7) (This equivalence means that robust FLDA is a special type of robust matched filtering problem studied in the 1980s; see, e.g., [8] for more on robust matched filtering.) We will prove a ‘nonconventional’ minimax theorem for a Rayleigh quotient of the form (5), which will establish the main result described in §1. To do this, we consider a problem of the form minimize aT B−1a subject to (a, B) ∈V, (8) with variables a ∈Rn, B ∈Sn ++, and V is a convex compact subset of Rn × Sn ++ such that for each (a, B) ∈V, a is not zero. The objective of this problem is a matrix fractional function and so is convex on Rn ×Sn ++; see [3, §3.1.7]. Our problem (3) is the same as (8), with (7). It follows that (3) is a convex optimization problem. The following theorem states the minimax theorem for the function R. While R is convex in (a, B) for fixed w, it is not concave in w for fixed (a, B), so conventional convex-concave minimax theorems do not apply here. Theorem 1. Let (a⋆, B⋆) be an optimal solution to the problem (8), and let w⋆= B⋆−1a⋆. Then (w⋆, a⋆, B⋆) satisfies the minimax property R(w⋆, a⋆, B⋆) = max w̸=0 min (a,B)∈V R(w, a, B) = min (a,B)∈V max w̸=0 R(w, a, B), (9) and the saddle point property R(w, a⋆, B⋆) ≤R(w⋆, a⋆, B⋆) ≤R(w⋆, a, B), ∀w ∈Rn\{0}, ∀(a, B) ∈V. (10) Proof. It suffices to prove (10), since the saddle point property (10) implies the minimax property (9) [1, §2.6]. We start by observing that R(w, a⋆, B⋆) is maximized over nonzero w ̸= 0 by w⋆= B⋆−1a⋆(by the Cauchy-Schwartz inequality). What remains is to show min (a,B)∈V R(w⋆, a, B) = R(w⋆, a⋆, B⋆). (11) Since a⋆and B⋆are optimal for the convex problem (8) (by definition), they must satisfy the optimality condition D ∇a(aT B−1a) (a⋆,B⋆) , (a −a⋆) E + D ∇B(aT B−1a) (a⋆,B⋆) , (B −B⋆) E ≥0, ∀(a, B) ∈V (see [3, §4.2.3]). Using ∇a(aT B−1a) = 2B−1a, ∇B(aT B−1a) = −B−1aaT B−1, and ⟨X, Y ⟩= Tr(XY ) for X, Y ∈Sn, where Tr denotes trace, we can express the optimality condition as 2a⋆T B⋆−1(a −a⋆) −TrB⋆−1a⋆a⋆T B⋆−1(B −B⋆) ≥0, ∀(a, B) ∈V, or equivalently, 2w⋆T (a −a⋆) −w⋆T (B −B⋆)w⋆≥0, ∀(a, B) ∈V. (12) Now we turn to the convex optimization problem minimize R(w⋆, a, B) subject to (a, B) ∈V, (13) with variables (a, B). We will show that (a⋆, B⋆) is optimal for this problem, which will establish (11). A pair (¯a, ¯B) is optimal for (13) if and only if * ∇a (w⋆T a)2 w⋆T Bw⋆ (¯a, ¯ B) , (a −¯a) + + * ∇B (w⋆T a)2 w⋆T Bw⋆ (¯a, ¯ B) , (B −¯B) + ≥0, ∀(a, B) ∈V. Using ∇a (w⋆T a)2 w⋆T Bw⋆= 2 aT w⋆ w⋆Bw⋆w⋆, ∇B (w⋆T a)2 w⋆T Bw⋆= − (aT w⋆)2 (w⋆T Bw⋆)2 w⋆w⋆T , the optimality condition can be written as 2 ¯aT w⋆ w⋆T ¯Bw⋆w⋆T (a −¯a) −Tr (¯aT w⋆)2 (w⋆T ¯Bw⋆)2 w⋆w⋆T (B −¯B) = 2 ¯aT w⋆ w⋆T ¯Bw⋆w⋆T (a −¯a) − (¯aT w⋆)2 (w⋆T ¯Bw⋆)2 w⋆T (B −¯B)w⋆ ≥ 0, ∀(a, B) ∈V. Substituting ¯a = a⋆, ¯B = B⋆, and noting that a⋆T w⋆/w⋆T B⋆w⋆= 1, the optimality condition reduces to 2w⋆T (a −a⋆) −w⋆T (B −B⋆)w⋆≥0, ∀(a, B) ∈V, which is precisely (12). Thus, we have shown that (a⋆, B⋆) is optimal for (13), which in turn establishes (11). 3 Robust Fisher LDA with product form uncertainty models In this section, we focus on robust Fisher LDA with the product form uncertainty model U = M × S, (14) where M is the set of possible means and S is the set of possible covariances. For this model, the worst-case Fisher discriminant ratio can be written as min (µx,µy,Σx,Σy)∈U f(a, µx, µy, Σx, Σy) = min (µx,µy)∈M (wT (µx −µy))2 max(Σx,Σy)∈S wT (Σx + Σy)w. If we can find an analytic expression for max(Σx,Σy)∈S wT (Σx + Σy)w (as a function of w), we can simplify the robust Fisher LDA problem. As a more specific example, we consider the case in which S is given by S = Sx × Sy, Sx = {Σx | Σx ⪰0, ∥Σx −¯Σx∥F ≤δx}, Sy = {Σy | Σy ⪰0, ∥Σy −¯Σy∥F ≤δy}, (15) where δx, δy are positive constants, ¯Σx, ¯Σy ∈Sn ++, and ∥A∥F denotes the Frobenius norm of A, i.e., ∥A∥F = (Pn i,j=1 A2 ij)1/2. For this case, we have max (Σx,Σy)∈S wT (Σx + Σy)w = wT (¯Σx + ¯Σy + (δx + δy)I)w. (16) Here we have used the fact that for given ¯Σ ∈Sn ++, max∥Σ−¯Σ∥F ≤δ xT Σx = xT (¯Σ + δI)x (see, e.g., [6]). The worst-case Fisher discriminant ratio can be expressed as min (µx,µy)∈M (wT (µx −µy))2 wT (¯Σx + ¯Σy + (δx + δy)I)w. This is the same worst-case Fisher discriminant ratio obtained for a problem in which the covariances are certain, i.e., fixed to be ¯Σx +δxI and ¯Σy +δyI, and the means lie in the set M. We conclude that a robust optimal Fisher discriminant with the uncertainty model (14) in which S has the form (15) can be found by solving a robust Fisher LDA problem with these fixed values for the covariances. From the general solution method described in §1, it is given by w⋆= ¯Σx + ¯Σy + (δx + δy)I −1 (µ⋆ x −µ⋆ y), where µ⋆ x and µ⋆ y solve the convex optimization problem minimize (µx −µy)T ¯Σx + ¯Σy + (δx + δy)I −1 (µx −µy) subject to (µx, µy) ∈M, (17) with variables µx and µy. The problem (17) is relatively simple: it involves minimizing a convex quadratic function over the set of possible µx and µy. For example, if M is a product of two ellipsoids, (e.g., µx and µy each lie in some confidence ellipsoid) the problem (17) is to minimize a convex quadratic subject to two convex quadratic constraints. Such a problem is readily solved in O(n3) flops, since the dual problem has two variables, and evaluating the dual function and its derivatives can be done in O(n3) flops [3]. Thus, the effort to solve the robust is the same order (i.e., n3) as solving the nominal Fisher LDA (but with a substantially larger constant). 4 Numerical results To demonstrate robust Fisher LDA, we use the sonar and ionosphere benchmark problems from the UCI repository (www.ics.uci.edu/∼mlearn/MLRepository.html). The two benchmark problems have 208 and 351 points, respectively, and the dimension of each data point is 60 and 34, respectively. Each data set is randomly partitioned into a training set and a test set. We use the training set to compute the optimal discriminant and then test its performance using the test set. A larger training set typically gives better test performance. We let α denote the size of the training set, as a fraction of the total number of data points. For example, α = 0.3 means that 30% of the data points are used for training, and 70% are used to test the resulting discriminant. For various values of α, we generate 100 random partitions of the data (for each of the two benchmark problems), and collect the results. We use the following uncertainty models for the means µx, µy and the covariances Σx, Σy: (µx −¯µx)T Px(µx −¯µx) ≤1, ∥Σx −¯Σx∥F ≤ρx, (µy −¯µy)T Py(µy −¯µy) ≤1, ∥Σy −¯Σy∥F ≤ρy, Here the vectors ¯µx, ¯µy represent the nominal means and the matrices ¯Σx, ¯Σy represent the nominal covariances, and the matrices Px, Py and the constants ρx and ρy represent the confidence regions. The parameters are estimated through a resampling technique [4] as follows. For a given training set we create 100 new sets by resampling the original training set with a uniform distribution over all the data points. For each of these sets we estimate its mean and covariance and then take their average values as the nominal mean and covariance. We also evaluate the covariance Σµ of all the means obtained with the resampling. We then take Px = Σ−1 µ /n and Py = Σ−1 µ /n. This choice corresponds to a 50% confidence ellipsoid in the case of a Gaussian distribution. The parameters ρx and ρy are taken to be the maximum deviations between the covariances and the average covariances in the Frobenius norm sense, over the resampling of the training set. α (%) TSA (%) sonar robust nominal 20 30 40 50 60 70 80 50 60 70 80 90 100 α (%) 0 10 20 30 40 50 60 50 60 70 80 90 100 ionosphere robust nominal Figure 1: Test-set accuracy (TSA) for sonar and ionosphere benchmark versus size of the training set. The solid line represents the robust Fisher LDA results and the dotted line the nominal Fisher LDA results. The vertical bars represent the standard deviation. Figure 1 summarizes the classification results. For each of our two problems, and for each value of α, we show the average test set accuracy (TSA), as well as the standard deviation (over the 100 instances of each problem with the given value of α). The plots show the robust Fisher LDA performs substantially better than the nominal Fisher LDA for small training sets, but this performance gap disappears as the training set becomes larger. 5 Robust kernel Fisher discriminant analysis In this section we show how to ‘kernelize’ the robust Fisher LDA. We will consider only a specific class of uncertainty models; the arguments we develop here can be extended to more general cases. In the kernel approach we map the problem to an higher dimensional space Rf via a mapping φ : Rn →Rf so that the new decision boundary is more general and possibly nonlinear. Let the data be mapped as x →φ(x) ∼(¯µφ(x), ¯Σφ(x)), y →φ(y) ∼(¯µφ(y), ¯Σφ(y)). The uncertainty model we consider has the form µφ(x) −µφ(y) = ¯µφ(x) −¯µφ(y) + Puf, ∥uf∥≤1, ∥Σφ(x) −¯Σφ(x)∥F ≤ρx, ∥Σφ(y) −¯Σφ(y)∥F ≤ρy. (18) Here the vectors ¯µφ(x), ¯µφ(y) represent the nominal means, the matrices ¯Σφ(x), ¯Σφ(y) represent the nominal covariances, and the (positive semidefinite) matrix P and the constants ρx and ρy represent the confidence regions in the feature space. The worst-case Fisher discriminant ratio in the feature space is then given by min ∥uf ∥≤1,∥Σφ(x)−¯Σφ(x)∥F ≤ρx,∥Σφ(y)−¯Σφ(y)∥F ≤ρy (wT f (¯µφ(x) −¯µφ(y) + Puf))2 wT f (Σφ(x) + Σφ(y))wf . The robust kernel Fisher discriminant analysis problem is to find the discriminant in the feature space that maximizes this ratio. Using the technique described in §3, we can see that the robust kernel Fisher discriminant analysis problem can be cast as maximize min ∥uf ∥≤1 (wT f (¯µφ(x) −¯µφ(y) + Puf))2 wT f (¯Σφ(x) + ¯Σφ(y) + (ρx + ρy)I)wf subject to wf ̸= 0, (19) where the discriminant wf ∈Rf is defined in the new feature space. To apply the kernel trick to the problem (19), the nonlinear decision boundary should be entirely expressed in terms of inner products of the mapped data only. The following proposition tells us a set of conditions to do so. Proposition 1. Given the sample points {xi}Nx i=1 and {yi}Ny i=1, suppose that ¯µφ(x),¯µφ(y), ¯Σφ(x),¯Σφ(y), and P can be written as ¯µφ(x) = PNx i=1 λiφ(xi), ¯µφ(y) = PNy i=1 λi+Nxφ(yi), P = UΥU T , ¯Σφ(x) = PNx i=1 Λi,i(φ(xi) −¯µφ(x))(φ(xi) −¯µφ(x))T , ¯Σφ(y) = PNy i=1 Λi+Nx,i+Nx(φ(yi) −¯µφ(y))(φ(yi) −¯µφ(y))T , where λ ∈RNx+Ny, Υ ∈SNx+Ny + , Λ ∈SNx+Ny + is a diagonal matrix, and U is a matrix whose columns are the vectors {φ(xi) −¯µφ(x)}Nx i=1 and {φ(yi) −¯µφ(y)}Ny i=1. Denote as Φ the matrix whose columns are the vectors {φ(xi)}Nx i=1, {φ(yi)}Ny i=1 and define D1 = Kβ, D2 = K(I −λ1T N)Υ(I −λ1T N)KT , D3 = K(I −λ1T N)Λ(I −λ1T N)KT + (ρx + ρy)K, D4 = K, where K is the kernel matrix Kij = (ΦT Φ)ij, 1N is a vector of ones of length Nx + Ny, and β ∈RNx+Ny is such that βi = λi for i = 1, . . . , Nx and βi = −λi for i = Nx + 1, . . . , Nx + Ny. Let ν⋆be an optimal solution of the problem maximize min ξT D4ξ≤1 νT (D1 + D2ξ)(D1 + D2ξ)T ν νT D3ν subject to ν ̸= 0. (20) Then, w⋆ f = Φν⋆is an optimal solution of the problem (19). Moreover, for every point z ∈Rn, w⋆T f φ(z) = Nx X i=1 ν⋆ i K(z, xi) + Ny X i=1 ν⋆ i+NxK(z, yi). (21) Along the lines of the proofs of Corollary 5 in [6], we can prove this proposition. References [1] D. Bertsekas, A. Nedi´c, and A. Ozdaglar. Convex Analysis and Optimization. Athena Scientific, 2003. [2] C. Bhattacharyya. Second order cone programming formulations for feature selection. Journal of Machine Learning Research, 5:1417–1433, 2004. [3] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004. [4] B. Efron and R.J. Tibshirani. An Introduction to Bootstrap. Chapman and Hall, London UK, 1993. [5] K. Huang, H. Yang, I. King, M. Lyu, and L. Chan. The minimum error minimax probability machine. Journal of Machine Learning Research, 5:1253–1286, 2004. [6] G. Lanckriet, L. El Ghaoui, C. Bhattacharyya, and M. Jordan. A robust minimax approach to classification. Journal of Machine Learning Research, 3:555–582, 2002. [7] S. Mika, G. R¨atsch, and K. M¨uller. A mathematical programming approach to the kernel Fisher algorithm, 2001. In Advances in Neural Information Processing Systems, 13, pp. 591-597, MIT Press. [8] S. Verd´u and H. Poor. On minimax robustness: A general approach and applications. IEEE Transactions on Information Theory, 30(2):328–340, 1984.
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Nested sampling for Potts models Iain Murray Gatsby Computational Neuroscience Unit University College London i.murray@gatsby.ucl.ac.uk David J.C. MacKay Cavendish Laboratory University of Cambridge mackay@mrao.cam.ac.uk Zoubin Ghahramani Gatsby Computational Neuroscience Unit University College London zoubin@gatsby.ucl.ac.uk John Skilling Maximum Entropy Data Consultants Ltd. skilling@eircom.net Abstract Nested sampling is a new Monte Carlo method by Skilling [1] intended for general Bayesian computation. Nested sampling provides a robust alternative to annealing-based methods for computing normalizing constants. It can also generate estimates of other quantities such as posterior expectations. The key technical requirement is an ability to draw samples uniformly from the prior subject to a constraint on the likelihood. We provide a demonstration with the Potts model, an undirected graphical model. 1 Introduction The computation of normalizing constants plays an important role in statistical inference. For example, Bayesian model comparison needs the evidence, or marginal likelihood of a model M Z = p(D|M) = Z p(D|θ, M)p(θ|M) dθ ≡ Z L(θ)π(θ) dθ, (1) where the model has prior π and likelihood L over parameters θ after observing data D. This integral is usually intractable for models of interest. However, given its importance in Bayesian model comparison, many approaches—both sampling-based and deterministic—have been proposed for estimating it. Often the evidence cannot be obtained using samples drawn from either the prior π, or the posterior p(θ|D, M) ∝L(θ)π(θ). Practical Monte Carlo methods need to sample from a sequence of distributions, possibly at different “temperatures” p(θ|β) ∝L(θ)βπ(θ) (see Gelman and Meng [2] for a review). These methods are sometimes cited as a gold standard for comparison with other approximate techniques, e.g. Beal and Ghahramani [3]. However, care is required in choosing intermediate distributions; appropriate temperature-based distributions may be difficult or impossible to find. Nested sampling provides an alternate standard, which makes no use of temperature and does not require tuning of intermediate distributions or other large sets of parameters. θ1 θ2 x 1 1 2 1 4 1 8 L(x) (a) θ1 θ2 x 1 L(x) x1 x2 x3 (b) θ1 θ2 x 1 L(x) x1 (c) Figure 1: (a) Elements of parameter space (top) are sorted by likelihood and arranged on the x-axis. An eighth of the prior mass is inside the innermost likelihood contour in this figure. (b) Point xi is drawn from the prior inside the likelihood contour defined by xi−1. Li is identified and p({xi}) is known, but exact values of xi are not known. (c) With N particles, the least likely one sets the likelihood contour and is replaced by a new point inside the contour ({Li} and p({xi}) are still known). Nested sampling uses a natural definition of Z, a sum over prior mass. The weighted sum over likelihood elements is expressed as the area under a monotonic one-dimensional curve “L vs x” (figure 1(a)), where: Z = Z L(θ)π(θ) dθ = Z 1 0 L(θ(x)) dx. (2) This is a change of variables dx(θ) = π(θ)dθ, where each volume element of the prior in the original θ-vector space is mapped onto a scalar element on the one-dimensional x-axis. The ordering of the elements on the x-axis is chosen to sort the prior mass in decreasing order of likelihood values (x1 < x2 ⇒L(θ(x1)) > L(θ(x2))). See appendix A for dealing with elements with identical likelihoods. Given some points {(xi, Li)}I i=1 ordered such that xi > xi+1, the area under the curve (2) is easily approximated. We denote by ˆZ estimates obtained using a trapezoidal rule. Rectangle rules upper and lower bound the error ˆZ −Z. Points with known x-coordinates are unavailable in general. Instead we generate points, {θi}, such that the distribution p(x) is known (where x ≡{xi}), and find their associated {Li}. A simple algorithm to draw I points is algorithm 1, see also figure 1(b). Algorithm 1 Initial point: draw θ1 ∼π(θ). for i = 2 to I: draw θi ∼˘π(θ|L(θi−1)), where ˘π(θ|L(θi−1)) ∝ π(θ) L(θ) > L(θi−1) 0 otherwise. (3) Algorithm 2 Initialize: draw N points θ(n) ∼π(θ) for i = 2 to I: • m = argminn L(θ(n)) • θi−1 = θ(m) • draw θm ∼˘π(θ|L(θi−1)), given by equation (3) We know p(x1) = Uniform(0, 1), because x is a cumulative sum of prior mass. Similarly p(xi|xi−1) = Uniform(0, xi−1), as every point is drawn from the prior subject to L(θi) > L(θi−1) ⇒xi < xi−1. This recursive relation allows us to compute p(x). A simple generalization, algorithm 2, uses multiple θ particles; at each step the least likely is replaced with a draw from a constrained prior (figure 1(c)). Now p(x1|N)= NxN−1 1 and subsequent points have p(xi/xi−1|xi−1, N) = N(xi/xi−1)N−1. This 1e-120 1e-100 1e-80 1e-60 1e-40 1e-20 1 0 200 400 600 800 1000 1200 1400 1600 1800 2000 xi i ⟨x⟩ exp(⟨log x⟩) error bars 1e-115 1e-110 1e-105 1e-100 1e-95 1e-90 1e-85 1e-80 1e-75 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950 2000 xi i ⟨x⟩ exp(⟨log x⟩) error bars Figure 2: The arithmetic and geometric means of xi against iteration number, i, for algorithm 2 with N = 8. Error bars on the geometric mean show exp(−i/N ± √ i/N). Samples of p(x|N) are superimposed (i = 1600 . . . 1800 omitted for clarity). distribution over x combined with observations {Li} gives a distribution over ˆZ: p( ˆZ|{Li}, N) ≈ Z δ( ˆZ(x) −ˆZ)p(x|N) dx. (4) Samples from the posterior over θ are also available, see Skilling [1] for details. Nested sampling was introduced by Skilling [1]. The key idea is that samples from the prior, subject to a nested sequence of constraints (3), give a probabilistic realization of the curve, figure 1(a). Related work can be found in McDonald and Singer [4]. Explanatory notes and some code are available online1. In this paper we present some new discussion of important issues regarding the practical implementation of nested sampling and provide the first application to a challenging problem. This leads to the first cluster-based method for Potts models with first-order phase transitions of which we are aware. 2 Implementation issues 2.1 MCMC approximations The nested sampling algorithm assumes obtaining samples from ˘π(θ|L(θi−1)), equation (3), is possible. Rejection sampling using π would slow down exponentially with iteration number i. We explore approximate sampling from ˘π using Markov chain Monte Carlo (MCMC) methods. In high-dimensional problems it is likely that the majority of ˘π’s mass is typically in a thin shell at the contour surface [5, p37]. This suggests finding efficient chains that sample at constant likelihood, a microcanonical distribution. In order to complete an ergodic MCMC method, we also need transition operators that can alter the likelihood (within the constraint). A simple Metropolis method may suffice. We must initialize the Markov chain for each new sample somewhere. One possibility is to start at the position of the deleted point, θi−1, on the contour constraint, which is independent of the other points and not far from the bulk of the required uniform distribution. However, if the Markov chain mixes slowly amongst modes, the new point starting at θi−1 may be trapped in an insignificant mode. In this case it would be better to start at one of the other N −1 existing points inside the contour constraint. They are all draws from the correct distribution, ˘π(θ|L(θi−1)), so represent modes fairly. However, this method may also require many Markov chain steps, this time to make the new point effectively independent of the point it cloned. 1http://www.inference.phy.cam.ac.uk/bayesys/ −5 0 5 0 10 20 30 (a) −5 0 5 0 100 200 300 (b) −5 0 5 0 50 100 150 200 250 (c) Figure 3: Histograms of errors in the point estimate log( ˜ Z) over 1000 random experiments for different approximations. The test system was a 40-dimensional hypercube of length 100 with uniform prior centered on the origin. The loglikelihood was L = −θ⊤θ/2. Nested sampling used N = 10, I = 2000. (a) Monte Carlo estimation (equation (5)) using S = 12 sampled trajectories (b) S = 1200 sampled trajectories. (c) Deterministic approximation using the geometric mean trajectory. In this example perfect integration over p(x|N) gives a distribution of width ≈3 over log( ˆ Z). Therefore, improvements over (c) for approximating equation (5) are unwarranted. 2.2 Integrating out x To estimate quantities of interest, we average over p(x|N), as in equation (4). The mean of a distribution over log( ˆZ) can be found by simple Monte Carlo estimation: log(Z) ≈ Z log( ˆZ(x))p(x|N) dx ≈1 S S X s=1 log( ˆZ(x(s))) x(s) ∼p(x|N). (5) This scheme is easily implemented for any expectation under p(x|N), including error bars from the variance of log( ˆZ). To reduce noise in comparisons between runs it is advisable to reuse the same samples from p(x|N) (e.g. clamp the seed used to generate them). A simple deterministic approximation is useful for understanding, and also provides fast to compute, low variance estimators. Figure 2 shows sampled trajectories of xi as the algorithm progresses. The geometric mean path, xi ≈ exp( R p(xi|N) log xi dxi) = e−i/N, follows the path of typical settings of x. Using this single x setting is a reasonable and very cheap alternative to averaging over settings (equation 5); see figure 3. Typically the trapezoidal estimate of the integral, ˆZ, is dominated by a small number of trapezoids, around iteration i∗say. Considering uncertainty on just log xi∗= −i∗/N ± √ i∗/N provides reasonable and convenient error bars. 3 Potts Models The Potts model, an undirected graphical model, defines a probability distribution over discrete variables s = (s1, . . . , sn), each taking on one of q distinct “colors”: P (s|J, q) = 1 ZP(J, q) exp X (ij)∈E J(δsisj −1) . (6) The variables exist as nodes on a graph where (ij) ∈E means that nodes i and j are linked by an edge. The Kronecker delta, δsisj is one when si and sj are the same color and zero otherwise. Neighboring nodes pay an “energy penalty” of J when they are different colors. Here we assume identical positive couplings J > 0 on each edge (section 4 discusses the extension to different Jij). The Ising model and Boltzmann machine are both special cases of the Potts model with q=2. Our goal is to compute the normalization constant ZP(J, q), where the discrete variables s are the θ variables that need to be integrated (i.e. summed) over. 3.1 Swendsen–Wang sampling We will take advantage of the “Fortuin-Kasteleyn-Swendsen-Wang” (FKSW) joint distribution identified explicitly in Edwards and Sokal [6] over color variables s and a bond variable for each edge in E, dij ∈{0, 1}: P (s, d) = 1 ZP(J, q) Y (ij)∈E (1 −p)δdij,0 + pδdij,1δsi,sj , p ≡(1 −e−J). (7) The marginal distribution over s in the FKSW model is the Potts distribution, equation (6). The marginal distribution over the bonds is the random cluster model of Fortuin and Kasteleyn [7]: P (d) = 1 ZP(J, q)pD(1−p)|E|−DqC(d) = 1 ZP(J, q) exp(D log(eJ −1))e−J|E|qC(d), (8) where C(d) is the number of connected components in a graph with edges wherever dij =1, and D=P (ij)∈E dij. As the partition functions of equations 6, 7 and 8 are identical, we should consider using any of these distributions to compute ZP(J, q). The algorithm of Swendsen and Wang [8] performs block Gibbs sampling on the joint model by alternately sampling from P(dij|s) and P(s|dij). This can convert a sample from any of the three distributions into a sample from one of the others. 3.2 Nested Sampling A simple approximate nested sampler uses a fixed number of Gibbs sampling updates of ˘π. Cluster-based updates are also desirable in these models. Focusing on the random cluster model, we rewrite equation (8): P (d) = 1 ZN L(d)π(d) where (9) ZN = ZP(J, q) Zπ exp(J|E|), L(d) = exp(D log(eJ −1)), π(d) = 1 Zπ qC(d). Likelihood thresholds are thresholds on the total number of bonds D. Many states have identical D, which requires careful treatment, see appendix A. Nested sampling on this system will give the ratio of ZP/Zπ. The prior normalization, Zπ, can be found from the partition function of a Potts system at J = log(2). The following steps give two MCMC operators to change the bonds d →d′: 1. Create a random coloring, s, uniformly from the qC(d) colorings satisfying the bond constraints d, as in the Swendsen–Wang algorithm. 2. Count sites that allow bonds, E = P (ij)∈E δsi,sj. 3. Either, operator 1: record the number of bonds D′ = P (ij)∈E dij Or, operator 2: draw D′ from Q(D′|E(s)) ∝ E(s) D′ . 4. Throw away the old bonds, d, and pick uniformly from one of the E(s) D′ ways of setting D′ bonds in the E available sites. The probability of proposing a particular coloring and new setting of the bonds is Q(s, d′|d) = Q(d′|s, D′)Q(D′|E(s))Q(s|d) = 1 E(s) D′ Q(D′|E(s)) 1 qC(d) . (10) Summing over colorings, the correct Metropolis-Hastings acceptance ratio is: a = π(d′) π(d) · P s Q(s, d|d′) P s Q(s, d′|d) = qC(d′) qC(d) · qC(d) qC(d′) P s Q(D|s)/ E(s) D P s Q(D′|s)/ E(s) D′ = 1, (11) Table 1: Partition function results for 16×16 Potts systems (see text for details). Method q = 2 (Ising), J = 1 q = 10, J = 1.477 Gibbs AIS 7.1 ± 1.1 (1.5) Swendsen–Wang AIS 7.4 ± 0.1 (1.2) Gibbs nested sampling 7.1 ± 1.0 12.2 ± 2.4 Random-cluster nested sampling 7.1 ± 0.7 14.1 ± 1.8 Acceptance ratio 7.3 11.2 regardless of the choice in step 3. The simple first choice solves the difficult problem of navigating at constant D. The second choice defines an ergodic chain2. 4 Results Table 1 shows results on two example systems: an Ising model, q = 1, and a q = 10 Potts model in an difficult parameter regime. We tested nested samplers using Gibbs sampling and the cluster-based algorithm, annealed importance sampling (AIS) [9] using both Gibbs sampling and Swendsen–Wang cluster updates. We also developed an acceptance ratio method [10] based on our representation in equation (9), which we ran extensively and should give nearly correct results. Annealed importance sampling (AIS) was run 100 times, with a geometric spacing of 104 settings of J as the annealing schedule. Nested sampling used N = 100 particles and 100 full-system MCMC updates to approximate each draw from ˘π. Each Markov chain was initialized at one of the N−1 particles satisfying the current constraint. In trials using the other alternative (section 2.1) the Gibbs nested sampler could get stuck permanently in a local maximum of the likelihood, while the cluster method gave erroneous answers for the Ising system. AIS performed very well on the Ising system. We took advantage of its performance in easy parameter regimes to compute Zπ for use in the cluster-based nested sampler. However, with a “temperature-based” annealing schedule, AIS was unable to give useful answers for the q = 10 system. While nested sampling appears to be correct within its error bars. It is known that even the efficient Swendsen–Wang algorithm mixes slowly for Potts models with q > 4 near critical values of J [11], see figure 4. Typical Potts model states are either entirely disordered or ordered; disordered states contain a jumble of small regions with different colors (e.g. figure 4(b)), in ordered states the system is predominantly one color (e.g. figure 4(d)). Moving between these two phases is difficult; defining a valid MCMC method that moves between distinct phases requires knowledge of the relative probability of the whole collections of states in those phases. Temperature-based annealing algorithms explore the model for a range of settings of J and fail to capture the correct behavior near the transition. Despite using closely related Markov chains to those used in AIS, nested sampling can work in all parameter regimes. Figure 4(e) shows how nested sampling can explore a mixture of ordered and disordered phases. By moving steadily through these states, nested sampling is able to estimate the prior mass associated with each likelihood value. 2Proof: with finite probability all si are given the same color, then any allowable D′ is possible, in turn all allowable d′ have finite probability. (a) ⇒ (b) (c) ⇒ (d) (e) Figure 4: Two 256 × 256, q = 10 Potts models with starting states (a) and (c) were simulated with 5 × 106 full-system Swendsen–Wang updates with J = 1.42577. The corresponding results, (b) and (d) are typical of all the intermediate samples: Swendsen– Wang is unable to take (a) into an ordered phase, or (c) into a disordered phase, although both phases are typical at this J. (e) in contrast shows an intermediate state of nested sampling, which succeeds in bridging the phases. This behaviour is not possible in algorithms that use J as a control parameter. The potentials on every edge of the Potts model in this paper were the same. Much of the formalism above generalizes to allow different edge weights Jij on each edge, and non-zero biases on each variable. Indeed Edwards and Sokal [6] gave a general procedure for constructing such auxiliary-variable joint distributions. This generalization would make the model more relevant to MRFs used in other fields (e.g. computer vision). The challenge for nested sampling remains the invention of effective sampling schemes that keep a system at or near constant energy. Generalizing step 4 in section 3.2 would be the difficult step. Other temperatureless Monte Carlo methods exist, e.g. Berg and Neuhaus [12] study the Potts model using the multicanonical ensemble. Nested sampling has some unique properties compared to the established method. Formally it has only one free parameter, N the number of particles. Unless problems with multiple modes demand otherwise, N = 1 often reveals useful information, and if the error bars on Z are too large further runs with larger N may be performed. 5 Conclusions We have applied nested sampling to compute the normalizing constant of a system that is challenging for many Monte Carlo methods. • Nested sampling’s key technical requirement, an ability to draw samples uniformly from a constrained prior, is largely solved by efficient MCMC methods. • No complex schedules are required; steady progress towards compact regions of large likelihood is controlled by a single free parameter, N, the number of particles. • Multiple particles, a built-in feature of this algorithm, are often necessary to obtain accurate results. • Nested sampling has no special difficulties on systems with first order phasetransitions, whereas all temperature-based methods fail. We believe that nested sampling’s unique properties will be found useful in a variety of statistical applications. A Degenerate likelihoods The description in section 1 assumed that the likelihood function provides a total ordering of elements of the parameter space. However, distinct elements dx and dx′ could have the same likelihood, either because the parameters are discrete, or because the likelihood is degenerate. One way to break degeneracies is through a joint model with variables of interest θ and an independent variable m ∈[0, 1]: P (θ, m) = P (θ) · P (m) = 1 Z L(θ)π(θ) · 1 Zm L(m)π(m) (12) where L(m) = 1 + ϵ(m −0.5), π(m) = 1 and Zm = 1. We choose ϵ such that log(ϵ) is smaller than the smallest difference in log(L(θ)) allowed by machine precision. Standard nested sampling is now possible. Assuming we have a likelihood constraint Li, we need to be able to draw from P (θ′, m′|θ, m, Li) ∝ π(θ′)π(m′) L(θ′)L(m′) > Li, 0 otherwise. (13) The additional variable can be ignored except for L(θ′) = L(θi), then only m′ > m are possible. Therefore, the probability of states with likelihood L(θi) are weighted by (1 −m′). References [1] John Skilling. Nested sampling. In R. Fischer, R. Preuss, and U. von Toussaint, editors, Bayesian inference and maximum entropy methods in science and engineering, AIP Conference Proceeedings 735, pages 395–405, 2004. [2] Andrew Gelman and Xiao-Li Meng. Simulating normalizing constants: from importance sampling to bridge sampling to path sampling. Statist. Sci., 13(2):163–185, 1998. [3] Matthew J. Beal and Zoubin Ghahramani. The variational Bayesian EM algorithm for incomplete data: with application to scoring graphical model structures. Bayesian Statistics, 7:453–464, 2003. [4] I. R. McDonald and K. Singer. Machine calculation of thermodynamic properties of a simple fluid at supercritical temperatures. J. Chem. Phys., 47(11):4766–4772, 1967. [5] David J.C. MacKay. Information Theory, Inference, and Learning Algorithms. CUP, 2003. www.inference.phy.cam.ac.uk/mackay/itila/. [6] Robert G. Edwards and Alan D. Sokal. Generalization of the Fortuin-KasteleynSwendsen-Wang representation and Monte Carlo algorithm. Phys.Rev. D, 38(6), 1988. [7] C. M. Fortuin and P. W. Kasteleyn. On the random-cluster model. I. Introduction and relation to other models. Physica, 57:536–564, 1972. [8] R. H. Swendsen and J. S. Wang. Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett., 58(2):86–88, January 1987. [9] Radford M. Neal. Annealed importance sampling. Statistics and Computing, 11: 125–139, 2001. [10] Charles H. Bennett. Efficient estimation of free energy differences from Monte Carlo data. Journal of Computational Physics, 22(2):245–268, October 1976. [11] Vivek K. Gore and Mark R. Jerrum. The Swendsen-Wang process does not always mix rapidly. In 29th ACM Symposium on Theory of Computing, pages 674–681, 1997. [12] Bernd A. Berg and Thomas Neuhaus. Multicanonical ensemble: A new approach to simulate first-order phase transitions. Phys. Rev. Lett., 68(1):9–12, January 1992.
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Correlated Topic Models David M. Blei Department of Computer Science Princeton University John D. Lafferty School of Computer Science Carnegie Mellon University Abstract Topic models, such as latent Dirichlet allocation (LDA), can be useful tools for the statistical analysis of document collections and other discrete data. The LDA model assumes that the words of each document arise from a mixture of topics, each of which is a distribution over the vocabulary. A limitation of LDA is the inability to model topic correlation even though, for example, a document about genetics is more likely to also be about disease than x-ray astronomy. This limitation stems from the use of the Dirichlet distribution to model the variability among the topic proportions. In this paper we develop the correlated topic model (CTM), where the topic proportions exhibit correlation via the logistic normal distribution [1]. We derive a mean-field variational inference algorithm for approximate posterior inference in this model, which is complicated by the fact that the logistic normal is not conjugate to the multinomial. The CTM gives a better fit than LDA on a collection of OCRed articles from the journal Science. Furthermore, the CTM provides a natural way of visualizing and exploring this and other unstructured data sets. 1 Introduction The availability and use of unstructured historical collections of documents is rapidly growing. As one example, JSTOR (www.jstor.org) is a not-for-profit organization that maintains a large online scholarly journal archive obtained by running an optical character recognition engine over the original printed journals. JSTOR indexes the resulting text and provides online access to the scanned images of the original content through keyword search. This provides an extremely useful service to the scholarly community, with the collection comprising nearly three million published articles in a variety of fields. The sheer size of this unstructured and noisy archive naturally suggests opportunities for the use of statistical modeling. For instance, a scholar in a narrow subdiscipline, searching for a particular research article, would certainly be interested to learn that the topic of that article is highly correlated with another topic that the researcher may not have known about, and that is not explicitly contained in the article. Alerted to the existence of this new related topic, the researcher could browse the collection in a topic-guided manner to begin to investigate connections to a previously unrecognized body of work. Since the archive comprises millions of articles spanning centuries of scholarly work, automated analysis is essential. Several statistical models have recently been developed for automatically extracting the topical structure of large document collections. In technical terms, a topic model is a generative probabilistic model that uses a small number of distributions over a vocabulary to describe a document collection. When fit from data, these distributions often correspond to intuitive notions of topicality. In this work, we build upon the latent Dirichlet allocation (LDA) [4] model. LDA assumes that the words of each document arise from a mixture of topics. The topics are shared by all documents in the collection; the topic proportions are document-specific and randomly drawn from a Dirichlet distribution. LDA allows each document to exhibit multiple topics with different proportions, and it can thus capture the heterogeneity in grouped data that exhibit multiple latent patterns. Recent work has used LDA in more complicated document models [9, 11, 7], and in a variety of settings such as image processing [12], collaborative filtering [8], and the modeling of sequential data and user profiles [6]. Similar models were independently developed for disability survey data [5] and population genetics [10]. Our goal in this paper is to address a limitation of the topic models proposed to date: they fail to directly model correlation between topics. In many—indeed most—text corpora, it is natural to expect that subsets of the underlying latent topics will be highly correlated. In a corpus of scientific articles, for instance, an article about genetics may be likely to also be about health and disease, but unlikely to also be about x-ray astronomy. For the LDA model, this limitation stems from the independence assumptions implicit in the Dirichlet distribution on the topic proportions. Under a Dirichlet, the components of the proportions vector are nearly independent; this leads to the strong and unrealistic modeling assumption that the presence of one topic is not correlated with the presence of another. In this paper we present the correlated topic model (CTM). The CTM uses an alternative, more flexible distribution for the topic proportions that allows for covariance structure among the components. This gives a more realistic model of latent topic structure where the presence of one latent topic may be correlated with the presence of another. In the following sections we develop the technical aspects of this model, and then demonstrate its potential for the applications envisioned above. We fit the model to a portion of the JSTOR archive of the journal Science. We demonstrate that the model gives a better fit than LDA, as measured by the accuracy of the predictive distributions over held out documents. Furthermore, we demonstrate qualitatively that the correlated topic model provides a natural way of visualizing and exploring such an unstructured collection of textual data. 2 The Correlated Topic Model The key to the correlated topic model we propose is the logistic normal distribution [1]. The logistic normal is a distribution on the simplex that allows for a general pattern of variability between the components by transforming a multivariate normal random variable. Consider the natural parameterization of a K-dimensional multinomial distribution: p(z | η) = exp{ηT z −a(η)}. (1) The random variable Z can take on K values; it can be represented by a K-vector with exactly one component equal to one, denoting a value in {1, . . . , K}. The cumulant generating function of the distribution is a(η) = log PK i=1 exp{ηi} . (2) The mapping between the mean parameterization (i.e., the simplex) and the natural parameterization is given by ηi = log θi/θK. (3) Notice that this is not the minimal exponential family representation of the multinomial because multiple values of η can yield the same mean parameter. Zd,n Wd,n N D K Σ µ ηd βk Figure 1: Top: Graphical model representation of the correlated topic model. The logistic normal distribution, used to model the latent topic proportions of a document, can represent correlations between topics that are impossible to capture using a single Dirichlet. Bottom: Example densities of the logistic normal on the 2-simplex. From left: diagonal covariance and nonzero-mean, negative correlation between components 1 and 2, positive correlation between components 1 and 2. The logistic normal distribution assumes that η is normally distributed and then mapped to the simplex with the inverse of the mapping given in equation (3); that is, f(ηi) = exp ηi/ P j exp ηj. The logistic normal models correlations between components of the simplicial random variable through the covariance matrix of the normal distribution. The logistic normal was originally studied in the context of analyzing observed compositional data such as the proportions of minerals in geological samples. In this work, we extend its use to a hierarchical model where it describes the latent composition of topics associated with each document. Let {µ, Σ} be a K-dimensional mean and covariance matrix, and let topics β1:K be K multinomials over a fixed word vocabulary. The correlated topic model assumes that an N-word document arises from the following generative process: 1. Draw η | {µ, Σ} ∼N(µ, Σ). 2. For n ∈{1, . . . , N}: (a) Draw topic assignment Zn | η from Mult(f(η)). (b) Draw word Wn | {zn, β1:K} from Mult(βzn). This process is identical to the generative process of LDA except that the topic proportions are drawn from a logistic normal rather than a Dirichlet. The model is shown as a directed graphical model in Figure 1. The CTM is more expressive than LDA. The strong independence assumption imposed by the Dirichlet in LDA is not realistic when analyzing document collections, where one may find strong correlations between topics. The covariance matrix of the logistic normal in the CTM is introduced to model such correlations. In Section 3, we illustrate how the higher order structure given by the covariance can be used as an exploratory tool for better understanding and navigating a large corpus of documents. Moreover, modeling correlation can lead to better predictive distributions. In some settings, such as collaborative filtering, the goal is to predict unseen items conditional on a set of observations. An LDA model will predict words based on the latent topics that the observations suggest, but the CTM has the ability to predict items associated with additional topics that are correlated with the conditionally probable topics. 2.1 Posterior inference and parameter estimation Posterior inference is the central challenge to using the CTM. The posterior distribution of the latent variables conditional on a document, p(η, z1:N | w1:N), is intractable to compute; once conditioned on some observations, the topic assignments z1:N and log proportions η are dependent. We make use of mean-field variational methods to efficiently obtain an approximation of this posterior distribution. In brief, the strategy employed by mean-field variational methods is to form a factorized distribution of the latent variables, parameterized by free variables which are called the variational parameters. These parameters are fit so that the Kullback-Leibler (KL) divergence between the approximate and true posterior is small. For many problems this optimization problem is computationally manageable, while standard methods, such as Markov Chain Monte Carlo, are impractical. The tradeoff is that variational methods do not come with the same theoretical guarantees as simulation methods. See [13] for a modern review of variational methods for statistical inference. In graphical models composed of conjugate-exponential family pairs and mixtures, the variational inference algorithm can be automatically derived from general principles [2, 14]. In the CTM, however, the logistic normal is not conjugate to the multinomial. We will therefore derive a variational inference algorithm by taking into account the special structure and distributions used by our model. We begin by using Jensen’s inequality to bound the log probability of a document: log p(w1:N | µ, Σ, β) ≥ (4) Eq [log p(η | µ, Σ)] + PN n=1(Eq [log p(zn | η)] + Eq [log p(wn | zn, β)]) + H (q) , where the expectation is taken with respect to a variational distribution of the latent variables, and H (q) denotes the entropy of that distribution. We use a factorized distribution: q(η1:K, z1:N | λ1:K, ν2 1:K, φ1:N) = QK i=1 q(ηi | λi, ν2 i ) QN n=1 q(zn | φn). (5) The variational distributions of the discrete variables z1:N are specified by the Kdimensional multinomial parameters φ1:N. The variational distribution of the continuous variables η1:K are K independent univariate Gaussians {λi, νi}. Since the variational parameters are fit using a single observed document w1:N, there is no advantage in introducing a non-diagonal variational covariance matrix. The nonconjugacy of the logistic normal leads to difficulty in computing the expected log probability of a topic assignment: Eq [log p(zn | η)] = Eq ηT zn −Eq h log(PK i=1 exp{ηi}) i . (6) To preserve the lower bound on the log probability, we upper bound the log normalizer with a Taylor expansion, Eq h log PK i=1 exp{ηi} i ≤ζ−1(PK i=1 Eq [exp{ηi}]) −1 + log(ζ), (7) where we have introduced a new variational parameter ζ. The expectation Eq [exp{ηi}] is the mean of a log normal distribution with mean and variance obtained from the variational parameters {λi, ν2 i }; thus, Eq [exp{ηi}] = exp{λi + ν2 i /2} for i ∈{1, . . . , K}. wild type mutant mutations mutants mutation gene yeast recombination phenotype genes p53 cell cycle activity cyclin regulation protein phosphorylation kinase regulated cell cycle progression amino acids cdna sequence isolated protein amino acid mrna amino acid sequence actin clone gene disease mutations families mutation alzheimers disease patients human breast cancer normal development embryos drosophila genes expression embryo developmental embryonic developmental biology vertebrate mantle crust upper mantle meteorites ratios rocks grains isotopic isotopic composition depth co2 carbon carbon dioxide methane water energy gas fuel production organic matter earthquake earthquakes fault images data observations features venus surface faults ancient found impact million years ago africa site bones years ago date rock climate ocean ice changes climate change north atlantic record warming temperature past genetic population populations differences variation evolution loci mtdna data evolutionary males male females female sperm sex offspring eggs species egg fossil record birds fossils dinosaurs fossil evolution taxa species specimens evolutionary synapses ltp glutamate synaptic neurons long term potentiation ltp synaptic transmission postsynaptic nmda receptors hippocampus ca2 calcium release ca2 release concentration ip3 intracellular calcium intracellular intracellular ca2 ca2 i ras atp camp gtp adenylyl cyclase cftr adenosine triphosphate atp guanosine triphosphate gtp gap gdp neurons stimulus motor visual cortical axons stimuli movement cortex eye ozone atmospheric measurements stratosphere concentrations atmosphere air aerosols troposphere measured brain memory subjects left task brains cognitive language human brain learning Figure 2: A portion of the topic graph learned from 16,351 OCR articles from Science. Each node represents a topic, and is labeled with the five most probable phrases from its distribution (phrases are found by the “turbo topics” method [3]). The interested reader can browse the full model at http://www.cs.cmu.edu/˜lemur/science/. Given a model {β1:K, µ, Σ} and a document w1:N, the variational inference algorithm optimizes equation (4) with respect to the variational parameters {λ1:K, ν1:K, φ1:N, ζ}. We use coordinate ascent, repeatedly optimizing with respect to each parameter while holding the others fixed. In variational inference for LDA, each coordinate can be optimized analytically. However, iterative methods are required for the CTM when optimizing for λi and ν2 i . The details are given in Appendix A. Given a collection of documents, we carry out parameter estimation in the correlated topic model by attempting to maximize the likelihood of a corpus of documents as a function of the topics β1:K and the multivariate Gaussian parameters {µ, Σ}. We use variational expectation-maximization (EM), where we maximize the bound on the log probability of a collection given by summing equation (4) over the documents. In the E-step, we maximize the bound with respect to the variational parameters by performing variational inference for each document. In the M-step, we maximize the bound with respect to the model parameters. This is maximum likelihood estimation of the topics and multivariate Gaussian using expected sufficient statistics, where the expectation is taken with respect to the variational distributions computed in the E-step. The E-step and M-step are repeated until the bound on the likelihood converges. In the experiments reported below, we run variational inference until the relative change in the probability bound of equation (4) is less than 10−6, and run variational EM until the relative change in the likelihood bound is less than 10−5. 3 Examples and Empirical Results: Modeling Science In order to test and illustrate the correlated topic model, we estimated a 100-topic CTM on 16,351 Science articles spanning 1990 to 1999. We constructed a graph of the latent topics and the connections among them by examining the most probable words from each topic and the between-topic correlations. Part of this graph is illustrated in Figure 2. In this subgraph, there are three densely connected collections of topics: material science, geology, and cell biology. Furthermore, an estimated CTM can be used to explore otherwise unstructured observed documents. In Figure 4, we list articles that are assigned to the cognitive science topic and articles that are assigned to both the cogNumber of topics Held−out log likelihood 5 10 20 30 40 50 60 70 80 90 100 110 120 −116400 −116000 −115600 −115200 −114800 −114400 −114000 −113600 −113200 −112800 G G G G G G G G G G G G G G G G G G G G G G G G G G CTM LDA G G G G G G G G G G G G G 10 20 30 40 50 60 70 80 90 100 110 120 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Number of topics L(CTM) − L(LDA) G G G G G G G G G G G G G Figure 3: (L) The average held-out probability; CTM supports more topics than LDA. See figure at right for the standard error of the difference. (R) The log odds ratio of the held-out probability. Positive numbers indicate a better fit by the correlated topic model. nitive science and visual neuroscience topics. The interested reader is invited to visit http://www.cs.cmu.edu/˜lemur/science/ to interactively explore this model, including the topics, their connections, and the articles that exhibit them. We compared the CTM to LDA by fitting a smaller collection of articles to models of varying numbers of topics. This collection contains the 1,452 documents from 1960; we used a vocabulary of 5,612 words after pruning common function words and terms that occur once in the collection. Using ten-fold cross validation, we computed the log probability of the held-out data given a model estimated from the remaining data. A better model of the document collection will assign higher probability to the held out data. To avoid comparing bounds, we used importance sampling to compute the log probability of a document where the fitted variational distribution is the proposal. Figure 3 illustrates the average held out log probability for each model and the average difference between them. The CTM provides a better fit than LDA and supports more topics; the likelihood for LDA peaks near 30 topics while the likelihood for the CTM peaks close to 90 topics. The means and standard errors of the difference in log-likelihood of the models is shown at right; this indicates that the CTM always gives a better fit. Another quantitative evaluation of the relative strengths of LDA and the CTM is how well the models predict the remaining words after observing a portion of the document. Suppose we observe words w1:P from a document and are interested in which model provides a better predictive distribution p(w | w1:P ) of the remaining words. To compare these distributions, we use perplexity, which can be thought of as the effective number of equally likely words according to the model. Mathematically, the perplexity of a word distribution is defined as the inverse of the per-word geometric average of the probability of the observations, Perp(Φ) = QD d=1 QNd i=P +1 p(wi | Φ, w1:P ) −1 PD d=1(Nd−P ) , where Φ denotes the model parameters of an LDA or CTM model. Note that lower numbers denote more predictive power. The plot in Figure 4 compares the predictive perplexity under LDA and the CTM. When a (1) Separate Neural Bases of Two Fundamental Memory Processes in the Human Medial Temporal Lobe (2) Inattentional Blindness Versus Inattentional Amnesia for Fixated but Ignored Words (3) Making Memories: Brain Activity that Predicts How Well Visual Experience Will be Remembered (4) The Learning of Categories: Parallel Brain Systems for Item Memory and Category Knowledge (5) Brain Activation Modulated by Sentence Comprehension (1) A Head for Figures (2) Sources of Mathematical Thinking: Behavioral and Brain Imaging Evidence (3) Natural Language Processing (4) A Romance Blossoms Between Gray Matter and Silicon (5) Computer Vision Top Articles with {brain, memory, human, visual, cognitive} Top Articles with {brain, memory, human, visual, cognitive} and {computer, data, information, problem, systems} % observed words Predictive perplexity 10 20 30 40 50 60 70 80 90 1800 2000 2200 2400 2600 G G G G G G G G G G G G G G G G G G CTM LDA Figure 4: (Left) Exploring a collection through its topics. (Right) Predictive perplexity for partially observed held-out documents from the 1960 Science corpus. small number of words have been observed, there is less uncertainty about the remaining words under the CTM than under LDA—the perplexity is reduced by nearly 200 words, or roughly 10%. The reason is that after seeing a few words in one topic, the CTM uses topic correlation to infer that words in a related topic may also be probable. In contrast, LDA cannot predict the remaining words as well until a large portion of the document as been observed so that all of its topics are represented. Acknowledgments Research supported in part by NSF grants IIS-0312814 and IIS0427206 and by the DARPA CALO project. References [1] J. Aitchison. The statistical analysis of compositional data. Journal of the Royal Statistical Society, Series B, 44(2):139–177, 1982. [2] C. Bishop, D. Spiegelhalter, and J. Winn. VIBES: A variational inference engine for Bayesian networks. In NIPS 15, pages 777–784. Cambridge, MA, 2003. [3] D. Blei, J. Lafferty, C. Genovese, and L. Wasserman. Turbo topics. In progress, 2006. [4] D. Blei, A. Ng, and M. Jordan. Latent Dirichlet allocation. Journal of Machine Learning Research, 3:993–1022, January 2003. [5] E. Erosheva. Grade of membership and latent structure models with application to disability survey data. PhD thesis, Carnegie Mellon University, Department of Statistics, 2002. [6] M. Girolami and A. Kaban. Simplicial mixtures of Markov chains: Distributed modelling of dynamic user profiles. In NIPS 16, pages 9–16, 2004. [7] T. Griffiths, M. Steyvers, D. Blei, and J. Tenenbaum. Integrating topics and syntax. In Advances in Neural Information Processing Systems 17, 2005. [8] B. Marlin. Collaborative filtering: A machine learning perspective. Master’s thesis, University of Toronto, 2004. [9] A. McCallum, A. Corrada-Emmanuel, and X. Wang. The author-recipient-topic model for topic and role discovery in social networks. 2004. [10] J. Pritchard, M. Stephens, and P. Donnelly. Inference of population structure using multilocus genotype data. Genetics, 155:945–959, June 2000. [11] M. Rosen-Zvi, T. Griffiths, M. Steyvers, and P. Smith. In UAI ’04: Proceedings of the 20th Conference on Uncertainty in Artificial Intelligence, pages 487–494. [12] J. Sivic, B. Rusell, A. Efros, A. Zisserman, and W. Freeman. Discovering object categories in image collections. Technical report, CSAIL, MIT, 2005. [13] M. Wainwright and M. Jordan. A variational principle for graphical models. In New Directions in Statistical Signal Processing, chapter 11. MIT Press, 2005. [14] E. Xing, M. Jordan, and S. Russell. A generalized mean field algorithm for variational inference in exponential families. In Proceedings of UAI, 2003. A Variational Inference We describe a coordinate ascent optimization algorithm for the likelihood bound in equation (4) with respect to the variational parameters. The first term of equation (4) is Eq [log p(η | µ, Σ)] = (1/2) log |Σ−1| −(K/2) log 2π −(1/2)Eq (η −µ)T Σ−1(η −µ) , (8) where Eq (η −µ)T Σ−1(η −µ) = Tr(diag(ν2)Σ−1) + (λ −µ)T Σ−1(λ −µ). (9) The second term of equation (4), using the additional bound in equation (7), is Eq [log p(zn | η)] = PK i=1 λiφn,i −ζ−1 PK i=1 exp{λi + ν2 i /2} + 1 −log ζ. (10) The third term of equation (4) is Eq [log p(wn | zn, β)] = PK i=1 φn,i log βi,wn. (11) Finally, the fourth term is the entropy of the variational distribution: PK i=1 1 2(log ν2 i + log 2π + 1) −PN n=1 Pk i=1 φn,i log φn,i. (12) We maximize the bound in equation (4) with respect to the variational parameters λ1:K, ν1:K, φ1:N, and ζ. We use a coordinate ascent algorithm, iteratively maximizing the bound with respect to each parameter. First, we maximize equation (4) with respect to ζ, using the second bound in equation (7). The derivative with respect to ζ is f ′(ζ) = N ζ−2 PK i=1 exp{λi + ν2 i /2} −ζ−1 , (13) which has a maximum at ˆζ = PK i=1 exp{λi + ν2 i /2}. (14) Second, we maximize with respect to φn. This yields a maximum at ˆφn,i ∝exp{λi}βi,wn, i ∈{1, . . . , K}. (15) Third, we maximize with respect to λi. Since equation (4) is not amenable to analytic maximization, we use a conjugate gradient algorithm with derivative dL/dλ = −Σ−1(λ −µ) + PN n=1 φn,1:K −(N/ζ) exp{λ + ν2/2} . (16) Finally, we maximize with respect to ν2 i . Again, there is no analytic solution. We use Newton’s method for each coordinate, constrained such that νi > 0: dL/dν2 i = −Σ−1 ii /2 −N/2ζ exp{λ + ν2 i /2} + 1/(2ν2 i ). (17) Iterating between these optimizations defines a coordinate ascent algorithm on equation (4).
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Learning in Silicon: Timing is Everything John V. Arthur and Kwabena Boahen Department of Bioengineering University of Pennsylvania Philadelphia, PA 19104 {jarthur, boahen}@seas.upenn.edu Abstract We describe a neuromorphic chip that uses binary synapses with spike timing-dependent plasticity (STDP) to learn stimulated patterns of activity and to compensate for variability in excitability. Specifically, STDP preferentially potentiates (turns on) synapses that project from excitable neurons, which spike early, to lethargic neurons, which spike late. The additional excitatory synaptic current makes lethargic neurons spike earlier, thereby causing neurons that belong to the same pattern to spike in synchrony. Once learned, an entire pattern can be recalled by stimulating a subset. 1 Variability in Neural Systems Evidence suggests precise spike timing is important in neural coding, specifically, in the hippocampus. The hippocampus uses timing in the spike activity of place cells (in addition to rate) to encode location in space [1]. Place cells employ a phase code: the timing at which a neuron spikes relative to the phase of the inhibitorytheta rhythm (5-12Hz) conveys information. As an animal approaches a place cell’s preferred location, the place cell not only increases its spike rate, but also spikes at earlier phases in the theta cycle. To implement a phase code, the theta rhythm is thought to prevent spiking until the input synaptic current exceeds the sum of the neuron threshold and the decreasing inhibition on the downward phase of the cycle [2]. However, even with identical inputs and common theta inhibition, neurons do not spike in synchrony. Variability in excitability spreads the activity in phase. Lethargic neurons (such as those with high thresholds) spike late in the theta cycle, since their input exceeds the sum of the neuron threshold and theta inhibition only after the theta inhibition has had time to decrease. Conversely, excitable neurons (such as those with low thresholds) spike early in the theta cycle. Consequently, variability in excitability translates into variability in timing. We hypothesize that the hippocampus achieves its precise spike timing (about 10ms) through plasticity enhanced phase-coding (PEP). The source of hippocampal timing precision in the presence of variability (and noise) remains unexplained. Synaptic plasticity can compensate for variability in excitability if it increases excitatory synaptic input to neurons in inverse proportion to their excitabilities. Recasting this in a phase-coding framework, we desire a learning rule that increases excitatory synaptic input to neurons directly related to their phases. Neurons that lag require additional synaptic input, whereas neurons that lead 190µm 120µm A B Figure 1: STDP Chip. A The chip has a 16-by-16 array of microcircuits; one microcircuit includes four principal neurons, each with 21 STDP circuits. B The STDP Chip is embedded in a circuit board including DACs, a CPLD, a RAM chip, and a USB chip, which communicates with a PC. require none. The spike timing-dependent plasticity (STDP) observed in the hippocampus satisfies this requirement [3]. It requires repeated pre-before-post spike pairings (within a time window) to potentiate and repeated post-before-pre pairings to depress a synapse. Here we validate our hypothesis with a model implemented in silicon, where variability is as ubiquitous as it is in biology [4]. Section 2 presents our silicon system, including the STDP Chip. Section 3 describes and characterizes the STDP circuit. Section 4 demonstrates that PEP compensates for variability and provides evidence that STDP is the compensation mechanism. Section 5 explores a desirable consequence of PEP: unconventional associative pattern recall. Section 6 discusses the implications of the PEP model, including its benefits and applications in the engineering of neuromorphic systems and in the study of neurobiology. 2 Silicon System We have designed, submitted, and tested a silicon implementation of PEP. The STDP Chip was fabricated through MOSIS in a 1P5M 0.25µm CMOS process, with just under 750,000 transistors in just over 10mm2 of area. It has a 32 by 32 array of excitatory principal neurons commingled with a 16 by 16 array of inhibitory interneurons that are not used here (Figure 1A). Each principal neuron has 21 STDP synapses. The address-event representation (AER) [5] is used to transmit spikes off chip and to receive afferent and recurrent spike input. To configure the STDP Chip as a recurrent network, we embedded it in a circuit board (Figure 1B). The board has five primary components: a CPLD (complex programmable logic device), the STDP Chip, a RAM chip, a USB interface chip, and DACs (digital-to-analog converters). The central component in the system is the CPLD. The CPLD handles AER traffic, mediates communication between devices, and implements recurrent connections by accessing a lookup table, stored in the RAM chip. The USB interface chip provides a bidirectional link with a PC. The DACs control the analog biases in the system, including the leak current, which the PC varies in real-time to create the global inhibitory theta rhythm. The principal neuron consists of a refractory period and calcium-dependent potassium circuit (RCK), a synapse circuit, and a soma circuit (Figure 2A). RCK and the synapse are Postsyn. Spike ISOMA Soma RCK Synapse AH STDP Presyn. Spike LPF PE Presyn. Spike 0 0.05 0.1 Raster 0 0.05 0.1 0 0.02 0.04 0.06 0.08 0.1 Spike probability Time(s) A B Figure 2: Principal neuron. A A simplified schematic is shown, including: the synapse, refractory and calcium-dependent potassium channel (RCK), soma, and axon-hillock (AH) circuits, plus their constituent elements, the pulse extender (PE) and the low-pass filter (LPF). B Spikes (dots) from 81 principal neurons are temporally dispersed, when excited by poisson-likeinputs (58Hz) and inhibitedby the common 8.3Hz theta rhythm (solid line). The histogram includes spikes from five theta cycles. composed of two reusable blocks: the low-pass filter (LPF) and the pulse extender (PE). The soma is a modified version of the LPF, which receives additional input from an axonhillock circuit (AH). RCK is inhibitory to the neuron. It consists of a PE, which models calcium influx during a spike, and a LPF, which models calcium buffering. When AH fires a spike, a packet of charge is dumped onto a capacitor in the PE. The PE’s output activates until the charge decays away, which takes a few milliseconds. Also, while the PE is active, charge accumulates on the LPF’s capacitor, lowering the LPF’s output voltage. Once the PE deactivates, this charge leaks away as well, but this takes tens of milliseconds because the leak is smaller. The PE’s and the LPF’s inhibitory effects on the soma are both described below in terms of the sum (ISHUNT) of the currents their output voltages produce in pMOS transistors whose sources are at Vdd (see Figure 2A). Note that, in the absence of spikes, these currents decay exponentially, with a time-constant determined by their respective leaks. The synapse circuit is excitatory to the neuron. It is composed of a PE, which represents the neurotransmitter released into the synaptic cleft, and a LPF, which represents the bound neurotransmitter. The synapse circuit is similar to RCK in structure but differs in function: It is activated not by the principal neuron itself but by the STDP circuits (or directly by afferent spikes that bypass these circuits, i.e., fixed synapses). The synapse’s effect on the soma is also described below in terms of the current (ISYN) its output voltage produces in a pMOS transistor whose source is at Vdd. The soma circuit is a leaky integrator. It receives excitation from the synapse circuit and shunting inhibition from RCK and has a leak current as well. Its temporal behavior is described by: τ dISOMA dt + ISOMA = ISYN I0 ISHUNT where ISOMA is the current the capacitor’s voltage produces in a pMOS transistor whose source is at Vdd (see Figure 2A). ISHUNT is the sum of the leak, refractory, and calciumdependent potassium currents. These currents also determine the time constant: τ = C Ut κISHUNT , where I0 and κ are transistor parameters and Ut is the thermal voltage. ~LTP ~LTD STDP circuit Decay Integrator SRAM Presynaptic spike Postsynaptic spike Inverse number of pairings -80 -40 0 40 80 0.1 0.05 0 0.05 0.1 Spike timing: tpre - tpost (ms) Presynaptic spike Postsynaptic spike Potentiation Depression A B Figure 3: STDP circuit design and characterization. A The circuit is composed of three subcircuits: decay, integrator, and SRAM. B The circuit potentiates when the presynaptic spike precedes the postsynaptic spike and depresses when the postsynaptic spike precedes the presynaptic spike. The soma circuit is connected to an AH, the locus of spike generation. The AH consists of model voltage-dependent sodium and potassium channel populations (modified from [6] by Kai Hynna). It initiates the AER signaling process required to send a spike off chip. To characterize principal neuron variability, we excited 81 neurons with poisson-like 58Hz spike trains (Figure 2B). We made these spike trains poisson-like by starting with a regular 200Hz spike train and dropping spikes randomly, with probability of 0.71. Thus spikes were delivered to neurons that won the coin toss in synchrony every 5ms. However, neurons did not lock onto the input synchrony due to filtering by the synaptic time constant (see Figure 2B). They also received a common inhibitory input at the theta frequency (8.3Hz), via their leak current. Each neuron was prevented from firing more than one spike in a theta cycle by its model calcium-dependent potassium channel population. The principal neurons’ spike times were variable. To quantify the spike variability, we used timing precision, which we define as twice the standard deviation of spike times accumulated from five theta cycles. With an input rate of 58Hz the timing precision was 34ms. 3 STDP Circuit The STDP circuit (related to [7]-[8]), for which the STDP Chip is named, is the most abundant, with 21,504 copies on the chip. This circuit is built from three subcircuits: decay, integrator, and SRAM (Figure 3A). The decay and integrator are used to implement potentiation, and depression, in a symmetric fashion. The SRAM holds the current binary state of the synapse, either potentiated or depressed. For potentiation, the decay remembers the last presynaptic spike. Its capacitor is charged when that spike occurs and discharges linearly thereafter. A postsynaptic spike samples the charge remaining on the capacitor, passes it through an exponential function, and dumps the resultant charge into the integrator. This charge decays linearly thereafter. At the time of the postsynaptic spike, the SRAM, a cross-coupled inverter pair, reads the voltage on the integrator’s capacitor. If it exceeds a threshold, the SRAM switches state from depressed to potentiated (∼LTD goes high and ∼LTP goes low). The depression side of the STDP circuit is exactly symmetric, except that it responds to postsynaptic activation followed by presynaptic activation and switches the SRAM’s state from potentiated to depressed (∼LTP goes high and ∼LTD goes low). When the SRAM is in the potentiated state, the presynaptic 0.2 0.4 0.6 50 Time(s) 0.2 0.4 0.6 Time(s) 58 67 75 83 92 100 Before STDP After STDP text 50 60 70 80 90 100 0 10 20 30 40 50 Timing precision(ms) Input rate(Hz) Before STDP After STDP B A C Figure 4: Plasticity enhanced phase-coding. A Spike rasters of 81 neurons (9 by 9 cluster) display synchrony over a two-fold range of input rates after STDP. B The degree of enhancement is quantified by timing precision. C Each neuron (center box) sends synapses to (dark gray) and receives synapses from (light gray) twenty-one randomly chosen neighbors up to five nodes away (black indicates both connections). spike activates the principal neuron’s synapse; otherwise the spike has no effect. We characterized the STDP circuit by activating a plastic synapse and a fixed synapse– which elicits a spike at different relative times. We repeated this pairing at 16Hz. We counted the number of pairings required to potentiate (or depress) the synapse. Based on this count, we calculated the efficacy of each pairing as the inverse number of pairings required (Figure 3B). For example, if twenty pairings were required to potentiate the synapse, the efficacy of that pre-before-post time-interval was one twentieth. The efficacy of both potentiation and depression are fit by exponentials with time constants of 11.4ms and 94.9ms, respectively. This behavior is similar to that observed in the hippocampus: potentiation has a shorter time constant and higher maximum efficacy than depression [3]. 4 Recurrent Network We carried out an experiment designed to test the STDP circuit’s ability to compensate for variability in spike timing through PEP. Each neuron received recurrent connections from 21 randomly selected neurons within an 11 by 11 neighborhood centered on itself (see Figure 4C). Conversely, it made recurrent connections to randomly chosen neurons within the same neighborhood. These connections were mediated by STDP circuits, initialized to the depressed state. We chose a 9 by 9 cluster of neurons and delivered spikes at a mean rate of 50 to 100Hz to each one (dropping spikes with a probability of 0.75 to 0.5 from a regular 200Hz train) and provided common theta inhibition as before. We compared the variability in spike timing after five seconds of learning with the initial distribution. Phase coding was enhanced after STDP (Figure 4A). Before STDP, spike timing among neurons was highly variable (except for the very highest input rate). After STDP, variability was virtually eliminated (except for the very lowest input rate). Initially, the variability, characterized by timing precision, was inversely related to the input rate, decreasing from 34 to 13ms. After five seconds of STDP, variability decreased and was largely independent of input rate, remaining below 11ms. after STDP Synaptic state 50 100 150 200 250 0 5 10 15 20 25 Spiking order Potentiated synapses A B Figure 5: Compensating for variability. A Some synapses (dots) become potentiated (light) while others remain depressed (dark) after STDP. B The number of potentiated synapses neurons make (pluses) and receive (circles) is negatively (r = -0.71) and positively (r = 0.76) correlated to their rank in the spiking order, respectively. Comparing the number of potentiated synapses each neuron made or received with its excitability confirmed the PEP hypothesis (i.e., leading neurons provide additional synaptic current to lagging neurons via potentiated recurrent synapses). In this experiment, to eliminate variability due to noise (as opposed to excitability), we provided a 17 by 17 cluster of neurons with a regular 200Hz excitatory input. Theta inhibition was present as before and all synapses were initialized to the depressed state. After 10 seconds of STDP, a large fraction of the synapses were potentiated (Figure 5A). When the number of potentiated synapses each neuron made or received was plotted versus its rank in spiking order (Figure 5B), a clear correlation emerged (r = -0.71 or 0.76, respectively). As expected, neurons that spiked early made more and received fewer potentiated synapses. In contrast, neurons that spiked late made fewer and received more potentiated synapses. 5 Pattern Completion After STDP, we found that the network could recall an entire pattern given a subset, thus the same mechanisms that compensated for variability and noise could also compensate for lack of information. We chose a 9 by 9 cluster of neurons as our pattern and delivered a poisson-like spike train with mean rate of 67Hz to each one as in the first experiment. Theta inhibition was present as before and all synapses were initialized to the depressed state. Before STDP, we stimulated a subset of the pattern and only neurons in that subset spiked (Figure 6A). After five seconds of STDP, we stimulated the same subset again. This time they recruited spikes from other neurons in the pattern, completing it (Figure 6B). Upon varying the fraction of the pattern presented, we found that the fraction recalled increased faster than the fraction presented. We selected subsets of the original pattern randomly, varying the fraction of neurons chosen from 0.1 to 1.0 (ten trials for each). We classified neurons as active if they spiked in the two second period over which we recorded. Thus, we characterized PEP’s pattern-recall performance as a function of the probability that the pattern in question’s neurons are activated (Figure 6C). At a fraction of 0.50 presented, nearly all of the neurons in the pattern are consistently activated (0.91±0.06), showing robust pattern completion. We fitted the recall performance with a sigmoid that reached 0.50 recall fraction with an input fraction of 0.30. No spurious neurons were activated during any trials. before STDP after STDP Network activity Network activity Rate(Hz) 0 1 2 3 4 5 6 7 8 Rate(Hz) 0 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Fraction of pattern stimulated Fraction of pattern actived A B C Figure 6: Associative recall. A Before STDP, half of the neurons in a pattern are stimulated; only they are activated. B After STDP, half of the neurons in a pattern are stimulated, and all are activated. C The fraction of the pattern activated grows faster than the fraction stimulated. 6 Discussion Our results demonstrate that PEP successfully compensates for graded variations in our silicon recurrent network using binary (on–off) synapses (in contrast with [8], where weights are graded). While our chip results are encouraging, variability was not eliminated in every case. In the case of the lowest input (50Hz), we see virtually no change (Figure 4A). We suspect the timing remains imprecise because, with such low input, neurons do not spike every theta cycle and, consequently, provide fewer opportunities for the STDP synapses to potentiate. This shortfall illustrates the system’s limits; it can only compensate for variability within certain bounds, and only for activity appropriate to the PEP model. As expected, STDP is the mechanism responsible for PEP. STDP potentiated recurrent synapses from leading neurons to lagging neurons, reducing the disparity among the diverse population of neurons. Even though the STDP circuits are themselves variable, with different efficacies and time constants, when using timing the sign of the weight-change is always correct (data not shown). For this reason, we chose STDP over other more physiological implementations of plasticity, such as membrane-voltage-dependent plasticity (MVDP), which has the capability to learn with graded voltage signals [9], such as those found in active dendrites, providing more computational power [10]. Previously, we investigated a MVDP circuit, which modeled a voltage-dependent NMDAreceptor-gated synapse [11]. It potentiated when the calcium current analog exceeded a threshold, which was designed to occur only during a dendritic action potential. This circuit produced behavior similar to STDP, implying it could be used in PEP. However, it was sensitive to variability in the NMDA and potentiation thresholds, causing a fraction of the populationto potentiate anytime the synapse received an input and another fraction to never potentiate, rendering both subpopulations useless. Therefore, the simpler, less biophysical STDP circuit won out over the MVDP circuit: In our system timing is everything. Associative storage and recall naturally emerge in the PEP network when synapses between neurons coactivated by a pattern are potentiated. These synapses allow neurons to recruit their peers when a subset of the pattern is presented, thereby completing the pattern. However, this form of pattern storage and completion differs from Hopfield’s attractor model [12] . Rather than forming symmetric, recurrent neuronal circuits, our recurrent network forms asymmetric circuits in which neurons make connections exclusively to less excitable neurons in the pattern. In both the poisson-like and regular cases (Figures 4 & 5), only about six percent of potentiated connections were reciprocated, as expected by chance. We plan to investigate the storage capacity of this asymmetric form of associative memory. Our system lends itself to modeling brain regions that use precise spike timing, such as the hippocampus. We plan to extend the work presented to store and recall sequences of patterns, as the hippocampus is hypothesized to do. Place cells that represent different locations spike at different phases of the theta cycle, in relation to the distance to their preferred locations. This sequential spiking will allow us to link patterns representing different locations in the order those locations are visited, thereby realizing episodic memory. We propose PEP as a candidate neural mechanism for information coding and storage in the hippocampal system. Observations from the CA1 region of the hippocampus suggest that basal dendrites (which primarily receive excitation from recurrent connections) support submillisecond timing precision, consistent with PEP [13]. We have shown, in a silicon model, PEP’s ability to exploit such fast recurrent connections to sharpen timing precision as well as to associatively store and recall patterns. Acknowledgments We thank Joe Lin for assistance with chip generation. The Office of Naval Research funded this work (Award No. N000140210468). References [1] O’Keefe J. & Recce M.L. (1993). Phase relationship between hippocampal place units and the EEG theta rhythm. Hippocampus 3(3):317-330. [2] Mehta M.R., Lee A.K. & Wilson M.A. (2002) Role of experience and oscillations in transforming a rate code into a temporal code. Nature 417(6890):741-746. [3] Bi G.Q. & Wang H.X. (2002) Temporal asymmetry in spike timing-dependent synaptic plasticity. Physiology & Behavior 77:551-555. [4] Rodriguez-Vazquez, A., Linan, G., Espejo S. & Dominguez-Castro R. (2003) Mismatch-induced trade-offs and scalability of analog preprocessing visual microprocessor chips. Analog Integrated Circuits and Signal Processing 37:73-83. [5] Boahen K.A. (2000) Point-to-point connectivity between neuromorphic chips using address events. IEEE Transactions on Circuits and Systems II 47:416-434. [6] Culurciello E.R., Etienne-Cummings R. & Boahen K.A. (2003) A biomorphic digital image sensor. IEEE Journal of Solid State Circuits 38:281-294. [7] Bofill A., Murray A.F & Thompson D.P. (2005) Citcuits for VLSI Implementation of Temporally Asymmetric Hebbian Learning. In: Advances in Neural Information Processing Systems 14, MIT Press, 2002. [8] Cameron K., Boonsobhak V., Murray A. & Renshaw D. (2005) Spike timing dependent plasticity (STDP) can ameliorate process variations in neuromorphic VLSI. IEEE Transactions on Neural Networks 16(6):1626-1627. [9] Chicca E., Badoni D., Dante V., D’Andreagiovanni M., Salina G., Carota L., Fusi S. & Del Giudice P. (2003) A VLSI recurrent network of integrate-and-fire neurons connected by plastic synapses with long-term memory. IEEE Transaction on Neural Networks 14(5):1297-1307. [10] Poirazi P., & Mel B.W. (2001) Impact of active dendrites and structural plasticity on the memory capacity of neural tissue. Neuron 29(3)779-796. [11] Arthur J.V. & Boahen K. (2004) Recurrently connected silicon neurons with active dendrites for one-shot learning. In: IEEE International Joint Conference on Neural Networks 3, pp.1699-1704. [12] Hopfield J.J. (1984) Neurons with graded response have collective computational properties like those of two-state neurons. Proceedings of the National Academy of Science 81(10):3088-3092. [13] Ariav G., Polsky A. & Schiller J. (2003) Submillisecond precision of the input-output transformation function mediated by fast sodium dendritic spikes in basal dendrites of CA1 pyramidal neurons. Journal of Neuroscience 23(21):7750-7758.
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Correcting sample selection bias in maximum entropy density estimation Miroslav Dud´ık, Robert E. Schapire Princeton University Department of Computer Science 35 Olden St, Princeton, NJ 08544 {mdudik,schapire}@princeton.edu Steven J. Phillips AT&T Labs −Research 180 Park Ave, Florham Park, NJ 07932 phillips@research.att.com Abstract We study the problem of maximum entropy density estimation in the presence of known sample selection bias. We propose three bias correction approaches. The first one takes advantage of unbiased sufficient statistics which can be obtained from biased samples. The second one estimates the biased distribution and then factors the bias out. The third one approximates the second by only using samples from the sampling distribution. We provide guarantees for the first two approaches and evaluate the performance of all three approaches in synthetic experiments and on real data from species habitat modeling, where maxent has been successfully applied and where sample selection bias is a significant problem. 1 Introduction We study the problem of estimating a probability distribution, particularly in the context of species habitat modeling. It is very common in distribution modeling to assume access to independent samples from the distribution being estimated. In practice, this assumption is violated for various reasons. For example, habitat modeling is typically based on known occurrence locations derived from collections in natural history museums and herbariums as well as biological surveys [1, 2, 3]. Here, the goal is to predict the species’ distribution as a function of climatic and other environmental variables. To achieve this in a statistically sound manner using current methods, it is necessary to assume that the sampling distribution and species distributions are not correlated. In fact, however, most sampling is done in locations that are easier to access, such as areas close to towns, roads, airports or waterways [4]. Furthermore, the independence assumption may not hold since roads and waterways are often correlated with topography and vegetation which influence species distributions. New unbiased sampling may be expensive, so much can be gained by using the extensive existing biased data, especially since it is becoming freely available online [5]. Although the available data may have been collected in a biased manner, we usually have some information available about the nature of the bias. For instance, in the case of habitat modeling, some factors influencing the sampling distribution are well known, such as distance from roads, towns, etc. In addition, a list of visited sites may be available and viewed as a sample of the sampling distribution itself. If such a list is not available, the set of sites where any species from a large group has been observed may be a reasonable approximation of all visited locations. In this paper, we study probability density estimation under sample selection bias. We assume that the sampling distribution (or an approximation) is known during training, but we require that unbiased models not use any knowledge of sample selection bias during testing. This requirement is vital for habitat modeling where models are often applied to a different region or under different climatic conditions. To our knowledge this is the first work addressing sample selection bias in a statistically sound manner and in a setup suitable for species habitat modeling from presence-only data. We propose three approaches that incorporate sample selection bias in a common density estimation technique based on the principle of maximum entropy (maxent). Maxent with ℓ1-regularization has been successfully used to model geographic distributions of species under the assumption that samples are unbiased [3]. We review ℓ1-regularized maxent with unbiased data in Section 2, and give details of the new approaches in Section 3. Our three approaches make simple modifications to unbiased maxent and achieve analogous provable performance guarantees. The first approach uses a bias correction technique similar to that of Zadrozny et al. [6, 7] to obtain unbiased confidence intervals from biased samples as required by our version of maxent. We prove that, as in the unbiased case, this produces models whose log loss approaches that of the best possible Gibbs distribution (with increasing sample size). In contrast, the second approach we propose first estimates the biased distribution and then factors the bias out. When the target distribution is a Gibbs distribution, the solution again approaches the log loss of the target distribution. When the target distribution is not Gibbs, we demonstrate that the second approach need not produce the optimal Gibbs distribution (with respect to log loss) even in the limit of infinitely many samples. However, we prove that it produces models that are almost as good as the best Gibbs distribution according to a certain Bregman divergence that depends on the selection bias. In addition, we observe good empirical performance for moderate sample sizes. The third approach is an approximation of the second approach which uses samples from the sampling distribution instead of the distribution itself. One of the challenges in studying methods for correcting sample selection bias is that unbiased data sets, though not required during training, are needed as test sets to evaluate performance. Unbiased data sets are difficult to obtain — this is the very reason why we study this problem! Thus, it is almost inevitable that synthetic data must be used. In Section 4, we describe experiments evaluating performance of the three methods. We use both fully synthetic data, as well as a biological dataset consisting of a biased training set and an independently collected reasonably unbiased test set. Related work. Sample selection bias also arises in econometrics where it stems from factors such as attrition, nonresponse and self selection [8, 9, 10]. It has been extensively studied in the context of linear regression after Heckman’s seminal paper [8] in which the bias is first estimated and then a transform of the estimate is used as an additional regressor. In the machine learning community, sample selection bias has been recently considered for classification problems by Zadrozny [6]. Here the goal is to learn a decision rule from a biased sample. The problem is closely related to cost-sensitive learning [11, 7] and the same techniques such as resampling or differential weighting of samples apply. However, the methods of the previous two approaches do not apply directly to density estimation where the setup is “unconditional”, i.e. there is no dependent variable, or, in the classification terminology, we only have access to positive examples, and the cost function (log loss) is unbounded. In addition, in the case of modeling species habitats, we face the challenge of sample sizes that are very small (2–100) by machine learning standards. 2 Maxent setup In this section, we describe the setup for unbiased maximum entropy density estimation and review performance guarantees. We use a relaxed formulation which will yield an ℓ1-regularization term in our objective function. The goal is to estimate an unknown target distribution π over a known sample space X based on samples x1, . . . , xm ∈X. We assume that samples are independently distributed according to π and denote the empirical distribution by ˜π(x) = |{1 ≤i ≤m : xi = x}|/m. The structure of the problem is specified by real valued functions fj : X →R, j = 1, . . . , n, called features and by a distribution q0 representing a default estimate. We assume that features capture all the relevant information available for the problem at hand and q0 is the distribution we would choose if we were given no samples. The distribution q0 is most often assumed uniform. For a limited number of samples, we expect that ˜π will be a poor estimate of π under any reasonable distance measure. However, empirical averages of features will not be too different from their expectations with respect to π. Let p[f] denote the expectation of a function f(x) when x is chosen randomly according to distribution p. We would like to find a distribution p which satisfies |p[fj] −˜π[fj]| ≤βj for all 1 ≤j ≤n, (1) for some estimates βj of deviations of empirical averages from their expectations. Usually there will be infinitely many distributions satisfying these constraints. For the case when the default distribution q0 is uniform, the maximum entropy principle tells us to choose the distribution of maximum entropy satisfying these constraints. In general, we should minimize the relative entropy from q0. This corresponds to choosing the distribution that satisfies the constraints (1) but imposes as little additional information as possible when compared with q0. Allowing for asymmetric constraints, we obtain the formulation min p∈∆RE(p ∥q0) subject to ∀1 ≤j ≤n : aj ≤p[fj] ≤bj. (2) Here, ∆⊆RX is the simplex of probability distributions and RE(p ∥q) is the relative entropy (or Kullback-Leibler divergence) from q to p, an information theoretic measure of difference between the two distributions. It is non-negative, equal to zero only when the two distributions are identical, and convex in its arguments. Problem (2) is a convex program. Using Lagrange multipliers, we obtain that the solution takes the form qλ(x) = q0(x)eλ·f(x)/Zλ (3) where Zλ = P x q0(x)eλ·f(x) is the normalization constant. Distributions qλ of the form (3) will be referred to as q0-Gibbs or just Gibbs when no ambiguity arises. Instead of solving (2) directly, we solve its dual: min λ∈Rn log Zλ −1 2 P j (bj + aj)λj + 1 2 P j (bj −aj)|λj| . (4) We can choose from a range of general convex optimization techniques or use some of the algorithms in [12]. For the symmetric case when [aj, bj] = ˜π[fj] −βj, ˜π[fj] + βj , (5) the dual becomes min λ∈Rn −˜π[log qλ] + P j βj|λj| . (6) The first term is the empirical log loss (negative log likelihood), the second term is an ℓ1regularization. Small values of log loss mean a good fit to the data. This is balanced by regularization forcing simpler models and hence preventing overfitting. When all the primal constraints are satisfied by the target distribution π then the solution ˆq of the dual is guaranteed to be not much worse an approximation of π than the best Gibbs distribution q∗. More precisely: Theorem 1 (Performance guarantees, Theorem 1 of [12]). Assume that the distribution π satisfies the primal constraints (2). Let ˆq be the solution of the dual (4). Then for an arbitrary Gibbs distribution q∗= qλ∗ RE(π ∥ˆq) ≤RE(π ∥q∗) + P j (bj −aj)|λ∗ j|. Input: finite domain X features f1, . . . , fn where fj : X →[0, 1] default estimate q0 regularization parameter β > 0 sampling distribution s samples x1, . . . , xm ∈X Output: qˆλ approximating the target distribution Let β0 = ` β/√m ´ · min {f σs[1/s], (max 1/s −min 1/s)/2} [c0, d0] = ˆ f πs[1/s] −β0, f πs[1/s] + β0 ˜ ∩ ˆ min 1/s, max 1/s] For j = 1, . . . , n: βj = ` β/√m ´ · min {f σs[fj/s], (max fj/s −min fj/s)/2} [cj, dj] = ˆ f πs[fj/s] −βj, f πs[fj/s] + βj ˜ ∩ ˆ min fj/s, max fj/s] [aj, bj] = ˆ cj/d0, dj/c0 ˜ ∩ ˆ min fj, max fj] Solve the dual (4) Algorithm 1: DEBIASAVERAGES. Table 1: Example 1. Comparison of distributions q∗and q∗∗minimizing RE(π ∥qλ) and RE(πs ∥qλs). x f(x) π(x) s(x) πs(x) q∗(x) q∗∗s(x) q∗∗(x) 1 (0, 0) 0.4 0.4 0.64 0.25 0.544 0.34 2 (0, 1) 0.1 0.4 0.16 0.25 0.256 0.16 3 (1, 0) 0.1 0.1 0.04 0.25 0.136 0.34 4 (1, 1) 0.4 0.1 0.16 0.25 0.064 0.16 When features are bounded between 0 and 1, the symmetric box constraints (5) with βj = O( p (log n)/m) are satisfied with high probability by Hoeffding’s inequality and the union bound. Then the relative entropy from ˆq to π will not be worse than the relative entropy from any Gibbs distribution q∗to π by more than O(∥λ∗∥1 p (log n)/m). In practice, we set βj = β/√m · min {˜σ[fj], σmax[fj]} (7) where β is a tuned constant, ˜σ[fj] is the sample deviation of fj, and σmax[fj] is an upper bound on the standard deviation, such as (maxx fj(x) −minx fj(x))/2. We refer to this algorithm for unbiased data as UNBIASEDMAXENT. 3 Maxent with sample selection bias In the biased case, the goal is to estimate the target distribution π, but samples do not come directly from π. For nonnegative functions p1, p2 defined on X, let p1p2 denote the distribution obtained by multiplying weights p1(x) and p2(x) at every point and renormalizing: p1p2(x) = p1(x)p2(x) P x′ p1(x′)p2(x′). Samples x1, . . . , xm come from the biased distribution πs where s is the sampling distribution. This setup corresponds to the situation when an event being observed occurs at the point x with probability π(x) while we perform an independent observation with probability s(x). The probability of observing an event at x given that we observe an event is then equal to πs(x). The empirical distribution of m samples drawn from πs will be denoted by f πs. We assume that s is known (principal assumption, see introduction) and strictly positive (technical assumption). Approach I: Debiasing Averages. In our first approach, we use the same algorithm as for the unbiased case but employ a different method to obtain confidence intervals [aj, bj]. Since we do not have direct access to samples from π, we use a version of the Bias Correction Theorem of Zadrozny [6] to convert expectations with respect to πs to expectations with respect to π. Theorem 2 (Bias Correction Theorem [6], Translation Theorem [7]). πs[f/s] πs[1/s] = π[f]. Hence, it suffices to give confidence intervals for πs[f/s] and πs[1/s] to obtain confidence intervals for π[f]. Corollary 3. Assume that for some sample-derived bounds cj, dj, 0 ≤j ≤n, with high probability 0 < c0 ≤πs[1/s] ≤d0 and 0 ≤cj ≤πs[fj/s] ≤dj for all 1 ≤j ≤n. Then with at least the same probability cj/d0 ≤π[fj] ≤dj/c0 for all 1 ≤j ≤n. If s is bounded away from 0 then Chernoff bounds may be used to determine cj, dj. Corollary 3 and Theorem 1 then yield guarantees that this method’s performance converges, with increasing sample sizes, to that of the “best” Gibbs distribution. In practice, confidence intervals [cj, dj] may be determined using expressions analogous to (5) and (7) for random variables fj/s, 1/s and the empirical distribution f πs. After first restricting the confidence intervals in a natural fashion, this yields Algorithm 1. Alternatively, we could use bootstrap or other types of estimates for the confidence intervals. Approach II: Factoring Bias Out. The second algorithm does not approximate π directly, but uses maxent to estimate the distribution πs and then converts this estimate into an approximation of π. If the default estimate of π is q0, then the default estimate of πs is q0s. Applying unbiased maxent to the empirical distribution f πs with the default q0s, we obtain a q0s-Gibbs distribution q0seˆλ·f approximating πs. We factor out s to obtain q0eˆλ·f as an estimate of π. This yields the algorithm FACTORBIASOUT. This approach corresponds to ℓ1-regularized maximum likelihood estimation of π by q0-Gibbs distributions. When π itself is q0-Gibbs then the distribution πs is q0s-Gibbs. Performance guarantees for unbiased maxent imply that estimates of πs converge to πs as the number of samples increases. Now, if infx s(x) > 0 (which is the case for finite X) then estimates of π obtained by factoring out s converge to π as well. When π is not q0-Gibbs then πs is not q0s-Gibbs either. We approximate π by a q0-Gibbs distribution ˆq = qˆλ which, with an increasing number of samples, minimizes RE(πs ∥qλs) rather than RE(π ∥qλ). Our next example shows that these two minimizers may be different. Example 1. Consider the space X = {1, 2, 3, 4} with two features f1, f2. Features f1, f2, target distribution π, sampling distribution s and the biased distribution πs are given in Table 1. We use the uniform distribution as a default estimate. The minimizer of RE(π ∥qλ) is the unique uniform-Gibbs distribution q∗such that q∗[f] = π[f]. Similarly, the minimizer q∗∗s of RE(πs ∥qλs) is the unique s-Gibbs distribution for which q∗∗s[f] = πs[f]. Solving for these exactly, we find that q∗and q∗∗are as given in Table 1, and that these two distributions differ. Even though FACTORBIASOUT does not minimize RE(π ∥qλ), we can show that it minimizes a different Bregman divergence. More precisely, it minimizes a Bregman divergence between certain projections of the two distributions. Bregman divergences generalize some common distance measures such as relative entropy or the squared Euclidean distance, and enjoy many of the same favorable properties. The Bregman divergence associated with a convex function F is defined as DF (u ∥v) = F(u)−F(v)−∇F(v)·(u−v). Proposition 4. Define F : RX + →R as F(u) = P x s(x)u(x) log u(x). Then F is a convex function and for all p1, p2 ∈∆, RE(p1s ∥p2s) = DF (p′ 1 ∥p′ 2), where p′ 1(x) = p1(x)/P x′ s(x′)p1(x′) and p′ 2(x) = p2(x)/P x′ s(x′)p2(x′) are projections of p1, p2 along lines tp, t ∈R onto the hyperplane P x s(x)p(x) = 1. Approach III: Approximating FACTORBIASOUT. As mentioned in the introduction, knowing the sampling distribution s exactly is unrealistic. However, we often have access to samples from s. In this approach we assume that s is unknown but that, in addition to samples x1, . . . , xm from πs, we are also given a separate set of samples x(1), x(2), . . . , x(N) from s. We use the algorithm FACTORBIASOUT with the sampling distribution s replaced by the corresponding empirical distribution ˜s. To simplify the algorithm, we note that instead of using q0˜s as a default estimate for πs, it suffices to replace the sample space X by X ′ = x(1), x(2), . . . , x(N) and use q0 10 100 1000 1.5 2 2.5 10 100 1000 1 1.5 2 10 100 1000 1.8 2 2.2 number of training samples (m) relative entropy to target unbiased maxent approximate factor bias out debias averages 1,000 samples factor bias out 10,000 samples target=π1 RE(π1||u)=4.5 target=π2 RE(π2||u)=5.0 target=π3 RE(π3||u)=3.3 Figure 1: Learning curves for synthetic experiments. We use u to denote the uniform distribution. For the sampling distribution s, RE(s ∥u) = 0.8. Performance is measured in terms of relative entropy to the target distribution as a function of an increasing number of training samples. The number of samples is plotted on a log scale. restricted to X ′ as a default. The last step of factoring out ˜s is equivalent to using ˆλ returned for space X ′ on the entire space X. When the sampling distribution s is correlated with feature values, X ′ might not cover all feature ranges. In that case, reprojecting on X may yield poor estimates outside of these ranges. We therefore do “clamping”, restricting values fj(x) to their ranges over X ′ and capping values of the exponent ˆλ · f(x) at its maximum over X ′. The resulting algorithm is called APPROXFACTORBIASOUT. 4 Experiments Conducting real data experiments to evaluate bias correction techniques is difficult, because bias is typically unknown and samples from unbiased distributions are not available. Therefore, synthetic experiments are often a necessity for precise evaluation. Nevertheless, in addition to synthetic experiments, we were also able to conduct experiments with real-world data for habitat modeling. Synthetic experiments. In synthetic experiments, we generated three target uniformGibbs distributions π1, π2, π3 over a domain X of size 10,000. These distributions were derived from 65 features indexed as fi, 0 ≤i ≤9 and fij, 0 ≤i ≤j ≤9. Values fi(x) were chosen independently and uniformly in [0, 1], and we set fij(x) = fi(x)fj(x). Fixing these features, we generated weights for each distribution. Weights λi and λii were generated jointly to capture a range of different behaviors for values of fi in the range [0, 1]. Let US denote a random variable uniform over the set S. Each instance of US corresponds to a new independent variable. We set λii = U{−1,0,1}U[1,5] and λi to be λiiU[−3,1] if λii ̸= 0, and U{−1,1}U[2,10] otherwise. Weights λij, i < j were chosen to create correlations between fi’s that would be observable, but not strong enough to dominate λi’s and λii’s. We set λij = −0.5 or 0 or 0.5 with respective probabilities 0.05, 0.9 and 0.05. In maxent, we used a subset of features specifying target distributions and some irrelevant features. We used features f ′ i, 0 ≤i ≤9 and their squares f ′ ii, where f ′ i(x) = fi(x) for 0 ≤i ≤5 (relevant features) and f ′ i(x) = U[0,1] for 6 ≤i ≤9 (irrelevant features). Once generated, we used the same set of features in all experiments. We generated a sampling distribution s correlated with target distributions. More specifically, s was a Gibbs distribution generated from features f (s) i , 0 ≤i ≤5 and their squares f (s) ii , where f (s) i (x) = U[0,1] for 0 ≤i ≤1 and f (s) i = fi+2 for 2 ≤i ≤5. We used weights λ(s) i = 0 and λ(s) ii = −1. For every target distribution, we evaluated the performance of UNBIASEDMAXENT, DEBIASAVERAGES, FACTORBIASOUT and APPROXFACTORBIASOUT with 1,000 and 10,000 samples from the sampling distribution. The performance was evaluated in terms of relative entropy to the target distribution. We used training sets of sizes 10 to 1000. We considered five randomly generated training sets and took the average performance over these five sets for settings of β from the range [0.05, 4.64]. We report results for the best β, chosen separately for each average. The rationale behind this approach is that we want to Table 2: Results of real data experiments. Average performance of unbiased maxent and three bias correction approaches over all species in six regions. The uniform distribution would receive the log loss of 14.2 and AUC of 0.5. Results of bias correction approaches are italicized if they are significantly worse and set in boldface if they are significantly better than those of the unbiased maxent according to a paired t-test at the level of significance 5%. average log loss average AUC awt can nsw nz sa swi awt can nsw nz sa swi unbiased maxent 13.78 12.89 13.40 13.77 13.14 12.81 0.69 0.58 0.71 0.72 0.78 0.81 debias averages 13.92 13.10 13.88 14.31 14.10 13.59 0.67 0.64 0.65 0.67 0.68 0.78 factor bias out 13.90 13.13 14.06 14.20 13.66 13.46 0.71 0.69 0.72 0.72 0.78 0.83 apx. factor bias out 13.89 13.40 14.19 14.07 13.62 13.41 0.72 0.72 0.73 0.73 0.78 0.84 explore the potential performance of each method. Figure 1 shows the results at the optimal β as a function of an increasing number of samples. FACTORBIASOUT is always better than UNBIASEDMAXENT. DEBIASAVERAGES is worse than UNBIASEDMAXENT for small sample sizes, but as the number of training samples increases, it soon outperforms UNBIASEDMAXENT and eventually also outperforms FACTORBIASOUT. APPROXFACTORBIASOUT improves as the number of samples from the sampling distribution increases from 1,000 to 10,000, but both versions of APPROXFACTORBIASOUT perform worse than UNBIASEDMAXENT for the distribution π2. Real data experiments. In this set of experiments, we evaluated maxent in the task of estimating species habitats. The sample space is a geographic region divided into a grid of cells and samples are known occurrence localities — cells where a given species was observed. Every cell is described by a set of environmental variables, which may be categorical, such as vegetation type, or continuous, such as altitude or annual precipitation. Features are real-valued functions derived from environmental variables. We used binary indicator features for different values of categorical variables and binary threshold features for continuous variables. The latter are equal to one when the value of a variable is greater than a fixed threshold and zero otherwise. Species sample locations and environmental variables were all produced and used as part of the “Testing alternative methodologies for modeling species’ ecological niches and predicting geographic distributions” Working Group at the National Center for Ecological Analysis and Synthesis (NCEAS). The working group compared modeling methods across a variety of species and regions. The training set contained presence-only data from unplanned surveys or incidental records, including those from museums and herbariums. The test set contained presence-absence data from rigorously planned independent surveys. We compared performance of our bias correction approaches with that of the unbiased maxent which was among the top methods in the NCEAS comparison [13]. We used the full dataset consisting of 226 species in 6 regions with 2–5822 training presences per species (233 on average) and 102–19120 test presences/absences. For more details see [13]. We treated training occurrence locations for all species in each region as sampling distribution samples and used them directly in APPROXFACTORBIASOUT. In order to apply DEBIASAVERAGES and FACTORBIASOUT, we estimated the sampling distribution using unbiased maxent. Sampling distribution estimation is also the first step of [6]. In contrast with that work, however, our experiments do not use the sampling distribution estimate during evaluation and hence do not depend on its quality. The resulting distributions were evaluated on test presences according to the log loss and on test presences and absences according to the area under an ROC curve (AUC) [14]. AUC quantifies how well the predicted distribution ranks test presences above test absences. Its value is equal to the probability that a randomly chosen presence will be ranked above a randomly chosen absence. The uniformly random prediction receives AUC of 0.5 while a perfect prediction receives AUC of 1.0. In Table 2 we show performance of our three approaches compared with the unbiased maxent. All three algorithms yield on average a worse log loss than the unbiased maxent. This can perhaps be attributed to the imperfect estimate of the sampling distribution or to the sampling distribution being zero over large portions of the sample space. In contrast, when the performance is measured in terms of AUC, FACTORBIASOUT and APPROXFACTORBIASOUT yield on average the same or better AUC as UNBIASEDMAXENT in all six regions. Improvements in regions awt, can and swi are dramatic enough so that both of these methods perform better than any method evaluated in [13]. 5 Conclusions We have proposed three approaches that incorporate information about sample selection bias in maxent and demonstrated their utility in synthetic and real data experiments. Experiments also raise several questions that merit further research: DEBIASAVERAGES has the strongest performance guarantees, but it performs the worst in real data experiments and catches up with other methods only for large sample sizes in synthetic experiments. This may be due to poor estimates of unbiased confidence intervals and could be possibly improved using a different estimation method. FACTORBIASOUT and APPROXFACTORBIASOUT improve over UNBIASEDMAXENT in terms of AUC over real data, but are worse in terms of log loss. This disagreement suggests that methods which aim to optimize AUC directly could be more successful in species modeling, possibly incorporating some concepts from FACTORBIASOUT and APPROXFACTORBIASOUT. APPROXFACTORBIASOUT performs the best on real world data, possibly due to the direct use of samples from the sampling distribution rather than a sampling distribution estimate. However, this method comes without performance guarantees and does not exploit the knowledge of the full sample space. Proving performance guarantees for APPROXFACTORBIASOUT remains open for future research. Acknowledgments This material is based upon work supported by NSF under grant 0325463. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NSF. The NCEAS data was kindly shared with us by the members of the “Testing alternative methodologies for modeling species’ ecological niches and predicting geographic distributions” Working Group, which was supported by the National Center for Ecological Analysis and Synthesis, a Center funded by NSF (grant DEB0072909), the University of California and the Santa Barbara campus. References [1] Jane Elith. Quantitative methods for modeling species habitat: Comparative performance and an application to Australian plants. In Scott Ferson and Mark Burgman, editors, Quantitative Methods for Conservation Biology, pages 39–58. Springer-Verlag, 2002. [2] A. Guisan and N. E. Zimmerman. Predictive habitat distribution models in ecology. Ecological Modelling, 135:147–186, 2000. [3] Steven J. Phillips, Miroslav Dud´ık, and Robert E. Schapire. A maximum entropy approach to species distribution modeling. In Proceedings of the Twenty-First International Conference on Machine Learning, 2004. [4] S. Reddy and L. M. D´avalos. Geographical sampling bias and its implications for conservation priorities in Africa. Journal of Biogeography, 30:1719–1727, 2003. [5] Barbara R. Stein and John Wieczorek. Mammals of the world: MaNIS as an example of data integration in a distributed network environment. Biodiversity Informatics, 1(1):14–22, 2004. [6] Bianca Zadrozny. Learning and evaluating classifiers under sample selection bias. In Proceedings of the Twenty-First International Conference on Machine Learning, 2004. [7] Bianca Zadrozny, John Langford, and Naoki Abe. Cost-sensitive learning by cost-proportionate example weighting. In Proceedings of the Third IEEE International Conference on Data Mining, 2003. [8] James J. Heckman. Sample selection bias as a specification error. Econometrica, 47(1):153–161, 1979. [9] Robert M. Groves. Survey Errors and Survey Costs. Wiley, 1989. [10] Roderick J. Little and Donald B. Rubin. Statistical Analysis with Missing Data. Wiley, second edition, 2002. [11] Charles Elkan. The foundations of cost-sensitive learning. In Proceedings of the Seventeenth International Joint Conference on Artificial Intelligence, 2001. [12] Miroslav Dud´ık, Steven J. Phillips, and Robert E. Schapire. Performance guarantees for regularized maximum entropy density estimation. In 17th Annual Conference on Learning Theory, 2004. [13] J. Elith, C. Graham, and NCEAS working group. Comparing methodologies for modeling species’ distributions from presence-only data. In preparation. [14] J. A. Hanley and B. S. McNeil. The meaning and use of the area under a receiver operating characteristic (ROC) curve. Radiology, 143:29–36, 1982.
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Rate Distortion Codes in Sensor Networks: A System-level Analysis Tatsuto Murayama and Peter Davis NTT Communication Science Laboratories Nippon Telegraph and Telephone Corporation “Keihanna Science City”, Kyoto 619-0237, Japan {murayama,davis}@cslab.kecl.ntt.co.jp Abstract This paper provides a system-level analysis of a scalable distributed sensing model for networked sensors. In our system model, a data center acquires data from a bunch of L sensors which each independently encode their noisy observations of an original binary sequence, and transmit their encoded data sequences to the data center at a combined rate R, which is limited. Supposing that the sensors use independent LDGM rate distortion codes, we show that the system performance can be evaluated for any given finite R when the number of sensors L goes to infinity. The analysis shows how the optimal strategy for the distributed sensing problem changes at critical values of the data rate R or the noise level. 1 Introduction Device and sensor networks are shaping many activities in our society. These networks are being deployed in a growing number of applications as diverse as agricultural management, industrial controls, crime watch, and military applications. Indeed, sensor networks can be considered as a promising technology with a wide range of potential future markets [1]. Still, for all the promise, it is often difficult to integrate the individual components of a sensor network in a smart way. Although we see many breakthroughs in component devices, advanced software, and power managements, system-level understanding of the emerging technology is still weak. It requires a shift in our notion of “what to look for”. It requires a study of collective behavior and resulting trade-offs. This is the issue that we address in this article. We demonstrate the usefulness of adopting new approaches by considering the following scenario. Consider that a data center is interested in the data sequence, {X(t)}∞ t=1, which cannot be observed directly. Therefore, the data center deploys a bunch of L sensors which each independently encodes its noisy observation of the sequence, {Yi(t)}∞ t=1, without sharing any information, i.e., the sensors are not permitted to communicate and decide what to send to the data center beforehand. The data center collects separate samples from all the L sensors and uses them to recover the original sequence. However, since {X(t)}∞ t=1 is not the only pressing matter which the data center must consider, the combined data rate R at which the sensors can communicate with it is strictly limited. A formulation of decentralized communication with estimation task, the “CEO problem”, was first proposed by Berger and Zhang [2], providing a new theoretical framework for large scale sensing systems. In this outstanding work, some interesting properties of such systems have been revealed. If the sensors were permitted to communicate on the basis of their pooled observations, then they would be able to smooth out their independent observation noises entirely as L goes to infinity. Therefore, the data center can achieve an arbitrary fidelity D(R), where D(·) denotes the distortion rate function of {X(t)}. In particular, the data center recovers almost complete information if R exceeds the entropy rate of {X(t)}. However, if the sensors are not allowed to communicate with each other, there does not exist a finite value of R for which even infinitely many sensors can make D arbitrarily small [2]. In this paper, we introduce a new analytical model for a massive sensing system with a finite data rate R. More specifically, we assume that the sensors use LDGM codes for rate distortion coding, while the data center recovers the original sequence by using optimal “majority vote” estimation [3]. We consider the distributed sensing problem of deciding the optimal number of sensors L given the combined data rate R. Our asymptotic analysis successfully provides the performance of the whole sensing system when L goes to infinity, where the data rate for an individual sensor information vanishes. Here, we exploit statistical methods which have recently been developed in the field of disordered statistical systems, in particular, the spin glass theory. The paper is organized as follows. In Section 2, we introduce a system model for the sensor network. Section 3 summarizes the results of our approach, where the following section provides the outline of our analysis. Conclusions are given in the last section. 2 System Model Let P (x) be a probability distribution common to {X(t)} ∈X , and W(y|x) be a stochastic matrix defined on X × Y, with Y denotes the common alphabet of {Yi(t)}, where i = 1, · · · , L and t ≥1. In the general setup, we assume that the instantaneous joint probability distribution in the form Pr[x, y1, · · · , yL] = P (x) L i=1 W(yi|x) for the temporally memoryless source {X(t)}∞ t=1. Here, the random variables Yi(t) are conditionally independent when X(t) is given, and the conditional probabilities W[yi(t)|x(t)] are identical for all i and t. In this paper, we impose the binary assumptions to the problem, i.e., the data sequence {X(t)} and its noisy observations {Yi(t)} are all assumed to be binary sequences. Therefore, the stochastic matrix can be parameterized as W(y|x) = 1 −p, if y = x p, otherwise , where p ∈[0, 1] represents the observation noise. Note also that the alphabets have been selected as X = Y. Furthermore, for simplicity, we also assume that P (x) = 1/2 always holds, implying that a purely random source is observed. At the encoding stage, a sensor i encodes a block yi = [yi(1), · · · , yi(n)]T of length n from the noisy observation {yi(t)}∞ t=1, into a block zi = [zi(1), · · · , zi(m)]T of length m defined on Z. Hereafter, we take the Boolean representation of the binary alphabet X = {0, 1}, therefore Y = Z = {0, 1} as well. Let ˆyi be a reproduction sequence for the block, and we have a known integer m < n. Then, making use of a Boolean matrix Ai of dimensionality n×m, we are to find an m bit codeword sequence zi = [zi(1), · · · , zi(m)]T which satisfies ˆyi = Aizi (mod 2) , (1) where the fidelity criterion D = 1 ndH(yi, ˆyi) (2) holds [4]. Here the Hamming distance dH(·, ·) is used for the distortion measure. Note that we have applied modulo-2 arithmetic for the additive operation in (1). Let A i be characterized by K ones per row and C per column. The finite, and usually small, numbers K and C define a particular LDGM code family. The data center then collects the L codeword sequences, z1, · · · , zL. Since all the L codewords are of the same length m, the combined data rate will be R = L × m/n. Therefore, in our scenario, the data center deploys exchangeable sensors with fixed quality reproductions, ˆy1, · · · , ˆyL. Lastly, the tth symbol of the estimate, ˆx = [ˆx(1), · · · , ˆx(n)]T , is to be calculated by majority vote [3], ˆx(t) = 0, if ˆy1(t) + · · · + ˆyL(t) ≤L/2 1, otherwise . (3) Therefore, overall performance of the system can be measured by the expected bit error frequency for decisions by the majority vote (3), Pe = Pr[x ̸= ˆx]. In this paper, we consider two limit cases of decentralization levels; (1) The extreme situation of L →∞, and (2) the case of L = R. The former case means that the data rate for an individual sensor information vanishes, while the latter case results in the transmission without coding techniques. In general, it is difficult to determine which level is optimal for the estimation, i.e., which scenario results in the smaller value of Pe. Indeed, by using the rate distortion codes, the data center could use as many sensors as possible for a given R. However, the quality of the individual reproduction would be less informative. The best choice seems to depend largely on R, as well as p. 3 Main Results For simplicity, we consider the following two solvable cases; K = 2 for C ≥K and the optimal case of K →∞. Let p be a given observation noise level, and R the finite real value of a given combined data rate. Letting L →∞, we find the expected bit error frequency to be Pe(p, R) = −(1−2p)cg √ R −∞ dr N(0, 1) (4) with the constant value cg = 1 √ 2 √α 2 + 2 ln 2 √α − √α 2 −σ2 √α ⟨tanh2 x⟩π(x) (K = 2) √ 2 ln2 (K →∞) (5) where the rescaled variance σ2 = α ⟨ˆx2⟩ˆπ(ˆx) and the first step RSB enforcement −1 2 + 2 α ln 2 + 1 2 −σ2 α ⟨tanh2 x (1 + 2x csch x sech x)⟩π(x) = 0 holds. Here N(X, Y ) denotes the normal distribution with the mean X and the variance Y . The rescaled variance σ2 and the scale invariant parameter α is determined numerically, where we use the following notations. ⟨· ⟩π(x) = ∞ −∞ dx √ 2πσ2 exp −x2 2σ2 ( · ) , ⟨· ⟩ˆπ(ˆx) = +1 −1 dˆx √ 2πσ2 (1 −ˆx2)−1 exp −(tanh−1 ˆx)2 2σ2 ( · ) . 0 0.1 0.2 0.3 0.4 0.5 −2 −1 0 1 2 p P (dB) e (p,R) (a) Narrow Band 0 0.1 0.2 0.3 0.4 0.5 −150 −100 −50 0 50 100 150 p P (dB) e (p,R) (b) Broadband R = 1 R = 2 R = 10 R = 100 R = 500 R = 1000 Figure 1: P (dB) e (p, R) for K = 2. (a) Narrow band (b) Broadband Therefore, it is straightforward to evaluate (4) with (5) for given parameters, p and R. For a given finite value of R, we see what happens to the quality of the estimate when the noise level p varies. Fig. 1 and Fig. 2 shows the typical behavior of the bit error frequency, Pe(p, R), in decibel (dB), where the reference level is chosen as P (0) e (p, R) =
(R−1)/2 l=0 R l (1 −p)lpR−l, (R is odd)
R/2−1 l=0 R l (1 −p)lpR−l + 1 2 R R/2 (1 −p)R/2pR/2 (R is even) (6) for a given integer R. The reference (6) denotes Pe for the case of L = R, i.e., the case when the sensors are not allowed to compress their observations. Here, in decibel, we have P (dB) e (p, R) = 10 log Pe(p, R) P (0) e (p, R) , where the log is to base 10. Note that the zero level in decibel occurs when the measured error frequency Pe(p, R) is equal to the reference level. Therefore, it is also possible to have negative levels, which would mean an expected bit error frequency much smaller than the reference level. In the case of small combined data rate R, the narrow band case, the numerical results in Fig. 1 (a) and Fig. 2 (a) show that the quality of the estimate is sensitive to the parity of the integer R. In particular, the R = 2 case has the lowest threshold level, pc = 0.0921 for Fig. 1 (a) and pc = 0.082 for Fig. 2 (a) respectively, beyond which the L →∞scenario outperforms the L = R scenario, while the R = 1 case does not have such a threshold. In contrast, if the bandwidth is wide enough, the difference of the expected bit error probabilities in decibel, P (dB) e (p, R), is proved to have similar qualitative characteristics as shown in Fig. 1 (b) and Fig. 2 (b). Moreover, our preliminary experiments for larger systems also indicate that the threshold pc seems to converge to the value, 0.165 and 0.146 respectively, as L goes to infinity; we are currently working on the theoretical derivation. 4 Outline of Derivation Since the predetermined matrices A1, · · · , AL are selected randomly, it is quite natural to say that the instantaneous series, defined by ˆy(t) = [ˆy1(t), · · · , ˆyL(t)]T , can be modeled 0 0.1 0.2 0.3 0.4 0.5 −2 −1 0 1 2 p P (dB) e (p,R) (a) Narrow Band 0 0.1 0.2 0.3 0.4 0.5 −150 −100 −50 0 50 100 150 p P (dB) e (p,R) (b) Broadband R = 1 R = 2 R = 10 R = 100 R = 500 R = 1000 Figure 2: P (dB) e (p, R) for K →∞. (a) Narrow band (b) Broadband using the Bernoulli trials. Here, the reproduction problem reduces to a channel model, where the stochastic matrix is defined as W(ˆy|x) = q, if ˆy = x 1 −q, otherwise , (7) where q denotes the quality of the reproductions, i.e., Pr[x ̸= ˆyi] = 1−q for i = 1, · · · , L. Letting the channel model (7) for the reproduction problem be valid, the expected bit error frequency can be well captured by using the cumulative probability distributions Pe = Pr[x ̸= ˆx] = B( L−1 2 : L, q), if L is odd B( L 2 −1 : L, q) + 1 2b( L 2 : L, q) otherwise (8) with B(L′ : L, q) = L′ l=0 b(l : L, q) , b(l : L, q) = L l ql(1 −q)L−l , where an integer l be the total number of non-flipped elements in ˆy(t), and the second term (1/2)b(L/2 : L, q) represents random guessing with l = L/2. Note that the reproduction quality q can be easily obtained by the simple algebra q = pD + (1 −p)(1 −D), where D is the distortion with respect to coding. Since the error probability (8) is given by a function of q, we firstly derive an analytical solution for the quality q in the limit L →∞, keeping R finite. In this approach, we apply the method of statistical mechanics to evaluate the typical performance of the codes [4]. As a first step, we translate the Boolean alphabets Z = {0, 1} to the “Ising” ones, S = {+1, −1}. Consequently, we need to translate the additive operations, such as, zi(s) + zi(s′) (mod 2) into their multiplicative representations, σ i(s) × σi(s′) ∈S for s, s′ = 1, · · · , m. Similarly, we translate the Boolean yi(t)s into the Ising Ji(t)s. For simplicity, we omit the subscript i, which labels the L agents, in the rest of this section. Following the prescription of Sourlas [5], we examine the Gibbs-Boltzmann distribution Pr[σ] = exp [−βH(σ|J)] Z(J) with Z(J) = σ e−βH(σ|J) , (9) where the Hamiltonian of the Ising system is defined as H(σ|J) = − s1<···<sK As1...sKJi[t(s1, . . ., sK)]σ(s1) . . . σ(sK) . (10) The observation index t(s1, . . ., sK) specifies the proper value of t given the set s1, . . ., sK, so that it corresponds to the parity check equation (1). Here the elements of the symmetric tensor As1...sK, representing dilution, is either zero or one depending on the set of indices (s1, . . ., sK). Since there are C non-zero elements randomly chosen for any given index s, we find
s2,...,sK Ass2...sK = C . The code rate is R/L = K/C because a reproduction sequence has C bits per index s and carries K bits of the codeword. It is easy to see that the Hamiltonian (10) is counting the reproduction errors, [1 −Jt(s1,...,sK) · σ(s1) . . .σ(sK)]/2. Moreover, according to the statistical mechanics, we can easily derive the “observable” quantities using the free energy defined as f = −1 β ⟨lnZ(J)⟩A,J which carries all information about the statistics of the system. Here, β denotes an “inverse temperature” for the Gibbs-Boltzmann distribution (9), and ⟨·⟩A,J represents the configurational average. Therefore, we have to average the logarithm of the partition function Z(J) over the given distribution ⟨·⟩A,J after the calculation of the partition function. Finally, to perform such a program, the replica trick is used [6]. The theory of replica symmetry breaking can provide the free energy resulting in the expression f = −1 βn ln cosh β −K ⟨ln[1 + tanh(βx) tanh(βˆx)]⟩π(x),ˆπ(ˆx) +1 2 J=±1 ln 1 + tanh(βJ) K l=1 tanh(βxl) π(x) + C K ln σ=±1 C l=1 [1 + σ tanh(βˆxl)] ˆπ(ˆx) , (11) where ⟨·⟩π(x) denotes the averaging over p(xl)s and so on. The variation of (11) by π(x) and ˆπ(ˆx) under the condition of normalization gives the saddle point condition π(x) = δ x − C−1 l=1 ˆxl ˆπ(ˆx) , ˆπ(ˆx) = 1 2 J=±1 δ [ˆx −µ(x1, . . ., xK−1; J)] π(x) , where µ(x1, . . ., xK−1; J) = 1 β tanh−1 tanh(βJ) K−1 l=1 tanh(βxl) . We now investigate the case of K = 2. Applying the central limit theorem to π(x) [7], we get π(x) = 1 √ 2πCσ2 e− x2 2Cσ2 , (12) where σ2 is the variance of ˆπ(ˆx). Here the resulting distribution (12) is a even function. The leading contribution to µ is then given by µ(x; J) ∼J · tanh(βx) as β goes to zero; The expression is valid in the asymptotic region L ≫1 for a fixed R. Then, the formula for the delta function yields [8] ˆπ(ˆx) = δ x −1 β tanh−1 ˆx ρ′ 1 β tanh−1 ˆx; ˆx −1 π(x) = (1 −ˆx2)−1 2πβ2Cσ2 exp −(tanh−1 ˆx)2 2β2Cσ2 , (13) where we have used ρ(x; ˆx) = ˆx −tanh(βx). Therefore, we have σ2 = ⟨ˆx2⟩ˆπ(ˆx) = +1 −1 dˆx 2πβ2Cσ2 ˆx2 1 −ˆx2 exp −(tanh−1 ˆx)2 2β2Cσ2 for given β2C. Inserting (12), (13) into (11), we get f = −β 2 −R β ln2 + 1 −2σ2 2 β tanh2 ˜x ˜π(˜x) with ˜π(˜x) = 1 2πβ2Cσ2 e− ˜x2 2β2Cσ2 , where we rewrite ˜x = βx. The theory of replica symmetry breaking tells us that relevant value of β should not be smaller than the “freezing point” βg, which implies the vanishing entropy condition: ∂f ∂β = −1 2 + 2 β2gC ln 2 + 1 −2σ2 2 tanh2 ˜x (1 + 2˜x csch ˜x sech ˜x) ˜π(˜x) = 0 . Accordingly, it is convenient for us to define a scaling invariant parameter α = β 2 gC, and to rewrite the variance ˜σ2 = ασ2 for simplicity. Introducing these newly defined parameters, the above results could be summarized as follows. Given R and L, we find f = R L −1 2 α 2 −ln2 2 α + α 2 1 2 −˜σ2 α ⟨tanh2 ˜x⟩˜π(˜x) with ˜σ2 = α ⟨ˆx2⟩ˆπ(ˆx), where the condition −1 2 + 2 α ln 2 + 1 2 −˜σ2 α tanh2 ˜x (1 + 2˜x csch ˜x sech ˜x) ˜π(˜x) = 0 (14) holds. Here we denote ⟨· ⟩˜π(˜x) = ∞ −∞ d˜x √ 2π˜σ2 exp −˜x2 2˜σ2 ( · ) , ⟨· ⟩ˆπ(ˆx) = +1 −1 dˆx √ 2π˜σ2 (1 −ˆx2)−1 exp −(tanh−1 ˆx)2 2˜σ2 ( · ) . Lastly, by using the cumulative probability distribution, we get Pe = L/2 l=0 L l ql(1 −q)L−l ∼ L/2 0 dr N(Lq, Lq(1 −q)) . (15) It is easy to see that (15) can be converted to a standard normal distribution by changing variables to ˜r = (r −Lq)/ Lq(1 −q) [7], so d˜r = dr/ Lq(1 −q), yielding Pe ∼ ˜rg − √ L d˜r N(0, 1) with ˜rg = 2 √ L(1 −2p) D −1 2 = R 2 (1 −2p) −1 2 √α −2 ln2 √α + √α 1 2 −˜σ2 α ⟨tanh2 ˜x⟩˜π(˜x) . Note that the relation D = (1 + f)/2 holds at the vanishing entropy condition (14) [4]. Finally, we obtain the main result (4) in Section 3 in the limit L →∞, when we use proper notations for the variables and the name of the function. We can investigate the asymptotic case of K →∞in a similar way. Since the leading contribution to ˆπ(ˆx) comes from the value of x in the vicinity of √ Cσ2, we find the expression ˆπ(ˆx) ≈ δ ˆx −yβK(Cσ2) K 2 by using the power counting. Therefore, within the Parisi RSB scheme, one obtain a set of equations √ Lf = − √αc 2 − R √αc ln2 , −1 2 + R αc ln 2 = 0 with the scale-invariant αc = β2L. This results in cg = √ 2 ln2, as is mentioned before. 5 Conclusion This paper provides a system-level perspective for massive sensor networks. The decentralized sensing problem argued in this paper was first addressed by Berger and his collaborators. However, this paper is the first work that gives a scheme to analyze practically tractable codes in the given finite data rate, and shows the existence of threshold level of noise of which the optimal levels of decentralization changes. Future work includes the theoretical derivation of the threshold level p c where R goes to infinity, as well as the implementation problem. Acknowledgments The authors thank Jun Muramatsu and Naonori Ueda for useful discussions. This work was supported by the Ministry of Education, Science, Sports and Culture (MEXT) of Japan, under the Grant-in-Aid for Young Scientists (B), 15760288. References [1] (2005) Intel@Mote. [Online]. Available: http://www.intel.com/research/exploratory/motes.htm [2] T. Berger, Z. Zhang, and H. Viswanathan, “The CEO problem,” IEEE Trans. Inform. Theory, vol. 42, pp. 887–902, May 1996. [3] D. J. C. MacKay, Information Theory, Inference and Learning Algorithms. Cambridge, UK: Cambridge University Press, 2003. [4] T. Murayama and M. Okada, “Rate distortion function in the spin glass state: a toy model,” in Advances in Neural Information Processing Systems 15 (NIPS’02), Denver, USA, Dec. 2002, pp. 423–430. [5] N. Sourlas, “Spin-glass models as error-correcting codes,” Nature, vol. 339, pp. 693–695, June 1989. [6] V. Dotsenko, Introduction to the Replica Theory of Disordered Statistical Systems. Cambridge, UK: Cambridge University Press, 2001. [7] W. Hays, Statistics (5th Edition). Belmont, CA: Wadsworth Publishing, 1994. [8] C. W. Wong, Introduction to Mathematical Physics: Methods and Concepts. Oxford, UK: Oxford University Press, 1991.
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Beyond Pair-Based STDP: a Phenomenogical Rule for Spike Triplet and Frequency Effects Jean-Pascal Pfister and Wulfram Gerstner School of Computer and Communication Sciences and Brain-Mind Institute, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne {jean-pascal.pfister, wulfram.gerstner}@epfl.ch Abstract While classical experiments on spike-timing dependent plasticity analyzed synaptic changes as a function of the timing of pairs of pre- and postsynaptic spikes, more recent experiments also point to the effect of spike triplets. Here we develop a mathematical framework that allows us to characterize timing based learning rules. Moreover, we identify a candidate learning rule with five variables (and 5 free parameters) that captures a variety of experimental data, including the dependence of potentiation and depression upon pre- and postsynaptic firing frequencies. The relation to the Bienenstock-Cooper-Munro rule as well as to some timing-based rules is discussed. 1 Introduction Most experimental studies of Spike-Timing Dependent Plasticity (STDP) have focused on the timing of spike pairs [1, 2, 3] and so do many theoretical models. The spike-pair based models can be divided into two classes: either all pairs of spikes contribute in a homogeneous fashion [4, 5, 6, 7, 8, 9, 10] (called ‘all-to-all’ interaction in the following) or only pairs of ‘neighboring’ spikes [11, 12, 13] (called ‘nearest-spike’ interaction in the following); cf. [14, 15]. Apart from these phenomenological models, there are also models that are somewhat closer to the biophysics of synaptic changes [16, 17, 18, 19]. Recent experiments have furthered our understanding of timing effects in plasticity and added at least two different aspects: firstly, it has been shown that the mechanism of potentiation in STDP is different from that of depression [20] and secondly, it became clear that not only the timing of pairs, but also of triplets of spikes contributes to the outcome of plasticity experiments [21, 22]. In this paper, we introduce a learning rule that takes these two aspects partially into account in a simple way. Depression is triggered by pairs of spikes with post-before-pre timing, whereas potentiation is triggered by triplets of spikes consisting of 1 pre- and 2 postsynaptic spikes. Moreover, in our model the pair-based depression includes an explicit dependence upon the mean postsynaptic firing rate. We show that such a learning rule accounts for two important stimulation paradigms: P1 (Relative Spike Timing): Both the pre- and postsynaptic spike trains consist of a burst of N spikes at regular intervals T, but the two spike trains are shifted by a time ∆t = tpost −tpre. The total weight change is a function of the relative timing ∆t (this gives the standard STDP function), but also a function of the firing frequency ρ = 1/T during the burst; cf. Fig. 1A (data from L5 pyramidal neurons in visual cortex). P2 (Poisson Firing): The pre- and postsynaptic spike trains are generated by two independent Poisson processes with rates ρx and ρy respectively. Protocol P2 has less experimental support but it helps to establish a relation to the Bienenstock-Cooper-Munro (BCM) model [23]. To see that relation, it is useful to plot the weight change as a function of the postsynaptic firing rate, i.e., ∆w ∝φ(ρy) (cf. Fig 1B). Note that the function φ has only been measured indirectly in experiments [24, 25]. We emphasize that in the BCM model, ∆w = ρxφ(ρy, ¯ρy) (1) the function φ depends not only on the current firing rate ρy, but also on the mean firing rate ¯ρy averaged over the recent past which has the effect that the threshold between depression and potentiation is not fixed but dynamic. More precisely, this threshold θ depends nonlinearly on the mean firing rate ¯ρy: θ = α¯ρp y, p > 1 (2) with parameters α and p. Previous models of STDP have already discussed the relation of STDP to the BCM rule [16, 12, 17, 26], but none of these seems to be completely satisfactory as discussed in Section 4. We will also compare our results to the rule of [21] which was together with the work of [16] amongst the first triplet rules to be proposed. A B 0 10 20 30 40 50 −100 −50 0 50 100 ∆w [%] ρ [Hz] 0 10 20 30 40 50 −5 0 5 10 15 20x 10 −4 ⟨˙w⟩[ms−1] ρy [Hz] Figure 1: A. Weight change in an experiment on cortical synapses using pairing protocol (P1) (solid line: ∆t = 10 ms, dot-dashed line ∆t = −10 ms) as a function of the frequency ρ. Figure redrawn from [11]. B. Weight change in protocol P2 according to the BCM rule for θ = 20, 30, 40 Hz. 2 A Framework for STDP Several learning rules in the modeling literature can be classified according to the two criteria introduced above: (i) all-to-all interaction vs. nearest spike interaction; (ii) pairbased vs. triplet based rules. Point (ii) can be elaborated further in the context of an expansion (pairs, triplets, quadruplets, ... of spikes) that we introduce now. 2.1 Volterra Expansion (‘all-to-all’) For the sake of simplicity, we assume that weight changes occur at the moment of presynaptic spike arrival or at the moment of postsynaptic firing. The direction and amplitude of the weight change depends on the configuration of spikes in the presynaptic spike train X(t) = P k δ(t −tk x) and the postsynaptic spike train Y (t) = P k δ(t −tk y). With some arbitrary functionals F[X, Y ] and G[X, Y ], we write (see also [8]) ˙w(t) = X(t)F[X, Y ] + Y (t)G[X, Y ] (3) Clearly, there can be other neuronal variables that influence the synaptic dynamics. For example, the weight change can depend on the current weight value w [8, 15, 10], the Ca2+ concentration [17, 19], the depolarization [25, 27, 28], the mean postsynaptic firing rate ¯ρy(t) [23],.... Here, we will consider only the dependence upon the history of the pre- and postsynaptic firing times and the mean postsynaptic firing rate ¯ρy. Note that even if ¯ρy depends via a low-pass filter τρ ˙¯ρy = −¯ρy + Y (t) on the past spike train Y of the postsynaptic neuron, the description of the problem will turn out to be simpler if the mean firing rate is considered as a separate variable. Therefore, let us write the instantaneous weight change as ˙w(t) = X(t)F([X, Y ], ¯ρy(t)) + Y (t)G([X, Y ], ¯ρy(t)) (4) The goal is now to determine the simplest functionals F and G that would be consistent with the experimental protocols P1 and P2 introduced above. Since the functionals are unknown, we perform a Volterra expansion of F and G in the hope that a small number of low-order terms are sufficient to explain a large body of experimental data. The Volterra expansion [29] of the functional G can be written as1 G([X, Y ]) = Gy 1 + Z ∞ 0 Gxy 2 (s)X(t −s)ds + Z ∞ 0 Gyy 2 (s)Y (t −s)ds + Z ∞ 0 Z ∞ 0 Gxxy 3 (s, s′)X(t −s)X(t −s′)ds′ds + Z ∞ 0 Z ∞ 0 Gxyy 3 (s, s′)X(t −s)Y (t −s′)ds′ds + Z ∞ 0 Z ∞ 0 Gyyy 3 (s, s′)Y (t −s)Y (t −s′)ds′ds + . . . (5) Similarly, the expansion of F yields F([X, Y ]) = F x 1 + Z ∞ 0 F xx 2 (s)X(t −s)ds + Z ∞ 0 Fxy 2 (s)Y (t −s)ds + . . . (6) Note that the upper index in functions represents the type of interaction. For example, Gxyy 3 (in bold face above) refers to a triplet interaction consisting of 1 pre- and 2 postsynaptic spikes. Note that the Gxyy 3 term could correspond to a pre-post-post sequence as well as a post-pre-post sequence. Similarly the term F xy 2 picks up the changes caused by arrival of a presynaptic spike after postsynaptic spike firing. Several learning rules with all-to-all interaction can be classified in this framework, e.g. [5, 6, 7, 8, 9, 10]. 2.2 Our Model Not all term in the expansion need to be non-zero. In fact, in the results section we will show that a learning rule with Gxyy 3 (s, s′) ≥0 for all s, s′ > 0 and F xy 2 (s) ≤0 for s > 0 and all other terms set to zero is sufficient to explain the results from protocols P1 and P2. Thus, in our learning rule an isolated pair of spikes in configuration post-before-pre will lead to depression. An isolated spike pair pre-before-post, on the other hand, would not be sufficient to trigger potentiation, whereas a triplet pre-post-post or post-pre-post will do so (see Fig. 2). 1For the sake of clarity we have omitted the dependence on ¯ρy. A B y τ + τ τ− Figure 2: A. Triplet interaction for LTP B. Pair interaction for LTD. To be specific, we consider F xy 2 (s) = −A−(¯ρy)e − s τ− and Gxyy 3 (s, s′) = A+e − s τ+ e−s′ τy . (7) Such an exponential model can be implemented by a mechanistic update involving three variables (the dot denotes a temporal derivative) ˙a = −a τ+ ; if t = tk x then a →a + 1 ˙b = −b τ− ; if t = tk y then b →b + 1 (8) ˙c = −c τy ; if t = tk y then c →c + 1 The weight update is then ˙w(t) = −A−(¯ρy)X(t)b(t) + A+Y (t)a(t)c(t). (9) 2.3 Nearest Spike Expansion (truncated model) Following ideas of [11, 12, 13], the expansion can also be restricted to neighboring spikes only. Let us denote by fy(t) the firing time of the last postsynaptic spike before time t. Similarly, fx(t′) denotes the timing of the last presynaptic spike preceding t′. With this notation the Volterra expansion of the preceding section can be repeated in a form that only nearest spikes play a role. A classification of the models [11, 12, 13] is hence possible. We focus immediately on the truncated version of our model ˙w(t) = X(t)F xy 2 (t −fy(t), ¯ρy(t)) + Y (t)Gxyy 3 (t −fx(t), t −fy(t)) (10) The mechanistic model that generates the truncated version of the model is similar to Eq. (8) except that under the appropriate update condition, the variable goes to one, i.e. a →1, b → 1 and c →1. The weight update is identical to that of the all-to-all model, Eq. (9). 3 Results One advantage of our formulation is that we can derive explicit formulas for the total weight changes induced by protocols P1 and P2. 3.1 All-to-all Interaction If we use protocol P1 with a total of N pre- and postsynaptic spikes at frequency ρ shifted by a time ∆t, then the total weight change ∆w is for our model with all-to-all interaction ∆w = A+ N−1 X k=0 N−1 X k′=1 (N −max(k, k′)) exp −k/ρ + ∆t τ+ exp −k′ τyρ λk(−∆t) − A−(¯ρy) N−1 X k=0 (N −k) exp −k/ρ −∆t τ− λk(∆t) (11) where λk(∆t) = 1−δk0Θ(∆t) with Θ the Heaviside step function. The results are plotted in Fig. 3 top-left for N = 60 spikes. P1 P2 All-to-all 0 10 20 30 40 50 −100 −50 0 50 100 150 200 ∆w [%] ρ [Hz] 0 10 20 30 40 50 0 10 20 x 10 −5 ⟨˙w⟩[ms−1] ρy [Hz] Nearest Spike 0 10 20 30 40 50 −50 0 50 100 ∆w [%] ρ [Hz] 0 10 20 30 40 50 0 5 10 15 x 10 −5 ⟨˙w⟩[ms−1] ρy [Hz] Figure 3: Triplet learning rule. Summary of all results of protocol P1 (left) and P2 (right) for an all-to-all (top) and nearest-spike (bottom) interaction scheme. For the left column, the upper thick lines correspond to positive timing (∆t > 0) while the lower thin lines to negative timing. Dashed line: ∆t = ±2 ms, solid line: ∆t = ±10 ms and dot-dashed line ∆t = ±30 ms. The error bars indicate the experimental data points of Fig. 1A. Right column: dashed-line ¯ρy = 8 Hz, solid line ¯ρy = 10 Hz and dot-dashed line ¯ρy = 12 Hz. Top: τy = 200 ms, bottom: τy = 40 ms. The mean firing rate ¯ρy reflects the firing activity during the recent past (i.e. before the start of the experiment) and is assumed as fixed during the experiment. The exact value does not matter. Overall, the frequency dependence of changes ∆w is very similar to that observed in experiments. If X and Y are independent Poisson process, the protocol P2 gives a total weight change that can be calculated using standard arguments [8] ⟨˙w⟩= −A−(¯ρy)ρxρyτ−+ A+ρxρ2 yτ+τy (12) As before, the mean firing rate ¯ρy reflects the firing activity during the recent past and is assumed as fixed during the experiment. In order to implement a sliding threshold as in the BCM rule, we take A−(¯ρy) = β−¯ρ2 y/ρ2 0 where we set ρ0 = 10 Hz. This yields a frequency dependent threshold θ(¯ρy) = β−τ−¯ρ2 y/(A+τ+τyρ2 0). As can be seen in Fig. 3 top-right our model exhibits all essential features of a BCM rule. 3.2 Nearest Spike Interaction We now apply protocols P1 and P2 to our truncated rule, i.e. restricted to the nearest-spike interaction; cf. Eq. (10) where the expression of F xy 2 and Gxyy 3 are taken from Eq. (7). The weight change ∆w for the protocol P1 can be calculated explicitly and is plotted in Fig. 3 bottom-left. For protocol P2 (see Fig. 3 bottom-right) we find ⟨˙w⟩= ρx −A−(¯ρy)ρy ρy + α− + A+ ρx + α+ ρ2 y ρy + αy ! (13) where αy = τ −1 y . If we assume that ρx ≪αx, Eq. (13) is a BCM learning rule. In summary, both versions of our learning rule (all-to-all or nearest-spike) yield a frequency dependence that is consistent with experimental results under protocol P1 and with the BCM rule tested under protocol P2. We note that our learning rule contains only two terms, i.e., a triplet term (1 pre and 2 post) for potentiation and a post-pre pair term for depression. The dynamics is formulated using five variables (a, b, c, ¯ρy, w) and five parameters (τ+, τ−, τy, A+, β−). τ+ = 16.8 ms and τ−= 33.7 ms are taken from [14]. A+ and β−are chosen such that the weight changes for ∆t = ±10 ms and ρ = 20 Hz fit the experimental data [11]. 4 Discussion - Comparison with Other Rules While we started out developing a general framework, we focused in the end on a simple model with only five parameters - why, then, this model and not some other combination of terms? To answer this question we apply our approach to a couple of other models, i.e., pair-based models (all-to-all or nearest spike), triplet-based models, and others. 4.1 STDP Models Based on Spike Pairs Pair-based models with all-to-all interaction [4, 5, 6, 7, 8, 9, 10] yield under Poisson stimulation (protocol P2) a total weight change that is linear in presynaptic and postsynaptic frequencies. Thus, as a function of postsynaptic frequency we always find a straight line with a slope that depends on the integral of the STDP function [5, 7]. Thus pair-based models with all-to-all interaction need to be excluded in view of BCM features of plasticity [25, 24]. P1 P2 Pair 0 10 20 30 40 50 −60 −40 −20 0 20 40 60 80 ∆w [%] ρ [Hz] 0 10 20 30 40 50 −1 0 1 2 3 x 10 −5 ρy [Hz] ⟨˙w⟩[ms−1] F-D 0 10 20 30 40 50 −150 −100 −50 0 50 100 ∆w [%] ρ [Hz] 0 10 20 30 40 50 −20 −15 −10 −5 0 5x 10 −6 ρy [Hz] ⟨˙w⟩[ms−1] Figure 4: Pair learning rule in a nearest spike interaction scheme (top) and Froemke-Dan rule (bottom). For the left column, the higher thick lines correspond to positive timing (∆t > 0) while the lower thin lines to negative timing. Dashed line: ∆t = ±2 ms, solid line: ∆t = ±10 ms and dot-dashed line ∆t = ±30 ms. Right column: dashed-line ¯ρy = 8 Hz, solid line ¯ρy = 10 Hz and dot-dashed line ¯ρy = 12 Hz. The parameters of the F-D model are taken from [21]. The dependence upon ¯ρy has been added to the original F-D rule (A−→β−¯ρ2 y/ρ2 0). A pair-based model with nearest-spike interaction, however, can give a non-linear dependence upon the postsynaptic frequency under protocol P2 with fixed threshold between depression and potentation [12]. We can go beyond the results of [12] by adding a suitable dependence of the parameter A−upon ¯ρy which yields a sliding threshold; cf. Fig. 4 top right. But even a pair rule restricted to nearest-spike interaction is unable to account for the results of protocol P1. An important feature of the experimental results with protocol P1 is that potentiation only occurs above a minimal firing frequency of the postsynaptic neuron (cf. Fig. 1A) whereas pair-based rules always exhibit potentiation with pre-before-post timing even in the limit of low frequencies; cf. Fig. 4 top left. The intuitive reason is that at low frequency the total weight change is proportional to the number of pre-post pairings and this argument can be directly transformed into a mathematical proof (details omitted). Thus, pair-based rules of potentiation (all-to-all or nearest spike) cannot account for results of protocol P1 and must be excluded. 4.2 Comparison with Triplet-Based Learning Rules The model of Senn et al. [16] can well account of the results under protocol P1. A classification of this rule within our framework reveals that the update algorithm generates pair terms of the form pre-post and post-pre, as well as triplet terms of the form pre-post-post and post-pre-pre. As explained in the previous paragraph, a pair term pre-post generated potentiation even at very low frequencies which is not realistic. In order to avoid this effect in their model, Senn et al. included additional threshold values which increased the number of parameters in their model to 9 [16] while the number of variables is 5 as in our model. Moreover, the mapping of the model of Senn et al. to the BCM rule is not ideal, since the sliding threshold is different for each individual synapse [16]. An explicit triplet rule has been proposed by Froemke and Dan [21]. In our framework, the rule can be classified as a combination of triplet terms for potentiation and depression. Following the same line or argument as in the preceding sections we can calculate the total weight change for protocols P1 and P2. The result is shown in Fig. 4 bottom. We can clearly see that the pairing experiment P1 yields a behavior opposite to the one found experimentally and the BCM behavior is not at all reproduced in protocol P2. 4.3 Summary We consider our model as a minimal model to account for results of protocol P1 and P2, but, of course, several factors are not captured by the model. First, our model has no dependence upon the current weight value, but, in principle, this could be included along the lines of [10]. Second, the model has no explicit dependence upon the membrane potential or calcium concentration, but the postsynaptic neuron enters only via its firing activity. 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Is Early Vision Optimized for Extracting Higher-order Dependencies? Yan Karklin yan+@cs.cmu.edu Michael S. Lewicki∗ lewicki@cnbc.cmu.edu Computer Science Department & Center for the Neural Basis of Cognition Carnegie Mellon University Abstract Linear implementations of the efficient coding hypothesis, such as independent component analysis (ICA) and sparse coding models, have provided functional explanations for properties of simple cells in V1 [1,2]. These models, however, ignore the non-linear behavior of neurons and fail to match individual and population properties of neural receptive fields in subtle but important ways. Hierarchical models, including Gaussian Scale Mixtures [3, 4] and other generative statistical models [5, 6], can capture higher-order regularities in natural images and explain nonlinear aspects of neural processing such as normalization and context effects [6,7]. Previously, it had been assumed that the lower level representation is independent of the hierarchy, and had been fixed when training these models. Here we examine the optimal lower-level representations derived in the context of a hierarchical model and find that the resulting representations are strikingly different from those based on linear models. Unlike the the basis functions and filters learned by ICA or sparse coding, these functions individually more closely resemble simple cell receptive fields and collectively span a broad range of spatial scales. Our work unifies several related approaches and observations about natural image structure and suggests that hierarchical models might yield better representations of image structure throughout the hierarchy. 1 Introduction Efficient coding hypothesis has been proposed as a guiding computational principle for the analysis of early visual system and motivates the search for good statistical models of natural images. Early work revealed that image statistics are highly non-Gaussian [8, 9], and models such as independent component analysis (ICA) and sparse coding have been developed to capture these statistics to form efficient representations of natural images. It has been suggested that these models explain the basic computational goal of early visual cortex, as evidenced by the similarity between the learned parameters and the measured receptive fields of simple cells in V1. ∗To whom correspondence should be addressed In fact, it is not clear exactly how well these methods predict the shapes of neural receptive fields. There has been no thorough characterization of ICA and sparse coding results for different datasets, pre-processing methods, and specific learning algorithms employed, although some of these factors clearly affect the resulting representation [10]. When ICA or sparse coding is applied to natural images, the resulting basis functions resemble Gabor functions [1, 2] — 2D sine waves modulated by Gaussian envelopes — which also accurately model the shapes of simple cell receptive fields [11]. Often, these results are visualized in a transformed space, by taking the logarithm of the pixel intensities, sphering (whitening) the image space, or filtering the images to flatten their spectrum. When analyzed in the original image space, the learned filters (the models’ analogues of neural receptive fields) do not exhibit the multi-scale properties of the visual system, as they tend to cluster at high spatial frequencies [10, 12]. Neural receptive fields, on the other hand, span a broad range of spatial scales, and exhibit distributions of spatial phase and other parameters unmatched by ICA and SC results [13,14]. Therefore, as models of early visual processing, these models fail to predict accurately either the individual or the population properties of cortical visual neurons. Linear efficient coding methods are also limited in the type of statistical structure they can capture. Applied to natural images, their coefficients contain significant residual dependencies that cannot be accounted for by the linear form of the models. Several solutions have been proposed, including multiplicative Gaussian Scale Mixtures [4] and generative hierarchical models [5, 6]. These models capture some of the observed dependencies; but their analysis so far has been focused on the higher-order structure learned by the model. Meanwhile, the lower-level representation is either chosen a priori [4] or adapted separately, in the absence of the hierarchy [6] or with a fixed hierarchical structure specified in advance [5]. Here we examine whether the optimal lower-level representation of natural images is different when trained in the context of such non-linear hierarchical models. We also illustrate how the model not only describes sparse marginal densities and magnitude dependencies, but captures a variety of joint density functions that are consistent with previous observations and theoretical conjectures. We show that learned lower-level representations are strikingly different from those learned by the linear models: they are more multi-scale, spanning a wide range of spatial scales and phases of the Gabor sinusoid relative to the Gaussian envelope. Finally, we place these results in the context of whitening, gain control, and non-linear neural processing. 2 Fully adaptable scale mixture model A simple and scalable model for natural image patches is a linear factor model, in which the data x are assumed to be generated as a linear combination of basis functions with additive noise x = Au + ϵ . (1) Typically, the noise is assumed to be Gaussian with variance σ2 ϵ, thus P(x|A, u) ∝exp − X i 1 2σ2ϵ |x −Au|2 i ! . (2) The coefficients u are assumed to be mutually independent, and often modeled with sparse distributions (e.g. Laplacian) that reflect the non-Gaussian statistics of natural scenes [8,9], P(u) = Y i P(ui) ∝exp(− X i |ui|) . (3) We can then adapt the basis functions A to maximize the expected log-likelihood of the data L = ⟨log P(x|A)⟩over the data ensemble, thereby learning a compact, efficient representation of structure in natural images. This is the model underlying the sparse coding algorithm [2] and closely related to independent component analysis (ICA) [1]. An alternative to fixed sparse priors for u (3) is to use a Gaussian Scale Mixture (GSM) model [3]. In these models, each observed coefficient ui is modeled as a product of random Gaussian variable yi and a multiplier λi, ui = p λiyi (4) Conditional on the value of the multiplier λi, the probability P(ui|λi) is Gaussian with variance λi, but the form of the marginal distribution P(ui) = Z N(0, λi)P(λi)dλi (5) depends on the probability function of λi and can assume a variety of shapes, including sparse heavy-tailed functions that fit the observed distributions of wavelet and ICA coefficients [4]. This type of model can also account for the observed dependencies among coefficients u, for example, by expressing them as pair-wise dependencies among the multiplier variables λ [4,15]. A more general model, proposed in [6, 16], employs a hierarchical prior for P(u) with adapted parameters tuned to the global patterns in higher-order dependencies. Specifically, the logarithm of the variances of P(u) is assumed to be a linear function of the higher-order random variables v, log σ2 u = Bv . (6) Conditional on the higher-order variables, the joint distribution of coefficients is factorisable, as in GSM. In fact, if the conditional density P(u|v) is Gaussian, this Hierarchical Scale Mixture (HSM) is equivalent to a GSM model, with λ = σ2 u and P(u|λ) = P(u|v) = N(0, exp(Bv)), with the added advantage of a more flexible representation of higher-order statistical regularities in B. Whereas previous GSM models of natural images focused on modeling local relationships between coefficients of fixed linear transforms, this general hierarchical formulation is fully adaptable, allowing us to recover the optimal lower-level representation A, as well as the higher-order components B. Parameter estimation in the HSM involves adapting model parameters A and B to maximize data log-likelihood L = ⟨log P(x|A, B)⟩. The gradient descent algorithm for the estimation of B has been previously described (see [6]). The optimal lower-level basis A is computed similarly to the sparse coding algorithm — the goal is to minimize reconstruction error of the inferred MAP estimate ˆu. However, ˆu is estimated not with a fixed sparsifying prior, but with a concurrently adapted hierarchical prior. If we assume a Gaussian conditional density P(u|v) and a standard-Normal prior P(v), the MAP estimates are computed as {ˆu, ˆv} = arg min u,v P(u, v|x, A, B) (7) = arg min u,v P(x|A, B, u, v)P(u|v)P(v) (8) = arg min u,v 1 2σ2ϵ X i |x −Au|2 i + X j [Bv]j 2 + u2 j 2e[Bv]j ! + X k v2 k 2 . (9) Marginalizing over the latent higher-order variables in the hierarchical models leads to sparse distributions similar to the Laplacian and other density functions assumed in ICA. Gaussian, qu = 2 Laplacian, qu = 1 Gen Gauss, qu = 0.7 HSM, B = [0;0] HSM, B = [1;1] HSM, B = [2;2] HSM, B = [1;−1] HSM, B = [2;−2] HSM, B = [1;−2] Figure 1: This model can describe a variety of joint density functions for coefficients u. Here we show example scatter plots and contour plots of some bivariate densities. Top row: Gaussian, Laplacian, and generalized Gaussian densities of the form p(u) ∝exp(−|u|q). Middle and bottom row: Hierarchical Scale Mixtures with different sets of parameters B. For illustration, in the hierarchical models the dimensionality of v is 1, and the matrix B is simply a column vector. These densities are computed by marginalizing over the latent variables v, here assumed to follow a standard normal distribution. Even with this simple hierarchy, the model can generate sparse star-shaped (bottom row) or radially symmetric (middle row) densities, as well as more complex non-symmetric densities (bottom right). In higher dimensions, it is possible to describe more complex joint distributions, with different marginals along different projections. However, although the model distribution for individual coefficients is similar to the fixed sparse priors of ICA and sparse coding, the model is fundamentally non-linear and might yield a different lower-level representation; the coefficients u are no longer mutually independent, and the optimal set of basis functions must account for this. Also, the shape of the joint marginal distribution in the space of all the coefficients is more complex than the i.i.d. joint density of the linear models. Bi-variate joint distributions of GSM coefficients can capture non-linear dependencies in wavelet coefficients [4]. In the fully adaptable HSM, however, the joint density can take a variety of shapes that depend on the learned parameters B (figure 1). Note that this model can produce sparse, star-shaped distributions as in the linear models, or radially symmetric distributions that cannot be described by the linear models. Such joint density profiles have been observed empirically in the responses of phase-offset wavelet coefficients to natural images and have inspired polar transformation and quadrature pair models [17] (as well as connections to phaseinvariant neural responses). The model described here can capture these joint densities and others, but rather than assume this structure a priori, it learns it automatically from the data. 3 Methods To examine how the lower-level representation is affected by the hierarchical model structure, we compared A learned by the sparse coding algorithm [2] and the HSM described above. The models were trained on 20 ×20 image patches sampled from 40 images of outdoor scenes in the Kyoto dataset [12]. We applied a low-pass radially symmetric filter to the full images to eliminate high corner frequencies (artifacts of the square sampling lattice), and removed the DC component from each image patch, but did no further pre-processing. All the results and analyses are reported in the original data space. Noise variance σ2 ϵ was set to 0.1, and the basis functions were initialized to small random values and adapted on stochastically sampled batches of 300 patches. We ran the algorithm for 10,000 iterations with a step size of 0.1 (tapered for the last 1,000 iterations, once model parameters were relatively unchanging). The parameters of the hierarchical model were estimated in a similar fashion. Gradient descent on A and B was performed in parallel using MAP estimates ˆu and ˆv. The step size for adapting B was gradually increased from .0001 to .01, because emergence of the variance patterns requires some stabilization in the basis functions in A. Because encoding in the sparse coding and in the hierarchical model is a non-linear process, it is not possible to compare the inverse of A to physiological data. Instead, we estimated the corresponding filters using reverse correlation to derive a linear approximation to a non-linear system, which is also a common method for characterizing V1 simple cells. We analyzed the resulting filters by fitting them with 2D Gabor functions, then examining the distribution of their frequencies, phase, and orientation parameters. 4 Results The shapes of basis functions and filters obtained with sparse coding have been previously analyzed and compared to neural receptive fields [10, 14]. However, some of the reported results were in the whitened space or obtained by training on filtered images. In the original space, sparse coding basis functions have very particular shapes: except for a few large, low frequency functions, all are localized, odd-symmetric, and span only a single period of the sinusoid (figure 2, top left). The estimated filters are similar but smaller (figure 2, bottom left), with peak spatial frequencies clustered at higher frequencies (figure 3). In the hierarchical model, the learned representation is strikingly different (figure 2, right panels). Both the basis and the filters span a wider range of spatial scales, a result previously unobserved for models trained on non-preprocessedimages, and one that is more consistent with physiological data [13, 14]. Also, the shapes of the basis functions are different — they more closely resemble Gabor functions, although they tend to be less smooth than the sparse coding basis functions. Both SC- and HSM-derived filters are well fit with Gabor functions. We also compared the distributions of spatial phases for filters obtained with sparse coding and the hierarchical model (figure 4). While sparse coding filters exhibit a strong tendency for odd-symmetric phase profiles, the hierarchical model results in a much more uniform distribution of spatial phases. Although some phase asymmetry has been observed in simple cell receptive fields, their phase properties tend to be much more uniform than sparse coding filters [14]. In the hierarchical model, the higher-order representation B is also adapted to the statistical structure of natural images. Although the choice of the prior density for v (e.g. sparse or Gaussian) can determine the type of structure captured in B, we discovered that it does not affect the nature of the lower-level representation. For the results reported here, we assumed a Gaussian prior on v. Thus, as in other multi-variate Gaussian models, the precise directions of B are not important; the learned vectors only serve to collectively describe the volume of the space. In this case, they capture the principal components of the logvariances. Because we were interested specifically in the lower-level representation, we did not analyze the matrix B in detail, though the principal components of this space seem to SC basis funcs HSM basis funcs SC filters HSM filters Figure 2: The lower-level representations learned by sparse coding (SC) and the hierarchical scale model (HSM). Shown are subsets of the learned basis functions and the estimates for the filters obtained with reverse correlation. These functions are displayed in the original image space. 0° 90° 180° 0 0.25 0.5 0° 90° 180° 0 0.25 0.5 SC filters HSM filters Figure 3: Scatter plots of peak frequencies and orientations of the Gabor functions fitted to the estimated filters. The units on the radial scale are cycles/pixel and the solid line is the Nyquist limit. Although both SC and HSM filters exhibit predominantly high spatial frequencies, the hierarchical model yields a representation that tiles the spatial frequency space much more evenly. 0 20 40 0 π/4 π/2 SC phase 0 20 40 0 π/4 π/2 HSM phase 0.06 0.13 0.25 0.50 0 25 50 SC freq 0.06 0.13 0.25 0.50 0 25 50 HSM freq Figure 4: The distributions of phases and frequencies for Gabor functions fitted to sparse coding (SC) and hierarchical scale model (HSM) filters. The phase units specify the phase of the sinusoid in relation to the peak of the Gaussian envelope of the Gabor function; 0 is even-symmetric, π/2 is odd-symmetric. The frequency axes are in cycles/pixel. group co-localized lower-level basis functions and separately represent spatial contrast and oriented image structure. As reported previously [6,16], with a sparse prior on v, the model learns higher-order components that individually capture complex spatial, orientation, and scale regularities in image data. 5 Discussion We have demonstrated that adapting a general hierarchical model yields lower-level representations that are significantly different than those obtained using fixed priors and linear generative models. The resulting basis functions and filters are multi-scale and more consistent with several observed characteristics of neural receptive fields. It is interesting that the learned representations are similar to the results obtained when ICA or sparse coding is applied to whitened images (i.e. with a flattened power spectrum). This might be explained by the fact that whitening “spheres” the input space, normalizing the scale of different directions in the space. The hierarchical model is performing a similar scaling operation through the inference of higher-order variables v that scale the priors on basis function coefficients u. Thus the model can rely on a generic “white” lower level representation, while employing an adaptive mechanism for normalizing the space, which accounts for non-stationary statistics on an image-by-image basis [6]. A related phenomenon in neural processing is gain control, which might be one specific type of a general adaptation process. The flexibility of the hierarchical model allows us to learn a lower-level representation that is optimal in the context of the hierarchy. Thus, we expect the learned parameters to define a better statistical model for natural images than other approaches in which the lower-level representation or the higher-order dependencies are fixed in advance. For example, the flexible marginal distributions, illustrated in figure 1, should be able to capture a wider range of statistical structure in natural images. One way to quantify the benefit of an adapted lowerlevel representation is to apply the model to problems like image de-noising and filling-in missing pixels. Related models have achieved state-of-the-art performance [15, 18], and we are currently investigating whether the added flexibility of the model discussed here confers additional advantages. Finally, although the results presented here are more consistent with the observed properties of neural receptive fields, several discrepancies remain. For example, our results, as well as those of other statistical models, fail to account for the prevalence of low spatial frequency receptive fields observed in V1. This could be a result of the specific choice of the distribution assumed by the model, although the described hierarchical framework makes few assumptions about the joint distribution of basis function coefficients. More likely, the non-stationary statistics of the natural scenes play a role in determining the properties of the learned representation. As suggested by previous results [10], different image data-sets can lead to different parameters. This provides a strong motivation for training models with an “over-complete” basis, in which the number of basis functions is greater than the dimensionality of the input data [19]. In this case, different subsets of the basis functions can adapt to optimally represent different image contexts, and the population properties of such over-complete representations could be significantly different. It would be particularly interesting to investigate representations learned in these models in the context of a hierarchical model. 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Schwartz and E. P. Simoncelli. Natural signal statistics and sensory gain control. Nat. Neurosci., 4:819–825, 2001. [8] D. Field. What is the goal of sensory coding. Neural Computation, 6:559–601, 1994. [9] D. R. Ruderman and W. Bialek. Statistics of natural images: Scaling in the woods. Physical Review Letters, 73(6):814–818, 1994. [10] J. H. van Hateren and A. van der Schaaf. Independent component filters of natural images compared with simple cells in primary visual cortex. Proceedings of the Royal Society, London B, 265:359–366, 1998. [11] J. P. Jones and L. A. Palmer. An evaluation of the two-dimensional gabor filter model of simple receptive fields in cat striate cortex. Journal of Neurophysiology, 58(6):1233–1258, 1987. [12] E. Doi and M. S. Lewicki. Sparse coding of natural images using an overcomplete set of limited capacity units. In Advances in Neural Processing Information Systems 18, 2004. [13] R. L. De Valois, D. G. Albrecht, and L. G. Thorell. Spatial frequency selectivity of cells in macaque visual cortex. Vision Research, 22:545–559, 1982. [14] D. L. Ringach. Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex. Journal of Neurophysiology, 88:455–463, 2002. [15] J. Portilla, V. Strela, M. J. Wainwright, and E.P. Simoncelli. Image denoising using Gaussian scale mixtures in the wavelet domain. IEEE Transactions on Image Processing, 12:1338–1351, 2003. [16] Y. Karklin and M.S. Lewicki. Learning higher-order structures in natural images. Network: Computation in Neural Systems, 14:483–499, 2003. [17] C. Zetzsche and G. Krieger. Nonlinear neurons and highorder statistics: New approaches to human vision and electronic image processing. In B. Rogowitz and T.V. Pappas, editors, Proc. SPIE on Human Vision and Electronic Imaging IV, volume 3644, pages 2–33, 1999. [18] M. S. Lewicki and B. A. Olshausen. A probabilistic framework for the adaptation and comparison of image codes. Journal of the Optical Society of America A, 16(7):1587–1601, 1999. [19] B. A. Olshausen and D. J. Field. Sparse coding with an overcomplete basis set: A strategy employed by V1? Vision Research, 37(23), 1997.
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An Approximate Inference Approach for the PCA Reconstruction Error Manfred Opper Electronics and Computer Science University of Southampton Southampton, SO17 1BJ mo@ecs.soton.ac.uk Abstract The problem of computing a resample estimate for the reconstruction error in PCA is reformulated as an inference problem with the help of the replica method. Using the expectation consistent (EC) approximation, the intractable inference problem can be solved efficiently using only two variational parameters. A perturbative correction to the result is computed and an alternative simplified derivation is also presented. 1 Introduction This paper was motivated by recent joint work with Ole Winther on approximate inference techniques (the expectation consistent (EC) approximation [1] related to Tom Minka’s EP [2] approach) which allows us to tackle high–dimensional sums and integrals required for Bayesian probabilistic inference. I was looking for a nice model on which I could test this approximation. It had to be simple enough so that I would not be bogged down by large numerical simulations. But it had to be nontrivial enough to be of at least modest interest to Machine Learning. With the somewhat unorthodox application of approximate inference to resampling in PCA I hope to be able to stress the following points: • Approximate efficient inference techniques can be useful in areas of Machine Learning where one would not necessarily assume that they are applicable. This can happen when the underlying probabilistic model is not immediately visible but shows only up as a result a of mathematical transformation. • Approximate inference methods can be highly robust allowing for analytic continuations of model parameters to the complex plane or even noninteger dimensions. • It is not always necessary to use a large number of variational parameters in order to get reasonable accuracy. • Inference methods could be systematically improved using perturbative corrections. The work was also stimulated by previous joint work with D¨orthe Malzahn [3] on resampling estimates for generalization errors of Gaussian process models and Supportvector– Machines. 2 Resampling estimators for PCA Principal Component Analysis (PCA) is a well known and widely applied tool for data analysis. The goal is to project data vectors y from a typically high (d-) dimensional space into an optimally chosen lower (q-) dimensional linear space with q << d, thereby minimizing the expected projection error ε = E||y −Pq[y]||2, where Pq[y] denotes the projection. E stands for an expectation over the distribution of the data. In practice where the distribution is not available, one has to work with a data sample D0 consisting of N vectors yk = (yk(1), yk(2), . . . , yk(d))T , k = 1, . . . , N. We arrange these vectors into a (d×N) data matrix Y = (y1, y2, . . . , yN). Assuming centered data, the optimal subspace is spanned by the eigenvectors ul of the d × d data covariance matrix C = 1 N YYT corresponding to the q largest eigenvalues λk. We will assume that these correspond to all eigenvectors λk > λ above some threshold value λ. After computing the PCA projection, one would be interested in finding out if the computed subspace represents the data well by estimating the average projection error on novel data y (ie not contained in D0) which are drawn from the same distribution. Fixing the projection Pq, the error can be rewritten as E = X λl<λ E Tr yyT uluT l (1) where the expectation is only over y and the training data are fixed. The training error Et = P λl<λ λ2 l can be obtained without knowledge of the distribution but will usually only give an optimistically biased estimate for E. 2.1 A resampling estimate for the error New artificial data samples D of arbitrary size can be created by resampling a number of data points from D0 with or without replacement. A simple choice would be to choose all data independently with the same probability 1/N, but other possibilities can also be implemented within our formalism. Thus, some yi in D0 may appear multiple times in D and others not at all. The idea of performing PCA on resampled data sets D and testing on the remaining data D0\D, motivates the following definition of a resample averaged reconstruction error Er = 1 N0 ED X yi /∈D;λl<λ Tr yiyT i uluT l (2) as a proxy for E. ED is the expectation over the resampling process. This is an estimator of the bootstrap type [3,4]. N0 is the expected number of data in D0 which are not contained in the random set D. The rest of the paper will discuss a method for efficiently approximating (2). 2.2 Basic formalism We introduce “occupation numbers” si which count how many times yi is containd in D. We also introduce two matrices D and C. D is a diagonal random matrix Dii = Di = 1 µΓ(si + ϵδsi,0) C(ϵ) = Γ N YDYT . (3) C(0) is proportional to the covariance matrix of the resampled data. µ is the sampling rate, i.e. µN = ED[P i si] is the expexted number of data in D (counting multiplicities). The role of Γ will be explained later. Using ϵ, we can generate expressions that can be used in (2) to sum over the data which are not contained in the set D C′(0) = 1 µN X j δsj,0yjyT j . (4) In the following λk and uk will always denote eigenvalues and eigenvectors of the data dependent (i.e. random) covariance matrix C(0). The desired averages can be constructed from the d × d matrix Green’s function G(Γ) = (C(0) + ΓI)−1 = X k ukuT k λk + Γ (5) Using the well known representation of the Dirac δ distribution given by δ(x) = limη→0+ ℑ 1 π(x−iη) where i = √−1 and ℑdenotes the imaginary part, we get lim η→0+ 1 π ℑG(Γ −iη) = X k ukuT k δ (λk + Γ) . (6) Hence, we have Er = E0 r + Z λ 0+ dλ′ εr(λ′) (7) where εr(λ) = 1 π lim η→0+ ℑ1 N0 ED X j δsj,0 Tr yjyT j G(−λ −iη) (8) defines the error density from all eigenvalues > 0 and E0 r is the contribution from the eigenspace with λk = 0. The latter can also be easily expressed from G as E0 r = lim Γ→0 1 N0 ED X j δsj,0 Tr yjyT j ΓG(Γ) (9) We can also compute the resample averaged density of eigenvalues using ρ(λ) = 1 πµN lim η→0+ ℑED [Tr G(−λ −iη)] (10) 3 A Gaussian probabilistic model The matrix Green’s function for Γ > 0 can be generated from a Gaussian partition function Z. This is a well known construction in statistical physics, and has also been used within the NIPS community to study the distribution of eigenvalues for an average case analysis of PCA [5]. Its use for computing the expected reconstruction error is to my knowledge new. With the (N × N) kernel matrix K = 1 N YT Y we define the Gaussian partition function Z = Z dx exp −1 2xT K−1 + D x (11) = |K| 1 2 Γd/2(2π)(N−d)/2 Z ddz exp −1 2zT (C(ϵ) + ΓI) z . (12) x is an N dimensional integration variable. The equality can be easily shown by expressing the integrals as determinants. 1 The first representation (11) is useful for computing the resampling average and the second one connects directly to the definition of the matrix Green’s function G. Note, that by its dependence on the kernel matrix K, a generalization to d = ∞dimensional feature spaces and kernel PCA is straightforward. The partition function can then be understood as a certain Gaussian process expectation. We will not discuss this point further. The free energy F = −ln Z enables us to generate the following quantities −2∂ln Z ∂ϵ ϵ=0 = 1 µN N X j=1 δsj,0 Tr yjyT j G(Γ) (13) −2∂ln Z ∂Γ = d Γ + Tr G(Γ) (14) where we have used (4) for (13). (13) will be used for the computation of (8) and (14) applies to the density of eigenvalues. Note that the definition of the partition function Z requires that Γ > 0, whereas the application to the reconstruction error (7) needs negative values Γ = −λ < 0. Hence, an analytic continuation of end results must be performed. 4 Resampling average and replicas (13) and (14) show that we can compute the desired resampling averages from the expected free energy −ED[ln Z]. This can be expressed using the “replica trick” of statistical physics (see e.g. [6]) using ED[ln Z] = lim n→0 1 n ln ED[Zn] , (15) where one attempts an approximate computation of ED[Zn] for integer n and uses a continuation to real numbers at the end. The n times replicated and averaged partition function (11) can be written in the form Z(n) .= ED[Zn] = Z dx ψ1(x) ψ2(x) (16) where we set x .= (x1, . . . , xn) and ψ1(x) = ED " exp ( −1 2 n X a=1 xT a Dxa )# ψ2(x) = exp " −1 2 n X a=1 xT a K−1xa # (17) The unaveraged partition function Z (11) is Gaussian, but the averaged Z(n) is not and usually intractable. 5 Approximate inference To approximate Z(n), we will use the EC approximation recently introduced by Opper & Winther [1]. For this method we need two auxiliary distributions p1(x) = 1 Z1 ψ1(x)e−Λ1xT x p0(x) = 1 Z0 e−1 2 Λ0xT x , (18) where Λ1 and Λ0 are “variational” parameters to be optimized. p1 tries to mimic the intractable p(x) ∝ψ1(x) ψ2(x), replacing the multivariate Gaussian ψ2 by a simpler, i.e. 1If K has zero eigenvalues, a division of Z by |K| 1 2 is necessary. This additive renormalization of the free energy −ln Z will not influence the subsequent computations. tractable diagonal one. One may think of using a general diagonal matrix Λ1, but we will restrict ourselves in the present case to the simplest case of a spherical Gaussian with a single parameter Λ1. The strategy is to split Z(n) into a product of Z1 and a term that has to be further approximated: Z(n) = Z1 Z dx p1(x) ψ2(x) eΛ1xT x (19) ≈ Z1 Z dx p0(x) ψ2(x) eΛ1xT x ≡Z(n) EC(Λ1, Λ0) . The approximation replaces the intractable average over p1 by a tractable one over p0. To optmize Λ1 and Λ0 we argue as follows: We try to make p0 as close as possible to p1 by matching the moments ⟨xT x⟩1 = ⟨xT x⟩0. The index denotes the distribution which is used for averaging. By this step, Λ0 becomes a function of Λ1. Second, since the true partition function Z(n) is independent of Λ1, we expect that a good approximation to Z(n) should be stationary with respect to variations of Λ1. Both conditions can be expressed by the requirement that ln Z(n) EC(Λ1, Λ0) must be stationary with respect to variations of Λ1 and Λ0. Within this EC approximation we can carry out the replica limit ED[ln Z] ≈ln ZEC = limn→0 1 n ln Z(n) EC and get after some calculations −ln ZEC = −ED ln Z dx e−1 2 xT (D+(Λ0−Λ)I)x − (20) −ln Z dx e−1 2 xT (K−1+ΛI)x + ln Z dx e−1 2 Λ0xT x where we have set Λ = Λ0 −Λ1. Since the first Gaussian integral factorises, we can now perform the resampling average in (20) relatively easy for the case when all sj’s in (3) are independent. Assuming e.g. Poisson probabilities p(s) = e−µ µs s! gives a good approximation for the case of resampling µN points with replacement. The variational equations which make (20) stationary are ED 1 Λ0 −Λ + Di = 1 Λ0 1 N X k ωi 1 + ωkΛ = 1 Λ0 (21) where ωk are the eigenvalues of the matrix K. The variational equations have to be solved in the region Γ = −λ < 0 where the original partition function does not exist. The resulting parameters Λ0 and Λ will usually come out as complex numbers. 6 Experiments By eliminating the parameter Λ0 from (21) it is possible to reduce the numerical computations to solving a nonlinear equation for a single complex parameter Λ which can be solved easily and fast by a Newton method. While the analytical results are based on Poisson statistics, the simulations of random resampling was performed by choosing a fixed number (equal to the expected number of the Poisson distribution) of data at random with replacement. The first experiment was for a set of data generated at random from a spherical Gaussian. To show that resampling maybe useful, we give on on the left hand side of Figure 1 the reconstruction error as a function of the value of λ below which eigenvalues are dicarded. 0 1 2 3 4 5 0 5 10 15 20 25 eigenvalue λ Figure 1: Left: Errors for PCA on N = 32 spherically Gaussian data with d = 25 and µ = 3. Smooth curve: approximate resampled error estimate, upper step function: true error. Lower step function: Training error. Right: Comparison of EC approximation (line) and simulation (histogramme) of the resampled density of eigenvalues for N = 50 spherically Gaussian data of dimensionality d = 25. The sampling rate was µ = 3. The smooth function is the approximate resampling error (3× oversampled to leave not many data out of the samples) from our method. The upper step function gives the true reconstruction error (easy to calculate for spherical data) from (1). The lower step function is the training error. The right panel demonstrates the accuracy of the approximation on a similar set of data. We compare the analytically approximated density of states with the results of a true resampling experiment, where eigenvalues for many samples are counted into small bins. The theoretical curve follows closely the experiment. Since the good accuracy might be attributed to the high symmetry of the toy data, we have also performed experiments on a set of N = 100 handwritten digits with d = 784. The results in Figure 2 are promising. Although the density of eigenvalues is more accurate than the resampling error, the latter comes still out reasonable. 7 Corrections I will show next that the EC approximation can be augmented by a perturbation expansion. Going back to (19), we can write Z(n) Z1 = Z dx p1(x) ψ2(x) eΛ1xT x = Z dx ψ2(x) e 1 2 ΛxT x Z dk (2π)Nn e−ikT xχ(k) where χ(k) .= R dx p1(x)eikT x is the characteristic function of the density p1 (18). ln χ(k) is the cumulant generating function. Using the symmetries of the density p1, we can perform a power series expansion of ln χ(k), which starts with a quadratic term (second cumulant) ln χ(k) = −M2 2 kT k + R(k) , (22) where M2 = ⟨xT a xa⟩1. It can be shown that if we neglect R(k) (containing the higher order cumulants) and carry out the integral over k, we end up replacing p1 by a simpler Gaussian p0 with matching moments M2, i.e. the EC approximation. Higher order corrections to the free energy −ED[ln Z] = −ln ZEC + ∆F1 + . . . can be obtained perturbatively by writing χ(k) = e−M2 2 kT k(1+R(k)+. . .). This expansion is similar in spirit to Edgeworth 0 0.5 1 1.5 0 5 10 15 20 eigenvalue λ Figure 2: Left: Resampling error (µ = 1) for PCA on a set of 100 handwritten digits (“5”) with d = 784. The approximation (line) for µ = 1 is compared with simulations of the random resampling. Right: Resampled density of eigenvalues for the same data set. Only the nonzero eigenvalues are shown. expansions in statistics. The present case is more complicated by the extra dimensions introduced by the replicating of variables and the limit n →0. After a lengthy calculation one finds for the lowest order correction (containing the monomials in k of order 4) to the free energy: ∆F1 = −1 4ED Λ0 Λ0 −Λ + Di −1 2 × X i Λ0 K−1 + ΛI −1 ii −1 2 (23) I illustrate the effect of ∆F1 on a correction to the reconstruction error in the “zero– subspace” using (9) and (13) for the digit data as a function of µ. Resampling used the Poisson approximation.The left panel of Figure 3 demonstrates that the true correction is fairly small. The right panel shows that the lowest order term ∆F1 accounts for a major part of the true correction when µ < 3. The strong underestimation for larger µ needs further investigation. 8 The calculation without replicas Knowing with hindsight how the final EC result (20) looks like, we can rederive it using another method which does not rely on the “replica trick”. We first write down an exact expression for −ln Z before averaging. Expressing Gaussian integrals by determinants yields −ln Z = −ln Z dx e−1 2 xT (D+(Λ0−Λ)I)x −ln Z dx e−1 2 xT (K−1+ΛI)x + (24) + ln Z dx e−1 2 Λ0xT x + 1 2 ln det(I + r) where the matrix r has elements rij = 1 − Λ0 Λ0−Λ+Di Λ0 K−1 + ΛI −1 −I ij . The EC approximation is obtained by simply neglecting r. Corrections to this are found by expanding ln det (I + r) = Tr ln (I + r) = ∞ X k=1 (−1)k+1 k Tr rk (25) 0 0.5 1 1.5 2 2.5 3 3.5 4 2 4 6 8 10 12 14 16 18 20 22 Resampling rate µ Resampled reconstruction error (λ = 0) 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 Correction to resampling error Resampling rate µ Figure 3: Left: Resampling error E0 r from the λ = 0 subspace as a function of resampling rate for the digits data. The approximation (lower line) is compared with simulations of the random resampling (upper line). Right: The difference between approximation and simulations (upper curve) and its estimate (lower curve) from the perturbative correction (23). The first order term in the expansion (25) vanishes after averaging (see (21)) and the second order term gives exactly the correction of the cumulant method (23). 9 Outlook It will be interesting to extend the perturbative framework for the computation of corrections to inference approximations to other, more complex models. However, our results indicate that the use and convergence of such perturbation expansion needs to be critically investigated and that the lowest order may not always give a clear indication of the accuracy of the approximation. The alternative derivation for our simple model could present an interesting ground for testing these ideas. Acknowledgments I would like to thank Ole Winther for the great collaboration on the EC approximation. References [1] Manfred Opper and Ole Winther. Expectation consistent free energies for approximate inference. In NIPS 17, 2005. [2] T. P. Minka. Expectation propagation for approximate Bayesian inference. In UAI 2001, pages 362–369, 2001. [3] D. Malzahn and M. Opper. An approximate analytical approach to resampling averages. Journal of Machine Learning Research, pages 1151–1173, 2003. [4] B. Efron, R. J. Tibshirani. An Introduction to the Bootstrap. Monographs on Statistics and Applied Probability 57, Chapman & Hall, 1993. [5] D. C. Hoyle and M. Rattray Limiting form of the sample covariance matrix eigenspectrum in PCA and kernel PCA. In NIPS 16, 2003. [6] A. Engel and C. Van den Broeck, Statistical Mechanics of Learning (Cambridge University Press, 2001).
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Active Bidirectional Coupling in a Cochlear Chip Bo Wen and Kwabena Boahen Department of Bioengineering University of Pennsylvania Philadelphia, PA 19104 {wenbo,boahen}@seas.upenn.edu Abstract We present a novel cochlear model implemented in analog very large scale integration (VLSI) technology that emulates nonlinear active cochlear behavior. This silicon cochlea includes outer hair cell (OHC) electromotility through active bidirectional coupling (ABC), a mechanism we proposed in which OHC motile forces, through the microanatomical organization of the organ of Corti, realize the cochlear amplifier. Our chip measurements demonstrate that frequency responses become larger and more sharply tuned when ABC is turned on; the degree of the enhancement decreases with input intensity as ABC includes saturation of OHC forces. 1 Silicon Cochleae Cochlear models, mathematical and physical, with the shared goal of emulating nonlinear active cochlear behavior, shed light on how the cochlea works if based on cochlear micromechanics. Among the modeling efforts, silicon cochleae have promise in meeting the need for real-time performance and low power consumption. Lyon and Mead developed the first analog electronic cochlea [1], which employed a cascade of second-order filters with exponentially decreasing resonant frequencies. However, the cascade structure suffers from delay and noise accumulation and lacks fault-tolerance. Modeling the cochlea more faithfully, Watts built a two-dimensional (2D) passive cochlea that addressed these shortcomings by incorporating the cochlear fluid using a resistive network [2]. This parallel structure, however, has its own problem: response gain is diminished by interference among the second-order sections’ outputs due to the large phase change at resonance [3]. Listening more to biology, our silicon cochlea aims to overcome the shortcomings of existing architectures by mimicking the cochlear micromechanics while includingouter hair cell (OHC) electromotility. Although how exactly OHC motile forces boost the basilar membrane’s (BM) vibration remains a mystery, cochlear microanatomy provides clues. Based on these clues, we previously proposed a novel mechanism, active bidirectional coupling (ABC), for the cochlear amplifier [4]. Here, we report an analog VLSI chip that implements this mechanism. In essence, our implementation is the first silicon cochlea that employs stimulus enhancement (i.e., active behavior) instead of undamping (i.e., high filter Q [5]). The paper is organized as follows. In Section 2, we present the hypothesized mechanism (ABC), first described in [4]. In Section 3, we provide a mathematical formulation of the Oval window Round window organ of Corti BM DC IHC OHC i -1 i i+1 PhP RL BM Basal Apical Stereocilia d A B Figure 1: The inner ear. A Cutaway showing cochlear ducts (adapted from [6]). B Longitudinal view of cochlear partition (CP) (modified from [7]-[8]). Each outer hair cell (OHC) tilts toward the base while the Deiter’s cell (DC) on which it sits extends a phalangeal process (PhP) toward the apex. The OHCs’ stereocilia and the PhPs’ apical ends form the reticular lamina (RL). d is the tilt distance, and the segment size. IHC: inner hair cell. model as the basis of cochlear circuit design. Then we proceed in Section 4 to synthesize the circuit for the cochlear chip. Last, we present chip measurements in Section 5 that demonstrate nonlinear active cochlear behavior. 2 Active Bidirectional Coupling The cochlea actively amplifies acoustic signals as it performs spectral analysis. The movement of the stapes sets the cochlear fluid into motion, which passes the stimulus energy onto a certain region of the BM, the main vibrating organ in the cochlea (Figure 1A). From the base to the apex, BM fibers increase in width and decrease in thickness, resulting in an exponential decrease in stiffness which, in turn, gives rise to the passive frequency tuning of the cochlea. The OHCs’ electromotility is widely thought to account for the cochlea’s exquisite sensitivity and discriminability. The exact way that OHC motile forces enhance the BM’s motion, however, remains unresolved. We propose that the triangular mechanical unit formed by an OHC, a phalangeal process (PhP) extended from the Deiter’s cell (DC) on which the OHC sits, and a portion of the reticular lamina (RL), between the OHC’s stereocilia end and the PhP’s apical tip, plays an active role in enhancing the BM’s responses (Figure 1B). The cochlear partition (CP) is divided into a number of segments longitudinally. Each segment includes one DC, one PhP’s apical tip and one OHC’s stereocilia end, both attached to the RL. Approximating the anatomy, we assume that when an OHC’s stereocilia end lies in segment i −1, its basolateral end lies in the immediately apical segment i. Furthermore, the DC in segment i extends a PhP that angles toward the apex of the cochlea, with its apical end inserted just behind the stereocilia end of the OHC in segment i + 1. Our hypothesis (ABC) includes both feedforward and feedbackward interactions. On one hand, the feedforward mechanism, proposed in [9], hypothesized that the force resulting from OHC contraction or elongation is exerted onto an adjacent downstream BM segment due to the OHC’s basal tilt. On the other hand, the novel insight of the feedbackward mechanism is that the OHC force is delivered onto an adjacent upstream BM segment due to the apical tilt of the PhP extending from the DC’s main trunk. In a nutshell, the OHC motile forces, through the microanatomy of the CP, feed forward and backward, in harmony with each other, resulting in bidirectional coupling between BM segments in the longitudinal direction. Specifically, due to the opposite action of OHC 0 5 10 15 20 25 Distance from stapes HmmL -0.2 0 0.5 1 ReHZmL!!!!!!!!!!!!!!!!!!!!!! S HxL M HxL A B Figure 2: Wave propagation (WP) and basilar membrane (BM) impedance in the active cochlear model with a 2kHz pure tone (α = 0.15, γ = 0.3). A WP in fluid and BM. B BM impedance Zm (i.e., pressure divided by velocity), normalized by p S(x)M(x). Only the resistive component is shown; dot marks peak location. forces on the BM and the RL, the motion of BM segment i −1 reinforces that of segment i while the motion of segment i + 1 opposes that of segment i, as described in detail in [4]. 3 The 2D Nonlinear Active Model To provide a blueprint for the cochlear circuit design, we formulate a 2D model of the cochlea that includes ABC. Both the cochlea’s length (BM) and height (cochlear ducts) are discretized into a number of segments, with the original aspect ratio of the cochlea maintained. In the following expressions, x represents the distance from the stapes along the CP, with x = 0 at the base (or the stapes) and x = L (uncoiled cochlear duct length) at the apex; y represents the vertical distance from the BM, with y = 0 at the BM and y = ±h (cochlear duct radius) at the bottom/top wall. Providing that the assumption of fluid incompressibility holds, the velocity potential φ of the fluids is required to satisfy ▽2φ(x, y, t) = 0, where ▽2 denotes the Laplacian operator. By definition, this potential is related to fluid velocities in the x and y directions: Vx = −∂φ/∂x and Vy = −∂φ/∂y. The BM is driven by the fluid pressure difference across it. Hence, the BM’s vertical motion (with downward displacement being positive) can be described as follows. Pd(x) + FOHC(x) = S(x)δ(x) + β(x) ˙δ(x) + M(x)¨δ(x), (1) where S(x) is the stiffness, β(x) is the damping, and M(x) is the mass, per unit area, of the BM; δ is the BM’s downward displacement. Pd = ρ ∂(φSV(x, y, t) −φST(x, y, t))/∂t is the pressure difference between the two fluid ducts (the scala vestibuli (SV) and the scala tympani (ST)), evaluated at the BM (y = 0); ρ is the fluid density. The FOHC(x) term combines feedforward and feedbackward OHC forces, described by FOHC(x) = s0 tanh(αγS(x)δ(x −d)/s0) −tanh(αS(x)δ(x + d)/s0), (2) where α denotes the OHC motility, expressed as a fraction of the BM stiffness, and γ is the ratio of feedforward to feedbackward coupling, representing relative strengths of the OHC forces exerted on the BM segment through the DC, directly and via the tilted PhP. d denotes the tilt distance, which is the horizontal displacement between the source and the recipient of the OHC force, assumed to be equal for the forward and backward cases. We use the hyperbolic tangent function to model saturation of the OHC forces, the nonlinearity that is evident in physiological measurements [8]; s0 determines the saturation level. We observed wave propagation in the model and computed the BM’s impedance (i.e., the ratio of driving pressure to velocity). Following the semi-analytical approach in [2], we simulated a linear version of the model (without saturation). The traveling wave transitions from long-wave to short-wave before the BM vibration peaks; the wavelength around the characteristic place is comparable to the tilt distance (Figure 2A). The BM impedance’s real part (i.e., the resistive component) becomes negative before the peak (Figure 2B). On the whole, inclusion of OHC motility through ABC boosts the traveling wave by pumping energy onto the BM when the wavelength matches the tilt of the OHC and PhP. 4 Analog VLSI Design and Implementation Based on our mathematical model, which produces realistic responses, we implemented a 2D nonlinear active cochlear circuit in analog VLSI, taking advantage of the 2D nature of silicon chips. We first synthesize a circuit analog of the mathematical model, and then we implement the circuit in the log-domain. We start by synthesizing a passive model, and then extend it to a nonlinear active one by including ABC with saturation. 4.1 Synthesizing the BM Circuit The model consists of two fundamental parts: the cochlear fluid and the BM. First, we design the fluid element and thus the fluid network. In discrete form, the fluids can be viewed as a grid of elements with a specific resistance that corresponds to the fluid density or mass. Since charge is conserved for a small sheet of resistance and so are particles for a small volume of fluid, we use current to simulate fluid velocity. At the transistor level, the current flowing through the channel of a MOS transistor, operating subthreshold as a diffusive element, can be used for this purpose. Therefore, following the approach in [10], we implement the cochlear fluid network using a diffusor network formed by a 2D grid of nMOS transistors. Second, we design the BM element and thus the BM. As current represents velocity, we rewrite the BM boundary condition (Equation 1, without the FOHC term): ˙Iin = S(x) R Imemdt+β(x)Imem +M(x) ˙Imem, (3) where Iin, obtained by applying the voltage from the diffusor network to the gate of a pMOS transistor, represents the velocity potential scaled by the fluid density. In turn, Imem drives the diffusor network to match the fluid velocity with the BM velocity, ˙δ. The FOHC term is dealt with in Section 4.2. Implementing this second-order system requires two state-space variables, which we name Is and Io. And with s = jω, our synthesized BM design (passive) is τ1Iss + Is = −Iin + Io, (4) τ2Ios + Io = Iin −bIs, (5) Imem = Iin + Is −Io, (6) where the two first-order systems are both low-pass filters (LPFs), with time constants τ1 and τ2, respectively; b is a gain factor. Thus, Iin can be expressed in terms of Imem as: Iins2 = (b + 1)/τ1τ2 + ((τ1 + τ2)/τ1τ2)s + s2 Imem. Comparing this expression with the design target (Equation 3) yields the circuit analogs: S(x) = (b + 1)/τ1τ2, β(x) = (τ1 + τ2)/τ1τ2, and M(x) = 1. Note that the mass M(x) is a constant (i.e., 1), which was also the case in our mathematical model simulation. These analogies require that τ1 and τ2 increase exponentially to Iin+ Vq IinIoutC+ Iout+ + + + + + + + + + + b b + + Iin+ IinIsIs+ IT+ IT+ + Io+ IoImem+ ImemLPF BM Half LPF Iout+) ( From neighbors To neighbors C A B Iin+ Iout+ IinIoutFigure 3: Low-pass filter (LPF) and second-order section circuit design. A Half-LPF circuit. B Complete LPF circuit formed by two half-LPF circuits. C Basilar membrane (BM) circuit. It consists of two LPFs and connects to its neighbors through Is and IT. simulate the exponentially decreasing BM stiffness (and damping); b allows us to achieve a reasonable stiffness for a practical choice of τ1 and τ2 (capacitor size is limited by silicon area). 4.2 Adding Active Bidirectional Coupling To include ABC in the BM boundary condition, we replace δ in Equation 2 with R Imemdt to obtain FOHC = rffS(x)T R Imem(x −d)dt −rfbS(x)T R Imem(x + d)dt , where rff= αγ and rfb = α denote the feedforward and feedbackward OHC motility factors, and T denotes saturation. The saturation is applied to the displacement, instead of the force, as this simplifies the implementation. We obtain the integrals by observing that, in the passive design, the state variable Is = −Imem/sτ1. Thus, R Imem(x −d)dt = −τ1fIsf and RImem(x + d)dt = −τ1bIsb. Here, Isf and Isb represent the outputs of the first LPF in the upstream and downstream BM segments, respectively; τ1f and τ1b represent their respective time constants. To reduce complexity in implementation, we use τ1 to approximate both τ1f and τ1b as the longitudinal span is small. We obtain the active BM design by replacing Equation 5 with the synthesis result: τ2Ios + Io = Iin −bIs + rfb(b + 1)T (−Isb) −rff(b + 1)T (−Isf). Note that, to implement ABC, we only need to add two currents to the second LPF in the passive system. These currents, Isf and Isb, come from the upstream and downstream neighbors of each segment. Base Apex IST ISV ITIs+ Is+ IsIT+ IsIT+ ITIT+ ITIs+ IsIT+ ITIs+ IsIin+ IinImem+ ImemBM BM Fluid Fluid A B Vsat Vsat Figure 4: Cochlear chip. A Architecture: Two diffusive grids with embedded BM circuits model the cochlea. B Detail. BM circuits exchange currents with their neighbors. 4.3 Class AB Log-domain Implementation We employ the log-domain filtering technique [11] to realize current-mode operation. In addition, following the approach proposed in [12], we implement the circuit in Class AB to increase dynamic range, reduce the effect of mismatch and lower power consumption. This differential signaling is inspired by the way the biological cochlea works—the vibration of BM is driven by the pressure difference across it. Taking a bottom-up strategy, we start by designing a Class AB LPF, a building block for the BM circuit. It is described by τ(I+ out −I− out)s + (I+ out −I− out) = I+ in −I− in and τI+ outI− outs + I+ outI− out = I2 q, where Iq sets the geometric mean of the positive and negative components of the output current, and τ sets the time constant. Combining the common-mode constraint with the differential design equation yields the nodal equation for the positive path (the negative path has superscripts + and −swapped): C ˙V + out = Iτ (I+ in −I− in) + (I2 q/I+ out −I+ out) /(I+ out + I− out). This nodal equation suggests the half-LPF circuit shown in Figure 3A. V + out, the voltage on the positive capacitor (C+), gates a pMOS transistor to produce the corresponding current signal, I+ out (V − out and I− out are similarly related). The bias Vq sets the quiescent current Iq while Vτ determines the current Iτ, which is related to the time constant by τ = CuT/κIτ (κ is the subthreshold slope coefficient and uT is the thermal voltage). Two of these subcircuits, connected in push–pull, form a complete LPF (Figure 3B). The BM circuit is implemented using two LPFs interacting in accordance with the synthesized design equations (Figure 3C). Imem is the combination of three currents, Iin, Is, and Io. Each BM sends out Is and receives IT, a saturated version of its neighbor’s Is. The saturation is accomplished by a current-limiting transistor (see Figure 4B), which yields IT = T (Is) = IsIsat/(Is + Isat), where Isat is set by a bias voltage Vsat. 4.4 Chip Architecture We fabricated a version of our cochlear chip architecture (Figure 4) with 360 BM circuits and two 4680-element fluid grids (360×13). This chip occupies 10.9mm2 of silicon area in 0.25µm CMOS technology. Differential input signals are applied at the base while the two fluid grids are connected at the apex through a fluid element that represents the helicotrema. 5 Chip Measurements We carried out two measurements that demonstrate the desired amplification by ABC, and the compressive growth of BM responses due to saturation. To obtain sinusoidal current as the input to the BM subcircuits, we set the voltages applied at the base to be the logarithm of a half-wave rectified sinusoid. We first investigated BM-velocity frequency responses at six linearly spaced cochlear positions (Figure 5). The frequency that maximally excites the first position (Stage 30), defined as its characteristic frequency (CF), is 12.1kHz. The remaining five CFs, from early to later stages, are 8.2k, 1.7k, 905, 366, and 218Hz, respectively. Phase accumulation at the CFs ranges from 0.56 to 2.67π radians, comparable to 1.67π radians in the mammalian cochlea [13]. Q10 factor (the ratio of the CF to the bandwidth 10dB below the peak) ranges from 1.25 to 2.73, comparable to 2.55 at mid-sound intensity in biology (computed from [13]). The cutoff slope ranges from -20 to -54dB/octave, as compared to -85dB/octave in biology (computed from [13]). 0.1 0.2 0.5 1 2 5 10 20 Frequency HkHzL -10 0 10 20 30 40 50 BM Velocity Amplitude HdBL 30 70 110 150 190 230 Stage 0.1 0.2 0.5 1 2 5 10 20 Frequency HkHzL -4 -2 0 BM Velocity Phase HΠ radiansL A B Figure 5: Measured BM-velocity frequency responses at six locations. A Amplitude. B Phase. Dashed lines: Biological data (adapted from [13]). Dots mark peaks. We then explored the longitudinalpattern of BM-velocity responses and the effect of ABC. Stimulating the chip using four different pure tones, we obtained responses in which a 4kHz input elicits a peak around Stage 85 while 500Hz sound travels all the way to Stage 178 and peaks there (Figure 6A). We varied the input voltage level and obtained frequency responses at Stage 100 (Figure 6B). Input voltage level increases linearly such that the current increases exponentially; the input current level (in dB) was estimated based on the measured κ for this chip. As expected, we observed linearly increasing responses at low frequencies in the logarithmic plot. In contrast, the responses around the CF increase less and become broader with increasing input level as saturation takes effect in that region (resembling a passive cochlea). We observed 24dB compression as compared to 27 to 47dB in biology [13]. At the highest intensities, compression also occurs at low frequencies. These chip measurements demonstrate that inclusion of ABC, simply through coupling neighboring BM elements, transforms a passive cochlea into an active one. This active cochlear model’s nonlinear responses are qualitatively comparable to physiological data. 6 Conclusions We presented an analog VLSI implementation of a 2D nonlinear cochlear model that utilizes a novel active mechanism, ABC, which we proposed to account for the cochlear amplifier. ABC was shown to pump energy into the traveling wave. Rather than detecting the wave’s amplitude and implementing an automatic-gain-control loop, our biomorphic model accomplishes this simply by nonlinear interactions between adjacent neighbors. Im0 50 100 150 200 Stage Number -10 0 10 20 BM Velocity Amplitude HdBL 500 Hz 1k 2k 4k Frequency 0.2 0.5 1 2 5 10 20 Frequency HkHzL 0 20 40 60 BM Velocity Amplitude HdBL 0 dB 16 dB 32 dB 48 dB Input Level Stage 100 A B Figure 6: Measured BM-velocity responses (cont’d). A Longitudinal responses (20-stage moving average). Peak shifts to earlier (basal) stages as input frequency increases from 500 to 4kHz. B Effects of increasing input intensity. Responses become broader and show compressive growth. plemented in the log-domain, with Class AB operation, our silicon cochlea shows enhanced frequency responses, with compressive behavior around the CF, when ABC is turned on. These features are desirable in prosthetic applications and automatic speech recognition systems as they capture the properties of the biological cochlea. References [1] Lyon, R.F. & Mead, C.A. (1988) An analog electronic cochlea. IEEE Trans. Acoust. Speech and Signal Proc., 36: 1119-1134. [2] Watts, L. (1993) Cochlear Mechanics: Analysis and Analog VLSI . Ph.D. thesis, Pasadena, CA: California Institute of Technology. [3] Fragni`ere, E. (2005) A 100-Channel analog CMOS auditory filter bank for speech recognition. IEEE International Solid-State Circuits Conference (ISSCC 2005), pp. 140-141. [4] Wen, B. & Boahen, K. (2003) A linear cochlear model with active bi-directional coupling. The 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC 2003), pp. 2013-2016. [5] Sarpeshkar, R., Lyon, R.F., & Mead, C.A. (1996) An analog VLSI cochlear model with new transconductance amplifier and nonlinear gain control. Proceedings of the IEEE Symposium on Circuits and Systems (ISCAS 1996), 3: 292-295. [6] Mead, C.A. (1989) Analog VLSI and Neural Systems. Reading, MA: Addison-Wesley. [7] Russell, I.J. & Nilsen, K.E. (1997)The locationof the cochlear amplifier: Spatial representation of a single tone on the guinea pig basilar membrane. Proc. Natl. Acad. Sci. USA, 94: 2660-2664. [8] Geisler, C.D. (1998) From sound to synapse: physiology of the mammalian ear . Oxford University Press. [9] Geisler, C.D. & Sang, C. (1995) A cochlear model using feed-forward outer-hair-cell forces. Hearing Research, 86: 132-146. [10] Boahen, K.A. & Andreou, A.G. (1992) A contrast sensitive silicon retina with reciprocal synapses. In Moody, J.E. and Lippmann, R.P. (eds.), Advances in Neural Information Processing Systems 4 (NIPS 1992), pp. 764-772, Morgan Kaufmann, San Mateo, CA. [11] Frey, D.R. (1993) Log-domain filtering: an approach to current-mode filtering. IEE Proc. G, Circuits Devices Syst., 140 (6): 406-416. [12] Zaghloul, K. & Boahen, K.A. (2005) An On-Off log-domain circuit that recreates adaptive filtering in the retina. IEEE Transactions on Circuits and Systems I: Regular Papers , 52 (1): 99-107. [13] Ruggero, M.A., Rich, N.C., Narayan, S.S., & Robles, L. (1997) Basilar membrane responses to tones at the base of the chinchilla cochlea. J. Acoust. Soc. Am., 101 (4): 2151-2163.
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Affine Structure From Sound Sebastian Thrun Stanford AI Lab Stanford University, Stanford, CA 94305 Email: thrun@stanford.edu Abstract We consider the problem of localizing a set of microphones together with a set of external acoustic events (e.g., hand claps), emitted at unknown times and unknown locations. We propose a solution that approximates this problem under a far field approximation defined in the calculus of affine geometry, and that relies on singular value decomposition (SVD) to recover the affine structure of the problem. We then define low-dimensional optimization techniques for embedding the solution into Euclidean geometry, and further techniques for recovering the locations and emission times of the acoustic events. The approach is useful for the calibration of ad-hoc microphone arrays and sensor networks. 1 Introduction Consider a set of acoustic sensors (microphones) for detecting acoustic events in the environment (e.g., a hand clap). The structure from sound (SFS) problem addresses the problem of simultaneously localizing a set of N sensors and a set of M external acoustic events, whose locations and emission times are unknown. The SFS problem is relevant to the spatial calibration problem for microphone arrays. Classically, microphone arrays are mounted on fixed brackets of known dimensions; hence there is no spatial calibration problem. Ad-hoc microphone arrays, however, involve a person placing microphones at arbitrary locations with limited knowledge as to where they are. Today’s best practice requires a person to measure the distance between the microphones by hand, and to apply algorithms such as multi-dimensional scaling (MDS) [1] for recovering their locations. When sensor networks are deployed from the air [4], manual calibration may not be an option. Some techniques rely on GPS receivers [8]. Others require a capability to emit and sense wireless radio signals [5] or sounds [9, 10], which are then used to estimate relative distances between microphones (directly or indirectly, as in [9]). Unfortunately, wireless signal strength is a poor estimator of range, and active acoustic and GPS localization techniques are uneconomical in that they consume energy and require additional hardware. In contrast, SFS relies on environmental acoustic events such as hand claps, which are not generated by the sensor network. The general SFS problem was previously treated in [2] under the name passive localization. A related paper [3] describes a technique for incrementally localizing a microphone relative to a well-calibrated microphone array through external sound events. In this paper, the structure from sound (SFS) problem is defined as the simultaneous localization problem of N sound sensors and M acoustic events in the environment detected by these sensors. Each event occurs at an unknown time and an unknown location. The sensors are able to measure the detection times of the event. We assume that the clocks of the sensors are synchronized (see [6]); that events are spaced sufficiently far apart in time to make the association between different sensors unambiguous; and we also assume absence of sound reverberation. For the ease of representation, the paper assumes a 2D world; although the technique is easily generalized to 3D. Under the assumption of independent and identically distributed (iid) Gaussian noise, the SFS problem can be formulated as a least squares problem in a space over three types of variables: the locations of the microphones, the locations of the acoustic events, and their emission times. However, this least squares problem is plagued by local minima, and the number of constraints is quite large. The gist of this paper transforms this optimization problem into a sequence of simpler problems, some of which can be solved optimally, without the danger of getting stuck in local minima. The key transformation involves a far field approximation, which presupposes that the sound sources are relatively far away from the sensors. This approximation reformulates the problem as one of recovering the incident angle of the acoustic signal, which is the same for all sensors for any fixed acoustic event. The resulting optimization problem is still non-linear; however, by relaxing the laws of Euclidean geometry into the more general calculus of affine geometry, the optimization problem can be solved by singular value decomposition (SVD). The resulting solution is mapped back into Euclidean space by optimizing a matrix of size 2 × 2, which is easily carried out using gradient descent. A subsequent non-linear optimization step overcomes the far field approximation and enables the algorithm to recover locations and emission times of the defining acoustic events. Experimental results illustrate that our approach reliably solves hard SFS problems where gradient-based techniques consistently fail. Our approach is similar in spirit to the affine solution to the structure from motion (SFM) problem proposed by a seminal paper by Tomasi&Kanade [11], which was later extended to the non-orthographic case [7]. Like us, these authors expressed the structure finding problem using affine geometry, and applied SVD for solving it. SFM is of course defined for cameras, not for microphone arrays. Camera measure angles, whereas microphones measure range. This paper establishes an affine solution to the structure from sound problem that tends to work well in practice. 2 Problem Definition 2.1 Setup We are given N sensors (microphones) located in a 2D plane. We shall denote the location of the i-th sensor by (xi yi), which defined the following sensor location matrix of size N × 2: X = x1 y1 x2 y2 ... ... xN yN (1) We assume that the sensor array detects M acoustic events. Each event has as unknown coordinate and an unknown emission time. The coordinate of the j-th event shall be denoted (aj bj), providing us with the event location matrix A of size M × 2. The emission time of the j-th acoustic event is denoted tj, resulting in the vector T of length M: A = a1 b1 a2 b2 ... ... aM bM T = t1 t2 ... tM (2) X, A, and T, comprise the set of unknown variables. In problems such as sensor calibration, only X is of interest. In general SFS applications, A and T might also be of interest. 2.2 Measurement Data In SFS, the variables X, A, and T are recovered from data. The data establishes the detection times of the acoustic events by the individual sensors. Specifically, the data matrix is of the form: D = d1,1 d1,2 · · · d1,M d2,1 d2,2 · · · d2,M ... ... ... ... dN,1 dN,2 · · · dN,M (3) Here each di,j denotes the detection time of acoustical event j by sensor i. Notice that we assume that there is no data association problem. Even if all acoustic events sound alike, the correspondence between different detections is easily established as long as there exists sufficiently long time gaps between any two sound events. The matrix D is a random field induced by the laws of sound propagation (without reverberation). In the absence of measurement noise, each di,j is the sum of the corresponding emission time tj, plus the time it takes for sound to travel from (aj bj) to (xi yi): di,j = tj + c−1 xi yi − aj bj (4) Here | · | denotes the L2 norm (Euclidean distance), and c denoted the speed of sound. 2.3 Relative Formulation Obviously, we cannot recover the global coordinates of the sensors. Hence, without loss of generality, we define the first sensor’s location as x1 = y1 = 0. This gives us the relative location matrix for the sensors: ¯ X = x2 −x1 y2 −y1 x3 −x1 y3 −y1 ... ... xN −x1 yN −y1 (5) This relative sensor location matrix is of dimension (N −1) × 2. It shall prove convenient to subtract from the arrival time di,j the arrival time d1,j measured by the first sensor i = 1. This relative arrival time is defined as ∆i,j := di,j − d1,j. In the relative arrival time, the absolute emission times tj cancel out: ∆i,j = tj + c−1 xi yi − aj bj −tj −c−1 aj bj = c−1 xi yi − aj bj − aj bj (6) We now define the matrix of relative arrival times: ∆ = d2,1 −d1,1 d2,2 −d1,2 · · · d2,M −d1,M d3,1 −d1,1 d3,2 −d1,2 · · · d3,M −d1,M ... ... ... ... dN,1 −d1,1 dN,2 −d1,2 · · · dN,M −d1,M (7) This matrix ∆is of dimension (N −1) × M. 2.4 Least Squares Formulation The relative sensor locations X and the corresponding locations of the acoustic events A can now be recovered through the following least squares problem. This optimization seeks to identify X and A so as to minimize the quadratic difference between the predicted relative measurements and the actual measurements. ⟨A∗, X∗⟩ = argmin X,A N X i=2 M X j=1 xi yi − aj bj − aj bj −∆i,j 2 (8) The minimum of this expression is a maximum likelihood solution for the SFS problem under the assumption of iid Gaussian measurement noise. If emission times are of interest, they are now easily recovered by the following weighted mean: T ∗ = 1 N N X i=1 di,j −c xi yi − aj bj (9) The minimum of Eq. 8 is not unique. This is because any solution can be rotated around the origin of the coordinate system, and mirrored through any axis intersecting the origin. This shall not concern us, as we shall be content with any solution of Eq. 8; others are then easily generated. What is of concern, however, is the fact that minimizing Eq. 8 is difficult. A straw man algorithm—which tends to work poorly in practice—involves starting with random guesses for X and A and then adjusting them in the direction of the negative gradient until convergence. As we shall show experimentally, such gradient algorithms work poorly in practice because of the large number of local minima. 3 The Far Field Approximation The essence of our approximation pertains to the fact that for far range acoustic events— i.e., events that are (infinitely) far away from the sensor array—the incoming sound wave hits each sensor at the same incident angle. Put differently, the rays connecting the location of an acoustic event (aj bj) with each of the perceiving sensors (xi yi) are approximately parallel for all i (but not for all j!). Under the far field approximation, these incident angles are entirely parallel. Thus, all that matters are the incident angle of the acoustic events. To derive an equation for this case, it shall prove convenient to write the Euclidean distance between a sensor and an acoustic event as a function of the incident angle α. This angle is given by the four-quadrant extension of the arctan function: αi,j = arctan2 bj −yi aj −xi (10) The Euclidean distance between (aj bj) and (xi yi) can now be written as xi yi − aj bj = (cos αi,j sin αi,j) aj −xi bj −yi (11) For far-away points (aj bj), we can safely assume that all incident angles for the j-th acoustic event are identical: αj := α1,j = α2,j = . . . = αN,j (12) Hence we substitute αj for αi,j in Eq. 11. Plugging this back into Eq. 6, this gives us the following expression for ∆i,j: ∆i,j = c−1 xi yi − aj bj − aj bj ≈ c−1 (cos αj sin αj) aj −xi bj −yi − aj bj = c−1 (cos αj sin αj) xi yi (13) This leads to the following non-linear least squares problem for the desired sensor locations: ⟨X∗, α∗ 1, . . . , α∗ M⟩ = argmin X,α1,...,αM X cos α1 cos α2 · · · cos αM sin α1 sin α2 · · · sin αM −∆ 2 (14) The reader many notice that in this formulation of the SFS problem, the locations of the sound events (aj, bj) have been replaced by αj, the incident angles of the sound waves. One might think of this as the “ortho-acoustic” model of sound propagation (in analogy to the orthographic camera model in computer vision). The ortho-acoustic projection reduces the number of variables in the optimization. However, the argument in the quadratic expression is still non-linear, due to the non-linear trigonometric functions involved. 4 Affine Solution for the Sensor Locations Eq. 14 is trivially solvable in the space of affine geometry. Following [11], in affine geometry projections can be arbitrary linear functions, not just rotations and translations. Specifically, let us replace the specialized matrix cos α1 cos α2 · · · cos αM sin α1 sin α2 · · · sin αM (15) by a general 2 × M matrix of the form Γ = γ1,1 γ1,2 · · · γ1,M γ2,1 γ2,2 · · · γ2,M (16) This leads to the least squares problem ⟨X∗, Γ∗⟩ = argmin X,Γ |XΓ −∆|2 (17) In the noise free-case case, we know that there must exist a X and a Γ for which XΓ = ∆. This suggests that the rank of ∆should be 2, since it is the product of a matrix of size (N −1) × 2 and a matrix of size 2 × M. Further, we can recover both X and Γ via singular value decomposition (SVD). Specifically, we know that the matrix ∆can be decomposed as into three other matrices, U, V , and W: UV W T = svd(∆) (18) where U is a matrix of size (N −1) × 2, V a diagonal matrix of eigenvalues of size 2 × 2, and W a matrix of size M × 2. In practice, ∆might be of higher rank because of noise or because of violations of the far field assumption, but it suffices to restrict the consideration to the first two eigenvalues. The decomposition in Eq. 18 leads to the optimal affine solution of the SFS problem: X = UV and Γ = W T (19) However, this solution is not yet Euclidean, since Γ might not be of the form of Eq. 15. Specifically, Eq. 15 is a function of angles, and each row in Eq. 15 must be of the form cos2 γj + sin2 γj = 1. Clearly, this constraint is not enforced in the SVD. However, there is an easy “trick” for recovering a X and Γ for which this constraint is at least approximately met. The key insight is that for any invertible 2 × 2 matrix C, X′ = UV C−1 and Γ′ = CW T (20) is equally a solution to the factorization problem in Eq.18. This is because X′Γ′ = UV C−1CW T = UV W T = XΓ (21) The remaining search problem, thus, is the problem of finding an appropriate matrix C for which Γ′ is of the form of Eq. 15. This is a non-linear optimization problem, but it is much lower-dimensional than the original SFS problem (it only involves 4 parameters!). Specifically, we seek a C for which Γ′ = CW T minimizes C∗ = argmin C (1 1) (Γ′ · Γ′) | {z } (∗) −(1 1 · · · 1) 2 (22) Here “·” denotes the dot product. The expression labeled (∗) evaluates to a vector of expressions of the form (γ2 1,1 + γ2 2,1 γ2 1,2 + γ2 2,2 · · · γ2 1,M + γ2 2,M) (23) 4 6 8 10 12 14 0 0.5 1 1.5 2 2.5 3 3.5 error (95% confidence intervals) N, M (here N=M) (a) Error grad. desc. @ @ R SVD ? SVD+grad. desc. ? 4 6 8 10 12 14 −5 −4 −3 −2 −1 0 1 2 log−error (95% confidence intervals) N, M (here N=M) (b) Log-error grad. desc. 6 SVD ? SVD+grad. desc. 6 Figure 1: (a) Error and (b) log error for three different algorithms: gradient descent (red), SVD (blue), and SVD followed by gradient descent (green). Performance is shown for different values of N and M, with N = M. The plot also shows 95% confidence bars. (a) ground truth sensors acoustic events (b) gradient descent (c) SVD (d) SVD + grad. desc. Figure 2: Typical SFS results for a simulated array of nine microphones spaced in a regular grid, surrounded by 9 sounds arranged on a circle. (a) Ground truth; (b) Result of plain gradient descent after convergence; the dashed lines visualize the residual error; (c) Result of the SVD with sound directions as indicated; and (d) Result of gradient descent initialized with our SVD result. The minimization in Eq. 22 is carried out through standard gradient descent. It involves only 4 variables (C is of the size 2 × 2), and each single iteration is linear in O(N + M) (instead of the O(NM) constraints that define Eq. 8). In (tens of thousands of) experiments with synthetic noise-free data, we find empirically that gradient descent reliably converges to the globally optimal solution. 5 Recovering the Acoustic Event Locations and Emission Times With regards to the acoustic events, the optimization for the far field case only yields the incident angles. In the near field setting, in which the incident angles tend to differ for different sensors, it may be desirable to recover the locations A of the acoustic event and the corresponding emission times T. To determine these variables, we use the vector X∗from the far field case as mere starting points in a subsequent gradient search. The event location matrix A is initialized by selecting points sufficiently far away along the estimated incident angle for the far field approximation to be sound: A = k Γ′∗ (24) Here Γ′∗= C∗W T with C∗defined in Eq. 22, and k is a multiple of the diameter of the locations in X. With this initial guess for A, we apply gradient descent to optimize Eq. 8, and finally use Eq. 9 to recover T. 6 Experimental Results We ran a series of simulation experiments to characterize the quality of our algorithm, especially in comparison with the obvious nonlinear least squares problem (Eq. 8) from which it is derived. Fig. 1 graphs the residual error as a function of the number of sensors 0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 3 3.5 error (95% confidence intervals) diameter ratio of events vs sensor array (a) Error grad. desc. @ @ R SVD SVD+grad. desc. 0 2 4 6 8 10 −4 −3 −2 −1 0 1 2 log−error (95% confidence intervals) diameter ratio of events vs sensor array (b) Log-error grad. desc. SVD SVD+grad. desc. Figure 3: (a) Error and (b) log-error for three different algorithms (gradient descent in red, SVD in blue, and SVD followed by gradient descent in green), graphed here for varying distances of the sound events to the sensor array. An error above 2 means the reconstruction has entirely failed. All diagrams also show the 95% confidence intervals, and we set N = M = 10. (a) One of our motes used to generate the data (b) Optimal vs. hand-measured sounds motes motes (c) Result of gradient descent (d) SVD and GD Figure 4: Results using our seven sensor motes as the sensor array, and a seventh mote to generate sound events. (a) A mote; (b) the globally optimal solution (big circles) compared to the handmeasures locations (small circles); (c) a typical result of vanilla gradient descent; and (d) the result of our approach, all compared to the optimal solution given the (noisy) data. N and acoustic events M (here N = M). Panel (a) plots the regular error along with 95% confidence intervals, and panel (b) the corresponding log-error. Clearly, as N and M increase, plain gradient descent tends to diverge, whereas our approach converges. Each data point in these graphs was obtained by averaging 1,000 random configurations, in which sensors were sampled uniformly within an interval of 1×1m; sounds were placed at varying ranges, from 2m to 10m. An example outcome (for a non-random configuration!) is shown in Fig. 2. This figure plots (a) a simulated sensor array consisting of 9 sensors with 9 sound sources arranged in a circle; and (b)-(d) the resulting reconstructions of our three methods. For the SVD result shown in (c), only the directions of the incoming sounds are shown. An interesting question pertains to the effect of the far field approximation in cases where it is clearly violated. To examine the robustness of our approach, we ran a series of experiments in which we varied the diameter of the acoustic events relative to the diameter of the sensors. If this parameter is 1, the acoustic events are emitted in the same region as the microphones; for values such as 10, the events are far away. Fig. 3 graphs the residual errors and log-errors. The further away the acoustic events, the better our results. However, even for nearby events, for which the far field assumption is clearly invalid, our approach generates results that are no worse than those of the plain gradient descent technique. We also implemented our approach using a physical sensor array. Fig. 4 plots empirical results using a microphone array comprised of seven Crossbow sensor motes, one of which is shown in Panel (a). Panels (b-d) compare the recovered structure with the one that globally minimizes the LMS error, which we obtain by running gradient descent using the hand-measured locations as starting point. Panel (a) in Fig. 4 shows the manually measured locations; the relatively high deviation to the LMS optimum is the result of measurement error, which is amplified by the fact that our motes are only spaced a few tens of centimeters apart from each other (the standard deviation in the timing error corresponds to a distance of 6.99cm, and the motes are placed between 14cm and 125cm apart). Panel (b) in Fig. 4 shows the solution of plain gradient descent applied to applied to Eq.8 and compares it to the optimal reconstruction; and Panel (c) illustrates our solution. In all plots the lines indicate residual error. This result shows that our method may work well on real-world data that is noisy and that does not adhere to the far field assumption. 7 Discussion This paper considered the structure from sound problem and presented an algorithm for solving it. Our approach makes is possible to simultaneously recover the location of a collection of microphones, the locations of external acoustic events detected by these microphones, and the emission times for these events. By resorting to affine geometry, our approach overcomes the problem of local minima in the structure from sound problem. There remain a number of open research issues. We believe the extension to 3-D is mathematically straightforward but requires empirical validation. The current approach also fails to address reverberation problems that are common in confined space. It shall further be interesting to investigate data association problems in the SFS framework, and to develop parallel algorithms that can be implemented on sensor networks with limited communication resources. Finally, of great interest should be the incomplete data case in which individual sensors may fail to detect acoustic events—a problem studied in [2]. Acknowledgement The motes data was made available by Rahul Biswas, which is gratefully acknowledged. We also acknowledge invaluable suggestions by three anonymous reviewers. References [1] S.T. Birchfield and A. Subramanya. Microphone array position calibration by basis-point classical multidimensional scaling. IEEE Trans. Speech and Audio Processing, forthcoming. [2] R. Biswas and S. Thrun. A passive approach to sensor network localization. IROS-04. [3] J.C. Chen, R.E. Hudson, and K. Yao. Maximum likelihod source localization and unknown sensor location estimation for wideband signals in the near-field. IEEE Trans. Signal Processing, 50, 2002. [4] P. Corke, S. Hrabar, R. Peterson, D. Rus, S. Saripalli, and G. Sukhatme. Deployment and connectivity repair of a sensor net with a flying robot. ISER-04. [5] E. Elnahrawy, X. Li, and R. Martin. The limits of localization using signal strength: A comparative study. SECON-04. [6] J. Elson and K. Romer. Wireless sensor networks: A new regime for time synchronization. HotNets-02. [7] S. Mahamud and M. Hebert. Iterative projective reconstruction from multiple views. CVPR-00. [8] D. Niculescu and B. Nath. Ad hoc positioning system (APS). GLOBECOM-01. [9] V.C. Raykar, I.V. Kozintsev, and R. Lienhart. Position calibration of microphones and loudspeakers in distributed computing platforms. IEEE transaction on Speech and Audio Processing, 13(1), 2005. [10] J. Sallai, G. Balogh, M. Maroti, and A. Ledeczi. Acoustic ranging in resource-constrained sensor networks. eCOTS-04. [11] C. Tomasi and T. Kanade. Shape and motion from image streams under orthography: A factorization method. IJCV, 9(2), 1992. [12] T.L. Tung, K. Yao, D. Chen, R.E. Hudson, and C.W. Reed. Source localization and spatial filtering using wideband music and maxiumum power beam forming for multimedia applications. In SIPS-99.
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Measuring Shared Information and Coordinated Activity in Neuronal Networks Kristina Lisa Klinkner Statistics Department University of Michigan Ann Arbor, MI 48109 kshalizi@umich.edu Cosma Rohilla Shalizi Statistics Department Carnegie Mellon University Pittsburgh, PA 15213 cshalizi@stat.cmu.edu Marcelo F. Camperi Physics Department University of San Francisco San Francisco, CA 94118 camperi@usfca.edu Abstract Most nervous systems encode information about stimuli in the responding activity of large neuronal networks. This activity often manifests itself as dynamically coordinated sequences of action potentials. Since multiple electrode recordings are now a standard tool in neuroscience research, it is important to have a measure of such network-wide behavioral coordination and information sharing, applicable to multiple neural spike train data. We propose a new statistic, informational coherence, which measures how much better one unit can be predicted by knowing the dynamical state of another. We argue informational coherence is a measure of association and shared information which is superior to traditional pairwise measures of synchronization and correlation. To find the dynamical states, we use a recently-introduced algorithm which reconstructs effective state spaces from stochastic time series. We then extend the pairwise measure to a multivariate analysis of the network by estimating the network multi-information. We illustrate our method by testing it on a detailed model of the transition from gamma to beta rhythms. Much of the most important information in neural systems is shared over multiple neurons or cortical areas, in such forms as population codes and distributed representations [1]. On behavioral time scales, neural information is stored in temporal patterns of activity as opposed to static markers; therefore, as information is shared between neurons or brain regions, it is physically instantiated as coordination between entire sequences of neural spikes. Furthermore, neural systems and regions of the brain often require coordinated neural activity to perform important functions; acting in concert requires multiple neurons or cortical areas to share information [2]. Thus, if we want to measure the dynamic network-wide behavior of neurons and test hypotheses about them, we need reliable, practical methods to detect and quantify behavioral coordination and the associated information sharing across multiple neural units. These would be especially useful in testing ideas about how particular forms of coordination relate to distributed coding (e.g., that of [3]). Current techniques to analyze relations among spike trains handle only pairs of neurons, so we further need a method which is extendible to analyze the coordination in the network, system, or region as a whole. Here we propose a new measure of behavioral coordination and information sharing, informational coherence, based on the notion of dynamical state. Section 1 argues that coordinated behavior in neural systems is often not captured by existing measures of synchronization or correlation, and that something sensitive to nonlinear, stochastic, predictive relationships is needed. Section 2 defines informational coherence as the (normalized) mutual information between the dynamical states of two systems and explains how looking at the states, rather than just observables, fulfills the needs laid out in Section 1. Since we rarely know the right states a prori, Section 2.1 briefly describes how we reconstruct effective state spaces from data. Section 2.2 gives some details about how we calculate the informational coherence and approximate the global information stored in the network. Section 3 applies our method to a model system (a biophysically detailed conductance-based model) comparing our results to those of more familiar second-order statistics. In the interest of space, we omit proofs and a full discussion of the existing literature, giving only minimal references here; proofs and references will appear in a longer paper now in preparation. 1 Synchrony or Coherence? Most hypotheses which involve the idea that information sharing is reflected in coordinated activity across neural units invoke a very specific notion of coordinated activity, namely strict synchrony: the units should be doing exactly the same thing (e.g., spiking) at exactly the same time. Investigators then measure coordination by measuring how close the units come to being strictly synchronized (e.g., variance in spike times). From an informational point of view, there is no reason to favor strict synchrony over other kinds of coordination. One neuron consistently spiking 50 ms after another is just as informative a relationship as two simultaneously spiking, but such stable phase relations are missed by strict-synchrony approaches. Indeed, whatever the exact nature of the neural code, it uses temporally extended patterns of activity, and so information sharing should be reflected in coordination of those patterns, rather than just the instantaneous activity. There are three common ways of going beyond strict synchrony: cross-correlation and related second-order statistics, mutual information, and topological generalized synchrony. The cross-correlation function (the normalized covariance function; this includes, for present purposes, the joint peristimulus time histogram [2]), is one of the most widespread measures of synchronization. It can be efficiently calculated from observable series; it handles statistical as well as deterministic relationships between processes; by incorporating variable lags, it reduces the problem of phase locking. Fourier transformation of the covariance function γXY (h) yields the cross-spectrum FXY (ν), which in turn gives the spectral coherence cXY (ν) = F 2 XY (ν)/FX(ν)FY (ν), a normalized correlation between the Fourier components of X and Y . Integrated over frequencies, the spectral coherence measures, essentially, the degree of linear cross-predictability of the two series. ([4] applies spectral coherence to coordinated neural activity.) However, such second-order statistics only handle linear relationships. Since neural processes are known to be strongly nonlinear, there is little reason to think these statistics adequately measure coordination and synchrony in neural systems. Mutual information is attractive because it handles both nonlinear and stochastic relationships and has a very natural and appealing interpretation. Unfortunately, it often seems to fail in practice, being disappointingly small even between signals which are known to be tightly coupled [5]. The major reason is that the neural codes use distinct patterns of activity over time, rather than many different instantaneous actions, and the usual approach misses these extended patterns. Consider two neurons, one of which drives the other to spike 50 ms after it does, the driving neuron spiking once every 500 ms. These are very tightly coordinated, but whether the first neuron spiked at time t conveys little information about what the second neuron is doing at t — it’s not spiking, but it’s not spiking most of the time anyway. Mutual information calculated from the direct observations conflates the “no spike” of the second neuron preparing to fire with its just-sitting-around “no spike”. Here, mutual information could find the coordination if we used a 50 ms lag, but that won’t work in general. Take two rate-coding neurons with base-line firing rates of 1 Hz, and suppose that a stimulus excites one to 10 Hz and suppresses the other to 0.1 Hz. The spiking rates thus share a lot of information, but whether the one neuron spiked at t is uninformative about what the other neuron did then, and lagging won’t help. Generalized synchrony is based on the idea of establishing relationships between the states of the various units. “State” here is taken in the sense of physics, dynamics and control theory: the state at time t is a variable which fixes the distribution of observables at all times ≥t, rendering the past of the system irrelevant [6]. Knowing the state allows us to predict, as well as possible, how the system will evolve, and how it will respond to external forces [7]. Two coupled systems are said to exhibit generalized synchrony if the state of one system is given by a mapping from the state of the other. Applications to data employ statespace reconstruction [8]: if the state x ∈X evolves according to smooth, d-dimensional deterministic dynamics, and we observe a generic function y = f(x), then the space Y of time-delay vectors [y(t), y(t −τ), ...y(t −(k −1)τ)] is diffeomorphic to X if k > 2d, for generic choices of lag τ. The various versions of generalized synchrony differ on how, precisely, to quantify the mappings between reconstructed state spaces, but they all appear to be empirically equivalent to one another and to notions of phase synchronization based on Hilbert transforms [5]. Thus all of these measures accommodate nonlinear relationships, and are potentially very flexible. Unfortunately, there is essentially no reason to believe that neural systems have deterministic dynamics at experimentally-accessible levels of detail, much less that there are deterministic relationships among such states for different units. What we want, then, but none of these alternatives provides, is a quantity which measures predictive relationships among states, but allows those relationships to be nonlinear and stochastic. The next section introduces just such a measure, which we call “informational coherence”. 2 States and Informational Coherence There are alternatives to calculating the “surface” mutual information between the sequences of observations themselves (which, as described, fails to capture coordination). If we know that the units are phase oscillators, or rate coders, we can estimate their instantaneous phase or rate and, by calculating the mutual information between those variables, see how coordinated the units’ patterns of activity are. However, phases and rates do not exhaust the repertoire of neural patterns and a more general, common scheme is desirable. The most general notion of “pattern of activity” is simply that of the dynamical state of the system, in the sense mentioned above. We now formalize this. Assuming the usual notation for Shannon information [9], the information content of a state variable X is H[X] and the mutual information between X and Y is I[X; Y ]. As is well-known, I[X; Y ] ≤min H[X], H[Y ]. We use this to normalize the mutual state information to a 0 −1 scale, and this is the informational coherence (IC). ψ(X, Y ) = I[X; Y ] min H[X], H[Y ] , with 0/0 = 0 . (1) ψ can be interpreted as follows. I[X; Y ] is the Kullback-Leibler divergence between the joint distribution of X and Y , and the product of their marginal distributions [9], indicating the error involved in ignoring the dependence between X and Y . The mutual information between predictive, dynamical states thus gauges the error involved in assuming the two systems are independent, i.e., how much predictions could improve by taking into account the dependence. Hence it measures the amount of dynamically-relevant information shared between the two systems. ψ simply normalizes this value, and indicates the degree to which two systems have coordinated patterns of behavior (cf. [10], although this only uses directly observable quantities). 2.1 Reconstruction and Estimation of Effective State Spaces As mentioned, the state space of a deterministic dynamical system can be reconstructed from a sequence of observations. This is the main tool of experimental nonlinear dynamics [8]; but the assumption of determinism is crucial and false, for almost any interesting neural system. While classical state-space reconstruction won’t work on stochastic processes, such processes do have state-space representations [11], and, in the special case of discretevalued, discrete-time series, there are ways to reconstruct the state space. Here we use the CSSR algorithm, introduced in [12] (code available at http://bactra.org/CSSR). This produces causal state models, which are stochastic automata capable of statistically-optimal nonlinear prediction; the state of the machine is a minimal sufficient statistic for the future of the observable process[13].1 The basic idea is to form a set of states which should be (1) Markovian, (2) sufficient statistics for the next observable, and (3) have deterministic transitions (in the automata-theory sense). The algorithm begins with a minimal, one-state, IID model, and checks whether these properties hold, by means of hypothesis tests. If they fail, the model is modified, generally but not always by adding more states, and the new model is checked again. Each state of the model corresponds to a distinct distribution over future events, i.e., to a statistical pattern of behavior. Under mild conditions, which do not involve prior knowledge of the state space, CSSR converges in probability to the unique causal state model of the data-generating process [12]. In practice, CSSR is quite fast (linear in the data size), and generalizes at least as well as training hidden Markov models with the EM algorithm and using cross-validation for selection, the standard heuristic [12]. One advantage of the causal state approach (which it shares with classical state-space reconstruction) is that state estimation is greatly simplified. In the general case of nonlinear state estimation, it is necessary to know not just the form of the stochastic dynamics in the state space and the observation function, but also their precise parametric values and the distribution of observation and driving noises. Estimating the state from the observable time series then becomes a computationally-intensive application of Bayes’s Rule [17]. Due to the way causal states are built as statistics of the data, with probability 1 there is a finite time, t, at which the causal state at time t is certain. This is not just with some degree of belief or confidence: because of the way the states are constructed, it is impossible for the process to be in any other state at that time. Once the causal state has been established, it can be updated recursively, i.e., the causal state at time t+1 is an explicit function of the causal state at time t and the observation at t + 1. The causal state model can be automatically converted, therefore, into a finite-state transducer which reads in an observation time series and outputs the corresponding series of states [18, 13]. (Our implementation of CSSR filters its training data automatically.) The result is a new time series of states, from which all non-predictive components have been filtered out. 2.2 Estimating the Coherence Our algorithm for estimating the matrix of informational coherences is as follows. For each unit, we reconstruct the causal state model, and filter the observable time series to produce a series of causal states. Then, for each pair of neurons, we construct a joint histogram of 1Causal state models have the same expressive power as observable operator models [14] or predictive state representations [7], and greater power than variable-length Markov models [15, 16]. a b Figure 1: Rastergrams of neuronal spike-times in the network. Excitatory, pyramidal neurons (numbers 1 to 1000) are shown in green, inhibitory interneurons (numbers 1001 to 1300) in red. During the first 10 seconds (a), the current connections among the pyramidal cells are suppressed and a gamma rhythm emerges (left). At t = 10s, those connections become active, leading to a beta rhythm (b, right). the state distribution, estimate the mutual information between the states, and normalize by the single-unit state informations. This gives a symmetric matrix of ψ values. Even if two systems are independent, their estimated IC will, on average, be positive, because, while they should have zero mutual information, the empirical estimate of mutual information is non-negative. Thus, the significance of IC values must be assessed against the null hypothesis of system independence. The easiest way to do so is to take the reconstructed state models for the two systems and run them forward, independently of one another, to generate a large number of simulated state sequences; from these calculate values of the IC. This procedure will approximate the sampling distribution of the IC under a null model which preserves the dynamics of each system, but not their interaction. We can then find p-values as usual. We omit them here to save space. 2.3 Approximating the Network Multi-Information There is broad agreement [2] that analyses of networks should not just be an analysis of pairs of neurons, averaged over pairs. Ideally, an analysis of information sharing in a network would look at the over-all structure of statistical dependence between the various units, reflected in the complete joint probability distribution P of the states. This would then allow us, for instance, to calculate the n-fold multi-information, I[X1, X2, . . . Xn] ≡ D(P||Q), the Kullback-Leibler divergence between the joint distribution P and the product of marginal distributions Q, analogous to the pairwise mutual information [19]. Calculated over the predictive states, the multi-information would give the total amount of shared dynamical information in the system. Just as we normalized the mutual information I[X1, X2] by its maximum possible value, min H[X1], H[X2], we normalize the multiinformation by its maximum, which is the smallest sum of n −1 marginal entropies: I[X1; X2; . . . Xn] ≤min k X i̸=k H[Xn] Unfortunately, P is a distribution over a very high dimensional space and so, hard to estimate well without strong parametric constraints. We thus consider approximations. The lowest-order approximation treats all the units as independent; this is the distribution Q. One step up are tree distributions, where the global distribution is a function of the joint distributions of pairs of units. Not every pair of units needs to enter into such a distribution, though every unit must be part of some pair. Graphically, a tree distribution corresponds to a spanning tree, with edges linking units whose interactions enter into the global probability, and conversely spanning trees determine tree distributions. Writing ET for the set of pairs (i, j) and abbreviating X1 = x1, X2 = x2, . . . Xn = xn by X = x, one has T(X = x) = Y (i,j)∈ET T(Xi = xi, Xj = xj) T(Xi = xi)T(Xj = xj) n Y i=1 T(Xi = xi) (2) where the marginal distributions T(Xi) and the pair distributions T(Xi, Xj) are estimated by the empirical marginal and pair distributions. We must now pick edges ET so that T best approximates the true global distribution P. A natural approach is to minimize D(P||T), the divergence between P and its tree approximation. Chow and Liu [20] showed that the maximum-weight spanning tree gives the divergence-minimizing distribution, taking an edge’s weight to be the mutual information between the variables it links. There are three advantages to using the Chow-Liu approximation. (1) Estimating T from empirical probabilities gives a consistent maximum likelihood estimator of the ideal ChowLiu tree [20], with reasonable rates of convergence, so T can be reliably known even if P cannot. (2) There are efficient algorithms for constructing maximum-weight spanning trees, such as Prim’s algorithm [21, sec. 23.2], which runs in time O(n2 + n log n). Thus, the approximation is computationally tractable. (3) The KL divergence of the Chow-Liu distribution from Q gives a lower bound on the network multi-information; that bound is just the sum of the mutual informations along the edges in the tree: I[X1; X2; . . . Xn] ≥D(T||Q) = X (i,j)∈ET I[Xi; Xj] (3) Even if we knew P exactly, Eq. 3 would be useful as an alternative to calculating D(P||Q) directly, evaluating log P(x)/Q(x) for all the exponentially-many configurations x. It is natural to seek higher-order approximations to P, e.g., using three-way interactions not decomposable into pairwise interactions [22, 19]. But it is hard to do so effectively, because finding the optimal approximation to P when such interactions are allowed is NP [23], and analytical formulas like Eq. 3 generally do not exist [19]. We therefore confine ourselves to the Chow-Liu approximation here. 3 Example: A Model of Gamma and Beta Rhythms We use simulated data as a test case, instead of empirical multiple electrode recordings, which allows us to try the method on a system of over 1000 neurons and compare the measure against expected results. The model, taken from [24], was originally designed to study episodes of gamma (30–80Hz) and beta (12–30Hz) oscillations in the mammalian nervous system, which often occur successively with a spontaneous transition between them. More concretely, the rhythms studied were those displayed by in vitro hippocampal (CA1) slice preparations and by in vivo neocortical EEGs. The model contains two neuron populations: excitatory (AMPA) pyramidal neurons and inhibitory (GABAA) interneurons, defined by conductance-based Hodgkin-Huxley-style equations. Simulations were carried out in a network of 1000 pyramidal cells and 300 interneurons. Each cell was modeled as a one-compartment neuron with all-to-all coupling, endowed with the basic sodium and potassium spiking currents, an external applied current, and some Gaussian input noise. The first 10 seconds of the simulation correspond to the gamma rhythm, in which only a group of neurons is made to spike via a linearly increasing applied current. The beta rhythm a b c d Figure 2: Heat-maps of coordination for the network, as measured by zero-lag cross-correlation (top row) and informational coherence (bottom), contrasting the gamma rhythm (left column) with the beta (right). Colors run from red (no coordination) through yellow to pale cream (maximum). (subsequent 10 seconds) is obtained by activating pyramidal-pyramidal recurrent connections (potentiated by Hebbian preprocessing as a result of synchrony during the gamma rhythm) and a slow outward after-hyper-polarization (AHP) current (the M-current), suppressed during gamma due to the metabotropic activation used in the generation of the rhythm. During the beta rhythm, pyramidal cells, silent during gamma rhythm, fire on a subset of interneurons cycles (Fig. 1). Fig. 2 compares zero-lag cross-correlation, a second-order method of quantifying coordination, with the informational coherence calculated from the reconstructed states. (In this simulation, we could have calculated the actual states of the model neurons directly, rather than reconstructing them, but for purposes of testing our method we did not.) Crosscorrelation finds some of the relationships visible in Fig. 1, but is confused by, for instance, the phase shifts between pyramidal cells. (Surface mutual information, not shown, gives similar results.) Informational coherence, however, has no trouble recognizing the two populations as effectively coordinated blocks. The presence of dynamical noise, problematic for ordinary state reconstruction, is not an issue. The average IC is 0.411 (or 0.797 if the inactive, low-numbered neurons are excluded). The tree estimate of the global informational multi-information is 3243.7 bits, with a global coherence of 0.777. The right half of Fig. 2 repeats this analysis for the beta rhythm; in this stage, the average IC is 0.614, and the tree estimate of the global multi-information is 7377.7 bits, though the estimated global coherence falls very slightly to 0.742. This is because low-numbered neurons which were quiescent before are now active, contributing to the global information, but the over-all pattern is somewhat weaker and more noisy (as can be seen from Fig. 1b.) So, as expected, the total information content is higher, but the overall coordination across the network is lower. 4 Conclusion Informational coherence provides a measure of neural information sharing and coordinated activity which accommodates nonlinear, stochastic relationships between extended patterns of spiking. It is robust to dynamical noise and leads to a genuinely multivariate measure of global coordination across networks or regions. Applied to data from multi-electrode recordings, it should be a valuable tool in evaluating hypotheses about distributed neural representation and function. Acknowledgments Thanks to R. Haslinger, E. Ionides and S. Page; and for support to the Santa Fe Institute (under grants from Intel, the NSF and the MacArthur Foundation, and DARPA agreement F30602-00-2-0583), the Clare Booth Luce Foundation (KLK) and the James S. McDonnell Foundation (CRS). References [1] L. F. Abbott and T. J. Sejnowski, eds. Neural Codes and Distributed Representations. MIT Press, 1998. [2] E. N. Brown, R. E. Kass, and P. P. Mitra. Nature Neuroscience, 7:456–461, 2004. [3] D. H. Ballard, Z. Zhang, and R. P. N. Rao. In R. P. N. Rao, B. A. Olshausen, and M. S. Lewicki, eds., Probabilistic Models of the Brain, pp. 273–284, MIT Press, 2002. [4] D. R. Brillinger and A. E. P. Villa. In D. R. Brillinger, L. T. Fernholz, and S. Morgenthaler, eds., The Practice of Data Analysis, pp. 77–92. Princeton U.P., 1997. [5] R. Quian Quiroga et al. Physical Review E, 65:041903, 2002. [6] R. F. Streater. Statistical Dynamics. Imperial College Press, London. [7] M. L. Littman, R. S. Sutton, and S. Singh. In T. G. Dietterich, S. Becker, and Z. Ghahramani, eds., Advances in Neural Information Processing Systems 14, pp. 1555–1561. MIT Press, 2002. [8] H. Kantz and T. Schreiber. Nonlinear Time Series Analysis. Cambridge U.P., 1997. [9] T. M. Cover and J. A. Thomas. Elements of Information Theory. Wiley, 1991. [10] M. Palus et al. Physical Review E, 63:046211, 2001. [11] F. B. Knight. Annals of Probability, 3:573–596, 1975. [12] C. R. Shalizi and K. L. Shalizi. In M. Chickering and J. Halpern, eds., Uncertainty in Artificial Intelligence: Proceedings of the Twentieth Conference, pp. 504–511. AUAI Press, 2004. [13] C. R. Shalizi and J. P. Crutchfield. Journal of Statistical Physics, 104:817–819, 2001. [14] H. Jaeger. Neural Computation, 12:1371–1398, 2000. [15] D. Ron, Y. Singer, and N. Tishby. Machine Learning, 25:117–149, 1996. [16] P. B¨uhlmann and A. J. Wyner. Annals of Statistics, 27:480–513, 1999. [17] N. U. Ahmed. Linear and Nonlinear Filtering for Scientists and Engineers. World Scientific, 1998. [18] D. R. Upper. PhD thesis, University of California, Berkeley, 1997. [19] E. Schneidman, S. Still, M. J. Berry, and W. Bialek. Physical Review Letters, 91:238701, 2003. [20] C. K. Chow and C. N. Liu. IEEE Transactions on Information Theory, IT-14:462–467, 1968. [21] T. H. Cormen et al. Introduction to Algorithms. 2nd ed. MIT Press, 2001. [22] S. Amari. IEEE Transacttions on Information Theory, 47:1701–1711, 2001. [23] S. Kirshner, P. Smyth, and A. Robertson. Tech. Rep. 04-04, UC Irvine, Information and Computer Science, 2004. [24] M. S. Olufsen et al. Journal of Computational Neuroscience, 14:33–54, 2003.
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Distance Metric Learning for Large Margin Nearest Neighbor Classification Kilian Q. Weinberger, John Blitzer and Lawrence K. Saul Department of Computer and Information Science, University of Pennsylvania Levine Hall, 3330 Walnut Street, Philadelphia, PA 19104 {kilianw, blitzer, lsaul}@cis.upenn.edu Abstract We show how to learn a Mahanalobis distance metric for k-nearest neighbor (kNN) classification by semidefinite programming. The metric is trained with the goal that the k-nearest neighbors always belong to the same class while examples from different classes are separated by a large margin. On seven data sets of varying size and difficulty, we find that metrics trained in this way lead to significant improvements in kNN classification—for example, achieving a test error rate of 1.3% on the MNIST handwritten digits. As in support vector machines (SVMs), the learning problem reduces to a convex optimization based on the hinge loss. Unlike learning in SVMs, however, our framework requires no modification or extension for problems in multiway (as opposed to binary) classification. 1 Introduction The k-nearest neighbors (kNN) rule [3] is one of the oldest and simplest methods for pattern classification. Nevertheless, it often yields competitive results, and in certain domains, when cleverly combined with prior knowledge, it has significantly advanced the state-ofthe-art [1, 14]. The kNN rule classifies each unlabeled example by the majority label among its k-nearest neighbors in the training set. Its performance thus depends crucially on the distance metric used to identify nearest neighbors. In the absence of prior knowledge, most kNN classifiers use simple Euclidean distances to measure the dissimilarities between examples represented as vector inputs. Euclidean distance metrics, however, do not capitalize on any statistical regularities in the data that might be estimated from a large training set of labeled examples. Ideally, the distance metric for kNN classification should be adapted to the particular problem being solved. It can hardly be optimal, for example, to use the same distance metric for face recognition as for gender identification, even if in both tasks, distances are computed between the same fixed-size images. In fact, as shown by many researchers [2, 6, 7, 8, 12, 13], kNN classification can be significantly improved by learning a distance metric from labeled examples. Even a simple (global) linear transformation of input features has been shown to yield much better kNN classifiers [7, 12]. Our work builds in a novel direction on the success of these previous approaches. In this paper, we show how to learn a Mahanalobis distance metric for kNN classification. The metric is optimized with the goal that k-nearest neighbors always belong to the same class while examples from different classes are separated by a large margin. Our goal for metric learning differs in a crucial way from those of previous approaches that minimize the pairwise distances between all similarly labeled examples [12, 13, 17]. This latter objective is far more difficult to achieve and does not leverage the full power of kNN classification, whose accuracy does not require that all similarly labeled inputs be tightly clustered. Our approach is largely inspired by recent work on neighborhood component analysis [7] and metric learning by energy-based models [2]. Though based on the same goals, however, our methods are quite different. In particular, we are able to cast our optimization as an instance of semidefinite programming. Thus the optimization we propose is convex, and its global minimum can be efficiently computed. Our approach has several parallels to learning in support vector machines (SVMs)—most notably, the goal of margin maximization and a convex objective function based on the hinge loss. In light of these parallels, we describe our approach as large margin nearest neighbor (LMNN) classification. Our framework can be viewed as the logical counterpart to SVMs in which kNN classification replaces linear classification. Our framework contrasts with classification by SVMs, however, in one intriguing respect: it requires no modification for problems in multiway (as opposed to binary) classification. Extensions of SVMs to multiclass problems typically involve combining the results of many binary classifiers, or they require additional machinery that is elegant but nontrivial [4]. In both cases the training time scales at least linearly in the number of classes. By contrast, our learning problem has no explicit dependence on the number of classes. 2 Model Let {(⃗xi, yi)}n i=1 denote a training set of n labeled examples with inputs ⃗xi ∈Rd and discrete (but not necessarily binary) class labels yi. We use the binary matrix yij ∈{0, 1} to indicate whether or not the labels yi and yj match. Our goal is to learn a linear transformation L:Rd →Rd, which we will use to compute squared distances as: D(⃗xi, ⃗xj) = ∥L(⃗xi −⃗xj)∥2. (1) Specifically, we want to learn the linear transformation that optimizes kNN classification when distances are measured in this way. We begin by developing some useful terminology. Target neighbors In addition to the class label yi, for each input ⃗xi we also specify k “target” neighbors— that is, k other inputs with the same label yi that we wish to have minimal distance to ⃗xi, as computed by eq. (1). In the absence of prior knowledge, the target neighbors can simply be identified as the k nearest neighbors, determined by Euclidean distance, that share the same label yi. (This was done for all the experiments in this paper.) We use ηij ∈{0, 1} to indicate whether input ⃗xj is a target neighbor of input ⃗xi. Like the binary matrix yij, the matrix ηij is fixed and does not change during learning. Cost function Our cost function over the distance metrics parameterized by eq. (1) has two competing terms. The first term penalizes large distances between each input and its target neighbors, while the second term penalizes small distances between each input and all other inputs that do not share the same label. Specifically, the cost function is given by: ε(L) = X ij ηij∥L(⃗xi−⃗xj)∥2 + c X ijl ηij(1−yil) 1 + ∥L(⃗xi−⃗xj)∥2−∥L(⃗xi−⃗xl)∥2 + , (2) where in the second term [z]+ = max(z, 0) denotes the standard hinge loss and c > 0 is some positive constant (typically set by cross validation). Note that the first term only penalizes large distances between inputs and target neighbors, not between all similarly labeled examples. Large margin ⃗xi ⃗xi margin local neighborhood ⃗xi ⃗xi margin BEFORE AFTER Similarly labeled Differently labeled Differently labeled target neighbor Figure 1: Schematic illustration of one input’s neighborhood ⃗xi before training (left) versus after training (right). The distance metric is optimized so that: (i) its k=3 target neighbors lie within a smaller radius after training; (ii) differently labeled inputs lie outside this smaller radius, with a margin of at least one unit distance. Arrows indicate the gradients on distances arising from the optimization of the cost function. The second term in the cost function incorporates the idea of a margin. In particular, for each input ⃗xi, the hinge loss is incurred by differently labeled inputs whose distances do not exceed, by one absolute unit of distance, the distance from input ⃗xi to any of its target neighbors. The cost function thereby favors distance metrics in which differently labeled inputs maintain a large margin of distance and do not threaten to “invade” each other’s neighborhoods. The learning dynamics induced by this cost function are illustrated in Fig. 1 for an input with k=3 target neighbors. Parallels with SVMs The competing terms in eq. (2) are analogous to those in the cost function for SVMs [11]. In both cost functions, one term penalizes the norm of the “parameter” vector (i.e., the weight vector of the maximum margin hyperplane, or the linear transformation in the distance metric), while the other incurs the hinge loss for examples that violate the condition of unit margin. Finally, just as the hinge loss in SVMs is only triggered by examples near the decision boundary, the hinge loss in eq. (2) is only triggered by differently labeled examples that invade each other’s neighborhoods. Convex optimization We can reformulate the optimization of eq. (2) as an instance of semidefinite programming [16]. A semidefinite program (SDP) is a linear program with the additional constraint that a matrix whose elements are linear in the unknown variables is required to be positive semidefinite. SDPs are convex; thus, with this reformulation, the global minimum of eq. (2) can be efficiently computed. To obtain the equivalent SDP, we rewrite eq. (1) as: D(⃗xi, ⃗xj) = (⃗xi −⃗xj)⊤M(⃗xi −⃗xj), (3) where the matrix M = L⊤L, parameterizes the Mahalanobis distance metric induced by the linear transformation L. Rewriting eq. (2) as an SDP in terms of M is straightforward, since the first term is already linear in M = L⊤L and the hinge loss can be “mimicked” by introducing slack variables ξij for all pairs of differently labeled inputs (i.e., for all ⟨i, j⟩ such that yij = 0). The resulting SDP is given by: Minimize P ij ηij(⃗xi −⃗xj)⊤M(⃗xi −⃗xj) + c P ij ηij(1 −yil)ξijl subject to: (1) (⃗xi −⃗xl)⊤M(⃗xi −⃗xl) −(⃗xi −⃗xj)⊤M(⃗xi −⃗xj) ≥1 −ξijl (2) ξijl ≥0 (3) M ⪰0. The last constraint M ⪰0 indicates that the matrix M is required to be positive semidefinite. While this SDP can be solved by standard online packages, general-purpose solvers tend to scale poorly in the number of constraints. Thus, for our work, we implemented our own special-purpose solver, exploiting the fact that most of the slack variables {ξij} never attain positive values1. The slack variables {ξij} are sparse because most labeled inputs are well separated; thus, their resulting pairwise distances do not incur the hinge loss, and we obtain very few active constraints. Our solver was based on a combination of sub-gradient descent in both the matrices L and M, the latter used mainly to verify that we had reached the global minimum. We projected updates in M back onto the positive semidefinite cone after each step. Alternating projection algorithms provably converge [16], and in this case our implementation worked much faster than generic solvers2. 3 Results We evaluated the algorithm in the previous section on seven data sets of varying size and difficulty. Table 1 compares the different data sets. Principal components analysis (PCA) was used to reduce the dimensionality of image, speech, and text data, both to speed up training and avoid overfitting. Except for Isolet and MNIST, all of the experimental results are averaged over several runs of randomly generated 70/30 splits of the data. Isolet and MNIST have pre-defined training/test splits. For the other data sets, we randomly generated 70/30 splits for each run. Both the number of target neighbors (k) and the weighting parameter (c) in eq. (2) were set by cross validation. (For the purpose of cross-validation, the training sets were further partitioned into training and validation sets.) We begin by reporting overall trends, then discussing the individual data sets in more detail. We first compare kNN classification error rates using Mahalanobis versus Euclidean distances. To break ties among different classes, we repeatedly reduced the neighborhood size, ultimately classifying (if necessary) by just the k = 1 nearest neighbor. Fig. 2 summarizes the main results. Except on the smallest data set (where over-training appears to be an issue), the Mahalanobis distance metrics learned by semidefinite programming led to significant improvements in kNN classification, both in training and testing. The training error rates reported in Fig. 2 are leave-one-out estimates. We also computed test error rates using a variant of kNN classification, inspired by previous work on energy-based models [2]. Energy-based classification of a test example ⃗xt was done by finding the label that minimizes the cost function in eq. (2). In particular, for a hypothetical label yt, we accumulated the squared distances to the k nearest neighbors of ⃗xt that share the same label in the training set (corresponding to the first term in the cost function); we also accumulated the hinge loss over all pairs of differently labeled examples that result from labeling ⃗xt by yt (corresponding to the second term in the cost function). Finally, the test example was classified by the hypothetical label that minimized the combination of these two terms: yt =argminyt X j ηtj∥L(⃗xt−⃗xj)∥2+c X j,i=t∨l=t ηij(1−yil) 1 + ∥L(⃗xi−⃗xj)∥2−∥L(⃗xi−⃗xl)∥2 + As shown in Fig. 2, energy-based classification with this assignment rule generally led to even further reductions in test error rates. Finally, we compared our results to those of multiclass SVMs [4]. On each data set (except MNIST), we trained multiclass SVMs using linear and RBF kernels; Fig. 2 reports the results of the better classifier. On MNIST, we used a non-homogeneous polynomial kernel of degree four, which gave us our best results. (See also [9].) 1A great speedup can be achieved by solving an SDP that only monitors a fraction of the margin conditions, then using the resulting solution as a starting point for the actual SDP of interest. 2A matlab implementation is currently available at http://www.seas.upenn.edu/∼kilianw/lmnn. Iris Wine Faces Bal Isolet News MNIST examples (train) 106 126 280 445 6238 16000 60000 examples (test) 44 52 120 90 1559 2828 10000 classes 3 3 40 3 26 20 10 input dimensions 4 13 1178 4 617 30000 784 features after PCA 4 13 30 4 172 200 164 constraints 5278 7266 78828 76440 37 Mil 164 Mil 3.3 Bil active constraints 113 1396 7665 3099 45747 732359 243596 CPU time (per run) 2s 8s 7s 13s 11m 1.5h 4h runs 100 100 100 100 1 10 1 Table 1: Properties of data sets and experimental parameters for LMNN classification. MNIST NEWS ISOLET BAL FACES IRIS WINE 2.1 17.6 13.4 13.0 12.4 8.6 4.7 3.7 3.3 14.4 9.7 8.4 7.8 5.9 2.6 2.7 19.0 4.3 4.7 5.8 4.4 2.6 2.7 1.7 1.3 1.2 1.9 1.2 20.0 11.0 9.4 4.7 14.1 10.0 8.2 0.3 30.0 1.1 4.3 3.5 2.2 30.1 Energy based classification kNN Mahalanobis distance kNN Euclidean distance Multiclass SVM testing error rate (%) training error rate (%) Figure 2: Training and test error rates for kNN classification using Euclidean versus Mahalanobis distances. The latter yields lower test error rates on all but the smallest data set (presumably due to over-training). Energy-based classification (see text) generally leads to further improvement. The results approach those of state-of-the-art multiclass SVMs. Small data sets with few classes The wine, iris, and balance data sets are small data sets, with less than 500 training examples and just three classes, taken from the UCI Machine Learning Repository3. On data sets of this size, a distance metric can be learned in a matter of seconds. The results in Fig. 2 were averaged over 100 experiments with different random 70/30 splits of each data set. Our results on these data sets are roughly comparable (i.e., better in some cases, worse in others) to those of neighborhood component analysis (NCA) and relevant component analysis (RCA), as reported in previous work [7]. Face recognition The AT&T face recognition data set4 contains 400 grayscale images of 40 individuals in 10 different poses. We downsampled the images from to 38 × 31 pixels and used PCA to obtain 30-dimensional eigenfaces [15]. Training and test sets were created by randomly sampling 7 images of each person for training and 3 images for testing. The task involved 40-way classification—essentially, recognizing a face from an unseen pose. Fig. 2 shows the improvements due to LMNN classification. Fig. 3 illustrates the improvements more graphically by showing how the k = 3 nearest neighbors change as a result of learning a Mahalanobis metric. (Though the algorithm operated on low dimensional eigenfaces, for clarity the figure shows the rescaled images.) 3Available at http://www.ics.uci.edu/∼mlearn/MLRepository.html. 4Available at http://www.uk.research.att.com/facedatabase.html Among 3 nearest neighbors before but not after training: Test Image: Among 3 nearest neighbors after but not before training: Figure 3: Images from the AT&T face recognition data base. Top row: an image correctly recognized by kNN classification (k = 3) with Mahalanobis distances, but not with Euclidean distances. Middle row: correct match among the k=3 nearest neighbors according to Mahalanobis distance, but not Euclidean distance. Bottom row: incorrect match among the k=3 nearest neighbors according to Euclidean distance, but not Mahalanobis distance. Spoken letter recognition The Isolet data set from UCI Machine Learning Repository has 6238 examples and 26 classes corresponding to letters of the alphabet. We reduced the input dimensionality (originally at 617) by projecting the data onto its leading 172 principal components—enough to account for 95% of its total variance. On this data set, Dietterich and Bakiri report test error rates of 4.2% using nonlinear backpropagation networks with 26 output units (one per class) and 3.3% using nonlinear backpropagation networks with a 30-bit error correcting code [5]. LMNN with energy-based classification obtains a test error rate of 3.7%. Text categorization The 20-newsgroups data set consists of posted articles from 20 newsgroups, with roughly 1000 articles per newsgroup. We used the 18828-version of the data set5 which has crosspostings removed and some headers stripped out. We tokenized the newsgroups using the rainbow package [10]. Each article was initially represented by the weighted word-counts of the 20,000 most common words. We then reduced the dimensionality by projecting the data onto its leading 200 principal components. The results in Fig. 2 were obtained by averaging over 10 runs with 70/30 splits for training and test data. Our best result for LMMN on this data set at 13.0% test error rate improved significantly on kNN classification using Euclidean distances. LMNN also performed comparably to our best multiclass SVM [4], which obtained a 12.4% test error rate using a linear kernel and 20000 dimensional inputs. Handwritten digit recognition The MNIST data set of handwritten digits6 has been extensively benchmarked [9]. We deskewed the original 28×28 grayscale images, then reduced their dimensionality by retaining only the first 164 principal components (enough to capture 95% of the data’s overall variance). Energy-based LMNN classification yielded a test error rate at 1.3%, cutting the baseline kNN error rate by over one-third. Other comparable benchmarks [9] (not exploiting additional prior knowledge) include multilayer neural nets at 1.6% and SVMs at 1.2%. Fig. 4 shows some digits whose nearest neighbor changed as a result of learning, from a mismatch using Euclidean distance to a match using Mahanalobis distance. 4 Related Work Many researchers have attempted to learn distance metrics from labeled examples. We briefly review some recent methods, pointing out similarities and differences with our work. 5Available at http://people.csail.mit.edu/jrennie/20Newsgroups/ 6Available at http://yann.lecun.com/exdb/mnist/ Test Image: Nearest neighbor before training: Nearest neighbor after training: Figure 4: Top row: Examples of MNIST images whose nearest neighbor changes during training. Middle row: nearest neighbor after training, using the Mahalanobis distance metric. Bottom row: nearest neighbor before training, using the Euclidean distance metric. Xing et al [17] used semidefinite programming to learn a Mahalanobis distance metric for clustering. Their algorithm aims to minimize the sum of squared distances between similarly labeled inputs, while maintaining a lower bound on the sum of distances between differently labeled inputs. Our work has a similar basis in semidefinite programming, but differs in its focus on local neighborhoods for kNN classification. Shalev-Shwartz et al [12] proposed an online learning algorithm for learning a Mahalanobis distance metric. The metric is trained with the goal that all similarly labeled inputs have small pairwise distances (bounded from above), while all differently labeled inputs have large pairwise distances (bounded from below). A margin is defined by the difference of these thresholds and induced by a hinge loss function. Our work has a similar basis in its appeal to margins and hinge loss functions, but again differs in its focus on local neighborhoods for kNN classification. In particular, we do not seek to minimize the distance between all similarly labeled inputs, only those that are specified as neighbors. Goldberger et al [7] proposed neighborhood component analysis (NCA), a distance metric learning algorithm especially designed to improve kNN classification. The algorithm minimizes the probability of error under stochastic neighborhood assignments using gradient descent. Our work shares essentially the same goals as NCA, but differs in its construction of a convex objective function. Chopra et al [2] recently proposed a framework for similarity metric learning in which the metrics are parameterized by pairs of identical convolutional neural nets. Their cost function penalizes large distances between similarly labeled inputs and small distances between differently labeled inputs, with penalties that incorporate the idea of a margin. Our work is based on a similar cost function, but our metric is parameterized by a linear transformation instead of a convolutional neural net. In this way, we obtain an instance of semidefinite programming. Relevant component analysis (RCA) constructs a Mahalanobis distance metric from a weighted sum of in-class covariance matrices [13]. It is similar to PCA and linear discriminant analysis (but different from our approach) in its reliance on second-order statistics. Hastie and Tibshirani [?] and Domeniconi et al [6] consider schemes for locally adaptive distance metrics that vary throughout the input space. The latter work appeals to the goal of margin maximization but otherwise differs substantially from our approach. In particular, Domeniconi et al [6] suggest to use the decision boundaries of SVMs to induce a locally adaptive distance metric for kNN classification. By contrast, our approach (though similarly named) does not involve the training of SVMs. 5 Discussion In this paper, we have shown how to learn Mahalanobis distance metrics for kNN classification by semidefinite programming. Our framework makes no assumptions about the structure or distribution of the data and scales naturally to large number of classes. Ongoing work is focused in three directions. First, we are working to apply LMNN classification to problems with hundreds or thousands of classes, where its advantages are most apparent. Second, we are investigating the kernel trick to perform LMNN classification in nonlinear feature spaces. As LMMN already yields highly nonlinear decision boundaries in the original input space, however, it is not obvious that “kernelizing” the algorithm will lead to significant further improvement. Finally, we are extending our framework to learn locally adaptive distance metrics [6, 8] that vary across the input space. Such metrics should lead to even more flexible and powerful large margin classifiers. References [1] S. Belongie, J. Malik, and J. Puzicha. Shape matching and object recognition using shape contexts. IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI), 24(4):509– 522, 2002. [2] S. Chopra, R. Hadsell, and Y. LeCun. Learning a similiarty metric discriminatively, with application to face verification. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR-05), San Diego, CA, 2005. [3] T. Cover and P. Hart. Nearest neighbor pattern classification. In IEEE Transactions in Information Theory, IT-13, pages 21–27, 1967. [4] K. Crammer and Y. Singer. On the algorithmic implementation of multiclass kernel-based vector machines. Journal of Machine Learning Research, 2:265–292, 2001. [5] T. G. Dietterich and G. Bakiri. Solving multiclass learning problems via error-correcting output codes. In Journal of Artificial Intelligence Research, number 2 in 263-286, 1995. [6] C. Domeniconi, D. Gunopulos, and J. Peng. Large margin nearest neighbor classifiers. IEEE Transactions on Neural Networks, 16(4):899–909, 2005. [7] J. Goldberger, S. Roweis, G. Hinton, and R. Salakhutdinov. Neighbourhood components analysis. In L. K. Saul, Y. Weiss, and L. Bottou, editors, Advances in Neural Information Processing Systems 17, pages 513–520, Cambridge, MA, 2005. MIT Press. [8] T. Hastie and R. Tibshirani. Discriminant adaptive nearest neighbor classification. IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI), 18:607–616, 1996. [9] Y. LeCun, L. Jackel, L. Bottou, A. Brunot, C. Cortes, J. Denker, H. Drucker, I. Guyon, U. Muller, E. Sackinger, P. Simard, and V. Vapnik. A comparison of learning algorithms for handwritten digit recognition. In F.Fogelman and P.Gallinari, editors, Proceedings of the 1995 International Conference on Artificial Neural Networks (ICANN-95), pages 53–60, Paris, 1995. [10] A. K. McCallum. Bow: A toolkit for statistical language modeling, text retrieval, classification and clustering. http://www.cs.cmu.edu/ mccallum/bow, 1996. [11] B. Sch¨olkopf and A. J. Smola. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press, Cambridge, MA, 2002. [12] S. Shalev-Shwartz, Y. Singer, and A. Y. Ng. Online and batch learning of pseudo-metrics. In Proceedings of the 21st International Conference on Machine Learning, Banff, Canada, 2004. [13] N. Shental, T. Hertz, D. Weinshall, and M. Pavel. Adjustment learning and relevant component analysis. In Proceedings of the Seventh European Conference on Computer Vision (ECCV-02), volume 4, pages 776–792, London, UK, 2002. Springer-Verlag. [14] P. Y. Simard, Y. LeCun, and J. Decker. Efficient pattern recognition using a new transformation distance. In Advances in Neural Information Processing Systems, volume 6, pages 50–58, San Mateo, CA, 1993. Morgan Kaufman. [15] M. Turk and A. Pentland. Eigenfaces for recognition. Journal of Cognitive Neuroscience, 3(1):71–86, 1991. [16] L. Vandenberghe and S. P. Boyd. Semidefinite programming. SIAM Review, 38(1):49–95, March 1996. [17] E. P. Xing, A. Y. Ng, M. I. Jordan, and S. Russell. Distance metric learning, with application to clustering with side-information. In T. G. Dietterich, S. Becker, and Z. Ghahramani, editors, Advances in Neural Information Processing Systems 14, Cambridge, MA, 2002. MIT Press.
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Estimating the “wrong” Markov random field: Benefits in the computation-limited setting Martin J. Wainwright Department of Statistics, and Department of Electrical Engineering and Computer Science UC Berkeley, Berkeley CA 94720 wainwrig@{stat,eecs}.berkeley.edu Abstract Consider the problem of joint parameter estimation and prediction in a Markov random field: i.e., the model parameters are estimated on the basis of an initial set of data, and then the fitted model is used to perform prediction (e.g., smoothing, denoising, interpolation) on a new noisy observation. Working in the computation-limited setting, we analyze a joint method in which the same convex variational relaxation is used to construct an M-estimator for fitting parameters, and to perform approximate marginalization for the prediction step. The key result of this paper is that in the computation-limited setting, using an inconsistent parameter estimator (i.e., an estimator that returns the “wrong” model even in the infinite data limit) is provably beneficial, since the resulting errors can partially compensate for errors made by using an approximate prediction technique. En route to this result, we analyze the asymptotic properties of M-estimators based on convex variational relaxations, and establish a Lipschitz stability property that holds for a broad class of variational methods. We show that joint estimation/prediction based on the reweighted sum-product algorithm substantially outperforms a commonly used heuristic based on ordinary sum-product. 1 Keywords: Markov random fields; variational method; message-passing algorithms; sum-product; belief propagation; parameter estimation; learning. 1 Introduction Consider the problem of joint learning (parameter estimation) and prediction in a Markov random field (MRF): in the learning phase, an initial collection of data is used to estimate parameters, and the fitted model is then used to perform prediction (e.g., smoothing, interpolation, denoising) on a new noisy observation. Disregarding computational cost, there exist optimal methods for solving this problem (Route A in Figure 1). For general MRFs, however, optimal methods are computationally intractable; consequently, many researchers have examined various types of message-passing methods for learning and prediction problems, including belief propagation [3, 6, 7, 14], expectation propagation [5], linear response [4], as well as reweighted message-passing algorithms [10, 13]. Accordingly, it is of considerable interest to understand and quantify the performance loss incurred 1Work partially supported by Intel Corporation Equipment Grant 22978, an Alfred P. Sloan Foundation Fellowship, and NSF Grant DMS-0528488. by using computationally tractable methods versus exact methods (i.e., Route B versus A in Figure 1). APPROXIMATE PARAMETER ESTIMATION ESTIMATION OPTIMAL PARAMETER APPROXIMATE PREDICTION OPTIMAL PREDICTION DATA SOURCE PREDICTION PREDICTION ROUTE A ROUTE B NEW OBSERVATIONS {xi} bθ θ∗ bz(y, µ; θ∗) bz(y, τ; bθ) y Error Figure 1. Route A: computationally intractable combination of parameter estimation and prediction. Route B: computationally efficient combination of approximate parameter estimation and prediction. It is now well known that many message-passing algorithms—including mean field, (generalized) belief propagation, expectation propagation and various convex relaxations—can be understood from a variational perspective; in particular, all of these message-passing algorithms are iterative methods solving relaxed forms of an exact variational principle [12]. This paper focuses on the analysis of variational methods based convex relaxations, which includes a broad range of extant algorithms—among them the tree-reweighted sum-product algorithm [11], reweighted forms of generalized belief propagation [13], and semidefinite relaxations [12]. Moreover, it is straightforward to modify other message-passing methods (e.g., expectation propagation [5]) so as to “convexify” them. At a high level, the key idea of this paper is the following: given that approximate methods can lead to errors at both the estimation and prediction phases, it is natural to speculate that these sources of error might be arranged to partially cancel one another. Our theoretical analysis confirms this intuition: we show that with respect to end-to-end performance, it is in fact beneficial, even in the infinite data limit, to learn the “wrong” the model by using an inconsistent parameter estimator. More specifically, we show how any convex variational method can be used to define a surrogate likelihood function. We then investigate the asymptotic properties of parameter estimators based maximizing such surrogate likelihoods, and establish that they are asymptotically normal but inconsistent in general. We then prove that any variational method that is based on a strongly concave entropy approximation is globally Lipschitz stable. Finally, focusing on prediction for a coupled mixture of Gaussians, we prove upper bounds on the increase in MSE of our computationally efficient method, relative to the unachievable Bayes optimum. We provide experimental results using the tree-reweighted (TRW) sumproduct algorithm that confirm the stability of our methods, and demonstrate its superior performance to a heuristic method based on standard sum-product. 2 Background We begin with necessary notation and background on multinomial Markov random fields, as well as variational representations and methods. Markov random fields: Given an undirected graph G = (V, E) with N = |V | vertices, we associate to each vertex s ∈V a discrete random variable Xs, taking values in Xs = {0, 1 . . . , m −1}. We assume that the vector X = {Xs | s ∈V } has a distribution that is Markov with respect to the graph G, so that its distribution can be represented in the form p(x; θ) = exp{ X s∈V θs(xs) + X (s,t)∈E θst(xs, xt) −A(θ)} (1) Here A(θ) := log P x∈X N exp P s∈V θs(xs) + P (s,t)∈E θst(xs, xt) is the cumulant generating function that normalizes the distribution, and θs(·) and θst(·, ·) are potential functions. In particular, we make use of the parameterization θs(xs) := P j∈Xs θs;jI j[xs], where I j[xs] is an indicator function for the event {xs = j}; the quantity θst is defined analogously. Overall, the family of MRFs (1) is an exponential family with canonical parameter θ ∈Rd. Note that the elements of the canonical parameters are associated with vertices {θs;j, s ∈V, j ∈Xs} and edges {θst;jk, (s, t) ∈E, (j, k) ∈Xs × Xt} of the underlying graph. Variational representation: We now describe how the cumulant generating function can be represented as the solution of an optimization problem. The constraint set is given by MARG(G; φ) := µ ∈Rd | µ = P x∈X N p(x)φ(x) for some p(·) , consisting of all globally realizable singleton µs(·) and pairwise µst(· , ·) marginal distributions on the graph G. For any µ ∈MARG(G; φ), we define A∗(µ) = −maxp H(p), where the maximum is taken over all distributions that have mean parameters µ. With these definitions, it can be shown [12] that A has the variational representation A(θ) = max µ∈MARG(G;φ) θT µ −A∗(µ) . (2) 3 From convex surrogates to joint estimation/prediction In general, solving the variational problem (2) is intractable for two reasons: (i) the constraint set MARG(G; φ) is extremely difficult to characterize; and (ii) the dual function A∗lacks a closed-form representation. These challenges motivate approximations to A∗ and MARG(G; φ); the resulting relaxed optimization problem defines a convex surrogate to the cumulant generating function. Convex surrogates: Let REL(G; φ) be a compact and convex outer bound to the marginal polytope MARG(G; φ), and let B∗be a strictly convex and twice continuously differentiable approximation to the dual function A∗. We use these approximations to define a convex surrogate B via the relaxed optimization problem B(θ) := max τ∈REL(G;φ) θT τ −B∗(τ) . (3) The function B so defined has several desirable properties. First, since B is defined by the maximum of a collection of functions linear in θ, it is convex [1]. Moreover, by the strict convexity of B∗and compactness of REL(G; φ), the optimum is uniquely attained at some τ(θ). Finally, an application of Danskin’s theorem [1] yields that B is differentiable, and that ∇B(θ) = τ(θ). Since τ(θ) has a natural interpretation as a pseudomarginal, this last property of B is analogous to the well-known cumulant generating property of A—namely, ∇A(θ) = µ(θ). One example of such a convex surrogate is the tree-reweighted Bethe free energy considered in our previous work [11]. For this surrogate, the relaxed constraint set REL(G; φ) takes the form LOCAL(G; φ) := τ ∈Rd + | P xs τs(xs) = 1, P xt τst(xs, xt) = τs(xs) , whereas the entropy approximation B∗is of the “convexified” Bethe form −B∗(τ) = X s∈V Hs(τs) − X (s,t)∈E ρstIst(τst). (4) Here Hs and Ist are the singleton entropy and edge-based mutual information, respectively, and the weights ρst are derived from the graph structure so as to ensure convexity (see [11] for more details). Analogous convex variational formulations underlie the reweighted generalized BP algorithm [13], as well as a log-determinant relaxation [12]. Approximate parameter estimation using surrogate likelihoods: Consider the problem of estimating the parameter θ using i.i.d. samples {x1, . . . , xn}. For an MRF of the form (1), the maximum likelihood estimate (MLE) is specified using the vector bµ of empirical marginal distributions (singleton bµs and pairwise bµst). Since the likelihood is intractable to optimize (due to the cumulant generating function A), it is natural to use the convex surrogate B to define an alternative estimator obtained by maximizing the regularized surrogate likelihood: bθn := arg max θ∈Rd LB(θ; bµ) −λnR(θ) = arg max θ∈Rd θT bµ −B(θ) −λnR(θ) . (5) Here R : Rd →R+ is a regularization function (e.g., R(θ) = ∥θ∥2), whereas λn > 0 is a regularization coefficient. For the tree-reweighted Bethe surrogate, we have shown in previous work [10] that in the absence of regularization, the optimal parameter estimates bθn have a very simple closed-form solution, specified in terms of the weights ρst and the empirical marginals bµ. If a regularizing term is added, these estimates no longer have a closed-form solution, but the optimization problem (5) can still be solved efficiently by message-passing methods. Joint estimation/prediction: Using such an estimator, we now consider the joint approach to estimation and prediction illustrated in Figure 2. Using an initial set of i.i.d. samples, we first use the surrogate likelihood (5) to construct a parameter estimate bθn. Given a new noisy or incomplete observation y, we wish to perform near-optimal prediction or data fusion using the fitted model (e.g., for smoothing or interpolation of a noisy image). In order to do so, we first incorporate the new observation into the model, and then use the message-passing algorithm associated with the convex surrogate B in order to compute approximate pseudomarginals τ. These pseudomarginals can then be used to construct a prediction bz(y; τ), where the specifics of the prediction depend on the observation model. We provide a concrete illustration in Section 5 using a mixture-of-Gaussians observation model. 4 Analysis Asymptotics of estimator: We begin by considering the asymptotic behavior of the parameter estmiator bθn defined by the surrogate likelihood (5). Since this parameter estimator is a particular type of M-estimator, the following result follows from standard techniques [8]: Proposition 1. For a general graph with cycles, bθn converges in probability to some fixed bθ ̸= θ∗; moreover, √n[bθn −bθ] is asymptotically normal. A key property of the estimator is its inconsistency—i.e., the estimated model bθ differs from the true model θ∗even in the limit of large data. Despite this inconsistency, we will see that bθn is useful for performing prediction. Algorithmic stability: A desirable property of any algorithm—particularly one applied to statistical data—is that it exhibit an appropriate form of stability with respect to its inputs. Not all message-passing algorithms have such stability properties. For instance, the standard BP algorithm, although stable for relatively weakly coupled MRFs [3, 6], can be highly unstable due to phase transitions. Previous experimental work has shown that methods based on convex relaxations, including reweighted belief propagation [10], Generic algorithm for joint parameter estimation and prediction: 1. Estimate parameters bθn from initial data x1, . . . , xn by maximizing surrogate likelihood LB. 2. Given a new set of observations y, incorporate them into the model: eθs( · ; ys) = bθn s ( · ) + log p(ys | · ). (6) 3. Compute approximate marginals τ by using the message-passing algorithm associated with the convex surrogate B. Use approximate marginals to construct prediction bz(y; τ) of z based on the observation y and pseudomarginals τ. Figure 2. Algorithm for joint parameter estimation and prediction. Both the learning and prediction steps are approximate, but the key is that they are both based on the same underlying convex surrogate B. Such a construction yields a provably beneficial cancellation of the two sources of error (learning and prediction). reweighted generalized BP [13], and log-determinant relaxations [12] appear to be very stable. Here we provide theoretical support for these empirical observations: in particular, we prove that, in sharp contrast to non-convex methods, any variational method based on a strongly convex entropy approximation is globally stable. A function f : Rn →R is strongly convex if there exists a constant c > 0 such that f(y) ≥f(x) + ∇f(x)T y −x) + c 2∥y −x∥2 for all x, y ∈Rn. For a twice continuously differentiable function, this condition is equivalent to the eigenspectrum of the Hessian ∇2f(x) being uniformly bounded away from zero by c. With this definition, we have: Proposition 2. Consider any variational method based on a strongly concave entropy approximation −B∗; moreover, for any parameter θ ∈Rd, let τ(θ) denote the associated set of pseudomarginals. If the optimum is attained interior of the constraint set, then there exists a constant R < +∞such that ∥τ(θ + δ) −τ(θ)∥ ≤ R∥δ∥ for all θ, δ ∈Rd. Proof. By our construction of the convex surrogate B, we have τ(θ) = ∇B(θ), so that the statement is equivalent to the assertion that the gradient ∇B is a Lipschitz function. Applying the mean value theorem to ∇B, we can write ∇B(θ + δ) −∇B(θ) = ∇2B(θ + tδ)δ where t ∈[0, 1]. Consequently, in order to establish the Lipschitz condition, it suffices to show that the spectral norm of ∇2B(γ) is uniformly bounded above over all γ ∈Rd. Differentiating the relation ∇B(θ) = τ(θ) yields ∇2B(θ) = ∇τ(θ). Now standard sensitivity analysis results [1] yield that ∇τ(θ) = [∇2B∗(τ(θ)]−1. Finally, our assumption of strong convexity of B∗yields that the spectral norm of ∇2B∗(τ) is uniformly bounded away from zero, which yields the claim. Many existing entropy approximations, including the convexifed Bethe entropy (4), can be shown to be strongly concave [9]. 5 Bounds on performance loss We now turn to theoretical analysis of the joint method for parameter estimation and prediction illustrated in Figure 2. Note that given our setting of limited computation, the Bayes optimum is unattainable for two reasons: (a) it has knowledge of the exact parameter value θ∗; and (b) the prediction step (7) involves computing exact marginal probabilities µ. Therefore, our ultimate goal is to bound the performance loss of our method relative to the unachievable Bayes optimum. So as to obtain a concrete result, we focus on the special case of joint learning/prediction for a mixture-of-Gaussians; however, the ideas and techniques described here are more generally applicable. Prediction for mixture of Gaussians: Suppose that the discrete random vector is a label vector for the components in a finite mixture of Gaussians: i.e., for each s ∈V , the random variable Zs is specified by p(Zs = zs | Xs = j; θ∗) ∼N(νj, σ2 j ), for j ∈{0, 1, . . . , m − 1}. Such models are widely used in statistical signal and image processing [2]. Suppose that we observe a noise-corrupted version of Zs—namely Ys = αZs + √ 1 −α2Ws, where Ws ∼N(0, 1) is additive Gaussian noise, and the parameter α ∈[0, 1] specifies the signalto-noise ratio (SNR) of the observation model. (Here α = 0 corresponds to pure noise, whereas α = 1 corresponds to completely uncorrupted observations.) With this set-up, it is straightforward to show that the optimal Bayes least squares estimator (BLSE) of Z takes the form bzs(y; µ) := m−1 X j=0 µs(j; θ∗) ωj(α) ys −νj + νj , (7) where µs(j; θ∗) is the exact marginal of the distribution p(y | x)p(x; θ∗); and ωj(α) := ασ2 j α2σ2 j +(1−α2) is the usual BLSE weighting for a Gaussian with variance σj. For this set-up, the approximate predictor bzs(y; τ) defined by our joint procedure in Figure 2 corresponds to replacing the exact marginals µ with the pseudomarginals τs(j; eθ) obtained by solving the variational problem with eθ. Bounds on performance loss: We now turn to a comparison of the mean-squared error (MSE) of the Bayes optimal predictor bz(Y ; µ) to the MSE of the surrogate-based predictor bz(Y ; τ). More specifically, we provide an upper bound on the increase in MSE, where the bound is specified in terms of the coupling strength and the SNR parameter α. Although results of this nature can be derived more generally, for simplicity we focus on the case of two mixture components (m = 2), and consider the asymptotic setting, in which the number of data samples n →+∞, so that the law of large numbers [8] ensures that the empirical marginals bµn converge to the exact marginal distributions µ∗. Consequently, the MLE converges to the true parameter value θ∗, whereas Proposition 1 guarantees that our approximate parameter estimate bθn converges to the fixed quantity bθ. By construction, we have the relations ∇B(bθ) = µ∗= ∇A(θ∗). An important factor in our bound is the quantity L(θ∗; bθ) := sup δ∈Rd σmax ∇2A(θ∗+ δ) −∇2B(bθ + δ) , (8) where σmax denotes the maximal singular value. Following the argument in the proof of Proposition 2, it can be seen that L(θ∗; bθ) is finite. Two additional quantities that play a role in our bound are the differences ∆ω(α) := ω1(α) −ω0(α), and ∆ν(α) := [1 −ω1(α)]ν1 −[1 −ω0(α)]ν0, where ν0, ν1 are the means of the two Gaussian components. Finally, we define γ(Y ; α) ∈ Rd with components log p(Ys|Xs=1) p(Ys|Xs=0) for s ∈V , and zeroes otherwise. With this notation, we state the following result (see the technical report [9] for the proof): Theorem 1. Let MSE(τ) and MSE(µ) denote the mean-squared prediction errors of the surrogate-based predictor bz(y; τ), and the Bayes optimal estimate bz(y; µ) respectively. The MSE increase I(α) := 1 N MSE(τ) −MSE(µ) is upper bounded by I(α) ≤ E ( Ω2(α)∆2 ν(α) + Ω(α) h ∆2 ω(α) rP s Y 4 s N + 2|∆ν(α)| |∆ω(α)| rP s Y 2 s N i) where Ω(α) := min{1, L(θ∗; bθ)∥γ(Y ;α) N ∥}. 0 0.5 1 0 0.5 1 0 10 20 30 40 50 Edge strength IND SNR Performance loss 0 0.5 1 0 0.5 1 0 10 20 30 40 50 Edge strength BP SNR Performance loss 0 0.5 1 0 0.5 1 0 10 20 30 40 50 Edge strength TRW SNR Performance loss (a) (b) (c) 0 0.5 1 0 0.5 1 0 1 2 3 4 5 Edge strength IND SNR Performance loss 0 0.5 1 0 0.5 1 0 1 2 3 4 5 Edge strength BP SNR Performance loss 0 0.5 1 0 0.5 1 0 1 2 3 4 5 Edge strength TRW SNR Performance loss (d) (e) (f) Figure 3. Surface plots of the percentage increase in MSE relative to Bayes optimum for different methods as a function of observation SNR and coupling strength. Top row: Gaussian mixture with components (ν0, σ2 0) = (−1, 0.5) and (ν1, σ2 1) = (1, 0.5). Bottom row: Gaussian mixture with components (ν0, σ2 0) = (0, 1) and (ν0, σ2 1) = (0, 9). Left column: independence model (IND). Center column: ordinary belief propagation (BP). Right column: tree-reweighted algorithm (TRW). It can be seen that I(α) →0 as α →0+ and as α →1−, so that the surrogate-based method is asymptotically optimal for both low and high SNR. The behavior of the bound in the intermediate regime is controlled by the balance between these two terms. Experimental results: In order to test our joint estimation/prediction procedure, we have applied it to coupled Gaussian mixture models on different graphs, coupling strengths, observation SNRs, and mixture distributions. Although our methods are more generally applicable, here we show representative results for m = 2 components, and two different mixture types. The first ensemble, constructed with mean and variance components (ν0, σ2 0) = (0, 1) and (ν1, σ2 1) = (0, 9), mimics heavy-tailed behavior. The second ensemble is bimodal, with components (ν0, σ2 0) = (−1, 0.5) and (ν1, σ2 1) = (1, 0.5). In both cases, each mixture component is equally weighted. Here we show results for a 2-D grid with N = 64 nodes. Since the mixture variables have m = 2 states, the coupling distribution can be written as p(x; θ) ∝exp P s∈V θsxs + P (s,t)∈E θstxsxt . where x ∈{−1, +1}N are spin variables indexing the mixture components. In all trials, we chose θs = 0 for all nodes s ∈V , which ensures uniform marginal distributions p(xs; θ) at each node. For each coupling strength γ ∈[0, 1], we chose edge parameters as θst ∼U[0, γ], and we varied the SNR parameter α controlling the observation model in [0, 1]. We evaluated the following three methods based on their increase in mean-squared error (MSE) over the Bayes optimal predictor (7): (a) As a baseline, we used the independence model for the mixture components: parameters are estimated θs(xs) = log bµs(xs), and setting coupling terms θst(xs, xt) equal to zero. The prediction step reduces to performing BLSE at each node independently. (b) The standard belief propagation (BP) approach is based on estimating parameters (see step (1) of Figure 2) using ρst = 1 for all edges (s, t), and using BP to compute the pseudomarginals. (c) The tree-reweighted method (TRW) is based on estimating parameters using the tree-reweighted surrogate [10] with weights ρst = 1 2 for all edges (s, t), and using the TRW sum-product algorithm to compute the pseudomarginals. Shown in Figure 3 are 2-D surface plots of the average percentage increase in MSE, taken over 100 trials, as a function of the coupling strength γ ∈[0, 1] and the observation SNR parameter α ∈[0, 1] for the independence model (left column), BP approach (middle column) and TRW method (right column). For weak coupling (γ ≈0), all three methods— including the independence model—perform quite well, as should be expected given the weak dependency. Although not clear in these plots, BP outperforms TRW for weak coupling; however, both methods lose than than 1% in this regime. As the coupling is increased, the BP method eventually deteriorates quite seriously; indeed, for large enough coupling and low/intermediate SNR, its performance can be worse than the independence model. Looking at alternative models (in which phase transitions are known), we have found that this rapid degradation co-incides with the appearance of multiple fixed points. In contrast, the behavior of the TRW method is extremely stable, consistent with our theory. 6 Conclusion We have described and analyzed joint methods for parameter estimation and prediction/smoothing using variational methods that are based on convex surrogates to the cumulant generating function. Our results—both theoretical and experimental—confirm the intuition that in the computation-limited setting, in which errors arise from approximations made both during parameter estimation and subsequent prediction, it is provably beneficial to use an inconsistent parameter estimator. Our experimental results on the coupled mixture of Gaussian model confirm the theory: the tree-reweighted sum-product algorithm yields prediction results close to the Bayes optimum, and substantially outperforms an analogous but heuristic method based on standard belief propagation. References [1] D. Bertsekas. Nonlinear programming. Athena Scientific, Belmont, MA, 1995. [2] M. Crouse, R. Nowak, and R. Baraniuk. Wavelet-based statistical signal processing using hidden Markov models. IEEE Trans. Signal Processing, 46:886–902, April 1998. [3] A. Ihler, J. Fisher, and A. S. Willsky. Loopy belief propagation: Convergence and effects of message errors. Journal of Machine Learning Research, 6:905–936, May 2005. [4] M. A. R. Leisink and H. J. Kappen. Learning in higher order Boltzmann machines using linear response. Neural Networks, 13:329–335, 2000. [5] T. P. Minka. A family of algorithms for approximate Bayesian inference. PhD thesis, MIT, January 2001. [6] S. Tatikonda and M. I. Jordan. Loopy belief propagation and Gibbs measures. In Proc. Uncertainty in Artificial Intelligence, volume 18, pages 493–500, August 2002. [7] Y. W. Teh and M. Welling. On improving the efficiency of the iterative proportional fitting procedure. In Workshop on Artificial Intelligence and Statistics, 2003. [8] A. W. van der Vaart. Asymptotic statistics. Cambridge University Press, Cambridge, UK, 1998. [9] M. J. Wainwright. Joint estimation and prediction in Markov random fields: Benefits of inconsistency in the computation-limited regime. Technical Report 690, Department of Statistics, UC Berkeley, 2005. [10] M. J. Wainwright, T. S. Jaakkola, and A. S. Willsky. Tree-reweighted belief propagation algorithms and approximate ML estimation by pseudomoment matching. In Workshop on Artificial Intelligence and Statistics, January 2003. [11] M. J. Wainwright, T. S. Jaakkola, and A. S. Willsky. A new class of upper bounds on the log partition function. IEEE Trans. Info. Theory, 51(7):2313–2335, July 2005. [12] M. J. Wainwright and M. I. Jordan. A variational principle for graphical models. In New Directions in Statistical Signal Processing. MIT Press, Cambridge, MA, 2005. [13] W. Wiegerinck. Approximations with reweighted generalized belief propagation. In Workshop on Artificial Intelligence and Statistics, January 2005. [14] J. Yedidia, W. T. Freeman, and Y. Weiss. Constructing free energy approximations and generalized belief propagation algorithms. IEEE Trans. Info. Theory, 51(7):2282–2312, July 2005.
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Gaussian Processes for Multiuser Detection in CDMA receivers Juan Jos´e Murillo-Fuentes, Sebastian Caro Dept. Signal Theory and Communications University of Seville {murillo,scaro}@us.es Fernando P´erez-Cruz Gatsby Computational Neuroscience University College London fernando@gatsby.ucl.ac.uk Abstract In this paper we propose a new receiver for digital communications. We focus on the application of Gaussian Processes (GPs) to the multiuser detection (MUD) in code division multiple access (CDMA) systems to solve the near-far problem. Hence, we aim to reduce the interference from other users sharing the same frequency band. While usual approaches minimize the mean square error (MMSE) to linearly retrieve the user of interest, we exploit the same criteria but in the design of a nonlinear MUD. Since the optimal solution is known to be nonlinear, the performance of this novel method clearly improves that of the MMSE detectors. Furthermore, the GP based MUD achieves excellent interference suppression even for short training sequences. We also include some experiments to illustrate that other nonlinear detectors such as those based on Support Vector Machines (SVMs) exhibit a worse performance. 1 Introduction One of the major issues in present wireless communications is how users share the resources. And particularly, how they access to a common frequency band. Code division multiple access (CDMA) is one of the techniques exploited in third generation communications systems and is to be employed in the next generation. In CDMA each user uses direct sequence spread spectrum (DS-SS) to modulate its bits with an assigned code, spreading them over the entire frequency band. While typical receivers deal only with interferences and noise intrinsic to the channel (i.e. Inter-Symbolic Interference, intermodulation products, spurious frequencies, and thermal noise), in CDMA we also have interference produced by other users accessing the channel at the same time. Interference limitation due to the simultaneous access of multiple users systems has been the stimulus to the development of a powerful family of Signal Processing techniques, namely Multiuser Detection (MUD). These techniques have been extensively applied to CDMA systems. Thus, most of last generation digital communication systems such as Global Positioning System (GPS), wireless 802.11b, Universal Mobile Telecommunication System (UMTS), etc, may take advantage of any improvement on this topic. In CDMA, we face the retrieval of a given user, the user of interest (UOI), with the knowledge of its associated code or even the whole set of users codes. Hence, we face the suppression of interference due to others users. If all users transmit with the same power, ĹM ĹM ĹM . . . h1(z) h2(z) hK(z) . . . bt(1) bt(2) bt(K) C(z) Noise nt Channel Code filters Chip rate sampler MUD Figure 1: Synchronous CDMA system but the UOI is far from the receiver, most users reach the receiver with a larger amplitude, making it more difficult to detect the bits of the UOI. This is well-known as the near-far problem. Simple detectors can be designed by minimizing the mean square error (MMSE) to linearly retrieve the user of interest [5]. However, these detectors need large sequences of training data. Besides, the optimal solution is known to be nonlinear. There has been several attempts to solve the problem using nonlinear techniques. There are solutions based on Neural Networks such as multilayer perceptron or radial basis functions [1, 3], but training times are long and unpredictable. Recently, support vector machines (SVM) have been also applied to CDMA MUD [4]. This solution need very long training sequences (a few hundreds bits) and they are only tested in toy examples with very few users and short spreading sequences (the code for each user). In this paper, we will present a multiuser detector based on Gaussian Processes [7]. The MUD detector is inspired by the linear MMSE criteria, which can be interpreted as a Bayesian linear regressor. In this sense, we can extend the linear MMSE criteria to nonlinear decision functions using the same ideas developed in [6] to present Gaussian Processes for regression. The rest of the paper is organised as follows. In Section 2, we present the multiuser detection problem in CDMA communication systems and the widely used minimum mean square error receiver. We propose a nonlinear receiver based on Gaussian Processes in Section 3. Section 4 is devoted to show, through computer experiments, the advantages of the GP-MUD receiver with short training sequences. We compare it to the linear MMSE and the nonlinear SVM MUD. We conclude the paper in Section 5 presenting some remarks and future work. 2 CDMA Communication System Model and MUD Consider a synchronous CDMA digital communication system [5] as depicted in Figure 1. Its main goal is to share the channel between different users, discriminating between them by different assigned codes. Each transmitted bit is upsampled and multiplied by the users’ spreading codes and then the chips for each bit are transmitted into the channel (each element of the spreading code is either +1 or −1 and they are known as chips). The channel is assumed to be linear and noisy, therefore the chips from different users are added together, plus Gaussian noise. Hence, the MUD has to recover from these chips the bits corresponding to each user. At each time step t, the signal in the receiver can be represented in matrix notation as: xt = HAbt + nt (1) where bt is a column vector that contains the bits (+1 or −1) for the K users at time k. The K × K diagonal matrix A contains the amplitude of each user, which represents the attenuation that each user’s transmission suffers through the channel (this attenuation depends on the distance between the user and the receiver). H is an L × K matrix which contains in each column the L-dimensional spreading code for each of the K users. The spreading codes are designed to present a low cross-correlation between them and between any shifted version of the codes, to guarantee that the bits from each user can be readily recovered. The codes are known as spreading sequences, because they augment the occupied bandwidth of the transmitted signal by L. Finally, xt represents the L received chips to which Gaussian noise has been added, which is denoted by nt. At reception, we aim to estimate the original transmitted symbols of any user i, bt(i), hereafter the user of interest. Linear MUDs estimate these bits as ˆbt(i) = sgn{w⊤ i xt} (2) The matched filter (MF) wi = hi, a simple correlation between xt and the ith spreading code, is the optimal receiver if there were no additional users in the system, i.e. the received signal is only corrupted by Gaussian noise. The near-far problem arises when remaining users, apart from the UOI, are received with significantly higher amplitude. While the optimal solution is known to be nonlinear [5], some linear receivers such as the minimum mean square error (MMSE) present good performances and are used in practice. The MMSE receiver for the ith user solves: w∗ i = arg min wi E (bt(i) −w⊤ i xt)2 = arg min wi E (bt(i) −w⊤ i (HAbt + νk))2 (3) where wi represents the decision function of the linear classifier. We can derive the MMSE receiver by taking derivatives with respect to wi and equating to zero, obtaining: wMMSEde i = R−1 xx hi (4) where Rxx = E[xtx⊤ t ] is the correlation between the received vectors and hi represents the spreading sequence of the UOI. This receiver is known as the decentralized MMSE receiver as it can be implemented without knowing the spreading sequences of the remaining users. Its main limitation is its performance, which is very low even for high signal to noise ratio, and it needs many examples (thousands) before it can recover the received symbols. If the spreading codes of all the users are available, as in the base station, this information can be used to improve the performance of the MMSE detector. We can define zk = H⊤xt, which is a vector of sufficient statistics for this problem [5]. The vector zk is the matched-filter output for each user and it reduces the dimensionality of our problem from the number of chips L to the number of users K, which is significantly lower in most applications. In this case the receiver is known as the centralized detector and it is defined as: wMMSEcent i = HR−1 zz H⊤hi (5) where Rzz = E[ztz⊤ t ] is the correlation matrix of the received chips after the MFs. These MUDs have good convergence properties and do not need a training sequence to decode the received bits, but they need large training sequences before their probability of error is low. Therefore the initially received bits will present a very high probability of error that will make impossible to send any information on them. Some improvements can be achieved by using higher order statistics [2], but still the training sequences are not short enough for most applications. 3 Gaussian Processes for Multiuser Detection The MMSE detector minimizes the functional in (3), which gives the best linear classifier. As we know, the optimal classifier is nonlinear [5], and the MMSE criteria can be readily extended to provide nonlinear models by mapping the received chips to a higher dimensional space. In this case we will need to solve: w∗ i = arg min wi ( N X k=1 bt(i) −w⊤ i φ(xt) 2 + λ∥wi∥2 ) (6) in which we have changed the expectation by the empirical mean over a training set and we have incorporated a regularizer to avoid overfitting. φ(·) represents the nonlinear mapping of the received chips. The wi that minimizes (6) can be interpreted as the mode of the parameters in a Bayesian linear regressor, as noted in [6], and since the likelihood and the prior are both Gaussians, so it will be the posterior. For any received symbol x∗, we know that it will be distributed as a Gaussian with mean: µ(x∗) = 1 λφ⊤(x∗)A−1Φ⊤b (7) and variance σ2(x∗) = φ⊤(x∗)A−1φ(x∗) (8) where Φ = [φ(x1), φ(x2), . . . , φ(xN)]⊤, b = [b1(i), b2(i), . . . , bN(i)]⊤and A = Φ⊤Φ + 1 λI. In the case the nonlinear mapping is unknown, we can still obtain the mean and variance for each received sample using the kernel of the transformation, being the mean: µ(x∗) = k⊤P−1b (9) and variance σ2(x∗) = k(x∗, x∗) + k⊤P−1k (10) where k(·, ·) = φ⊤(·)φ(·) is the kernel of the nonlinear transformation, k = [k(x∗, x1), k(x∗, x2), . . . , k(x∗, xN)], and P = ΦΦ⊤+ λI = K + λI (11) where (K)kℓ= k(xt, xℓ). The kernel that we will use in our experiments are: k(xt, xℓ) = eθ[1] exp(−eθ[4]∥xt −xℓ∥2) + eθ[3]x⊤ t xℓ+ eθ[2]δr,ℓ (12) The covariance function in (12) is a good kernel for solving the GP-MUD, because it contains a linear and a nonlinear part. The optimal decision surface for MUD is nonlinear, unless the spreading codes are orthogonal to each other, and its deviation from the linear solution depends on how strong the correlations between codes are. In most cases, a linear detector is very close to the optimal decision surface, as spreading codes are almost orthogonal, and only a minor correction is needed to achieve the optimal decision boundary. In this sense the proposed GP covariance function is ideal for the problem. The linear part can mimic the best linear decision boundary and the nonlinear part modifies it, where the linear explanation is not optimal. Also using a radial basis kernel for the nonlinear part is a good choice to achieve nonlinear decisions. Because, the received chips form a constellation of 2K clouds of points with Gaussian spread around its centres. Picturing the receiver as a Gaussian Process for regression, instead of a Regularised Least Square functional, allows us to either obtain the hyperparameters by maximizing the likelihood or marginalised them out using Monte Carlo techniques, as explained in [6]. For the −2 0 2 4 6 8 10 12 14 10 −4 10 −3 10 −2 10 −1 10 0 SNR(dB) BER BER vs SNR Figure 2: Bit Error Rate versus Signal to Noise ratio for the MF (▽), MMSECentralized (□), MMSE-Decentralized (◦), SVM-centralized (⊲), GP-Centralized (⋄) and GP-Decentralized (∗) with k = 8 users and n = 30 training samples. The powers of the interfering users is distributed homogeneously between 0 and 30 dB above that of the UOI. problem at hand speed is a must and we will be using the maximum likelihood hyperparameters. We have just shown above how we can make predictions in the nonlinear case (9) using the received symbols from the channel. In an analogy with the MMSE receiver, this will correspond to the decentralized GP-MUD detector as we will not need to know the other users’ codes to detect the bits sent to us. It is also relevant to notice that we do not need our spreading code for detection, as the decentralized MMSE detector did. We can also obtain a centralized GP-MUD detector using as input vectors zt = H⊤xt. 4 Experiments In this section we include the typical evaluation of the performance in a digital communications system, i.e., Bit Error Rate (BER). The test environment is a synchronous CDMA system in which the users are spread using Gold sequences with spreading factor L = 31 and K = 8 users, which are typical values in CDMA based mobile communication systems. We consider the same amplitude matrix in all experiments. These amplitudes are random values to achieve an interferer to signal ratio of 30 dB. Hence, the interferers are 30 dB over the UOI. We study the worse scenario and hence we will detect the user which arrives to the receiver with the lowest amplitude. We compare the performance of the GP centralized and decentralized MUDs to the performance of the MMSE detectors, the Matched Filter detector and the (centralized) SVMMUD in [4]. The SVM-MUD detector uses a Gaussian kernel and its width is adapted incorporating knowledge of the noise variance in the channel. We found that this setting −2 0 2 4 6 8 10 12 14 10 −4 10 −3 10 −2 10 −1 10 0 SNR(dB) BER BER vs SNR Figure 3: Bit Error Rate versus Signal to Noise ratio for the MF (▽), MMSECentralized (□), MMSE-Decentralized (◦), SVM-centralized (⊲), GP-Centralized (⋄) and GP-Decentralized (∗) with k = 8 users and n = 80 training samples. The powers of the interfering users is distributed homogeneously between 0 and 30 dB above that of the UOI. usually does not perform well for this experimental specification and we have set them using validation. We believe this might be due to either the reduced number of users in their experiments (2 or 3) or because they used the same amplitude for all the users, so they did not encounter the near-far problem. We have included three experiments in which we have defined the number of training experiments equal to 30, 80 and 160. For each training set we have computed the BER for 106 bits. The reported results are mean curves for 50 different trials. The results in Figure 2 show that the detectors based on GPs are able to reduce the probability of error as the signal to noise ratio in the channel decreases with only 30 samples in the training sequence. The GP centralized MUD is only 1.5-2dB worse than the best achievable probability of error, which is obtained in absence of interference (indicated by the dashed line). The GP decentralized MUD reduces the probability of error as the signal to noise increases, but it remains between 3-4dB from the optimal performance. The other detectors are not able to decrease the BER even for a very high signal to noise ratio in the channel. These figures show that the GP based MUD can outperform the other MUD when very short training sequences are available. Figure 3 highlights that the SVM-MUD (centralized) and the MSSE centralized detectors are able to reduce the BER as the SNR increases, but they are still far from the performance of the GP-MUD. The centralized GP-MUD basically provides optimal performance as it is less than 0.3db from the possible achieved BER when there is no interference in the channel. The decentralized GP-MUD outperforms the other two centralized detectors (SVM and MMSE) since it is able to provide lower BER without needing to know the code of the remaining users. −2 0 2 4 6 8 10 12 14 10 −4 10 −3 10 −2 10 −1 10 0 SNR(dB) BER BER vs SNR Figure 4: Bit Error Rate versus Signal to Noise ratio for the MF (▽), MMSECentralized (□), MMSE-Decentralized (◦), SVM-centralized (⊲), GP-Centralized (⋄) and GP-Decentralized (∗) with k = 8 users and n = 160 training samples. The powers of the interfering users is distributed homogeneously between 0 and 30 dB above that of the UOI. Finally, in Figure 4 we include the results for 160 training samples. In this case, the centralized GP-MUD lies above the optimal BER curve and the decentralized GP-MUD performs as the SVM-MUD detector. The centralized MMSE detector still presents very high probability of error for high signal to noise ratios and we need over 500 samples to obtain a performance similar to the centralized GP with 80 samples. For 160 samples the MMSE decentralized is already able to slightly reduce the bit error rate for very high signal to noise ratios. But to achieve the performance showed by the decentralized GP-MUD it needs several thousands samples. 5 Conclusions and Further Work We propose a novel approach based on Gaussian Processes for regression to solve the nearfar problem in CDMA receivers. Since the optimal solution is known to be nonlinear the Gaussian Processes are able to obtain this nonlinear decision surface with very few training examples. This is the main advantage of this method as it only requires a few tens training examples instead of the few hundreds needed by other nonlinear techniques as SVMs. This will allow its application in real communication systems, as training sequence of 26 samples are typically used in the GSM standard for mobile Telecommunications. The most relevant result of this paper is the performance shown by the decentralized GPMUD receiver, since it can be directly used over any CDMA system. The decentralized GP-MUD receiver does not need to know the codes from the other users and does not require the users to be aligned, as the other methods do. While the other receiver will degrade its performance if the users are not aligned, the decentralized GP-MUD receiver will not, providing a more robust solution to the near far problem. We have presented some preliminary work, which shows that GPs for regression are suitable for the near-far problem in MUD. We have left for further work a more extensive set of experiments changing other parameters of the system such as: the number of users, the length of the spreading code, and the interferences with other users. But still, we believe the reported results are significant since we obtain low bit error rates for training sequences as short as 30 bits. Acknowledgements Fernando P´erez-Cruz is Supported by the Spanish Ministry of Education Postdoctoral Fellowships EX2004-0698. This work has been partially funded by research grants TIC200302602 and TIC2003-03781 by the Spanish Ministry of Education. References [1] G. C. Orsak B. Aazhang, B. P. Paris. Neural networks for multiuser detection in codedivision multiple-access communications. IEEE Transactions on Communications, 40:1212–1222, 1992. [2] Antonio Caama˜no-Fernandez, Rafael Boloix-Tortosa, Javier Ramos, and Juan J. Murillo-Fuentes. High order statistics in multiuser detection. IEEE Trans. on Man and Cybernetics C. Accepted for publication, 2004. [3] U. Mitra and H. V. Poor. Neural network techniques for adaptive multiuser demodulation. IEEE Journal Selected Areas on Communications, 12:14601470, 1994. [4] L. Hanzo S. Chen, A. K. Samingan. Support vector machine multiuser receiver for DS-CDMA signals in multipath channels. IEEE Transactions on Neural Network, 12(3):604–611, December 2001. [5] S. Verd´u. Multiuser Detection. Cambridge University Press, 1998. [6] C. Williams. Prediction with gaussian processes: From linear regression to linear prediction and beyond. [7] Christopher K. I. Williams and Carl Edward Rasmussen. Gaussian processes for regression. In David S. Touretzky, Michael C. Mozer, and Michael E. Hasselmo, editors, Proc. Conf. Advances in Neural Information Processing Systems, NIPS, volume 8. MIT Press, 1995.
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Metric Learning by Collapsing Classes Amir Globerson School of Computer Science and Engineering, Interdisciplinary Center for Neural Computation The Hebrew University Jerusalem, 91904, Israel gamir@cs.huji.ac.il Sam Roweis Machine Learning Group Department of Computer Science University of Toronto, Canada roweis@cs.toronto.edu Abstract We present an algorithm for learning a quadratic Gaussian metric (Mahalanobis distance) for use in classification tasks. Our method relies on the simple geometric intuition that a good metric is one under which points in the same class are simultaneously near each other and far from points in the other classes. We construct a convex optimization problem whose solution generates such a metric by trying to collapse all examples in the same class to a single point and push examples in other classes infinitely far away. We show that when the metric we learn is used in simple classifiers, it yields substantial improvements over standard alternatives on a variety of problems. We also discuss how the learned metric may be used to obtain a compact low dimensional feature representation of the original input space, allowing more efficient classification with very little reduction in performance. 1 Supervised Learning of Metrics The problem of learning a distance measure (metric) over an input space is of fundamental importance in machine learning [10, 9], both supervised and unsupervised. When such measures are learned directly from the available data, they can be used to improve learning algorithms which rely on distance computations such as nearest neighbour classification [5], supervised kernel machines (such as GPs or SVMs) and even unsupervised clustering algorithms [10]. Good similarity measures may also provide insight into the underlying structure of data (e.g. inter-protein distances), and may aid in building better data visualizations via embedding. In fact, there is a close link between distance learning and feature extraction since whenever we construct a feature f (x) for an input space X, we can measure distances between x 1 ; x 2 2 X using a simple distance function (e.g. Euclidean) d[f (x 1 ); f (x 2 )℄in feature space. Thus by fixing d, any feature extraction algorithm may be considered a metric learning method. Perhaps the simplest illustration of this approach is when the f (x) is a linear projection of x 2 < r so that f (x) = W x. The Euclidean distance between f (x 1 ) and f (x 2 ) is then the Mahalanobis distance kf (x 1 ) f (x 2 )k 2 = (x 1 x 2 ) T A(x 1 x 2 ), where A = W T W is a positive semidefinite matrix. Much of the recent work on metric learning has indeed focused on learning Mahalanobis distances, i.e. learning the matrix A. This is also the goal of the current work. A common approach to learning metrics is to assume some knowledge in the form of equivalence relations, i.e. which points should be close and which should be far (without specifying their exact distances). In the classification setting there is a natural equivalence relation, namely whether two points are in the same class or not. One of the classical statistical methods which uses this idea for the Mahalanobis distance is Fisher’s Linear Discriminant Analysis (see e.g. [6]). Other more recent methods are [10, 9, 5] which seek to minimize various separation criteria between the classes under the new metric. In this work, we present a novel approach to learning such a metric. Our approach, the Maximally Collapsing Metric Learning algorithm (MCML), relies on the simple geometric intuition that if all points in the same class could be mapped into a single location in feature space and all points in other classes mapped to other locations, this would result in an ideal approximation of our equivalence relation. Our algorithm approximates this scenario via a stochastic selection rule, as in Neighborhood Component Analysis (NCA) [5]. However, unlike NCA, the optimization problem is convex and thus our method is completely specified by our objective function. Different initialization and optimization techniques may affect the speed of obtaining the solution but the final solution itself is unique. We also show that our method approximates the local covariance structure of the data, as opposed to Linear Discriminant Analysis methods which use only global covariance structure. 2 The Approach of Collapsing Classes Given a set of n labeled examples (x i ; y i ), where x i 2 < r and y i 2 f1 : : : k g, we seek a similarity measure between two points in X space. We focus on Mahalanobis form metrics d(x i ; x j jA) = d A ij = (x i x j ) T A(x i x j ) ; (1) where A is a positive semidefinite (PSD) matrix. Intuitively, what we want from a good metric is that it makes elements of X in the same class look close whereas those in different classes appear far. Our approach starts with the ideal case when this is true in the most optimistic sense: same class points are at zero distance, and different class points are infinitely far. Alternatively this can be viewed as mapping x via a linear projection W x ( A = W T W), such that all points in the same class are mapped into the same point. This intuition is related to the analysis of spectral clustering [8], where the ideal case analysis of the algorithm results in all same cluster points being mapped to a single point. To learn a metric which approximates the ideal geometric setup described above, we introduce, for each training point, a conditional distribution over other points (as in [5]). Specifically, for each x i we define a conditional distribution over points i 6= j such that p A (j ji) = 1 Z i e d A ij = e d A ij P k 6=i e d A ik i 6= j : (2) If all points in the same class were mapped to a single point and infinitely far from points in different classes, we would have the ideal “bi-level” distribution: p 0 (j ji) / n 1 y i = y j 0 y i 6= y j : (3) Furthermore, under very mild conditions, any set of points which achieves the above distribution must have the desired geometry. In particular, assume there are at least ^ r + 2 points in each class, where ^ r = rank [A℄(note that ^ r r). Then p A (j ji) = p 0 (j ji) ( 8i; j) implies that under A all points in the same class will be mapped to a single point, infinitely far from other class points 1. 1Proof sketch: The infinite separation between points of different classes follows simply from Thus it is natural to seek a matrix A such that p A (j ji) is as close as possible to p 0 (j ji). Since we are trying to match distributions, we minimize the KL divergence KL [p 0 jp℄: min A X i KL[p 0 (j ji)jp A (j ji)℄ s:t: A 2 P S D (4) The crucial property of this optimization problem is that it is convex in the matrix A. To see this, first note that any convex linear combination of feasible solutions A = A 0 + (1 )A 1 s.t. 0 1 is still a feasible solution, since the set of PSD matrices is convex. Next, we can show that f (A) alway has a greater cost than either of the endpoints. To do this, we rewrite the objective function f (A) = P i KL [p 0 (j ji)jp(j ji)℄in the form 2: f (A) = X i;j :y j =y i log p(j ji) = X i;j :y j =y i d A ij + X i log Z i where we assumed for simplicity that classes are equi-probable, yielding a multiplicative constant. To see why f (A) is convex, first note that d A ij = (x i x j ) T A(x i x j ) is linear in A, and thus convex. The function log Z i is a log P exp function of affine functions of A and is therefore also convex (see [4], page 74). 2.1 Convex Duality Since our optimization problem is convex, it has an equivalent convex dual. Specifically, the convex dual of Eq. (4) is the following entropy maximization problem: max p(j ji) X i H [p(j ji)℄ s:t: X i E p 0 (j ji) [v j i v T j i ℄ X i E p(j ji) [v j i v T j i ℄ 0 (5) where v j i = x j x i, H [℄is the entropy function and we require P j p(j ji) = 1 8i. To prove this duality we start with the proposed dual and obtain the original problem in Equation 4 as its dual. Write the Lagrangian for the above problem (where is PSD) 3 L(p; ; ) = X i H (p(j ji)) T r (( X i (E p 0 [v j i v T j i ℄ E p [v j i v T j i ℄))) X i i ( X j p(j ji) 1) The dual function is defined as g (; ) = min p L(p; ; ). To derive it, we first solve for the minimizing p by setting the derivative of L(p; ; ) w.r.t. p(j ji) equal to zero. 0 = 1 + log p(j ji) + T r (v j i v T j i ) i ) p(j ji) = e i 1 T r (v j i v T j i ) Plugging this solution to L(p; ; ) we get g (; ) = T r ( P i E p 0 [v j i v T j i ℄) + P i i P i;j p(j ji). The dual problem is to maximize g (; ). We can do this analytically w.r.t. i, yielding 1 i = log P j e T r (v j i v T j i ). Now note that T r (v j i v T j i ) = v T j i v j i = d j i, so we can write g () = X i;j :y i =y j d j i X i log X j e d j i which is minus our original target function. Since g () should be maximized, and 0 we have the desired duality result (identifying with A). p 0 (j ji) = 0 when y j 6= y i. For a given point x i, all the points j in its class satisfy p(j ji) / 1. Due to the structure of p(j ji) in Equation 2, and because it is obeyed for all points in x 0 i s class, this implies that all the points in that class are equidistant from each other. However, it is easy to show that the maximum number of different equidistant points (also known as the equilateral dimension [1]) in ^ r dimensions is ^ r + 1. Since by assumption we have at least ^ r + 2 points in the class of x i, and A maps points into < ^ r, it follows that all points are identical. 2Up to an additive constant P i H [p 0 (j ji)℄. 3We consider the equivalent problem of minimizing minus entropy. 2.1.1 Relation to covariance based and embedding methods The convex dual derived above reveals an interesting relation to covariance based learning methods. The sufficient statistics used by the algorithm are a set of n “spread” matrices. Each matrix is of the form E p 0 (j ji) [v j i v T j i ℄. The algorithm tries to find a maximum entropy distribution which matches these matrices when averaged over the sample. This should be contrasted with the covariance matrices used in metric learning such as Fisher’s Discriminant Analysis. The latter uses the within and between class covariance matrices. The within covariance matrix is similar to the covariance matrix used here, but is calculated with respect to the class means, whereas here it is calculated separately for every point, and is centered on this point. This highlights the fact that MCML is not based on Gaussian assumptions where it is indeed sufficient to calculate a single class covariance. Our method can also be thought of as a supervised version of the Stochastic Neighbour Embedding algorithm [7] in which the “target” distribution is p 0 (determined by the class labels) and the embedding points are not completely free but are instead constrained to be of the form W x i. 2.2 Optimizing the Convex Objective Since the optimization problem in Equation 4 is convex, it is guaranteed to have only a single minimum which is the globally optimal solution4. It can be optimized using any appropriate numerical convex optimization machinery; all methods will yield the same solution although some may be faster than others. One standard approach is to use interior point Newton methods. However, these algorithms require the Hessian to be calculated, which would require O (d 4 ) resources, and could be prohibitive in our case. Instead, we have experimented with using a first order gradient method, specifically the projected gradient approach as in [10]. At each iteration we take a small step in the direction of the negative gradient of the objective function5, followed by a projection back onto the PSD cone. This projection is performed simply by taking the eigen-decomposition of A and removing the components with negative eigenvalues. The algorithm is summarized below: Input: Set of labeled data points (x i ; y i ), i = 1 : : : n Output: PSD metric which optimally collapses classes. Initialization: Initialize A 0 to some PSD matrix (randomly or using some initialization heuristic). Iterate: Set A t+1 = A t 5 f (A t ) where 5f (A) = P ij (p 0 (j ji) p(j ji))(x j x i )(x j x i ) T Calculate the eigen-decomposition of A t+1 A t+1 = P k k u k u T k , then set A t+1 = P k max( k ; 0)u k u T k Of course in principle it is possible to optimize over the dual instead of the primal but in our case, if the training data consists of n points in r-dimensional space then the primal has only O (r 2 =2) variables while the dual has O (n 2 ) so it will almost always be more efficient to operate on the primal A directly. One exception to this case may be the kernel version (Section 4) where the primal is also of size O (n 2 ). 4When the data can be exactly collapsed into single class points, there will be multiple solutions at infinity. However, this is very unlikely to happen in real data. 5In the experiments, we used an Armijo like step size rule, as described in [3]. 3 Low Dimensional Projections for Feature Extraction The Mahalanobis distance under a metric A can be interpreted as a linear projection of the original inputs by the square root of A, followed by Euclidean distance in the projected space. Matrices A which have less than full rank correspond to Mahalanobis distances based on low dimensional projections. Such metrics and the induced distances can be advantageous for several reasons [5]. First, low dimensional projections can substantially reduce the storage and computational requirements of a supervised method since only the projections of the training points must be stored and the manipulations at test time all occur in the lower dimensional feature space. Second, low dimensional projections re-represent the inputs, allowing for a supervised embedding or visualization of the original data. If we consider matrices A with rank at most q, we can always represent them in the form A = W T W for some projection matrix W of size q r. This corresponds to projecting the original data into a q-dimensional space specified by the rows of W. However, rank constraints on a matrix are not convex [4], and hence the rank constrained problem is not convex and is likely to have local minima which make the optimization difficult and illdefined since it becomes sensitive to initial conditions and choice of optimization method. Luckily, there is an alternative approach to obtaining low dimensional projections, which does specify a unique solution by sequentially solving two globally tractable problems. This is the approach we follow here. First we solve for a (potentially) full rank metric A using the convex program outlined above, and then obtain a low rank projection from it via spectral decomposition. This is done by diagonalizing A into the form A = P r i=1 i v i v T i where 1 2 : : : r are eigenvalues of A and v i are the corresponding eigenvectors. To obtain a low rank projection we constrain the sum above to include only the q terms corresponding to the q largest eigenvalues: A q = P q i=1 i v i v T i . The resulting projection is uniquely defined (up to an irrelevant unitary transformation) as W = diag( p 1 ; : : : p q )[v T 1 ; : : : ; v T q ℄. In general, the projection returned by this approach is not guaranteed to be the same as the projection corresponding to minimizing our objective function subject to a rank constraint on A unless the optimal metric A is of rank less than or equal to q. However, as we show in the experimental results, it is often the case that for practical problems the optimal A has an eigen-spectrum which is rapidly decaying, so that many of its eigenvalues are indeed very small, suggesting the low rank solution will be close to optimal. 4 Learning Metrics with Kernels It is interesting to consider the case where x i are mapped into a high dimensional feature space (x i ) and a Mahalanobis distance is sought in this space. We focus on the case where dot products in the feature space may be expressed via a kernel function, such that (x i ) (x j ) = k (x i ; x j ) for some kernel k. We now show how our method can be changed to accommodate this setting, so that optimization depends only on dot products. Consider the regularized target function: f Reg (A) = X i KL [p 0 (j ji)jp(j ji)℄ + T r (A) ; (6) where the regularizing factor is equivalent to the Frobenius norm of the projection matrix W since T r (A) = kW k 2. Deriving w.r.t. W we obtain W = U X, where U is some matrix which specifies W as a linear combination of sample points, and the i th row of the matrix X is x i. Thus A is given by A = X T U T U X. Defining the PSD matrix ^ A = U T U, we can recast our optimization as looking for a PSD matrix ^ A, where the Mahalanobis distance is (x i x j ) T X T ^ AX (x i x j ) = (k i k j ) T ^ A (k i k j ), where we define k i = X x i. This is exactly our original distance, with x i replaced by k i, which depends only on dot products in X space. The regularization term also depends solely on the dot products since T r (A) = T r (X T ^ AX ) = T r (X X T ^ A) = T r (K ^ A ), where K is the kernel matrix given by K = X X T. Note that the trace is a linear function of ^ A, keeping the problem convex. Thus, as long as dot products can be represented via kernels, the optimization can be carried out without explicitly using the high dimensional space. To obtain a low dimensional solution, we follow the approach in Section 3: obtain a decomposition A = V T D V 6, and take the projection matrix to be the first q rows of D 0:5 V . As a first step, we calculate a matrix B such that ^ A = B T B, and thus A = X T B T B X. Since A is a correlation matrix for the rows of B X it can be shown (as in Kernel PCA) that its (left) eigenvectors are linear combinations of the rows of B X. Denoting by V = B X the eigenvector matrix, we obtain, after some algebra, that B K B T = D . We conclude that is an eigenvector of the matrix B K B T . Denote by ^ the matrix whose rows are orthonormal eigenvectors of B K B T . Then V can be shown to be orthonormal if we set V = D 0:5 ^ B X. The final projection will then be D 0:5 V x i = ^ B k i. Low dimensional projections will be obtained by keeping only the first q components of this projection. 5 Experimental Results We compared our method to several metric learning algorithms on a supervised classification task. Training data was first used to learn a metric over the input space. Then this metric was used in a 1-nearest-neighbor algorithm to classify a test set. The datasets we investigated were taken from the UCI repository and have been used previously in evaluating supervised methods for metric learning [10, 5]. To these we added the USPS handwritten digits (downsampled to 8x8 pixels) and the YALE faces [2] (downsampled to 31x22). The algorithms used in the comparative evaluation were Fisher’s Linear Discriminant Analysis (LDA), which projects on the eigenvectors of S 1 W S B where S W ; S B are the within and between class covariance matrices. The method of Xing et al [10] which minimizes the mean within class distance, while keeping the mean between class distance larger than one. Principal Component Analysis (PCA). There are several possibilities for scaling the PCA projections. We tested several, and report results of the empirically superior one (PCAW), which scales the projection components so that the covariance matrix after projection is the identity. PCAW often performs poorly on high dimensions, but globally outperforms all other variants. We also evaluated the kernel version of MCML with an RBF kernel (denoted by KMCML)7. Since all methods allow projections to lower dimensions we compared performance for different projection dimensions 8. The out-of sample performance results (based on 40 random splits of the data taking 70% for training and 30% for testing9) are shown in Figure 1. It can be seen that when used in a simple nearest-neighbour classifier, the metric learned by MCML almost always performs as well as, or significantly better than those learned by all other methods, across most dimensions. Furthermore, the kernel version of MCML outperforms the linear one on most datasets. 6Where V is orthonormal, and the eigenvalues in D are sorted in decreasing order. 7The regularization parameter and the width of the RBF kernel were chosen using 5 fold crossvalidation. KMCML was only evaluated for datasets with less than 1000 training points. 8To obtain low dimensional mappings we used the approach outlined in Section 3. 9Except for the larger datasets where 1000 random samples were used for training. 2 4 6 8 10 0.1 0.2 0.3 Projection Dimension Error Rate Wine MCML PCAW LDA XING KMCML 10 20 30 0.05 0.1 0.15 0.2 0.25 Projection Dimension Error Rate Ion 1 2 3 4 0.1 0.2 0.3 0.4 0.5 Balance 5 10 15 20 0 0.2 0.4 Soybean−small 5 10 15 20 0.3 0.4 0.5 0.6 Protein 10 20 30 40 50 0.1 0.15 0.2 0.25 Spam 10 20 30 40 50 0.1 0.2 0.3 0.4 Yale7 5 10 15 0.25 0.3 0.35 0.4 Housing 10 20 30 40 50 0.2 0.4 0.6 Digits Figure 1: Classification error rate on several UCI datasets, USPS digits and YALE faces, for different projection dimensions. Algorithms are our Maximally Collapsing Metric Learning (MCML), Xing et.al.[10], PCA with whitening transformation (PCAW) and Fisher’s Discriminant Analysis (LDA). Standard errors of the means shown on curves. No results given for XING on YALE and KMCML on Digits and Spam due to the data size. 5.1 Comparison to non convex procedures The methods in the previous comparison are all well defined, in the sense that they are not susceptible to local minima in the optimization. They also have the added advantage of obtaining projections to all dimensions using one optimization run. Below, we also compare the MCML results to the results of two non-convex procedures. The first is the Non Convex variant of MCML (NMCML): The objective function of MCML can be optimized w.r.t the projection matrix W, where A = W T W. Although this is no longer a convex problem, it is not constrained and is thus easier to optimize. The second non convex method is Neighbourhood Components Analysis (NCA) [5], which attempts to directly minimize the error incurred by a nearest neighbor classifier. For both methods we optimized the matrix W by restarting the optimization separately for each size of W. Minimization was performed using a conjugate gradient algorithm, initialized by LDA or randomly. Figure 2 shows results on a subset of the UCI datasets. It can be seen that the performance of NMCML is similar to that of MCML, although it is less stable, possibly due to local minima, and both methods usually outperform NCA. The inset in each figure shows the spectrum of the MCML matrix A, revealing that it often drops quickly after a few dimensions. This illustrates the effectiveness of our two stage optimization procedure, and suggests its low dimensional solutions are close to optimal. 6 Discussion and Extensions We have presented an algorithm for learning maximally collapsing metrics (MCML), based on the intuition of collapsing classes into single points. MCML assumes that each class 2 4 6 8 10 0 0.1 0.2 Projection Dimension Error Rate Wine MCML NMCML NCA 1 2 3 4 0.1 0.2 0.3 0.4 Balance 5 10 15 20 0 0.05 0.1 Soybean 5 10 15 20 0.3 0.4 0.5 0.6 Protein 10 20 30 0.1 0.2 0.3 Ion 5 10 15 0.25 0.3 0.35 0.4 Housing Figure 2: Classification error for non convex procedures, and the MCML method. Eigen-spectra for the MCML solution are shown in the inset. may be collapsed to a single point, at least approximately, and thus is only suitable for unimodal class distributions (or for simply connected sets if kernelization is used). However, if points belonging to a single class appear in several disconnected clusters in input (or feature) space, it is unlikely that MCML could collapse the class into a single point. It is possible that using a mixture of distributions, an EM-like algorithm can be constructed to accommodate this scenario. The method can also be used to learn low dimensional projections of the input space. We showed that it performs well, even across a range of projection dimensions, and consistently outperforms existing methods. Finally, we have shown how the method can be extended to projections in high dimensional feature spaces using the kernel trick. The resulting nonlinear method was shown to improve classification results over the linear version. References [1] N. Alon and P. Pudlak. Equilateral sets in l n p . Geom. Funct. Anal., 13(3), 2003. [2] P. N. Belhumeur, J. Hespanha, and D. J. Kriegman. Eigenfaces vs. Fisherfaces: Recognition using class specific linear projection. In ECCV (1), 1996. [3] D.P. Bertsekas. On the Goldstein-Levitin-Polyak gradient projection method. IEEE Transaction on Automatic Control, 21(2):174–184, 1976. [4] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge Univ. Press, 2004. [5] J. Goldberger, S. Roweis, G. Hinton, and R. Salakhutdinov. Neighbourhood components analysis. In Advances in Neural Information Processing Systems (NIPS), 2004. [6] T. Hastie, R. Tibshirani, and J.H. Friedman. The elements of statistical learning: data mining, inference, and prediction. New York: Springer-Verlag, 2001. [7] G. Hinton and S. Roweis. Stochastic neighbor embedding. In Advances in Neural Information Processing Systems (NIPS), 2002. [8] A. Ng, M. Jordan, and Y. Weiss. On spectral clustering: Analysis and an algorithm. In Advances in Neural Information Processing Systems (NIPS), 2001. [9] N. Shental, T. Hertz, D. Weinshall, and M. Pavel. Adjustment learning and relevant component analysis. In Proc. of ECCV, 2002. [10] E. Xing, A. Ng, M. Jordan, and S. Russell. Distance metric learning, with application to clustering with side-information. In Advances in Neural Information Processing Systems (NIPS), 2004.
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Walk-Sum Interpretation and Analysis of Gaussian Belief Propagation Jason K. Johnson, Dmitry M. Malioutov and Alan S. Willsky Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology Cambridge, MA 02139 {jasonj,dmm,willsky}@mit.edu Abstract This paper presents a new framework based on walks in a graph for analysis and inference in Gaussian graphical models. The key idea is to decompose correlations between variables as a sum over all walks between those variables in the graph. The weight of each walk is given by a product of edgewise partial correlations. We provide a walk-sum interpretation of Gaussian belief propagation in trees and of the approximate method of loopy belief propagation in graphs with cycles. This perspective leads to a better understanding of Gaussian belief propagation and of its convergence in loopy graphs. 1 Introduction We consider multivariate Gaussian distributions defined on graphs. The nodes of the graph denote random variables and the edges indicate statistical dependencies between variables. The family of all Gauss-Markov models defined on a graph is naturally represented in the information form of the Gaussian density which is parameterized by the inverse covariance matrix, i.e., the information matrix. This information matrix is sparse, reflecting the structure of the defining graph such that only the diagonal elements and those off-diagonal elements corresponding to edges of the graph are non-zero. Given such a model, we consider the problem of computing the mean and variance of each variable, thereby determining the marginal densities as well as the mode. In principle, these can be obtained by inverting the information matrix, but the complexity of this computation is cubic in the number of variables. More efficient recursive calculations are possible in graphs with very sparse structure – e.g., in chains, trees and in graphs with “thin” junction trees. For these models, belief propagation (BP) or its junction tree variants efficiently compute the marginals [1]. In more complex graphs, even this approach can become computationally prohibitive. Then, approximate methods such as loopy belief propagation (LBP) provide a tractable alternative to exact inference [1, 2, 3, 4]. We develop a “walk-sum” formulation for computation of means, variances and correlations that holds in a wide class of Gauss-Markov models which we call walk-summable. In particular, this leads to a new interpretation of BP in trees and of LBP in general. Based on this interpretation we are able to extend the previously known sufficient conditions for convergence of LBP to the class of walk-summable models (which includes all of the following: trees, attractive models, and pairwise-normalizable models). Our sufficient condition is tighter than that given in [3] as the class of diagonally-dominant models is a strict subset of the class of pairwise-normalizable models. Our results also explain why no examples were found in [3] where LBP did not converge. The reason is that they presume a pairwisenormalizable model. We also explain why, in walk-summable models, LBP converges to the correct means but not to the correct variances (proving “walk-sum” analogs of results in [3]). In general, walk-summability is not necessary for LBP convergence. Hence, we also provide a tighter (essentially necessary) condition for convergence of LBP variances based on walk-summability of the LBP computation tree. This provides deeper insight into why LBP can fail to converge – because the LBP computation tree is not always well-posed – which suggests connections to [5]. This paper presents the key ideas and outlines proofs of the main results. A more detailed presentation will appear in a technical report [6]. 2 Preliminaries A Gauss-Markov model (GMM) is defined by a graph G = (V, E) with edge set E ⊂ V 2 , i.e., some set of two-element subsets of V , and a collection of random variables x = (xi, i ∈V ) with probability density given in information form1: p(x) ∝exp{−1 2x′Jx + h′x} (1) where J is a symmetric positive definite (J ≻0) matrix which is sparse so as to respect the graph G: if {i, j} ̸∈E then Ji,j = 0. We call J the information matrix and h the potential vector. Let N(i) = {j|{i, j} ∈E} denote the neighbors of i in the graph. The mean µ ≡E{x} and covariance P ≡E{(x −µ)(x −µ)′} are given by: µ = J−1h and P = J−1 (2) The partial correlation coefficients are given by: ρi,j ≡ cov(xi; xj|xV \{i,j}) p var(xi|xV \{i,j})var(xj|xV \{i,j}) = − Ji,j p Ji,iJj,j (3) Thus, Jij = 0 if and only if xi and xj are independent given the other variables xV \{i,j}. We say that this model is attractive if all partial correlations are non-negative. It is pairwisenormalizable if there exists a diagonal matrix D ≻0 and a collection of non-negative definite matrices {Je ⪰0, e ∈E}, where (Je)i,j is zero unless i, j ∈e, such that: J = D + X e∈E Je (4) It is diagonally-dominant if for all i ∈V : P j̸=i |Ji,j| < Ji,i. The class of diagonallydominant models is a strict subset of the class of pairwise-normalizable models [6]. Gaussian Elimination and Belief Propagation Integrating (1) over all possible values of xi reduces to Gaussian elimination (GE) in the information form (see also [7]), i.e., p(x\i) ≡ Z p(x\i, xi)dxi ∝exp{−1 2x′ \i ˆJ\ix\i + ˆh′ \ix\i} (5) where \i ≡V \ {i}, i.e. all variables except i, and ˆJ\i = J\i,\i −J\i,iJ−1 i,i Ji,\i and ˆh\i = h\i −J\i,iJ−1 i,i hi (6) 1The work also applies to p(x|y), i.e. where some variables y are observed. However, the observations y are fixed, and we redefine p(x) ≜p(x|y) (conditioning on y is implicit throughout). With local observations p(x|y) ∝p(x) Q i p(yi|xi), conditioning does not change the graph structure. −ρ ρ ρ 2 ρ ρ 3 4 1 2 3 4 3 2 4 3 1 1 2 −ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ 4 1 1 1 −ρ ρ n = 1 n = 2 n = 3 3 4 3 2 4 1 1 4 3 2 4 2 −ρ ρ ρ ρ ρ ρ ρ ρ −ρ ρ ρ ρ 1 (a) (b) (c) Figure 1: (a) Graph of a GMM with nodes {1, 2, 3, 4} and with edge weights (partial correlations) as shown. In (b) and (c) we illustrate the first three levels of the LBP computation tree rooted at nodes 1 and 2. After 3 iterations of LBP in (a), the marginals at nodes 1 and 2 are identical to the marginals at the root of (b) and (c) respectively. In trees, the marginal of any given node can be efficiently computed by sequentially eliminating leaves of the tree until just that node remains. BP may be seen as a message-passing form of GE in which a message passed from node i to node j ∈N(i) captures the effect of eliminating the subtree rooted at i. Thus, by a two-pass procedure, BP efficiently computes the marginals at all nodes of the tree. The equations for LBP are identical except that messages are updated iteratively and in parallel. There are two messages per edge, one for each ordered pair (i, j) ∈E. We specify each message in information form with parameters: ∆h(n) i→j, ∆J(n) i→j (initialized to zero for n = 0). These are iteratively updated as follows. For each (i, j) ∈E, messages from N(i) \ j are fused at node i: ˆh(n) i\j = hi + X k∈N(i)\j ∆h(n) k→i and ˆJ(n) i\j = Ji,i + X k∈N(i)\j ∆J(n) k→i (7) This fused information at node i is predicted to node j: ∆h(n+1) i→j = −Jj,i( ˆJ(n) i\j )−1ˆh(n) i\j and ∆J(n+1) i→j = −Jj,i( ˆJ(n) i\j )−1Ji,j (8) After n iterations, the marginal of node i is obtained by fusing all incoming messages: ˆh(n) i = hi + X k∈N(i) ∆h(n) k→i and ˆJ(n) i = Ji,i + X k∈N(i) ∆J(n) k→i (9) The mean and variance are given by ( ˆJ(n) i )−1ˆh(n) i and ( ˆJ(n) i )−1. In trees, this is the marginal at node i conditioned on zero boundary conditions at nodes (n + 1) steps away and LBP converges to the correct marginals after a finite number of steps equal to the diameter of the tree. In graphs with cycles, LBP may not converge and only yields approximate marginals when it does. A useful fact about LBP is the following [2, 3, 5]: the marginal computed at node i after n iterations is identical to the marginal at the root of the n-step computation tree rooted at node i. This tree is obtained by “unwinding” the loopy graph for n steps (see Fig. 1). Note that each node of the graph may be replicated many times in the computation tree. Also, neighbors of a node in the computation tree correspond exactly with neighbors of the associated node in the original graph (except at the last level of the tree where some neighbors are missing). The corresponding J matrix defined on the computation tree has the same node and edge values as in the original GMM. 3 Walk-Summable Gauss-Markov Models In this section we present the walk-sum formulation of inference in GMMs. Let ϱ(A) denote the spectral radius of a symmetric matrix A, defined to be the maximum of the absolute values of the eigenvalues of A. The geometric series (I +A+A2+. . . ) converges if and only if ϱ(A) < 1. If it converges, it converges to (I −A)−1. Now, consider a GMM with information matrix J. Without loss of generality, let J be normalized (by rescaling variables) to have Ji,i = 1 for all i. Then, ρi,j = −Ji,j and the (zero-diagonal) matrix of partial correlations is given by R = I −J. If ϱ(R) < 1, then we have a geometric series for the covariance matrix: ∞ X l=0 Rl = (I −R)−1 = J−1 = P (10) Let ¯R = (|rij|) denote the matrix of element-wise absolute values. We say that the model is walk-summable if ϱ( ¯R) < 1. Walk-summability implies ϱ(R) < 1 and J ≻0. Example 1. Consider a 5-node cycle with normalized information matrix J, which has all partial correlations on the edges set to ρ. If ρ = −.45, then the model is valid (i.e. positive definite) with minimum eigenvalue λmin(J) ≈.2719 > 0, and walk-summable with ϱ( ¯R) = .9 < 1. However, when ρ = −.55, then the model is still valid with λmin(J) ≈.1101 > 0, but no longer walk-summable with ϱ( ¯R) = 1.1 > 1. Walk-summability allows us to interpret (10) as computing walk-sums in the graph. Recall that the matrix R reflects graph structure: ρi,j = 0 if {i, j} ̸∈E. These act as weights on the edges of the graph. A walk w = (w0, w1, ..., wl) is a sequence of nodes wi ∈V connected by edges {wi, wi+1} ∈E where l is the length of the walk. The weight ρ(w) of walk w is the product of edge weights along the walk: ρ(w) = lY s=1 ρws−1,ws (11) At each node i ∈V , we also define a zero-length walk w = (i) for which ρ(w) = 1. Walk-Sums. Given a set of walks W, we define the walk-sum over W by ρ(W) = X w∈W ρ(w) (12) which is well-defined (i.e., independent of summation order) because ϱ( ¯R) < 1 implies absolute convergence. Let Wi l→j denote the set of l-length walks from i to j and let Wi→j = ∪∞ l=0Wi l→j. The relation between walks and the geometric series (10) is that the entries of Rl correspond to walk-sums over l-length walks from i to j in the graph, i.e., (Rl)i,j = ρ(Wi l→j). Hence, Pi,j = ∞ X l=0 (Rl)i,j = X l ρ(Wi l→j) = ρ(∪lWi l→j) = ρ(Wi→j) (13) In particular, the variance σ2 i ≡Pi,i of variable i is the walk-sum taken over the set Wi→i of self-return walks that begin and end at i (defined so that (i) ∈Wi→i). The means can be computed as reweighted walk-sums, i.e., where each walk is scaled by the potential at the start of the walk: ρ(w; h) = hw0ρ(w), and ρ(W; h) = P w∈W ρ(w; h). Then, µi = X j∈V Pi,jhj = X j ρ(Wj→i)hj = ρ(W∗→i; h) (14) where W∗→i ≡∪j∈V Wj→i is the set of all walks which end at node i. We have found that a wide class of GMMs are walk-summable: Proposition 1 (Walk-Summable GMMs) All of the following classes of GMMs are walksummable:2 (i) attractive models, (ii) trees and (iii) pairwise-normalizable3 models. 2That is if we take a valid model (with J ≻0) in these classes then it automatically has ϱ( ¯R) < 1. 3In [6], we also show that walk-summability is actually equivalent to pairwise-normalizability. Proof Outline. (i) R = ¯R and J = I −¯R ≻0 implies λmax( ¯R) < 1. Because ¯R has non-negative elements, ϱ( ¯R) = λmax( ¯R) < 1. In (ii) & (iii), negating any ρij, it still holds that J = I −R ≻0 : (ii) negating ρij doesn’t affect the eigenvalues of J (remove edge {i, j} and, in each eigenvector, negate all entries in one subtree); (iii) negating ρij preserves J{i,j} ⪰0 in (4) so J ≻0. Thus, making all ρij > 0, we find I −¯R ≻0 and ¯R ≺I. Similarly, making all ρij < 0, −¯R ≺I. Therefore, ϱ( ¯R) < 1. ⋄ 4 Recursive Walk-Sum Calculations on Trees In this section we derive a recursive algorithm which accrues the walk-sums (over infinite sets of walks) necessary for exact inference on trees and relate this to BP. Walk-summability guarantees correctness of this algorithm which reorders walks in a non-trivial way. We start with a chain of N nodes: its graph G has nodes V = {1, . . . , N} and edges E = {e1, .., eN−1} where ei = {i, i + 1}. The variance at node i is σ2 i = ρ(Wi→i). The set Wi→i can be partitioned according to the number of times that walks return to node i: Wi→i = ∪∞ r=0W(r) i→i where W(r) i→i is the set of all self-return walks which return to i exactly r times. In particular, W(0) i→i = {(i)} for which ρ(W(0) i→i) = 1. A walk which starts at node i and returns r times is a concatenation of r single-revisit self-return walks, so ρ(W(r) i→i) = ρ(W(1) i→i)r. This means: ρ(Wi→i) = ρ(∪∞ r=0W(r) i→i) = ∞ X r=0 ρ(W(r) i→i) = ∞ X r=0 ρ(W(1) i→i)r = 1 1 −ρ(W(1) i→i) (15) This geometric series converges since the model is walk-summable. Hence, calculating the single-revisit self-return walk-sum ρ(W(1) i→i) determines the variance σ2 i . The single-revisit walks at node i consist of walks in the left subchain, and walks in the right subchain. Let Wi→i\j be the set of self-return walks of i which never visit j, so e.g. all w ∈Wi→i\i+1 are contained in the subgraph {1, . . . , i}. With this notation: ρ(W(1) i→i) = ρ(W(1) i→i\i+1) + ρ(W(1) i→i\i−1) (16) The left single-revisit self-return walk-sums ρ(W(1) i→i\i+1) can be computed recursively starting from node 1. At node 1, ρ(W(1) 1→1\2) = 0 and ρ(W1→1\2) = 1. A single-revisit self-return walk from node i in the left subchain consists of a step to node i −1, then some number of self-return walks in the subgraph {1, . . . , i −1}, and a step from i −1 back to i: ρ(W(1) i→i\i+1) = ρ2 i,i−1ρ(Wi−1→i−1\i) = ρ2 i,i−1 1 −ρ(W(1) i−1→i−1\i) (17) Thus single-revisit (and multiple revisit) walk-sums in the left subchain of every node i can be calculated in one forward pass through the chain. The same can be done for the right subchain walk-sums at every node i, by starting at node N, and going backwards. Using equations (15) and (16) these quantities suffice to calculate the variances at all nodes of the chain. A similar forwards-backwards procedure computes the means as reweighted walksums over the left and right single-visit walks for node i, which start at an arbitrary node (in the left or right subchain) and end at i, never visiting i before that [6]. In fact, these recursive walk-sum calculations map exactly to operations in BP – e.g., in a normalized chain ∆Ji−1→i = −ρ(W(1) i→i\i+1) and ∆hi−1→i = −ρ(W(1) ∗→i\i+1; h). The same strategy applies for trees: both single-revisit and single-visit walks at node i can be partitioned according to which subtree (rooted at a neighbor j ∈N(i) of i) the walk lives in. This leads to a two-pass walks-sum calculation on trees (from the leaves to the root, and back) to calculate means and variances at all nodes. 5 Walk-sum Analysis of Loopy Belief Propagation First, we analyze LBP in the case that the model is walk-summable and show that LBP converges and includes all the walks for the means, but only a subset of the walks for the variances. Then, we consider the case of non-walksummable models and relate convergence of the LBP variances to walk-summability of the computation tree. 5.1 LBP in walk-summable models To compute means and variances in a walk-summable model, we need to calculate walksums for certain sets of walks in the graph G. Running LBP in G is equivalent to exact inference in the computation tree for G, and hence calculating walk-sums for certain walks in the computation tree. In the computation tree rooted at node i, walks ending at the root have a one-to-one correspondence with walks ending at node i in G. Hence, LBP captures all of the walks necessary to calculate the means. For variances, the walks captured by LBP have to start and end at the root in the computation tree. However, some of the selfreturn walks in G translate to walks in the computation tree that end at the root but start at a replica of the root, rather than at the root itself. These walks are not captured by the LBP variances. For example, in Fig. 1(a), the walk (1, 2, 3, 1) is a self-return walk in the original graph G but is not a self-return walk in the computation tree shown in Fig. 1(b). LBP variances capture only those self-return walks of the original graph G which also are self-return walks in the computation tree – e.g., the walk (1, 3, 2, 3, 4, 3, 1) is a selfreturn walk in both Figs. 1(a) and (b). We call these backtracking walks. These simple observations lead to our main result: Proposition 2 (Convergence of LBP for walk-summable GMMs) If the model is walksummable, then LBP converges: the means converge to the true means and the LBP variances converge to walk-sums over just the backtracking self-return walks at each node. Proof Outline. All backtracking walks have positive weights, since each edge is traversed an even number of times. For a walk-summable model, LBP variances are walks-sums over the backtracking walks and are therefore monotonically increasing with the iterations. They also are bounded above by the absolute self-return walk-sums (diagonal elements of P l ¯Rl) and hence converge. For the means: the series P∞ l=0 Rlh converges absolutely since |Rlh| ≤¯Rl|h|, and the series P l ¯Rl|h| is a linear combination of terms of the absolutely convergent series P l ¯Rl. The LBP means are a rearrangement of the absolutely convergent series P∞ l=0 Rlh, so they converge to the same values. ⋄ As a corollary, LBP converges for all of the model classes listed in Proposition 1. Also, in attractive models, the LBP variances are less than or equal to the true variances. Correctness of the means was also shown in [3] for pairwise-normalizable models.4 They also show that LBP variances omit some terms needed for the correct variances. These terms correspond to correlations between the root and its replicas in the computation tree. In our framework, each such correlation is a walk-sum over the subset of non-backtracking self-return walks in G which, in the computation tree, begin at a particular replica of the root. Example 2. Consider the graph in Fig. 1(a). For ρ = .39, the model is walk-summable with ϱ( ¯R) ≈.9990. For ρ = .395 and ρ = .4, the model is still valid but is not walk-summable, with ϱ( ¯R) ≈1.0118 and 1.0246 respectively. In Fig. 2(a) we show LBP variances for node 1 (the other nodes are similar) vs. the iteration number. As ρ increases, first the model is walk-summable and LBP converges, then for a small interval the model is not walk-summable but LBP still converges,5 and for larger ρ LBP does not converge. Also, 4However, they only prove convergence for the subset of diagonally dominant models. 5Hence, walk-summability is sufficient but not necessary for convergence of LBP. 0 10 20 30 40 0 5 0 50 100 150 200 −200 0 200 ρ = 0.4 ρ = 0.39 ρ = 0.395 0 10 20 30 0.7 0.8 0.9 1 1.1 LBP converges LBP does not converge (a) (b) Figure 2: (a) LBP variances vs. iteration. (b) ϱ(Rn) vs. iteration. for ρ = .4, we note that ϱ(R) = .8 < 1 and the series P l Rl converges (but P l ¯Rl does not) and LBP does not converge. Hence, ϱ(R) < 1 is not sufficient for LBP convergence showing the importance of the stricter walk-summability condition ϱ( ¯R) < 1. 5.2 LBP in non-walksummable models We extend our analysis to develop a tighter condition for convergence of LBP variances based on walk-summability of the computation tree (rather than walk-summability on G).6 For trees, walk-summability and validity are equivalent, i.e. J ≻0 ⇔ϱ( ¯R) < 1, hence our condition is equivalent to validity of the computation tree. First, we note that when a model on G is valid (J is positive-definite) but not walksummable, then some finite computation trees may be invalid (indefinite). This turns out to be the reason why LBP variances can fail to converge. Walk-summability of the original GMM implies walk-summability (and hence validity) of all of its computation trees. But if the GMM is not walk-summable, then its computation tree may or may not be walksummable. In Example 2, for ρ = .395 the computation tree is still walk-summable (even though the model on G is not) and LBP converges. For ρ = .4, the computation tree is not walk-summable and LBP does not converge. Indeed, LBP is not even well-posed in this case (because the computation tree is indefinite) which explains its strange behavior seen in the bottom plot of Fig. 2(a) (e.g., non-monotonicity and negative variances). We characterize walk-summability of the computation tree as follows. Let Tn be the nstep computation tree rooted at some node i and define Rn ≜In −Jn where Jn is the normalized information matrix on Tn and In is the n × n identity matrix. The n-step computation tree Tn is walk-summable (valid) if and only if ϱ(Rn) < 1 (in trees, ϱ( ¯Rn) = ϱ(Rn)). The sequence {ϱ(Rn)} is monotonically increasing and bounded above by ϱ( ¯R) (see [6]) and hence converges. We are interested in the quantity ϱ∞≡limn→∞ϱ(Rn). Proposition 3 (LBP validity/variance convergence) (i) If ϱ∞< 1, then all finite computation trees are valid and the LBP variances converge. (ii) If ϱ∞> 1, then the computation tree eventually becomes invalid and LBP is ill-posed. Proof Outline. (i) For some δ > 0, ϱ(Rn) ≤1 −δ for all n which implies: all computation trees are walk-summable and variances monotonically increase; λmax(Rn) ≤1 −δ, λmin(Jn) ≥δ, and (Pn)i,i ≤λmax(Pn) ≤1 δ . The variances are monotonically increasing 6We can focus on one tree: if the computation tree rooted at node i is walk-summable, then so is the computation tree rooted at any node j. Also, if a finite computation tree rooted at node i is not walk-summable, then some finite tree at node j also becomes non-walksummable for n large enough. and bounded above, hence they converge. (ii) If limn→∞ϱ(Rn) > 1, then there exists an m for which ϱ(Rn) > 1 for all n ≥m and the computation tree is invalid. ⋄ As discussed in [6], LBP is well-posed if and only if the information numbers computed on the right in (7) and (9) are strictly positive for all n. Hence, it is easily detected if the LBP computation tree becomes invalid. In this case, continuing to run LBP is not meaningful and will lead to division by zero and/or negative variances. Example 3. Consider a 4-node cycle with edge weights (−ρ, ρ, ρ, ρ). In Fig. 2(b), for ρ = .49 we plot ϱ(Rn) vs. n (lower curve) and observe that limn→∞ϱ(Rn) ≈.98 < 1, and LBP converges (similar to the upper plot of Fig. 2(a)). For ρ = .51 (upper curve), the model defined on the 4-node cycle is still valid but limn→∞ϱ(Rn) ≈1.02 > 1 so LBP is ill-posed and does not converge (similar to the lower plot of Fig. 2(a)). In non-walksummable models, the series LBP computes for the means is not absolutely convergent and may diverge even when variances converge (e.g., in Example 2 with ρ = .39867). However, in all cases where variances converge we have observed that with enough damping of BP messages7 we also obtain convergence of the means. Apparently, it is the validity of the computation tree that is critical for convergence of Gaussian LBP. 6 Conclusion We have presented a walk-sum interpretation of inference in GMMs and have applied this framework to analyze convergence of LBP extending previous results. In future work, we plan to develop extended walk-sum algorithms which gather more walks than LBP. Another approach is to estimate variances by sampling random walks in the graph. We also are interested to explore possible connections between results in [8] – based on selfavoiding walks in Ising models – and sufficient conditions for convergence of discrete LBP [9] which have some parallels to our walk-sum analysis in the Gaussian case. Acknowledgments This research was supported by the Air Force Office of Scientific Research under Grant FA9550-04-1, the Army Research Office under Grant W911NF-051-0207 and by a grant from MIT Lincoln Laboratory. References [1] J. Pearl. Probabilistic inference in intelligent systems. Morgan Kaufmann, 1988. [2] J. Yedidia, W. Freeman, and Y. Weiss. Understanding belief propagation and its generalizations. Exploring AI in the new millennium, pages 239–269, 2003. [3] Y. Weiss and W. Freeman. Correctness of belief propagation in Gaussian graphical models of arbitrary topology. Neural Computation, 13:2173–2200, 2001. [4] P. Rusmevichientong and B. Van Roy. An analysis of belief propagation on the turbo decoding graph with Gaussian densities. IEEE Trans. Information Theory, 48(2):745–765, Feb. 2001. [5] S. Tatikonda and M. Jordan. Loopy belief propagation and Gibbs measures. UAI, 2002. [6] J. Johnson, D. Malioutov, and A. Willsky. Walk-Summable Gaussian Networks and Walk-Sum Interpretation of Gaussian Belief Propagation. TR-2650, LIDS, MIT, 2005. [7] K. Plarre and P. Kumar. Extended message passing algorithm for inference in loopy Gaussian graphical models. Ad Hoc Networks, 2004. [8] M. Fisher. Critical temperatures of anisotropic Ising lattices II, general upper bounds. Physical Review, 162(2), 1967. [9] A. Ihler, J. Fisher III, and A. Willsky. Message Errors in Belief Propagation. NIPS, 2004. 7Modify (8) as follows: ∆h(n+1) i→j = (1 −α)∆h(n) i→j + α(−Ji,j( ˆJ(n) i\j )−1ˆh(n) i\j) with 0 < α ≤1. In Example 2, with ρ = .39867 and α = .9 the means converge.
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Analyzing Coupled Brain Sources: Distinguishing True from Spurious Interaction Guido Nolte1, Andreas Ziehe3, Frank Meinecke1 and Klaus-Robert M¨uller1,2 1 Fraunhofer FIRST.IDA, Kekul´estr. 7, 12489 Berlin, Germany 2 Dept. of CS, University of Potsdam, August-Bebel-Strasse 89, 14482 Potsdam, Germany 3 TU Berlin, Inst. for Software Engineering, Franklinstr. 28/29, 10587 Berlin, Germany {nolte,ziehe,meinecke,klaus}@first.fhg.de Abstract When trying to understand the brain, it is of fundamental importance to analyse (e.g. from EEG/MEG measurements) what parts of the cortex interact with each other in order to infer more accurate models of brain activity. Common techniques like Blind Source Separation (BSS) can estimate brain sources and single out artifacts by using the underlying assumption of source signal independence. However, physiologically interesting brain sources typically interact, so BSS will—by construction— fail to characterize them properly. Noting that there are truly interacting sources and signals that only seemingly interact due to effects of volume conduction, this work aims to contribute by distinguishing these effects. For this a new BSS technique is proposed that uses anti-symmetrized cross-correlation matrices and subsequent diagonalization. The resulting decomposition consists of the truly interacting brain sources and suppresses any spurious interaction stemming from volume conduction. Our new concept of interacting source analysis (ISA) is successfully demonstrated on MEG data. 1 Introduction Interaction between brain sources, phase synchrony or coherent states of brain activity are believed to be fundamental for neural information processing (e.g. [2, 6, 5]). So it is an important topic to devise new methods that can more reliably characterize interacting sources in the brain. The macroscopic nature and the high temporal resolution of electroencephalography (EEG) and magnetoencephalography (MEG) in the millisecond range makes these measurement technologies ideal candidates to study brain interactions. However, interpreting data from EEG/MEG channels in terms of connections between brain sources is largely hampered by artifacts of volume conduction, i.e. the fact that activities of single sources are observable as superposition in all channels (with varying amplitude). So ideally one would like to discard all—due to volume conduction—seemingly interacting signals and retain only truly linked brain source activity. So far neither existing source separation methods nor typical phase synchronization analysis (e.g. [1, 5] and references therein) can adequately handle signals when the sources are both superimposed and interacting i.e. non-independent (cf. discussions in [3, 4]). It is here where we contribute in this paper by proposing a new algorithm to distinguish true from spurious interaction. A prerequisite to achieve this goal was recently established by [4]: as a consequence of instantaneous and linear volume conduction, the cross-spectra of independent sources are real-valued, regardless of the specifics of the volume conductor, number of sources or source configuration. Hence, a non-vanishing imaginary part of the cross-spectra must necessarily reflect a true interaction. Drawbacks of Nolte’s method are: (a) cross-spectra for all frequencies in multi-channel systems contain a huge amount of information and it can be tedious to find the interesting structures, (b) it is very much possible that the interacting brain consists of several subsystems which are independent of each other but are not separated by that method, and (c) the method is well suited for rhythmic interactions while wide-band interactions are not well represented. A recent different approach by [3] uses BSS as preprocessing step before phase synchronization is measured. The drawback of this method is the assumption that there are not more sources than sensors, which is often heavily violated because, e.g., channel noise trivially consists of as many sources as channels, and, furthermore, brain noise can be very well modelled by assuming thousands of randomly distributed and independent dipoles. To avoid the drawbacks of either method we will formulate an algorithm called interacting source analysis (ISA) which is technically based on BSS using second order statistics but is only sensitive to interacting sources and, thus, can be applied to systems with arbitrary noise structure. In the next section, after giving a short introduction to BSS as used for this paper, we will derive some fundamental properties of our new method. In section 3 we will show in simulated data and real MEG examples that the ISA procedure finds the interacting components and separates interacting subsystems which are independent of each other. 2 Theory The fundamental assumption of ICA is that a data matrix X, without loss of generality assumed to be zero mean, originates from a superposition of independent sources S such that X = AS (1) where A is called the mixing matrix which is assumed to be invertible. The task is to find A and hence S (apart from meaningless ordering and scale transformations of the columns of A and the rows of S) by merely exploiting statistical independence of the sources. Since independence implies that the sources are uncorrelated we may choose W, the estimated inverse mixing matrix, such that the covariance matrix of ˆS ≡WX (2) is equal to the identity matrix. This, however, does not uniquely determine W because for any such W also UW, where U is an arbitrary orthogonal matrix, leads to a unit covariance matrix of ˆS. Uniqueness can be restored if we require that W not only diagonalizes the covariance matrix but also cross-correlation matrices for various delays τ, i.e. we require that WCX(τ)W † = diag (3) with CX(τ) ≡⟨x(t)x†(t + τ)⟩ (4) where x(t) is the t.th column of X and ⟨.⟩means expectation value which is estimated by the average over t. Although at this stage all expressions are real-valued we introduce a complex formulation for later use. Note, that since under the ICA assumption the cross-correlation matrices CS(τ) of the source signals are diagonal CS ij(τ) = ⟨si(t)si(t + τ)⟩δij = CS ji, (5) the cross-correlation matrices of the mixtures are symmetric: CX(τ) = ACS(τ)A† = ACS(τ)A†† = CX†(τ) (6) Hence, the antisymmetric part of CX(τ) can only arise due to meaningless fluctuations and can be ignored. In fact, the above TDSEP algorithm uses symmetrized versions of CX(τ) [8]. Now, the key and new point of our method is that we will turn the above argument upside down. Since non-interacting sources do not contribute (systematically) to the antisymmetrized correlation matrices D(τ) ≡CX(τ) −CX†(τ) (7) any (significant) non-vanishing elements in D(τ) must arise from interacting sources, and hence the analysis of D(τ) is ideally suited to study the interacting brain. In doing so we exploit that neuronal interactions necessarily take some time which is well above the typical time resolution of EEG/MEG measurements. It is now our goal to identify one or many interacting systems from a suitable spatial transformation which corresponds to a demixing of the systems rather than individual sources. Although we concentrate on those components which explicitly violate the independence assumption we will use the technique of simultaneous diagonalization to achieve this goal. We first note that a diagonalization of D(τ) using a real-valued W is meaningless since with D(τ) also WD(τ)W † is anti-symmetric and always has vanishing diagonal elements. Hence D(τ) can only be diagonalized with a complex-valued W with subsequent interpretation of it in terms of a real-valued transformation. We will here discuss the case where all interacting systems consist of pairs of neuronal sources. Properties of systems with more than two interacting systems will be discussed below. Furthermore, for simplicity we assume an even number of channels. Then a realvalued spatial transformation W1 exists such that the set of D(τ) becomes decomposed into K = N/2 blocks of size 2 × 2 W1D(τ)W † 1 = α1(τ) 0 1 −1 0 0 0 0 ... 0 0 0 αK(τ) 0 1 −1 0 (8) Each block can be diagonalized e.g. with ˜W2 = 1 −i 1 i (9) and with W2 = idK×K ⊗˜W2 (10) we get W2W1D(τ)W † 1 W † 2 = diag (11) From a simultaneous diagonalization of D(τ) we obtain an estimate of the demixing matrix ˆW of the true demixing matrix W = W2W1. We are interested in the columns of W −1 1 which correspond to the spatial patterns of the interacting sources. Let us denote the N ×2 submatrix of a matrix B consisting of the (2k −1).th and the 2k.th column as (B)k. Then we can write (W −1 1 )k ∼(W −1)k ˜W2 (12) and hence the desired spatial patterns of the k.th system are a complex linear superposition of the (2k −1).th and the 2k.th column of W. The subspace spanned in channel-space by the two interacting sources, denoted as span((A)k), can now be found by separating real and imaginary part of W −1 span((A)k) = span ℜ((W −1)k), ℑ((W −1)k) (13) According to (13) we can calculate from W just the 2D-subspaces spanned by the interacting systems but not the patterns of the sources themselves. The latter would indeed be impossible because all we analyze are anti-symmetric matrices which are, for each system, constructed as anti-symmetric outer products of the two respective field patterns. These anti-symmetric matrices are, apart from an irrelevant global scale, invariant with respect to a linear and real-valued mixing of the sources within each system. The general procedure can now be outlined as follows. 1. From the data construct anti-symmetric cross-correlation matrices as defined in Eq.(7) for reasonable set of delays τ. 2. Find a complex matrix W such that WD(τ)W † is approximately diagonal for all τ. 3. If the system consists of subsystems of paired interactions (and indeed, according to our own experience, very much in practice) the diagonal elements in WD(τ)W † come in pairs in the form ±iλ. Each pair constitutes one interacting system. The corresponding two columns in W −1, with separated real and imaginary parts, form an N × 4 matrix V with rank 2. The span of V coincides with the space spanned by the respective system. In practice, V will have two singular values which are just very small rather than exactly zero. The corresponding singular vectors should then be discarded. Instead of analyzing V in the above way it is also possible to simply take the real and imaginary part of either one of the two columns. 4. Similar to the spatial analysis, it is not possible to separate the time-courses of two interacting sources within one subsystem. In general, two estimated time-courses, say ˆs1(t) and ˆs2(t), are an unknown linear combination of the true source activations s1(t) and s2(t). To understand the type of interaction it is still recommended to look at the power and autocorrelation functions. Invariant with respect to linear mixing with one subsystem is the anti-symmetrized cross-correlation between ˆs1(t) and ˆs2(t) and, equivalently, the imaginary part of the cross-spectral density. For the k.th system, these quantities are given by the k.th diagonal λk(τ) and their respective Fourier transforms. While (approximate) simultaneous diagonalization of D(τ) using complex demixing matrices is always possible with pairwise interactions we can expect only block-diagonal structure if a larger number of sources are interacting within one or more subsystems. We will show below for simulated data that the algorithm still finds these blocks although the actual goal, i.e. diagonal WD(τ)W †, is not reachable. 3 Results 3.1 Simulated data Matrices were approximately simultaneously diagonalized with the DOMUNG-algorithm [7], which was generalized to the complex domain. Here, an initial guess for the demixing matrix W is successively optimized using a natural gradient approach combined with line search according to the requirement that the off-diagonals are minimal under the constraint det(W) = 1. Special care has to be taken in the choice of the initial guess. Due to the complex-conjugation symmetry of our problem (i.e., W ∗diagonalizes as well as W) the initial guess may not be set to a real-valued matrix because then the component of the gradient in imaginary direction will be zero and W will converge to a real-valued saddle point. We simulated two random interacting subsystems of dimensions NA and NB which were assumed to be mutually independent. The two subsystems were mapped into N = NA +NB channels with a random mixture matrix. The anti-symmetrized cross-correlation matrices read D(τ) = A DA(τ) 0 0 DB(τ) A† (14) where A is a random real-valued N × N matrix, and DA(τ) (DB(τ)), with τ = 1...20, are a set of random anti-symmetric NA × NA (NB × NB) matrices. Note, that in this context, τ has no physical meaning. As expected, we have found that if one of the subsystems is two-dimensional the respective block can always be diagonalized exactly for any number of τs. We have also seen, that the diagonalization procedure always perfectly separates the two subsystems even if a diagonalization within a subsystem is not possible. A typical result for NA = 2 and NB = 3 is presented in Fig.1. In the left panel we show the average of the absolute value of correlation matrices before spatial mixing. In the middle panel we show the respective result after random spatial mixture and subsequent demixing, and in the right panel we show W1A where W1 is the estimated real version of the demixing matrix as explained in the preceding section. We note again, that also for the two-dimensional block, which can always be diagonalized exactly, one can only recover the corresponding two-dimensional subspace and not the source components themselves. 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Figure 1: Left: average of the absolute values of correlation matrices before spatial mixing; middle: same after random spatial mixture and subsequent demixing; right: product of the estimated demixing matrix and the true mixing matrix (W1A). White indicates zero and black the maximum value for each matrix. 3.2 Real MEG data We applied our method to real data gathered in 93 MEG channels during triggered finger movements of the right or left hand. We recall that for each interacting component we get two results: a) the 2D subspace spanned by the two components and b) the diagonals of the demixed system, say ±iλk(τ). To visualize the 2D subspace in a unique way we construct from the two patterns of the k.th system, say x1 and x2, the anti-symmetric outer product Dk ≡x1xT 2 −x2xT 1 (15) Indeed, the k.th subsystem contributes this matrix to the anti-symmetrized crosscorrelations D(τ) with varying amplitude for all τ. The matrix Dk is now visualized as shown in Figs.3. The i.th row of Dk corresponds to the interaction of the i.th channel to all others and this interaction is represented by the contour-plot within the i.th circle located at the respective channel location. In this example, the observed structure clearly corresponds to the interaction between eye-blinks and visual cortex since occipital channels interact with channels close to the eyes and vice versa. In the upper panels of Fig.2 we show the corresponding temporal and spectral structures of this interaction, represented by λk(τ), and its Fourier transform, respectively. We observe in the temporal domain a peak at a delay around 120 ms (indicated by the arrow) which corresponds well to the response time of the primary visual cortex to visual input. In the lower panels of Fig.2 we show the temporal and spectral pattern of another interacting component with a clear peak in the alpha range (10 Hz). The corresponding spatial pattern (Fig.4) clearly indicates an interacting system in occipital-parietal areas. 0 200 400 600 800 1000 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 time in msec power in a.u. 0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 frequency in Hz power in a.u. 0 200 400 600 800 −0.5 0 0.5 1 time in msec power in a.u. 0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 frequency in Hz power in a.u. Figure 2: Diagonals of demixed antisymmetric correlation matrices as a function of delay τ (left panels) and, after Fourier transformation, as a function of frequency (right panels). Top: interaction of eye-blinks and visual cortex; bottom: interaction of alpha generators. Figure 3: Spatial pattern corresponding to the interaction between eye-blinks and visual cortex. 4 Conclusion When analyzing interaction between brain sources from macroscopic measurements like EEG/MEG it is important to distinguish physiologically reasonable patterns of interaction and spurious ones. In particular, volume conduction effects make large parts of the cortex seemingly interact although in reality such contributions are purely artifactual. Existing BSS methods that have been used with success for artifact removal and for estimation of brain sources will by construction fail when attempting to separate interacting i.e. nonindependent brain sources. In this work we have proposed a new BSS algorithm that uses anti-symmetrized cross-correlation matrices and subsequent diagonalization and can thus reliably extract meaningful interaction while ignoring all spurious effects. Experiments using our interacting source analysis (ISA) reveal interesting relationships that are found blindly, e.g. inferring a component that links both eyes with visual cortex activity in a self-paced finger movement experiment. A more detailed look at the spectrum exhibits a peak at the typing frequency, and, in fact going back to the original MEG traces, eye-blinks were strongly coupled with the typing speed. This simple finding exemplifies that ISA is a powerful new technique for analyzing dynamical correlations in macroscopic brain measurements. Future studies will therefore apply ISA to other neurophysiological paradigms in order to gain insights into the coherence and synchronicity patterns of cortical dynamics. It is especially of high interest to explore the possibilities of using true brain interactions as revealed by the imaginary part of cross-spectra as complementing information to improve the performance of brain computer interfaces. Acknowledgements. We thank G. Curio for valuable discussions. This work was supported in part by the IST Programme of the European Community, under PASCAL Network Figure 4: Spatial pattern corresponding to the interaction between alpha generators. of Excellence, IST-2002-506778 and the BMBF in the BCI III project (grant 01BE01A). This publication only reflects the author’s views. References [1] A. Hyvarinen, J. Karhunen, and E. Oja. Independent Component Analysis. Wiley, 2001. [2] V.K. Jirsa. Connectivity and dynamics of neural information processing. Neuroinformatics, (2):183–204, 2004. [3] Frank Meinecke, Andreas Ziehe, J¨urgen Kurths, and Klaus-Robert M¨uller. Measuring Phase Synchronization of Superimposed Signals. Physical Review Letters, 94(8), 2005. [4] G. Nolte, O. Bai, L. Wheaton, Z. Mari, S. Vorbach, and M. Hallet. Identifying true brain interaction from eeg data using the imaginary part of coherency. Clinical Neurophysiology, 115:2292–2307, 2004. [5] A. Pikovsky, M. Rosenblum, and J. Kurths. Synchronization – A Universal Concept in Nonlinear Sciences. Cambridge University Press, 2001. [6] W. Singer. Striving for coherence. Nature, 397(6718):391–393, Feb 1999. [7] A. Yeredor, A. Ziehe, and K.-R. M¨uller. Approximate joint diagonalization using a natural-gradient approach. In Carlos G. Puntonet and Alberto Prieto, editors, Lecture Notes in Computer Science, volume 3195, pages 89–96, Granada, 2004. SpringerVerlag. Proc. ICA 2004. [8] A. Ziehe and K.-R. M¨uller. TDSEP – an efficient algorithm for blind separation using time structure. In L. Niklasson, M. Bod´en, and T. Ziemke, editors, Proceedings of the 8th International Conference on Artificial Neural Networks, ICANN’98, Perspectives in Neural Computing, pages 675 – 680, Berlin, 1998. Springer Verlag.
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Group and Topic Discovery from Relations and Their Attributes Xuerui Wang, Natasha Mohanty, Andrew McCallum Department of Computer Science University of Massachusetts Amherst, MA 01003 {xuerui,nmohanty,mccallum}@cs.umass.edu Abstract We present a probabilistic generative model of entity relationships and their attributes that simultaneously discovers groups among the entities and topics among the corresponding textual attributes. Block-models of relationship data have been studied in social network analysis for some time. Here we simultaneously cluster in several modalities at once, incorporating the attributes (here, words) associated with certain relationships. Significantly, joint inference allows the discovery of topics to be guided by the emerging groups, and vice-versa. We present experimental results on two large data sets: sixteen years of bills put before the U.S. Senate, comprising their corresponding text and voting records, and thirteen years of similar data from the United Nations. We show that in comparison with traditional, separate latent-variable models for words, or Blockstructures for votes, the Group-Topic model’s joint inference discovers more cohesive groups and improved topics. 1 Introduction The field of social network analysis (SNA) has developed mathematical models that discover patterns in interactions among entities. One of the objectives of SNA is to detect salient groups of entities. Group discovery has many applications, such as understanding the social structure of organizations or native tribes, uncovering criminal organizations, and modeling large-scale social networks in Internet services such as Friendster.com or LinkedIn.com. Social scientists have conducted extensive research on group detection, especially in fields such as anthropology and political science. Recently, statisticians and computer scientists have begun to develop models that specifically discover group memberships [5, 2, 7]. One such model is the stochastic Blockstructures model [7], which discovers the latent groups or classes based on pair-wise relation data. A particular relation holds between a pair of entities (people, countries, organizations, etc.) with some probability that depends only on the class (group) assignments of the entities. This model is extended in [4] to support an arbitrary number of groups by using a Chinese Restaurant Process prior. The aforementioned models discover latent groups by examining only whether one or more relations exist between a pair of entities. The Group-Topic (GT) model presented in this paper, on the other hand, considers both the relations between entities and also the attributes of the relations (e.g., the text associated with the relations) when assigning group memberships. The GT model can be viewed as an extension of the stochastic Blockstructures model [7] with the key addition that group membership is conditioned on a latent variable, which in turn is also associated with the attributes of the relation. In our experiments, the attributes of relations are words, and the latent variable represents the topic responsible for generating those words. Our model captures the (language) attributes associated with interactions, and uses distinctions based on these attributes to better assign group memberships. Consider a legislative body and imagine its members forming coalitions (groups), and voting accordingly. However, different coalitions arise depending on the topic of the resolution up for a vote. In the GT model, the discovery of groups is guided by the emerging topics, and the forming of topics is shaped by emerging groups.Resolutions that would have been assigned the same topic in a model using words alone may be assigned to different topics if they exhibit distinct voting patterns. Topics may be merged if the entities vote very similarly on them. Likewise, multiple different divisions of entities into groups are made possible by conditioning them on the topics. The importance of modeling the language associated with interactions between people has recently been demonstrated in the Author-Recipient-Topic (ART) model [6]. It can measure role similarity by comparing the topic distributions for two entities. However, the ART model does not explicitly discover groups formed by entities. When forming latent groups, the GT model simultaneously discovers salient topics relevant to relationships between entities—topics which the models that only examine words are unable to detect. We demonstrate the capabilities of the GT model by applying it to two large sets of voting data: one from US Senate and the other from the General Assembly of the UN. The model clusters voting entities into coalitions and simultaneously discovers topics for word attributes describing the relations (bills or resolutions) between entities. We find that the groups obtained from the GT model are significantly more cohesive (p-value < 0.01) than those obtained from the Blockstructures model. The GT model also discovers new and more salient topics that help better predict entities’ behaviors. 2 Group-Topic Model The Group-Topic model is a directed graphical model that clusters entities with relations between them, as well as attributes of those relations. The relations may be either symmetric or asymmetric and have multiple attributes. In this paper, we focus on symmetric relations and have words as the attributes on relations. The graphical model representation of the model and our notation are shown in Figure 1. tb ∼Uniform(1/T) wit|φt ∼Multinomial(φt) φt|η ∼Dirichlet(η) git|θt ∼Multinomial(θt) θt|α ∼Dirichlet(α) v(b) ij |γ(b) gigj ∼Binomial(γ(b) gigj) γ(b) gh |β ∼Beta(β). Without considering the topics of events, or by treating all events in a corpus as reflecting a single topic, the simplified model becomes equivalent to the stochastic Blockstructures model [7]. Here, each event defines a relationship, e.g., whether in the event two entities’ group(s) behave the same way or not. On the other hand, in our model a relation may also have multiple attributes. When we consider the complete model, the dataset is dynamically divided into T sub-blocks each of which corresponds to a topic. The generative process of the GT model is as right. We want to perform joint inference on (text) attributes and relations to obtain topic-wise group memberships. We employ Gibbs sampling to conduct inference. Note that we adopt conjugate priors in our setting, and thus we can easily integrate out θ, φ and γ to decrease η φ w v γ β t Nb Sb 2 T BG2 S T B g α θ SYMBOL DESCRIPTION gst entity s’s group assignment in topic t tb topic of an event b w(b) k the kth token in the event b v(b) ij entity i and j’s group(s) behaved same (1) or differently (2) on the event b S # of entities T # of topics G # of groups B # of events V # of unique words Nb # of word tokens in the event b Sb # of entities who participated in the event b Figure 1: The Group-Topic model and notations used in this paper the uncertainty associated with them.. In our case we need to compute the conditional distribution P(gst|w, v, g−st, t, α, β, η) and P(tb|w, v, g, t−b, α, β, η), where g−st denotes the group assignments for all entities except entity s in topic t, and t−b represents the topic assignments for all events except event b. Beginning with the joint probability of a dataset, and using the chain rule, we can obtain the conditional probabilities conveniently. In our setting, the relationship we are investigating is always symmetric, so we do not distinguish Rij and Rji in our derivations (only Rij(i ≤j) remain). Thus P(gst|v, g−st, w, t, α, β, η) ∝ αgst+ntgst−1 PG g=1(αg+ntg)−1 QB b=1 I(tb = t) QG h=1 Q2 k=1 Qd(b) gsthk x=1 βk+m(b) gsthk−x QP2 k=1 d(b) gsthk x=1 (P2 k=1(βk+m(b) gsthk)−x ! , where ntg represents how many entities are assigned into group g in topic t, ctv represents how many tokens of word v are assigned to topic t, m(b) ghk represents how many times group g and h vote same (k = 1) and differently (k = 2) on event b, I(tb = t) is an indicator function, and d(b) gsthk is the increase in m(b) gsthk if entity s were assigned to group gst than without considering s at all (if I(tb = t) = 0, we ignore the increase in event b). P(tb|v, g, w, t−b, α, β, η) ∝ QV v=1 Qe(b) v x=1(ηv+ctbv−x) QPV v=1 e(b) v x=1 PV v=1(ηv+ctbv)−x QG g=1 QG h=g Q2 k=1 Γ(βk+m(b) ghk) Γ(P2 k=1(βk+m(b) ghk)), where e(b) v is the number of tokens of word v in event b. The GT model uses information from two different modalities whose likelihoods are generally not directly comparable, since the number of occurrences of each type may vary greatly. Thus we raise the first term in the above formula to a power, as is common in speech recognition when the acoustic and language models are combined. 3 Related Work There has been a surge of interest in models that describe relational data, or relations between entities viewed as links in a network, including recent work in group discovery [2, 5]. The GT model is an enhancement of the stochastic Blockstructures model [7] and Datasets Avg. AI for GT Avg. AI for Baseline p-value Senate 0.8294 0.8198 < .01 UN 0.8664 0.8548 < .01 Table 1: Average AI for GT and Baseline for both Senate and UN datasets. The group cohesion in GT is significantly better than in baseline. the extended model of Kemp et al. [4] as it takes advantage of information from different modalities by conditioning group membership on topics. In this sense, the GT model draws inspiration from the Role-Author-Recipient-Topic (RART) model [6]. As an extension of ART model, RART clusters together entities with similar roles. In contrast, the GT model presented here clusters entities into groups based on their relations to other entities. There has been a considerable amount of previous work in understanding voting patterns. Exploring the notion that the behavior of an entity can be explained by its (hidden) group membership, Jakulin and Buntine [3] develop a discrete PCA model for discovering groups, where each entity can belong to each of the k groups with a certain probability, and each group has its own specific pattern of behaviors. They apply this model to voting data in the 108th US Senate where the behavior of an entity is its vote on a resolution. We apply our GT model also to voting data. However, unlike [3], since our goal is to cluster entities based on the similarity of their voting patterns, we are only interested in whether a pair of entities voted the same or differently, not their actual yes/no votes. This “content-ignorant” feature is similarly found in work on web log clustering [1]. 4 Experimental Results We present experiments applying the GT model to the voting records of members of two legislative bodies: the US Senate and the UN General Assembly. For comparison, we present the results of a baseline method that first uses a mixture of unigrams to discover topics and associate a topic with each resolution, and then runs the Blockstructures model [7] separately on the resolutions assigned to each topic. This baseline approach is similar to the GT model in that it discovers both groups and topics, and has different group assignments on different topics. However, whereas the baseline model performs inference serially, GT performs joint inference simultaneously. We are interested in the quality of both the groups and the topics. In the political science literature, group cohesion is quantified by the Agreement Index (AI) [3], which, based on the number of group members that vote Yes, No or Abstain, measures the similarity of votes cast by members of a group during a particular roll call. Higher AI means better cohesion. The group cohesion using the GT model is found to be significantly greater than the baseline group cohesion under pairwise t-test, as shown in Table 1 for both datasets, which indicates that the GT model is better able to capture cohesive groups. 4.1 The US Senate Dataset Our Senate dataset consists of the voting records of Senators in the 101st-109th US Senate (1989-2005) obtained from the Library of Congress THOMAS database. During a roll call for a particular bill, a Senator may respond Yea or Nay to the question that has been put to vote, else the vote will be recorded as Not Voting. We do not consider Not Voting as a unique vote since most of the time it is a result of a Senator being absent from the session of the US Senate. The text associated with each resolution is composed of its index terms provided in the database. There are 3423 resolutions in our experiments (we excluded roll calls that were not associated with resolutions). Since there are far fewer words than Economic Education Military Misc. Energy federal education government energy labor school military power insurance aid foreign water aid children tax nuclear tax drug congress gas business students aid petrol employee elementary law research care prevention policy pollution Table 2: Top words for topics generated with the mixture of unigrams model on the Senate dataset. The headers are our own summary of the topics. Economic Education + Domestic Foreign Social Security + Medicare labor education foreign social insurance school trade security tax federal chemicals insurance congress aid tariff medical income government congress care minimum tax drugs medicare wage energy communicable disability business research diseases assistance Table 3: Top words for topics generated with the GT model on the Senate dataset. The topics are influenced by both the words and votes on the bills. pairs of votes, we raise the text likelihood to the 5th power (mentioned in Section 2) in the experiments with this dataset so as to balance its influence during inference. We cluster the data into 4 topics and 4 groups (cluster sizes are chosen somewhat arbitrarily) and compare the results of GT with the baseline. The most likely words for each topic from the traditional mixture of unigrams model is shown in Table 2, whereas the topics obtained using GT are shown in Table 3. The GT model collapses the topics Education and Energy together into Education and Domestic, since the voting patterns on those topics are quite similar. The new topic Social Security + Medicare did not have strong enough word coherence to appear in the baseline model, but it has a very distinct voting pattern, and thus is clearly found by the GT model. Thus, importantly, GT discovers topics that help predict people’s behavior and relations, not simply word co-occurrences. Examining the group distribution across topics in the GT model, we find that on the topic Economic the Republicans form a single group whereas the Democrats split into 3 groups indicating that Democrats have been somewhat divided on this topic. On the other hand, in Education + Domestic and Social Security + Medicare, Democrats are more unified whereas the Republicans split into 3 groups. The group membership of Senators on Education + Domestic issues is shown in Table 4. We see that the first group of Republicans include a Democratic Senator from Texas, a state that usually votes Republican. Group 2 (majority Democrats) includes Sen. Chafee who has been involved in initiatives to improve education, as well as Sen. Jeffords who left the Republican Party to become an Independent and has championed legislation to strengthen education and environmental protection. Nearly all the Republican Senators in Group 4 (in Table 4) are advocates for education and many of them have been awarded for their efforts. For instance, Sen. Voinovich and Sen. Symms are strong supporters of early education and vocational education, respectively; and Group 1 Group 3 Group 4 73 Republicans Cohen (R-ME) Armstrong (R-CO) Brown (R-CO) Krueger (D-TX) Danforth (R-MO) Garn (R-UT) DeWine (R-OH) Group 2 Durenberger (R-MN) Humphrey (R-NH) Thompson (R-TN) 90 Democrats Hatfield (R-OR) McCain (R-AZ) Fitzgerald (R-IL) Chafee (R-RI) Heinz (R-PA) McClure (R-ID) Voinovich (R-OH) Jeffords (I-VT) Kassebaum (R-KS) Roth (R-DE) Miller (D-GA) Packwood (R-OR) Symms (R-ID) Coleman (R-MN) Specter (R-PA) Wallop(R-WY) Snowe (R-ME) Collins (R-ME) Table 4: Senators in the four groups corresponding to Education + Domestic in Table 3. Everything Nuclear Human Rights Security in Middle East nuclear rights occupied weapons human israel use palestine syria implementation situation security countries israel calls Table 5: Top words for topics generated from mixture of unigrams model with the UN dataset. Only text information is utilized to form the topics, as opposed to Table 6 where our GT model takes advantage of both text and voting information. Sen. Roth has voted for tax deductions for education. It is also interesting to see that Sen. Miller (D-GA) appears in a Republican group; although he is in favor of educational reforms, he is a conservative Democrat and frequently criticizes his own party—even backing Republican George W. Bush over Democrat John Kerry in the 2004 Presidential Election. Many of the Senators in Group 3 have also focused on education and other domestic issues such as energy, however, they often have a more liberal stance than those in Group 4, and come from states that are historically less conservative. For example, Sen. Danforth has presented bills for a more fair distribution of energy resources. Sen. Kassebaum is known to be uncomfortable with many Republican views on domestic issues such as education, and has voted against voluntary prayer in school. Thus, both Groups 3 and 4 differ from the Republican core (Group 2) on domestic issues, and also differ from each other. We also inspect the Senators that switch groups the most across topics in the GT model. The top 5 Senators are Shelby (D-AL), Heflin (D-AL), Voinovich (R-OH), Johnston (D-LA), and Armstrong (R-CO). Sen. Shelby (D-AL) votes with the Republicans on Economic, with the Democrats on Education + Domestic and with a small group of maverick Republicans on Foreign and Social Security + Medicare. Sen. Shelby, together with Sen. Heflin, is a Democrat from a fairly conservative state (Alabama) and are found to side with the Republicans on many issues. 4.2 The United Nations Dataset The second dataset involves the voting record of the UN General Assembly1. We focus on the resolutions discussed from 1990-2003, which contain votes of 192 countries on 931 resolutions. If a country is present during the roll call, it may choose to vote Yes, No or 1http://home.gwu.edu/∼voeten/UNVoting.htm G Nuclear Nonproliferation Nuclear Arms Race Human Rights R nuclear nuclear rights O states arms human U united prevention palestine P weapons race occupied ↓ nations space israel Brazil UK Brazil Columbia France Mexico 1 Chile Spain Columbia Peru Monaco Chile Venezuela... East-Timor Peru... USA India Nicaragua Japan Russia Papua 2 Germany Micronesia Rwanda UK... Swaziland Russia... Fiji... China Japan USA India Germany Japan 3 Mexico Italy... Germany Iran Poland UK... Pakistan... Hungary... Russia... Kazakhstan China China Belarus Brazil India 4 Yugoslavia Mexico Indonesia Azerbaijan Indonesia Thailand Cyprus... Iran... Philippines... Thailand USA Belarus Philippines Israel Turkmenistan 5 Malaysia Palau Azerbaijan Nigeria Uruguay Tunisia... Kyrgyzstan... Table 6: Top words for topics generated from the GT model with the UN dataset as well as the corresponding groups for each topic (column). The countries listed for each group are ordered by their 2005 GDP (PPP). Abstain. Unlike the Senate dataset, a country’s vote can have one of three possible values instead of two. Because we parameterize agreement and not the votes themselves, this 3value setting does not require any change to our model. In experiments with this dataset, we use a weighting factor 500 for text (adjusting the likelihood of text by a power of 500 so as to make it comparable with the likelihood of pairs of votes for each resolution). We cluster this dataset into 3 topics and 5 groups (chosen somewhat arbitrarily). The most probable words in each topic from the mixture of unigrams model is shown in Table 5. For example, Everything Nuclear constitutes all resolutions that have anything to do with the use of nuclear technology, including nuclear weapons. Comparing these with topics generated from the GT model shown in Table 6, we see that the GT model splits the discussion about nuclear technology into two separate topics, Nuclear Nonproliferation (generally about countries obtaining nuclear weapons and management of nuclear waste), and Nuclear Arms Race (focused on the historic arms race between Russia and the US, and preventing a nuclear arms race in outer space). These two issues had drastically different voting patterns in the UN, as can be seen in the contrasting group structure for those topics in Table 6. Thus, again, the GT model is able to discover more salient topics—topics that reflect the voting patterns and coalitions, not simply word co-occurrence alone. The countries in Table 6 are ranked by their GDP in 2005.2 As seen in Table 6, groups formed in Nuclear Arms Race are unlike the groups formed in other topics. These groups map well to the global political situation of that time when, despite the end of the Cold War, there was mutual distrust between Russia and the US with regard to the continued manufacture of nuclear weapons. For missions to outer space and nuclear arms, India was a staunch ally of Russia, while Israel was an ally of the US. 5 Conclusions We introduce the Group-Topic model that jointly discovers latent groups in a network as well as clusters of attributes (or topics) of events that influence the interaction between entities. The model extends prior work on latent group discovery by capturing not only pair-wise relations between entities but also multiple attributes of the relations (in particular, words describing the relations). In this way the GT model obtains more cohesive groups as well as salient topics that influence the interaction between groups. This paper demonstrates that the Group-Topic model is able to discover topics capturing the group based interactions between members of a legislative body. The model can be applied not just to voting data, but any data having relations with attributes. We are now using the model to analyze the citations in academic papers capturing the topics of research papers and discovering research groups. The model can be altered suitably to consider other categorical, multi-dimensional, and continuous attributes characterizing relations. Acknowledgments This work was supported in part by the CIIR, the Central Intelligence Agency, the National Security Agency, the National Science Foundation under NSF grant #IIS-0326249, and by the Defense Advanced Research Projects Agency, through the Department of the Interior, NBC, Acquisition Services Division, under contract #NBCHD030010. We would also like to thank Prof. Vincent Moscardelli, Chris Pal and Aron Culotta for helpful discussions. References [1] Doug Beeferman and Adam Berger. Agglomerative clustering of a search engine query log. In The 6th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining, 2000. [2] Indrajit Bhattacharya and Lise Getoor. Deduplication and group detection using links. In The 10th SIGKDD Conference Workshop on Link Analysis and Group Detection (LinkKDD), 2004. [3] Aleks Jakulin and Wray Buntine. Analyzing the US Senate in 2003: Similarities, networks, clusters and blocs, 2004. http://kt.ijs.si/aleks/Politics/us senate.pdf. [4] Charles Kemp, Thomas L. Griffiths, and Joshua Tenenbaum. Discovering latent classes in relational data. Technical report, AI Memo 2004-019, MIT CSAIL, 2004. [5] Jeremy Kubica, Andrew Moore, Jeff Schneider, and Yiming Yang. Stochastic link and group detection. In The 17th National Conference on Artificial Intelligence (AAAI), 2002. [6] Andrew McCallum, Andres Corrada-Emanuel, and Xuerui Wang. Topic and role discovery in social networks. In The 19th International Joint Conference on Artificial Intelligence, 2005. [7] Krzysztof Nowicki and Tom A.B. Snijders. Estimation and prediction for stochastic blockstructures. Journal of the American Statistical Association, 96(455):1077–1087, 2001. 2http://en.wikipedia.org/wiki/List of countries by GDP %28PPP%29. In Table 6, we omit some countries (represented by ...) in order to show other interesting but relatively low-ranked countries (for example, Russia) in the GDP list.
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Factorial Switching Kalman Filters for Condition Monitoring in Neonatal Intensive Care Christopher K. I. Williams and John Quinn School of Informatics, University of Edinburgh Edinburgh EH1 2QL, UK c.k.i.williams@ed.ac.uk john.quinn@ed.ac.uk Neil McIntosh Simpson Centre for Reproductive Health, Edinburgh EH16 4SB, UK neil.mcintosh@ed.ac.uk Abstract The observed physiological dynamics of an infant receiving intensive care are affected by many possible factors, including interventions to the baby, the operation of the monitoring equipment and the state of health. The Factorial Switching Kalman Filter can be used to infer the presence of such factors from a sequence of observations, and to estimate the true values where these observations have been corrupted. We apply this model to clinical time series data and show it to be effective in identifying a number of artifactual and physiological patterns. 1 Introduction In a neonatal intensive care unit (NICU), an infant’s vital signs, including heart rate, blood pressures, blood gas properties and temperatures, are continuously monitored and displayed at the cotside. The levels of these measurements and the way they vary give an indication of the baby’s health, but they can be affected by many different things. The potential factors include handling of the baby, different cardiovascular and respiratory conditions, the effects of drugs which have been administered, and the setup of the monitoring equipment. Each factor has an effect on the dynamics of the observations, some by affecting the physiology of the baby (such as an oxygen desaturation), and some by overwriting the measurements with artifactual values (such as a probe dropout). We use a Factorial Switching Kalman Filter (FSKF) to model such data. This consists of three sets of variables which we call factors, state and observations, as indicated in Figure 1(a). There are a number of hidden factors; these are discrete variables, modelling for example if the baby is in a normal respiratory state or not, or if a probe is disconnected or not. The state of baby denotes continuous-valued quantities; this models the true values of infant’s physiological variables, but also has dimensions to model certain artifact processes (see below). The observations are those readings obtained from the monitoring equipment, and are subject to corruption by artifact etc. By describing the dynamical regime associated with each combination of factors as a linear Gaussian model we obtain a FSKF, which extends the Switching Kalman Filter (see e.g. [10, 3]) to incorporate multiple independent factors. With this method we can infer the value of each factor and estimate the true values of vital signs during the times that the measurements are obscured by artifact. By using an interpretable hidden state structure for this application, domain knowledge can be used to set some of the parameters. This paper demonstrates an application of the FSKF to NICU monitoring data. In Section 2 we introduce the model, and discuss the links to previous work in the field. In Section 3 we describe an approach for setting the parameters of the model and in Section 4 we show results from the model when applied to NICU data. Finally we close with a discussion in Section 5. 2 Model description The Factorial Switching Kalman Filter is shown in Figure 1(a). In this model, M factors f (1) t . . . f (M) t affect the hidden continuous state xt and the observations yt. The factor f (m) can take on K(m) different values. For example, a simple factor is ‘ECG probe dropout’, taking on two possible values, ‘dropped out’ or ‘normal’. As factors in this application can affect the observations either by altering the baby’s physiology or overwriting them with artifactual values, the hidden state vector xt contains information on both the ‘true’ physiological condition of the baby and on the levels of any artifactual processes. The dynamical regime at time t is controlled by the ‘switch’ variable st, which is the cross product of the individual factors, st = f (1) t ⊗. . . ⊗f (M) t . (1) For a given setting of st, the hidden continuous state and the observations are related by: xt ∼N(A(st)xt−1 + d(st), Q(st)), yt ∼N(H(st)xt, R(st)), (2) where as in the SKF the system dynamics and observation process are dependent on the switch variable. Here A(st) is a square system matrix, d(st) is a drift vector, H(st) is the state-observations matrix, and Q(st) and R(st) are noise covariance matrices. The factors are taken to be a priori independent and first-order Markovian, so that p(st|st−1) = M Y m=1 p(f (m) t |f (m) t−1 ) . (3) 2.1 Application-specific setup The continuous hidden state vector x contains two types of values, the true physiological values, xp, and those of artifactual processes, xa. The true values are modelled as independent autoregressive processes, described in more detail in section 3. To represent this as a state space, the vector xt has to contain the value of the current state and store the value of the states at previous times. Note that artifact state values can be affected by physiological state, but not the other way round. For example, one factor we model is the arterial blood sample, seen in Figure 1(b), lower panel. This occurs when a three-way valve is closed in the baby’s arterial line, in order for a clinician to draw blood for a sample. While the valve is closed a pump works against the pressure sensor, causing the systolic and diastolic blood pressure measurements to rise artificially. The artifactual values in this case always start at around the value of the baby’s diastolic blood pressure. The factors modelled in these experiments are listed in Table 1. The dropout factors represent the case where probes are disconnected and measurements fall to zero on the channels supplied by that probe. In this case, the true physiological values are completely hidden. Artifactual state True state Observations Factor 2 (physiological) Factor 1 (artifactual) HR HR 0 50 100 150 200 Sys. BP Sys. BP 40 50 60 70 80 Blood sample Blood sample ECG dropout ECG dropout 0 200 400 600 800 1000 1200 (a) (b) Figure 1: (a) shows a graphical representation of a Factorial Switching Kalman Filter, with M = 2 factors. Squares are discrete values, circles are continuous and shaded nodes are observed. Panel (b) shows ECG dropout and arterial blood sample events occurring simultaneously. HR denotes heart rate, Sys. BP denotes the systolic blood pressure, and times are in seconds. The dashed line indicates the estimate of true values and the greyscale denotes two standard deviation error bars. We see uncertainty increasing while observations are artifactual. The traces at the bottom show the inferred duration of the arterial blood sample and ECG dropout events. The transcutaneous probe (TCP) provides measurements of the partial pressure of oxygen (TcPO2) and carbon dioxide (TcPCO2) in the baby’s blood, and is recalibrated every few hours. This process has three stages: firstly calibration, where TcPO2 and TcPCO2 are set to known values by applying a gas to the probe, secondly a stage where the probe is in air and TcPCO2 drops to zero, and finally an equilibration phase where both values slowly return to the physiological baseline when the probe is replaced. As explained above, when an arterial blood sample is being taken one sees a characteristic ramp in the blood pressure measurements. Temperature probe disconnection frequently occurs in conjunction with handling. The core temperature probe is under the baby and can come off when the baby is turned over for an examination, causing the readings to drop to the ambient temperature level of the incubator over the course of a few minutes. When the probe is reapplied, the measurements gradually return to the true level of the baby’s core temperature. Bradycardia is a genuine physiological occurrence where the heart rate temporarily drops, often with a characteristic curve, then a systemic reaction brings the measurements back to the baseline. The final factor models opening of the portals on the baby’s incubator. Because the environment within the incubator is closely regulated, an intervention can be inferred from a fall in the incubator humidity measurements. While the portals are open and a clinician is handling the baby, we expect increased variability in the measurements from the probes that are still attached. 2.2 Inference For the application of real time clinical monitoring, we are interested in filtering, inferring xt and st from the observations y1:t. However, the time taken for exact inference of the posterior p(xt, st|y1:t) scales exponentially with t, making it intractable. This is because the probabilities of having moved between every possible combination of switch settings in times t −1 and t are needed to calculate the posterior at time t. Hence the number of FACTOR POSSIBLE SETTINGS 5 Probe dropout factors: pulse oximeter, ECG, arterial line, temperature probe, transcutaneous probe 1. Dropped out 2. Normal TCP recalibration 1. O2 high, CO2 low 2. CO2 →0 3. Equilibration 4. Normal Arterial blood sample 1. Blood sample 2. Normal Temperature probe disconnection 1. Temperature probe disconnection 2. Reconnection 3. Normal Bradycardia 1. Bradycardia onset 2. HR restabilisation 3. Normal Incubator open 1. Incubator portals opened 2. Normal Table 1: Description of factors. Gaussians needed to represent the posterior exactly at each time step increases by a factor of K, the number of cross-product switch settings, where K = QM m=1 K(m). In this experiment we use the Gaussian Sum approximation [1]. At each time step we maintain an approximation of p(xt|st, y1:t) as a mixture of K Gaussians. Calculating the Kalman updates and likelihoods for every possible setting of st+1 will result in the posterior p(xt+1|st+1, y1:t+1) having K2 mixture components, which can be collapsed back into K components by matching means and variances of the distribution, as described in [6]. For comparison we also use Rao-Blackwellised particle filtering (RBPF) [7] for approximate inference. In this technique a number of particles are propagated through each time step, each with a switch state st and an estimate of the mean and variance of xt. A value for the switch state st+1 is obtained for each particle by sampling from the transition probabilities, after which Kalman updates are performed and a likelihood value can be calculated. Based on this likelihood, particles can be either discarded or multiplied. Because Kalman updates are not calculated for every possible setting of st+1, this method can give a significant increase in speed when there are many factors, with some tradeoff in accuracy. Both inference methods can be speeded up by considering the dropout factors. Because a probe dropout always results in an observation of zero on the corresponding measurement channels, the value of yt can be examined at each step. If it is not equal to zero then we know that the likelihood of a dropout factor being active will be very low, so there is no need to calculate it explicitly. Similarly, if any of the observations are zero then we only perform Kalman updates and calculate likelihoods for those switch states with the appropriate dropout setting. 2.3 Relation to previous work The SKF and various approximations for inference have been described by many authors, see e.g. [10, 3]. In [5], the authors used a 2-factor FSKF in a speech recognition application; the two factors corresponded to (i) phones and (ii) the phone-to-spectrum transformation. There has also been much prior work on condition monitoring in intensive care; here we give a brief review of some of these studies and the relationship to our own work. The specific problem of artifact detection in physiological time series data has been approached in a number of ways. For example Tsien [9] used machine learning techniques, notably decision trees and logistic regression, to classify each observation yt as genuine or artifactual. Hoare and Beatty [4] describe the use of time series analysis techniques (ARIMA models, moving average and Kalman filters) to predict the next point in a patient’s monitoring trace. If the difference between the observed value and the predicted value was outside a predetermined range, the data point was classified as artifactual. Our application of a model with factorial state extends this work by explaining the specific cause of an artifact, rather than just the fact that a certain data point is artifactual or not. We are not aware of other work in condition monitoring using a FSKF. 3 Parameter estimation We use hand-annotated training data from a number of babies to estimate the parameters of the model. Factor dynamics: Using equation 3 we can calculate the state transition probabilities from the transition probabilities for individual state variables, P(f (m) t = a|f (m) t−1 = b). The estimates for these are given by P(f (m) t = a|f (m) t−1 = b) = (nba + c) / PK(m) c=1 (nbc + c) , where nba is the number of transitions from state b to state a in the training data. The smoothing constant c (in our experiments we set c = 1) is added to stop any of the transition probabilities being zero or very small. While a zero probability could be useful for a sequence of states that we know are impossible, in general we want to avoid it. This solution can be justified theoretically as a maximum a posteriori estimate where the prior is given by a Dirichlet distribution. The factor dynamics can be used to create left-to-right models, e.g. for passing through the sequence O2 high, CO2 low; CO2 →0; equilibration in the TCP recalibration case. System dynamics: When no factor is active (i.e. non-normal), the baby is said to be in a stable condition and has some capacity for self-regulation. In this condition we consider each observation channel separately, and use standard methods to fit AR or ARIMA models to each channel. Most channels vary around reference ranges when the baby is stable and are well fitted by AR(2) models. Heart rate and blood pressure observation channels are more volatile and stationarity is improved after differencing. Heart rate dynamics, for example, are well fitted with an ARIMA(2,1,0) process. Representing trained AR or ARIMA processes in state space form is then straightforward. The observational data tends to have some high frequency noise on it (see e.g. Fig. 1(b), lower panel) due to probe error and quantization effects. Thus we smooth sections of stable data with a 21-point moving average in order to obtain training data for the system dynamics. The Yule-Walker equations are then used to set parameters for this moving-averaged data. The fit can be verified for each observation channel by comparing the spectrum of new data with the theoretical spectrum of the AR process (or the spectrum of the differenced data for ARIMA processes), see e.g. [2]. The measurement noise matrix R is estimated by calculating the variance of the differences between the original and averaged training data for each measurement channel. Above we have modelled the dynamics for a baby in the stable condition; we now describe some of the system models used when the factors are active (i.e. non-normal). The drop and rise in temperature measurements caused by a temperature probe disconnection closely resemble exponential decay and can be therefore be fitted with an AR(1) process. This also applies to the equilibration stage of a TCP recalibration. The dynamics corresponding to the bradycardia factor are set by finding the mean slope of the fall and rise in heart rate, which is used for the drift term d, then fitting an AR(1) process to the residuals. The arterial blood sample dynamics are modelled with linear drift; note that the variable in xa corresponding to the value of the arterial blood sample is tied Blood sample TCP recal. Bradycardia TC disconnect Incu. open AUC EER AUC EER AUC EER AUC EER AUC EER FHMM 0.97 0.02 0.78 0.25 0.67 0.42 0.75 0.35 0.97 0.07 GS 0.99 0.01 0.91 0.12 0.72 0.39 0.88 0.19 0.97 0.06 RBPF 0.62 0.46 0.90 0.14 0.76 0.37 0.85 0.32 0.95 0.08 Table 2: Inference results on evaluation data. FHMM denotes the Factorial Hidden Markov Model, GS denotes the Gaussian Sum approximation, and RBPF denotes RaoBlackwellised particle filtering with 560 particles. AUC denotes area under ROC curve and EER denotes the equal error rate. to the diastolic blood pressure value while the factor is inactive. We also use linear drift to model the drop in incubator humidity measurements corresponding to a clinician opening the incubator portals. We assume that the measurement noise from each probe is the same for physiological and artifactual readings, for example if the core temperature probe is attached to the baby’s skin or is reading ambient incubator temperature. Combining factors: The parameters {A, H, Q, R, d} have to be supplied for every combination of factors. It might be thought that training data would be needed for each of these possible combinations, but in practice parameters can be trained for factors individually and then combined, as we know that some of the phenomena we want to model only affect a subset of the channels, or override other phenomena [8]. This process of setting parameters for each combination of factor settings can be automated. The factors are arranged in a partially ordered set, where later factors overwrite the dynamics A, Q, d or observations H, R on at least one channel of their predecessor. For example, the ‘bradycardia’ factor overwrites the heart rate dynamics of the normal state, while the ‘ECG dropout’ factor overwrites the heart rate observations; if both these things are happening simultaneously then we expect the same observations as if there was only an ECG dropout, but the dynamics of the true state xp are propagated as though there was only a bradycardia. Having found this ordering it is straightforward to merge the trained parameters for every combination of factors. 4 Results Monitoring data was obtained from eight infants of 28 weeks gestation during their first week of life, from the NICU at Edinburgh Royal Infirmary. The data for each infant was collected every second for 24 hours, on nine channels: heart rate, systolic and diastolic blood pressures, TcPO2, TcPCO2, O2 saturation, core temperature and incubator temperature and humidity. These infants were the first 8 in the NICU database who satisfied the age criteria and were monitored on all 8 channels for some 24 hour period within their first week. Four infants were used for training the model and four for evaluation. The test data was annotated with the times of occurrences of the factors in Table 1 by a clinical expert and one of the authors. Some examples of inference under the model are shown in Figures 1(b) and 2. In Figure 1(b) two factors, arterial blood sample and ECG dropout are simultaneously active, and the inference works nicely in this case, with growing uncertainty about the true value of the heart-rate and blood pressure channels when artifactual readings are observed. The upper panel in figure 2(a) shows two examples of bradycardia being detected. In the lower panel, the model correctly infers the times that a clinician enters the incubator and replaces a disconnected core temperature probe. Figure 2(b) illustrates the simultaneous detection of a TCP artifact (the TCP recal state plotted is obtained by summing the probabilities of 50 100 150 200 HR Bradycardia 0 20 40 60 80 35 36 37 38 Core temp. 55 60 65 70 Incu humidity Incu open TC probe off 0 1000 2000 3000 40 60 80 Sys. BP 30 40 50 60 70 Dia. BP 0 5 10 TcPCO2 0 10 20 30 TcPO2 TCP recal Blood sample 0 500 1000 1500 2000 (a) (b) Figure 2: Inferred durations of physiological and artifactual states: (a) shows two episodes of bradycardia (top), and a clinician entering the incubator and replacing the core temperature probe (bottom). Plot (b) shows the inference of two simultaneous artifact processes, arterial blood sampling and TCP recalibration. Times are in seconds. the three non-normal TCP states) and a blood sample spike. In Table 2 we show the performance of the model on the test data. The inferred probabilities for each factor were compared with the gold standard which has a binary value for each factor setting at each time point. Inference was done using the Gaussian sum approximation and RBPF, where the number of particles was set so that the two inference methods had the same execution time. As a baseline we also used a Factorial Hidden Markov Model (FHMM) to infer when each factor was active. This model has the same factor structure as the FSKF, without any hidden continuous state. The FHMM parameters were set using the same training data as the FSKF. It can be seen that the FSKF generalised well to the data from the test set. Inferences using the Gaussian Sum approximation had consistently higher area under the ROC curve and lower equal error rates than the FHMM. In particular, the inferred times of blood samples and incubator opening were reliably detected. The lower performance of the FHMM, which has no knowledge of the dynamics, illustrates the difficulty caused by baseline physiological levels changing over time and between babies. Inference results using Rao-Blackwellised particle filtering were less consistent. For blood sampling and opening of the incubator the performance was worse than the baseline model, though in detecting bradycardia the performance was marginally higher than for inferences made using either the FHMM or the Gaussian Sum approximation. Execution times for inference on 24 hours of monitoring data with the set of factors listed in Table 1 on a 3.2GHz processor were approximately 7 hours 10 minutes for the FSKF inference, and 100 seconds for the FHMM. 5 Discussion In this paper we have shown that the FSKF model can be applied successfully to complex monitoring data from a neonatal intensive care unit. There are a number of directions in which this work can be extended. Firstly, for simplicity we have used univariate autoregressive models for each component of the observations; it would be interesting to fit a multivariate model to this data instead, as we expect that there will be correlations between the channels. Also, there are additional factors that can be incorporated into the model, for example to model a pneumothorax event, where air becomes trapped inside the chest between the chest wall and the lung, causing the lung to collapse. Fortunately this event is relatively rare so it was not seen in the data we have analyzed in this experiment. Acknowledgements We thank Birgit Wefers for providing expert annotation of the evaluation data set, and the anonymous referees for their comments which helped improve the paper. This work was funded in part by a grant from the premature baby charity BLISS. The work was also supported in part by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778. This publication only reflects the authors’ views. References [1] D. L. Alspach and H. W. Sorenson. Nonlinear Bayesian Estimation Using Gaussian Sum Approximations. IEEE Transactions on Automatic Control, 17(4):439–448, 1972. [2] C. Chatfield. The Analysis of Time Series: An Introduction. Chapman and Hall, London, 4th edition, 1989. [3] Z. Ghahramani and G. E. Hinton. Variational Learning for Switching State-Space Models. Neural Computation, 12(4):963–996, 1998. [4] S.W. Hoare and P.C.W. Beatty. Automatic artifact identification in anaesthesia patient record keeping: a comparison of techniques. Medical Engineering and Physics, 22:547–553, 2000. [5] J. Ma and L. Deng. A mixed level switching dynamic system for continuous speech recognition. Computer Speech and Language, 18:49–65, 2004. [6] K. Murphy. Switching Kalman filters. Technical report, U.C. Berkeley, 1998. [7] K. Murphy and S. Russell. Rao-Blackwellised particle filtering for dynamic Bayesian networks. In A. Doucet, N. de Freitas, and N. Gordon, editors, Sequential Monte Carlo in Practice. Springer-Verlag, 2001. [8] A. Spengler. Neonatal baby monitoring. Master’s thesis, School of Informatics, University of Edinburgh, 2003. [9] C. Tsien. TrendFinder: Automated Detection of Alarmable Trends. PhD thesis, MIT, 2000. [10] M. West and P. J. Harrison. Bayesian Forecasting and Dynamic Models. SpringerVerlag, 1997. Second edition.
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A Criterion for the Convergence of Learning with Spike Timing Dependent Plasticity Robert Legenstein and Wolfgang Maass Institute for Theoretical Computer Science Technische Universitaet Graz A-8010 Graz, Austria {legi,maass}@igi.tugraz.at Abstract We investigate under what conditions a neuron can learn by experimentally supported rules for spike timing dependent plasticity (STDP) to predict the arrival times of strong “teacher inputs” to the same neuron. It turns out that in contrast to the famous Perceptron Convergence Theorem, which predicts convergence of the perceptron learning rule for a simplified neuron model whenever a stable solution exists, no equally strong convergence guarantee can be given for spiking neurons with STDP. But we derive a criterion on the statistical dependency structure of input spike trains which characterizes exactly when learning with STDP will converge on average for a simple model of a spiking neuron. This criterion is reminiscent of the linear separability criterion of the Perceptron Convergence Theorem, but it applies here to the rows of a correlation matrix related to the spike inputs. In addition we show through computer simulations for more realistic neuron models that the resulting analytically predicted positive learning results not only hold for the common interpretation of STDP where STDP changes the weights of synapses, but also for a more realistic interpretation suggested by experimental data where STDP modulates the initial release probability of dynamic synapses. 1 Introduction Numerous experimental data show that STDP changes the value wold of a synaptic weight after pairing of the firing of the presynaptic neuron at time tpre with a firing of the postsynaptic neuron at time tpost = tpre + ∆t to wnew = wold + ∆w according to the rule wnew = min{wmax, wold + W+ · e−∆t/τ+} , if ∆t > 0 max{0, wold −W−· e∆t/τ−} , if ∆t ≤0 , (1) with some parameters W+, W−, τ+, τ−> 0 (see [1]). If during training a teacher induces firing of the postsynaptic neuron, this rule becomes somewhat analogous to the well-known perceptron learning rule for McCulloch-Pitts neurons (= “perceptrons”). The Perceptron Convergence Theorem states that this rule enables a perceptron to learn, starting from any initial weights, after finitely many errors any transformation that it could possibly implement. However, we have constructed examples of input spike trains and teacher spike trains (omitted in this abstract) such that although a weight vector exists which produces the desired firing and which is stable under STDP, learning with STDP does not converge to a stable solution. On the other hand experiments in vivo have shown that neurons can be taught by suitable teacher input to adopt a given firing response [2, 3] (although the spiketiming dependence is not exploited there). We show in section 2 that such convergence of learning can be explained by STDP in the average case, provided that a certain criterion is met for the statistical dependence among Poisson spike inputs. The validity of the proposed criterion is tested in section 3 for more realistic models for neurons and synapses. 2 An analytical criterion for the convergence of STDP The average case analysis in this section is based on the linear Poisson neuron model (see [4, 5]). This neuron model outputs a spike train Spost(t) which is a realization of a Poisson process with the underlying instantaneous firing rate Rpost(t). We represent a spike train S(t) as a sum of Dirac-δ functions S(t) = P k δ(t −tk), where tk is the kth spike time of the spike train. The effect of an input spike at input i at time t′ is modeled by an increase in the instantaneous firing rate of an amount wi(t′)ϵ(t −t′), where ϵ is a response kernel and wi(t′) is the synaptic efficacy of synapse i at time t′. We assume ϵ(s) = 0 for s < 0 (causality), R ∞ 0 ds ϵ(s) = 1 (normalization of the response kernel), and ϵ(s) ≥0 for all s as well as wi ≥0 for all i (excitatory inputs). In the linear model, the contributions of all inputs are summed up linearly: Rpost(t) = n X j=1 Z ∞ 0 ds wj(t −s) ϵ(s) Sj(t −s) , (2) where S1, . . . , Sn are the n presynaptic spike trains. Note that in this spike generation process, the generation of an output spike is independent of previous output spikes. The STDP-rule (1) avoids the growth of weights beyond bounds 0 and wmax by simple clipping. Alternatively one can make the weight update dependent on the actual weight value. In [5] a general rule is suggested where the weight dependence has the form of a power law with a non-negative exponent µ. This weight update rule is defined by ∆w = W+ · (1 −w)µ · e−∆t/τ+ , if ∆t > 0 −W−· wµ · e∆t/τ− , if ∆t ≤0 , (3) where we assumed for simplicity that wmax = 1. Instead of looking at specific input spike trains, we consider the average behavior of the weight vector for (possibly correlated) homogeneous Poisson input spike trains. Hence, the change ∆wi is a random variable with a mean drift and fluctuations around it. We will in the following focus on the drift by assuming that individual weight changes are very small and only averaged quantities enter the learning dynamics, see [6]. Let Si be the spike train of input i and let S∗be the output spike train of the neuron. The mean drift of synapse i at time t can be approximated as ˙wi(t) = W+(1 −wi)µ Z ∞ 0 ds e−s/τCi(s; t) −W−wµ i Z 0 −∞ ds es/τCi(s; t) , (4) where Ci(s; t) = ⟨Si(t)S∗(t+s)⟩E is the ensemble averaged correlation function between input i and the output of the neuron (see [5, 6]). For the linear Poisson neuron model, inputoutput correlations can be described by means of correlations in the inputs. We define the normalized cross correlation between input spike trains Si and Sj with a common rate r > 0 as C0 ij(s) = ⟨Si(t) Sj(t + s)⟩E r2 −1 , (5) which assumes value 0 for uncorrelated Poisson spike trains. We assume in this article that C0 ij is constant over time. In our setup, the output of the neuron during learning is clamped to the teacher spike train S∗which is the output of a neuron with the target weight vector w∗. Therefore, the input-output correlations Ci(s; t) are also constant over time and we denote them by Ci(s) in the following. In our neuron model, correlations are shaped by the response kernel ϵ(s) and they enter the learning equation (4) with respect to the learning window. This motivates the definition of window correlations c+ ij and c− ij for the positive and negative learning window respectively: c± ij = 1 + 1 τ Z ∞ 0 ds e−s/τ Z ∞ 0 ds′ ϵ(s′)C0 ij(±s −s′) . (6) We call the matrices C± = {c± ij}i,j=1,...,n the window correlation matrices. Note that window correlations are non-negative and that for homogeneous Poisson input spike trains and for a non-negative response kernel, they are positive. For soft weight bounds and µ > 0, a synaptic weight can converge to a value arbitrarily close to 0 or 1, but not to one of these values directly. This motivates the following definition of learnability. Definition 2.1 We say that a target weight vector w∗∈{0, 1}n can approximately be learned in a supervised paradigm by STDP with soft weight bounds on homogeneous Poisson input spike trains (short: “w∗can be learned”) if and only if there exist W+, W−> 0, such that for µ →0 the ensemble averaged weight vector ⟨w(t)⟩E with learning dynamics given by Equation 4 converges to w∗for any initial weight vector w(0) ∈[0, 1]n. We are now ready to formulate an analytical criterion for learnability: Theorem 2.1 A weight vector w∗can be learned (when being teached with S∗) for homogeneous Poisson input spike trains with window correlation matrices C+ and C−to a linear Poisson neuron with non-negative response kernel if and only if w∗̸= 0 and Pn k=1 w∗ kc+ ik Pn k=1 w∗ kc− ik > Pn k=1 w∗ kc+ jk Pn k=1 w∗ kc− jk for all pairs ⟨i, j⟩∈{1, . . . , n}2 with w∗ i = 1 and w∗ j = 0. Proof idea: The correlation between an input and the teacher induced output is (by Eq. 2): Ci(s) = ⟨Si(t) S∗(t + s)⟩E = n X j=1 w∗ j Z ∞ 0 ds′ ϵ(s′) ⟨Si(t) Sj(t + s −s′)⟩E . Substitution of this equation into Eq. 4 yields the synaptic drift ˙wi = τr2 W+(1 −wi)µ n X j=1 w∗ j c+ ij −W−wµ i n X j=1 w∗ j c− ij . (7) We find the equilibrium points wµi of synapse i by setting ˙wi = 0 in Eq. 7. This yields wµi = 1 + 1 Λ1/µ i −1 , where Λi denotes W+ W− Pn j=1 w∗ j c+ ij Pn j=1 w∗ j c− ij . Note that the drift is zero if w∗= 0 which implies that w∗= 0 cannot be learned. For w∗̸= 0, one can show that wµ = (wµ1, . . . , wµn) is the only equilibrium point of the system and that it is stable. Since the system decomposes into n independent one-dimensional systems, convergence to w∗is guaranteed for all initial conditions. Furthermore, one sees that limµ→0 wµi = 1 if and only if Λi > 1, and limµ→0 wµi = 0 if and only if Λi < 1. Therefore, limµ→0 wµ = w∗holds if and only if Λi > 1 for all i with w∗ i = 1 and Λi < 1 for all i with w∗ i = 0. The theorem follows from the definition of Λi. For a wide class of cross-correlation functions, one can establish a relationship between learnability by STDP and the well-known concept of linear separability from linear algebra.1 Because of synaptic delays, the response of a spiking neuron to an input spike is delayed by some time t0. One can model such a delay in the response kernel by the restriction ϵ(s) = 0 for all s ≤t0. In the following Corollary we consider the case where input correlations C0 ij(s) appear only in a time window smaller than the delay: Corollary 2.1 If there exists a t0 ≥0 such that ϵ(s) = 0 for all s ≤t0 and C0 ij(s) = 0 for all s < −t0, i, j ∈{1, . . . , n}, then the following holds for the case of homogeneous Poisson input spike trains to a linear Poisson neuron with positive response kernel ϵ: A weight vector w∗can be learned if and only if w∗̸= 0 and w∗linearly separates the list L = ⟨⟨c+ 1 , w∗ 1⟩, . . . , ⟨c+ n , w∗ n⟩⟩, where c+ 1 , . . . , c+ n are the rows of C+. Proof idea: From the assumptions of the corollary it follows that c− ij = 1. In this case, the condition in Theorem 2.1 is equivalent to the statement that w∗linearly separates the list L = ⟨⟨c+ 1 , w∗ 1⟩, . . . , ⟨c+ n , w∗ n⟩⟩. Corollary 2.1 can be viewed as an analogon of the Perceptron Convergence Theorem for the average case analysis of STDP. Its formulation is tight in the sense that linear separability of the list L alone (as opposed to linear separability by the target vector w∗) is not sufficient to imply learnability. For uncorrelated input spike trains of rate r > 0, the normalized cross correlation functions are given by C0 ij(s) = δij r δ(s), where δij is the Kronecker delta function. The positive window correlation matrix C+ is therefore essentially a scaled version of the identity matrix. The following corollary then follows from Corollary 2.1: Corollary 2.2 A target weight vector w∗∈{0, 1}n can be learned in the case of uncorrelated Poisson input spike trains to a linear Poisson neuron with positive response kernel ϵ such that ϵ(s) = 0 for all s ≤0 if and only if w∗̸= 0. 3 Computer simulations of supervised learning with STDP In order to make a theoretical analysis feasible, we needed to make in section 2 a number of simplifying assumptions on the neuron model and the synapse model. In addition a number of approximations had to be used in order to simplify the estimates. We consider in this section the more realistic integrate-and-fire model2 for neurons and a model for synapses which are subject to paired-pulse depression and paired-pulse facilitation, in addition to the long term plasticity induced by STDP [7]. This model describes synapses with parameters U (initial release probability), D (depression time constant), and F (facilitation time constant) in addition to the synaptic weight w. The parameters U, D, and F were randomly 1Let c1, . . . , cm ∈Rn and y1, . . . , ym ∈{0, 1}. We say that a vector w ∈Rn linearly separates the list ⟨⟨c1, y1⟩, . . . , ⟨cm, ym⟩⟩if there exists a threshold Θ such that yi = sign(ci · w −Θ) for i = 1, . . . , m. We define sign(z) = 1 if z ≥0 and sign(z) = 0 otherwise. 2The membrane potential Vm of the neuron is given by τm dVm dt = −(Vm −Vresting) + Rm · (Isyn(t)+Ibackground +Iinject(t)) where τm = Cm ·Rm = 30ms is the membrane time constant, Rm = 1MΩis the membrane resistance, Isyn(t) is the current supplied by the synapses, Ibackground is a constant background current, and Iinject(t) represents currents induced by a “teacher”. If V m exceeds the threshold voltage Vthresh it is reset to Vreset = 14.2mV and held there for the length Trefract = 3ms of the absolute refractory period.Neuron parameters: Vresting = 0V , Ibackground randomly chosen for each trial from the interval [13.5nA, 14.5nA]. Vthresh was set such that each neuron spiked at a rate of about 25 Hz. This resulted in a threshold voltage slightly above 15mV . Synaptic parameters: Synaptic currents were modeled as exponentially decaying currents with decay time constants τS = 3ms (τS = 6ms) for excitatory (inhibitory) synapses. chosen from Gaussian distributions that were based on empirically found data for such connections. We also show that in some cases a less restrictive teacher forcing suffices, that tolerates undesired firing of the neuron during training. The results of section 2 predict that the temporal structure of correlations has a strong influence on the outcome of a learning experiment. We used input spike trains with cross correlations that decay exponentially with a correlation decay constant τcc.3 In experiment 1 we consider temporal correlations with τcc=10ms. Since such “broader” correlations are not problematic for STDP, sharper correlations (τcc=6ms) are considered in experiment 2. Experiment 1 (correlated input with τcc=10ms): In this experiment, a leaky integrateand-fire neuron received inputs from 100 dynamic synapses. 90% of these synapses were excitatory and 10% were inhibitory. For each excitatory synapse, the maximal efficacy wmax was chosen from a Gaussian distribution with mean 54 and SD 10.8, bounded by 54 ± 3SD. The 90 excitatory inputs were divided into 9 groups of 10 synapses per group. Spike trains were correlated within groups with correlation coefficients between 0 and 0.8, whereas there were virtually no correlations between spike trains of different groups.4 Target weight vectors w∗were chosen in the most adverse way: half of the weights of w∗ within each group was set to 0, the other half to its maximal value wmax (see Fig. 1C). 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time [sec] target trained A 0 500 1000 1500 2000 2500 0 0.2 0.4 0.6 0.8 1 time [sec] B angular error [rad] spike correlation [σ=5ms] 0 20 40 60 80 0 20 40 60 Synapse w trained D 0 20 40 60 80 0 20 40 60 Synapse w target C Figure 1: Learning a target weight vector w∗on correlated Poisson inputs. A) Output spike train on test data after one hour of training (trained) compared to the target output (target). B) Evolution of the angle between weight vector w(t) and the vector w∗that implements F in radiant (angular error, solid line), and spike correlation (dashed line). C) Target weight vector w∗consisting of elements with value 0 or the value wmax assigned to that synapse. D) Corresponding weights of the learned vector w(t) after 40 minutes of training. (All time data refer to simulated biological time) Before training, the weights of all excitatory synapses were initialized by randomly chosen small values. Weights of inhibitory synapses remained fixed throughout the experiment. Information about the target weight vector w∗was given to the neuron only in the form of short current injections (1 µA for 0.2 ms) at those times when the neuron with the weight vector w∗would have produced a spike. Learning was implemented as standard STDP (see rule 1) with parameters τ+ = τ−= 20ms, W+ = 0.45, W−/W+ = 1.05. Additional inhibitory input was given to the neuron during training that reduced the occurrence of non3We constructed input spike trains with normalized cross correlations (see Equation 5) approximately given by C0 ij(s) = ccij 2τccr e−|s|/τcc between inputs i and j for a mean input rate of r = 20Hz, a correlation coefficient cij, and a correlation decay constant of τcc = 10ms. 4The correlation coefficient cij for spike trains within group k consisting of 10 spike trains was set to cij = cck = 0.1 ∗(k −1) for k = 1, . . . , 9. teacher-induced firing of the neuron (see text below).5 Two different performance measures were used for analyzing the learning progress. The “spike correlation” measures for test inputs that were not used for training (but had been generated by the same process) the deviation between the output spike train produced by the target weight vector w∗for this input, and the output spike train produced for the same input by the neuron with the current weight vector w(t)6. The angular error measures the angle between the current weight vector w(t) and the target weight vector w∗. The results are shown in Fig. 1. One can see that the deviation of the learned weight vector shown in panel D from the target weight vector w∗(panel C) is very small, even for highly correlated groups of synapses with heterogeneous target weights. No significant changes in the results were observed for longer simulations (4 hours simulated biological time), showing stability of learning. On 20 trials (each with a new random distribution of maximal weights wmax, different initializations w(0) of the weight vector before learning, and new Poisson spike trains), a spike correlation of 0.83±0.06 was achieved (angular error 6.8±4.7 degrees). Note that learning is not only based on teacher spikes but also on non teacher-induced firing. Therefore, strongly correlated groups of inputs tend to cause autonomous (i.e., not teacher-induced) firing of the neuron which results in weight increases for all weights within the corresponding group of synapses according to well-known results for STDP [8, 5]. Obviously this effect makes it quite hard to learn a target weight vector w∗where half of the weights for each correlated group have value 0. The effect is reduced by the additional inhibitory input during training which reduces undesired firing. However, without this input a spike correlation of 0.79 ± 0.09 could still be achieved (angular error 14.1 ± 10 degrees). 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 input correlation cc spike correlation A 0 5 10 15 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 weight error [°] 1−spike correlation B predicted to be learnable predicted to be not learnable Figure 2: A) Spike correlation achieved for correlated inputs (solid line). Some inputs were correlated with cc plotted on the x-axis. Also, as a control the spike correlation achieved by randomly drawn weight vectors is shown (dashed line, where half of the weights were set to wmax and the other weights were set to 0). B) Comparison between theory and simulation results for a leaky integrateand-fire neuron and input correlations between 0.1 and 0.5 (τcc = 6ms). Each cross (open circle) marks a trial where the target vector was learnable (not learnable) according to Theorem 2.1. The actual learning performance of STDP is plotted for each trial in terms of the weight error (x-axis) and 1 minus the spike correlation (y-axis). Experiment 2 (testing the theoretical predictions for τcc=6ms): In order to evaluate the dependence of correlation among inputs we proceeded in a setup similar to experiment 1. 4 input groups consisting each of 10 input spike trains were constructed for which the correlations within each group had the same value cc while the input spike train to the other 50 excitatory synapses were uncorrelated. Again, half of the weights of w∗within 5We added 30 inhibitory synapses with weights drawn from a gamma distribution with mean 25 and standard deviation 7.5, that received additional 30 uncorrelated Poisson spike trains at 20 Hz. 6For that purpose each spike in these two output spike trains was replaced by a Gaussian function with an SD of 5 ms. The spike correlation between both output spike trains was defined as the correlation between the resulting smooth functions of time (for segments of length 100 s). each correlated group (and within the uncorrelated group) was set to 0, the other half to a randomly chosen maximal value. The learning performance after 1 hour of training for 20 trials is plotted in Fig. 2A for 7 different values of the correlation cc (τcc = 6ms) that is applied in 4 of the input groups (solid line). In order to test the approximate validity of Theorem 2.1 for leaky integrate-and-fire neurons and dynamic synapses, we repeated the above experiment for input correlations cc = 0.1, 0.2, 0.3, 0.4, and 0.5. For each correlation value, 20 learning trials (with different target vectors) were simulated. For each trial we first checked whether the (randomly chosen) target vector w∗was learnable according to the condition given in Theorem 2.1 (65% of the 100 learning trials were classified as being learnable).7 The actual performance of learning with STDP was evaluated after 50 minutes of training.8 The result is shown in Fig. 2B. It shows that the theoretical prediction of learnability or non-learnability for the case of simpler neuron models and synapses from Theorem 2.1 translates in a biologically more realistic scenario into a quantitative grading of the learning performance that can ultimately be achieved with STDP. 0 500 1000 1500 2000 2500 0 0.2 0.4 0.6 0.8 1 time [sec] A angular error [rad] weight deviation spike correlation 5 10 15 20 0 0.1 0.2 Synapse U trained C 5 10 15 20 0 0.1 0.2 Synapse U target B Figure 3: Results of modulation of initial release probabilities U. A) Performance of U-learning for a generic learning task (see text). B) Twenty values of the target U vector (each component assumes its maximal possible value or the value 0). C) Corresponding U values after 42 minutes of training. Experiment 3 (Modulation of initial release probabilities U by STDP): Experimental data from [9] suggest that synaptic plasticity does not change the uniform scaling of the amplitudes of EPSPs resulting from a presynaptic spike train (i.e., the parameter w), but rather redistributes the sum of their amplitudes. If one assumes that STDP changes the parameter U that determines the synaptic release probability for the first spike in a spike train, whereas the weight w remains unchanged, then the same experimental data that support the classical rule for STDP, support the following rule for changing U: Unew = min{Umax, Uold + U+ · e−∆t/τ+} , if ∆t > 0 max{0, Uold −U−· e∆t/τ−} , if ∆t ≤0 , (8) with suitable nonnegative parameters Umax, U+, U−, τ+, τ−. Fig. 3 shows results of an experiment where U was modulated with rule (8) (similar to experiment 1, but with uncorrelated inputs). 20 repetitions of this experiment yielded after 42 minutes of training the following results: spike correlation 0.88 ± 0.036, angular error 27.9 ± 3.7 degrees, for U+ = 0.0012, U−/U+ = 1.055. Apparently the output spike train is less sensitive to changes in the values of U than to changes in w. Consequently, since 7We had chosen a response kernel of the form ϵ(s) = 1 τ1−τ2 (e−s/τ1 −e−s/τ2) with τ1 = 2ms and τ2 = 1ms (Least mean squares fit of the double exponential to the peri-stimulus-time histogram (PSTH) of the neuron, which reflects the probability of spiking as a function of time s since an input spike), and calculated the window correlations c+ ij and c− ij numerically. 8To guarantee the best possible performance for each learning trial, training was performed on 27 different values for W−/W+ between 1.02 and 1.15. only the behavior of a neuron with vector U∗but not the vector U∗is made available to the neuron during training, the resulting correlation between target- and actual output spike trains is quite high, whereas angular error between U∗and U(t), as well as the average deviation in U, remain rather large. We also repeated experiment 1 (correlated Poisson inputs) with rule (8) for U-learning. 20 repetitions with different target weights and different initial conditions yielded after 35 minutes of training: spike correlation 0.75 ± 0.08, angular error 39.3 ± 4.8 degrees, for U+ = 8 · 10−4, U−/U+ = 1.09. 4 Discussion The main conclusion of this article is that for many common distributions of input spikes a spiking neuron can learn with STDP and teacher-induced input currents any map from input spike trains to output spike trains that it could possibly implement in a stable manner. We have shown in section 2 that a mathematical average case analysis can be carried out for supervised learning with STDP. This theoretical analysis produces the first criterion that allows us to predict whether supervised learning with STDP will succeed in spite of correlations among Poisson input spike trains. For the special case of “sharp correlations” (i.e. when the cross correlations vanish for time shifts larger than the synaptic delay) this criterion can be formulated in terms of linear separability of the rows of a correlation matrix related to the spike input, and its mathematical form is therefore reminiscent of the wellknown condition for learnability in the case of perceptron learning. In this sense Corollary 2.1 can be viewed as an analogon of the Perceptron Convergence Theorem for spiking neurons with STDP. Furthermore we have shown that an alternative interpretation of STDP where one assumes that it modulates the initial release probabilities U of dynamic synapses, rather than their scaling factors w, gives rise to very satisfactory convergence results for learning. Acknowledgment: We would like to thank Yves Fregnac, Wulfram Gerstner, and especially Henry Markram for inspiring discussions. References [1] L. F. Abbott and S. B. Nelson. Synaptic plasticity: taming the beast. Nature Neurosci., 3:1178– 1183, 2000. [2] Y. Fregnac, D. Shulz, S. Thorpe, and E. Bienenstock. A cellular analogue of visual cortical plasticity. Nature, 333(6171):367–370, 1988. [3] D. Debanne, D. E. Shulz, and Y. Fregnac. Activity dependent regulation of on- and off-responses in cat visual cortical receptive fields. Journal of Physiology, 508:523–548, 1998. [4] R. Kempter, W. Gerstner, and J. L. van Hemmen. Intrinsic stabilization of output rates by spikebased hebbian learning. Neural Computation, 13:2709–2741, 2001. [5] R. G¨utig, R. Aharonov, S. Rotter, and H. Sompolinsky. Learning input correlations through non-linear temporally asymmetric hebbian plasticity. Journal of Neurosci., 23:3697–3714, 2003. [6] R. Kempter, W. Gerstner, and J. L. van Hemmen. Hebbian learning and spiking neurons. Phys. Rev. E, 59(4):4498–4514, 1999. [7] H. Markram, Y. Wang, and M. Tsodyks. Differential signaling via the same axon of neocortical pyramidal neurons. PNAS, 95:5323–5328, 1998. [8] S. Song, K. D. Miller, and L. F. Abbott. Competitive hebbian learning through spike-timing dependent synaptic plasticity. Nature Neuroscience, 3:919–926, 2000. [9] H. Markram and M. Tsodyks. Redistribution of synaptic efficacy between neocortical pyramidal neurons. Nature, 382:807–810, 1996.
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Q-Clustering Mukund Narasimhan† Nebojsa Jojic‡ Jeff Bilmes† †Dept of Electrical Engineering, University of Washington, Seattle WA ‡Microsoft Research, Microsoft Corporation, Redmond WA {mukundn,bilmes}@ee.washington.edu and jojic@microsoft.com Abstract We show that Queyranne’s algorithm for minimizing symmetric submodular functions can be used for clustering with a variety of different objective functions. Two specific criteria that we consider in this paper are the single linkage and the minimum description length criteria. The first criterion tries to maximize the minimum distance between elements of different clusters, and is inherently “discriminative”. It is known that optimal clusterings into k clusters for any given k in polynomial time for this criterion can be computed. The second criterion seeks to minimize the description length of the clusters given a probabilistic generative model. We show that the optimal partitioning into 2 clusters, and approximate partitioning (guaranteed to be within a factor of 2 of the the optimal) for more clusters can be computed. To the best of our knowledge, this is the first time that a tractable algorithm for finding the optimal clustering with respect to the MDL criterion for 2 clusters has been given. Besides the optimality result for the MDL criterion, the chief contribution of this paper is to show that the same algorithm can be used to optimize a broad class of criteria, and hence can be used for many application specific criterion for which efficient algorithm are not known. 1 Introduction The clustering of data is a problem found in many pattern recognition tasks, often in the guises of unsupervised learning, vector quantization, dimensionality reduction, etc. Formally, the clustering problem can be described as follows. Given a finite set S, and a criterion function Jk defined on all partitions of S into k parts, find a partition of S into k parts {S1, S2, . . . , Sk} so that Jk ({S1, S2, . . . , Sk}) is maximized. The number of k-clusters for a size n > k data set is roughly kn/k! [5] so exhaustive search is not an efficient solution. The problem, in fact, is NP-complete for most desirable measures. Broadly speaking there are two classes of criteria for clustering. There are distance based criteria, for which a distance measure is specified between each pair of elements, and the criterion somehow combines either intercluster or intracluster distances into an objective function. The other class of criteria are model based, and for these, a probabilistic (generative) model is specified. There is no universally accepted criterion for clustering. The appropriate criterion is typically application dependent, and therefore, we do not claim that the two criteria considered in this paper are inherently better or more generally applicable than other criteria. However, we can show that for the single-linkage criterion, we can compute the optimal clustering into k parts (for any k), and for the MDL criterion, we can compute the optimal clustering into 2 parts using Queyranne’s algorithm. More generally, any criterion from a broad class of criterion can be solved by the same algorithm, and this class of criteria is closed under linear combinations. In addition to the theoretical elegance of a single algorithm solving a number of very different criterion, this means that we can optimize (for example) for the sum of single-linkage and MDL criterions (or positively scaled versions thereof). The two criterion we consider are quite different. The first, “discriminative”, criterion we consider is the single-linkage criterion. In this case, we are given distances d(s1, s2) between all elements s1, s2 ∈S, and we try and find clusters that maximize the minimum distance between elements of different clusters (i.e., maximize the separation of the clusters). This criterion has several advantages. Since we are only comparing distances, the distance measure can be chosen from any ordered set (addition/squaring/multiplication of distances need not be defined as is required for K-means, spectral clustering etc.). Further, this criterion only depends on the rank ordering of the distances, and so is completely insensitive to any monotone transformation of the distances. This gives a lot of flexibility in constructing a distance measure appropriate for an application. For example, it is a very natural candidate when the distance measure is derived from user studies (since users are more likely to be able to provide rankings than exact distances). On the other hand, this criterion is sensitive to outliers and may not be appropriate when there are a large number of outliers in the data set. The kernel based criterion considered in [3] is similar in spirit to this one. However, their algorithm only provides approximate solutions, and the extension to more than 2 clusters is not given. However, since they optimize the distance of the clusters to a hyperplane, it is more appropriate if the clusters are to be classified using a SVM. The second criterion we consider is “generative” in nature and is based on the Minimum Description Length principle. In this case we are given a (generative) probability model for the elements, and we attempt to find clusters so that describing or encoding the clusters (separately) can be done using as few bits as possible. This is also a very natural criterion grouping together data items that can be highly compressed translates to grouping elements that share common characteristics. This criterion has also been widely used in the past, though the algorithms given do not guarantee optimal solutions (even for 2 clusters). Since these criteria seem quite different in nature, it is surprising that the same algorithm can be used to find the optimal partitions into two clusters in both cases. The key principle here is the notion of submodularity (and its variants) [1, 2]. We will show that the problem of finding the optimal clusterings minimizing the description length is equivalent to the problem of minimizing a symmetric submodular function, and the problem of maximizing the cluster separation is equivalent to minimizing a symmetric function which, while not submodular, is closely related, and can be minimized by the same algorithm. 2 Background and Notation A clustering of a finite set S is a partition {S1, S2, . . . , Sk} of S. We will call the individual elements of the partition the clusters of the partition. If there are k clusters in the partition, then we say that the partition is a k-clustering. Let Ck(S) be the set of all k-clusterings for 1 ≤k ≤|S|. For the first criterion, we assume we are given a function d : S × S →R that represents the “distance” between objects. Intuitively, we expect that d(s, t) is large when the objects are dissimilar. We will assume that d(·, ·) is symmetric, but make no further assumptions. In particular we do not assume that d(·, ·) is a metric (Later on in this paper, we will not even assume that d(s, t) is a (real) number, but instead will allow the range of d to be a ordered set ). The distance between sets T and R is often defined to be the smallest distance between elements from these different clusters: D(R, T ) = minr∈R,t∈T d(r, t). The single-linkage criterion tries to maximize this distance, and hence an optimal 2-clustering is in arg max{S1,S2}∈C2(S) D(S1, S2). We let Ok(S) be the set of all optimal k-clusterings for 1 ≤k ≤|S| with respect to D(·, ·). It is known that an algorithm based on the Minimum Spanning Tree can be used to find optimal clusterings for the single-linkage criterion[8]. For the second criterion, we assume S is a collection of random variables, and for any subset T = {s1, s2, . . . , sm} of S, we let H(T ) be the entropy of the set of random variables {s1, s2, . . . , sm}. Now, the (expected) total cost of encoding or describing the set T is H(T ). So a partition {S1, S2} of S that minimizes the description length (DL) is in arg min {S1,S2}∈C2(S) DL(S1, S2) = arg min {S1,S2}∈C2(S) H(S1) + H(S2) We will denote by 2S the set of all subsets of S. A set function f : 2S →R assigns a (real) number to every subset of S. We say that f is submodular if f(A) + f(B) ≥ f(A ∪B) + f(A ∩B) for every A, B ⊆S. f is symmetric if f(A) = f(S \ A). In [1], Queyranne gives a polynomial time algorithm that finds a set A ∈2S \ {S, φ} that minimizes any symmetric submodular set function (specified in the form of an oracle). That is, Queyranne’s algorithm finds a non-trivial partition {S1, S \ S1} of S so that f(S1) (= f(S \S1)) minimizes f over all non-trivial subsets of S. The problem of finding non-trivial minimizers of a symmetric submodular function can be thought of a a generalization of the graph-cut problem. For a symmetric set function f, we can think of f(S1) as f(S1, S \S1), and if we can extend f to be defined on all pairs of disjoint subsets of S, then Rizzi showed in [2] that Queyranne’s algorithm works even when f is not submodular, as long as f is monotone and consistent, where f is monotone if for R, T, T ′ ⊆S with T ′ ⊆T and R∩T = φ we have f(R, T ′) ≤f(R, T ) and f is consistent if f(A, W ∪B) ≥f(B, A∪W) whenever A, B, W ⊆S are disjoint sets satisfying f(A, W) ≥f(B, W). The rest of this paper is organized as follows. In Section 3, we show that Queyranne’s algorithm can be used to find the optimal k-clustering (for any k) in polynomial time for the single-linkage criterion. In Section 4, we give an algorithm for finding the optimal clustering into 2 parts that minimizes the description length. In Section 5, we present some experimental results. 3 Single-Linkage: Maximizing the separation between clusters In this section, we show that Queyranne’s algorithm can be used for finding k-clusters (for any given k) that maximize the separation between elements of different clusters. We do this in two steps. First in Subsection 3.1, we show that Queyranne’s algorithm can partition the set S into two parts to maximize the distance between these parts in polynomial time. Then in Subsection 3.2, we show how this subroutine can be used to find optimal k clusters, also in polynomial time. 3.1 Optimal 2-clusterings In this section, we will show that the function −D(·, ·) is monotone and consistent. Therefore, by Rizzi’s result, it follows that we can find a 2-clustering {S1, S2} = {S1, S \ S1} that minimizes −D(S1, S2), and hence maximizes D(S1, S2). Lemma 1. If R ⊆T , then D(U, T) ≤D(U, R) (and hence −D(U, R) ≤−D(U, T)). This would imply that −D is monotone. To see this, observe that D(U, T) = min u∈U,t∈T d(u, t) = min min u∈U,r∈R d(u, r), min u∈U,t∈T \R d(u, t) ≤D(U, R) Lemma 2. Suppose that A, B, W are disjoint subsets of S and D(A, W) ≤D(B, W). Then D(A, W ∪B) ≤D(B, A ∪W). To see this first observe that D(A, B ∪W) = min(D(A, B), D(A, W)) because D(A, W ∪B) = min a∈A,x∈W∪B D(a, x) = min min a∈A,w∈W D(a, w), min a∈A,b∈B D(A, b) It follows that D(A, B ∪W) = min (D(A, B), D(A, W)) ≤min (D(A, B), D(B, W)) = min (D(B, A), D(B, W)) = D(B, A ∪W). Therefore, if −D(A, W) ≥−D(B, W), then −D(A, W ∪B) ≥−D(B, A ∪W). Hence −D(·, ·) is consistent. Therefore, −D(·, ·) is symmetric, monotone and consistent. Hence it can be minimized using Queyranne’s algorithm [2]. Therefore, we have a procedure to compute optimal 2clusterings. We now extend this to compute optimal k-clusterings. 3.2 Optimal k-clusterings We start off by extending our objective function for k-clusterings in the obvious way. The function D(R, T ) can be thought of as defining the separation or margin between the clusters R and T . We can generalize this notion to more than two clusters as follows. Let seperation({S1, S2, . . . , Sk}) = min i̸=j D(Si, Sj) = min Si̸=Sj si∈Si,sj∈Sj d(si, sj) Note that seperation({R, T }) = D(R, T ) for a 2-clustering. The function seperation : ∪|S| k=1Ck(S) →R takes a single clustering as its argument. However, D(·, ·) takes two disjoint subsets of S as its arguments the union of which need not be S in general. The margin is the distance between the closest elements of different clusters, and hence we will be interested in finding k-clusters that maximize the margin. Therefore, we seek an element in Ok(S) = arg max{S1,S2,...,Sk}∈Ck(S) seperation({S1, S2, . . . , Sk}). Let vk(S) be the margin of an element in Ok(S). Therefore, vk(S) is the best possible margin of any kclustering of S. An obvious approach to generating optimal k-clusterings given a method of generating optimal 2-clusterings is the following. Start off with an optimal 2-clustering {S1, S2}. Then apply the procedure to find 2-clusterings of S1 and S2, and stop when you have enough clusters. There are two potential problems with this approach. First, it is not clear that an optimal k-clustering can be a refinement of an optimal 2-clustering. That is, we need to be sure that there is an optimal k-clustering in which S1 is the union of some of the clusters, and S2 is the union of the remaining. Second, we need to figure out how many of the clusters S1 is the union of and how many S2 is the union of. In this section, we will show that for any k ≥3, there is always an optimal k-clustering that is a refinement of any given optimal 2-clustering. A simple dynamic programming algorithm takes care of the second potential problem. We begin by establishing some relationships between the separation of clusterings of different sizes. To compare the separation of clusterings with different number of clusters, we can try and merge two of the clusters from the clustering with more clusters. Say that S = {S1, S2, . . . , Sk} ∈Ck(S) is any k-clustering of S, and S′ is a (k −1)-clustering of S obtained by merging two of the clusters (say S1 and S2). Then S′ = {S1 ∪S2, S3, . . . , Sk} ∈Ck−1(S). Lemma 3. Suppose that S = {S1, S2, . . . , Sk} ∈ Ck(S) and S′ = {S1 ∪S2, S3, . . . , Sk} ∈Ck−1(S). Then seperation(S) ≤seperation(S′). In other words, refining a partition can only reduce the margin. Therefore, refining a clustering (i.e., splitting a cluster) can only reduce the separation. An immediate corollary is the following. Corollary 4. If Tl ∈Cl(S) is a refinement of Tk ∈Ck(S) (for k < l) then seperation(Tl) ≤ seperation(Tk). It follows that vk(S) ≥vl(S) if 1 ≤k < l ≤n. Proof. It suffices to prove the result for k = l −1. The first assertion follows immediately from Lemma 3. Let S ∈Ol(S) be an optimal l-clustering. Merge any two clusters to get S′ ∈Ck(S). By Lemma 3, vk(S) ≥seperation(S′) ≥seperation(S) = vl(S). Next, we consider the question of constructing larger partitions (i.e., partitions with more clusters) from smaller partitions. Given two clusterings S = {S1, S2, . . . , Sk} ∈Ck(S) and T = {T1, T2, . . . , Tl} ∈Cl(S) of S, we can create a new clustering U = {U1, U2, . . . , Um} ∈Cm(S) to be their common refinement. That is, the clusters of U consist of those elements that are in the same clusters of both S and T . Formally, U = {Si ∩Tj : 1 ≤i ≤k, 1 ≤j ≤l} Lemma 5. Let S = {S1, S2, . . . , Sk} ∈Ck(S) and T = {T1, T2, . . . , Tl} ∈Cl(S) be any two partitions. Let U = {U1, U2, . . . , Um} ∈Cm(S) be their common refinement. Then seperation(U) = min (seperation(S), seperation(T )). Proof. It is clear that seperation(U) ≤min (seperation(S), seperation(T )). To show equality, note that if a, b are in different clusters of U, then a, b must have been in different clusters of either S or T . This result can be thought of as expressing a relationship between seperation and the lattice of partitions of S which will be important to our later robustness extension Lemma 6. Suppose that S = {S1, S2} ∈O2(S) is an optimal 2-clustering. Then there is always an optimal k-clustering that is a refinement of S. Proof. Suppose that this is not the case. If T = {T1, T2, . . . , Tk} ∈Ok(S) is an optimal k-clustering, let r be the number of clusters of T that “do not respect” the partition {S1, S2}. That is, r is the number of clusters of T that intersect both S1 and S2 : r = |{1 ≤i ≤k : Ti ∩S1 ̸= φ and Ti ∩S2 ̸= φ}|. Pick T ∈Ok(S) to have the smallest r. If r = 0, then T is a refinement of S and there is nothing to show. Otherwise, r ≥1. Assume WLOG that T (1) 1 = T1 ∩S1 ̸= φ and T (2) 1 = T1 ∩S2 ̸= φ. Then T ′ = n T (1) 1 , T (2) 1 , T2, T3, . . . , Tk o ∈Ck+1(S) is a refinement of T and satisfies seperation(T ′) = seperation (T ). This follows from Lemma 3 along with the fact that (1) D(Ti, Tj) ≥seperation(T ) for any 2 ≤i < j ≤k, (2) D(T (i) 1 , Tj) ≥seperation(T ) for any i ∈{1, 2} and 2 ≤j ≤k, (3) D(T (1) 1 , T (2) 1 ) ≥seperation({S1, S2}) = v2(S) ≥ vk(S) = seperation(T ). Now, pick two clusters of T ′ that are either both contained in the same cluster of S or both “do not respect” S. Clearly this can always be done. Merge these clusters together to get an element T ′′ ∈Ck(S). By Lemma 3 merging clusters cannot decrease the margin. Therefore, seperation(T ′′) = seperation(T ′) = seperation(T ). However, T ′′ has fewer clusters that do not respect S hand T has, and hence we have a contradiction. This lemma implies that Queyranne’s algorithm, along with a simple dynamic programming algorithm can be used to find the best k clustering with time complexity O(k |S|3). Observe that in fact this problem can be solved in time O(|S|2) ([8]). Even though using Queyranne’s algorithm is not the fastest algorithm for this problem, the fact that it optimizes this criterion implies that it can be used to optimize conic combinations of submodular criteria and the single-linkage criterion. 3.3 Generating robust clusterings One possible issue with the metric we defined is that it is very sensitive to outliers and noise. To see this, note that if we have two very well separated clusters, then adding a few points “between” the clusters could dramatically decrease the separation. To increase the robustness of the algorithm, we can try to maximize the n smallest distances instead of maximizing just the smallest distance between clusters. If we give the nth smallest distance more importance than the smallest distance, this increases the noise tolerance by ignoring the effects of a few outliers. We will take n ∈N to be some fixed positive integer specified by the user. This will represent the desired degree of noise tolerance (larger gives more noise tolerance). Let Rn be the set of decreasing n-tuples of elements in R ∪{∞}. Given disjoint sets R, T ⊆S, let D(R, T ) be the element of Rn obtained as follows. Let L(R, T ) = ⟨d1, d2, . . . , d|R|·|T |⟩be an ordered list of distances between elements of R and T arranged in decreasing order. So for example, if R = {1, 2} and T = {3, 4}, with d(r, t) = r · t, then L(R, T ) = ⟨8, 6, 4, 3⟩. We define D(R, T ) as follows. If |R| · |T | ≥n, then D(R, T ) is the last (and thus least) n elements of L(R, T ). Otherwise, if |R|·|T | < n, then the first n−|R|·|T | elements of D(R, T ) are ∞, while the remaining elements are the elements of L(R, T ). So for example, if n = 2, then D(R, T ) in the above example would be ⟨4, 3⟩, if n = 3 then D(R, T ) = ⟨6, 4, 3⟩and if n = 6, then D(R, T ) = ⟨∞, ∞, 8, 6, 4, 3⟩. We define an operation ⊕on Rn as follows. To get ⟨l1, l2, . . . , ln⟩⊕⟨r1, r2, . . . , rn⟩, order the elements of ⟨l1, l2, . . . , ln, r1, r2, . . . , rn⟩in decreasing order, and let ⟨s1, s2, . . . , sn⟩ be the last n elements. For example, ⟨∞, 3, 2⟩⊕⟨∞, 6, 5⟩= ⟨5, 3, 2⟩and ⟨4, 3, 1⟩⊕ ⟨5, 4, 3⟩= ⟨3, 3, 1⟩. So, the ⊕operation picks off the n smallest elements. It is clear that this operation is commutative (symmetric), associative and that ⟨∞, ∞, . . . , ∞⟩ acts as an identity. Therefore, Rn forms a commutative semigroup. In fact, we can describe D(R, T ) as follows. For any pair of distinct elements r, t ∈S, let d′(r, t) = ⟨∞, ∞, . . . , d(r, t)⟩. Then D(R, T ) = L r∈R,t∈T d′(r, t). Notice the similarity to D(R, T ) = minr∈R,t∈T d(r, t). In fact, if we take n = 1, then the ⊕operation reduces to the minimum operation and we get back our original definitions. We can order Rn lexicographically. Therefore, Rn becomes an ordered semigroup. It is entirely straightforward to check that if R ⊆T , then D(U, T) ≺D(U, R), and that if A, B, W are disjoint sets with D(A, W) ≺D(B, W), then D(A, W ∪B) ≺D(B, A ∪W). It is also straightforward to extend Rizzi’s proof to see that Queyranne’s algorithm (with the obvious modifications) will generate a 2-clustering that minimizes this metric. It can also be verified that the results of Section 3.2 can be extended to this framework (also with the obvious modifications). In our experiments, we observed that selecting the parameter n is quite tricky. Now, Queyranne’s algorithm actually produces a (Gomory-Hu) tree [1] whose edges represent the cost of separating elements. In practice we noticed that restricting our search to only edges whose deletion results in clusters of at least certain sizes produces very good results. Other heuristics such as running the algorithm a number of times to eliminate outliers are also reasonable approaches. Modifying the algorithm to yield good results while retaining the theoretical guarantees is an open question. 4 MDL Clustering We assume that S is a collection of random variables for which we have a (generative) probability model. Since we have the joint probabilities of all subsets of the random variables, the entropy of any collection of the variables is well defined. The expected coding (or description) length of any collection T of random variables using an optimal coding scheme (or a random coding scheme) is known to be H(T ). The partition {S1, S2} of S that minimizes the coding length is therefore arg min{S1,S2}∈C2(S) H(S1) + H(S2). Now, arg min {S1,S2}∈C2(S) H(S1) + H(S2) = arg min {S1,S2}∈C2(S) H(S1) + H(S2) −H(S) = arg min {S1,S2}∈C2(S) I(S1; S2) where I(S1; S2) is the mutual information between S1 and S2 because S1 ∪S2 = S for all {S1, S2} ∈C2(S), Therefore, the problem of partitioning S into two parts to minimize the description length is equivalent to partitioning S into two parts to minimize the mutual information between the parts. It is shown in [9] that the function f : 2S →R defined by f(T ) = I(T ; S \ T ) is symmetric and submodular. Clearly the minima of this function correspond to partitions that minimize the mutual information between the parts. Therefore, the problem of partitioning in order to minimize the mutual information between the parts can be reduced to a symmetric submodular minimization problem, which can be solved using Queyranne’s algorithm in time O(|S|3) assuming oracle queries to a mutual information oracle. While implementing such a mutual information oracle is not trivial, for many realistic applications (including one we consider in this paper), the cost of computing a mutual information query is bounded above by the size of the data set, and so the entire algorithm is polynomial in the size of the data set. Symmetric submodular functions generalize notions like graph-cuts, and indeed, Queyranne’s algorithm generalizes an algorithm for computing graph-cuts. Since graph-cut based techniques are extensively used in many engineering applications, it might be possible to develop criteria that are more appropriate for these specific applications, while still retaining producing optimal partitions of size 2. It should be noted that, in general, we cannot use the dynamic programming algorithm to produce optimal clusterings with k > 2 clusters for the MDL criterion (or for general symmetric submodular functions). The key reason is that we cannot prove the equivalent of Lemma 6 for the MDL criterion. However, such an algorithm seems reasonable, and it does produce reasonable results. Another approach (which is computationally cheaper) is to compute k clusters by deleting k −1 edges of the Gomory-Hu tree produced by Queyranne’s algorithm. It can be shown [9] that this will yield a factor 2 approximation to the optimal k-clustering. More generally, if we have an arbitrary increasing submodular function (such as entropy) f : 2S →R, and we seek a clustering {S1, S2, . . . , Sk} to minimize the sum Pk i=1 f(Si), then we have an exact algorithm for 2-clusterings and a factor 2 approximation guarantee. Therefore, this generalizes approximation guarantees for graph k-cuts because for any graph G = (V, E), the function f : 2V →R where f(A) is the number of edges adjacent to the vertex set A is a submodular function. The finding a clustering to minimize Pk i=1 f(Si) is equivalent to finding a partition of the vertex set of size k to minimize the number of edges disconnected (i.e., to the graph k-cut problem). Another criterion which we can define similarly can be applied to clustering genomic sequences. Intuitively, two genomes are more closely related if they share more common subsequences. Therefore, a natural clustering criterion for sequences is to partition the sequences into clusters so that the sequences from different clusters share as few subsequences as possible. This problem too can be solved using this generic framework. 5 Results Table 1 compares Q-Clustering with various other algorithms. The left part of the table shows the error rates (in percentages) of the (robust) single-linkage criterion and some other techniques on the same data set as is reported in [3]. The data sets are images (of digits and faces), and the distance function we used was the Euclidean distance between the vector of the pixels in the images. The right part of the table compares the Q-Clustering using MDL criterion with other state of the art algorithms for haplotype tagging of SNPs (single nucleotide polymorphisms) in the ACE gene on the data set reported in [4]. In this problem, the goal is to identify a set of SNPs that can accurately predict at least 90% of the SNPs in ACE gene. Typically the SNPs are highly correlated, and so it is necessary to cluster SNPs to identify the correlated SNPs. Note it is very important to identify as few SNPs as possible because the number of clinical trials required grows exponentially with the number of SNPs. As can be seen Q-Clustering does very well on this data set. 6 Conclusions The maximum-separation (single-linkage) metric is a very natural “discriminative” criterion, and it has several advantages, including insensitivity to any monotone transformation of the distances. However, it is quite sensitive to outliers. The robust version does help Robust Max-Separation (Single-Linkage) MDL Error rate Error rate on Digits on Faces Q-Clustering 1.4 0 Max-Margin† 3 0 Spectral Clust.† 6 16.7 K-means† 7 24.4 #SNPs required Q-Clustering 3 EigenSNP‡ 5 Sliding Window‡ 15 htStep (up)‡ 7 htStep (down)‡ 7 Table 1: Comparing (robust) max-separation and MDL Q-Clustering with other techniques. Results marked by † and ‡ are from [3] and [4] respectively. a little, but it does require some additional knowledge (about the approximate number of outliers) and considerable tuning. It is possible that we could develop additional heuristics to automatically determine the parameters of the robust version. The MDL criterion is also a very natural one, and the results on haplotype tagging are quite promising. The MDL criterion can be seen as a generalization of graph cuts, and so it seems like Q-clustering can also be applied to optimize other criteria arising in problems like image segmentation, especially when there is a generative model. Another natural criterion for clustering strings is to partition the strings/sequences to minimize the number of common subsequences. This could have interesting applications in genomics. The key novelty of this paper is the guarantees of optimality produced by the algorithm, and the generaly framework into which a number of natural criterion fall. 7 Acknowledgments The authors acknowledge the assistance of Linli Xu in obtaining the data to test the algorithm and for providing the code used in [3]. Gilles Blanchard pointed out that the MST algorithm finds the optimal solution for the single-linkage criterion. The first and third authors were supported by NSF grant IIS-0093430 and an Intel Corporation Grant. References [1] M. Queyranne. “Minimizing symmetric submodular functions”, Math. Programming, 82, pages 3–12. 1998. [2] R. Rizzi, “On Minimizing symmetric set functions”, Combinatorica 20(3), pages 445–450, 2000. [3] L. Xu, J. Neufeld, B. Larson and D. Schuurmans. “Maximum Margin Clustering”, in Advances in Neural Information Processing Systems 17, pages 1537-1544, 2005. [4] Z. Lin and R. B. Altman. “Finding Haplotype Tagging SNPs by Use of Principal Components Analysis”, Am. J. Hum. Genet. 75, pages 850-861, 2004. [5] Jain, A.K. and R.C. Dubes, “Algorithms for Clustering Data.” Englewood Cliffs, N.J.: Prentice Hall, 1988. [6] P. Brucker, “On the complexity of clustering problems,” in R. Henn, B. Korte, and W. Oletti (eds.), Optimization and Operations Research, Lecture Notes in Economics and Mathematical Systems, Springer, Berlin 157. [7] P. Kontkanen, P. Myllym¨aki, W. Buntine, J. Rissanen and H. Tirri. “An MDL framework for data clustering”, HIIT Technical Report 2004. [8] M. Delattre and P. Hansen. “Bicriterion Cluster Analysis”, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol-2, No. 4, 1980 [9] M. Narasimhan, N. Jojic and J. Bilmes. “Q-Clustering”, Technical Report, Dept. of Electrical Engg., University of Washington, UWEETR-2006-0001, 2005
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Computing the Solution Path for the Regularized Support Vector Regression Lacey Gunter Department of Statistics University of Michigan Ann Arbor, MI 48109 lgunter@umich.edu Ji Zhu∗ Department of Statistics University of Michigan Ann Arbor, MI 48109 jizhu@umich.edu Abstract In this paper we derive an algorithm that computes the entire solution path of the support vector regression, with essentially the same computational cost as fitting one SVR model. We also propose an unbiased estimate for the degrees of freedom of the SVR model, which allows convenient selection of the regularization parameter. 1 Introduction The support vector regression (SVR) is a popular tool for function estimation problems, and it has been widely used on many real applications in the past decade, for example, time series prediction [1], signal processing [2] and neural decoding [3]. In this paper, we focus on the regularization parameter of the SVR, and propose an efficient algorithm that computes the entire regularized solution path; we also propose an unbiased estimate for the degrees of freedom of the SVR, which allows convenient selection of the regularization parameter. Suppose we have a set of training data (x1, y1), . . . , (xn, yn), where the input xi ∈ Rp and the output yi ∈R. Many researchers have noted that the formulation for the linear ϵ-SVR can be written in a loss + penalty form [4]: min β0,β n i=1 yi −β0 −βTxi ϵ + λ 2 βTβ (1) where |ξ|ϵ is the so called ϵ-insensitive loss function: |ξ|ϵ = 0 if |ξ| ≤ϵ |ξ| −ϵ otherwise The idea is to disregard errors as long as they are less than ϵ. Figure 1 plots the loss function. Notice that it has two non-differentiable points at ±ϵ. The regularization parameter λ controls the trade-offbetween the ϵ-insensitive loss and the complexity of the fitted model. ∗To whom the correspondence should be addressed. −3 −2 −1 0 1 2 3 0.0 0.5 1.0 1.5 2.0 2.5 y−f Loss Elbow R Right Center Left Elbow L Figure 1: The ϵ-insensitive loss function. In practice, one often maps x into a high (often infinite) dimensional reproducing kernel Hilbert space (RKHS), and fits a nonlinear kernel SVR model [4]: min β0,θ n i=1 |yi −f(xi)|ϵ + 1 2λ n i=1 n i′=1 θiθi′K(xi, xi′) (2) where f(x) = β0 + 1 λ n i=1 θiK(x, xi), and K(·, ·) is a positive-definite reproducing kernel that generates a RKHS. Notice that we write f(x) in a way that involves λ explicitly, and we will see later that θi ∈[−1, 1]. Both (1) and (2) can be transformed into a quadratic programming problem, hence most commercially available packages can be used to solve the SVR. In the past years, many specific algorithms for the SVR have also been developed, for example, interior point algorithms [4-5], subset selection algorithms [6–7], and sequential minimal optimization [4, 8–9]. All these algorithms solve the SVR for a pre-fixed regularization parameter λ, and it is well known that an appropriate value of λ is crucial for achieving small prediction error of the SVR. In this paper, we show that the solution θ(λ) is piecewise linear as a function of λ, which allows us to derive an efficient algorithm that computes the exact entire solution path {θ(λ), 0 ≤λ ≤∞}. We acknowledge that this work was inspired by one of the authors’ earlier work on the SVM setting [10]. Before delving into the technical details, we illustrate the concept of piecewise linearity of the solution path with a simple example. We generate 10 training observations using the famous sinc(·) function: y = sin(πx) πx + e, where x ∼U(−2π, 2π) and e ∼N(0, 0.192) We use the SVR with a 1-dimensional spline kernel K(x, x′) = 1 + k1(x)k1(x′) + k2(x)k2(x′) −k4(|x −x′|) (3) where k1(·) = · −1/2, k2 = (k2 1 −1/12)/2, k4 = (k4 1 −k2 1/2 + 7/240)/24. Figure 2 shows a subset of the piecewise linear solution path θ(λ) as a function of λ. In section 2, we describe the algorithm that computes the entire solution path of the SVR. In section 3, we propose an unbiased estimate for the degrees of freedom of the SVR, which can be used to select the regularization parameter λ. In section 4, we present numerical results on simulation data. We conclude the paper with a discussion section. 1 2 3 4 5 −1.0 −0.5 0.0 0.5 1.0 λ θ Figure 2: A subset of the solution path θ(λ) as a function of λ. 2 Algorithm For simplicity in notation, we describe the problem setup using the linear SVR, and the algorithm using the kernel SVR. 2.1 Problem Setup The linear ϵ-SVR (1) can be re-written in an equivalent way: min β0,β n i=1 (ξi + δi) + λ 2 βTβ subject to −(δi + ϵ) ≤yi −f(xi) ≤(ξi + ϵ), ξi, δi ≥0; f(xi) = β0 + βTxi, i = 1, . . . n This gives us the Lagrangian primal function LP : n i=1 (ξi + δi) + λ 2 βTβ + n i=1 αi(yi −f(xi) −ξi −ϵ) − n i=1 γi(yi −f(xi) + δi + ϵ) − n i=1 ρiξi − n i=1 τiδi. Setting the derivatives to zero we arrive at: ∂ ∂β : β = 1 λ n i=1 (αi −γi)xi (4) ∂ ∂β0 : n i=1 αi = n i=1 γi (5) ∂ ∂ξi : αi = 1 −ρi (6) ∂ ∂δi : γi = 1 −τi (7) where the Karush-Kuhn-Tucker conditions are αi(yi −f(xi) −ξi −ϵ) = 0 (8) γi(yi −f(xi) + δi + ϵ) = 0 (9) ρiξi = 0 (10) τiδi = 0 (11) Along with the constraint that our Lagrange multipliers must be non-negative, we can conclude from (6) and (7) that both 0 ≤αi ≤1 and 0 ≤γi ≤1. We also see from (8) and (9) that if αi is positive, then γi must be zero, and vice versa. These lead to the following relationships: yi −f(xi) > ϵ ⇒ αi = 1, ξi > 0, γi = 0, δi = 0; yi −f(xi) < −ϵ ⇒ αi = 0, ξi = 0, γi = 1, δi > 0; yi −f(xi) ∈(−ϵ, ϵ) ⇒ αi = 0, ξi = 0, γi = 0, δi = 0; yi −f(xi) = ϵ ⇒ αi ∈[0, 1], ξi = 0, γi = 0, δi = 0; yi −f(xi) = −ϵ ⇒ αi = 0, ξi = 0, γi ∈[0, 1], δi = 0. Using these relationships, we define the following sets that will be used later on when we are calculating the regularization path of the SVR: • R = {i : yi −f(xi) > ϵ, αi = 1, γi = 0} (Right of the elbows) • ER = {i : yi −f(xi) = ϵ, 0 ≤αi ≤1, γi = 0} (Right elbow) • C = {i : −ϵ < yi −f(xi) < ϵ, αi = 0, γi = 0} (Center) • EL = {i : yi −f(xi) = −ϵ, αi = 0, 0 ≤γi ≤0} (Left elbow) • L = {i : yi −f(xi) < −ϵ, αi = 0, γi = 1} (Left of the elbows) Notice from (4) that for every λ, β is fully determined by the values of αi and γi. For points in R, L and C, the values of αi and γi are known; therefore, the algorithm will focus on points resting at the two elbows ER and EL. 2.2 Initialization Initially, when λ = ∞we can see from (4) that β = 0. We can determine the value of β0 via a simple 1-dimensional optimization. For lack of space, we focus on the case that all the values of yi are distinct, and furthermore, the initial sets ER and EL have at most one point combined (which is the usual situation). In this case β0 will not be unique and each of the αi and γi will be either 0 or 1. Since β0 is not unique, we can focus on one particular solution path, for example, by always setting β0 equal to one of its boundary values (thus keeping one point at an elbow). As λ decreases, the range of β0 shrinks toward zero and reaches zero when we have two points at the elbows, and the algorithm proceeds from there. 2.3 The Path The formalized setup above can be easily modified to accommodate non-linear kernels; in fact, θi in (2) is equal to αi −γi. For the remaining portion of the algorithm we will use the kernel notation. The algorithm focuses on the sets of points ER and EL. These points have either f(xi) = yi −ϵ with αi ∈[0, 1], or f(xi) = yi + ϵ with γi ∈[0, 1]. As we follow the path we will examine these sets until one or both of them change, at which point we will say an event has occurred. Thus events can be categorized as: 1. The initial event, for which two points must enter the elbow(s) 2. A point from R has just entered ER, with αi initially 1 3. A point from L has just entered EL, with γi initially 1 4. A point from C has just entered ER, with αi initially 0 5. A point from C has just entered EL, with γi initially 0 6. One or more points in ER and/or EL have just left the elbow(s) to join either R, L, or C, with αi and γi initially 0 or 1 Until another event has occurred, all sets will remain the same. As a point passes through ER or EL, its respective αi or γi must change from 0 →1 or 1 →0. Relying on the fact that f(xi) = yi−ϵ or f(xi) = yi+ϵ for all points in ER or EL respectively, we can calculate αi and γi for these points. We use the subscript ℓto index the sets above immediately after the ℓth event has occurred, and let αℓ i, γℓ i , βℓ 0 and λℓbe the parameter values immediately after the ℓth event. Also let f ℓbe the function at this point. We define for convenience β0,λ = λ · β0 and hence βℓ 0,λ = λℓ· βℓ 0. Then since f(x) = 1 λ n i=1 (αi −γi)K(x, xi) + β0,λ for λℓ+1 < λ < λℓwe can write f(x) = f(x) −λℓ λ f ℓ(x) + λℓ λ f ℓ(x) = 1 λ ⎡ ⎣ i∈Eℓ R νiK(x, xi) − j∈Eℓ L ωjK(x, xj) + ν0 + λℓf ℓ(x) ⎤ ⎦, where νi = αi −αℓ i, ωj = γj −γℓ j and ν0 = β0,λ −βℓ 0,λ, and we can do the reduction in the second line since the αi and γi are fixed for all points in Rℓ, Lℓ, and Cℓand all points remain in their respective sets. Suppose |Eℓ R| = nℓ R and |Eℓ L| = nℓ L, so for the nℓ R + nℓ L points staying at the elbows we have (after some algebra) that 1 yk −ϵ ⎡ ⎣ i∈Eℓ R νiK(xk, xi) − j∈Eℓ L ωjK(xk, xj) + ν0 ⎤ ⎦ = λ −λℓ, ∀k ∈Eℓ R 1 ym + ϵ ⎡ ⎣ i∈Eℓ R νiK(xm, xi) − j∈Eℓ L ωjK(xm, xj) + ν0 ⎤ ⎦ = λ −λℓ, ∀m ∈Eℓ L Also, by condition (5) we have that i∈Eℓ R νi − j∈Eℓ L ωj = 0 This gives us nℓ R + nℓ L + 1 linear equations we can use to solve for each of the nℓ R + nℓ L + 1 unknown variables νi, ωj and ν0. Notice this system is linear in λ −λℓ, which implies that αi, γj and β0,λ change linearly in λ −λℓ. So we can write: αi = αℓ i + (λ −λℓ)bi ∀i ∈Eℓ R (12) γj = γℓ j + (λ −λℓ)bj ∀j ∈Eℓ L (13) β0,λ = βℓ 0,λ + (λ −λℓ)b0 (14) f(x) = λℓ λ f ℓ(x) −hℓ(x) + hℓ(x) (15) where (bi, bj, b0) is the solution when λ −λℓis equal to 1, and hℓ(x) = i∈Eℓ R biK(x, xi) − j∈Eℓ L bjK(x, xj) + b0. Given λℓ, equations (12), (13) and (15) allow us to compute the λ at which the next event will occur, λℓ+1. This will be the largest λ less than λℓ, such that either αi for i ∈Eℓ R reaches 0 or 1, or γj for j ∈Eℓ L reaches 0 or 1, or one of the points in R, L or C reaches an elbow. We terminate the algorithm either when the sets R and L become empty, or when λ has become sufficiently close to zero. In the later case we must have f ℓ−hℓ sufficiently small as well. 2.4 Computational cost The major computational cost for updating the solutions at any event ℓinvolves two things: solving the system of (nℓ R +nℓ L) linear equations, and computing hℓ(x). The former takes O((nℓ R +nℓ L)2) calculations by using inverse updating and downdating since the elbow sets usually differ by only one point between consecutive events, and the latter requires O(n(nℓ R + nℓ L)) computations. According to our experience, the total number of steps taken by the algorithm is on average some small multiple of n. Letting m be the average size of Eℓ R ∪Eℓ L, then the approximate computational cost of the algorithm is O cn2m + nm2 , which is comparable to a single SVR fitting algorithm that uses quadratic programming. 3 The Degrees of Freedom The degrees of freedom is an informative measure of the complexity of a fitted model. In this section, we propose an unbiased estimate for the degrees of freedom of the SVR, which allows convenient selection of the regularization parameter λ. Since the usual goal of regression analysis is to minimize the predicted squared-error loss, we study the degrees of freedom using Stein’s unbiased risk estimation (SURE) theory [11]. Given x, assuming y is generated according to a homoskedastic model: y ∼(μ(x), σ2) where μ is the true mean and σ2 is the common variance. Then the degrees of freedom of a fitted model f(x) can be defined as df(f) = n i=1 cov(f(xi), yi)/σ2 Stein showed that under mild conditions, n i=1 ∂fi/∂yi is an unbiased estimate of df(f). It turns out that for the SVR model, for every fixed λ, n i=1 ∂fi/∂yi has an extremely simple formula: df ≡ n i=1 ∂fi ∂yi = |ER| + |EL| (16) Therefore, |ER| + |EL| is a convenient unbiased estimate for the degrees of freedom of f(x). Due to the space restriction, we omit the proof here, but make a note that the proof relies on our SVR algorithm. In applying (16) to select the regularization parameter λ, we plug it into the GCV criterion [12] for model selection: n i=1(yi −f(xi))2 (n −df)2 The advantages of this criterion are that it does not assume a known σ2, and it avoids cross-validation, which is computationally intensive. In practice, we can first use our efficient algorithm to compute the entire solution path, then identify the appropriate value of λ that minimizes the GCV criterion. 4 Numerical Results To demonstrate our algorithm and the selection of λ using the GCV criterion, we show numerical results on simulated data. We consider both additive and multiplicative kernels using the 1-dimensional spline kernel (3), which are respectively K(x, x′) = p j=1 K(xj, x′ j) and K(x, x′) = p j=1 K(xj, x′ j) Simulations were based on the following four functions [13]: 1. f(x) = sin(πx) πx + e1, x ∈(−2π, 2π) 2. f(x) = 0.1e4x1 + 1 1+e−20(x2−.5) + 3x3 + 2x4 + x5 + e2, x ∈(0, 1)2 3. f(R, ω, L, C) = R2 + ωL + 1 ωC 21/2 + e3, 4. f(R, ω, L, C) = tan−1 ωL+ 1 ωC R + e4, where (R, ω, L, C) ∈(0, 100) × (2π(20, 280)) × (0, 1) × (1, 11) ei are distributed as N(0, σ2 i ), where σ1 = 0.19, σ2 = 1, σ3 = 218.5, σ4 = 0.18. We generated 300 training observations from each function along with 10,000 validation observations and 10,000 test observations. For the first two simulations we used the additive 1-dimensional spline kernel and for the second two simulations the multiplicative 1-dimensional spline kernel. We then found the λ that minimized the GCV criterion. The validation set was used to select the gold standard λ which minimized the prediction MSE. Using these λ’s we calculated the prediction MSE with the test data for each criterion. After repeating this for 20 times, the average MSE and standard deviation for the MSE can be seen in Table 1, which indicates the GCV criterion performs closely to optimal. Table 1: Simulation results of λ selection for SVR f(x) MSE-Gold Standard MSE-GCV 1 0.0385 (0.0011) 0.0389 (0.0011) 2 1.0999 (0.0367) 1.1120 (0.0382) 3 50095 (1358) 50982 (2205) 4 0.0459 (0.0023) 0.0471 (0.0028) 5 Discussion In this paper, we have proposed an efficient algorithm that computes the entire regularization path of the SVR. We have also proposed the GCV criterion for selecting the best λ given the entire path. The GCV criterion seems to work sufficiently well on the simulation data. However, we acknowledge that according to our experience on real data sets (not shown here due to lack of the space), the GCV criterion sometimes tends to over-fit the model. We plan to explore this issue further. Due to the difficulty of also selecting the best ϵ for the SVR, an alternate algorithm exists that automatically adjusts the value of ϵ, called the ν-SVR [4]. In this scenario, ϵ is treated as another free parameter. Using arguments similar to those for β0 in our above algorithm, one can show that ϵ is piecewise linear in 1/λ and its path can be calculated similarly. Acknowledgments We would like to thank Saharon Rosset for helpful comments. Gunter and Zhu are partially supported by grant DMS-0505432 from the National Science Foundation. References [1] M¨uler K, Smola A, R¨atsch G, Sch¨olkopf B, Kohlmorgen J & Vapnik V (1997) Predicting time series with support vector machines. Artificial Neural Networks, 999-1004. [2] Vapnik V, Golowich S & Smola A (1997) Support vector method for function approximation, regression estimation, and signal processing. NIPS 9. [3] Shpigelman L, Crammer K, Paz R, Vaadia E & Singer Y (2004) A temporal kernel-based model for tracking hand movements from neural activities. NIPS 17, 1273-1280. [4] Smola A & Sch¨olkopf B (2004) A tutorial on support vector regression. Statistics and Computing 14: 199-222. [5] Vanderbei, R. (1994) LOQO: An interior point code for quadratic programming. Technical Report SOR-94-15, Princeton University. [6] Osuna E, Freund R & Girosi F (1997) An improved training algorithm for support vector machines. Neural Networks for Signal Processing, 276-284. [7] Joachims T (1999) Making large-scale SVM learning practical. Advances in Kernel Methods – Support Vector Learning, 169-184. [8] Platt J (1999) Fast training of support vector machines using sequential minimal optimization. Advances in Kernel Methods – Support Vector Learning, 185-208. [9] Keerthi S, Shevade S, Bhattacharyya C & Murthy K (1999) Improvements to Platt’s SMO algorithm for SVM classifier design. Technical Report CD-99-14, NUS. [10] Hastie, T., Rosset, S., Tibshirani, R. & Zhu, J. (2004) The Entire Regularization Path for the Support Vector Machine. JMLR, 5, 1391-1415. [11] Stein, C. (1981) Estimation of the mean of a multivariate normal distribution. Annals of Statistics 9: 1135-1151. [12] Craven, P. & Wahba, G. (1979) Smoothing noisy data with spline function. Numerical Mathematics 31: 377-403. [13] Friedman, J. (1991) Multivariate Adaptive Regression Splines. Annals of Statistics 19: 1-67.
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Generalization Error Bounds for Aggregation by Mirror Descent with Averaging Anatoli Juditsky Laboratoire de Mod´elisation et Calcul - Universit´e Grenoble I B.P. 53, 38041 Grenoble, France anatoli.iouditski@imag.fr Alexander Nazin Institute of Control Sciences - Russian Academy of Science 65, Profsoyuznaya str., GSP-7, Moscow, 117997, Russia nazine@ipu.rssi.ru Alexandre Tsybakov Laboratoire de Probabilit´es et Mod`eles Al´eatoires - Universit´e Paris VI 4, place Jussieu, 75252 Paris Cedex, France tsybakov@ccr.jussieu.fr Nicolas Vayatis Laboratoire de Probabilit´es et Mod`eles Al´eatoires - Universit´e Paris VI 4, place Jussieu, 75252 Paris Cedex, France vayatis@ccr.jussieu.fr Abstract We consider the problem of constructing an aggregated estimator from a finite class of base functions which approximately minimizes a convex risk functional under the ℓ1 constraint. For this purpose, we propose a stochastic procedure, the mirror descent, which performs gradient descent in the dual space. The generated estimates are additionally averaged in a recursive fashion with specific weights. Mirror descent algorithms have been developed in different contexts and they are known to be particularly efficient in high dimensional problems. Moreover their implementation is adapted to the online setting. The main result of the paper is the upper bound on the convergence rate for the generalization error. 1 Introduction We consider the aggregation problem (cf. [16]) where we have at hand a finite class of M predictors which are to be combined linearly under an ℓ1 constraint ∥θ∥1 = λ on the vector θ ∈RM that determines the coefficients of the linear combination. In order to exhibit such a combination, we focus on the strategy of penalized convex risk minimization which is motivated by recent statistical studies of boosting and SVM algorithms [11, 14, 18]. Moreover, we take a stochastic approximation approach which is particularly relevant in the online setting since it leads to recursive algorithms where the update uses a single data observation per iteration step. In this paper, we consider a general setting for which we propose a novel stochastic gradient algorithm and show tight upper bounds on its expected accuracy. Our algorithm builds on the ideas of mirror descent methods, first introduced by Nemirovski and Yudin [12], which consider updates of the gradient in the dual space. The mirror descent algorithm has been successfully applied in high dimensional problems both in deterministic and stochastic settings [2, 7]. In the present work, we describe a particular instance of the algorithm with an entropy-like proxy function. This method presents similarities with the exponentiated gradient descent algorithm which was derived under different motivations in [10]. A crucial distinction between the two is the additional averaging step in our version which guarantees statistical performance. The idea of averaging recursive procedures is well-known (see e.g. [13] and the references therein) and it has been invoked recently by Zhang [19] for the standard stochastic gradient descent (taking place in the initial parameter space). Also it is worth noticing that most of the existing online methods are evaluated in terms of relative loss bounds which are related to the empirical risk while we focus on generalization error bounds (see [4, 5, 10] for insights on connections between the two types of criteria). The rest of the paper is organized as follows. We first introduce the setup (Section 2), then we describe the algorithm and state the main convergence result (Section 3). Further we provide the intuition underlying the proposed algorithm, and compare it to other methods (Section 4). We end up with a technical section dedicated to the proof of our main result (Section 5). 2 Setup and notations Let Z be a random variable with values in a measurable space (Z, A). We set a parameter λ > 0, and an integer M ≥2. The unknown parameter is a vector θ ∈RM which is compelled to stay in the decision set Θ = ΘM,λ defined by: ΘM,λ = θ = (θ(1), . . . , θ(M))T ∈RM + : XM i=1 θ(i) = λ . (1) Now we introduce the loss function Q : Θ × Z →R+ such that the random function Q(· , Z) : Θ →R+ is convex for almost all Z and define the convex risk function A : Θ → R+ to be minimized as follows: A(θ) = E Q(θ, Z) . (2) Assume a training sample is given in the form of a sequence (Z1, . . . , Zt−1), where each Zi has the same distribution as Z. We assume for simplicity that the training sequence is i.i.d. though this assumption can be weakened. We propose to minimize the convex target function A over the decision set Θ on the basis of the stochastic sub-gradients of Q: ui(θ) = ∇θQ(θ, Zi) , i = 1, 2, . . . , (3) Note that the expectations E ui(·) belong to the sub-differential of A(·). In the sequel, we will characterize the accuracy of an estimate bθt = bθt(Z1, . . . , Zt−1) ∈Θ of the minimizer of A by the excess risk: E A(bθt) −min θ∈Θ A(θ) (4) where the expectation is taken over the sample (Z1, . . . , Zt−1). We now introduce the notation that is necessary to present the algorithm in the next section. For a vector z = z(1), . . . , z(M)T ∈RM, define the norms ∥z∥1 def = XM j=1 |z(j)| , ∥z∥∞ def = max ∥θ∥1=1 zT θ = max j=1,...,M |z(j)| . The space RM equipped with the norm ∥· ∥1 is called the primal space E and the same space equipped with the dual norm ∥· ∥∞is called the dual space E∗. Introduce a so-called entropic proxy function: ∀θ ∈Θ, V (θ) = λ ln (M/λ) + XM j=1 θ(j) ln θ(j) , (5) which has its minimum at θ0 = (λ/M, . . . , λ/M)T . It is easy to check that this function is α-strongly convex with respect to the norm ∥· ∥1 with parameter α = 1/λ , i.e., V (sx + (1 −s)y) ≤sV (x) + (1 −s)V (y) −α 2 s(1 −s)∥x −y∥2 1 (6) for all x, y ∈Θ and any s ∈[0, 1]. Let β > 0 be a parameter. We call β-conjugate of V the following convex transform: ∀z ∈RM, Wβ(z) def = sup θ∈Θ −zT θ −βV (θ) . As it straightforwardly follows from (5), the β-conjugate is given here by: Wβ(z) = λ β ln 1 M XM k=1 e−z(k)/β , ∀z ∈RM, (7) which has a Lipschitz-continuous gradient w.r.t. ∥· ∥1 , namely, ∥∇Wβ(z) −∇Wβ( ˜z )∥1 ≤λ β ∥z −˜z∥∞, ∀z, ˜z ∈RM. (8) Though we will focus on a particular algorithm based on the entropic proxy function, our results apply for a generic algorithmic scheme which takes advantage of the general properties of convex transforms (see [8] for details). The key property in the proof is the inequality (8). 3 Algorithm and main result The mirror descent algorithm is a stochastic gradient algorithm in the dual space. At each iteration i, a new data point (Xi, Yi) is observed and there are two updates: one is the value ζi as the result of the stochastic gradient descent in the dual space, the other is the update of the parameter θi which is the ”mirror image” of ζi. In order to tune the algorithm properly, we need two fixed positive sequences (γi)i≥1 (stepsize) and (βi)i≥1 (temperature) such that βi ≥βi−1. The mirror descent algorithm with averaging is as follows: Algorithm. • Fix the initial values θ0 ∈Θ and ζ0 = 0 ∈RM. • For i = 1, . . . , t −1, do ζi = ζi−1 + γiui(θi−1) , θi = −∇Wβi(ζi) . (9) • Output at iteration t the following convex combination: ˆθt = Xt i=1 γiθi−1 .Xt j=1 γj . (10) At this point, we actually have described a class of algorithms. Given the observations of the stochastic sub-gradient (3), particular choices of the proxy function V , of the stepsize and temperature parameters, will determine the algorithm completely. We discuss these choices with more details in [8]. In this paper, we focus on the entropic proxy function and consider a nearly optimal choice for the stepsize and temperature parameters which is the following: γi ≡1 , βi = β0 √ i + 1 , i = 1, 2, . . . , β0 > 0 . (11) We can now state our rate of convergence result. Theorem. Assume that the loss function Q satisfies the following boundedness condition: sup θ∈Θ E ∥∇θQ(θ, Z)∥2 ∞≤L2 < ∞. (12) Fix also β0 = L/ √ ln M. Then, for any integer t ≥1, the excess risk of the estimate bθt described above satisfies the following bound: E A(bθt) −min θ∈Θ A(θ) ≤2 Lλ ( ln M)1/2 √t + 1 t . (13) Example. Consider the setting of supervised learning where the data are modelled by a pair (X, Y ) with X ∈X being an observation vector and Y a label, either integer (classification) or real-valued (regression). Boosting and SVM algorithms are related to the minimization of a functional R(f) = Eϕ(Y f(X)) where ϕ is a convex non-negative cost function (typically exponential, logit or hinge loss) and f belongs to a given class of combined predictors. The aggregation problem consists in finding the best linear combination of elements from a finite set of predictors {h1, . . . , hM} with hj : X →[−K, K]. Taking compact notations, it means that we search for f of the form f = θT H with H denoting the vector-valued function whose components are these base predictors: H(x) = (h1(x), . . . , hM(x))T , and θ belonging in a decision set Θ = ΘM,λ. Take for instance ϕ to be non-increasing. It is easy to see that this problem can be interpreted in terms of our general setting with Z = (X, Y ), Q(Z, θ) = ϕ(Y θT H(X)) and L = Kϕ′(Kλ). 4 Discussion In this section, we provide some insights on the method and the result of the previous section. 4.1 Heuristics Suppose that we want to minimize a convex function θ 7→A(θ) over a convex set Θ. If θ0, . . . , θt−1 are the available search points at iteration t, we can provide the affine approximations φi of the function A defined, for θ ∈Θ, by φi(θ) = A(θi−1) + (θ −θi−1)T ∇A(θi−1), i = 1, . . . , t . Here θ 7→∇A(θ) is a vector function belonging to the sub-gradient of A(·). Taking a convex combination of the φi’s, we obtain an averaged approximation of A(θ): ¯φt(θ) = Pt i=1 γi A(θi−1) + (θ −θi−1)T ∇A(θi−1) Pt i=1 γi . At first glance, it would seem reasonable to choose as the next search point a vector θ ∈Θ minimizing the approximation ¯φt, i.e., θt = arg min θ∈Θ ¯φt(θ) = arg min θ∈Θ θT t X i=1 γi∇A(θi−1) ! . (14) However, this does not make any progress, because our approximation is “good” only in the vicinity of search points θ0, . . . , θt−1. Therefore, it is necessary to modify the criterion, for instance, by adding a special penalty Bt(θ, θt−1) to the target function in order to keep the next search point θt in the desired region. Thus, one chooses the point: θt = arg min θ∈Θ " θT t X i=1 γi∇A(θi−1) ! + Bt(θ, θt−1) # . (15) Our algorithm corresponds to a specific type of penalty Bt(θ, θt−1) = βtV (θ), where V is the proxy function. Also note that in our problem the vector-function ∇A(·) is not available. Therefore, we replace in (15) the unknown gradients ∇A(θi−1) by the observed stochastic sub-gradients ui(θi−1). This yields a new definition of the t-th search point: θt = arg min θ∈Θ " θT t X i=1 γiui(θi−1) ! + βtV (θ) # = arg max θ∈Θ −ζT t θ −βtV (θ) , (16) where ζt = Pt i=1 γiui(θi−1). By a standard result of convex analysis (see e.g. [3]), the solution to this problem reads as −∇Wβt(ζt) and it is now easy to deduce the iterative scheme (9) of the mirror descent algorithm. 4.2 Comparison with previous work The versions of mirror descent method proposed in [12] are somewhat different from our iterative scheme (9). One of them, closest to ours, is studied in detail in [3]. It is based on the recursive relation θi = −∇W1 −∇V (θi−1) + γiui(θi−1) , i = 1, 2, . . . , (17) where the function V is strongly convex with respect to the norm of initial space E (which is not necessarily the space ℓM 1 ) and W1 is the 1-conjugate function to V . If Θ = RM and V (θ) = 1 2∥θ∥2 2, the scheme of (17) coincides with the ordinary gradient method. For the unit simplex Θ = ΘM,1 and the entropy type proxy function V from (5) with λ = 1, the coordinates θ(j) i of vector θi from (17) are: ∀j = 1, . . . , M, θ(j) i = θ(j) 0 exp − i X m=1 γmum, j(θm−1) ! M X k=1 θ(k) 0 exp − i X m=1 γmum, k(θm−1) ! . (18) The algorithm is also known as the exponentiated gradient (EG) method [10]. The differences between the algorithm (17) and ours are the following: • the initial iterative scheme of the Algorithm is different than that of (17), particularly, it includes the second tuning parameter βi ; moreover, the algorithm (18) uses initial value θ0 in a different manner; • our algorithm contains the additional averaging step of the updates (10). The convergence properties of the EG method (18) have been studied in a deterministic setting [6]. Namely, it has been shown that, under some assumptions, the difference At(θt) −minθ∈ΘM,1 At(θ), where At is the empirical risk, is bounded by a constant depending on M and t. If this constant is small enough, these results show that the EG method provides good numerical minimizers of the empirical risk At. The averaging step allows the use of the results provided in [5] to derive generalization error bounds from relative loss bounds. This technique leads to rates of convergence of the order p (ln M)/t as well but with suboptimal multiplicative factor in λ. Finally, we point out that the algorithm (17) may be deduced from the ideas mentioned in Subsection 4.1 and which are studied in the literature on proximal methods within the field of convex optimization (see, e.g., [9, 1] and the references therein). Namely, under rather general conditions, the variable θi from (17) solves the the minimization problem θi = arg min θ∈Θ θT γiui(θi−1) + B(θ, θi−1) , (19) where the penalty B(θ, θi−1) = V (θ) −V (θi−1) −(θ −θi−1)T ∇V (θi−1) represents the Bregman divergence between θ and θi−1 related to the function V . 4.3 General comments Performance and efficiency. The rate of convergence of order √ ln M/ √ t is typical without low noise assumptions (as they are introduced in [17]). Batch procedures based on minimization of the empirical convex risk functional present a similar rate. From the statistical point of view, there is no remarkable difference between batch and our mirror-descent procedure. On the other hand, from the computational point of view, our procedure is quite comparable with the direct stochastic gradient descent. However, the mirror-descent algorithm presents two major advantages as compared both to batch and to direct stochastic gradient: (i) its behavior with respect to the cardinality of the base class is better than for direct stochastic gradient descent (of the order of √ ln M in the Theorem, instead of M or √ M for direct stochastic gradient); (ii) mirror-descent presents a higher efficiency especially in high-dimensional problems as its algorithmic complexity and memory requirements are of strictly smaller order than for corresponding batch procedures (see [7] for a comparison). Optimality of the rate of convergence. Using the techniques of [7] and [16] it is not hard to prove minimax lower bound on the excess risk E A(bθt) −minθ∈ΘM,λ A(θ) having the order (ln M)1/2/ √ t for M ≥t1/2+δ with some δ > 0. This indicates that the upper bound of the Theorem is rate optimal for such values of M. Choice of the base class. We point out that the good behaviour of this method crucially relies on the choice of the base class of functions {hj}1≤j≤M. As far as theory is concerned, in order to provide a complete statistical analysis, one should establish approximation error bounds on the quantity inff∈FM,λ A(f) −inff A(f) showing that the richness of the base class is reflected both by diversity (orthogonality or independence) of the hj’s and by its cardinality M. For example, one can take hj’s as the eigenfunctions associated to some positive definite kernel. We refer to [14], [15], for related results. The choice of λ can be motivated by similar considerations. In fact, to minimize the approximation error it might be useful to take λ depending on the sample size t and tending to infinity with some slow rate as in [11]. A balance between the stochastic error as given in the Theorem and the approximation error would then determine the optimal choice of λ. 5 Proof of the Theorem Introduce the notation ∇A(θ) = Eui(θ) and ξi(θ) = ui(θ) −∇A(θ). Put vi = ui(θi−1) which gives ζi−ζi−1 = γivi. By continuous differentiability of Wβt−1 and by (8) we have: Wβi−1(ζi) = Wβi−1(ζi−1) + γivT i ∇Wβi−1(ζi−1) +γi Z 1 0 vT i ∇Wβi−1(τζi + (1 −τ)ζi−1) −∇Wβi−1(ζi−1) dτ ≤ Wβi−1(ζi−1) + γivT i ∇Wβi−1(ζi−1) + λγ2 i ∥vi∥2 ∞ 2βi−1 . Then, using the fact that (βi)i≥1 is a non-decreasing sequence and that, for z fixed, β 7→ Wβ(z) is a non-increasing function, we get Wβi(ζi) ≤Wβi−1(ζi) ≤Wβi−1(ζi−1) −γiθT i−1vi + λγ2 i ∥vi∥2 ∞ 2βi−1 . Summing up over the i’s and using the representation ζt = Pt i=1 γivi, we get: ∀θ ∈Θ, Xt i=1 γi(θi−1 −θ)T vi ≤−Wβt(ζt) −ζT t θ + Xt i=1 λγ2 i ∥vi∥2 ∞ 2βi−1 since Wβ0(ζ0) = 0. From definition of Wβ, we have, ∀ζ ∈RM and ∀θ ∈Θ, −Wβt(ζ) − ζT θ ≤βtV (θ). Finally, since vi = ∇A(θi−1) + ξi(θi−1), we get t X i=1 γi(θi−1 −θ)T ∇A(θi−1) ≤βtV (θ) − t X i=1 γi(θi−1 −θ)T ξi(θi−1) + t X i=1 λγ2 i ∥vi∥2 ∞ 2βi−1 . As we are to take expectations, we note that, conditioning on θi−1 and using the independence between θi−1 and (Xi, Yi), we have: E (θi−1 −θ)T ξi(θi−1) = 0. Now, convexity of A and the previous display lead to: ∀θ ∈Θ , E A(bθt) −A(θ) ≤ Pt i=1 γiE [(θi−1 −θ)T ∇A(θi−1)] Pt i=1 γi = 1 t t X i=1 E [(θi−1 −θ)T ∇A(θi−1)] ≤ √t + 1 t β0V ∗+ λL2 β0 , where we have set V ∗= maxθ∈Θ V (θ) and made use of the boundedness assumption E ∥ui(θ)∥2 ∞≤L2 and of the particular choice for the stepsize and temperature parameters. Noticing that V ∗= λ ln M and optimizing this bound in β0 > 0, we obtain the result. Acknowledgments We thank Nicol`o Cesa-Bianchi for sharing with us his expertise on relative loss bounds. References [1] Beck, A. & Teboulle, M. (2003) Mirror descent and nonlinear projected subgradient methods for convex optimization. Operations Research Letters, 31:167–175. [2] Ben-Tal, A., Margalit, T. & Nemirovski, A. (2001) The Ordered Subsets Mirror Descent optimization method and its use for the Positron Emission Tomography reconstruction problem. SIAM J. on Optimization, 12:79–108. [3] Ben-Tal, A. & Nemirovski, A.S. (1999) The conjugate barrier mirror descent method for non-smooth convex optimization. MINERVA Optimization Center Report, Technion Institute of Technology. Available at http://iew3.technion.ac.il/Labs/Opt/opt/Pap/CP MD.pdf [4] Cesa-Bianchi, N. & Gentile, C. (2005) Improved risk tail bounds for on-line algorithms. Submitted. [5] Cesa-Bianchi, N., Conconi, A. & Gentile, C. (2004) On the generalization ability of on-line learning algorithms. IEEE Transactions on Information Theory, 50(9):2050– 2057. [6] Helmbold, D.P., Kivinen, J. & Warmuth, M.K. (1999) Relative loss bounds for single neurons. IEEE Trans. on Neural Networks, 10(6):1291–1304. [7] Juditsky, A. & Nemirovski, A. (2000) Functional aggregation for nonparametric estimation. Annals of Statistics, 28(3): 681–712. [8] Juditsky, A.B., Nazin, A.V., Tsybakov, A.B. & Vayatis N. (2005) Recursive Aggregation of Estimators via the Mirror Descent Algorithm with Averaging. Technical Report LPMA, Universit´e Paris 6. Available at http://www.proba.jussieu.fr/pageperso/vayatis/publication.html [9] Kiwiel, K.C. (1997) Proximal minimization methods with generalized Bregman functions. SIAM J. Control Optim., 35:1142–1168. [10] Kivinen J. & Warmuth M.K. (1997) Additive versus exponentiated gradient updates for linear prediction. Information and Computation, Vol.132(1): 1–64. [11] Lugosi, G. & Vayatis, N. (2004) On the Bayes-risk consistency of regularized boosting methods (with discussion). Annals of Statitics, 32(1): 30–55. [12] Nemirovski, A.S. & Yudin, D.B. (1983) Problem Complexity and Method Efficiency in Optimization. Wiley-Interscience. [13] Polyak, B.T. & Juditsky, A.B. (1992) Acceleration of stochastic approximation by averaging. SIAM J. Control Optim., 30:838–855. [14] Scovel, J.C. & Steinwart, I. (2005) Fast Rates for Support Vector Machines. In Proceedings of the 18th Conference on Learning Theory (COLT 2005), Bertinoro, Italy. [15] Tarigan, B. & van de Geer, S. (2004) Adaptivity of Support Vector Machines with ℓ1 Penalty. Preprint, University of Leiden. [16] Tsybakov, A. (2003) Optimal Rates of Aggregation. Proceedings of COLT’03, LNCS, Springer, Vol. 2777:303–313. [17] Tsybakov, A. (2004) Optimal aggregation of classifiers in statistical learning. Annals of Statistics, 32(1):135–166. [18] Zhang, T. (2004) Statistical behavior and consistency of classification methods based on convex risk minimization (with discussion). Annals of Statistics, 32(1):56–85. [19] Zhang, T. (2004) Solving large scale linear prediction problems using stochastic gradient descent algorithms. In Proceedings of ICML’04.
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Asymptotics of Gaussian Regularized Least-Squares Ross A. Lippert M.I.T., Department of Mathematics 77 Massachusetts Avenue Cambridge, MA 02139-4307 lippert@math.mit.edu Ryan M. Rifkin Honda Research Institute USA, Inc. 145 Tremont Street Boston, MA 02111 rrifkin@honda-ri.com Abstract We consider regularized least-squares (RLS) with a Gaussian kernel. We prove that if we let the Gaussian bandwidth σ →∞while letting the regularization parameter λ →0, the RLS solution tends to a polynomial whose order is controlled by the rielative rates of decay of 1 σ2 and λ: if λ = σ−(2k+1), then, as σ →∞, the RLS solution tends to the kth order polynomial with minimal empirical error. We illustrate the result with an example. 1 Introduction Given a data set (x1, y1), (x2, y2), . . . , (xn, yn), the inductive learning task is to build a function f(x) that, given a new x point, can predict the associated y value. We study the Regularized Least-Squares (RLS) algorithm for finding f, a common and popular algorithm [2, 5] that can be used for either regression or classification: min f∈H 1 n n X i=1 (f(xi) −yi)2 + λ||f||2 K. Here, H is a Reproducing Kernel Hilbert Space (RKHS) [1] with associated kernel function K, ||f||2 K is the squared norm in the RKHS, and λ is a regularization constant controlling the tradeoff between fitting the training set accurately and forcing smoothness of f. The Representer Theorem [7] proves that the RLS solution will have the form f(x) = Pn i=1 ciK(xi, x), and it is easy to show [5] that we can find the coefficients c by solving the linear system (K + λnI)c = y, (1) where K is the n by n matrix satisfying Kij = K(xi, xj). We focus on the Gaussian kernel K(xi, xj) = exp(−||xi −xj||2/2σ2). Our work was originally motivated by the empirical observation that on a range of benchmark classification tasks, we achieved surprisingly accurate classification using a Gaussian kernel with a very large σ and a very small λ (Figure 1; additional examples in [6]). This prompted us to study the large-σ asymptotics of RLS. As σ →∞, K(xi, xj) →1 for arbitrary xi and xj. Consider a single test point x0. RLS will first find c using Equation 1, 1e−04 1e−01 1e+02 1e+05 0.4 0.6 0.8 1.0 1.2 1e−11 1e−08 1e−05 0.01 10 10000 RLSC Results for GALAXY Dataset Sigma Accuracy m=1.0d−249 m=0.9 m=0.99999 Fig. 1. RLS classification accuracy results for the UCI Galaxy dataset over a range of σ (along the x-axis) and λ (different lines) values. The vertical labelled lines show m, the smallest entry in the kernel matrix for a given σ. We see that when λ = 1e −11, we can classify quite accurately when the smallest entry of the kernel matrix is .99999. then compute f(x0) = ctk where k is the kernel vector, ki = K(xi, x0). Combining the training and testing steps, we see that f(x0) = yt(K + λnI)−1k. Both K and k are close to 1 for large σ, i.e. Kij = 1 + ϵij and ki = 1 + ϵi. If we directly compute c = (K + λnI)−1y, we will tend to wash out the effects of the ϵij term as σ becomes large. If, instead, we compute f(x0) by associating to the right, first computing point affinities (K + λnI)−1k, then the ϵij and ϵj interact meaningfully; this interaction is crucial to our analysis. Our approach is to Taylor expand the kernel elements (and thus K and k) in 1/σ, noting that as σ →∞, consecutive terms in the expansion differ enormously. In computing (K + λnI)−1k, these scalings cancel each other out, and result in finite point affinities even as σ →∞. The asymptotic affinity formula can then be “transposed” to create an alternate expression for f(x0). Our main result is that if we set σ2 = s2 and λ = s−(2k+1), then, as s →∞, the RLS solution tends to the kth order polynomial with minimal empirical error. The main theorem is proved in full. Due to space restrictions, the proofs of supporting lemmas and corollaries are omitted; an expanded version containing all proofs is available [4]. 2 Notation and definitions Definition 1. Let xi be a set of n + 1 points (0 ≤i ≤n) in a d dimensional space. The scalar xia denotes the value of the ath vector component of the ith point. The n × d matrix, X is given by Xia = xia. We think of X as the matrix of training data x1, . . . , xn and x0 as an 1×d matrix consisting of the test point. Let 1m, 1lm denote the m dimensional vector and l × m matrix with components all 1, similarly for 0m, 0lm. We will dispense with such subscripts when the dimensions are clear from context. Definition 2 (Hadamard products and powers). For two l × m matrices, N, M, N ⊙M denotes the l × m matrix given by (N ⊙M)ij = NijMij. Analogously, we set (N ⊙c)ij = N c ij. Definition 3 (polynomials in the data). Let I ∈Zd ≥0 (non-negative multi-indices) and Y be a k × d matrix. Y I is the k dimensional vector given by Y I i = Qd a=1 Y Ia ia . If h : Rd →R then h(Y ) is the k dimensional vector given by (h(Y ))i = h(Yi1, . . . , Yid). The d canonical vectors, ea ∈Zd ≥0, are given by (ea)b = δab. Any scalar function, f : R →R, applied to any matrix or vector, A, will be assumed to denote the elementwise application of f. We will treat y →ey as a scalar function (we have no need of matrix exponentials in this work, so the notation is unambiguous). We can re-express the kernel matrix and kernel vector in this notation: K = e 1 2σ2 Pd a=1 2Xea(Xea)t−X2ea1t n−1n(X2ea) t (2) = diag e− 1 2σ2 ||X||2 e 1 σ2 XXtdiag e− 1 2σ2 ||X||2 (3) k = e 1 2σ2 Pd a=1 2Xeaxea 0 −X2ea11−1nx2ea 0 (4) = diag e− 1 2σ2 ||X||2 e 1 σ2 Xxt 0e− 1 2σ2 ||x0||2. (5) 3 Orthogonal polynomial bases Let Vc = span{XI : |I| = c} and V≤c = Sc a=0 Vc which can be thought of as the set of all d variable polynomials of degree c, evaluated on the training data. Since the data are finite, there exists b such that V≤c = V≤b for all c ≥b. Generically, b is the smallest c such that c + d d ≥n. Let Q be an orthonormal matrix in Rn×n whose columns progressively span the V≤c spaces, i.e. Q = ( B0 B1 · · · Bb ) where QtQ = I and colspan{( B0 · · · Bc )} = V≤c. We might imagine building such a Q via the Gramm-Schmidt process on the vectors X0, Xe1, . . . , Xed, . . . XI, . . . taken in order of non-decreasing |I|. Letting CI = |I| I1 . . . Id be multinomial coefficients, the following relations between Q, X, and x0 are easily proved. (Xxt 0)⊙c = X |I|=c CIXI(xI 0)t hence (Xxt 0)⊙c ∈Vc (XXt)⊙c = X |I|=c CIXI(XI)t hence colspan{(XXt)⊙c} = Vc and thus, Bt i(Xxt 0)⊙c = 0 if i > c, Bt i(XXt)⊙cBj = 0 if i > c or j > c, and Bt c(XXt)⊙cBc is non-singular. Finally, we note that argminv∈V≤c{||y −v||} = P a≤c Ba(Bt ay). 4 Taking the σ →∞limit We will begin with a few simple lemmas about the limiting solutions of linear systems. At the end of this section we will arrive at the limiting form of suitably modified RLSC equations. Lemma 1. Let i1 < · · · < iq be positive integers. Let A(s), y(s) be a block matrix and block vector given by A(s) = A00(s) si1A01(s) · · · siqA0q(s) si1A10(s) si1A11(s) · · · siqA1q(s) · · · · · · · · · · · · siqAq0(s) siqAq1(s) · · · siqAqq(s) , y(s) = b0(s) si1b1(s) · · · siqbq(s) where Aij(s) and bi(s) are continuous matrix-valued and vector-valued functions of s with Aii(0) non-singular for all i. lim s→0 A−1(s)y(s) = A00(0) 0 · · · 0 A10(0) A11(0) · · · 0 · · · · · · · · · · · · Aq0(0) Aq1(0) · · · Aqq(0) −1 b0(0) b1(0) · · · bq(0) We are now ready to state and prove the main result of this section, characterizing the limiting large-σ solution of Gaussian RLS. Theorem 1. Let q be an integer satisfying q < b, and let p = 2q + 1. Let λ = Cσ−p for some constant C. Define A(c) ij = 1 c!Bt i(XXt)⊙cBj, and b(c) i = 1 c!Bt i (Xxt 0)⊙c. lim σ→∞ K + nCσ−pI −1 k = v where v = ( B0 · · · Bq ) w (6) b(0) 0 b(1) 1 · · · b(q) q = A(0) 00 0 · · · 0 A(1) 10 A(1) 11 · · · 0 · · · · · · · · · · · · A(q) q0 A(q) q1 · · · A(q) qq w (7) We first manipulate the equation (K + nλI)y = k according to the factorizations in (3) and (5). K = diag e− 1 2σ2 ||X||2 e 1 σ2 XXtdiag e− 1 2σ2 ||X||2 = NPN k = diag e− 1 2σ2 ||X||2 e 1 σ2 Xxt 0e− 1 2σ2 ||x0||2 = Nwα Noting that limσ→∞e− 1 2σ2 ||x0||2diag e 1 2σ2 ||X||2 = limσ→∞αN −1 = I, we have v ≡lim σ→∞(K + nCσ−pI)−1k = lim σ→∞(NPN + βI)−1Nwα = lim σ→∞αN −1(P + βN −2)−1w = lim σ→∞ e 1 σ2 XXt + nCσ−pdiag e 1 σ2 ||X||2−1 e 1 σ2 Xxt 0. Changing bases with Q, Qtv = lim σ→∞ Qte 1 σ2 XXtQ + nCσ−pQtdiag e 1 σ2 ||X||2 Q −1 Qte 1 σ2 Xxt 0. Expanding via Taylor series and writing in block form (in the b × b block structure of Q), Qte 1 σ2 XXtQ = Qt(XXt)⊙0Q + 1 1!σ2 Qt(XXt)⊙1Q + 1 2!σ4 Qt(XXt)⊙2Q + · · · = A(0) 00 0 · · · 0 0 0 · · · 0 · · · · · · · · · · · · 0 0 · · · 0 + 1 σ2 A(1) 00 A(1) 01 · · · 0 A(1) 10 A(1) 11 · · · 0 · · · · · · · · · · · · 0 0 · · · 0 + · · · Qte 1 σ2 Xxt 0 = Qt(Xxt 0)⊙0 + 1 σ2 Qt(Xxt 0)⊙1 + 1 σ4 Qt(Xxt 0)⊙2 + · · · = b(0) 0 0 · · · 0 + 1 σ2 b(1) 0 b(1) 1 · · · 0 + · · · nCσ−pQtdiag e 1 σ2 ||X||2 Q = nCσ−pI + · · · . Since the A(c) cc are non-singular, Lemma 3 applies, giving our result. ⊓⊔ 5 The classification function When performing RLS, the actual prediction of the limiting classifier is given via f∞(x0) ≡lim σ→∞yt(K + nCσ−pI)−1k. Theorem 1 determines v = limσ→∞(K +nCσ−pI)−1k,showing that f∞(x0) is a polynomial in the training data X. In this section, we show that f∞(x0) is, in fact, a polynomial in the test point x0. We continue to work with the orthonormal vectors Bi as well as the auxilliary quantities A(c) ij and b(c) i from Theorem 1. Theorem 1 shows that v ∈V≤q: the point affinity function is a polynomial of degree q in the training data, determined by (7). X i,j≤c c!BiA(c) ij Bt j = (XXt)⊙c hence X j≤c c!BcA(c) cj Bt j = BcBt c(XXt)⊙c X i≤c c!Bib(c) i = (Xxt 0)⊙c hence c!Bcb(c) i = BcBt c(Xxt 0)⊙c we can restate Equation 7 in an equivalent form: Bt 0 · · · Bt q t 0!b(0) 0 1!b(1) 1 · · · q!b(q) q − 0!A(0) 00 0 · · · 0 1!A(1) 10 1!A(1) 11 · · · 0 · · · · · · · · · · · · q!A(q) q0 q!A(q) q1 · · · q!A(q) qq Bt 0 · · · Bt q v = 0 (8) X c≤q c!Bcb(c) c − X c≤q X j≤c c!BcA(c) cj Bt jv = 0 (9) X c≤q BcBt c (Xxt 0)⊙c −(XXt)⊙cv = 0. (10) Up to this point, our results hold for arbitrary training data X. To proceed, we require a mild condition on our training set. Definition 4. X is called generic if XI1, . . . , XIn are linearly independent for any distinct multi-indices {Ii}. Lemma 2. For generic X, the solution to Equation 7 (or equivalently, Equation 10) is determined by the conditions ∀I : |I| ≤q, (XI)tv = xI 0, where v ∈V≤q. Theorem 2. For generic data, let v be the solution to Equation 10. For any y ∈Rn, f(x0) = ytv = h(x0), where h(x) = P |I|≤q aIxI is a multivariate polynomial of degree q minimizing ||y −h(X)||. We see that as σ →∞, the RLS solution tends to the minimum empirical error kth order polynomial. 6 Experimental Verification In this section, we present a simple experiment that illustrates our results. We consider a fith-degree polynomial function. Figure 2 plots f, along with a 150 point dataset drawn by choosing xi uniformly in [0, 1], and choosing y = f(x) + ϵi, where ϵi is a Gaussian random variable with mean 0 and standard deviation .05. Figure 2 also shows (in red) the best polynomial approximations to the data (not to the ideal f) of various orders. (We omit third order because it is nearly indistinguishable from second order.) According to Theorem 1, if we parametrize our system by a variable s, and solve a Gaussian regularized least-squares problem with σ2 = s2 and λ = Cs−(2k+1) for some integer k, then, as s →∞, we expect the solution to the system to tend to the kth-order databased polynomial approximation to f. Asymptotically, the value of the constant C does not matter, so we (arbitrarily) set it to be 1. Figure 3 demonstrates this result. We note that these experiments frequently require setting λ much smaller than machineϵ. As a consequence, we need more precision than IEEE double-precision floating-point, and our results cannot be obtained via many standard tools (e.g., MATLAB(TM)) We performed our experiments using CLISP, an implementation of Common Lisp that includes arithmetic operations on arbitrary-precision floating point numbers. 7 Discussion Our result provides insight into the asymptotic behavior of RLS, and (partially) explains Figure 1: in conjunction with additional experiments not reported here, we believe that 0.0 0.2 0.4 0.6 0.8 1.0 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 x y G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G 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 f 0th order 1st order 2nd order 4th order 5th order f(x), Random Sample of f(x), and Polynomial Approximations Fig. 2. f(x) = .5(1 −x) + 150x(x −.25)(x −.3)(x −.75)(x −.95), a random dataset drawn from f(x) with added Gaussian noise, and data-based polynomial approximations to f. we are recovering second-order polynomial behavior, with the drop-off in performance at various λ’s occurring at the transition to third-order behavior, which cannot be accurately recovered in IEEE double-precision floating-point. Although we used the specific details of RLS in deriving our solution, we expect that in practice, a similar result would hold for Support Vector Machines, and perhaps for Tikhonov regularization with convex loss more generally. An interesting implication of our theorem is that for very large σ, we can obtain various order polynomial classifications by sweeping λ. In [6], we present an algorithm for solving for a wide range of λ for essentially the same cost as using a single λ. This algorithm is not currently practical for large σ, due to the need for extended-precision floating point. Our work also has implications for approximations to the Gaussian kernel. Yang et al. use the Fast Gauss Transform (FGT) to speed up matrix-vector multiplications when performing RLS [8]. In [6], we studied this work; we found that while Yang et al. used moderate-tosmall values of σ (and did not tune λ), the FGT sacrificed substantial accuracy compared to the best achievable results on their datasets. We showed empirically that the FGT becomes much more accurate at larger values of σ; however, at large-σ, it seems likely we are merely recovering low-order polynomial behavior. We suggest that approximations to the Gaussian kernel must be checked carefully, to show that they produce sufficiently good results are moderate values of σ; this is a topic for future work. References 1. Aronszajn. Theory of reproducing kernels. Transactions of the American Mathematical Society, 68:337–404, 1950. 2. Evgeniou, Pontil, and Poggio. Regularization networks and support vector machines. Advances In Computational Mathematics, 13(1):1–50, 2000. 0.0 0.2 0.4 0.6 0.8 1.0 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 0th order solution, and successive approximations. Deg. 0 polynomial s = 1.d+1 s = 1.d+2 s = 1.d+3 0.0 0.2 0.4 0.6 0.8 1.0 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1st order solution, and successive approximations. Deg. 1 polynomial s = 1.d+1 s = 1.d+2 0.0 0.2 0.4 0.6 0.8 1.0 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 4th order solution, and successive approximations. Deg. 4 polynomial s = 1.d+1 s = 1.d+2 s = 1.d+3 s = 1.d+4 0.0 0.2 0.4 0.6 0.8 1.0 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 5th order solution, and successive approximations. Deg. 5 polynomial s = 1.d+1 s = 1.d+3 s = 1.d+5 s = 1.d+6 Fig. 3. As s →∞, σ2 = s2 and λ = s−(2k+1), the solution to Gaussian RLS approaches the kth order polynomial solution. 3. Keerthi and Lin. Asymptotic behaviors of support vector machines with gaussian kernel. Neural Computation, 15(7):1667–1689, 2003. 4. Ross Lippert and Ryan Rifkin. Asymptotics of gaussian regularized least-squares. Technical Report MIT-CSAIL-TR-2005-067, MIT Computer Science and Artificial Intelligence Laboratory, 2005. 5. Rifkin. Everything Old Is New Again: A Fresh Look at Historical Approaches to Machine Learning. PhD thesis, Massachusetts Institute of Technology, 2002. 6. Rifkin and Lippert. Practical regularized least-squares: λ-selection and fast leave-one-outcomputation. In preparation, 2005. 7. Wahba. Spline Models for Observational Data, volume 59 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial & Applied Mathematics, 1990. 8. Yang, Duraiswami, and Davis. Efficient kernel machines using the improved fast Gauss transform. In Advances in Neural Information Processing Systems, volume 16, 2004.
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Large scale networks fingerprinting and visualization using the k-core decomposition J. Ignacio Alvarez-Hamelin∗ LPT (UMR du CNRS 8627), Universit´e de Paris-Sud, 91405 ORSAY Cedex France Ignacio.Alvarez-Hamelin@lri.fr Luca Dall’Asta LPT (UMR du CNRS 8627), Universit´e de Paris-Sud, 91405 ORSAY Cedex France Luca.Dallasta@th.u-psud.fr Alain Barrat LPT (UMR du CNRS 8627), Universit´e de Paris-Sud, 91405 ORSAY Cedex France Alain.Barrat@th.u-psud.fr Alessandro Vespignani School of Informatics, Indiana University, Bloomington, IN 47408, USA alexv@indiana.edu Abstract We use the k-core decomposition to develop algorithms for the analysis of large scale complex networks. This decomposition, based on a recursive pruning of the least connected vertices, allows to disentangle the hierarchical structure of networks by progressively focusing on their central cores. By using this strategy we develop a general visualization algorithm that can be used to compare the structural properties of various networks and highlight their hierarchical structure. The low computational complexity of the algorithm, O(n + e), where n is the size of the network, and e is the number of edges, makes it suitable for the visualization of very large sparse networks. We show how the proposed visualization tool allows to find specific structural fingerprints of networks. 1 Introduction In recent times, the possibility of accessing, handling and mining large-scale networks datasets has revamped the interest in their investigation and theoretical characterization along with the definition of new modeling frameworks. In particular, mapping projects of the World Wide Web and the physical Internet offered the first chance to study topology and traffic of large-scale networks. Other studies followed describing population networks of practical interest in social science, critical infrastructures and epidemiology [1, 2, 3]. The study of large scale networks, however, faces us with an array of new challenges. The definitions of centrality, hierarchies and structural organizations are hindered by the large size of these networks and the complex interplay of connectivity patterns, traffic flows and geographical, social and economical attributes characterizing their basic elements. In this ∗Further author information: J.I.A-H. is also with Facultad de Ingenier´ıa, Universidad de Buenos Aires, Paseo Col´on 850, C 1063 ACV Buenos Aires, Argentina. context, a large research effort is devoted to provide effective visualization and analysis tools able to cope with graphs whose size may easily reach millions of vertices. In this paper, we propose a visualization algorithm based on the k-core decomposition able to uncover in a two-dimensional layout several topological and hierarchical properties of large scale networks. The k-core decomposition [4] consists in identifying particular subsets of the graph, called k-cores, each one obtained by recursively removing all the vertices of degree smaller than k, until the degree of all remaining vertices is larger than or equal to k. Larger values of the index k clearly correspond to vertices with larger degree and more central position in the network’s structure. This visualization tool allows the identification of real or computer-generated networks’ fingerprints, according to properties such as hierarchical arrangement, degree correlations and centrality. The distinction between networks with seemingly similar properties is achieved by inspecting the different layouts generated by the visualization algorithm. In addition, the running time of the algorithm grows only linearly with the size of the network, granting the scalability needed for the visualization of very large sparse networks. The proposed (publicly available [5]) algorithm appears therefore as a convenient method for the general analysis of large scale complex networks and the study of their architecture. The paper is organized as follows: after a brief survey on k-core studies (section 2), we present the basic definitions and the graphical algorithms in section 3 along with the basic features of the visualization layout. Section 4 shows how the visualizations obtained with the present algorithm may be used for network fingerprinting, and presents two examples of visualization of real networks. 2 Related work While a large number of algorithms aimed at the visualization of large scale networks have been developed (e.g., see [6]), only a few consider explicitly the k-core decomposition. Vladimir Batagelj et al. [7] studied the k-core decomposition applied to visualization problems, introducing some graphical tools to analyse the cores, mainly based on the visualization of the adjacency matrix of certain k-cores. To the best of our knowledge, the algorithm presented by Baur et al. in [8] is the only one completely based on a k-core analysis and directly targeted at the study of large information networks. This algorithm uses a spectral layout to place vertices having the largest shell index. A combination of barycentric and iteratively directed-forces allows to place the vertices of each k-shell, in decreasing order. Finally, the network is drawn in three dimensions, using the z axis to place each shell in a distinct horizontal layer. Note that the spectral layout is not able to distinguish two or more disconnected components. The algorithm by Baur et al. is also tuned for representing AS graphs and its total complexity depends on the size of the highest k-core (see [9] for more details on spectral layout), making the computation time of this proposal largely variable. In this respect, the algorithm presented here is different in that it can represent networks in which k-cores are composed by several connected components. Another difference is that representations in 2D are more suited for information visualization than other representations (see [10] and references therein). Finally, the algorithm parameters can be universally defined, yielding a fast and general tool for analyzing all types of networks. It is interesting to note that the notion of k-cores has been recently used in biologically related contexts, where it was applied to the analysis of protein interaction networks [11] or in the prediction of protein functions [12, 13]. Further applications in Internet-related areas can be found in [14], where the k-core decomposition is used for filtering out peripheral Autonomous Systems (ASes), and in [15] where the scale invariant structure of degree correlations and mapping biases in AS maps is shown. Finally in [16, 17], an interesting approach based on the k-core decomposition has been used to provide a conceptual and structural model of the Internet; the so-called medusa model for the Internet. 3 Graphical representation Let us consider a graph G = (V, E) of |V | = n vertices and |E| = e edges; a k-core is defined as follows [4]: -A subgraph H = (C, E|C) induced by the set C ⊆V is a k-core or a core of order k iff ∀v ∈C : degreeH(v) ≥k, and H is the maximum subgraph with this property. A k-core of G can therefore be obtained by recursively removing all the vertices of degree less than k, until all vertices in the remaining graph have at least degree k. Furthermore, we will use the following definitions: -A vertex i has shell index c if it belongs to the c-core but not to (c + 1)-core. We denote by ci the shell index of vertex i. -A shell Cc is composed by all the vertices whose shell index is c. The maximum value c such that Cc is not empty is denoted cmax. The k-core is thus the union of all shells Cc with c ≥k. -Each connected set of vertices having the same shell index c is a cluster Qc. Each shell Cc is thus composed by clusters Qc m, such that Cc = ∪1≤m≤qc maxQc m, where qc max is the number of clusters in Cc. The visualization algorithm we propose places vertices in 2 dimensions, the position of each vertex depending on its shell index and on the index of its neighbors. A color code allows for the identification of shell indices, while the vertex’s original degree is provided by its size that depends logarithmically on the degree. For the sake of clarity, our algorithm represents a small percentage of the edges, chosen uniformly at random. As mentioned, a central role in our visualization method is played by multi-components representation of kcores. In the most general situation, indeed, the recursive removal of vertices having degree less than a given k can break the original network into various connected components, each of which might even be once again broken by the subsequent decomposition. Our method takes into account this possibility, however we will first present the algorithm in the simplified case, in which none of the k-cores is fragmented. Then, this algorithm will be used as a subroutine for treating the general case (Table 1). 3.1 Drawing algorithm for k-cores with single connected component k-core decomposition. The shell index of each vertex is computed and stored in a vector C, along with the shells Cc and the maximum index cmax. Each shell is then decomposed into clusters Qc m of connected vertices, and each vertex i is labeled by its shell index ci and by a number qi representing the cluster it belongs to. The two dimensional graphical layout. The visualization is obtained assigning to each vertex i a couple of polar coordinates (ρi, αi): the radius ρi is a function of the shell index of the vertex i and of its neighbors; the angle αi depends on the cluster number qi. In this way, k-shells are displayed as layers with the form of circular shells, the innermost one corresponding to the set of vertices with highest shell index. A vertex i belongs to the cmax −ci layer from the center. More precisely, ρi is computed according to the following formula: ρi = (1 −ϵ)(cmax −ci) + ϵ |Vcj≥ci(i)| X j∈Vcj ≥ci(i) (cmax −cj) , (1) Vcj≥ci(i) is the set of neighbors of i having shell index cj larger or equal to ci. The parameter ϵ controls the possibility of rings overlapping, and is one of the only three external parameters required to tune image’s rendering. Inside a given shell, the angle αi of a vertex i is computed as follow: αi = 2π X 1≤m<qi |Qm| |Cci| + N |Qqi| 2|Cci| , π · |Qqi| |Cci| , (2) where Qqi and Cci are respectively the cluster qi and ci-shell the vertex belongs to, N is a normal distribution of mean |Qqi|/(2|Cci|) and width 2π|Qqi|/|Cci|. Since we are interested in distinguishing different clusters in the same shell, the first term on the right side of Eq. 2, referring to clusters with m < qi, allows to allocate a correct partition of the angular sector to each cluster. The second term on the right side of Eq. 2, on the other hand, specifies a random position for the vertex i in the sector assigned to the cluster Qqi. Colors and size of vertices. Colors depend on the shell index: vertices with shell index 1 are violet, and the maximum shell index vertices are red, following the rainbow color scale. The diameter of each vertex corresponds to the logarithm of its degree, giving a further information on vertex’s properties. The vertices with largest shell index are placed uniformly in a disk of radius u, which is the unit length (u = 1 for this reduced algorithm). 3.2 Extended algorithm for networks with many k-cores components The algorithm presented in the previous section can be used as the basic routine to define an extended algorithm aimed at the visualization of networks for which some k-cores are fragmented; i.e. made by more than one connected component. This issue is solved by assigning to each connected component of a k-core a center and a size, which depends on the relative sizes of the various components. Larger components are put closer to the global center of the representation (which has Cartesian coordinates (0, 0)), and have larger sizes. The algorithm begins with the center at the origin (0, 0). Whenever a connected component of a k-core, whose center p had coordinates (Xp, Yp), is broken into several components by removing all vertices of degree k, i.e. by applying the next decomposition step, a new center is computed for each new component. The center of the component h has coordinates (Xh, Yh), defined by Xh = Xp+δ(cmax−ch)·up·ϱh·cos(φh) ; Yh = Yp+δ(cmax−ch)·up·ϱh·sin(φh) , (3) where δ scales the distance between components, cmax is the maximum shell index and ch is the core number of component h (the components are numbered by h = 1, · · · , hmax in an arbitrary order), up is the unit length of its parent component, ϱh and φh are the radial and angular coordinates of the new center with respect to the parent center (Xp, Yp). We define ϱh and φh as follows: ϱh = 1 − |Sh| P 1≤j≤hmax |Sj| ; φh = φini + 2π P 1≤j≤hmax |Sj| X 1≤j≤h |Sj| , (4) where Sh is the set of vertices in the component h, P j |Sj| is the sum of the sizes of all components having the same parent component. In this way, larger components will be closer to the original parent component’s center p. The angle φh has two contributions. The initial angle φini is chosen uniformly at random1, while the angle sector is the sum of component angles whose number is less than or equal to the actual component number h. 1Note that if φini is fixed, all the centers of the various components are aligned in the final representation. Algorithm 1 1 k := 1 and end := false 2 while not end do 3 (end, C)←make core k 4 (Q, T )←compute clusters k −1, if k > 1 5 S←compute components k 6 (X, Y )←compute origin coordinates cmp k (Eqs. from 3 to 4) 7 U←compute unit size cmp k (Eq. 5) 8 k := k + 1 9 for each node i do 10 if ci == cmax then 11 set ρi and αi according to a uniform distribution in the disk of radius u (u is the core representation unit size) 12 else 13 set ρi and αi according to Eqs. 1 and 2 14 (X, Y)←compute final coordinates ρ α U X Y (Eq. 6) Table 1: Algorithm for the representation of networks using k-cores decomposition Finally, the unit length uh of a component h is computed as uh = |Sh| P 1≤j≤hmax |Sj| · up , (5) where up is the unit length of its parent component. Larger unit length and size are therefore attributed to larger components. For each vertex i, radial and angular coordinates are computed by equations 1 and 2 as in the previous algorithm. These coordinates are then considered as relative to the center (Xh, Yh) of the component to which i belongs. The position of i is thus given by xi = Xh + γ · uh · ρi · cos(αi); yi = Yh + γ · uh · ρi · sin(αi) (6) where γ is a parameter controlling the component’s diameter. The global algorithm is formally presented in Table 1. The main loop is composed by the following functions. First, the function {(end, C) ←make core k} recursively removes all vertices of degree k −1, obtaining the k-core, and stores into C the shell index k −1 of the removed vertices. The boolean variable end is set to true if the k-core is empty, otherwise it is set to false. The function {(Q, T ) ←compute clusters k −1} operates the decomposition of the (k −1)shell into clusters, storing for each vertex the cluster label into the vector Q, and filling table T , which is indexed by the shell index c and cluster label q: T (c, q) = (P 1≤m<q |Qm|/|Cc|, |Qq|/|Cc|). The possible decomposition of the k-core into connected components is determined by function {S ←compute components k}, that also collects into a vector S the number of vertices contained in each component. At the following step, functions {(X, Y ) ←compute origin coordinates cmp k} and {U ←compute unit size cmp k} get, respectively, the center and size of each component of the k-core, gathering them in vectors X, Y and U. Finally, the coordinates of each vertex are computed and stored in the vectors X and Y. Algorithm complexity. Batagelj and Zversnik [18] present an algorithm to perform the k-core decomposition, and show that its time complexity is O(e) (where e is the number of edges) for a connected graph. For a general graph it is O(n + e), where n is the number of nodes, which makes the algorithm very efficient for sparse graphs where e is of order n. max k −1 min k +1 min k max k shell index degree 10 d_max 3 max k −1 min k +1 min k max k shell index degree 10 d_max 3 Figure 1: Structure of a typical layout in two important cases: on the left, all k-cores are connected; on the right, some k-cores are composed by more than one connected component. The vertices are arranged in a series of concentric shells corresponding to the various k-shells. The diameter of each shell depends on both the shell index and, in case of multiple components (right) also on the relative fraction of vertices belonging to the different components. 3.3 Basic features of the visualization’s layout The main features of the layout’s structure obtained with the above algorithms are visible in Fig.1 where, for the sake of simplicity, we do not show any edge. The two-dimensional layout is composed of a series of concentric circular shells. Each shell corresponds to a single shell index and all vertices in it are therefore drawn with the same color. A color scale allows to distinguish different shell indices: the violet is used for the minimum shell index kmin, then we use a graduated rainbow scale for higher and higher shell indices up to the maximum value kmax that is colored in red. The diameter of each k-shell depends on the shell index k, and is proportional to kmax −k (In Fig.1, the position of each shell is identified by a circle having the corresponding diameter). The presence of a trivial order relation in the shell indices ensures that all shells are placed in a concentric arrangement. On the other hand, when a k-core is fragmented in two or more components, the diameters of the different components depend also on the relative number of vertices belonging to each of them, i.e. the fraction between the number of vertices belonging to that component and the total number of vertices in that core. This is a very important information, providing a way to distinguish between multiple components at a given shell index. Finally, the size of each node is proportional to the original degree of that vertex; we use a logarithmic scale for the size of the drawn bullets. 4 Network fingerprinting The k-core decomposition peels the network layer by layer, revealing the structure of the different shells from the outmost one to the more internal ones. The algorithm provides a direct way to distinguish the network’s different hierarchies and structural organization by means of some simple quantities: the radial width of the shells, the presence and size of clusters of vertices in the shells, the correlations between degree and shell index, the distribution of the edges interconnecting vertices of different shells, etc. 1) Shells Width: The thickness of a shell depends on the shell index properties of the neighbors of the vertices in the corresponding shell. For a given shell-diameter (black circle in the median position of shells in Fig.2), each vertex can be placed more internal or more external with respect to this reference. Nodes with more neighbors in higher shells are closer to the center and viceversa: in Fig.2, node y is more internal than node x because it Node y has more neighbors in the higher cores than node x. max k −1 min k +1 min k max k shell index degree 10 d_max 3 node x node y The thickness of the shell depends on the neighbors with higher coreness. max k −1 min k +1 min k max k shell index degree 10 d_max 3 Isolated nodes nodes in the same shell. Clusters: nodes connected with Figure 2: Left: each shell has a certain radial width. This width depends on the correlation’s properties of the vertices in the shell. In the second shell, we have pinpointed two nodes x and y. Node y is more internal than x because a larger part of its neighbors belongs to higher k-shells compared to x’s neighbors. The figure on the right shows the clustering properties of nodes in the same k-shell. In each k-shell, nodes that are directly connected between them (in the original graph) are drawn close one to the other, as in a cluster. Some of these sets of nodes are circled and highlighted in gray. Three examples of isolated nodes are also indicated; these nodes have no connections with the others of the same shell. has three edges towards higher index nodes, while x has only one. The maximum thickness of the shells is controlled by the ϵ parameter (Eq. 1). 2) Shell Clusters: The angular distribution of vertices in the shells is not completely homogeneous. Fig.2 shows that clusters of vertices can be observed. The idea is to group together all nodes of the same shell that are directly linked in the original graph and to represent them close one to another. Thus, a shell is divided in many angular sectors, each containing a cluster of vertices. This feature allows to figure out at a glance if the shells are composed of a single large connected component rather than divided into many small clusters, or even if there are isolated vertices (i.e. disconnected from all other nodes in the shell, not from the rest of the k-core!). 3) Degree-Shell index Correlation: Another property that can be studied from the obtained layouts is the correlation between the degree of the nodes and the shell index. Both quantities are centrality measures and the nature of their correlations is a very important feature characterizing a network’s topology. The nodes displayed in the most internal shells are those forming the central core of the network; the presence of degree-index correlations then corresponds to the fact that the central nodes are most likely high-degree hubs of the network. This effect is observed in many real communication networks with a clear hierarchical structure, such as the Internet at the Autonomous System level or the World Wide Air-transportation network [5]. On the contrary, the presence of hubs in external shells is typical of less hierarchically structured networks such as the World-Wide Web or the Internet Router Level. In this case, star-like configurations appear with high degree vertices connected only to very low degree vertices. These vertices are rapidly pruned out in the k-core decomposition even if they have a very high degree, leading to the presence of local hubs in external shells, as in Fig. 3. 4) Edges: The visualization shows only a homogeneously randomly sampled fraction of the edges, which can be tuned in order to get the better trade-off between the clarity of visualization and the necessity of giving information on the way the nodes are mainly connected. Edge-reduction techniques can be implemented to improve the algorithm’s capacity in representing edges; however, a homogeneous sampling does not alter the extraction of topological information, ensuring a low computational cost. Finally, the two halves of each max k −1 min k +1 min k max k shell index degree 10 d_max 3 The degree is strongly correlated with the shell index. max k −1 min k +1 min k max k shell index degree 10 d_max 3 Degree and shell index are correlated but with large fluctuations. Figure 3: Correlations between shell index and degree. On the left, we report a graph with strong correlation: the size of the nodes grows from the periphery to the center, in correspondence with the shell index. In the right-hand case, the degree-index correlations are blurred by large fluctuations, as stressed by the presence of hubs in the external shells. edge are colored with the color of the corresponding extremities to emphasize the connection among vertices in different shells. 5) Disconnected components: The fragmentation of any given k-core in two or more disconnected components is represented by the presence of a corresponding number of circular shells with different centers (Fig. 1). The diameter of these circles is related with the number of nodes of each component and modulated by the γ parameter (Eq. 6). The distance between components is controlled by the δ parameter (Eq. 3). In summary, the proposed algorithm makes possible a direct, visual investigation of a series of properties: hierarchical structures of networks, connectivity and clustering properties inside a given shell; relations and interconnectivity between different levels of the hierarchy, correlations between degree and shell index, i.e. between different measures of centrality. Numerous examples of the application of this tool to the visualization of real and computer generated networks can be found on the web page of the publicly available tool [5]. For example, the lack of hierarchy and structure of the Erd¨os-R´enyi random graph is clearly identified. Similarly the time correlations present in the Barab´asi-Albert network find a clear fingerprint in our visualization layout. Here we display another interesting illustration of the use and capabilities of the proposed algorithm in the analysis of large sparse graphs: the identification of the different hierarchical arrangement of the Internet network when visualized at the Autonomous system (AS) and the Router (IR) levels 2. The AS level is represented by collected routes of Oregon route-views [19] project, from May 2001. For the IR level, we use the graph obtained by an exploration of Govindan and Tangmunarunkit [20] in 2000. These networks are composed respectively by about 11500 and 200000 nodes. Figures 4 and 5 display the representations of these two different maps of Internet. At the AS level, all shells are populated, and, for any given shell, the vertices are distributed on a relatively large range of the radial coordinate, which means that their neighborhoods are variously composed. The shell index and the degree are very correlated, with a clear hierarchical structure, and links go principally from one shell to another. The hierarchical structure exhibited by our analysis of the AS level is a striking property; for instance, one might exploit it for showing that in the Internet high-degree vertices are naturally (as an implicit result of the self-organizing growth) placed in the innermost structure. At higher resolution, i.e. at the IR level, Internet’s properties are less structured: external layers, of 2The parameters are here set to the values ϵ = 0.18, δ = 1.3 and γ = 1.5. lowest shell index, contain vertices with large degree. For instance, we find 20 vertices with degree larger than 100 but index smaller than 6. The correlation between shell index and degree is thus clearly of a very different nature in the maps of Internet obtained at different granularities. Figure 4: Graphical representation of the AS network. The three snapshots correspond to the full network (top left), with the color scale of the shell index and the size scale for the nodes’ degrees, and to two magnifications showing respectively a more central part (top right) and a radial slice of the layout (bottom). 5 Conclusions Exploiting k-core decomposition, and the corresponding natural hierarchical structures, we develop a visualization algorithm that yields a layout encoding a considerable amount of the information needed for network fingerprinting in the simplicity of a 2D representation. One can easily read basic features of the graph (degree, hierarchical structure, etc.) as well as more entangled features, e.g. the relation between a vertex and the hierarchical position of its neighbors. The present visualization strategy is a useful tool to discriminate between networks with different topological properties and structural arrangement, and may be also used for comparison of models with real data, providing a further interesting tool for model Figure 5: Same as Fig. 4, for the graphical representation of the IR network. validation. Finally, we also provide a publicly available tool for visualizing networks [5]. Acknowledgments: This work has been partially funded by the European Commission Fet Open project COSIN IST-2001-33555 and contract 001907 (DELIS). References [1] R. Albert and A.-L. Barab´asi, “Statistical mechanics of complex networks,” Rev. Mod. Phys. 74, pp. 47–97, 2000. [2] S. N. Dorogovtsev and J. F. F. Mendes, Evolution of networks: From biological nets to the Internet and WWW, Oxford University Press, 2003. [3] R. Pastor-Satorras and A. Vespignani, Evolution and structure of the Internet: A statistical physics approach, Cambridge University Press, 2004. [4] V. Batagelj and M. Zaversnik, “Generalized Cores,” cs.DS/0202039 , 2002. [5] LArge NETwork VIsualization tool. http://xavier.informatics.indiana.edu/lanet-vi/. [6] http://http://i11www.ira.uka.de/cosin/tools/index.php. [7] V. Batagelj, A. Mrvar, and M. Zaversnik, “Partitioning Approach to Visualization of Large Networks,” in Graph Drawing ’99, Castle Stirin, Czech Republic, LNCS 1731, pp. 90–98, 1999. [8] M. Baur, U. Brandes, M. Gaertler, and D. Wagner, “Drawing the AS Graph in 2.5 Dimensions,” in ”12th International Symposium on Graph Drawing, Springer-Verlag editor”, pp. 43–48, 2004. [9] U. Brandes and S. Cornelsen, “Visual Ranking of Link Structures,” Journal of Graph Algorithms and Applications 7(2), pp. 181–201, 2003. [10] B. Shneiderman, “Why not make interfaces better than 3d reality?,” IEEE Computer Graphics and Applications 23, pp. 12–15, November/December 2003. [11] G. D. Bader and C. W. V. Hogue, “An automated method for finding molecular complexes in large protein interaction networks,” BMC Bioinformatics 4(2), 2003. [12] M. Altaf-Ul-Amin, K. Nishikata, T. Koma, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, and S. Kanaya, “Prediction of Protein Functions Based on K-Cores of Protein-Protein Interaction Networks and Amino Acid Sequences,” Genome Informatics 14, pp. 498–499, 2003. [13] S. Wuchty and E. Almaas, “Peeling the Yeast protein network,” Proteomics. 2005 Feb;5(2):4449. 5(2), pp. 444–449, 2005. [14] M. Gaertler and M. Patrignani, “Dynamic Analysis of the Autonomous System Graph,” in IPS 2004, International Workshop on Inter-domain Performance and Simulation, Budapest, Hungary, pp. 13–24, 2004. [15] I. Alvarez-Hamelin, L. Dall’Asta, A. Barrat, and A. Vespignani, “k-core decomposition: a tool for the analysis of large scale internet graphs,” cs.NI/0511007 . [16] S. Carmi, S. Havlin, S. Kirkpatrick, Y. Shavitt, and E. Shir, 2005. http://www.cs.huji.ac.il/˜kirk/Jellyfish_Dimes.ppt. [17] S. Carmi, S. Havlin, S. Kirkpatrick, Y. Shavitt, and E. Shir, “Medusa - new model of internet topology using k-shell decomposition,” cond-mat/0601240 . [18] V. Batagelj and M. Zaversnik, “An O(m) Algorithm for Cores Decomposition of Networks,” cs.DS/0310049 , 2003. [19] University of Oregon Route Views Project. http://www.routeviews.org/. [20] R. Govindan and H. Tangmunarunkit, “Heuristics for Internet Map Discovery,” in IEEE INFOCOM 2000, pp. 1371–1380, IEEE, (Tel Aviv, Israel), March 2000.
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Norepinephrine and Neural Interrupts Peter Dayan Angela J. Yu Gatsby Computational Neuroscience Unit Center for Brain, Mind & Behavior University College London Green Hall, Princeton University 17 Queen Square, London WC1N 3AR, UK Princeton, NJ 08540, USA dayan@gatsby.ucl.ac.uk ajyu@princeton.edu Abstract Experimental data indicate that norepinephrine is critically involved in aspects of vigilance and attention. Previously, we considered the function of this neuromodulatory system on a time scale of minutes and longer, and suggested that it signals global uncertainty arising from gross changes in environmental contingencies. However, norepinephrine is also known to be activated phasically by familiar stimuli in welllearned tasks. Here, we extend our uncertainty-based treatment of norepinephrine to this phasic mode, proposing that it is involved in the detection and reaction to state uncertainty within a task. This role of norepinephrine can be understood through the metaphor of neural interrupts. 1 Introduction Theoretical approaches to understanding neuromodulatory systems are plagued by the latter’s neural ubiquity, evolutionary longevity, and temporal promiscuity. Neuromodulators act in potentially different ways over many different time-scales [14]. There are various general notions about their roles, such as regulating sleeping and waking [13] and changing the signal to noise ratios of cortical neurons [11]. However, these are slowly giving way to more specific computational ideas [20, 7, 10, 24, 25, 5], based on such notions as optimal gain scheduling, prediction error and uncertainty. In this paper, we focus on the short term activity of norepinephrine (NE) neurons in the locus coeruleus [18, 1, 2, 3, 16, 4]. These neurons project NE to subcortical structures and throughout the entire cortex, with individual neurons having massive axonal arborizations [12]. Over medium and short time-scales, norepinephrine is implicated in various ways in attention, vigilance, and learning. Given the widespread distribution and effects of NE in key cognitive tasks, it is very important to understand what it is in a task that drives the activity of NE neurons, and thus what computational effects it may be exerting. Figure 1 illustrates some of the key data that has motivated theoretical treatments of NE. Figure 1A;B;C show more tonic responses operating around a time-scale of minutes. Figures 1D;E;F show the short-term effects that are our main focus here. Briefly, Figures 1A;B show that when the rules of a task are reversed, NE influences the speed of adaptation to the changed contingency (Figure 1A) and the activity of noradrenergic cells is tonically elevated (Figure 1B). Based on these data, we suggested [24, 25] that medium-term NE reports unexpected uncertainty arising from unpredicted changes in an environment or task. This signal is a key part of a strategy for inference in potentially labile contexts. It operates in collaboration with a putatively cholinergic signal which reports on expected uncertainty that arises, for instance, from known variability or noise. 1 5 10 15 0 20 40 60 80 100 Idazoxan Saline target non-target A % rats reaching criterion # days after spatial →visual shift B 0 15 30 Spikes/sec Time (min) C Time (sec) FA rate (Hz) Spikes/sec D Spikes/sec Time E Spikes/sec Time (sec) F Time Figure 1: NE activity and effects. (A) Rats solve a sequential decision problem in a linear maze. When the relevant cues are switched after a few days of learning (from spatial to visual), rats with pharmacologically boosted NE (“idazoxan”) learn to use the new set of cues faster than the controls. Adapted from [9]. (B) In a vigilance task, monkeys respond to rare targets and ignore common distractor stimuli. The trace shows the activity of a single NE neuron in the locus coeruleus (LC) around the time of a target-distractor reversal (vertical line). Tonic activity is elevated for a considerable period. Adapted from [2]. (C) Correlation between the gross fluctuations in the tonic activity of a single NE neuron (upper) and performance in the task (lower, measured by false alarm rate). Adapted from [20]. (D) Single NE cells are activated on a phasic time-scale stimulus locked (vertical line) to the target (upper plot) and not the distractor (lower plot). Adapted from [16]. (E) The average responses of a large number of norepinephrine cells (over a total of 41,454 trials) stimulus locked (vertical line) to targets or distractors, sorted by the nature and rectitude of the response. The asterisk marks (similar) early activation of the neurons by the stimulus. Adapted from [16]. (F) In a GO/NO-GO olfactory discrimination task for rats, single units are activated by the target odor (and not by the distractor odor), but are temporally much more tightly locked to the response (right) than the stimulus (left). Trials are ordered according to the time between stimulus (blue) and response (red). Adapted from [4]. However, Figures 1D;E;F, along with other substantial neurophysiological data on the activity of NE neurons [18, 4], show NE neurons have phasic response properties that lie outside this model. The data in Figure 1D;E come from a vigilance task [1], in which subjects can gain reward by reacting to a rare target (a rectangle oriented one way), while ignoring distractors (a rectangle oriented in the orthogonal direction). Under these circumstances, NE is consistently activated by the target and not the distractor (Figure 1D). There are also clear correlations in the magnitude of the NE activity and the nature of a trial: hit, miss, false alarm, correct reject (Figure 1E). It is known that the activity is weaker if the targets are more common [17] (though the lack of response to rare distractors shows that NE is not driven by mere rarity), and disappears if no action need be taken in response to the target [18]. In fact, the signal is more tightly related in time to the subsequent action than the preceding stimulus (Figure 1F). The signal has been qualitatively described in terms of influencing or controlling the allocation of behavioral or cognitive resources [20, 4]. Since it arises on every trial in an extremely well-learned task with stable stimulus contingencies, this NE signal clearly cannot be indicating unpredicted task changes. Brown et al [5] have recently made the seminal suggestion that it reports changes in the statistical structure of the input (stimulus-present versus stimulus-absent) to decision-making circuits that are involved in initiating differential responding to distinct target stimuli. A statistically necessary consequence of the change in the input structure is that afferent information should be integrated differently: sensory responses should be ignored if no target is present, but taken seriously otherwise. Their suggestion is that NE, by changing the gain of neurons in the decision-making circuit, has exactly this optimizing effect. In this paper, we argue for a related, but distinct, notion of phasic NE, suggesting that it reports on unexpected state changes within a task. This is a significant, though natural, extension of its role in reporting unexpected task changes [25]. We demonstrate that it accounts well for the neurophysiological data. In agreement with the various accounts of the effects of phasic NE, we consider its role as a form of internal interrupt signal [6]. Computers use interrupts to organize the correct handling of internal and external events such as timers or peripheral input. Higher-level programs specify what interrupts are allowed to gain control, and the consequences thereof. We argue that phasic NE is the medium for a somewhat similar neural interrupt, allowing the correct handling of statistically atypical events. This notion relates comfortably to many existing views of phasic NE, and provides a computational correlate for quantitative models. 2 The Model Figure 2A illustrates a simple hidden Markov generative model (HMM) of the vigilance task in Figure 1B-E. The (start) state models the condition established when the monkey fixates the light and initiates a trial. Following a somewhat variable delay, either the target (target) or the distractor (distractor) is presented, and the monkey must respond appropriately (release a continuously depressed bar for target and continue pressing for distractor) The transition out of start is uniformly distributed between timesteps 6 and 10, implemented by a time-varying transition function for this node: P(st|st−1 = start) = 1 −qt st = start 0.8qt st = distractor 0.2qt st = target (1) where qt =1/(11−t) for (6 ≤t ≤10) and qt =0 otherwise. The start and target states are assumed to be absorbing states (self-transition probability = 1). This transition function ensures that the stimulus onset has a uniform distribution between 6 and 10 timesteps (and 0 otherwise). Given that a transition out of start (into either target or distractor) takes place, the probability is .2 for entering target and .8 for start, as in the actual task. In addition, it is assumed that the node start does not emit observations, while target emits xt = t with probability η > 0.5 and d with probability 1 −η, and distractor emits xt = d with probability η and t with probability 1 −η. The transition out of start is evident as soon as the first d or t is observed, while the magnitude of η controls the “confusability” of the target and distractor states. Figure 2B shows a typical run from this generative model. The transition into target happens on step 10 (top), and the outputs generated are a mixture of t and d(middle), with an overall prevalence of t (bottom). Exact inference on this model can be performed in a manner similar to the forward pass in a standard HMM: P(st|x1, . . . , xt) ∝p(xt|st) X st−1 P(st|st−1)P(st−1|x1, . . . , xt−1) . (2) Because start does not produce outputs, as soon as the first t is observed, the probability of start plummets to 0. There then ensues an inferential battle between target and distractor, with the latter having the initial advantage, since its prior probability is 80%. 1−q(t) D 0.8 q(t) 0.2 q(t) target start distract 1.0 1.0 outputs η 1−η 1−η η T s d t s d t 10 20 30 0 10 20 0 0.5 1 0 0.5 1 10 20 30 0 0.5 1 0 5 0 5 0 5 10 20 30 0 5 A B timestep state output cumulative outputs C Probability timestep P(start) P(distract) P(target) D NE activity timestep hit stim resp fa miss cr Figure 2: The model. (A) Task is modeled as a hidden Markov model (HMM), with transitions from start to either distractor (probability .8) or target (probability .2). The transitions happen between timesteps 6 and 10 with uniform probability; distractor and target are absorbing states. The only outputs are from the absorbing states, and the two have overlapping output distributions over t and d with probabilities η > .5 for their “own” output (t for target, and d for distractor), and 1−η for the other output. (B) Sample run with a transition from start to target at timestep 10 (upper). The outputs favor target once the state has changed (middle), more clearly shown in the cumulative plot (bottom). (C) Correct probabilistic inference in the task leads to the probabilities for the three states as shown. The distractor’s initial advantage arises from a base rate effect, as it is the more likely default transition. (D) Model NE signal for four trials including one for hit (top; same trials as in B;C), a false alarm (fa), a miss (miss) and a correct rejection (cr). The second vertical line represents the point at which the decision was taken (target vs. distractor). Because of the preponderance of transitions to distractor over target, the distractor state can be thought of as the reference or default state. Evidence against that default state is a form of unexpected uncertainty within a task, and we propose that phasic NE reports this uncertainty. More specifically, NE signals P(target|x1, . . . , xt)/P(target), where P(target) = .2 is the prior probability of observing a target trial. We assume that a target-response is initiated when P(st|x1, . . . , xt) exceeds 0.95, or equivalently, when the NE signal exceeds 0.95/P(target). This implies the following intuitive relationship: the smaller the probability of the non-default state target the greater the NE-mediated “surprise” signal has to be in order to convince the inferential system that an anomalous stimulus has been observed. We also assume that if the posterior probability of target reaches 0.01, then the trial ends with no action (either a cr or a miss). The asymmetry in the thresholds arises from the asymmetry in the response contingencies of the task. Further, to model non-inferential errors, we assume that there is probability of 0.0005 per timestep of releasing the bar after the transition out of start. Once a decision is reached, the NE signal is set back to baseline (1, for equal prior and posterior) after a delay of 5 timesteps. Note that the precise form of the mapping from unexpected uncertainty to NE spikes is rather arbitrary. In particular, there may be a strong non-linearity, such as a thresholded response profile. For simplicity, we assume a linear mapping between the two. The NE activity during the start state is also rather arbitrary. Activity is at baseline before the stimulus comes on, since prior and posterior match when there is no explicit information from the world. When the stimulus comes on, the divisive normalization makes the activity go above baseline because although the transition was expected, its occurrence was not predicted with perfect precision. The magnitude of this activity depends on the precision of the model of the time of the transition; and the uncertainty in the interval timer. We set it to a small super-baseline level to match the data. 10 20 30 40 0 1 2 3 4 10 20 30 40 0 1 2 3 4 5 A stim hit fa miss cr Timestep NE activity B resp Timestep Figure 3: NE activity. (A) NE activity locked to the stimulus onset (ie the transition out of start). (B) NE activity response-locked to the decision to act, just for hit and fa trials. Note the difference in scale between the two figures. 3 Results Figure 2C illustrates the inferential performance of the model for the sample run in Figure 2B;C. When the first t is observed on timestep 10, the probability of start drops to 0 and the probability of distractor, which has an initial advantage over target due to its higher probability, eventually loses out to target as the evidence overwhelms the prior. Figure 2D shows the model’s NE signal for one example each of hit, fa, miss, and cr trials. Figure 3 presents our main results. Figure 3A shows the average NE signal for the four classes of responses (hit, false alarm, miss, and correct rejection), time-locked to the start of the stimulus. These traces should be compared with those in Figure 1E. The basic form of the rise of the signal in the model is broadly similar to that in the data; as we have argued, the fall is rather arbitrary. Figure 3B shows the average signal locked to the time of reaction (for hit and false alarm trials) rather than stimulus onset. As in the data (Figure 1F), response-locked activities are much more tightly clustered, although this flatters the model somewhat, since we do not allow for any variability in the response time as a function of when the probability of state target reaches the threshold. Since the decay of the signal following a response is unconstrained, the trace terminates when the response is determined, usually when the probability of target reaches threshold, but also sometimes when there is an accidental erroneous response. Figure 4 shows some additional features of the NE signal in this case. Figure 4A compares the effect of making the discrimination between target and distractor more or less difficult in the model (upper) and in the data (lower; [16]). As in the data, the stimulus-locked NE signal is somewhat broader for the more difficult case, since information has to build up over a longer period. Also as in the data, correct rejections are much less affected than hits. Figure 4B shows response locked NE. Although it is correctly slightly broader for the more difficult discrimination, the timing is not quite the same. This is largely due to the lack of a realistic model tying the defeat of the default state assumption to a behavioral response. For the easy task (η = 0.675), there were 19% hits, 1.5% false alarms, 1% misses and 77% correct rejections. For the difficult task (η = 0.65) the main difference was an increase in the number of misses to 1.5%, largely at the expense of hits. Note that since the NE signal is calculated relative to the prior likelihood, making target more likely would reduce the NE signal exactly proportionally. The data certainly hint at such a reduction [17] although the precise proportionality is not clear. 4 Discussion The present model of the phasic activity of NE cells is a direct and major extension of our previous model of tonic aspects of this neuromodulator. The key difference is that 10 20 30 40 0 1 2 3 4 10 20 30 40 0 1 2 3 4 5 A Time (sec) Spikes/sec B Time (sec) C NE activity Timestep hit cr D Timestep Figure 4: NE activities and task difficulty. (A) Stimulus-locked LC responses are slower and broader for a more difficult discrimination; where difficulty is controlled by the similarity of target and distractor stimuli. (B) When aligned to response, LC activities for easy and difficult discriminations are more similar, although their response in the more difficult condition is still somewhat attenuated compared to the easy one. Data in A;B adapted from [16]. (C) Discrimination difficulty in the model is controlled by the parameter η. When η is reduced from 0.675 (easy; solid) to 0.65 (hard; dashed), simulated NE activity also becomes slower and broader when aligned to stimulus. (D) Same traces aligned to response indicate NE activity in the difficult condition is attenuated in the model. unexpected uncertainty is now about the state within a current characterization of the task rather than about the characterization as a whole. These aspects of NE functionality are likely quite widespread, and allow us to account for a much wider range of data on this neuromodulator. In the model, NE activity is explicitly normalized by prior probabilities arising from the default state transitions in tasks. This is necessary to measure specifically unexpected uncertainty, and explains the decrement in NE phasic response as a function of the target probability [17]. It is also associated with the small activation to the stimulus onset, although the precise form of this deserves closer scrutiny. For instance, if subjects were to build a richer model of the statistics of the time of the transition out of the start state, then we should see this reflected directly in the NE signal even before the stimulus comes on, for instance related to the inverse of the survival function for the transition. We would also expect this transition to effect a different NE signature if stimuli were expected during start that could also be confused with those expected during target and distractor. If NE indeed reports on the failure of the current state within the model of the task to account successfully for the observations, then what effect should it have? One useful way to think about the signal is in terms of an interrupt signal in computers. In these, a control program establishes a set of conditions (eg keyboard input) under which normal processing should be interrupted, in order that the consequence of the interrupt (eg a keystroke) can be appropriately handled. Computers have highly centralized processing architecture, and therefore the interrupt signal only needs a very limited spatial extent to exert a widespread effect on the course of computation. By contrast, processing in the brain is highly distributed, and therefore it is necessary for the interrupt signal to have a widespread distribution, so that the full ramifications of the failure of the current state can be felt. Neuromodulatory systems are ideal vehicles for the signal. The interrupt signal should engage mechanisms for establishing the new state, which then allows a new set of conditions to be established as to which interrupts will be allowed to occur, and also to take any appropriate action (as in the task we modeled). The interrupt signal can be expected to be beneficial, for instance, when there is competition between tasks for the use of neural resources such as receptive fields [8]. Apart from interrupts such as these under sophisticated top-down control, there are also more basic contingencies from things such as critical potential threats and stressors that should exert a rapid and dramatic effect on neural processing (these also have computational analogues in signals such as that power is about to fail). The NE system is duly subject to what might be considered as bottom-up as well as top-down influences [21]. The interrupt-based account is a close relative of existing notions of phasic NE. For instance, NE has been implicated in the process of alerting [23]. The difference from our account is perhaps the stronger tie in the latter to actual behavioral output. A task with second-order contingencies may help to differentiate the two accounts. There are also close relations with theories [20, 5] that suggest how NE may be an integral part of an optimal decisional strategy. These propose that NE controls the gain in competitive decisionmaking networks that implement sequential decision-making [22], essentially by reporting on the changes in the statistical structure of the inputs induced by stimulus onset. It is also suggested that a more extreme change in the gain, destabilizing the competitive networks through explosive symmetry breaking, can be used to freeze or lock-in any small difference in the competing activities. The idea that NE can signal the change in the input statistics occasioned by the (temporallyunpredictable) occurrence of the target is highly appealing. However, the statistics of the input change when either the target or the distractor appears, and so the preference for responding to the target at the expense of the distractor is strange. The effect of forcing the decision making network to become unstable, and therefore enforcing a speeded decision is much closer to an interrupt; but then it is not clear why this signal should decrease as the target becomes more common. Further, since in the unstable regime, the statistical optimality of integration is effectively abandoned, the computational appeal of the signal is somewhat weakened. However, this alternative theory does make an important link to sequential statistical analysis [22], raising issues about things like thresholds for deciding target and distractor that should be important foci of future work here too. Figure 1C shows an additional phenomenon that has arisen in a task when subjects were not even occasionally taxed with difficult discrimination problems. The overall performance fluctuates dramatically (shown by the changing false alarm rate), in a manner that is tightly correlated with fluctuations in tonic NE activity. Periods of high tonic activity are correlated with low phasic activation to the targets (data not shown). Aston-Jones, Cohen and their colleagues [20, 3] have suggested that NE regulates the balance between exploration and exploitation. The high tonic phase is associated with the former, with subjects failing to concentrate on the contingencies that lead to their current rewards in order to search for stimuli or actions that might be associated with better rewards. Increasing the ease of interruptability to either external cues or internal state changes, could certainly lead to apparently exploratory behavior. However, there is little evidence as to how this sort of exploration is being actively determined, since, for instance, the macroscopic fluctuations evident in Figure 1C do not arise in response to any experimental contingency. Given the relationship between phasic and tonic firing, further investigation of these periodic fluctuations and their implications would be desirable. Finally, in our previous model [24, 25], tonic NE was closely coupled with tonic acetylcholine (ACh), with the latter reporting expected rather than unexpected uncertainty. The account of ACh should transfer somewhat directly into the short-term contingencies within a task – we might expect it to be involved in reporting on aspects of the known variability associated with each state, including each distinct stimulus state as well as the no-stimulus state. As such, this ACh signal might be expected to be relatively more tonic than NE (an effect that is also apparent in our previous work on more tonic interactions between ACh and NE (eg Figure 2 of [24]). One attractive target for an account along these lines is the sustained attention task studied by Sarter and colleagues, which involves temporal uncertainty. Performance in this task is exquisitely sensitive to cholinergic manipulation [19], but unaffected by gross noradrenergic manipulation [15]. We may again expect there to be interesting part-opponent and part-synergistic interactions between the neuromodulators. Acknowledgements We are grateful to Gary Aston-Jones, Sebastien Bouret, Jonathan Cohen, Peter Latham, Susan Sara, and Eric Shea-Brown for helpful discussions. Funding was from the Gatsby Charitable Foundation, the EU BIBA project and the ACI Neurosciences Int´egratives et Computationnelles of the French Ministry of Research. References [1] Aston-Jones, G, Rajkowski, J, Kubiak, P & Alexinsky, T (1994). Locus coeruleus neurons in monkey are selectively activated by attended cues in a vigilance task. J. Neurosci. 14:44674480. [2] Aston-Jones, G, Rajkowski, J & Kubiak, P (1997). Conditioned responses of monkey locus coeruleus neurons anticipate Acquisition of discriminative behavior in a vigilance task. Neuroscience 80:697-715. [3] Aston-Jones, G, Rajkowski, J & Cohen, J (2000). Locus coeruleus and regulation of behavioral flexibility and attention. Prog. Brain Res. 126:165-182. [4] Bouret, S & Sara, SJ (2004). Reward expectation, orientation of attention and locus coeruleusmedial frontal cortex interplay during learning. Eur. J. Neurosci. 20:791-802. [5] Brown, E, Gao, J, Holmes, P, Bogacz, R, Gilzenrat, M & Cohen, JD (2005). Simple neural networks that optimize decisions. Int. J. Bif. & Chaos, in press. [6] David Johnson, J (2003). Noradrenergic control of cognition: global attenuation and an interrupt function. Med. Hypoth. 60:689-692. [7] Dayan, P & Yu, AJ (2001). ACh, uncertainty, and cortical inference. NIPS 2001. [8] Desimone, R & Duncan, J (1995). Neural mechanisms of selective visual attention. Annual Reviews in Neuroscience 18:193-222. [9] Devauges, V & Sara, SJ (1990). Activation of the noradrenergic system facilitates an attentional shift in the rat. Beh. Brain Res. 39:19-28. [10] Doya, K (2002). Metalearning and neuromodulation. Neur. Netw. 15:495-506. [11] Foote, SL, Freedman, R & Oliver, AP (1975). Effects of putative neurotransmitters on neuronal activity in monkey auditory cortex. Brain Res. 86:229-242. [12] Freedman, R, Foote, SL & Bloom, FE (1975) Histochemical characterization of a neocortical projection of the nucleus locus coeruleus in the squirrel monkey. J. Comp. Neurol. 164:209-231. [13] Jouvet, M (1969). Biogenic amines and the states of sleep. Science 163:32-41. [14] Marder, E & Thirumalai, V (2002). Cellular, synaptic and network effects of neuromodulation. Neur. Netw. 15:479-493. [15] McGaughy, J, Sandstrom, M, Ruland, S, Bruno JP & Sarter, M (1997). Lack of effects of lesions of the dorsal noradrenergic bundle on behavioral vigilance. Beh. Neurosci. 111:646-652. [16] Rajkowski, J, Majczynski, H, Clayton, E & Aston-Jones, G (2004). Activation of monkey locus coeruleus neurons varies with difficulty and performance in a target detection task. J. Neurophysiol. 92:361-371. [17] Rajkowski, J, Majczynski, H, Clayton, E, Cohen, JD & Aston-Jones, G (2002). Phasic activation of monkey locus coeruleus (LC) neurons with recognition of motivationally relevant stimuli. Society for Neuroscience, Abstracts 86.10. [18] Sara, SJ & Segal, M (1991). Plasticity of sensory responses of locus coeruleus neurons in the behaving rat: implications for cognition. Prog. Brain Res. 88:571-585. [19] Turchi, J & Sarter, M (2001). Bidirectional modulation of basal forebrain NMDA receptor function differentially affects visual attention but not visual discrimination performance. Neuroscience 104:407-417. [20] Usher, M, Cohen, JD, Servan-Schreiber, D, Rajkowski, J & Aston-Jones, G (1999). The role of locus coeruleus in the regulation of cognitive performance. Science 283:549-554. [21] Van Bockstaele, EJ, Chan, J & Pickel, VM (1996). Input from central nucleus of the amygdala efferents to pericoerulear dendrites, some of which contain tyrosine hydroxylase immunoreactivity. Journal of Neuroscience Research 45:289-302. [22] Wald, A (1947). Sequential Analysis. New York, NY: John Wiley & Sons. [23] Witte, EA & Marrocco, RT (1997). Alteration of brain noradrenergic activity in rhesus monkeys affects the alerting component of covert orienting. Psychopharmacology 132:315-323. [24] Yu, AJ & Dayan, P (2003). Expected and unexpected uncertainty. ACh and NE in the neocortex. NIPS 2002. [25] Yu, AJ & Dayan, P (2005). Uncertainty, neuromodulation, and attention. Neuron 46, 681-692.
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Consensus Propagation Ciamac C. Moallemi Stanford University Stanford, CA 95014 USA ciamac@stanford.edu Benjamin Van Roy Stanford University Stanford, CA 95014 USA bvr@stanford.edu Abstract We propose consensus propagation, an asynchronous distributed protocol for averaging numbers across a network. We establish convergence, characterize the convergence rate for regular graphs, and demonstrate that the protocol exhibits better scaling properties than pairwise averaging, an alternative that has received much recent attention. Consensus propagation can be viewed as a special case of belief propagation, and our results contribute to the belief propagation literature. In particular, beyond singly-connected graphs, there are very few classes of relevant problems for which belief propagation is known to converge. 1 Introduction Consider a network of n nodes in which the ith node observes a number yi ∈[0, 1] and aims to compute the average Pn i=1 yi/n. The design of scalable distributed protocols for this purpose has received much recent attention and is motivated by a variety of potential needs. In both wireless sensor and peer-to-peer networks, for example, there is interest in simple protocols for computing aggregate statistics (see, for example, the references in [1]), and averaging enables computation of several important ones. Further, averaging serves as a primitive in the design of more sophisticated distributed information processing algorithms. For example, a maximum likelihood estimate can be produced by an averaging protocol if each node’s observations are linear in variables of interest and noise is Gaussian [2]. As another example, averaging protocols are central to policy-gradient-based methods for distributed optimization of network performance [3]. In this paper we propose and analyze a new protocol – consensus propagation – for asynchronous distributed averaging. As a baseline for comparison, we will also discuss another asychronous distributed protocol – pairwise averaging – which has received much recent attention. In pairwise averaging, each node maintains its current estimate of the average, and each time a pair of nodes communicate, they revise their estimates to both take on the mean of their previous estimates. Convergence of this protocol in a very general model of asynchronous computation and communication was established in [4]. Recent work [5, 6] has studied the convergence rate and its dependence on network topology and how pairs of nodes are sampled. Here, sampling is governed by a certain doubly stochastic matrix, and the convergence rate is characterized by its second-largest eigenvalue. Consensus propagation is a simple algorithm with an intuitive interpretation. It can also be viewed as an asynchronous distributed version of belief propagation as applied to approximation of conditional distributions in a Gaussian Markov random field. When the network of interest is singly-connected, prior results about belief propagation imply convergence of consensus propagation. However, in most cases of interest, the network is not singlyconnected and prior results have little to say about convergence. In particular, Gaussian belief propagation on a graph with cycles is not guaranteed to converge, as demonstrated by examples in [7]. In fact, there are very few relevant cases where belief propagation on a graph with cycles is known to converge. Some fairly general sufficient conditions have been established [8, 9, 10], but these conditions are abstract and it is difficult to identify interesting classes of problems that meet them. One simple case where belief propagation is guaranteed to converge is when the graph has only a single cycle [11, 12, 13]. Recent work proposes the use of belief propagation to solve maximum-weight matching problems and proves convergence in that context [14]. [15] proves convergence in the application of belief propogation to a classification problem. In the Gaussian case, [7, 16] provide sufficient conditions for convergence, but these conditions are difficult to interpret and do not capture situations that correspond to consensus propagation. With this background, let us discuss the primary contributions of this paper: (1) we propose consensus propagation, a new asynchronous distributed protocol for averaging; (2) we prove that consensus propagation converges even when executed asynchronously. Since there are so few classes of relevant problems for which belief propagation is known to converge, even with synchronous execution, this is surprising; (3) We characterize the convergence time in regular graphs of the synchronous version of consensus propagation in terms of the the mixing time of a certain Markov chain over edges of the graph; (4) we explain why the convergence time of consensus propagation scales more gracefully with the number of nodes than does that of pairwise averaging, and for certain classes of graphs, we quantify the improvement. 2 Algorithm Consider a connected undirected graph (V, E) with |V | = n nodes. For each node i ∈ V , let N(i) = {j : (i, j) ∈E} be the set of neighbors of i. Each node i ∈V is assigned a number yi ∈[0, 1]. The goal is for each node to obtain an estimate of ¯y = P i∈V yi/n through an asynchronous distributed protocol in which each node carries out simple computations and communicates parsimonious messages to its neighbors. We propose consensus propagation as an approach to the aforementioned problem. In this protocol, if a node i communicates to a neighbor j at time t, it transmits a message consisting of two numerical values. Let µt ij ∈R and Kt ij ∈R+ denote the values associated with the most recently transmitted message from i to j at or before time t. At each time t, node j has stored in memory the most recent message from each neighbor: {µt ij, Kt ij|i ∈N(j)}. The initial values in memory before receiving any messages are arbitrary. Consensus propagation is parameterized by a scalar β > 0 and a non-negative matrix Q ∈Rn×n + with Qij > 0 if and only if i ̸= j and (i, j) ∈E. Let ⃗E ⊆V × V be a set consisting of two directed edges (i, j) and (j, i) per undirected edge (i, j) ∈E. For each (i, j) ∈⃗E, it is useful to define the following three functions: Fij(K) = 1 + P u∈N(i)\j Kui 1 + 1 βQij 1 + P u∈N(i)\j Kui , (1) Gij(µ, K) = yi + P u∈N(i)\j Kuiµui 1 + P u∈N(i)\j Kui , Xi(µ, K) = yi + P u∈N(i) Kuiµui 1 + P u∈N(i) Kui . (2) For each t, denote by Ut ⊆⃗E the set of directed edges along which messages are transmitted at time t. Consensus propagation is presented below as Algorithm 1. Algorithm 1 Consensus propagation. 1: for time t = 1 to ∞do 2: for all (i, j) ∈Ut do 3: Kt ij ←Fij(Kt−1) 4: µt ij ←Gij(µt−1, Kt−1) 5: end for 6: for all (i, j) /∈Ut do 7: Kt ij ←Kt−1 ij 8: µt ij ←µt−1 ij 9: end for 10: xt ←X(µt, Kt) 11: end for Consensus propagation is a distributed protocol because computations at each node require only information that is locally available. In particular, the messages Fij(Kt−1) and Gij(Kt−1) transmitted from node i to node j depend only on {µt−1 ui , Kt−1 ui |u ∈N(i)}, which node i has stored in memory. Similarly, xt i, which serves as an estimate of y, depends only on {µt ui, Kt ui|u ∈N(i)}. Consensus propagation is an asynchronous protocol because only a subset of the potential messages are transmitted at each time. Our convergence analysis can also be extended to accommodate more general models of asynchronism that involve communication delays, as those presented in [17]. In our study of convergence time, we will focus on the synchronous version of consensus propagation. This is where Ut = ⃗E for all t. Note that synchronous consensus propagation is defined by: Kt = F(Kt−1), µt = G(µt−1, Kt−1), xt = X(µt−1, Kt−1). (3) 2.1 Intuitive Interpretation Consider the special case of a singly connected graph. For any (i, j) ∈⃗E, there is a set Sij ⊂V of nodes that can transmit information to Sji = V \ Sij only through (i, j). In order for nodes in Sji to compute y, they must at least be provided with the average µ∗ ij among observations at nodes in Sij and the cardinality K∗ ij = |Sij|. The messages µt ij and Kt ij can be viewed as estimates. In fact, when β = ∞, µt ij and Kt ij converge to µ∗ ij and K∗ ij, as we will now explain. Suppose the graph is singly connected, β = ∞, and transmissions are synchronous. Then, Kt ij = 1 + X u∈N(i)\j Kt−1 ui , (4) for all (i, j) ∈⃗E. This is a recursive characterization of |Sij|, and it is easy to see that it converges in a number of iterations equal to the diameter of the graph. Now consider the iteration µt ij = yi + P u∈N(i)\j Kt−1 ui µt−1 ui 1 + P u∈N(i)\j Kt−1 ui , for all (i, j) ∈⃗E. A simple inductive argument shows that at each time t, µt ij is an average among observations at Kt ij nodes in Sij, and after a number of iterations equal to the diameter of the graph, µt = µ∗. Further, for any i ∈V , y = yi + P u∈N(i) Kuiµui 1 + P u∈N(i) Kui , so xt i converges to y. This interpretation can be extended to the asynchronous case where it elucidates the fact that µt and Kt become µ∗and K∗after every pair of nodes in the graph has established bilateral communication through some sequence of transmissions among adjacent nodes. Suppose now that the graph has cycles. If β = ∞, for any (i, j) ∈⃗E that is part of a cycle, Kt ij →∞whether transmissions are synchronous or asynchronous, so long as messages are transmitted along each edge of the cycle an infinite number of times. A heuristic fix might be to compose the iteration (4) with one that attenuates: ˜Kt ij ←1+P u∈N(i)\j Kt−1 ui , and Kt ij ←˜Kt ij/(1+ϵij ˜Kt ij). Here, ϵij > 0 is a small constant. The message is essentially unaffected when ϵij ˜Kt ij is small but becomes increasingly attenuated as ˜Kt ij grows. This is exactly the kind of attenuation carried out by consensus propagation when βQij = 1/ϵij < ∞. Understanding why this kind of attenuation leads to desirable results is a subject of our analysis. 2.2 Relation to Belief Propagation Consensus propagation can also be viewed as a special case of belief propagation. In this context, belief propagation is used to approximate the marginal distributions of a vector x ∈Rn conditioned on the observations y ∈Rn. The mode of each of the marginal distributions approximates y. Take the prior distribution over (x, y) to be the normalized product of potential functions {ψi(·)|i ∈V } and compatibility functions {ψβ ij(·)|(i, j) ∈E}, given by ψi(xi) = exp(−(xi −yi)2), and ψβ ij(xi, xj) = exp(−βQij(xi −xj)2), where Qij, for each (i, j) ∈⃗E, and β are positive constants. Note that β can be viewed as an inverse temperature parameter; as β increases, components of x associated with adjacent nodes become increasingly correlated. Let Γ be a positive semidefinite symmetric matrix such that xT Γx = P (i,j)∈E Qij(xi − xj)2. Note that when Qij = 1 for all (i, j) ∈E, Γ is the graph Laplacian. Given the vector y of observations, the conditional density of x is pβ(x) ∝ Y i∈V ψi(xi) Y (i,j)∈E ψβ ij(xi, xj) = exp −∥x −y∥2 2 −βxT Γx . Let xβ denote the mode of pβ(·). Since the distribution is Gaussian, each component xβ i is also the mode of the corresponding marginal distribution. Note that xβ it is the unique solution to the positive definite quadratic program minimize x ∥x −y∥2 2 + βxT Γx. (5) The following theorem, whose proof can be found in [1], suggests that if β is sufficiently large each component xβ i can be used as an estimate of the mean value ¯y. Theorem 1. P i xβ i /n = ¯y and limβ↑∞xβ i = ¯y, for all i ∈V . In belief propagation, messages are passed along edges of a Markov random field. In our case, because of the structure of the distribution pβ(·), the relevant Markov random field has the same topology as the graph (V, E). The message Mij(·) passed from node i to node j is a distribution on the variable xj. Node i computes this message using incoming messages from other nodes as defined by the update equation M t ij(xj) = κ Z ψij(x′ i, xj)ψi(x′ i) Y u∈N(i)\j M t−1 ui (x′ i) dx′ i. (6) Here, κ is a normalizing constant. Since our underlying distribution pβ(·) is Gaussian, it is natural to consider messages which are Gaussian distributions. In particular, let (µt ij, Kt ij) ∈R × R+ parameterize Gaussian message M t ij(·) according to M t ij(xj) ∝ exp −Kt ij(xj −µt ij)2 . Then, (6) is equivalent to synchronous consensus propagation iterations for Kt and µt. The sequence of densities pt j(xj) ∝ψj(xj) Y i∈N(j) M t ij(xj) = exp −(xj −yj)2 − X i∈N(j) Kt ij(xj −µt ij)2 , is meant to converge to an approximation of the marginal conditional distribution of xj. As such, an approximation to xβ j is given by maximizing pt j(·). It is easy to show that, the maximum is attained by xt j = Xj(µt, Kt). With this and aforementioned correspondences, we have shown that consensus propagation is a special case of belief propagation. Readers familiar with belief propagation will notice that in the derivation above we have used the sum product form of the algorithm. In this case, since the underlying distribution is Gaussian, the max product form yields equivalent iterations. 3 Convergence The following theorem is our main convergence result. Theorem 2. (i) There are unique vectors (µβ, Kβ) such that Kβ = F(Kβ), and µβ = G(µβ, Kβ). (ii) Assume that each edge (i, j) ∈⃗E appears infinitely often in the sequence of communication sets {Ut}. Then, independent of the initial condition (µ0, K0), limt→∞Kt = Kβ, and limt→∞µt = µβ. (iii) Given (µβ, Kβ), if xβ = X(µβ, Kβ), then xβ is the mode of the distribution pβ(·). The proof of this theorem can be found in [1], but it rests on two ideas. First, notice that, according to the update equation (1), Kt evolves independently of µt. Hence, we analyze Kt first. Following the work of [7], we prove that the functions {Fij(·)} are monotonic. This property is used to establish convergence to a unique fixed point. Next, we analyze µt assuming that Kt has already converged. Given fixed K, the update equations for µt are linear, and we establish that they induce a contraction with respect to the maximum norm. This allows us to establish existence of a fixed point and asynchronous convergence. 4 Convergence Time for Regular Graphs In this section, we will study the convergence time of synchronous consensus propagation. For ϵ > 0, we will say that an estimate ˜x of ¯y is ϵ-accurate if ∥˜x −¯y1∥2,n ≤ϵ. Here, for integer m, ∥· ∥2,m is the norm on Rm defined by ∥x∥2,m = ∥x∥2/√m. We are interested in the number of iterations required to obtain an ϵ-accurate estimate of the mean ¯y. 4.1 The Case of Regular Graphs We will restrict our analysis of convergence time to cases where (V, E) is a d-regular graph, for d ≥2. Extension of our analysis to broader classes of graphs remains an open issue. We will also make simplifying assumptions that Qij = 1, µ0 ij = yi, and K0 = [k0]ij for some scalar k0 ≥0. In this restricted setting, the subspace of constant K vectors is invariant under F. This implies that there is some scalar kβ > 0 so that Kβ = [kβ]ij. This kβ is the unique solution to the fixed point equation kβ = (1+(d−1)kβ)/((1+(1+(d−1)kβ)/β). Given a uniform initial condition K0 = [k0]ij, we can study the sequence of iterates {Kt} by examining the scalar sequence {kt}, defined by kt = (1 + (d −1)kt−1)(1 + (1 + (d −1)kt−1)/β). In particular, we have Kt = [kt]ij, for all t ≥0. Similarly, in this setting, the equations for the evolution of µt take the special form µt ij = yi 1 + (d −1)kt−1 + 1 − 1 1 + (d −1)kt−1 X u∈N(i)\j µt−1 ui d −1. Defining γt = 1/(1 + (d −1)kt), we have, in vector form, µt = γt−1ˆy + (1 −γt−1) ˆPµt−1, (7) where ˆy ∈Rnd is a vector with ˆyij = yi and ˆP ∈Rnd×nd + is a doubly stochastic matrix. The matrix ˆP corresponds to a Markov chain on the set of directed edges ⃗E. In this chain, an edge (i, j) transitions to an edge (u, i) with u ∈N(i)\j, with equal probability assigned to each such edge. As in (3), we associate each µt with an estimate xt of xβ according to xt = y/(1 + dkβ) + dkβAµt/(1 + dkβ), where A ∈Rn×nd + is a matrix defined by (Aµ)j = P i∈N(j) µij/d. The update equation (7) suggests that the convergence of µt is intimately tied to a notion of mixing time associated with ˆP. Let ˆP ⋆be the Ces`aro limit ˆP ⋆= limt→∞ Pt−1 τ=0 ˆP τ/t. Define the Ces`aro mixing time τ ⋆by τ ⋆= supt≥0 ∥Pt τ=0( ˆP τ −ˆP ⋆)∥2,nd. Here, ∥·∥2,nd is the matrix norm induced by the corresponding vector norm ∥·∥2,nd. Since ˆP is a stochastic matrix, ˆP ⋆is well-defined and τ ⋆< ∞. Note that, in the case where ˆP is aperiodic, irreducible, and symmetric, τ ⋆corresponds to the traditional definition of mixing time: the inverse of the spectral gap of ˆP. A time t∗is said to be an ϵ-convergence time if estimates xt are ϵ-accurate for all t ≥t∗. The following theorem, whose proof can be found in [1], establishes a bound on the ϵconvergence time of synchronous consensus propagation given appropriately chosen β, as a function of ϵ and τ ⋆. Theorem 3. Suppose k0 ≤kβ. If d = 2 there exists a β = Θ((τ ⋆/ϵ)2) and if d > 2 there exists a β = Θ(τ ⋆/ϵ) such that some t∗= O((τ ⋆/ϵ) log(τ ⋆/ϵ)) is an ϵ-convergence time. Alternatively, suppose k0 = kβ. If d = 2 there exists a β = Θ((τ ⋆/ϵ)2) and if d > 2 there exists a β = Θ(τ ⋆/ϵ) such that some t∗= O((τ ⋆/ϵ) log(1/ϵ)) is an ϵ-convergence time. In the first part of the above theorem, k0 is initialized arbitrarily so long as k0 ≤kβ. Typically, one might set k0 = 0 to guarantee this. The second case of interest is when k0 = kβ, so that kt = kβ for all t ≥0 Theorem 3 suggests that initializing with k0 = kβ leads to an improvement in convergence time. However, in our computational experience, we have found that an initial condition of k0 = 0 consistently results in faster convergence than k0 = kβ. Hence, we suspect that a convergence time bound of O((τ ⋆/ϵ) log(1/ϵ)) also holds for the case of k0 = 0. Proving this remains an open issue. Theorems 3 posits choices of β that require knowledge of τ ⋆, which may be both difficult to compute and also requires knowledge of the graph topology. This is not a major restriction, however. It is not difficult to imagine variations of Algorithm 1 which use a doubling sequence of guesses for the Ces´aro mixing time τ ⋆. Each guess leads to a choice of β and a number of iterations t∗ to run with that choice of β. Such a modified algorithm would still have an ϵ-convergence time of O((τ ⋆/ϵ) log(τ ⋆/ϵ)). 5 Comparison with Pairwise Averaging Using the results of Section 4, we can compare the performance of consensus propagation to that of pairwise averaging. Pairwise averaging is usually defined in an asynchronous setting, but there is a synchronous counterpart which works as follows. Consider a doubly stochastic symmetric matrix P ∈Rn×n such that Pij = 0 if (i, j) /∈E. Evolve estimates according to xt = Pxt−1, initialized with x0 = y. Clearly xt = P ty →¯y1 as t ↑∞. In the case of a singly-connected graph, synchronous consensus propagation converges exactly in a number of iterations equal to the diameter of the graph. Moreover, when β = ∞, this convergence is to the exact mean, as discussed in Section 2.1. This is the best one can hope for under any algorithm, since the diameter is the minimum amount of time required for a message to travel between the two most distant nodes. On the other hand, for a fixed accuracy ϵ, the worst-case number of iterations required by synchronous pairwise averaging on a singly-connected graph scales at least quadratically in the diameter [18]. The rate of convergence of synchronous pairwise averaging is governed by the relation ∥xt−¯y1∥2,n ≤λt 2, where λ2 is the second largest eigenvalue of P. Let τ2 = 1/ log(1/λ2), and call it the mixing time of P. In order to guarantee ϵ-accuracy (independent of y), t > τ2 log(1/ϵ) suffices and t = Ω(τ2 log(1/ϵ)) is required [6]. Consider d-regular graphs and fix a desired error tolerance ϵ. The number of iterations required by consensus propagation is Θ(τ ⋆log τ ⋆), whereas that required by pairwise averaging is Θ(τ2). Both mixing times depend on the size and topology of the graph. τ2 is the mixing time of a process on nodes that transitions along edges whereas τ ⋆is the mixing time of a process on directed edges that transitions towards nodes. An important distinction is that the former process is allowed to “backtrack” where as the latter is not. By this we mean that a sequence of states {i, j, i} can be observed in the vertex process, but the sequence {(i, j), (j, i)} cannot be observed in the edge process. As we will now illustrate through an example, it is this difference that makes τ2 larger than τ ⋆and, therefore, pairwise averaging less efficient than consensus propagation. In the case of a cycle (d = 2) with an even number of nodes n, minimizing the mixing time over P results in τ2 = Θ(n2) [19]. For comparison, as demonstrated in the following theorem (whose proof can be found in [1]), τ ⋆is linear in n. Theorem 4. For the cycle with n nodes, τ ⋆≤n/ √ 2. Intuitively, the improvement in mixing time arises from the fact that the edge process moves around the cycle in a single direction and therefore explores the entire graph within n iterations. The vertex process, on the other hand, randomly transitions back and forth among adjacent nodes, relying on chance to eventually explore the entire cycle. The cycle example demonstrates a Θ(n/ log n) advantage offered by consensus propagation. Comparisons of mixing times associated with other graph topologies remains an issue for future analysis. But let us close by speculating on a uniform grid of n nodes over the m-dimensional unit torus. Here, n1/m is an integer, and each vertex has 2m neighbors, each a distance n−1/m away. With P optimized, it can be shown that τ2 = Θ(n2/m) [20]. We put forth a conjecture on τ ⋆. Conjecture 1. For the m-dimensional torus with n nodes, τ ⋆= Θ(n(2m−1)/m2). Acknowledgments The authors wish to thank Balaji Prabhakar and Ashish Goel for their insights and comments. The first author was supported by a Benchmark Stanford Graduate Fellowship. This research was supported in part by the National Science Foundation through grant IIS-0428868 and a supplement to grant ECS-9985229 provided by the Management of Knowledge Intensive Dynamic Systems Program (MKIDS). References [1] C. C. Moallemi and B. Van Roy. Consensus propagation. Technical report, Management Science & Engineering Deptartment, Stanford University, 2005. URL: http://www. moallemi.com/ciamac/papers/cp-2005.pdf. [2] L. Xiao, S. Boyd, and S. Lall. A scheme for robust distributed sensor fusion based on average consensus. To appear in the proceedings of IPSN, 2005. [3] C. C. Moallemi and B. Van Roy. Distributed optimization in adaptive networks. In Advances in Neural Information Processing Systems 16, 2004. [4] J. N. Tsitsiklis. Problems in Decentralized Decision-Making and Computation. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, 1984. [5] D. Kempe, A. Dobra, and J. Gehrke. Gossip-based computation of aggregate information. In ACM Symposium on Theory of Computing, 2004. [6] S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah. Gossip algorithms: Design, analysis and applications. To appear in the proceedings of INFOCOM, 2005. [7] P. Rusmevichientong and B. Van Roy. An analysis of belief propagation on the turbo decoding graph with Gaussian densities. IEEE Transactions on Information Theory, 47(2):745–765, 2001. [8] S. Tatikonda and M. I. Jordan. Loopy belief propagation and Gibbs measures. In Proceedings of the 18th Conference on Uncertainty in Artificial Intelligence, 2002. [9] T. Heskes. On the uniqueness of loopy belief propagation fixed points. Neural Computation, 16(11):2379–2413, 2004. [10] A. T. Ihler, J. W. Fisher III, and A. S. Willsky. Message errors in belief propagation. In Advances in Neural Information Processing Systems, 2005. [11] G. Forney, F. Kschischang, and B. Marcus. Iterative decoding of tail-biting trelisses. In Proceedings of the 1998 Information Theory Workshop, 1998. [12] S. M. Aji, G. 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CMOL CrossNets: Possible Neuromorphic Nanoelectronic Circuits Jung Hoon Lee Xiaolong Ma Konstantin K. Likharev Stony Brook University Stony Brook, NY 11794-3800 klikharev@notes.cc.sunysb.edu Abstract Hybrid “CMOL” integrated circuits, combining CMOS subsystem with nanowire crossbars and simple two-terminal nanodevices, promise to extend the exponential Moore-Law development of microelectronics into the sub-10-nm range. We are developing neuromorphic network (“CrossNet”) architectures for this future technology, in which neural cell bodies are implemented in CMOS, nanowires are used as axons and dendrites, while nanodevices (bistable latching switches) are used as elementary synapses. We have shown how CrossNets may be trained to perform pattern recovery and classification despite the limitations imposed by the CMOL hardware. Preliminary estimates have shown that CMOL CrossNets may be extremely dense (~107 cells per cm2) and operate approximately a million times faster than biological neural networks, at manageable power consumption. In Conclusion, we discuss in brief possible short-term and long-term applications of the emerging technology. 1 Introduction: CMOL Circuits Recent results [1, 2] indicate that the current VLSI paradigm based on CMOS technology can be hardly extended beyond the 10-nm frontier: in this range the sensitivity of parameters (most importantly, the gate voltage threshold) of silicon field-effect transistors to inevitable fabrication spreads grows exponentially. This sensitivity will probably send the fabrication facilities costs skyrocketing, and may lead to the end of Moore’s Law some time during the next decade. There is a growing consensus that the impending Moore’s Law crisis may be preempted by a radical paradigm shift from the purely CMOS technology to hybrid CMOS/nanodevice circuits, e.g., those of “CMOL” variety (Fig. 1). Such circuits (see, e.g., Ref. 3 for their recent review) would combine a level of advanced CMOS devices fabricated by the lithographic patterning, and two-layer nanowire crossbar formed, e.g., by nanoimprint, with nanowires connected by simple, similar, two-terminal nanodevices at each crosspoint. For such devices, molecular single-electron latching switches [4] are presently the leading candidates, in particular because they may be fabricated using the self-assembled monolayer (SAM) technique which already gave reproducible results for simpler molecular devices [5]. (b) βFCMOS Fnano α (a) nanodevices nanowiring and nanodevices interface pins upper wiring level of CMOS stack In order to overcome the CMOS/nanodevice interface problems pertinent to earlier proposals of hybrid circuits [6], in CMOL the interface is provided by pins that are distributed all over the circuit area, on the top of the CMOS stack. This allows to use advanced techniques of nanowire patterning (like nanoimprint) which do not have nanoscale accuracy of layer alignment [3]. The vital feature of this interface is the tilt, by angle α = arcsin(Fnano/βFCMOS), of the nanowire crossbar relative to the square arrays of interface pins (Fig. 1b). Here Fnano is the nanowiring half-pitch, FCMOS is the half-pitch of the CMOS subsystem, and β is a dimensionless factor larger than 1 that depends on the CMOS cell complexity. Figure 1b shows that this tilt allows the CMOS subsystem to address each nanodevice even if Fnano << βFCMOS. By now, it has been shown that CMOL circuits can combine high performance with high defect tolerance (which is necessary for any circuit using nanodevices) for several digital applications. In particular, CMOL circuits with defect rates below a few percent would enable terabit-scale memories [7], while the performance of FPGA-like CMOL circuits may be several hundred times above that of overcome purely CMOL FPGA (implemented with the same FCMOS), at acceptable power dissipation and defect tolerance above 20% [8]. In addition, the very structure of CMOL circuits makes them uniquely suitable for the implementation of more complex, mixed-signal information processing systems, including ultradense and ultrafast neuromorphic networks. The objective of this paper is to describe in brief the current status of our work on the development of so-called Distributed Crossbar Networks (“CrossNets”) that could provide high performance despite the limitations imposed by CMOL hardware. A more detailed description of our earlier results may be found in Ref. 9. 2 Synapses The central device of CrossNet is a two-terminal latching switch [3, 4] (Fig. 2a) which is a combination of two single-electron devices, a transistor and a trap [3]. The device may be naturally implemented as a single organic molecule (Fig. 2b). Qualitatively, the device operates as follows: if voltage V = Vj – Vk applied between the external electrodes (in CMOL, nanowires) is low, the trap island has no net electric charge, and the single-electron transistor is closed. If voltage V approaches certain threshold value V+ > 0, an additional electron is inserted into the trap island, and its field lifts the Coulomb blockade of the single-electron transistor, thus connecting the nanowires. The switch state may be reset (e.g., wires disconnected) by applying a lower voltage V < V- < V+. Due to the random character of single-electron tunneling [2], the quantitative description of the switch is by necessity probabilistic: actually, V determines only the rates Γ↑↓ of device Fig. 1. CMOL circuit: (a) schematic side view, and (b) top-view zoom-in on several adjacent interface pins. (For clarity, only two adjacent nanodevices are shown.) switching between its ON and OFF states. The rates, in turn, determine the dynamics of probability p to have the transistor opened (i.e. wires connected): dp/dt = Γ↑(1 - p) - Γ↓p. (1) The theory of single-electron tunneling [2] shows that, in a good approximation, the rates may be presented as Γ↑↓ = Γ0 exp{±e(V - S)/kBT} , (2) N O O N O O R N N O O O O R R N R C R R N R R C O R N R C R R O O O R = hexyl Fig. 2. (a) Schematics and (b) possible molecular implementation of the two-terminal single-electron latching switch where Γ0 and S are constants depending on physical parameters of the latching switches. Note that despite the random character of switching, the strong nonlinearity of Eq. (2) allows to limit the degree of the device “fuzziness”. 3 CrossNets Figure 3a shows the generic structure of a CrossNet. CMOS-implemented somatic cells (within the Fire Rate model, just nonlinear differential amplifiers, see Fig. 3b,c) apply their output voltages to “axonic” nanowires. If the latching switch, working as an elementary synapse, on the crosspoint of an axonic wire with the perpendicular “dendritic” wire is open, some current flows into the latter wire, charging it. Since such currents are injected into each dendritic wire through several (many) open synapses, their addition provides a natural passive analog summation of signals from the corresponding somas, typical for all neural networks. Examining Fig. 3a, please note the open-circuit terminations of axonic and dendritic lines at the borders of the somatic cells; due to these terminations the somas do not communicate directly (but only via synapses). The network shown on Fig. 3 is evidently feedforward; recurrent networks are achieved in the evident way by doubling the number of synapses and nanowires per somatic cell (Fig. 3c). Moreover, using dual-rail (bipolar) representation of the signal, and hence doubling the number of nanowires and elementary synapses once again, one gets a CrossNet with (a) single-electron transistor single-electron trap tunnel junction Vj Vk (b) diimide acceptor groups OPE wires clipping group somas coupled by compact 4-switch groups [9]. Using Eqs. (1) and (2), it is straightforward to show that that the average synaptic weight wjk of the group obeys the “quasi-Hebbian” rule: (3) Fig. 3. (a) Generic structure of the simplest, (feedforward, non-Hebbian) CrossNet. Red lines show “axonic”, and blue lines “dendritic” nanowires. Gray squares are interfaces between nanowires and CMOS-based somas (b, c). Signs show the dendrite input polarities. Green circles denote molecular latching switches forming elementary synapses. Bold red and blue points are open-circuit terminations of the nanowires, that do not allow somas to interact in bypass of synapses In the simplest cases (e.g., quasi-Hopfield networks with finite connectivity), the tri-level synaptic weights of the generic CrossNets are quite satisfactory, leading to just a very modest (~30%) network capacity loss. However, some applications (in particular, pattern classification) may require a larger number of weight quantization levels L (e.g., L ≈ 30 for a 1% fidelity [9]). This may be achieved by using compact square arrays (e.g., 4×4) of latching switches (Fig. 4). Various species of CrossNets [9] differ also by the way the somatic cells are distributed around the synaptic field. Figure 5 shows feedforward versions of two CrossNet types most explored so far: the so-called FlossBar and InBar. The former network is more natural for the implementation of multilayered perceptrons (MLP), while the latter system is preferable for recurrent network implementations and also allows a simpler CMOS design of somatic cells. The most important advantage of CrossNets over the hardware neural networks suggested earlier is that these networks allow to achieve enormous density combined with large cell connectivity M >> 1 in quasi-2D electronic circuits. 4 CrossNet training CrossNet training faces several hardware-imposed challenges: soma j soma k jk+ jk- + + - RL + - RL + RL RL (a) (b) (c) - ( ) ( ) ( ) 0 4 sinh sinh sinh . jk j k d w S V V dt γ γ γ = −Γ (i) The synaptic weight contribution provided by the elementary latching switch is binary, so that for most applications the multi-switch synapses (Fig. 4) are necessary. (ii) The only way to adjust any particular synaptic weight is to turn ON or OFF the corresponding latching switch(es). This is only possible to do by applying certain voltage V = Vj – Vk between the two corresponding nanowires. At this procedure, other nanodevices attached to the same wires should not be disturbed. (iii) As stated above, synapse state switching is a statistical progress, so that the degree of its “fuzziness” should be carefully controlled. Fig. 4. Composite synapse for providing L = 2n2+1 discrete levels of the weight in (a) operation and (b) weight adjustment modes. The dark-gray rectangles are resistive metallic strips at soma/nanowire interfaces Fig. 5. Two main CrossNet species: (a) FlossBar and (b) InBar, in the generic (feedforward, non-Hebbian, ternary-weight) case for the connectivity parameter M = 9. Only the nanowires and nanodevices coupling one cell (indicated with red dashed lines) to M post-synaptic cells (blue dashed lines) are shown; actually all the cells are similarly coupled We have shown that these challenges may be met using (at least) the following training methods [9]: Vw– A/2 Vw+ A/2 (b) ±(Vt +A/2) ±(Vt –A/2) RS i' = 1 2 … n i = 1 2 … n (a) RS RL i = 1 2 … n i' = 1 2 … n Vj Vj (a) (b) (i) Synaptic weight import. This procedure is started with training of a homomorphic “precursor” artificial neural network with continuous synaptic weighs wjk, implemented in software, using one of established methods (e.g., error backpropagation). Then the synaptic weights wjk are transferred to the CrossNet, with some “clipping” (rounding) due to the binary nature of elementary synaptic weights. To accomplish the transfer, pairs of somatic cells are sequentially selected via CMOS-level wiring. Using the flexibility of CMOS circuitry, these cells are reconfigured to apply external voltages ±VW to the axonic and dendritic nanowires leading to a particular synapse, while all other nanowires are grounded. The voltage level VW is selected so that it does not switch the synapses attached to only one of the selected nanowires, while voltage 2VW applied to the synapse at the crosspoint of the selected wires is sufficient for its reliable switching. (In the composite synapses with quasi-continuous weights (Fig. 4), only a part of the corresponding switches is turned ON or OFF.) (ii) Error backpropagation. The synaptic weight import procedure is straightforward when wjk may be simply calculated, e.g., for the Hopfield-type networks. However, for very large CrossNets used, e.g., as pattern classifiers the precursor network training may take an impracticably long time. In this case the direct training of a CrossNet may become necessary. We have developed two methods of such training, both based on “Hebbian” synapses consisting of 4 elementary synapses (latching switches) whose average weight dynamics obeys Eq. (3). This quasi-Hebbian rule may be used to implement the backpropagation algorithm either using a periodic time-multiplexing [9] or in a continuous fashion, using the simultaneous propagation of signals and errors along the same dual-rail channels. As a result, presently we may state that CrossNets may be taught to perform virtually all major functions demonstrated earlier with the usual neural networks, including the corrupted pattern restoration in the recurrent quasi-Hopfield mode and pattern classification in the feedforward MLP mode [11]. 5 CrossNet performance estimates The significance of this result may be only appreciated in the context of unparalleled physical parameters of CMOL CrossNets. The only fundamental limitation on the half-pitch Fnano (Fig. 1) comes from quantum-mechanical tunneling between nanowires. If the wires are separated by vacuum, the corresponding specific leakage conductance becomes uncomfortably large (~10-12 Ω-1m-1) only at Fnano = 1.5 nm; however, since realistic insulation materials (SiO2, etc.) provide somewhat lower tunnel barriers, let us use a more conservative value Fnano= 3 nm. Note that this value corresponds to 1012 elementary synapses per cm2, so that for 4M = 104 and n = 4 the areal density of neural cells is close to 2×107 cm-2. Both numbers are higher than those for the human cerebral cortex, despite the fact that the quasi-2D CMOL circuits have to compete with quasi-3D cerebral cortex. With the typical specific capacitance of 3×10-10 F/m = 0.3 aF/nm, this gives nanowire capacitance C0 ≈ 1 aF per working elementary synapse, because the corresponding segment has length 4Fnano. The CrossNet operation speed is determined mostly by the time constant τ0 of dendrite nanowire capacitance recharging through resistances of open nanodevices. Since both the relevant conductance and capacitance increase similarly with M and n, τ0 ≈ R0C0. The possibilities of reduction of R0, and hence τ0, are limited mostly by acceptable power dissipation per unit area, that is close to Vs 2/(2Fnano)2R0. For room-temperature operation, the voltage scale V0 ≈ Vt should be of the order of at least 30 kBT/e ≈ 1 V to avoid thermally-induced errors [9]. With our number for Fnano, and a relatively high but acceptable power consumption of 100 W/cm2, we get R0 ≈ 1010Ω (which is a very realistic value for single-molecule single-electron devices like one shown in Fig. 3). With this number, τ0 is as small as ~10 ns. This means that the CrossNet speed may be approximately six orders of magnitude (!) higher than that of the biological neural networks. Even scaling R0 up by a factor of 100 to bring power consumption to a more comfortable level of 1 W/cm2, would still leave us at least a four-orders-of-magnitude speed advantage. 6 Discussion: Possible applications These estimates make us believe that that CMOL CrossNet chips may revolutionize the neuromorphic network applications. Let us start with the example of relatively small (1-cm2-scale) chips used for recognition of a face in a crowd [11]. The most difficult feature of such recognition is the search for face location, i.e. optimal placement of a face on the image relative to the panel providing input for the processing network. The enormous density and speed of CMOL hardware gives a possibility to time-and-space multiplex this task (Fig. 6). In this approach, the full image (say, formed by CMOS photodetectors on the same chip) is divided into P rectangular panels of h×w pixels, corresponding to the expected size and approximate shape of a single face. A CMOS-implemented communication channel passes input data from each panel to the corresponding CMOL neural network, providing its shift in time, say using the TV scanning pattern (red line in Fig. 6). The standard methods of image classification require the network to have just a few hidden layers, so that the time interval Δt necessary for each mapping position may be so short that the total pattern recognition time T = hwΔt may be acceptable even for online face recognition. Fig. 6. Scan mapping of the input image on CMOL CrossNet inputs. Red lines show the possible time sequence of image pixels sent to a certain input of the network processing image from the upper-left panel of the pattern Indeed, let us consider a 4-Megapixel image partitioned into 4K 32×32-pixel panels (h = w = 32). This panel will require an MLP net with several (say, four) layers with 1K cells each in order to compare the panel image with ~103 stored faces. With the feasible 4-nm nanowire half-pitch, and 65-level synapses (sufficient for better than 99% fidelity [9]), each interlayer crossbar would require chip area about (4K×64 nm)2 = 64×64 μm2, fitting 4×4K of them on a ~0.6 cm2 chip. (The CMOS somatic-layer and communication-system overheads are negligible.) With the acceptable power consumption of the order of 10 W/cm2, the input-to-output signal propagation in such a network will take only about 50 ns, so that Δt may be of the order of 100 ns and the total time T = hwΔt of processing one frame of the order of 100 microseconds, much shorter than the typical TV frame time of ~10 milliseconds. The remaining w h image network input two-orders-of-magnitude time gap may be used, for example, for double-checking the results via stopping the scan mapping (Fig. 6) at the most promising position. (For this, a simple feedback from the recognition output to the mapping communication system is necessary.) It is instructive to compare the estimated CMOL chip speed with that of the implementation of a similar parallel network ensemble on a CMOS signal processor (say, also combined on the same chip with an array of CMOS photodetectors). Even assuming an extremely high performance of 30 billion additions/multiplications per second, we would need ~4×4K×1K×(4K)2/(30×109) ≈ 104 seconds ~ 3 hours per frame, evidently incompatible with the online image stream processing. Let us finish with a brief (and much more speculative) discussion of possible long-term prospects of CMOL CrossNets. Eventually, large-scale (~30×30 cm2) CMOL circuits may become available. According to the estimates given in the previous section, the integration scale of such a system (in terms of both neural cells and synapses) will be comparable with that of the human cerebral cortex. Equipped with a set of broadband sensor/actuator interfaces, such (necessarily, hierarchical) system may be capable, after a period of initial supervised training, of further self-training in the process of interaction with environment, with the speed several orders of magnitude higher than that of its biological prototypes. Needless to say, the successful development of such self-developing systems would have a major impact not only on all information technologies, but also on the society as a whole. Acknowledgments This work has been supported in part by the AFOSR, MARCO (via FENA Center), and NSF. Valuable contributions made by Simon Fölling, Özgür Türel and Ibrahim Muckra, as well as useful discussions with P. Adams, J. Barhen, D. Hammerstrom, V. Protopopescu, T. Sejnowski, and D. Strukov are gratefully acknowledged. References [1] Frank, D. J. et al. (2001) Device scaling limits of Si MOSFETs and their application dependencies. Proc. IEEE 89(3): 259-288. [2] Likharev, K. K. (2003) Electronics below 10 nm, in J. Greer et al. (eds.), Nano and Giga Challenges in Microelectronics, pp. 27-68. Amsterdam: Elsevier. [3] Likharev, K. K. and Strukov, D. B. (2005) CMOL: Devices, circuits, and architectures, in G. Cuniberti et al. (eds.), Introducing Molecular Electronics, Ch. 16. Springer, Berlin. [4] Fölling, S., Türel, Ö. & Likharev, K. K. (2001) Single-electron latching switches as nanoscale synapses, in Proc. of the 2001 Int. Joint Conf. on Neural Networks, pp. 216-221. Mount Royal, NJ: Int. Neural Network Society. [5] Wang, W. et al. (2003) Mechanism of electron conduction in self-assembled alkanethiol monolayer devices. Phys. Rev. B 68(3): 035416 1-8. [6] Stan M. et al. (2003) Molecular electronics: From devices and interconnect to circuits and architecture, Proc. IEEE 91(11): 1940-1957. [7] Strukov, D. B. & Likharev, K. K. (2005) Prospects for terabit-scale nanoelectronic memories. Nanotechnology 16(1): 137-148. [8] Strukov, D. B. & Likharev, K. K. (2005) CMOL FPGA: A reconfigurable architecture for hybrid digital circuits with two-terminal nanodevices. Nanotechnology 16(6): 888-900. [9] Türel, Ö. et al. (2004) Neuromorphic architectures for nanoelectronic circuits”, Int. J. of Circuit Theory and Appl. 32(5): 277-302. [10] See, e.g., Hertz J. et al. (1991) Introduction to the Theory of Neural Computation. Cambridge, MA: Perseus. [11] Lee, J. H. & Likharev, K. K. (2005) CrossNets as pattern classifiers. Lecture Notes in Computer Sciences 3575: 434-441.
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A General and Efficient Multiple Kernel Learning Algorithm S¨oren Sonnenburg∗ Fraunhofer FIRST Kekul´estr. 7 12489 Berlin Germany sonne@first.fhg.de Gunnar R¨atsch Friedrich Miescher Lab Max Planck Society Spemannstr. 39 T¨ubingen, Germany raetsch@tue.mpg.de Christin Sch¨afer Fraunhofer FIRST Kekul´estr. 7 12489 Berlin Germany christin@first.fhg.de Abstract While classical kernel-based learning algorithms are based on a single kernel, in practice it is often desirable to use multiple kernels. Lankriet et al. (2004) considered conic combinations of kernel matrices for classification, leading to a convex quadratically constraint quadratic program. We show that it can be rewritten as a semi-infinite linear program that can be efficiently solved by recycling the standard SVM implementations. Moreover, we generalize the formulation and our method to a larger class of problems, including regression and one-class classification. Experimental results show that the proposed algorithm helps for automatic model selection, improving the interpretability of the learning result and works for hundred thousands of examples or hundreds of kernels to be combined. 1 Introduction Kernel based methods such as Support Vector Machines (SVMs) have proven to be powerful for a wide range of different data analysis problems. They employ a so-called kernel function k(xi, xj) which intuitively computes the similarity between two examples xi and xj. The result of SVM learning is a α-weighted linear combination of kernel elements and the bias b: f(x) = sign N X i=1 αiyik(xi, x) + b ! , (1) where the xi’s are N labeled training examples (yi ∈{±1}). Recent developments in the literature on the SVM and other kernel methods have shown the need to consider multiple kernels. This provides flexibility, and also reflects the fact that typical learning problems often involve multiple, heterogeneous data sources. While this so-called “multiple kernel learning” (MKL) problem can in principle be solved via cross-validation, several recent papers have focused on more efficient methods for multiple kernel learning [4, 5, 1, 7, 3, 9, 2]. One of the problems with kernel methods compared to other techniques is that the resulting decision function (1) is hard to interpret and, hence, is difficult to use in order to extract rel∗For more details, datasets and pseudocode see http://www.fml.tuebingen.mpg.de /raetsch/projects/mkl silp. evant knowledge about the problem at hand. One can approach this problem by considering convex combinations of K kernels, i.e. k(xi, xj) = K X k=1 βkkk(xi, xj) (2) with βk ≥0 and P k βk = 1, where each kernel kk uses only a distinct set of features of each instance. For appropriately designed sub-kernels kk, the optimized combination coefficients can then be used to understand which features of the examples are of importance for discrimination: if one would be able to obtain an accurate classification by a sparse weighting βk, then one can quite easily interpret the resulting decision function. We will illustrate that the considered MKL formulation provides useful insights and is at the same time is very efficient. This is an important property missing in current kernel based algorithms. We consider the framework proposed by [7], which results in a convex optimization problem - a quadratically-constrained quadratic program (QCQP). This problem is more challenging than the standard SVM QP, but it can in principle be solved by general-purpose optimization toolboxes. Since the use of such algorithms will only be feasible for small problems with few data points and kernels, [1] suggested an algorithm based on sequential minimization optimization (SMO) [10]. While the kernel learning problem is convex, it is also non-smooth, making the direct application of simple local descent algorithms such as SMO infeasible. [1] therefore considered a smoothed version of the problem to which SMO can be applied. In this work we follow a different direction: We reformulate the problem as a semi-infinite linear program (SILP), which can be efficiently solved using an off-the-shelf LP solver and a standard SVM implementation (cf. Section 2 for details). Using this approach we are able to solve problems with more than hundred thousand examples or with several hundred kernels quite efficiently. We have used it for the analysis of sequence analysis problems leading to a better understanding of the biological problem at hand [16, 13]. We extend our previous work and show that the transformation to a SILP works with a large class of convex loss functions (cf. Section 3). Our column-generation based algorithm for solving the SILP works by repeatedly using an algorithm that can efficiently solve the single kernel problem in order to solve the MKL problem. Hence, if there exists an algorithm that solves the simpler problem efficiently (like SVMs), then our new algorithm can efficiently solve the multiple kernel learning problem. We conclude the paper by illustrating the usefulness of our algorithms in several examples relating to the interpretation of results and to automatic model selection. 2 Multiple Kernel Learning for Classification using SILP In the Multiple Kernel Learning (MKL) problem for binary classification one is given N data points (xi, yi) (yi ∈{±1}), where xi is translated via a mapping Φk(x) 7→RDk, k = 1 . . . K from the input into K feature spaces (Φ1(xi), . . . , ΦK(xi)) where Dk denotes the dimensionality of the k-th feature space. Then one solves the following optimization problem [1], which is equivalent to the linear SVM for K = 1:1 min wk∈RDk ,ξ∈RN + ,β∈RK + ,b∈R 1 2 K X k=1 βk∥wk∥2 !2 + C N X i=1 ξi (3) s.t. yi K X k=1 βkw⊤ k Φk(xi) + b ! ≥1 −ξi and K X k=1 βk = 1. 1[1] used a slightly different but equivalent (assuming tr(Kk) = 1, k = 1, . . . , K) formulation without the β’s, which we introduced for illustration. Note that the ℓ1-norm of β is constrained to one, while one is penalizing the ℓ2-norm of wk in each block k separately. The idea is that ℓ1-norm constrained or penalized variables tend to have sparse optimal solutions, while ℓ2-norm penalized variables do not [11]. Thus the above optimization problem offers the possibility to find sparse solutions on the block level with non-sparse solutions within the blocks. Bach et al. [1] derived the dual for problem (3), which can be equivalently written as: min γ∈R,1C≥α∈RN + γ s.t. 1 2 N X i,j=1 αiαjyiyjkk(xi, xj) − N X i=1 αi | {z } =:Sk(α) ≤γ and X i=1 αiyi = 0 (4) for k = 1, . . . , K, where kk(xi, xj) = (Φk(xi), Φk(xj)). Note that we have one quadratic constraint per kernel (Sk(α) ≤γ). In the case of K = 1, the above problem reduces to the original SVM dual. In order to solve (4), one may solve the following saddle point problem (Lagrangian): L := γ + K X k=1 βk(Sk(α) −γ) (5) minimized w.r.t. α ∈RN +, γ ∈R (subject to α ≤C1 and P i αiyi = 0) and maximized w.r.t. β ∈RK + . Setting the derivative w.r.t. to γ to zero, one obtains the constraint P k βk = 1 and (5) simplifies to: L = S(α, β) := PK k=1 βkSk(α) and leads to a min-max problem: max β∈Rk + min 1C≥α∈RN + K X k=1 βkSk(α) s.t. N X i=1 αiyi = 0 and K X k=1 βk = 1. (6) Assume α∗would be the optimal solution, then θ∗:= S(α∗, β) is minimal and, hence, S(α, β) ≥θ∗for all α (subject to the above constraints). Hence, finding a saddle-point of (5) is equivalent to solving the following semi-infinite linear program: max θ∈R,β∈RM + θ s.t. X k βk = 1 and K X k=1 βkSk(α) ≥θ (7) for all α with 0 ≤α ≤C1 and X i yiαi = 0 Note that this is a linear program, as θ and β are only linearly constrained. However there are infinitely many constraints: one for each α ∈RN satisfying 0 ≤α ≤C and PN i=1 αiyi = 0. Both problems (6) and (7) have the same solution. To illustrate that, consider β is fixed and we maximize α in (6). Let α∗be the solution that maximizes (6). Then we can decrease the value of θ in (7) as long as no α-constraint (7) is violated, i.e. down to θ = PK k=1 βkSk(α∗). Similarly, as we increase θ for a fixed α the maximizing β is found. We will discuss in Section 4 how to solve such semi infinite linear programs. 3 Multiple Kernel Learning with General Cost Functions In this section we consider the more general class of MKL problems, where one is given an arbitrary strictly convex differentiable loss function, for which we derive its MKL SILP formulation. We will then investigate in this general MKL SILP using different loss functions, in particular the soft-margin loss, the ǫ-insensitive loss and the quadratic loss. We define the MKL primal formulation for a strictly convex and differentiable loss function L as: (for simplicity we omit a bias term) min wk∈RDk 1 2 K X k=1 ∥wk∥ !2 + N X i=1 L(f(xi), yi) s.t. f(xi) = K X k=1 (Φk(xi), wk) (8) In analogy to [1] we treat problem (8) as a second order cone program (SOCP) leading to the following dual (see Supplementary Website or [17] for details): min γ∈R,α∈RN γ − N X i=1 L(L′−1(αi, yi), yi) + N X i=1 αiL′−1(αi, yi) (9) s.t. : 1 2
N X i=1 αiΦk(xi)
2 2 ≤γ, ∀k = 1 . . . K To derive the SILP formulation we follow the same recipe as in Section 2: deriving the Lagrangian leads to a max-min problem formulation to be eventually reformulated to a SILP: max θ∈R,β∈RK θ s.t. K X k=1 βk = 1 and PK k=1 βkSk(α) ≥θ, ∀α ∈RN, where Sk(α) = − N X i=1 L(L′−1(αi, yi), yi) + N X i=1 αiL′−1(αi, yi) + 1 2
N X i=1 αiΦk(xi)
2 2 . We assumed that L(x, y) is strictly convex and differentiable in x. Unfortunately, the soft margin and ǫ-insensitive loss do not have these properties. We therefore consider them separately in the sequel. Soft Margin Loss We use the following loss in order to approximate the soft margin loss: Lσ(x, y) = C σ log(1 + exp((1 −xy)σ)). It is easy to verify that limσ→∞Lσ(x, y) = C(1−xy)+. Moreover, Lσ is strictly convex and differentiable for σ < ∞. Using this loss and assuming yi ∈{±1}, we obtain : Sk(α) = − N X i=1 C σ „ log „ Cyi αi + Cyi « + log „ − αi αi + Cyi «« + N X i=1 αiyi+ 1 2 ‚‚‚‚‚ N X i=1 αiΦk(xi) ‚‚‚‚‚ 2 2 . If σ →∞, then the first two terms vanish provided that −C ≤αi ≤0 if yi = 1 and 0 ≤αi ≤C if yi = −1. Substituting α = −˜αiyi, we then obtain Sk(˜α) = −PN i=1 ˜αi + 1 2
PN i=1 ˜αiyiΦk(xi)
2 2 , with 0 ≤˜αi ≤C (i = 1, . . . , N), which is very similar to (4): only the P i αiyi = 0 constraint is missing, since we omitted the bias. One-Class Soft Margin Loss The one-class SVM soft margin (e.g. [15]) is very similar to the two class case and leads to Sk(α) = 1 2
PN i=1 αiΦk(xi)
2 2 subject to 0 ≤α ≤ 1 νN 1 and PN i=1 αi = 1. ǫ-insensitive Loss Using the same technique for the epsilon insensitive loss L(x, y) = C(1 −|x −y|)+, we obtain Sk(α, α∗) = 1 2
N X i=1 (αi −α∗ i )Φk(xi)
2 2 − N X i=1 (αi + α∗ i )ǫ − N X i=1 (αi −α∗ i )yi, with 0 ≤α, α∗≤C1. When including a bias term, we additionally have the constraint PN i=1(αi −α∗ i )yi = 0. It is straightforward to derive the dual problem for other loss functions such as the quadratic loss. Note that the dual SILP’s only differ in the definition of Sk and the domains of the α’s. 4 Algorithms to solve SILPs The SILPs considered in this work all have the following form: max θ∈R,β∈RM + θ s.t. PK k=1 βk = 1 and PM k=1 βkSk(α) ≥θ for all α ∈C (10) for some appropriate Sk(α) and the feasible set C ⊆RN of α depending on the choice of the cost function. Using Theorem 5 in [12] one can show that the above SILP has a solution if the corresponding primal is feasible and bounded. Moreover, there is no duality gap, if M = co{[S1(α), . . . , SK(α)]⊤| α ∈C} is a closed set. For all loss functions considered in this paper this holds true. We propose to use a technique called Column Generation to solve (10). The basic idea is to compute the optimal (β, θ) in (10) for a restricted subset of constraints. It is called the restricted master problem. Then a second algorithm generates a new constraint determined by α. In the best case the other algorithm finds the constraint that maximizes the constraint violation for the given intermediate solution (β, θ), i.e. αβ := argmin α∈C X k βkSk(α). (11) If αβ satisfies the constraint PK k=1 βkSk(αβ) ≥θ, then the solution is optimal. Otherwise, the constraint is added to the set of constraints. Algorithm 1 is a special case of the set of SILP algorithms known as exchange methods. These methods are known to converge (cf. Theorem 7.2 in [6]). However, no convergence rates for such algorithm are so far known.2 Since it is often sufficient to obtain an approximate solution, we have to define a suitable convergence criterion. Note that the problem is solved when all constraints are satisfied. Hence, it is a natural choice to use the normalized maximal constraint violation as a convergence criterion, i.e. ǫ := 1 − PK k=1 βt kSk(αt) θt , where (βt, θt) is the optimal solution at iteration t −1 and αt corresponds to the newly found maximally violating constraint of the next iteration. We need an algorithm to identify unsatisfied constraints, which, fortunately, turns out to be particularly simple. Note that (11) is for all considered cases exactly the dual optimization problem of the single kernel case for fixed β. For instance for binary classification, (11) reduces to the standard SVM dual using the kernel k(xi, xj) = P k βkkk(xi, xj): min α∈RN N X i,j=1 αiαjyiyjk(xi, xj) − N X i=1 αi with 0 ≤α ≤C1 and N X i=1 αiyi = 0. We can therefore use a standard SVM implementation in order to identify the most violated constraint. Since there exist a large number of efficient algorithms to solve the single kernel problems for all sorts of cost functions, we have therefore found an easy way to extend their applicability to the problem of Multiple Kernel Learning. In some cases it is possible to extend existing SMO based implementations to simultaneously optimize β and α. In [16] we have considered such an algorithm for the binary classification case that frequently recomputes the β’s.3 Empirically it is a few times faster than the column generation algorithm, but it is on the other hand much harder to implement. 5 Experiments In this section we will discuss toy examples for binary classification and regression, demonstrating that MKL can recover information about the problem at hand, followed by a brief review on problems for which MKL has been successfully used. 5.1 Classifications In Figure 1 we consider a binary classification problem, where we used MKL-SVMs with five RBF-kernels with different widths, to distinguish the dark star-like shape from the 2It has been shown that solving semi-infinite problems like (7), using a method related to boosting (e.g. [8]) one requires at most T = O(log(M)/ˆǫ2) iterations, where ˆǫ is the remaining constraint violation and the constants may depend on the kernels and the number of examples N [11, 14]. At least for not too small values of ˆǫ this technique produces reasonably fast good approximate solutions. 3Simplex based LP solvers often offer the possibility to efficient restart the computation when adding only a few constraints. Algorithm 1 The column generation algorithm employs a linear programming solver to iteratively solve the semi-infinite linear optimization problem (10). The accuracy parameter ǫ is a parameter of the algorithm. Sk(α) and C are determined by the cost function. S0 = 1, θ1 = −∞, β1 k = 1 K for k = 1, . . . , K for t = 1, 2, . . . do Compute αt = argmin α∈C K X k=1 βt kSk(α) by single kernel algorithm with K = K X k=1 βt kKk St = K X k=1 βt kSk(αt) if |1 −St θt | ≤ǫ then break (βt+1, θt+1) = argmax θ w.r.t. β ∈RK + , θ ∈R with K X k=1 βk = 1 and K X k=1 βkSr k ≥θ for r = 1, . . . , t end for light star. (The distance between the stars increases from left to right.) Shown are the obtained kernel weightings for the five kernels and the test error which quickly drops to zero as the problem becomes separable. Note that the RBF kernel with largest width was not appropriate and thus never chosen. Also with increasing distance between the stars kernels with greater widths are used. This illustrates that MKL one can indeed recover such tendencies. 5.2 Regression We applied the newly derived MKL support vector regression formulation, to the task of learning a sine function using three RBF-kernels with different widths. We then increased the frequency of the sine wave. As can be seen in Figure 2, MKL-SV regression abruptly switches to the width of the RBF-kernel fitting the regression problem best. In another regression experiment, we combined a linear function with two sine waves, one of lower frequency and one of high frequency, i.e. f(x) = c · x + sin(ax) + sin(bx). Using ten RBF-kernels of different width (see Figure 3) we trained a MKL-SVR and display the learned weights (a column in the figure). The largest selected width (100) models the linear component (since RBF with large widths are effectively linear) and the medium width (1) corresponds to the lower frequency sine. We varied the frequency of the high frequency sine wave from low to high (left to right in the figure). One observes that MKL determines Figure 1: A 2-class toy problem where the dark grey star-like shape is to be distinguished from the light grey star inside of the dark grey star. For details see text. 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 width 0.005 width 0.05 width 0.5 width 1 width 10 frequency kernel weight Figure 2: MKL-Support Vector Regression for the task of learning a sine wave (please see text for details). an appropriate combination of kernels of low and high widths, while decreasing the RBFkernel width with increased frequency. This shows that MKL can be more powerful than cross-validation: To achieve a similar result with cross-validation one has to use 3 nested loops to tune 3 RBF-kernel sigmas, e.g. train 10·9·8/6 = 120 SVMs, which in preliminary experiments was much slower then using MKL (800 vs. 56 seconds). frequency RBF kernel width 2 4 6 8 10 12 14 16 18 20 0.001 0.005 0.01 0.05 0.1 1 10 50 100 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Figure 3: MKL support vector regression on a linear combination of three functions: f(x) = c · x + sin(ax) + sin(bx). MKL recovers that the original function is a combination of functions of low and high complexity. For more details see text. 5.3 Applications in the Real World MKL has been successfully used on real-world datasets in the field of computational biology [7, 16]. It was shown to improve classification performance on the task of ribosomal and membrane protein prediction, where a weighting over different kernels each corresponding to a different feature set was learned. Random channels obtained low kernel weights. Moreover, on a splice site recognition task we used MKL as a tool for interpreting the SVM classifier [16], as is displayed in Figure 4. Using specifically optimized string kernels, we were able to solve the classification MKL SILP for N = 1.000.000 examples and K = 20 kernels, as well as for N = 10.000 examples and K = 550 kernels. −50 −40 −30 −20 −10 Exon Start +10 +20 +30 +40 +50 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Figure 4: The figure shows an importance weighting for each position in a DNA sequence (around a so called splice site). MKL was used to learn these weights, each corresponding to a sub-kernel which uses information at that position to discriminate true splice sites from fake ones. Different peaks correspond to different biologically known signals (see [16] for details). We used 65.000 examples for training with 54 sub-kernels. 6 Conclusion We have proposed a simple, yet efficient algorithm to solve the multiple kernel learning problem for a large class of loss functions. The proposed method is able to exploit the existing single kernel algorithms, whereby extending their applicability. In experiments we have illustrated that the MKL for classification and regression can be useful for automatic model selection and for obtaining comprehensible information about the learning problem at hand. It is future work to evaluate MKL algorithms for unsupervised learning such as Kernel PCA and one-class classification. Acknowledgments The authors gratefully acknowledge partial support from the PASCAL Network of Excellence (EU #506778), DFG grants JA 379 / 13-2 and MU 987/2-1. We thank Guido Dornhege, Olivier Chapelle, Olaf Weiss, Joaquin Qui˜no˜nero Candela, Sebastian Mika and K.-R. M¨uller for great discussions. References [1] Francis R. Bach, Gert R. G. Lanckriet, and Michael I. Jordan. Multiple kernel learning, conic duality, and the SMO algorithm. In Twenty-first international conference on Machine learning. ACM Press, 2004. [2] Kristin P. Bennett, Michinari Momma, and Mark J. Embrechts. Mark: a boosting algorithm for heterogeneous kernel models. KDD, pages 24–31, 2002. [3] Jinbo Bi, Tong Zhang, and Kristin P. Bennett. Column-generation boosting methods for mixture of kernels. KDD, pages 521–526, 2004. [4] O. Chapelle, V. Vapnik, O. Bousquet, and S. Mukherjee. Choosing multiple parameters for support vector machines. Machine Learning, 46(1-3):131–159, 2002. [5] I. Grandvalet and S. Canu. Adaptive scaling for feature selection in SVMs. In In Advances in Neural Information Processing Systems, 2002. [6] R. Hettich and K.O. Kortanek. Semi-infinite programming: Theory, methods and applications. SIAM Review, 3:380–429, September 1993. [7] G.R.G. Lanckriet, T. De Bie, N. Cristianini, M.I. Jordan, and W.S. Noble. A statistical framework for genomic data fusion. Bioinformatics, 2004. [8] R. Meir and G. R¨atsch. An introduction to boosting and leveraging. In S. Mendelson and A. Smola, editors, Proc. of the first Machine Learning Summer School in Canberra, LNCS, pages 119–184. Springer, 2003. in press. [9] C.S. Ong, A.J. Smola, and R.C. Williamson. Hyperkernels. In In Advances in Neural Information Processing Systems, volume 15, pages 495–502, 2003. [10] J. Platt. Fast training of support vector machines using sequential minimal optimization. In B. Sch¨olkopf, C.J.C. Burges, and A.J. Smola, editors, Advances in Kernel Methods — Support Vector Learning, pages 185–208, Cambridge, MA, 1999. MIT Press. [11] G. R¨atsch. Robust Boosting via Convex Optimization. PhD thesis, University of Potsdam, Computer Science Dept., August-Bebel-Str. 89, 14482 Potsdam, Germany, 2001. [12] G. R¨atsch, A. Demiriz, and K. Bennett. Sparse regression ensembles in infinite and finite hypothesis spaces. Machine Learning, 48(1-3):193–221, 2002. Special Issue on New Methods for Model Selection and Model Combination. Also NeuroCOLT2 Technical Report NC-TR-2000085. [13] G. R¨atsch, S. Sonnenburg, and C. Sch¨afer. Learning interpretable svms for biological sequence classification. BMC Bioinformatics, Special Issue from NIPS workshop on New Problems and Methods in Computational Biology Whistler, Canada, 18 December 2004, 7(Suppl. 1:S9), February 2006. [14] G. R¨atsch and M.K. Warmuth. Marginal boosting. NeuroCOLT2 Technical Report 97, Royal Holloway College, London, July 2001. [15] B. Sch¨olkopf and A. J. Smola. Learning with Kernels. MIT Press, Cambridge, MA, 2002. [16] S. Sonnenburg, G. R¨atsch, and C. Sch¨afer. Learning interpretable SVMs for biological sequence classification. In RECOMB 2005, LNBI 3500, pages 389–407. Springer-Verlag Berlin Heidelberg, 2005. [17] S. Sonnenburg, G. R¨atsch, S. Sch¨afer, and B. Sch¨olkopf. Large scale multiple kernel learning. Journal of Machine Learning Research, 2006. accepted.
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Stimulus Evoked Independent Factor Analysis of MEG Data with Large Background Activity S.S. Nagarajan Biomagnetic Imaging Laboratory Department of Radiology University of California, San Francisco San Francisco, CA 94122 sri@radiology.ucsf.edu H.T. Attias Golden Metallic, Inc. P.O. Box 475608 San Francisco, CA 94147 htattias@goldenmetallic.com K.E. Hild Biomagnetic Imaging Laboratory Department of Radiology University of California, San Francisco San Francisco, CA 94122 hild@mrsc.ucsf.edu K. Sekihara Dept. of Systems Design and Engineering Tokyo Metropolitan University Asahigaoka 6-6, Hino, Tokyo 191-0065 ksekiha@cc.tmit.ac.jp Abstract This paper presents a novel technique for analyzing electromagnetic imaging data obtained using the stimulus evoked experimental paradigm. The technique is based on a probabilistic graphical model, which describes the data in terms of underlying evoked and interference sources, and explicitly models the stimulus evoked paradigm. A variational Bayesian EM algorithm infers the model from data, suppresses interference sources, and reconstructs the activity of separated individual brain sources. The new algorithm outperforms existing techniques on two real datasets, as well as on simulated data. 1 Introduction Electromagnetic source imaging, the reconstruction of the spatiotemporal activation of brain sources from MEG and EEG data, is currently being used in numerous studies of human cognition, both in normal and in various clinical populations [1]. A major advantage of MEG/EEG over other noninvasive functional brain imaging techniques, such as fMRI, is the ability to obtain valuable information about neural dynamics with high temporal resolution on the order of milliseconds. An experimental paradigm that is very popular in imaging studies is the stimulus evoked paradigm. In this paradigm, a stimulus, e.g., a tone at a particular frequency and duration, is presented to the subject at a series of equally spaced time points. Each presentation (or trial) produces activity in a set of brain sources, which generates an electromagnetic field captured by the sensor array. These data constitute the stimulus evoked response, and analyzing them can help to gain insights into the mechanism used by the brain to process the stimulus and similar sensory inputs. This paper presents a new technique for analyzing stimulus evoked electromagnetic imaging data. An important problem in analyzing such data is that MEG/EEG signals, which are captured by sensors located outside the brain, contain not only signals generated by brain sources evoked by the stimulus, but also interference signals, generated by other sources such as spontaneous brain activity, eye blinks and other biological and non-biological sources of artifacts. Interference signals overlap spatially and temporally with the stimulus evoked signals, making it difficult to obtain accurate reconstructions of evoked brain sources. A related problem is that signals from different evoked sources themselves overlap with each other, making it difficult to localize individual sources and reconstruct their separate responses. Figure 1: Simulation example (see text) Many approaches have been taken to the problem of suppressing interference signals. One method is averaging over multiple trials, which reduces the contributions from interference sources, assuming that they are uncorrelated with the stimulus and that their autocorrelation time scale is shorter than the trial length. However, a successful application of this method requires a large number of trials, effectively limiting the number of stimulus conditions per experiment. It usually also requires manual rejection of trials containing conspicuous artifacts. A set of methods termed subspace techniques computes a projection of the sensor data onto the signal subspace, which corresponds to brain sources of interest. However, these methods rely on thresholding to determine the noise level, and tend to discard information below threshold. Consequently, those methods perform well only when the interference level is low. Independent component analysis (ICA) techniques [4-8], introduced more recently, attempt to decompose the sensor data into a set of signals that are mutually statistically independent. Artifacts such as eye blinks are independent of brain source activity and ICA has been able in many cases to successfully separate the two types of signals into distinct groups of output variables. However, ICA techniques have several shortcomings. First, they require pre-processing the sensor data to reduce dimensionality from, which causes loss of information on brain sources with relatively low amplitude. This is because, for K sensors, ICA must learn a square K × K unmixing matrix from N data points; typical values such as K = 275, N = 700 can lead to poor performance due to local maxima, overfitting, and slow convergence. Second, ICA assumes L + M = K′, where L, M are the number of evoked and interference sources and K′ < K is the reduced input dimensionality. However, many cases have L + M > K′, which leads to suboptimal and sometime failed separation. Third, ICA requires post-processing of its output signals, usually via manual examination by experts (though sometime by thresholding), to determine which signals correspond evoked brain sources of interest. The fourth drawback of ICA techniques is that, by design, they cannot exploit the advantage offered by the evoked stimulus paradigm. Whereas interference sources are continuously active, evoked sources become active at each trial only near the time of stimulus presentation, termed stimulus onset time. Hence, knowledge of the onset times can help separate the evoked sources. However, the onset times, which are determined by the experimental design and available during data analysis, are ignored by ICA. In this paper we present a novel technique for suppressing interference signals and separating signals from individual evoked sources. The technique is based on a new probabilistic graphical model termed stimulus evoked independent factor analysis (SEIFA). This model, an extension of [2], describes the observed sensor data in terms of two sets of independent variables, termed factors, which are not directly observable. The factors in the first set represent evoked sources, and the factors in the second set represent interference sources. The sensor data are generated by linearly combining the factors in the two sets using two mixing matrices, followed by adding sensor noise. The mixing matrices and the precision matrix of the sensor noise constitute the SEIFA model parameters, and are inferred from data using a variational Bayesian EM algorithm [3], which computes their posterior distribution. Separation of the evoked sources is achieved in the course of processing by the algorithm. Figure 2: Performance on simulated data (see text) The SEIFA model is free from the above four shortcomings. It can be applied directly to the sensor data without dimensionality reduction, therefore no information is lost. Rather than learning a square K × K unmixing matrix, it learns a K × (L + M) mixing matrix, where the number of interference factors M is minimized using automatic Bayesian model selection which is part of the algorithm. In addition, SEIFA is designed to explicitly model the stimulus evoked paradigm, hence it optimally exploits the knowledge of stimulus onset times. Consequently, evoked sources are automatically identified and no post-processing is required. 2 SEIFA Probabilistic Graphical Model This section presents the SEIFA probabilistic graphical model, which is the focus of this paper. The SEIFA model describes observed MEG sensor data in terms of three types of underlying, unobserved signals: (1) signals arising from stimulus evoked sources, (2) signals arising from interference sources, and (2) sensor noise signals. The model is inferred from data by an algorithm presented in the next section. Following inference, the model is used to separate the evoked source signals from those of the interference sources and from sensor noise, thus providing a clean version of the evoked response. The model further separates the evoked response into statistically independent factors. In addition, it produces a regularized correlation matrix of the clean evoked response and of each independent factors, which facilitates localization. Let yin denote the signal recorded by sensor i = 1 : K at time n = 1 : N. We assume that these signals arise from L evoked factors and M interference factors that are combined linearly. Let xjn denote the signal of evoked factor j = 1 : L, and let ujn denote the signal of interference factor j = 1 : M, both at time n. We use the term factor rather than source for a reason explained below. Let Aij denote the evoked mixing matrix, and let Bij denote the interference mixing matrix. Those matrices contain the coefficients of the linear combination of the factors that produces the data. They are analogous to the factor loading matrix in the factor analysis model. Let vin denote the noise signal on sensor i. We use an evoked stimulus paradigm, where a stimulus is presented at a specific time, termed the stimulus onset time, and is absent beforehand. The stimulus onset time is defined as n = N0 + 1. The period preceding the onset n = 1 : N0 is termed pre-stimulus period, and the period following the onset n = N0 + 1 : N is termed post-stimulus period. We assume the evoked factors are active only post stimulus and satisfy xjn = 0 before its onset. Hence yn = Bun + vn, n = 1 : N0 Axn + Bun + vn, n = N0 + 1 : N (1) To turn (1) into a probabilistic model, each signal must be modelled by a probability distribution. Here, each evoked factor is modelled by a mixture of Gaussian (MOG) distributions. For factor j we have a MOG model with Sj components, also termed states, p(xn) = L Y j=1 p(xjn) , p(xjn) = Sj X sj=1 N(xjn | µj,sj, νj,sj)πj,sj (2) State sj is a Gaussian with mean µj,sj and precision νj,sj, and its probability is πj,sj. We model the factors as mutually statistically independent. There are three reasons for using MOG distributions, rather than Gaussians, to describe the evoked factors. First, evoked brain sources are often characterized by spikes or by modulated harmonic functions, leading to non-Gaussian distributions. Second, previous work on ICA has shown that independent Gaussian sources that are linearly mixed cannot be separated. Since we aim to separate the evoked response into contributions from individual factors, we must therefore use independent non-Gaussian factor distributions. Third, as is well known, a MOG model with a suitably chosen number of states can describe arbitrary distributions at the desired level of accuracy. For interference signals and sensor noise we employ a Gaussian model. Each interference factor is modelled by an independent, zero-mean Gaussian distribution with unit precision, p(un) = M Y j=1 N(ujn | 0, 1) = N(un | 0, I) (3) The Gaussian model implies that we exploit only second order statistics of the interference signals. This contrasts with the evoked signals, whose MOG model facilitates exploiting higher order statistics, leading to more accurate reconstruction and to separation. The sensor noise is modelled by a zero-mean Gaussian distribution with a diagonal precision matrix λ, p(vn) = N(vn | 0, λ). From (1) we obtain p(yn | xn, un) = p(vn) where we substitute vn = yn −Axn −Bun with xn = 0 for n = 1 : N0. Hence, we obtain the distribution of the sensor signals conditioned on the evoked and interference factors, p(yn | xn, un, A, B) = N(yn | Bun, λ), n = 1 : N0 N(yn | Axn + Bun, λ), n = N0 + 1 : N (4) SEIFA also makes an i.i.d. assumption, meaning the signals at different time points are independent. Hence p(y, x, u | A, B) = Q n p(yn | xn, un, A, B)p(xn)p(un). where y, x, u denote collectively the signals yn, xn, un at all time points. The i.i.d. assumption is made for simplicity, and implies that the algorithm presented below can exploit the spatial statistics of the data but not their temporal statistics. To complete the definition of SEIFA, we must specify prior distributions over the model parameters. For the noise precision matrix λ we choose a flat prior, p(λ) = const. For the mixing matrices A, B we choose to use a conjugate prior p(A) = Y ij N(Aij | 0, λiαj) , p(B) = Y ij N(Bij | 0, λiβj) (5) where all matrix elements are independent zero-mean Gaussians and the precision of the ijth matrix element is proportional to the noise precision λi on sensor i. It is the λ dependence which makes this prior conjugate. The proportionality constants αj and βj constitute the parameters of the prior, a.k.a. hyperparameters. Eqs. (2,3,4,5) fully define the SEIFA model. 3 Inferring the SEIFA Model from Data: A VB-EM Algorithm This section presents an algorithm that infers the SEIFA model from data. SEIFA is a probabilistic model with hidden variables, since the evoked and interference factors are not directly observable, hence it must be treated in the EM framework. We use variational Bayesian EM (VB-EM), which has two relevant advantages over standard EM. First, it is more robust to overfitting, which can be a significant problem when working with high-dimensional but relatively short time series (here we analyze N < 1000 point long, K = 275 dimensional data sequences). To achieve this robustness, VB-EM computes (using a variational approximation) a full posterior distribution over model parameters, rather than a single MAP estimate. This means that VB-EM considers all possible parameters values, and computes the probability of each value conditioned on the observed data. It also performs automatic model order selection by optimizing the hyperparameters, and consequently uses the minimum number of parameters needed to explain the data. Second, VB-EM produces automatically regularized estimators for the evoked response correlation matrices (required for source localization), where standard EM produces poorly conditioned ones. This is also a result of computing a parameter posterior. VB-EM is an iterative algorithm, where each iteration consists of an E- and an M-step. E-step. For the pre-stimulus period n = 1 : N0 we compute the posterior over the interference factors un only. It is a Gaussian distribution with posterior mean ¯un and covariance Φ given by ¯un = Φ ¯BT λyn , Φ = ¯BT λ ¯B + I + KΨBB −1 (6) where ¯B are ΨBB are the posterior mean and covariance of the interference mixing matrix B computed in the M-step below (more precisely, the posterior covariance of the ith row of B is ΨBB/λi). For the post-stimulus period n = N0 + 1 : N we compute the posterior over the evoked and interference factors xn, un, and the collective state sn of the evoked factors. The latter is defined by the L-dimensional vector sn = (s1n, s2n, ..., sLn), where sjn = 1 : Sj is the state of evoked factor j at time n. The total number of collective states is S = Q j Sj. To simplify the notation, we combine the evoked and interference factors into a single L′ × 1 vector x′ n = (xn, un), where L′ = L + M, and their mixing matrices into a single K × L′ matrix A′ = (A, B). Now, at time n, let r run over all the S collective states. For each r, the posterior over the factors conditioned on sn = r is Gaussian, with posterior mean ¯xrn, ¯urn and covariance Γr given by ¯x′ rn = Γr ¯A′T λyn + ν′ rµ′ r , Γr = ¯A′T λ ¯A′ + ν′ r + KΨ −1 (7) We have defined ¯x′ rn = (¯xrn, ¯urn) and ¯A′ = ( ¯A, ¯B). The L × 1 vector µ′ r and the diagonal L×L matrix ν′ r contain the means and precisions of the individual states (see (2)) composing r. The posterior mean and covariance ¯A′, Ψ are computed in the M-step. Next, compute the posterior probability that sn = r by ¯πrn = 1 zn πr p | νr || Γr | exp −1 2yT n λyn + 1 2µT r νrµr −1 2 ¯x′ rnΓ−1 r ¯x′ rn (8) where zn is a normalization constant and µr, νr, πr are the MOG parameters of (2). M-step. We divide the model parameters into two sets. The first set includes the mixing matrices A, B, for which we compute full posterior distributions. The second set includes the noise precision λ and the diagonal hyperparameters matrices α, β, for which we compute MAP estimates. The posterior over A, B is Gaussian factorized over their rows, where the mean is ¯A = RyxΨ ¯B = RyuΨ , Ψ = Rxx + α Rxu RT xu Ruu + β −1 (9) and where the ith row of A′ = (A, B) has covariance Ψ/λi. The hyperparameters αj, βj are diagonal entries of diagonal matrices α, β. Ryx, Ryu, Rxx, Rxu, Ruu are posterior correlations between the factors and the data and among the factors themselves, e.g., Ryx = P n⟨ynxn⟩, Rxx = P n⟨xnxn⟩, where ⟨·⟩denotes posterior averaging. They are easily computed in terms of the E-step quantities ¯un, ¯x′ rn, Φ, Γr, ¯πrn and are omitted. Next, the hyperparameter matrices α, β are updated by α−1 = diag ¯AT λ ¯A/K + ΨAA , β−1 = diag ¯BT λ ¯B/K + ΨBB (10) and the noise precision matrix by λ−1 = diag(Ryy −¯ART yx −¯BRT yu)/N. ΨAA and ΨBB are the appropriate blocks of Ψ in (9). The interference mixing matrix and the noise precision are initialized from pre-stimulus data. We used MOG parameters corresponding to peaky (super-Gaussian) distributions. Estimating and Localizing Clean Evoked Responses. Let zj in = ⟨Aijxjn⟩denote the inferred individual contribution from evoked factor j to sensor signal i. It is given via posterior averaging by ¯zj in = ¯Aij ¯xjn (11) where ¯xn = P r ¯πr¯xrn. Computing this estimate amounts to obtaining a clean version of the individual contribution from each factor and of their combined contribution, and removing contributions from interference factors and sensor noise. The localization of individual evoked factors using sensor signals zj n can be achieved by many algorithms. In this paper, we use adaptive spatial filters that take data correlation matrices as inputs for localization, because these methods have been shown to have superior spatial resolution and non-zero localization bias [6]. Let Cj = P n⟨zj n(zj n)T ⟩denote the inferred sensor data correlation matrix corresponding to the individual contribution from evoked factor j. Then, Cj = ¯Aj( ¯Aj)T + λ−1(ΨAA)jj (Rxx)jj (12) where ¯Aj is a K ×1 vector denoting the jth column of ¯A. Notice that the VB-EM approach has produced a correlation matrix that is automatically regularized (due to the ΨAA term) and can be used for localization in its current form. In contrast, computing it from the signal estimates obtained by other methods, such as PCA or ICA, yields a poorly conditioned matrix that requires post-processing. 4 Experiments on Real and Simulated Data Simulations. Fig. 1 shows a simulation with two evoked sources and three interference sources with N = 10000, signal-to-interference (SIR) of 0 dB and signal-to-sensor-noise (SNR) of 5dB. The true locations of the evoked sources, each of which is denoted by •, and the true locations of the background sources, each of which is denoted by × are shown in the top left panel. The right column in the top row shows the time courses of the evoked sources as they appear at the sensors. The time courses of the actual sensor signals, which also include the effects of background sources and sensor noise, are shown in the middle row (right column). The bottom row shows the localization and time-course of cleaned Figure 3: Estimating auditory-evoked responses from small trial averages (see text) evoked sources estimated using SEIFA, which agrees with the true location and timecourse. Fig. 2 shows the mean performance as a function of SIR, across 50 Monte Carlo trials for N = 1000 and SNR of 10 dB, for different locations of evoked and interference sources. Denoising performance is quantified by the output signal-to-(noise+interference) ratio (SNIR) and shown in the top panel. SEIFA outperforms both our benchmark methods, providing a 5-10 dB improvement over JADE [7] and SVD. Separation performance of individual evoked factors is quantified by (separated-signal)-to-(noise+interference) ratio (SSNIR) (definition omitted) and is shown in the middle panel. SEIFA far outperforms JADE for this set of examples. JADE is able to separate the background sources from the evoked sources (hence gives good denoising performance), but it is not always able to separate the evoked sources from each other. The Infomax algorithm [4] (results not shown) exhibited poor separation performance similar to JADE. Finally, localization performance is quantified by the mean distance in cm between the true evoked source locations and the estimated locations, as shown in the bottom panel. Here too, SEIFA far outperforms all other methods, especially for low SIR. Notably, SEIFA performance appears to be quite robust to the i.i.d. assumption of the evoked and background sources, because in these simulations evoked sources were assumed to be damped sinusoids and interference sources were sinusoids. 4.1 Real Data Denoising averages from small number of trials. Auditory evoked responses from a particular subject obtained by averaging different number of trials are shown in figure 3 (left panel). SEIFA is able to clearly recover responses even from small trial averages. To quantify the performance of the different methods, a filtered version of the raw data for Navg = 250 was assumed as “ground-truth”, and is shown in the inset of the right panel. The output SNIR as a function of Navg is also shown in figure 3 (right panel).SEIFA exhibits the best performance especially for small trial averages. Separation of evoked sources. To highlight SEIFA’s ability to separately localize evoked sources, we conducted an experiment involving simultaneous presentation of auditory and somatosensory stimuli. We expected the activation of contralateral auditory and somatosensory cortices to overlap in time. A pure tone (400ms duration, 1kHz, 5 ms ramp up/down) was presented binaurally with a delay of 50 ms following a pneumatic tap on the left index finger. Averaging is performed over Navg = 100 trials triggered on the onset of the tap. Results from SEIFA for this experiment are shown in Figure 4. In these figures, one panel shows a contour map that shows the polarity and magnitude of the denoised and raw sensor signals in sensor space. The contour plot of the magnetic field on the sensor array, corresponding to the mapping of three-dimensional sensor surface array to points within a circle, shows the magnetic field profile at a particular instant of time relative to the stimulus presentation. Other panels show localization of a particular evoked factor overlaid on the subjects’ MRI. Three orthogonal projections - axial, sagittal and coronal MRI slices, that highlight all voxels having activity that is > 80% of maximum are shown. Results are based on the right hemisphere channels above contralateral somatosensory and auditory cortices. Localization of time-course of the first two factors estimated by SEIFA are shown in left and middle panels of figure 4. The first two factors localize to primary somatosensory cortex (SI), however with differential latencies. The first factor shows a peak response at a latency of 50 ms, whereas the second factor shows the response at a later latency. Interestingly, the third factor localizes to auditory cortex and the extracted timecourse corresponds well to an auditory evoked response that is well-separated from the somatosensory response (figure 3 right panels). Figure 4: Estimated SEIFA factors for auditory-somatosensory experiment 5 Extensions Whereas this paper uses fixed values for the number of evoked and interference sources L, M (though the effective number of interference sources was determined via optimizing the hyperparameter β), VB-EM facilitates inferring them from data, and we plan to investigate the effectiveness of this procedure. We also plan to infer the distribution of evoked sources (MOG parameters) from data rather than using a fixed distribution. Another extension that could enhance performance is exploiting temporal correlation in the data. We plan to do it by incorporating temporal (e.g., autoregressive) models into the source distributions and infer their parameters from data. References [1] S. Baillet, J. C. Mosher, and R. M. Leahy. Electromagnetic brain mapping.Signal Processing Magazine, 18:14-30, 2001. [2] H. Attias (1999). Independent Factor Analysis. Neur. Comp. 11, 803-851. [3] H. Attias (2000). A variational Bayes framework for graphical models. Adv. Neur. Info. Proc. Sys. 12, 209-215. [4] T.-P. Jung, S. Makeig, M. Westerfield, J. Townsend, E. Courchesne, T.J. Sejnowski (2000). Removal of eye artifacts from visual event related potentials in normal and clinical subjects. J. Clin. Neurophys. 40, 516-520. [5] S. Makeig, S. Debener, J. Onton, A. Delorme (2004). Mining event related brain dynamics. Trends Cog. Sci. 8, 204-210. [6] K. Sekihara, S. Nagarajan, D. Poeppel, A. Marantz, Y. Miyashita (2001). Reconstructing spatiotemporal activities of neural sources using a MEG vector beamformer technique. IEEE Trans. Biomed. Eng. 48, 760-771. [7] J.F.Cardoso (1999) High-order contrasts for independent component analysis, Neural Computation, 11(1):157–192. [8] R. Vigario, J. Sarela, V. Jousmaki, M. Hamalainen, E. Oja (2000). Independent component approach to the analysis of EEG and MEG recordings. IEEE Trans. Biomed. Eng. 47, 589-593.
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Augmented Rescorla-Wagner and Maximum Likelihood estimation. Alan Yuille Department of Statistics University of California at Los Angeles Los Angeles, CA 90095 yuille@stat.ucla.edu Abstract We show that linear generalizations of Rescorla-Wagner can perform Maximum Likelihood estimation of the parameters of all generative models for causal reasoning. Our approach involves augmenting variables to deal with conjunctions of causes, similar to the agumented model of Rescorla. Our results involve genericity assumptions on the distributions of causes. If these assumptions are violated, for example for the Cheng causal power theory, then we show that a linear Rescorla-Wagner can estimate the parameters of the model up to a nonlinear transformtion. Moreover, a nonlinear Rescorla-Wagner is able to estimate the parameters directly to within arbitrary accuracy. Previous results can be used to determine convergence and to estimate convergence rates. 1 Introduction It is important to understand the relationship between the Rescorla-Wagner (RW) algorithm [1,2] and theories of learning based on maximum likelihood (ML) estimation of the parameters of generative models [3,4,5]. The Rescorla-Wagner algorithm has been shown to account for many experimental findings. But maximum likelihood offers the promise of a sound statistical basis including the ability to learn sophisticated probabilistic models for causal learning [6,7,8]. Previous work, summarized in section (2), showed a direct relationship between the basic Rescorla-Wagner algorithm and maximum likelihood for the ∆P model of causal learning [4,9]. More recently, a generalization of Rescorla-Wagner was shown to perform maximum likelihood estimation for both the ∆P and the noisy-or models [10]. Throughout the paper, we follow the common practice of studying the convergence of the expected value of the weights and ignoring the fluctuations. The size of these fluctuations can be calculated analytically and precise convergence quantified [10]. In this paper, we greatly extend the connections between Rescorla-Wagner and ML estimation. We show that two classes of generalized Rescorla-Wagner algorithms can perform ML estimation for all generative models provided genericity assumptions on the causes are satisfied. These generalizations include augmenting the set of variables to represent conjunctive causes and are related to the augmented Rescorla-Wagner algorithm [2]. We also analyze the case where the genericity assumption breaks down and pay particular attention to Chengs’ causal power model [4,5]. We demonstrate that Rescorla-Wagner can perform ML estimation for this model up to a nonlinear transformation of the model parameters (i.e. Rescorla-Wagner does ML but in a different coordinate system). We sketch how a nonlinear Rescorla-Wagner can estimate the parameters directly. Convergence analysis from previous work [10] can be directly applied to these new Rescorla-Wagner algorithms. This gives convergence conditions and put bounds on the convergence rate. The analysis assumes that the data consists of i.i.d. samples from the (unknown) causal distribution. But the results can also be applied in the piecewise iid case (such as forward and backward blocking [11]). 2 Summary of Previous Work We summarize pervious work relating maximum likelihood estimation of generative models with the Rescorla-Wagner algorithm [4,9,10]. This work assumes that there is a binaryvalued event E which can be caused by one or more of two binary-valued causes C1, C2. The ∆P and Noisy-or theories use generative models of form: P∆P (E = 1|C1, C2, ω1, ω2) = ω1C1 + ω2C2 (1) PNoisy−or(E = 1|C1, C2, ω1, ω2) = ω1C1 + ω2C2 −ω1ω2C1C2, (2) where {ω1, ω2} are the model parameters. The training data consists of examples {Eµ, Cµ 1 , Cµ 2 }. The parameters {ω1, ω2} are estimated by Maximum Likelihood {ω∗ 1, ω∗ 2} = arg max {ω1,ω2} Y µ P(Eµ|Cµ 1 , Cµ 2 ; ω1, ω2)P(Cµ 1 , Cµ 2 ), (3) where P(C1, C2) is the distribution on the causes. It is independent of {ω1, ω2} and does not affect the Maximum Likelihood estimation, except for some non-generic cases to be discussed in section (5). An alternative approach to learning causal models is the Rescorla-Wagner algorithm which updates weights V1, V2 as follows: V t+1 1 = V t 1 + ∆V t 1 , V t+1 2 = V t 2 + ∆V t 2 , (4) where the update rule ∆V can take forms like: ∆V1 = α1C1(E −C1V1 −C2V2), ∆V2 = α2C2(E −C1V1 −C2V2), basic rule (5) ∆V1 = α1C1(1 −C2)(E −V1), ∆V2 = α2C2(1 −C1)(E −V2), variant rule. (6) It is known that if the basic update rule (5) is used then the weights converge to the ML estimates of the parameters {ω1, ω2} provided the data is generated by the ∆P model (1) [4,9] (but not for the noisy-or model). If the variant update rule (6) is used, then the weights converge to the parameters {ω1, ω2} of the ∆P model or the noisy-or model (2) depending on which model generates the data [10]. 3 Basic Ingredients This section describes three basic ingredients of this work: (i) the generative models, (ii) maximum likelihood, and (iii) the generalized Rescorla-Wagner algorithms. Representing the generative models. We represent the distribution P(E|⃗C; ⃗α) by the function: P(E = 1|⃗C; ⃗α) = X i αihi(⃗C), (7) where the {hi(⃗C)} are a set of basis functions and the {αi} are parameters. If the dimension of ⃗C is n, then the number of basis functions is 2n. All distributions of binary variables can be represented in this form. For example, if n = 2 we can use the basis: h1(⃗C) = 1, h2(⃗C) = C1, h3(⃗C) = C2, h4(⃗C) = C1C2, (8) Then the noisy-or model P(E = 1|C1, C2) = ω1C1 + ω2C2 −ω1ω2C1C2 corresponds to setting α1 = 0, α2 = ω1, α3 = ω2, α4 = −ω1ω2. Data Generation Assumption and Maximum Likelihood We assume that the observed data {Eµ, ⃗Cµ : µ ∈Λ} are i.i.d. samples from P(E|⃗C)P(⃗C). It is possible to adapt our results to cases where the data is piecewise i.i.d., such as blocking experiments, but we have no space to describe this here. Maximum Likelihood (ML) estimates the ⃗α by solving: ⃗α∗= arg min ⃗α − X µ∈Λ log{P(Eµ|⃗Cµ; ⃗α)P(⃗Cµ)} = arg min ⃗α − X µ∈Λ log P(Eµ|⃗Cµ; ⃗α). (9) Observe that the estimate of ⃗α is independent of P(⃗C) provided the distribution is generic. Important non-generic cases are treated in section (5). Generalized Rescorla-Wagner. The Rescorla-Wagner (RW) algorithm updates weights {Vi : i = 1, ..., n} by a discrete iterative algorithm: V t+1 i = V t i + ∆V t i , i = 1, ..., n. (10) We assume a generalized form: ∆Vi = X j Vjfij(⃗C) + Egi(⃗C), i, j = 1, ..., n (11) for functions {fij(⃗C)}, {gi(⃗C)}. It is easy to see that equations (5,6) are special cases. 4 Theoretical Results We now gives sufficient conditions which ensure that the only fixed points of generalized Rescorla-Wagner correspond to ML estimates of the parameters ⃗α of generative models P(E|⃗C, ⃗α). We then obtain two classes of generalized Rescorla-Wagner which satisfy these conditions. For one class, convergence to the fixed points follow directly. For the other class we need to adapt results from [10] to guarantee convergence to the fixed points. Our results assume genericity conditions on the distribution P(⃗C) of causes. We relax these conditions in section (5). The number of weights {Vi} used by the Rescorla-Wagner algorithm is equal to the number of parameters {αi} that specify the model. But many weights will remain zero unless conjunctions of causes occur, see section (6). Theorem 1. A sufficient condition for generalized Rescorla-Wagner (11), to have a unique fixed point at the maximum likelihood estimates of the parameters of a generative model P(E|⃗C; ⃗α) (7), is that < fij(⃗C) >P ( ⃗C)= −< gi(⃗C)hj(⃗C) >P ( ⃗C) ∀i, j and the matrix < fij(⃗C) >P ( ⃗C) is invertible. Proof. We calculate the expectation < ∆Vi >P (E| ⃗C)P ( ⃗C). This is zero if, and only if, P j Vj < fij(⃗C) >P ( ⃗C) + P j αj < gi(⃗C)hj(⃗C) >P ( ⃗C)= 0. The result follows. We use notation that < . >P ( ⃗C) is the expectation with respect to the probability distribution P(⃗C) on the causes. For example, < fij(⃗C) >P ( ⃗C)= P ⃗C P(⃗C)fij(⃗C). Hence the requirement that the matrix < fij(⃗C) >P ( ⃗C) is invertible usually requires that P(⃗C) is generic. See examples in sections (4.1,4.2). Convergence may still occur if the matrix < fij(⃗C) >P ( ⃗C) is non-invertible. Linear combinations of the weights will remained fixed (in the directions of the zero eigenvectors of the matrix) and the remaining linear co,mbinations will converge. Additional conditions to ensure convergence to the fixed point, and to determine the convergence rate, can be found using Theorems 3,4,5 in [10]. 4.1 Generalized RW class I We now give prove a corollary of Theorem 1 which will enable us to obtain our first class of generalized RW algorithms. Corollary 1. A sufficient condition for generalized RW to have fixed points at ML estimates of the model parameters is fij(⃗C) = −hi(⃗C)hj(⃗C), gi(⃗C) = hi(⃗C) ∀i, j and the matrix < hi(⃗C)hj(⃗C) >P ( ⃗C) is invertible. Moreover, convergence to the fixed point is guaranteed. Proof. Direct verification. Convergence to the fixed point follows from the gradient descent nature of the algorithm, see equation (12). These conditions define generalized RW class I (GRW-I) which is a natural extension of basic Rescorla-Wagner (5): ∆Vi = hi(⃗C){E − X j hj(⃗C)Vj} = −∂ ∂Vi (E − X j hj(⃗C)Vj)2, i = 1, ..., n (12) This GRW-I algorithm ia guaranteed to converge to the fixed point because it performs stochastic steepest descent. This is essentially the Widrow-Huff algorithm [12,13]. To illustrate Corollary 1, we show the relationships between GRW-I and ML for three different generative models: (i) the ∆P model, (ii) the noisy-or model, and (iii) the most general form of P(E|⃗C) for two causes. It is important to realize that these generative models form a hierarchy and GRW-I algorithms for the later models will also perform ML on the simpler ones. 1. The ∆P model. Set n = 2, h1(⃗C) = C1 and h2(⃗C) = C2. Then equation (12) reduces to the basic RW algorithm (5) with two weights V1, V2. By Corollary 1, we see that it performs ML estimation for the ∆P model (1). This rederives the known relationship between basic RW, ML, and the ∆P model [4,9]. Observe that Corollary 1 requires that the matrix < C1 >P ( ⃗C) < C1C2 >P ( ⃗C) < C1C2 >P ( ⃗C) < C2 >P ( ⃗C) be invertible. This is equivalent to the genericity condition < C1C2 >2 P ( ⃗C)̸=< C1 >P ( ⃗C)< C2 >P ( ⃗C). 2. The Noisy-Or model. Set n = 3 with h1(⃗C) = C1, h2(⃗C) = C2, h3(⃗C) = C1C2. Then Corollary 1 proves that the following algorithm will converge to estimate V ∗ 1 = ω1, V ∗ 2 = ω2 and V ∗ 3 = −ω1ω2 for the noisy-or model. ∆V1 = C1(E −C1V1 −C2V2 −C1C2V3) = C1(E −V1 −C2V2 −C2V3) ∆V2 = C2(E −C1V1 −C2V2 −C1C2V3) = C2(E −C1V1 −V2 −C1V3) ∆V3 = C1C2(E −C1V1 −C2V2 −C1C2V3) = C1C2(E −V1 −V2 −V3). (13) This algorithm is a minor variant of basic RW. Observe that this has more weights (n = 3) than the total number of causes. The first two weights V1 and V2 yield ω1, ω2 while the third weight V3 gives a (redundant) estimate of ω1ω2. The matrix < hi(⃗C)hj(⃗C) >P ( ⃗C) has determinant (< C1C2 > −< C1 >)(< C1C2 > −< C2 >) < C1C2 > and is invertible provided < C1 ≯= 0, 1, < C2 ≯= 0, 1 and < C1C2 ≯=< C1 >< C2 >. This rules out the special case in Cheng’s experiments [4,5] where C1 = 1 always, see discussion in section (5). It is known that basic RW is unable to do ML estimation for the noisy-or model if there are only two weights [4,5,9,10]. The differences here is that three weights are used. 3. The general two-cause model. Thirdly, we consider the most general model P(E|⃗C) for two causes. This can be written in the form: P(E = 1|C1, C2) = α1 + α2C1 + α3C2 + α4C1C2. (14) This corresponds to h1(⃗C) = 1, h2(⃗C) = C1, h3(⃗C) = C2, h4(⃗C) = C1C2. Corollary 1 gives us the most general algorithm: ∆V1 = (E −V1 −C1V2 −C2V3 −C1C2V4) = (E −V1 −C1V2 −C2V3 −C1C2V4) ∆V2 = C1(E −V1 −C1V2 −C2V3 −C1C2V4) = C1(E −V1 −V2 −C2V3 −C2V4) ∆V3 = C2(E −V1 −C1V2 −C2V3 −C1C2V4) = C2(E −V1 −C1V2 −V3 −C1V4) ∆V4 = C1C2(E −V1 −C1V2 −C2V3 −C1C2V4) = C1C2(E −V1 −V2 −V3 −V4). By Corollary 1, this algorithm will converge to V ∗ 1 = α1, V ∗ 2 = α2, V ∗ 3 = α3, V ∗ 4 = α4, provided the matrix is invertible. The determinant of the matrix < hi(⃗C)hj(⃗C) >P ( ⃗C) is < C1C2 > (< C1C2 > −< C1 >)(< C1C2 > −< C2 >)(1−< C1 > −< C2 > + < C1C2 >). This will be zero for special cases, for example if C1 = 1 always. It is important to realize that the most general GRW-I algorithm will converge if P(E|⃗C) is the ∆P or the noisy-or model. For ∆P it will converge to V ∗ 1 = 0, V ∗ 2 = ω1, V ∗ 3 = ω2, V ∗ 4 = 0. For noisy-or, it converges to V ∗ 1 = 0, V ∗ 2 = ω1, V ∗ 3 = ω2, V ∗ 4 = −ω1ω2. The learning system which implements the GRW-I algorithm will not know a priori whether the data is generated by ∆P, noisy-or, or the general model for P(E|C1, C2). It is therefore better to implement the most general algorithm because this works whatever model generated the data. Note: other functions {hi(⃗C)} will lead to different ways to parameterize the probability distribution P(E|⃗C). They will correspond to different RW algorithms. But their basic properties will be similar to those discussed in this section. 4.2 Generalized RW Class II We can obtain a second class of generalized RW algorithms which perform ML estimation. Corollary 2. A sufficient condition for RW to have unique fixed point at the ML estimate of the generative model P(E|⃗C) is that fij(⃗C) = −gi(⃗C)hj(⃗C), provided the matrix < hi(⃗C)hj(⃗C) >P ( ⃗C) is invertible. Proof. Direct verification. Corollary 2 defines GRW-II to be of form: ∆Vi = gi(⃗C){E − X j hj(⃗C)Vj}. (15) We illustrate GRW-II by applying it to the noisy-or model (2). It gives an algorithm very similar to equation (6). Set h1(⃗C) = C1, h2(⃗C) = C2, h3(⃗C) = C1C2 and g1(⃗C) = C1(1 −C2), g2(⃗C) = C2(1 −C1), g3(⃗C) = C1C2. Corollary 2 yields the update rule: ∆V1 = C1(1 −C2){E −C1V1 −C2V2 −C1C2V3} = C1(1 −C2){E −V1}, ∆V2 = C2(1 −C1){E −C1V1 −C2V2 −C1C2V3} = C2(1 −C1){E −V2}, ∆V3 = C1C2{E −C1V1 −C2V2 −C1C2V3} = C1C2{E −V1 −V2 −V3}. (16) The matrix < hi(⃗C)hj(⃗C) >P ( ⃗C) has determinant < C1C2 > (< C1 > −< C1C2 >)(< C2 > −< C1C2 >) and so is invertible for generic P(⃗C). The algorithm will converge to weights V ∗ 1 = ω1, V ∗ 2 = ω2, V ∗ 3 = −ω1ω2. If we change the model to ∆P, then we get convergence to V ∗ 1 = ω1, V ∗ 2 = ω2, V ∗ 3 = 0. Observe that the equations (16) are largely decoupled. In particular, the updates for V1 and V2 do not depend on the third weight V3. It is possible to remove the update equation for V3 by setting g3(⃗C) = 0. The remaining update equations for V1&V2 will converge to ω1, ω2 for both the noisy-or and the ∆P model. These reduced update equations are identical to those given by equation (6) which were proven to converge to ω1, ω2 [10]. We note that the matrix < hi(⃗C)hj(⃗C) >P ( ⃗C) now has a zero eigenvalue (because g3(⃗C) = 0) but this does not matter because it corresponds to the third weight V3. The matrix remains invertible if we restrict it to i, j = 1, 2. A limitation of GRW-II algorithm of equation (16) is that it only updates the weights if only one cause is active. So it would fail to explain effects such as blocking where both causes are on for part of the stimuli (Dayan personal communication). 5 Non-generic, coordinate transformations, and non-linear RW Our results have assumed genericity constraints on the distribution P(⃗C) of causes. They usually correspond to cases where one cause is always present. We now briefly discuss what happens when these constraints are violated. For simplicity, we concentrate on an important special case. Cheng’s PC theory [4,5] uses the noisy-or model for generating the data but cause C1 is a background cause which is on all the time (i.e. C1 = 1 always). This implies that < C2 >=< C1C2 > and so we cannot apply RW algorithms (13), the most general algorithm, or (16) because the matrix determinant will be zero in all three cases. Since C1 = 1 we can drop it as a variable and re-express the noisy-or model as: P(E = 1|⃗C) = ω1 + ω2(1 −ω1)C2. (17) Theorem 1 shows that we can define generalized RW algorithms to find ML estimates of ω1 and ω2(1 −ω1) (assuming ω1 ̸= 1). But, conversely, it is impossible to estimate ω2 directly by any linear generalized RW. The problem is simply a matter of different coordinate systems. RW estimates the parameters of the generative model in a different coordinate system than the one used to specify the model. There is a non-linear transformation between the coordinates systems relating {ω1, ω2} to {ω1, ω2(1 −ω1)}. So RW can estimate the ML parameters provided we allow for an additional non-linear transformation. From this perspective, the inability to RW to perfrom ML estimation for Cheng’s model is merely an artifact. If we reparameterize the generative model to be P(E = 1|⃗C) = ω1 + ˆω2C2, where ˆω2 = ω2(1 −ω1), then we can design an RW to estimate {ω1, ˆω2}. The non-linear transformation breaks down if ω1 = 1. In this case, the generative model P(E|⃗C) becomes independent of ω2 and so it is impossible to estimate it. But suppose we want to really estimate ω1 and ω2 directly (for Cheng’s model, the value of ω2 is the causal power and hence is a meaningful quantity [4,5]). To do this we first define a linear RW to estimate ω1 and ˆω2 = ω2(1 −ω1). The equations are: V t+1 1 = V t 1 + γ1∆V t 1 , V t+1 2 = V t 2 + γ2∆V t 2 . (18) with < V1 >7→ω1 and < V2 >7→ω2 for large t. The fluctuations (variances) are scaled by the parameters γ1, γ2 and hence can be made arbitrarily small, see [10]. To estimate ω2, we replace the variable V2 by a new variable V3 = V2/(1 −V1) which is updated by a nonlinear equation (V1 is updated as before): V t+1 3 = V t 3 + V t 3 1 −V t 1 δV t 1 + ∆V t 2 1 −V t 1 , (19) where we use V3 = V2/(1−V1) to re-express ∆V1 and ∆V2 in terms of functions of V1 and V3. Provided the fluctuations are small, by controlling the size of the γ’s, we can ensure that V3 converges arbitrarily close to ˆω2/(1 −ω1) = ω2. 6 Conclusion This paper shows that we can obtain linear generalizations of the Rescorla-Wagner algorithm which can learn the parameters of generative models by Maximum Likelihood. For one class of RW generalizations we have only shown that the fixed points are unique and correspond to ML estimates of the parameters of the generative models. But Theorems 3,4 & 5 of Yuille (2004) can be applied to determine convergence conditions. Convergence rates can be determined by these Theorems provided that the data is generated as i.i.d. samples from the generative model. These theorems can also be used to obtain convergence results for piecewise i.i.d. samples as occurs in foreward and backward blocking experiments. These generalizations of Rescorla-Wagner require augmenting the number of weight variables. This was already proposed, on experimental grounds, so that new weights get created if causes occur in conjunction, [2]. Note that this happens naturally in the algorithms presented (13, the most general algorithm,16) – weights remain at zero until we get an event C1C2 = 1. It is straightforward to extend the analysis to models with conjunctions of many causes. We conjecture that these generalizations converge to good approaximation to ML estimates if we truncate the conjunction of causes at a fixed order. Finally, many of our results have involved a genericity assumption on the distribution of causes P(⃗C). We have argued that when these assumptions are violated, for example in Cheng’s experiments, then generalized RW still performs ML estimation, but with a nonlinear transform. Alternatively we have shown how to define a nonlinear RW that estimates the parameters directly. Acknowledgement I acknowledge helpful conversations with Peter Dayan, Rich Shiffrin, and Josh Tennenbaum. I thank Aaron Courville for describing augmented Rescorla-Wagner. I thank the W.M. Keck Foundation for support and NSF grant 0413214. References [1]. R.A. Rescorla and A.R. Wagner. “A Theory of Pavlovian Conditioning”. In A.H. Black andW.F. Prokasy, eds. Classical Conditioning II: Current Research and Theory. New York. Appleton-Century-Crofts, pp 64-99. 1972. [2] R.A. Rescorla. Journal of Comparative and Physiological Psychology. 79, 307. 1972. [3]. B. A. Spellman. “Conditioning Causality”. In D.R. Shanks, K.J. Holyoak, and D.L. Medin, (eds). Causal Learning: The Psychology of Learning and Motivation, Vol. 34. San Diego, California. Academic Press. pp 167-206. 1996. [4]. P. Cheng. “From Covariance to Causation: A Causal Power Theory”. Psychological Review, 104, pp 367-405. 1997. [5]. M. Buehner and P. Cheng. “Causal Induction: The power PC theory versus the Rescorla-Wagner theory”. In Proceedings of the 19th Annual Conference of the Cognitive Science Society”. 1997. [6]. J.B. Tenenbaum and T.L. Griffiths. “Structure Learning in Human Causal Induction”. Advances in Neural Information Processing Systems 12. MIT Press. 2001. [7]. D. Danks, T.L. Griffiths, J.B. Tenenbaum. “Dynamical Causal Learning”. Advances in Neural Information Processing Systems 14. 2003. [8] A.C. Courville, N.D. Dew, and D.S. Touretsky. “Similarity and discrimination in classical conditioning”. NIPS. 2004. [9]. D. Danks. “Equilibria of the Rescorla-Wagner Model”. Journal of Mathematical Psychology. Vol. 47, pp 109-121. 2003. [10] A.L. Yuille. “The Rescorla-Wagner algorithm and Maximum Likelihood estimation of causal parameters”. NIPS. 2004. [11]. P. Dayan and S. Kakade. “Explaining away in weight space”. In Advances in Neural Information Processing Systems 13. 2001. [12] B. Widrow and M.E. Hoff. “Adapting Switching Circuits”. 1960 IRE WESCON Conv. Record., Part 4, pp 96-104. 1960. [13] A.G. Barto and R.S. Sutton. “Time-derivative Models of Pavlovian Conditioning”. In Learning and Computational Neuroscience: Foundations of Adaptive Networks. M. Gabriel and J. Moore (eds.). pp 497-537. MIT Press. Cambridge, MA. 1990.
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Laplacian Score for Feature Selection Xiaofei He1 Deng Cai2 Partha Niyogi1 1 Department of Computer Science, University of Chicago {xiaofei, niyogi}@cs.uchicago.edu 2 Department of Computer Science, University of Illinois at Urbana-Champaign dengcai2@uiuc.edu Abstract In supervised learning scenarios, feature selection has been studied widely in the literature. Selecting features in unsupervised learning scenarios is a much harder problem, due to the absence of class labels that would guide the search for relevant information. And, almost all of previous unsupervised feature selection methods are “wrapper” techniques that require a learning algorithm to evaluate the candidate feature subsets. In this paper, we propose a “filter” method for feature selection which is independent of any learning algorithm. Our method can be performed in either supervised or unsupervised fashion. The proposed method is based on the observation that, in many real world classification problems, data from the same class are often close to each other. The importance of a feature is evaluated by its power of locality preserving, or, Laplacian Score. We compare our method with data variance (unsupervised) and Fisher score (supervised) on two data sets. Experimental results demonstrate the effectiveness and efficiency of our algorithm. 1 Introduction Feature selection methods can be classified into “wrapper” methods and “filter” methods [4]. The wrapper model techniques evaluate the features using the learning algorithm that will ultimately be employed. Thus, they “wrap” the selection process around the learning algorithm. Most of the feature selection methods are wrapper methods. Algorithms based on the filter model examine intrinsic properties of the data to evaluate the features prior to the learning tasks. The filter based approaches almost always rely on the class labels, most commonly assessing correlations between features and the class label. In this paper, we are particularly interested in the filter methods. Some typical filter methods include data variance, Pearson correlation coefficients, Fisher score, and Kolmogorov-Smirnov test. Most of the existing filter methods are supervised. Data variance might be the simplest unsupervised evaluation of the features. The variance along a dimension reflects its representative power. Data variance can be used as a criteria for feature selection and extraction. For example, Principal Component Analysis (PCA) is a classical feature extraction method which finds a set of mutually orthogonal basis functions that capture the directions of maximum variance in the data. Although the data variance criteria finds features that are useful for representing data, there is no reason to assume that these features must be useful for discriminating between data in different classes. Fisher score seeks features that are efficient for discrimination. It assigns the highest score to the feature on which the data points of different classes are far from each other while requiring data points of the same class to be close to each other. Fisher criterion can be also used for feature extraction, such as Linear Discriminant Analysis (LDA). In this paper, we introduce a novel feature selection algorithm called Laplacian Score (LS). For each feature, its Laplacian score is computed to reflect its locality preserving power. LS is based on the observation that, two data points are probably related to the same topic if they are close to each other. In fact, in many learning problems such as classification, the local structure of the data space is more important than the global structure. In order to model the local geometric structure, we construct a nearest neighbor graph. LS seeks those features that respect this graph structure. 2 Laplacian Score Laplacian Score (LS) is fundamentally based on Laplacian Eigenmaps [1] and Locality Preserving Projection [3]. The basic idea of LS is to evaluate the features according to their locality preserving power. 2.1 The Algorithm Let Lr denote the Laplacian Score of the r-th feature. Let fri denote the i-th sample of the r-th feature, i = 1, · · · , m. Our algorithm can be stated as follows: 1. Construct a nearest neighbor graph G with m nodes. The i-th node corresponds to xi. We put an edge between nodes i and j if xi and xj are ”close”, i.e. xi is among k nearest neighbors of xj or xj is among k nearest neighbors of xi. When the label information is available, one can put an edge between two nodes sharing the same label. 2. If nodes i and j are connected, put Sij = e− ∥xi−xj ∥2 t , where t is a suitable constant. Otherwise, put Sij = 0. The weight matrix S of the graph models the local structure of the data space. 3. For the r-th feature, we define: fr = [fr1, fr2, · · · , frm]T , D = diag(S1), 1 = [1, · · · , 1]T , L = D −S where the matrix L is often called graph Laplacian [2]. Let efr = fr −fT r D1 1T D1 1 4. Compute the Laplacian Score of the r-th feature as follows: Lr = ef T r Lefr ef T r Defr (1) 3 Justification 3.1 Objective Function Recall that given a data set we construct a weighted graph G with edges connecting nearby points to each other. Sij evaluates the similarity between the i-th and j-th nodes. Thus, the importance of a feature can be thought of as the degree it respects the graph structure. To be specific, a ”good” feature should the one on which two data points are close to each other if and only if there is an edge between these two points. A reasonable criterion for choosing a good feature is to minimize the following object function: Lr = P ij(fri −frj)2Sij V ar(fr) (2) where V ar(fr) is the estimated variance of the r-th feature. By minimizing P ij(fri − frj)2Sij, we prefer those features respecting the pre-defined graph structure. For a good feature, the bigger Sij, the smaller (fri −frj), and thus the Laplacian Score tends to be small. Following some simple algebraic steps, we see that X ij (fri −frj)2 Sij = X ij f 2 ri + f 2 rj −2frifrj Sij = 2 X ij f 2 riSij −2 X ij friSijfrj = 2fT r Dfr −2fT r Sfr = 2fT r Lfr By maximizing V ar(fr), we prefer those features with large variance which have more representative power. Recall that the variance of a random variable a can be written as follows: V ar(a) = Z M (a −µ)2dP(a), µ = Z M adP(a) where M is the data manifold, µ is the expected value of a and dP is the probability measure. By spectral graph theory [2], dP can be estimated by the diagonal matrix D on the sample points. Thus, the weighted data variance can be estimated as follows: V ar(fr) = P i(fri −µr)2Dii µr = P i fri Dii P i Dii = 1 ( P i Dii) (P i friDii) = fT r D1 1T D1 To remove the mean from the samples, we define: efr = fr −fT r D1 1T D1 1 Thus, V ar(fr) = X i ef 2 riDii = ef T r Defr Also, it is easy to show that ef T r Lefr = fT r Lfr (please see Proposition 1 in Section 4.2 for detials). We finally get equation (1). It would be important to note that, if we do not remove the mean, the vector fr can be a nonzero constant vector such as 1. It is easy to check that, 1T L1 = 0 and 1T D1 > 0. Thus, Lr = 0. Unfortunately, this feature is clearly of no use since it contains no information. With mean being removed, the new vector efr is orthogonal to 1 with respect to D, i.e. ef T r D1 = 0. Therefore, efr can not be any constant vector other than 0. If efr = 0, ef T r Lefr = ef T r Defr = 0. Thus, the Laplacian Score Lr becomes a trivial solution and the r-th feature is excluded from selection. While computing the weighted variance, the matrix D models the importance (or local density) of the data points. We can also simply replace it by the identity matrix I, in which case the weighted variance becomes the standard variance. To be specific, efr = fr −fT r I1 1T I1 1 = fr −fT r 1 n 1 = fr −µ1 where µ is the mean of fri, i = 1, · · · , n. Thus, V ar(fr) = ef T r Iefr = 1 n (fr −µ1)T (fr −µ1) (3) which is just the standard variance. In fact, the Laplacian scores can be thought of as the Rayleigh quotients for the features with respect to the graph G, please see [2] for details. 3.2 Connection to Fisher Score In this section, we provide a theoretical analysis of the connection between our algorithm and the canonical Fisher score. Given a set of data points with label, {xi, yi}n i=1, yi ∈{1, · · · , c}. Let ni denote the number of data points in class i. Let µi and σ2 i be the mean and variance of class i, i = 1, · · · , c, corresponding to the r-th feature. Let µ and σ2 denote the mean and variance of the whole data set. The Fisher score is defined below: Fr = Pc i=1 ni(µi −µ)2 Pc i=1 niσ2 i (4) In the following, we show that Fisher score is equivalent to Laplacian score with a special graph structure. We define the weight matrix as follows: Sij = 1 nl , yi = yj = l; 0, otherwise. (5) Without loss of generality, we assume that the data points are ordered according to which class they are in, so that {x1, · · · , xn1} are in the first class, {xn1+1, · · · , xn1+n2} are in the second class, etc. Thus, S can be written as follows: S = S1 0 0 0 ... 0 0 0 Sc where Si = 1 ni 11T is an ni × ni matrix. For each Si, the raw (or column) sum is equal to 1, so Di = diag(Si1) is just the identity matrix. Define f1 r = [fr1, · · · , frn1]T , f2 r = [fr,n1+1, · · · , fr,n1+n2]T , etc. We now make the following observations. Observation 1 With the weight matrix S defined in (5), we have ef T r Lefr = fT r Lfr = P i niσ2 i , where L = D −S. To see this, define Li = Di −Si = Ii −Si, where Ii is the ni × ni identity matrix. We have fT r Lfr = c X i=1 (fi r)T Lifi r = c X i=1 (fi r)T (Ii −1 ni 11T )fi r = c X i=1 nicov(fi r, fi r) = c X i=1 niσ2 i Note that, since uT L1 = 1T Lu = 0, ∀u ∈Rn, the value of fT r Lfr remains unchanged by subtracting a constant vector (= α1) from fr. This shows thatef T r Lefr = fT r Lfr = P i niσ2 i . Observation 2 With the weight matrix S defined in (5), we haveef T r Defr = nσ2. To see this, by the definition of S, we have D = I. Thus, this is a immediate result from equation (3). Observation 3 With the weight matrix S defined in (5), we have Pc i=1 ni(µi −µ)2 = ef T r Defr −ef T r Lefr. To see this, notice c X i=1 ni(µi −µ)2 = c X i=1 niµ2 i −2niµiµ + niµ2 = c X i=1 1 ni (niµi)2 −2µ c X i=1 niµi + µ2 c X i=1 ni = c X i=1 1 ni (fi r)T 11T fi r −2nµ2 + nµ2 = c X i=1 fi rSifi r −1 n(nµ)2 = fT r Sfr −fT r ( 1 n11T )fr = fT r (I −S)fr −fT r (I −1 n11T )fr = fT r Lfr −nσ2 = ef T r Lef T r −ef T r Def T r This completes the proof. We therefore get the following relationship between the Laplacian score and Fisher score: Theorem 1 Let Fr denote the Fisher score of the r-th feature. With the weight matrix S defined in (5), we have Lr = 1 1+Fr . Proof From observations 1,2,3, we see that Fr = Pc i=1 ni(µi −µ)2 Pc i=1 niσ2 i = ef T r Def T r −ef T r Lef T r ef T r Lef T r = 1 Lr −1 Thus, Lr = 1 1+Fr . 4 Experimental Results Several experiments were carried out to demonstrate the efficiency and effectiveness of our algorithm. Our algorithm is a unsupervised filter method, while almost all the existing filter methods are supervised. Therefore, we compared our algorithm with data variance which can be performed in unsupervised fashion. 4.1 UCI Iris Data Iris dataset, popularly used for testing clustering and classification algorithms, is taken from UCI ML repository. It contains 3 classes of 50 instances each, where each class refers to a type of Iris plant. Each instance is characterized by four features, i.e. sepal length, sepal width, petal length, and petal width. One class is linearly separable from the other two, but the other two are not linearly separable from each other. Out of the four features it is known that the features F3 (petal length) and F4 (petal width) are more important for the underlying clusters. The class correlation for each feature is 0.7826, -0.4194, 0.9490 and 0.9565. We also used leave-one-out strategy to do classification by using each single feature. We simply used the nearest neighbor classifier. The classification error rates for the four features are 0.41, 0.52, 0.12 and 0.12, respectively. Our analysis indicates that F3 and F4 are better than F1 and F2 in the sense of discrimination. In figure 1, we present a 2-D visualization of the Iris data. We compared three methods, i.e. Variance, Fisher score and Laplacian Score for feature selection. All of them are filter methods which are independent to any learning tasks. However, Fisher score is supervised, while the other two are unsupervised. 40 50 60 70 80 20 25 30 35 40 45 Feature 1 Feature 2 Class 1 Class 2 Class 3 10 20 30 40 50 60 70 0 5 10 15 20 25 Feature 3 Feature 4 Class 1 Class 2 Class 3 Figure 1: 2-D visualization of the Iris data. By using variance, the four features are sorted as F3, F1, F4, F2. Laplacian score (with k ≥15) sorts these four features as F3, F4, F1, F2. Laplacian score (with 3 ≤k < 15) sorts these four features as F4, F3, F1, F2. With a larger k, we see more global structure of the data set. Therefore, the feature F3 is ranked above F4 since the variance of F3 is greater than that of F4. By using Fisher score, the four features are sorted as F3, F4, F1, F2. This indicates that Laplacian score (unsupervised) achieved the same result as Fisher score (supervised). 4.2 Face Clustering on PIE In this section, we apply our feature selection algorithm to face clustering. By using Laplacian score, we select a subset of features which are the most useful for discrimination. Clustering is then performed in such a subspace. 4.2.1 Data Preparation The CMU PIE face database is used in this experiment. It contains 68 subjects with 41,368 face images as a whole. Preprocessing to locate the faces was applied. Original images were normalized (in scale and orientation) such that the two eyes were aligned at the same position. Then, the facial areas were cropped into the final images for matching. The size of each cropped image is 32 × 32 pixels, with 256 grey levels per pixel. Thus, each image is represented by a 1024-dimensional vector. No further preprocessing is done. In this experiment, we fixed the pose and expression. Thus, for each subject, we got 24 images under different lighting conditions. For each given number k, k classes were randomly selected from the face database. This process was repeated 20 times (except for k = 68) and the average performance was computed. For each test (given k classes), two algorithms, i.e. feature selection using variance and Laplacian score are used to select the features. The K-means was then performed in the selected feature subspace. Again, the K-means was repeated 10 times with different initializations and the best result in terms of the objective function of K-means was recorded. 4.2.2 Evaluation Metrics The clustering result is evaluated by comparing the obtained label of each data point with that provided by the data corpus. Two metrics, the accuracy (AC) and the normalized mutual information metric (MI) are used to measure the clustering performance [6]. Given a data point xi, let ri and si be the obtained cluster label and the label provided by the data corpus, respectively. The AC is defined as follows: AC = Pn i=1 δ(si, map(ri)) n (6) where n is the total number of data points and δ(x, y) is the delta function that equals one if x = y and equals zero otherwise, and map(ri) is the permutation mapping function that 0 200 400 600 800 1000 0.4 0.5 0.6 0.7 0.8 0.9 Accuracy Number of features Laplacian Score Variance 0 200 400 600 800 1000 0.4 0.5 0.6 0.7 0.8 0.9 Mutual Information Number of features Laplacian Score Variance (a) 5 classes 0 200 400 600 800 1000 0.4 0.5 0.6 0.7 0.8 Accuracy Number of features Laplacian Score Variance 0 200 400 600 800 1000 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 Mutual Information Number of features Laplacian Score Variance (b) 10 classes 0 200 400 600 800 1000 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Accuracy Number of features Laplacian Score Variance 0 200 400 600 800 1000 0.55 0.6 0.65 0.7 0.75 0.8 0.85 Mutual Information Number of features Laplacian Score Variance (c) 30 classes 0 200 400 600 800 1000 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 Accuracy Number of features Laplacian Score Variance 0 200 400 600 800 1000 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 Mutual Information Number of features Laplacian Score Variance (d) 68 classes Figure 2: Clustering performance versus number of features maps each cluster label ri to the equivalent label from the data corpus. The best mapping can be found by using the Kuhn-Munkres algorithm [5]. Let C denote the set of clusters obtained from the ground truth and C′ obtained from our algorithm. Their mutual information metric MI(C, C′) is defined as follows: MI(C, C′) = X ci∈C,c′ j∈C′ p(ci, c′ j) · log2 p(ci, c′ j) p(ci) · p(c′ j) (7) where p(ci) and p(c′ j) are the probabilities that a data point arbitrarily selected from the corpus belongs to the clusters ci and c′ j, respectively, and p(ci, c′ j) is the joint probability that the arbitrarily selected data point belongs to the clusters ci as well as c′ j at the same time. In our experiments, we use the normalized mutual information MI as follows: MI(C, C′) = MI(C, C′) max(H(C), H(C′)) (8) where H(C) and H(C′) are the entropies of C and C′, respectively. It is easy to check that MI(C, C′) ranges from 0 to 1. MI = 1 if the two sets of clusters are identical, and MI = 0 if the two sets are independent. 4.2.3 Results We compared Laplacian score with data variance for clustering. Note that, we did not compare with Fisher score because it is supervised and the label information is not available in the clustering experiments. Several tests were performed with different numbers of clusters (k=5, 10, 30, 68). In all the tests, the number of nearest neighbors in our algorithm is taken to be 5. The experimental results are shown in Figures 2 and Table 1. As can be seen, in all these cases, our algorithm performs much better than using variance for feature selection. The clustering performance varies with the number of features. The best performance is obtained at very low dimensionality (less than 200). This indicates that feature selection is capable of enhancing clustering performance. In Figure 3, we show the selected features in the image domain for each test (k=5, 10, 30, 68), using our algorithm, data variance and Fisher score. The brightness of the pixels indicates their importance. That is, the more bright the pixel is, the more important. As can be seen, Laplacian score provides better approximation to Fisher score than data variance. Both Laplacian score (a) Variance (b) Laplacian Score (c) Fisher Score Figure 3: Selected features in the image domain, k = 5, 10, 30, 68. The brightness of the pixels indicates their importance. Table 1: Clustering performance comparisons (k is the number of clusters) Accuracy k Feature Number 20 50 100 200 300 500 1024 5 Laplacian Score 0.727 0.806 0.831 0.849 0.837 0.644 0.479 Variance 0.683 0.698 0.602 0.503 0.482 0.464 0.479 10 Laplacian Score 0.685 0.743 0.787 0.772 0.711 0.585 0.403 Variance 0.494 0.500 0.456 0.418 0.392 0.392 0.403 30 Laplacian Score 0.591 0.623 0.671 0.650 0.588 0.485 0.358 Variance 0.399 0.393 0.390 0.365 0.346 0.340 0.358 68 Laplacian Score 0.479 0.554 0.587 0.608 0.553 0.465 0.332 Variance 0.328 0.362 0.334 0.316 0.311 0.312 0.332 Mutual Information k Feature Number 20 50 100 200 300 500 1024 5 Laplacian Score 0.807 0.866 0.861 0.862 0.85 0.652 0.484 Variance 0.662 0.697 0.609 0.526 0.495 0.482 0.484 10 Laplacian Score 0.811 0.849 0.865 0.842 0.796 0.705 0.538 Variance 0.609 0.632 0.6 0.563 0.538 0.529 0.538 30 Laplacian Score 0.807 0.826 0.849 0.831 0.803 0.735 0.624 Variance 0.646 0.649 0.649 0.624 0.611 0.608 0.624 68 Laplacian Score 0.778 0.83 0.833 0.843 0.814 0.76 0.662 Variance 0.639 0.686 0.661 0.651 0.642 0.643 0.662 and Fisher score have the brightest pixels in the area of two eyes, nose, mouth, and face contour. This indicates that even though our algorithm is unsupervised, it can discover the most discriminative features to some extent. 5 Conclusions In this paper, we propose a new filter method for feature selection which is independent to any learning tasks. It can be performed in either supervised or unsupervised fashion. The new algorithm is based on the observation that local geometric structure is crucial for discrimination. Experiments on Iris data set and PIE face data set demonstrate the effectiveness of our algorithm. References [1] M. Belkin and P. Niyogi, “Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering,” Advances in Neural Information Processing Systems, Vol. 14, 2001. [2] Fan R. K. Chung, Spectral Graph Theory, Regional Conference Series in Mathematics, number 92, 1997. [3] X. He and P. Niyogi, “Locality Preserving Projections,” Advances in Neural Information Processing Systems, Vol. 16, 2003. [4] R. Kohavi and G. John, “Wrappers for Feature Subset Selection,” Artificial Intelligence, 97(12):273-324, 1997. [5] L. Lovasz and M. Plummer, Matching Theory, Akad´emiai Kiad´o, North Holland, 1986. [6] W. Xu, X. Liu and Y. Gong, “Document Clustering Based on Non-negative Matrix Factorization ,” ACM SIGIR Conference on Information Retrieval, 2003.
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Soft Clustering on Graphs Kai Yu1, Shipeng Yu2, Volker Tresp1 1Siemens AG, Corporate Technology 2Institute for Computer Science, University of Munich kai.yu@siemens.com, volker.tresp@siemens.com spyu@dbs.informatik.uni-muenchen.de Abstract We propose a simple clustering framework on graphs encoding pairwise data similarities. Unlike usual similarity-based methods, the approach softly assigns data to clusters in a probabilistic way. More importantly, a hierarchical clustering is naturally derived in this framework to gradually merge lower-level clusters into higher-level ones. A random walk analysis indicates that the algorithm exposes clustering structures in various resolutions, i.e., a higher level statistically models a longer-term diffusion on graphs and thus discovers a more global clustering structure. Finally we provide very encouraging experimental results. 1 Introduction Clustering has been widely applied in data analysis to group similar objects. Many algorithms are either similarity-based or model-based. In general, the former (e.g., normalized cut [5]) requires no assumption on data densities but simply a similarity function, and usually partitions data exclusively into clusters. In contrast, model-based methods apply mixture models to fit data distributions and assign data to clusters (i.e. mixture components) probabilistically. This soft clustering is often desired, as it encodes uncertainties on datato-cluster assignments. However, their density assumptions can sometimes be restrictive, e.g. clusters have to be Gaussian-like in Gaussian mixture models (GMMs). In contrast to flat clustering, hierarchical clustering makes intuitive senses by forming a tree of clusters. Despite of its wide applications, the technique is usually achieved by heuristics (e.g., single link) and lacks theoretical backup. Only a few principled algorithms exist so far, where a Gaussian or a sphere-shape assumption is often made [3, 1, 2]. This paper suggests a novel graph-factorization clustering (GFC) framework that employs data’s affinities and meanwhile partitions data probabilistically. A hierarchical clustering algorithm (HGFC) is further derived by merging lower-level clusters into higher-level ones. Analysis based on graph random walks suggests that our clustering method models data affinities as empirical transitions generated by a mixture of latent factors. This view significantly differs from conventional model-based clustering since here the mixture model is not directly for data objects but for their relations. Clusters with arbitrary shapes can be modeled by our method since only pairwise similarities are considered. Interestingly, we prove that the higher-level clusters are associated with longer-term diffusive transitions on the graph, amounting to smoother and more global similarity functions on the data manifold. Therefore, the cluster hierarchy exposes the observed affinity structure gradually in different resolutions, which is somehow similar to the wavelet method that analyzes signals in different bandwidths. To the best of our knowledge, this property has never been considered by other agglomerative hierarchical clustering algorithms (e.g., see [3]). The paper is organized as follows. In the following section we describe a clustering algorithm based on similarity graphs. In Sec. 3 we generalize the algorithm to hierarchical clustering, followed by a discussion from the random walk point of view in Sec. 4. Finally we present the experimental results in Sec. 5 and conclude the paper in Sec. 6. 2 Graph-factorization clustering (GFC) Data similarity relations can be conveniently encoded by a graph, where vertices denote data objects and adjacency weights represent data similarities. This section introduces graph factorization clustering, which is a probabilistic partition of graph vertices. Formally, let G(V, E) be a weighted undirected graph with vertices V = {vi}n i=1 and edges E ⊆{(vi, vj)}. Let W = {wij} be the adjacency matrix, where wij = wji, wij > 0 if (vi, vj) ∈E and wij = 0 otherwise. For instances, wij can be computed by the RBF similarity function based on the features of objects i and j, or by a binary indicator (0 or 1) of the k-nearest neighbor affinity. 2.1 Bipartite graphs Before presenting the main idea, it is necessary to introduce bipartite graphs. Let K(V, U, F) be the bipartite graph (e.g., Fig. 1–(b)), where V = {vi}n i=1 and U = {up}m p=1 are the two disjoint vertex sets and F contains all the edges connecting V and U. Let B = {bip} denote the n × m adjacency matrix with bip ≥0 being the weight for edge [vi, up]. The bipartite graph K induces a similarity between v1 and vj [6] wij = m X p=1 bipbjp λp = BΛ−1B⊤ ij , Λ = diag(λ1, . . . , λm) (1) where λp = Pn i=1 bip denotes the degree of vertex up ∈U. We can interpret Eq. (1) from the perspective of Markov random walks on graphs. wij is essentially a quantity proportional to the stationary probability of direct transitions between vi and vj, denoted by p(vi, vj). Without loss of generality, we normalize W to ensure P ij wij = 1 and wij = p(vi, vj). For a bipartite graph K(V, U, F), there is no direct links between vertices in V, and all the paths from vi to vj must go through vertices in U. This indicates p(vi, vj) = p(vi)p(vj|vi) = di X p p(up|vi)p(vj|up) = X p p(vi, up)p(up, vj) λp , where p(vj|vi) is the conditional transition probability from vi to vj, and di = p(vi) the degree of vi. This directly leads to Eq. (1) with bip = p(vi, up). 2.2 Graph factorization by bipartite graph construction For a bipartite graph K, p(up|vi) = bip/di tells the conditional probability of transitions from vi to up. If the size of U is smaller than that of V, namely m < n, then p(up|vi) indicates how likely data point i belongs to vertex p. This property suggests that one can construct a bipartite graph K(V, U, F) to approximate a given G(V, E), and then obtain a soft clustering structure, where U corresponds to clusters (see Fig. 1–(a) (b)). (a) (b) (c) Figure 1: (a) The original graph representing data affinities; (b) The bipartite graph representing data-to-cluster relations; (c) The induced cluster affinities. Eq. (1) suggests that this approximation can be done by minimizing ℓ(W, BΛ−1B⊤), given a distance ℓ(·, ·) between two adjacency matrices. To make the problem easy to solve, we remove the coupling between B and Λ via H = BΛ−1 and then have min H,Λ ℓ W, HΛH⊤ , s. t. n X i=1 hip = 1, H ∈Rn×m + , Λ ∈Dm×m + , (2) where Dm×m + denotes the set of m × m diagonal matrices with positive diagonal entries. This problem is a symmetric variant of non-negative matrix factorization [4]. In this paper we focus on the divergence distance between matrices. The following theorem suggests an alternating optimization approach to find a local minimum: Theorem 2.1. For divergence distance ℓ(X, Y) = P ij(xij log xij yij −xij + yij), the cost function in Eq. (2) is non-increasing under the update rule ( ˜· denote updated quantities) ˜hip ∝hip X j wij (HΛH⊤)ij λphjp, normalize s.t. X i ˜hip = 1; (3) ˜λp ∝λp X ij wij (HΛH⊤)ij hiphjp, normalize s.t. X p ˜λp = X ij wij. (4) The distance is invariant under the update if and only if H and Λ are at a stationary point. See Appendix for all the proofs in this paper. Similar to GMM, p(up|vi) = bip/ P q biq is the soft probabilistic assignment of vertex vi to cluster up. The method can be seen as a counterpart of mixture models on graphs. The time complexity is O(m2N) with N being the number of nonzero entries in W. This can be very efficient if W is sparse (e.g., for k-nearest neighbor graph the complexity O(m2nk) scales linearly with sample size n). 3 Hierarchical graph-factorization clustering (HGFC) As a nice property of the proposed graph factorization, a natural affinity between two clusters up and uq can be computed as p(up, uq) = n X i=1 bipbiq di = B⊤D−1B pq , D = diag(d1, . . . , dn) (5) This is similar to Eq. (1), but derived from another way of two-hop transitions U →V → U. Note that the similarity between clusters p and q takes into account a weighted average of contributions from all the data (see Fig. 1–(c)). Let G0(V0, E0) be the initial graph describing the similarities of totally m0 = n data points, with adjacency matrix W0. Based on G0 we can build a bipartite graph K1(V0, V1, F1), with m1 < m0 vertices in V1. A hierarchical clustering method can be motivated from the observation that the cluster similarity in Eq. (5) suggests a new adjacency matrix W1 for graph G1(V1, E1), where V1 is formed by clusters, and E1 contains edges connecting these clusters. Then we can group those clusters by constructing another bipartite graph K2(V1, V2, F2) with m2 < m1 vertices in V2, such that W1 is again factorized as in Eq. (2), and a new graph G2(V2, E2) can be built. In principal we can repeat this procedure until we get only one cluster. Algorithm 1 summarizes this algorithm. Algorithm 1 Hierarchical Graph-Factorization Clustering (HGFC) Require: given n data objects and a similarity measure 1: build the similarity graph G0(V0, E0) with adjacency matrix W0, and let m0 = n 2: for l = 1, 2, . . . , do 3: choose ml < ml−1 4: factorize Gl−1 to obtain Kl(Vl−1, Vl, Fl) with the adjacency matrix Bl 5: build a graph Gl(Vl, El) with the adjacency matrix Wl = B⊤ l D−1 l Bl, where Dl’s diagonal entries are obtained by summation over Bl’s columns 6: end for The algorithm ends up with a hierarchical clustering structure. For level l, we can assign data to the obtained ml clusters via a propagation from the bottom level of clusters. Based on the chain rule of Markov random walks, the soft (i.e., probabilistic) assignment of vi ∈ V0 to cluster v(l) p ∈Vl is given by p v(l) p |vi = X v(l−1)∈Vl−1 · · · X v(1)∈V1 p v(l) p |v(l−1) · · · p v(1)|vi = D−1 1 ¯Bl ip , (6) where ¯Bl = B1D−1 2 B2D−1 3 B3 . . . D−1 l Bl. One can interpret this by deriving an equivalent bipartite graph ¯Kl(V0, Vl, ¯Fl), and treating ¯Bl as the equivalent adjacency matrix attached to the equivalent edges ¯Fl connecting data V0 and clusters Vl. 4 Analysis of the proposed algorithms 4.1 Flat clustering: statistical modeling of single-hop transitions In this section we provide some insights to the suggested clustering algorithm, mainly from the perspective of random walks on graphs. Suppose that from a stationary stage of random walks on G(V, E), one observes πij single-hop transitions between vi and vj in a unitary time frame. As an intuition of graph-based view to similarities, if two data points are similar or related, the transitions between them are likely to happen. Thus we connect the observed similarities to the frequency of transitions via wij ∝πij. If the observed transitions are i.i.d. sampled from a true distribution p(vi, vj) = (HΛH⊤)ij where a bipartite graph is behind, then the log likelihood with respect to the observed transitions is L(H, Λ) = log Y ij p(vi, vj)πij ∝ X ij wij log(HΛH⊤)ij. (7) Then we have the following conclusion Proposition 4.1. For a weighted undirected graph G(V, E) and the log likelihood defined in Eq. (7), the following results hold: (i) Minimizing the divergence distance l(W, HΛH⊤) is equivalent to maximizing the log likelihood L(H, Λ); (ii) Updates Eq. (3) and Eq. (4) correspond to a standard EM algorithm for maximizing L(H, Λ). Figure 2: The similarities of vertices to a fixed vertex (marked in the left panel) on a 6nearest-neighbor graph, respectively induced by clustering level l = 2 (the middle panel) and l = 6 (the right panel). A darker color means a higher similarity. 4.2 Hierarchical clustering: statistical modeling of multi-hop transitions The adjacency matrix W0 of G0(V0, E0) only models one-hop transitions that follow direct links from vertices to their neighbors. However, the random walk is a process of diffusion on the graph. Within a relatively longer period, a walker starting from a vertex has the chance to reach vertices faraway through multi-hop transitions. Obviously, multihop transitions induce a slowly decaying similarity function on the graph. Based on the chain rule of Markov process, the equivalent adjacency matrix for t-hop transitions is At = W0(D−1 0 W0)t−1 = At−1D−1 0 W0. (8) Generally speaking, a slowly decaying similarity function on the similarity graph captures a global affinity structure of data manifolds, while a rapidly decaying similarity function only tells the local affinity structure. The following proposition states that in the suggested HGFC, a higher-level clustering implicitly employs a more global similarity measure caused by multi-hop Markov random walks: Proposition 4.2. For a given hierarchical clustering structure that starts from a bottom graph G0(V0, E0) to a higher level Gk(Vk, Ek), the vertices Vl at level 0 < l ≤k induces an equivalent adjacency matrix of V0, which is At with t = 2l−1 as defined in Eq. (8). Therefore the presented hierarchical clustering algorithm HGFC applies different sizes of time windows to examine random walks, and derives different scales of similarity measures to expose the local and global clustering structures of data manifolds. Fig. 2 illustrates the employed similarities of vertices to a fixed vertex in clustering levels l = 2 and 6, which corresponds to time periods t = 2 and 32. It can be seen that for a short period t = 2, the similarity is very local and helps to uncover low-level clusters, while in a longer period t = 32 the similarity function is rather global. 5 Empirical study We apply HGFC on USPS handwritten digits and Newsgroup text data. For USPS data we use the images of digits 1, 2, 3 and 4, with respectively 1269, 929, 824 and 852 images per class. Each image is represented as a 256-dimension vector. The text data contain totally 3970 documents covering 4 categories, autos, motorcycles, baseball, and hockey. Each document is represented by an 8014-dimension TFIDF feature vector. Our method employs a 10-nearest-neighbor graph, with the similarity measure RBF for USPS and cosine for Newsgroup. We perform 4-level HGFC, and set the cluster number, respectively from bottom to top, to be 100, 20, 10 and 4 for both data sets. We compare HGFC with two popular agglomerative hierarchical clustering algorithms, single link and complete link (e.g., [3]). Both methods merge two closest clusters at each step. Figure 3: Visualization of HGFC for USPS data set. Left: mean images of the top 3 clustering levels, along with a Hinton graph representing the soft (probabilistic) assignments of randomly chosen 10 digits (shown on the left) to the top 3rd level clusters; Middle: a Hinton graph showing the soft cluster assignments from top 3rd level to top 2nd level; Right: a Hinton graph showing the soft assignments from top 2nd level to top 1st level. Figure 4: Comparison of clustering methods on USPS (left) and Newsgroup (right), evaluated by normalized mutual information (NMI). Higher values indicate better qualities. Single link defines the cluster distance to be the smallest point-wise distance between two clusters, while complete link uses the largest one. A third compared method is normalized cut [5], which partitions data into two clusters. We apply the algorithm recursively to produce a top-down hierarchy of 2, 4, 8, 16, 32 and 64 clusters. We also compare with the k-means algorithm, k = 4, 10, 20 and 100. Before showing the comparison, we visualize a part of clustering results for USPS data in Fig. 3. On top of the left figure, we show the top three levels of the hierarchy with respectively 4, 10 and 20 clusters, where each cluster is represented by its mean image via an average over all the images weighted by their posterior probabilities of belonging to this cluster. Then 10 randomly sampled digits with soft cluster assignments to the top 3rd level clusters are illustrated with a Hinton graph. The middle and right figures in Fig. 3 show the assignments between clusters across the hierarchy. The clear diagonal block structure in all the Hinton graphs indicates a very meaningful cluster hierarchy. Normalized cut HGFC K-means “1” 635 630 1 3 1254 3 8 4 1265 1 0 3 “2” 2 4 744 179 1 886 33 9 17 720 95 97 “3” 2 1 817 4 1 4 816 3 10 9 796 9 “4” 10 6 1 835 4 8 2 838 58 20 0 774 Table 1: Confusion matrices of clustering results, 4 clusters, USPS data. In each confusion matrix, rows correspond true classes and columns correspond the found clusters. Normalized cut HGFC K-means autos 858 98 30 2 772 182 13 21 977 7 4 0 motor. 79 893 16 5 42 934 5 12 985 3 5 0 baseball 44 33 875 40 15 33 843 101 39 835 114 4 hockey 11 8 893 85 7 21 11 958 16 4 900 77 Table 2: Confusion matrices of clustering results, 4 clusters, Newsgroup data. In each confusion matrix, rows correspond true classes and columns correspond the found clusters. We compare the clustering methods by evaluating the normalized mutual information (NMI) in Fig. 4. It is defined to be the mutual information between clusters and true classes, normalized by the maximum of marginal entropies. Moreover, in order to more directly assess the clustering quality, we also illustrate the confusion matrices in Table 1 and Table 2, in the case of producing 4 clusters. We drop out the confusion matrices of single link and complete link in the tables, for saving spaces and also due to their clearly poor performance compared with others. The results show that single link performs poorly, as it greedily merges nearby data and tends to form a big cluster with some outliers. Complete link is more balanced but unsatisfactory either. For the Newsgroup data it even gets stuck at the 3601-th merge because all the similarities between clusters are 0. Top-down hierarchical normalized cut obtains reasonable results, but sometimes cannot split one big cluster (see the tables). The confusion matrices indicates that k-means does well for digit images but relatively worse for high-dimension textual data. In contrast, Fig. 4 shows that HGFC gives significantly higher NMI values than competitors on both tasks. It also produces confusion matrices with clear diagonal structures (see tables 1 and 2), which indicates a very good clustering quality. 6 Conclusion and Future Work In this paper we have proposed a probabilistic graph partition method for clustering data objects based on their pairwise similarities. A novel hierarchical clustering algorithm HGFC has been derived, where a higher level in HGFC corresponds to a statistical model of random walk transitions in a longer period, giving rise to a more global clustering structure. Experiments show very encouraging results. In this paper we have empirically specified the number of clusters in each level. In the near future we plan to investigate effective methods to automatically determine it. Another direction is hierarchical clustering on directed graphs, as well as its applications in web mining. Appendix Proof of Theorem 2.1. We first notice that P p λp = P ij wij under constraints P i hip = 1. Therefore we can normalize W by P ij wij and after convergence multiply all λp by this quantity to get the solution. Under this assumption we are maximizing L(H, Λ) = P ij wij log(HΛH⊤)ij with an extra constraint P p λp = 1. We first fix λp and show update Eq. (3) will not decrease L(H) ≡ L(H, Λ). We prove this by constructing an auxiliary function f(H, H∗) such that f(H, H∗) ≤ L(H) and f(H, H) = L(H). Then we know the update Ht+1 = arg maxH f(H, Ht) will not decrease L(H) since L(Ht+1) ≥f(Ht+1, Ht) ≥f(Ht, Ht) = L(Ht). Define f(H, H∗) = P ij wij P p h∗ ipλph∗ jp P l h∗ ilλlh∗ jl log hipλphjp −log h∗ ipλph∗ jp P l h∗ ilλlh∗ jl . f(H, H) = L(H) can be easily verified, and f(H, H∗) ≤L(H) also follows if we use concavity of log function. Then it is straightforward to verify Eq. (3) by setting the derivative of f with respect to hip to be zero. The normalization is due to the constraints and can be formally derived from this procedure with a Lagrange formalism. Similarly we can define an auxiliary function for Λ with H fixed, and verify Eq. (4). Proof of Proposition 4.1. (i) follows directly from the proof of Theorem 2.1. To prove (ii) we take up as the missing data and follow the standard way to derive the EM algorithm. In the E-step we estimate the a posteriori probability of taking up for pair (vi, vj) using Bayes’ rule: ˆp(up|vi, vj) ∝ p(vi|up)p(vj|up)p(up). And then in the M-step we maximize the “complete” data likelihood ˆL(G) = P ij wij P p ˆp(up|vi, vj) log p(vi|up)p(vj|up)p(up) with respect to model parameters hip = p(vi|up) and λp = p(up), with constraints P i hip = 1 and P p λp = 1. By setting the corresponding derivatives to zero we obtain hip ∝P j wij ˆp(up|vi, vj) and λp ∝P ij wij ˆp(up|vi, vj). It is easy to check that they are equivalent to updates Eq. (3) and Eq. (4) respectively. Proof of Proposition 4.2. We give a brief proof. Suppose that at level l the data-cluster relationship is described by ¯Kl(V0, Vl, ¯Fl) (see Eq. (6)) with adjacency matrix ¯Bl, degrees D0 for V0, and degrees Λl for Vl. In this case the induced adjacency matrix of V0 is ¯ Wl = ¯BlΛ−1 l ¯B ⊤ l , and the adjacency matrix of Vl is Wl = ¯B ⊤ l D−1 0 ¯Bl. Let Kl(Vl, Vl+1, Fl+1) be the bipartite graph connecting Vl and Vl+1, with the adjacency Bl+1 and degrees Λl+1 for Vl+1. Then the adjacency matrix of V0 induced by level l + 1 is ¯ Wl+1 = ¯BlΛ−1 l Bl+1Λ−1 l+1B⊤ l+1Λ−1 l ¯B ⊤ l = ¯ WlD−1 0 ¯ Wl, where relations Bl+1Λ−1 l+1B⊤ l+1 = ¯B ⊤ l D−1 0 ¯Bl and ¯ Wl = BlΛ−1 l B⊤ l are applied. Given the initial condition from the bottom level ¯ W1 = W0, it is not difficult to obtain ¯ Wl = At with t = 2l−1. References [1] J. Goldberger and S. Roweis. Hierarchical clustering of a mixture model. In L.K. Saul, Y. Weiss, and L. Bottou, editors, Neural Information Processing Systems 17 (NIPS*04), pages 505–512, 2005. [2] K.A. Heller and Z. Ghahramani. Bayesian hierarchical clustering. In Proceedings of the 22nd International Conference on Machine Learning, pages 297–304, 2005. [3] S. D. Kamvar, D. Klein, and C. D. Manning. Interpreting and extending classical agglomerative clustering algorithms using a model-based approach. In Proceedings of the 19th International Conference on Machine Learning, pages 283–290, 2002. [4] Daniel D. Lee and H. Sebastian Seung. Algorithms for non-negative matrix factorization. In T. K. Leen, T. G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems 13 (NIPS*00), pages 556–562, 2001. [5] Jianbo Shi and Jitendra Malik. Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(8):888–905, 2000. [6] D. Zhou, B. Sch¨olkopf, and T. Hofmann. Semi-supervised learning on directed graphs. In L.K. Saul, Y. Weiss, and L. 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Variational Bayesian Stochastic Complexity of Mixture Models Kazuho Watanabe∗ Department of Computational Intelligence and Systems Science Tokyo Institute of Technology Mail Box:R2-5, 4259 Nagatsuta, Midori-ku, Yokohama, 226-8503, Japan kazuho23@pi.titech.ac.jp Sumio Watanabe P& I Lab. Tokyo Institute of Technology swatanab@pi.titech.ac.jp Abstract The Variational Bayesian framework has been widely used to approximate the Bayesian learning. In various applications, it has provided computational tractability and good generalization performance. In this paper, we discuss the Variational Bayesian learning of the mixture of exponential families and provide some additional theoretical support by deriving the asymptotic form of the stochastic complexity. The stochastic complexity, which corresponds to the minimum free energy and a lower bound of the marginal likelihood, is a key quantity for model selection. It also enables us to discuss the effect of hyperparameters and the accuracy of the Variational Bayesian approach as an approximation of the true Bayesian learning. 1 Introduction The Variational Bayesian (VB) framework has been widely used as an approximation of the Bayesian learning for models involving hidden (latent) variables such as mixture models[2][4]. This framework provides computationally tractable posterior distributions with only modest computational costs in contrast to Markov chain Monte Carlo (MCMC) methods. In many applications, it has performed better generalization compared to the maximum likelihood estimation. In spite of its tractability and its wide range of applications, little has been done to investigate the theoretical properties of the Variational Bayesian learning itself. For example, questions like how accurately it approximates the true one remained unanswered until quite recently. To address these issues, the stochastic complexity in the Variational Bayesian learning of gaussian mixture models was clarified and the accuracy of the Variational Bayesian learning was discussed[10]. ∗This work was supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for JSPS Fellows 4637 and for Scientific Research 15500130, 2005. In this paper, we focus on the Variational Bayesian learning of more general mixture models, namely the mixtures of exponential families which include mixtures of distributions such as gaussian, binomial and gamma. Mixture models are known to be non-regular statistical models due to the non-identifiability of parameters caused by their hidden variables[7]. In some recent studies, the Bayesian stochastic complexities of non-regular models have been clarified and it has been proven that they become smaller than those of regular models[12][13]. This indicates an advantage of the Bayesian learning when it is applied to non-regular models. As our main results, the asymptotic upper and lower bounds are obtained for the stochastic complexity or the free energy in the Variational Bayesian learning of the mixture of exponential families. The stochastic complexity is important quantity for model selection and giving the asymptotic form of it also contributes to the following two issues. One is the accuracy of the Variational Bayesian learning as an approximation method since the stochastic complexity shows the distance from the variational posterior distribution to the true Bayesian posterior distribution in the sense of Kullback information. Indeed, we give the asymptotic form of the stochastic complexity as F(n) ≃λ log n where n is the sample size, by comparing the coefficient λ with that of the true Bayesian learning, we discuss the accuracy of the VB approach. Another is the influence of the hyperparameter on the learning process. Since the Variational Bayesian algorithm is a procedure of minimizing the functional that finally gives the stochastic complexity, the derived bounds indicate how the hyperparameters influence the process of the learning. Our results have an implication for how to determine the hyperparameter values before the learning process. We consider the case in which the true distribution is contained in the learner model. Analyzing the stochastic complexity in this case is most valuable for comparing the Variational Bayesian learning with the true Bayesian learning. This is because the advantage of the Bayesian learning is typical in this case[12]. Furthermore, this analysis is necessary and essential for addressing the model selection problem and hypothesis testing. The paper is organized as follows. In Section 2, we introduce the mixture of exponential family model. In Section 3, we describe the Bayesian learning. In Section 4, the Variational Bayesian framework is described and the variational posterior distribution for the mixture of exponential family model is derived. In Section 5, we present our main result. Discussion and conclusion follow in Section 6. 2 Mixture of Exponential Family Denote by c(x|b) a density function of the input x ∈RN given an M-dimensional parameter vector b = (b(1), b(2), · · ·, b(M))T ∈B where B is a subset of RM. The general mixture model p(x|θ) with a parameter vector θ is defined by p(x|θ) = K k=1 akc(x|bk), where integer K is the number of components and {ak|ak ≥0, K k=1 ak = 1} is the set of mixing proportions. The model parameter θ is {ak, bk}K k=1. A mixture model is called a mixture of exponential family (MEF) model or exponential family mixture model if the probability distribution c(x|b) for each component is given by the following form, c(x|b) = exp{b · f(x) + f0(x) −g(b)}, (1) where b ∈B is called the natural parameter, b · f(x) is its inner product with the vector f(x) = (f1(x), · · ·, fM(x))T , f0(x) and g(b) are real-valued functions of the input x and the parameter b, respectively[3]. Suppose functions f1, · · ·, fM and a constant function are linearly independent, which means the effective number of parameters in a single component distribution c(x|b) is M. The conjugate prior distribution ϕ(θ) for the MEF model is given by the product of the following two distributions on a = {ak}K k=1 and b = {bk}K k=1, ϕ(a) = Γ(Kφ0) Γ(φ0)K K k=1 aφ0−1 k , (2) ϕ(b) = K k=1 ϕ(bk) = K k=1 exp{ξ0(bk · ν0 −g(bk))} C(ξ0, ν0) , (3) where ξ0 > 0, ν0 ∈RM and φ0 > 0 are constants called hyperparameters and C(ξ, µ) = exp{ξ(µ · b −g(b))}db (4) is a function of ξ ∈R and µ ∈RM. The mixture model can be rewritten as follows by using a hidden variable y = (y1, · · ·, yK) ∈{(1, 0, · · ·, 0), (0, 1, · · ·, 0), · · ·, (0, 0, · · ·, 1)}, p(x, y|θ) = K k=1 akc(x|bk) yk . If and only if the datum x is generated from the kth component, yk = 1. 3 The Bayesian Learning Suppose n training samples Xn = {x1, · · ·, xn} are independently and identically taken from the true distribution p0(x). In the Bayesian learning of a model p(x|θ) whose parameter is θ, first, the prior distribution ϕ(θ) on the parameter θ is set. Then the posterior distribution p(θ|Xn) is computed from the given dataset and the prior by p(θ|Xn) = 1 Z(Xn) exp(−nHn(θ))ϕ(θ), (5) where Hn(θ) is the empirical Kullback information, Hn(θ) = 1 n n i=1 log p0(xi) p(xi|θ), (6) and Z(Xn) is the normalization constant that is also known as the marginal likelihood or the evidence of the dataset Xn[6]. The Bayesian predictive distribution p(x|Xn) is given by averaging the model over the posterior distribution as follows, p(x|Xn) = p(x|θ)p(θ|Xn)dθ. (7) The stochastic complexity F (Xn) is defined by F (Xn) = −log Z(Xn), (8) which is also called the free energy and is important in most data modelling problems. Practically, it is used as a criterion by which the model is selected and the hyperparameters in the prior are optimized[1][9]. Define the average stochastic complexity F (n) by F (n) = EXn F (Xn) , (9) where EXn[·] denotes the expectation value over all sets of training samples. Recently, it was proved that F (n) has the following asymptotic form[12], F (n) ≃λ log n −(m −1) log log n + O(1), (10) where λ and m are the rational number and the natural number respectively which are determined by the singularities of the set of true parameters. In regular statistical models, 2λ is equal to the number of parameters and m = 1, whereas in non-regular models such as mixture models, 2λ is not larger than the number of parameters and m ≥1. This means an advantage of the Bayesian learning. However, in the Bayesian learning, one computes the stochastic complexity or the predictive distribution by integrating over the posterior distribution, which typically cannot be performed analytically. As an approximation, the VB framework was proposed[2][4]. 4 The Variational Bayesian Learning 4.1 The Variational Bayesian Framework In the VB framework, the Bayesian posterior p(Y n, θ|Xn) of the hidden variables and the parameters is approximated by the variational posterior q(Y n, θ|Xn), which factorizes as q(Y n, θ|Xn) = Q(Y n|Xn)r(θ|Xn), (11) where Q(Y n|Xn) and r(θ|Xn) are posteriors on the hidden variables and the parameters respectively. The variational posterior q(Y n, θ|Xn) is chosen to minimize the functional F [q] defined by F[q] = Y n q(Y n, θ|Xn) log q(Y n, θ|Xn)p0(Xn) p(Xn, Y n, θ) dθ, (12) = F (Xn) + K(q(Y n, θ|Xn)||p(Y n, θ|Xn)), (13) where K(q(Y n, θ|Xn)||p(Y n, θ|Xn)) is the Kullback information between the true Bayesian posterior p(Y n, θ|Xn) and the variational posterior q(Y n, θ|Xn) 1. This leads to the following theorem. The proof is well known[8]. Theorem 1 If the functional F[q] is minimized under the constraint (11) then the variational posteriors, r(θ|Xn) and Q(Y n|Xn), satisfy r(θ|Xn) = 1 Cr ϕ(θ) exp log p(Xn, Y n|θ) Q(Y n|Xn), (14) Q(Y n|Xn) = 1 CQ exp log p(Xn, Y n|θ) r(θ|Xn), (15) 1K(q(x)||p(x)) denotes the Kullback information from a distribution q(x) to a distribution p(x), that is, K(q(x)||p(x)) = q(x) log q(x) p(x)dx. where Cr and CQ are the normalization constants2. We define the stochastic complexity in the VB learning F(Xn) by the minimum value of the functional F[q] , that is , F (Xn) = min r,Q F[q], which shows the accuracy of the VB approach as an approximation of the Bayesian learning. F(Xn) is also used for model selection since it gives an upper bound of the true Bayesian stochastic complexity F (Xn). 4.2 Variational Posterior for MEF Model In this subsection, we derive the variational posterior r(θ|Xn) for the MEF model based on (14) and then define the variational parameter for this model. Using the complete data {Xn, Y n} = {(x1, y1), · · ·, (xn, yn)}, we put yk i = ⟨yk i ⟩Q(Y n), nk = n i=1 yk i , and νk = 1 nk n i=1 yk i f(xi), where yk i = 1 if and only if the ith datum xi is from the kth component. The variable nk is the expected number of the data that are estimated to be from the kth component. From (14) and the respective prior (2) and (3), the variational posterior r(θ) is obtained as the product of the following two distributions3, r(a) = Γ(n + Kφ0) K k=1 Γ(nk + φ0) K k=1 ank+φ0−1 k , (16) r(b) = K k=1 r(bk) = K k=1 1 C(γk, µk) exp{γk(µk · bk −g(bk))}, (17) where µk = nkνk+ξ0ν0 nk+ξ0 and γk = nk + ξ0. Let ak = ⟨ak⟩r(a) = nk + φ0 n + Kφ0 , (18) bk = ⟨bk⟩r(bk) = 1 γk ∂log C(γk, µk) ∂µk , (19) and define the variational parameter θ by θ = ⟨θ⟩r(θ) = {ak, bk}K k=1. Then it is noted that the variational posterior r(θ) and CQ in (15) are parameterized by the variational parameter θ. Therefore, we denote them as r(θ|θ) and CQ(θ) henceforth. We define the variational estimator θvb by the variational parameter θ that attains the minimum value of the stochastic complexity F(Xn). Then, putting (15) into (12), we obtain F (Xn) = min θ {K(r(θ|θ)||ϕ(θ)) −(log CQ(θ) + S(Xn))}, (20) = K(r(θ|θvb)||ϕ(θ)) −(log CQ(θvb) + S(Xn)), (21) where S(Xn) = −n i=1 log p0(x). Therefore, our aim is to evaluate the minimum value of (20) as a function of the variational parameter θ. 2⟨·⟩p(x) denotes the expectation over p(x). 3Hereafter, we omit the condition Xn of the variational posteriors, and abbreviate them to q(Y n, θ), Q(Y n) and r(θ). 5 Main Result The average stochastic complexity F (n) in the VB learning is defined by F (n) = EXn[F(Xn)]. (22) We assume the following conditions. (i) The true distribution p0(x) is an MEF model p(x|θ0) which has K0 components and the parameter θ0 = {a∗ k, b∗ k}K0 k=1, p(x|θ0) = K0 k=1 a∗ k exp{b∗ k · f(x) + f0(x) −g(b∗ k)}, where b∗ k ∈RM and b∗ k ̸= b∗ j(k ̸= j). And suppose that the model p(x|θ) has K components, p(x|θ) = K k=1 ak exp{bk · f(x) + f0(x) −g(bk)}, and K ≥K0 holds. (ii) The prior distribution of the parameters is ϕ(θ) = ϕ(a)ϕ(b) given by (2) and (3) with ϕ(b) bounded. (iii) Regarding the distribution c(x|b) of each component, the Fisher information matrix I(b) = ∂2g(b) ∂b∂b satisfies 0 < |I(b)| < +∞, for arbitrary b ∈B 4. The function µ · b −g(b) has a stationary point at ˆb in the interior of B for each µ ∈{ ∂g(b) ∂b |b ∈B}. Under these conditions, we prove the following. Theorem 2 (Main Result) Assume the conditions (i),(ii) and (iii). Then the average stochastic complexity F(n) defined by (22) satisfies λ log n + EXn nHn(θvb) + C1 ≤F(n) ≤λ log n + C2, (23) for an arbitrary natural number n, where C1, C2 are constants independent of n and λ = (K −1)φ0 + M 2 , MK+K−1 2 , λ = (K −K0)φ0 + MK0+K0−1 2 (φ0 ≤M+1 2 ), MK+K−1 2 (φ0 > M+1 2 ). (24) This theorem shows the asymptotic form of the average stochastic complexity in the Variational Bayesian learning. The coefficients λ, λ of the leading terms are identified by K,K0, that are the numbers of components of the learner and the true distribution, the number of parameters M of each component and the hyperparameter φ0 of the conjugate prior given by (2). In this theorem, nHn(θvb) = −n i=1 log p(xi|θvb)−S(Xn), and −n i=1 log p(xi|θvb) is a training error which is computable during the learning. If the term EXn nHn(θvb) is a bounded function of n, then it immediately follows from this theorem that λ log n + O(1) ≤F 0(n) ≤λ log n + O(1), 4 ∂2g(b) ∂b∂b denotes the matrix whose ijth entry is ∂2g(b) ∂b(i)∂b(j) and |·| denotes the determinant of a matrix. where O(1) is a bounded function of n. In certain cases, such as binomial mixtures and mixtures of von-Mises distributions, it is actually a bounded function of n. In the case of gaussian mixtures, if B = RN, it is conjectured that the minus likelihood ratio minθ nHn(θ), a lower bound of nHn(θvb), is at most of the order of log log n[5]. Since the dimension of the parameter θ is MK + K −1, the average stochastic complexity of regular statistical models, which coincides with the Bayesian information criterion (BIC)[9] is given by λBIC log n where λBIC = MK+K−1 2 . Theorem 2 claims that the coefficient λ of log n is smaller than λBIC when φ0 ≤(M + 1)/2. This implies that the advantage of non-regular models in the Bayesian learning still remains in the VB learning. (Outline of the proof of Theorem 2) From the condition (iii), calculating C(γk, µk) in (17) by the saddle point approximation, K(r(θ|θ)||ϕ(θ)) in (20) is evaluated as follows 5, K(r(θ|θ)||ϕ(θ)) = G(a) − K k=1 log ϕ(bk) + Op(1), (25) where the function G(a) of a = {ak}K k=1 is given by G(a) = MK + K −1 2 log n + {M 2 −(φ0 −1 2)} K k=1 log ak. (26) Then log CQ(θ) in (20) is evaluated as follows. nHn(θ) + Op(1) ≤−(log CQ(θ) + S(Xn)) ≤nHn(θ) + Op(1) (27) where Hn(θ) = 1 n n i=1 log p(xi|θ0) K k=1 akc(xi|¯bk) exp
− C′ nk+min{φ0,ξ0} , and C′ is a constant. Thus, from (20), evaluating the right-hand sides of (25) and (27) at specific points near the true parameter θ0, we obtain the upper bound in (23). The lower bound in (23) is obtained from (25) and (27) by Jensen’s inequality and the constraint K k=1 ak = 1. (Q.E.D) 6 Discussion and Conclusion In this paper, we showed the upper and lower bounds of the stochastic complexity for the mixture of exponential family models in the VB learning. Firstly, we compare the stochastic complexity shown in Theorem 2 with the one in the true Bayesian learning. On the mixture models with M parameters in each component, the following upper bound for the coefficient of F (n) in (10) is known [13], λ ≤ (K + K0 −1)/2 (M = 1), (K −K0) + (MK0 + K0 −1)/2 (M ≥2). (28) By the certain conditions about the prior distribution under which the above bound was derived, we can compare the stochastic complexity when φ0 = 1. Putting φ0 = 1 in (24), we have λ = K −K0 + (MK0 + K0 −1)/2. (29) 5Op(1) denotes a random variable bounded in probability. Since we obtain F (n) ≃λ log n+O(1) under certain assumptions[11], let us compare λ of the VB learning to λ in (28) of the true Bayesian learning. When M = 1, that is, each component has one parameter, λ ≥λ holds since K0 ≤K. This means that the more redundant components the model has, the more the VB learning differs from the true Bayesian learning. In this case, 2λ is equal to the number of the parameters of the model. Hence the BIC[9] corresponds to λ log n when M = 1. If M ≥2, the upper bound of λ is equal to λ. This implies that the variational posterior is close to the true Bayesian posterior when M ≥2. More precise discussion about the accuracy of the approximation can be done for models on which tighter bounds or exact values of the coefficient λ in (10) are given[10]. Secondly, we point out that Theorem 2 shows how the hyperparameter φ0 influence the process of the VB learning. The coefficient λ in (24) indicates that only when φ0 ≤(M + 1)/2, the prior distribution (2) works to eliminate the redundant components that the model has and otherwise it works to use all the components. And lastly, let us give examples of how to use the theoretical bounds in (23). One can examine experimentally whether the actual iterative algorithm converges to the optimal variational posterior instead of local minima by comparing the stochastic complexity with our theoretical result. The theoretical bounds would also enable us to compare the accuracy of the VB learning with that of the Laplace approximation or the MCMC method. As mentioned in Section 4, our result will be important for developing effective model selection methods using F(Xn) in the future work. References [1] H.Akaike, “Likelihood and Bayes procedure,” Bayesian Statistics, (Bernald J.M. eds.) University Press, Valencia, Spain, pp.143-166, 1980. [2] H.Attias, ”Inferring parameters and structure of latent variable models by variational bayes,” Proc. of UAI, 1999. [3] L.D.Brown, “Fundamentals of statistical exponential families,” IMS Lecture NotesMonograph Series, 1986. [4] Z.Ghahramani, M.J.Beal, “Graphical models and variational methods,” Advanced Mean Field Methods , MIT Press, 2000. [5] J.A.Hartigan, “A Failure of likelihood asymptotics for normal mixtures,” Proc. of the Berkeley Conference in Honor of J.Neyman and J.Kiefer, Vol.2, 807-810, 1985. [6] D.J. Mackay, “Bayesian interpolation,” Neural Computation, 4(2), pp.415-447, 1992. [7] G.McLachlan, D.Peel,”Finite mixture models,” Wiley, 2000. [8] M.Sato, “Online model selection based on the variational bayes,” Neural Computation, 13(7), pp.1649-1681, 2001. [9] G.Schwarz, “Estimating the dimension of a model,” Annals of Statistics, 6(2), pp.461464, 1978. [10] K.Watanabe, S.Watanabe, ”Lower bounds of stochastic complexities in variational bayes learning of gaussian mixture models,” Proc. of IEEE CIS04, pp.99-104, 2004. [11] K.Watanabe, S.Watanabe, ”Stochastic complexity for mixture of exponential families in variational bayes,” Proc. of ALT05, pp.107-121, 2005. [12] S.Watanabe,“Algebraic analysis for non-identifiable learning machines,” Neural Computation, 13(4), pp.899-933, 2001. [13] K.Yamazaki, S.Watanabe, ”Singularities in mixture models and upper bounds of stochastic complexity,” Neural Networks, 16, pp.1029-1038, 2003.
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Hierarchical Linear/Constant Time SLAM Using Particle Filters for Dense Maps Austin I. Eliazar Ronald Parr Department of Computer Science Duke University Durham, NC 27708 {eliazar,parr}@cs.duke.edu Abstract We present an improvement to the DP-SLAM algorithm for simultaneous localization and mapping (SLAM) that maintains multiple hypotheses about densely populated maps (one full map per particle in a particle filter) in time that is linear in all significant algorithm parameters and takes constant (amortized) time per iteration. This means that the asymptotic complexity of the algorithm is no greater than that of a pure localization algorithm using a single map and the same number of particles. We also present a hierarchical extension of DP-SLAM that uses a two level particle filter which models drift in the particle filtering process itself. The hierarchical approach enables recovery from the inevitable drift that results from using a finite number of particles in a particle filter and permits the use of DP-SLAM in more challenging domains, while maintaining linear time asymptotic complexity. 1 Introduction The ability to construct and use a map of the environment is a critical enabling technology for many important applications, such as search and rescue or extraterrestrial exploration. Probabilistic approaches have proved successful at addressing the basic problem of localization using particle filters [6]. Expectation Maximization (EM) has been used successfully to address the problem of mapping [1] and Kalman filters [2, 10] have shown promise on the combined problem of simultaneous localization and mapping (SLAM). SLAM algorithms ought to produce accurate maps with bounded resource consumption per sensor sweep. To the extent that it is possible, it is desirable to avoid explicit map correcting actions, which are computationally intensive and would be symptomatic of accumulating error in the map. One family of approaches to SLAM assumes relatively sparse, relatively unambiguous landmarks and builds a Kalman filter over landmark positions [2, 9, 10] . Other approaches assume dense sensor data which individually are not very distinctive, such as those available from a laser range finder [7, 8]. An advantage of the latter group is that they are capable of producing detailed maps that can be used for path planning. In earlier work, we presented an algorithm called DP-SLAM [4], which produced extremely accurate, densely populated maps by maintaining a joint distribution over robot maps and poses using a particle filter. DP-SLAM uses novel data structures that exploit shared structure between maps, permitting efficient use of many joint map/pose particles. This gives DP-SLAM the ability to resolve map ambiguities automatically, as a natural part of the particle filtering process, effectively obviating the explicit loop closing phase needed for other approaches [7, 12]. A known limitation of particle filters is that they can require a very large number of particles to track systems with diffuse posterior distributions. This limitation strongly affected earlier versions of DP-SLAM, which had a worst-case run time that scaled quadratically with the number of particles. In this paper, we present a significant improvement to DPSLAM which reduces the run time to linear in the number of particles, giving multiple map hypothesis SLAM the same asymptotic complexity per particle as localization with a single map. The new algorithm also has a more straightforward analysis and implementation. Unfortunately, even with linear time complexity, there exist domains which require infeasibly large numbers of particles for accurate mapping. The cumulative effect of very small errors (resulting from sampling or discretization) can cause drift. To address the issue of drift in a direct and principled manner, we propose a hierarchical particle filter method which can specifically model and recover from small amounts of drift, while maintaining particle diversity longer than in typical particle filters. The combined result is an algorithm that can produce extraordinarily detailed maps of large domains at close to real time speeds. 2 Linear Time Algorithm A DP-SLAM ancestry tree contains all of the current particles as leaves. The parent of a given node represents the particle of the previous iteration from which that particle was resampled. An ancestry tree is minimal if the following two properties hold: 1. A node is a leaf node if and only if it corresponds to a current generation particle. 2. All interior nodes have at least two children. The first property is ensured by simply removing particles that are not resampled from the ancestry tree. The second property is ensured by merging parents with only-child nodes. It is easy to see that for a particle filter with P particles, the corresponding minimal ancestry tree will have a branching factor of at least two and depth of no more than O(P). The complexity of maintaining a minimal ancestry tree will depend upon the manner in which observations, and thus maps, are associated with nodes in the tree. DP-SLAM distributes this information in the following manner: All map updates for all nodes in the ancestry tree are stored in a single global grid, while each node in the ancestry tree also maintains a list of all grid squares updated by that node. The information contained in these two data structures is integrated for efficient access at each cycle of the particle filter through a new data structure called an map cache. 2.1 Core Data Structures The DP-SLAM map is a global occupancy grid-like array. Each grid cell contains an observation vector with one entry for each ancestry tree node that has made an observation of the grid cell. Each vector entry is an observation node containing the following fields: opacity a data structure storing sufficient statistics for the current estimate of the opacity of the grid cell to the laser range finder. See Eliazar and Parr [4] for details. parent a pointer to a parent observation node for which this node is an update. (If an ancestor of a current particle has seen this square already, then the opacity value for this square is considered an update to the previous value stored by the ancestor. However, both the update and the original observation are stored, since it may not be the case that all successors of the ancestor have made updates to this square.) anode a pointer to the ancestry tree node associated with the current opacity estimate. In previous versions of DP-SLAM, this information was stored using a balanced tree. This added significant overhead to the algorithm, both conceptual and computational, and is no longer required in the current version. The DP-SLAM ancestry tree is a basic tree data structure with pointers to parents and children. Each node in the ancestry tree also contains an onodes vector, which contains pointers to observation nodes in the grid cells updated by the ancestry tree node. 2.2 Map cache The main sacrifice that was made when originally designing DP-SLAM was that map accesses no longer took constant time, due to the need to search the observation vector at a given grid square. The map cache provides a way of returning to this constant time access, by reconstructing a separate local map which is consistent with the history of map updates for each particle. Each local map is only as large as the area currently observed, and therefore is of a manageable size. For a localization procedure using P particles and observing an area of A grid squares, there is a total of O(AP) map accesses. For the constant time accesses provided by the map cache to be useful, the time complexity to build the map cache needs to be O(AP). This result can be achieved by constructing the cache in two passes. The first pass is to iterate over all grid squares in the global map which could be within sensor range of the robot. For each of these grid squares, the observation vector stores all observations made of that grid square by any particle. This vector is traversed, and for each observation, we update the corresponding local map with a pointer back to the corresponding observation node. This creates a set of partial local maps that store pointers to map updates, but no inherited map information. Since the size of the observation vector can be no greater than the size of the ancestry tree, which has O(P) nodes, the first pass takes O(P) time per grid square. In the second pass we fill holes in the local maps by propagating inherited map information. The entire ancestry tree is traced, depth first, and the local map is checked for each ancestor node encountered. If the local map for the current ancestor node was not filled during the first pass, then the hole is patched by inheritance from the ancestor node’s parent. This will fill any gaps in the local maps for grid squares that have been seen by any current particle. As this pass is directly based on the size of the ancestry tree, it is also O(P) per grid square. Therefore, the total complexity of building the map cache is O(AP). For each particle, the algorithm constructs a grid of pointers to observation nodes. This provides constant time access to the opacity values consistent with each particle’s map. Localization now becomes trivial with this representation: Laser scans are traced through the corresponding local map, and the necessary opacity values are extracted via the pointers. With the constant time accesses afforded by the local maps, the total localization cost in DP-SLAM is now O(AP). 2.3 Updates and Deletions When the observations associated with a new particle’s sensor sweep are integrated into the map, two basic steps are performed. First, a new observation is added to the observation vector of each grid square which was visited by the particle’s laser casts. Next, a pointer to each new observation is added to this particle’s onodes vector. The cost of this operation is obviously no more than that of localization. There are two situations which require deleting nodes from the ancestry tree. The first is the simple case of removing a node from which the particle filter has not resampled. Each ancestor node maintains a vector of pointers to all observations attributed to it. Therefore, these entries can be removed from the observation vectors in the global grid in constant time. Since there can be no more deletions than there are updates, this process has an amortized cost of O(AP). The second case for deleting a node occurs when a node in the ancestry tree which has an only child is merged with that child. This involves replacing the opacity value for the parent with that of the child, and then removing that child’s entry from the associated grid cell’s observation vector. Therefore, this process is identical to the first case, except that each removal of an entry from the global map is preceded by a single update to the same grid square. Since the observation vector at each grid square is not ordered, additions to the vector can be done in constant time, and does not change the complexity from O(AP). 3 Drift A significant problem faced by current SLAM algorithms is that of drift. Small errors can accumulate over several iterations, and while the resulting map may seem locally consistent, there could be large total errors, which become apparent after the robot closes a large loop. In theory, drift can be avoided by some algorithms in situations where strong linear Gaussian assumptions hold [10]. In practice, it is hard to avoid drift, either as a consequence of violated assumptions or as a consequence of particle filtering. The best algorithms can only extend the distance that the robot travels before experiencing drift. Errors come from (at least) three sources: insufficient particle coverage, coarse precision, and resampling itself (particle depletion). The first problem is a well known issue with particle filters. Given a finite number of particles, there will be unsampled gaps in the particle coverage of the state space and the proximity to the true state can be as coarse as the size of these gaps. This is exacerbated by the fact that particle filters are often applied to high dimensional state spaces with Gaussian noise, making it impossible to cover unlikely (but still possible) events in the tails of distribution with high particle density. The second issue is coarse precision. This can occur as a result of explicit discretization through an occupancy grid, or implicit discretization through the use of a sensor with finite precision. Coarse precision can make minor perturbations in the state appear identical from the perspective of the sensors and the particle weights. Finally, resampling itself can lead to drift by shifting a finite population of particles away from low probability regions of the state space. While this behavior of a particle filter is typically viewed as a desirable reallocation of computational resources, it can shift particles away from the true state in some cases. The net effect of these errors can be the gradual accumulation of small errors resulting from failure to sample, differentiate, or remember a state vector that is sufficiently close to the true state. In practice, we have found that there exist large domains where high precision mapping is essentially impossible with any reasonable number of particles. 4 Hierarchical SLAM In the first part of the paper, we presented an approach to SLAM that reduced the asymptotic complexity per particle to that of pure localization. This is likely as low as can reasonably be expected and should allow the use of large numbers of particles for mapping. However, the discussion of drift in the previous section underscores that the ability to use large numbers of particles may not be sufficient, and we would like techniques that delay the onset of drift as long as possible. We therefore propose a hierarchical approach to SLAM that is capable of recognizing, representing, and recovering from drift. The basic idea is that the main sources of drift can be modeled as the cumulative effect of a sequence of random events. Through experimentation, we can quantify the expected amount of drift over a certain distance for a given algorithm, much in the same way that we create a probabilistic motion model for the noise in the robot’s odometry. Since the total drift over a trajectory is assumed to be a summation of many small, largely independent sources of error, it will be close to a Gaussian distribution. If we view the act of completing a small map segment as a random process with noise, we can then apply a higher level filter to the output of the map segment process in an attempt to track the underlying state more accurately. There are two benefits to this approach. First, it explicitly models and permits the correction of drift. Second, the coarser time granularity of the high level process implies fewer resampling steps and fewer opportunities for particle depletion. Thus, if we can model how much drift is expected to occur over a small section of the robot’s trajectory, we can maintain this extra uncertainty longer, and resolve inaccuracies or ambiguities in the map in a natural fashion. There are some special properties of the SLAM problem that make it particularly well suited to this approach. In the full generality of an arbitrary tracking problem, one should view drift as a problem that affects entire trajectories through state space and the complete belief state at any time. Sampling the space of drifts would then require sampling perturbations to the entire state vector. In this fully general case, the benefit of the hierarchical view would be unclear, as the end result would be quite similar to adding additional noise to the low level process. In SLAM, we can make two assumptions that simplify things. The first is that the robot state vector is highly correlated with the remaining state variables, and the second is that we have access to a low level mapping procedure with moderate accuracy and local consistency. Under these assumptions, the the effects of drift on low level maps can be accurately approximated by perturbations to the endpoints of the robot trajectory used to construct a low level map. By sampling drift only at endpoints, we will fail to sample some of the internal structure that is possible in drifts, e.g., we will fail to distinguish between a linear drift and a spiral pattern with the same endpoints. However, the existence of significant, complicated drift patterns within a map segment would violate our assumption of moderate accuracy and local consistency within our low level mapper. To achieve a hierarchical approach to SLAM, we use a standard SLAM algorithm using a small portion of the robot’s trajectory as input for the low level mapping process. The output is not only a distribution over maps, but also a distribution over robot trajectories. We can treat the distribution over trajectories as a distribution over motions in the higher level SLAM process, to which additional noise from drift is added. This allows us to use the output from each of our small mapping efforts as the input for a new SLAM process, working at a much higher level of time granularity. For the high level SLAM process, we need to be careful to avoid double counting evidence. Each low level mapping process runs as an independent process intialized with an empty map. The distribution over trajectories returned by the low level mapping process incorporates the effects of the observations used by the low level mapper. To avoid double counting, the high level SLAM process can only weigh the match between the new observations and the existing high level maps. In other words, all of the observations for a single high level motion step (single low level trajectory) must be evaluated against the high level map, before any of those observations are used to update the map. We summarize the high level SLAM loop for each high level particle as follows: 1. Sample a high level SLAM state (high level map and robot state). 2. Perturb the sampled robot state by adding random drift. 3. Sample a low level trajectory from the distribution over trajectories returned by the low level SLAM process. 4. Compute a high level weight by evaluating the trajectory and robot observations against the sampled high level map, starting from the perturbed robot state. 5. Update the high level map based upon the new observations. In practice this can give a much greater improvement in accuracy over simply doubling the resources allocated to a single level SLAM algorithm because the high level is able to model and recover from errors much longer than would be otherwise possible with only a single particle filter. In our implementation we used DP-SLAM at both levels of the hierarchy to ensure a total computational complexity of O(AP). However, there is reason to believe that this approach could be applied to any other sampling-based SLAM method just as effectively. We also implemented this idea with only one level of hierarchy, but multiple levels could provide additional robustness. We felt that the size of the domains on which we tested did not warrant any further levels. 5 Implementation and Empirical Results Our description of the algorithm and complexity analysis assumes constant time updates to the vectors storing information in the core DP-SLAM data structures. This can be achieved in a straightforward manner using doubly linked lists, but a somewhat more complicated implementation using adjustable arrays is dramatically more efficient in practice. A careful implementation can also avoid caching maps for interior nodes of the ancestry tree. As with previous versions of DP-SLAM, we generate many more particles than we keep at each iteration. Evaluating a particle requires line tracing 181 laser casts. However, many particles will have significantly lower probability than others and this can be discovered before they are fully evaluated. Using a technique we call particle culling we use partial scan information to identify and discard lower probability particles before they are evaluated fully. In practice, this leads to large reduction in the number of laser casts that are fully traced through the grid. Typically, less than one tenth of the particles generated are resampled. For a complex algorithm like DP-SLAM, asymptotic analysis may not always give a complete picture of real world performance. Therefore, we provide a comparison of actual run times for each method on three different data logs. The particle counts provided are the minimum number of particles needed (at each level) to produce high-quality maps reliably. The improved run time for the linear algorithm also reflects the benefits of some improvements in our culling technique and a cleaner implementation permitted by the linear time algorithm. The quadratic code is simply too slow to run on the Wean Hall data. Log files for these runs are available from the DP-SLAM web page: http://www.cs.duke.edu/˜parr/dpslam/. The results show a significant practical advantage for the linear code, and vast improvement, both in terms of time and number of particles, for the hierarchical implementation. Quadratic Linear Hierarchical Log Particles Minutes Particles Minutes Particles (high/low) Minutes loop5 1500 55 1500 14 200/250 12 loop25 11000 1345 11000 690 2000/3000 289 Wean Hall 120000 N/A 120000 2535 2000/3000 293 Finally, in Figure 1 we include sample output from the hierarchical mapper on the Wean Hall data shown in our table. In this domain, the robot travels approximately 220m before returning to its starting position. Each low level SLAM process was run for 75 time steps, with an average motion of 12cm for each time step. The nonhierarchical approach can produce a very similar result, but requires at least 120,000 particles to do so reliably. (Smaller numbers of particles produced maps with noticeable drifts and errors.) This extreme difference in particle counts and computation time demonstrates the great improvement that can be realized with the hierarchical approach. (The Wean Hall dataset has been mapped Figure 1: CMU’s Wean Hall at 4cm resolution, using hierarchical SLAM. Please zoom in on the map using a software viewer to appreciate some of the fine detail. successfully before at low resolution using a non-hierarchical approach with run time per iteration that grows with the number of iterations [8].) 6 Related Work Other methods have attempted to preserve uncertainty for longer numbers of time steps. One approach seeks to delay the resampling step for several iterations, so as to address the total noise in a certain number of steps as one Gaussian with a larger variance [8]. In general, look-ahead methods can “peek” at future observations to use the information from later time steps to influence samples at a previous time step [3]. The HYMM approach[11] combines different types of maps. Another way to interpret hierarchical SLAM is in terms of a hierarchical hidden Markov model framework [5]. In a hierarchical HMM, each node in the HMM has the potential to invoke sub-HMMs to produce a series of observations. The main difference is that in hierarchical HMMs, there is assumed to be a single process that can be represented in different ways. In our hierarchical SLAM approach, only the lowest level models a physical process, while higher levels model the errors in lower levels. 7 Conclusions and Future Research We have presented a SLAM algorithm which is the culmination of our efforts to make multiple hypothesis mapping practical for densely populated maps. Our first algorithmic accomplishment is to show that this requires no more effort, asymptotically, than pure localization using a particle filter. However, for mapping, the number of particles needed can be large and can still grow to be unmanageable for large domains due to drift. We therefore developed a method to improve the accuracy achieveable with a reasonable number of particles. This is accomplished through the use of a hierarchical particle filter. By allowing an additional level of sampling on top of a series of small particle filters, we can successfully maintain the necessary uncertainty to produce very accurate maps. This is due to the explicit modeling of the drift, a key process which differentiates this approach from previous attempts to preserve uncertainty in particle filters. The hierarchical approach to SLAM has been shown to be very useful in improving DPSLAM performance. This would lead us to believe that similar improvements could also be realized in applying this to other sampling based SLAM methods. SLAM is perhaps not the only viable application for hierarchical framework for particle filters. However, one of the key aspects of SLAM is that the drift can easily be represented by a very low dimensional descriptor. Other particle filter applications which have drift that must be modeled in many more dimensions could benefit much less from this hierarchical approach. The work of Hahnel et al. [8] has made progress in increasing efficiency and reducing drift by using scan matching rather than pure sampling from a noisy proposal distribution. Since much of the computation time used by DP-SLAM is spent evaluating bad particles, a combination of DP-SLAM with scan matching could yield significant practical speedups. Acknowledgments This research was supported by SAIC, the Sloan foundation, and the NSF. The Wean Hall data were gracriously provided by Dirk Hahnel and Dieter Fox. References [1] W. Burgard, D. Fox, H. Jans, C. Matenar, and S. Thrun. Sonar-based mapping with mobile robots using EM. In Proc. of the International Conference on Machine Learning, 1999. [2] P. Cheeseman, P. Smith, and M. Self. Estimating uncertain spatial relationships in robotics. In Autonomous Robot Vehicles, pages 167–193. Springer-Verlag, 1990. [3] N. de Freitas, R. Dearden, F. Hutter, R. Morales-Menendez, J. Mutch, and D. Poole. Diagnosis by a waiter and a Mars explorer. In IEEE Special Issue on Sequential State Estimation, pages 455–468, 2003. [4] A. Eliazar and R. Parr. DP-SLAM 2.0. In IEEE International Conference on Robotics and Automation (ICRA), 2004. [5] Shai Fine, Yoram Singer, and Naftali Tishby. The hierarchical hidden markov model: Analysis and applications. Machine Learning, 32(1):41–62, 1998. [6] Dieter Fox, Wolfram Burgard, Frank Dellaert, and Sebastian Thrun. Monte carlo localization: Efficient position estimation for mobile robots. In AAAI-99, 1999. [7] J. Gutmann and K. Konolige. Incremental mapping of large cyclic environments. In IEEE International Symposium on Computational Intelligence in Robotics and Automation (ICRA), pages 318–325, 2000. [8] Dirk Hahnel, Wolfram Burgard, Dieter Fox, and Sebastian Thrun. An efficient fastslam algorithm for generating maps of large-scale cyclic environments from raw laser range measurements. In Proceedings of the International Conference on Intelligent Robots and Systems, 2003. [9] John H. Leonard, , and Hugh F. Durrant-Whyte. Mobile robot localization by tracking geometric beacons. In IEEE Transactions on Robotics and Automation, pages 376–382. IEEE, June 1991. [10] M. Montemerlo, S. Thrun, D. Koller, and B. Wegbreit. FastSLAM 2.0: An improved particle filtering algorithm for simultaneous localization and mapping that provably converges. In IJCAI-03, Morgan Kaufmann, 2003. 1151–1156. [11] J. Nieto, J. Guivant, and E. Nebot. The HYbrid Metric Maps (HYMMS): A novel map representation for denseSLAM. In IEEE International Conference on Robotics and Automation (ICRA), 2004. [12] S. Thrun. A probabilistic online mapping algorithm for teams of mobile robots. International Journal of Robotics Research, 20(5):335–363, 2001.
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An exploration-exploitation model based on norepinepherine and dopamine activity Samuel M. McClure*, Mark S. Gilzenrat, and Jonathan D. Cohen Center for the Study of Brain, Mind, and Behavior Princeton University Princeton, NJ 08544 smcclure@princeton.edu; mgilzen@princeton.edu; jdc@princeton.edu Abstract We propose a model by which dopamine (DA) and norepinepherine (NE) combine to alternate behavior between relatively exploratory and exploitative modes. The model is developed for a target detection task for which there is extant single neuron recording data available from locus coeruleus (LC) NE neurons. An exploration-exploitation trade-off is elicited by regularly switching which of the two stimuli are rewarded. DA functions within the model to change synaptic weights according to a reinforcement learning algorithm. Exploration is mediated by the state of LC firing, with higher tonic and lower phasic activity producing greater response variability. The opposite state of LC function, with lower baseline firing rate and greater phasic responses, favors exploitative behavior. Changes in LC firing mode result from combined measures of response conflict and reward rate, where response conflict is monitored using models of anterior cingulate cortex (ACC). Increased long-term response conflict and decreased reward rate, which occurs following reward contingency switch, favors the higher tonic state of LC function and NE release. This increases exploration, and facilitates discovery of the new target. 1 Introduction A central problem in reinforcement learning is determining how to adaptively move between exploitative and exploratory behaviors in changing environments. We propose a set of neurophysiologic mechanisms whose interaction may mediate this behavioral shift. Empirical work on the midbrain dopamine (DA) system has suggested that this system is particularly well suited for guiding exploitative behaviors. This hypothesis has been reified by a number of studies showing that a temporal difference (TD) learning algorithm accounts for activity in these neurons in a wide variety of behavioral tasks [1,2]. DA release is believed to encode a reward prediction error signal that acts to change synaptic weights relevant for producing behaviors [3]. Through learning, this allows neural pathways to predict future expected reward through the relative strength of their synaptic connections [1]. Decision-making procedures based on these value estimates are necessarily greedy. Including reward bonuses for exploratory choices supports non-greedy actions [4] and accounts for additional data derived from DA neurons [5]. We show that combining a DA learning algorithm with models of response conflict detection [6] and NE function [7] produces an effective annealing procedure for alternating between exploration and exploitation. NE neurons within the LC alternate between two firing modes [8]. In the first mode, known as the phasic mode, NE neurons fire at a low baseline rate but have relatively robust phasic responses to behaviorally salient stimuli. The second mode, called the tonic mode, is associated with a higher baseline firing and absent or attenuated phasic responses. The effects of NE on efferent areas are modulatory in nature, and are well captured as a change in the gain of efferent inputs so that neuronal responses are potentiated in the presence of NE [9]. Thus, in phasic mode, the LC provides transient facilitation in processing, time-locked to the presence of behaviorally salient information in motor or decision areas. Conversely, in tonic mode, higher overall LC discharge rate increases gain generally and hence increases the probability of arbitrary responding. Consistent with this account, for periods when NE neurons are in the phasic mode, monkey performance is nearly perfect. However, when NE neurons are in the tonic mode, performance is more erratic, with increased response times and error rate [8]. These findings have led to a recent characterization of the LC as a dynamic temporal filter, adjusting the system's relative responsivity to salient and irrelevant information [8]. In this way, the LC is ideally positioned to mediate the shift between exploitative and exploratory behavior. The parameters that underlie changes in LC firing mode remain largely unexplored. Based on data from a target detection task by Aston-Jones and colleagues [10], we propose that LC firing mode is determined in part by measures of response conflict and reward rate as calculated by the ACC and OFC, respectively [8]. Together, the ACC and OFC are the principle sources of cortical input to the LC [8]. Activity in the ACC is known, largely through human neuroimaging experiments, to change in accord with response conflict [6]. In brief, relatively equal activity in competing behavioral responses (reflecting uncertainty) produces high conflict. Low conflict results when one behavioral response predominates. We propose that increased long-term response conflict biases the LC towards a tonic firing mode. Increased conflict necessarily follows changes in reward contingency. As the previously rewarded target no longer produces reward, there will be a relative increase in response ambiguity and hence conflict. This relationship between conflict and LC firing is analogous to other modeling work [11], which proposes that increased tonic firing reflects increased environmental uncertainty. As a final component to our model, we hypothesize that the OFC maintains an ongoing estimate in reward rate, and that this estimate of reward rate also influences LC firing mode. As reward rate increases, we assume that the OFC tends to bias the LC in favor of phasic firing to target stimuli. We have aimed to fix model parameters based on previous work using simpler networks. We use parameters derived primarily from a previous model of the LC by Gilzenrat and colleagues [7]. Integration of response conflict by the ACC and its influence on LC firing was borrowed from unpublished work by Gilzenrat and colleagues in which they fit human behavioral data in a diminishing utilities task. Given this approach, we interpret our observed improvement in model performance with combined NE and DA function as validation of a mechanism for automatically switching between exploitative and exploratory action selection. 2 Go-No-Go Task and Core Model We have modeled an experiment in which monkeys performed a target detection task [10]. In the task, monkeys were shown either a vertical bar or a horizontal bar and were required to make or omit a motor response appropriately. Initially, the vertical bar was the target stimulus and correctly responding was rewarded with a squirt of fruit juice (r=1 in the model). Responding to the non-target horizontal stimulus resulted in time out punishment (r=-.1; Figure 1A). No responses to either the target or non-target gave zero reward. After the monkeys had fully acquired the task, the experimenters periodically switched the reward contingency such that the previously rewarded stimulus (target) became the distractor, and vice versa. Following such reversals, LC neurons were observed to change from emitting phasic bursts of firing to the target, to tonic firing following the switch, and slowly back to phasic firing for the new target as the new response criteria was obtained [10]. Figure 1: Task and model design. (A) Responses were required for targets in order to obtain reward. Responses to distractors resulted in a minor punishment. No responses gave zero reward. (B) In the model, vertical and horizontal bar inputs (I1 and I2) fed to integrator neurons (X1 and X2) which then drove response units (Y1 and Y2). Responses were made if Y1 or Y2 crossed a threshold while input units were active. We have previously modeled this task [7,12] with a three-layer connectionist network in which two input units, I1 and I2, corresponding to the vertical and horizontal bars, drive two mutually inhibitory integrator units, X1 and X2. The integrator units subsequently feed two response units, Y1 and Y2 (Figure 1B). Responses are made whenever output from Y1 or Y2 crosses a threshold level of activity, θ. Relatively weak cross connections from each input unit to the opposite integrator unit (I1 to X2 and I2 to X1) are intended to model stimulus similarity. Both the integrator and response units were modeled as noisy, leaky accumulators: ! ˙ X i = "Xi +wX i I i Ii + wX i I j I j " wX i X j f (X j) +#i (1) ! ˙ Y i = "Yi +wYi X i f (Xi) " wYiY j f (Y j) +#i. (2) The ξi terms represent stochastic noise variables. The response function for each unit is sigmoid with gain, gt, determined by current LC activity (Eq. 9, below) ! f (X) = 1+ e"gt X "b ( ) ( ) "1 . (3) Response units, Y, were given a positive bias, b, and integrator units were unbiased. All weight values, biases, and variance of noise are as reported in [7]. Integration was done with a Euler method at time steps of 0.02. Simulation of stimulus presentations involved setting one of the input units to a value of 1.0 for 20 units of model time. Activation of I1 and I2 were alternated and 20 units of model time were allowed between presentations for the integrator and response units to relax to baseline levels of activity. Input 1 was initially set to be the target and input 2 the distractor. After 50 presentations of I1 and I2 the reward contingencies were switched; the model was run through 6 such blocks and reversals. The response during each stimulus presentation was determined by which of the two response units first crossed a threshold of output activity (i.e. f(Y1) > θ), or was a no response if neither unit crossed threshold. 3 Performance of model with DA-mediated learning In order to obtain a benchmark level of performance to compare against, we first determined how learning progresses with DA-mediated reinforcement learning alone. A reward unit, r, was included that had activity 0 except at the end of each stimulus presentation when its activity was set equal to the obtained reward outcome. Inhibitory inputs from the response units served as measures of expected reward. At the end of every trial, the DA unit, δ, obtained a value given by ! "(t) = r(t) # w"Y1Z(Y1(t)) # w"Y2Z(Y2(t)) (4) where Z(Y) is a threshold function that is 1 if f(Y)≥θ and is 0 otherwise. The output of dopamine neurons was used to update the weights along the pathway that lead to the response. Thus, at the end of every stimulus presentation, the weights between response units and DA neurons were updated according to ! w"Yi (t +1) = w"Yi (t) + #"(t)Z(Yi) (5) where the learning rate, λ, was set to 0.3 for all simulations. This learning rule allowed the weights to converge to the expected reward for selecting each of the two actions. Weights between integrator and response units were updated using the same rule as in Eq. 5, except the weights were restricted to a minimum value of 0.8. When the weight values were allowed to decrease below 0.8, sufficient activity never accumulated in the response units to allow discovery to new reward contingencies. As the model learned, the weights along the target pathway obtained a maximum value while those along the distractor pathway obtained a minimum value. After reversals, the model initially adapted by reducing the weights along the pathway associated with the previous target. The only way the model was able to obtain the new target was by noise pushing the new target response unit above threshold. Because of this, the performance of the model was greatly dependent of the value of the threshold used in the simulation (Figure 2B). When the threshold was low relative to noise, the model was able to quickly adapt to reversals. However, this also resulted in a high rate of responding to non-target stimuli even after learning. In order to reduce responding to the distractor, the threshold had to be raised, which also increased the time required to adapt following reward reversals. The network was initialized with equal preference for responding to input 1 or 2, and generally acquired the initial target faster than after reversals (see Figure 2B). Because of this, all subsequent analyses ignore this first learning period. For each value of threshold studied, we ran the model 100 times. Plots shown in Figures 2 and 3 show the probability that the model responded, when each input was activated, as a function of trial number (i.e. P(f(Yi)≥θ | Ii=1)). Figure 2: Model performance with DA alone. (A) DA neurons, δ, modulated weights from integrator to response units in order to modulate the probability of responding to each input. (B) The model successfully increases and decreases responding to inputs 1 and 2 as reward contingencies reverse. However, the model is unable to simultaneously obtain the new response quickly and maintain a low error rate once the response is learned. When threshold is relatively low (left plot), the model adapts quickly but makes frequent responses to the distractor. At higher threshold, responses are correctly omitted to the distractor, but the model acquires the new response slowly. 4 Improvement with NE-mediated annealing We used the FitzHugh-Nagumo set of differential equations to model LC activity. (These equations are generally used to model individual neurons, but we use them to model the activity in the nucleus as a whole.) Previous work has shown that these equations, with simple modifications, capture the fundamental aspects of tonic and phasic mode activity in the LC [7]. The FitzHugh-Nagumo equations involve two interacting variables v and u, where v is an activity term and u is an inhibitory dampening term. The output of the LC is given by the value of u, which conveniently captures the fact that the LC is self-inhibitory and that the postsynaptic effect of NE release is somewhat delayed [7]. The model included two inputs to the LC from the integrator units (X1 and X2) with modifiable weights. The state of the LC is then given by ! " v ˙ v = v(# $ v)(v $1) $ u + wvX1 f (X1) + wvX 2 f (X2) (6) ! " u ˙ u = h(v) # u (7) where the function h is defined by ! h(v) = Cv + (1" C)d (8) and governs the firing mode of the LC. In order to change firing mode, h can be modified so that the dynamics of u depend entirely on the state of the LC or so that the dynamics are independent of state. This alternation is governed by the parameter C. When C is equal to 1.0, the model is appropriately dampened and can burst sharply and return to a relatively low baseline level of activity (phasic mode). When C is small, the LC receives a fixed level of inhibition, which simultaneously reduces bursting activity and increases baseline activity (tonic mode) [7]. The primary function of the LC in the model is to modify the gain, g, of the response function of the integrator and response units as in equation 3. We let gain be a linear function of u with base value G and dependency on u given by k ! gt = G + kut . (9) The value of C was updated after every trial by measures of response conflict and reward rate. Response conflict was calculated as a normalized measure of the energy in the response units during the trial. For convenience, define Y1 to be a vector of the activity in unit Y1 at each point of time during a trial, f(Y1(t)). Let Y2 be defined similarly. The conflict during the trial is ! K = Y1 "Y2 Y1 Y2 (10) which correctly measures energy since Y1 and Y2 are connected with weight –1. This normalization procedure was necessary to account for changes in the magnitude of Y1 and Y2 activity due to learning. Based on previous work [8], we let conflict modify C separately based on a shortterm, KS, and long-term, KL, measure. The variable KS was updated at the end of every Tth trial according to ! KS (T +1) = (1"#S)KS (T ) +#SK(T ). (11) where εS was 0.2 and KS(T+1) was used to calculate the value of C used for the T+1th trial. KL was update with the same rule as KS except εL was 0.05. We let short- and long-term conflict have opposing effect on the firing mode of the LC. This was developed previously to capture human behavior in a diminishing utilities task. When short-term conflict increases, the LC is biased towards phasic firing (increased C). This allows the model to recover from occasional errors. However, when long-term conflict increases this is taken to indicate that the current decision strategy is not working. Therefore, increased long-term conflict biases the LC to the tonic mode so as to increase response volatility. Figure 3: Model performance with DA and NE. (A) The full model includes a conflict detection unit, K, and a reward rate measure, R, which combine to modify activity in the LC. The LC modifies the gain in the integrator and response units. (B) The benefit of including the LC in the model is insignificant when the response threshold is regularly crossed by noise alone, and hence when the error rate is high. (C) However, when the threshold is greater and error rate lower, NE dramatically improves the rate at which the new reward contingencies are learned after reversal. Reward rate, R, was updated at the end of every trial according to ! R(T +1) = (1"#R)R(T ) +#Rr (12) where r is the reward earned on the Tth trial. Increased reward rate was assumed to bias the LC to phasic firing. Reward rate, short-term conflict, and long-term conflict updated C according to ! C = " (KS) 1#" (KL) ( )" (R) (13) where each σ is a sigmoid function with a gain of 6.0 and no bias as determined by fitting to behavior with previous models. As with the model with DA alone, the effect of NE depended significantly on the value of the threshold θ. When θ was small, the improvement afforded by the LC was negligible (Figure 3B). However, when the threshold was significantly greater than noise, the improvement was substantial (Figure 3C). Monkeys were able to perform this task with accuracy greater than 90% and simultaneously were able to adapt to reversals within 50 trials [10]. While it is impossible to compare the output of our model with monkey behavior, we can make the qualitative assertion that, as with monkeys, our NE-based annealing model allows for high accuracy (and high threshold) decision-making while preserving adaptability to changes in reward contingencies. In order to better demonstrate this improvement, we fit single exponential curves to the plots of probability of accurately responding to the new target by trial number (as in Figure 3B,C). Shown in Figure 4 is the time constant for these exponential fits, which we term the discovery time constant, for different values of the threshold. As can be seen, the model with NE-mediated annealing maintains a relatively fast discovery time even as the threshold becomes relatively large. Figure 4: Summary of model performance with and without NE. 5 Discussion We have demonstrated that a model incorporating behavioral and learning effects previously ascribed to DA and NE produces an adaptive mechanism for switching between exploratory and exploitative decision-making. Our model uses measures of response conflict and reward rate to modify LC firing mode, and hence to change network dynamics in favor of more or less volatile behavior. In essence, combining previous models of DA and NE function produces a performance-based autoannealing algorithm. There are several limitations to this model that can be remedied by greater sophistication in the learning algorithm. The primary limitation is that the model varies between more or less volatile action selection only over the range of reward relevant to our studied task. Model parameters could be altered on a task-by-task basis to correct this; however, a more general scheme may be accomplished with a mean reward learning algorithm [13]. It has previously been argued that DA neurons may actually emit an average reward TD error [14]. This change may require allowing both short- and long-term reward rate control the LC firing mode (Eq. 13). Another limitation of this model is that, while exploration is increased as performance measures wane, exploration is not managed intelligently. This does not significantly affect the performance of our model since there are only two available actions. As the number of alternatives increases, rapid learning may require something akin to reward bonuses [4,5]. Understanding the interplay between DA and NE function in learning and decisionmaking is also relevant for understanding disease. Numerous psychiatric disorders are known to involve dysregulation of NE and DA release. Furthermore, hallmark features of ADHD and schizophrenia include cognitive disorders in which behavior appears either too volatile (ADHD) or too inflexible (schizophrenia) [15,16]. Improved models of DA-NE interplay during learning and decision-making, coupled with empirical data, may simultaneously improve knowledge of how the brain handles the exploration-exploitation dilemma and how this goes awry in disease. Acknowledg ments This work was supported by NIH grants P50 MH62196 and MH065214. Referen ces [1] Montague, P.R. Dayan, P., Sejnowski, T.J. (1996) A framework for mesencephalic dopamine systems based on predictive Hebbian learning. J. Neurosci. 16: 1936-1947. [2] Schultz, W. Dayan, P. & Montague, P.R. (1997) A neural substrate for prediction and reward. Science 275: 1593-1599. [3] Reynolds, J.N., Hyland, B.I., Wickens, J.R. (2001) A cellular mechanism of rewardrelated learning. Nature 413: 67-70. [4] Sutton, R.S. (1990) Integrated architectures for learning, planning, and reacting based on approximated dynamic programming. Mach. Learn., Proc. 7th International Conf. 216-224. [5] Kakade, S., Dayan, P. (2002) Dopamine: generalization and bonuses. Neural Networks 15: 549-559. [6] Botvinick, M.M., Braver, T.S., Barch, D.M., Carter, C.S., Cohen, J.D. (2001) Conflict monitoring and cognitive control. Psychol. Rev. 108: 624-652. [7] Gilzenrat, M.S., Holmes, B.D., Rajkowski, J., Aston-Jones, G., Cohen, J.D. (2002) Simplified dynamics in a model of noradrenergic modulation of cognitive performance. Neural Networks 15: 647-663. [8] Aston-Jones, G., Cohen, J.D. (2005) An integrative theory of locus coeruleusnorepinepherine function. Ann. Rev. Neurosci. 28: 403-450. [9] Servan-Schreiber, D., Printz, H., Cohen, J.D. (1990) A network model of catecholamine effects: gain, signal-to-noise ratio and behavior. Science 249: 892-895. [10] Aston-Jones, G., Rajkowski, J., Kubiak, P. (1997) Conditioned responses of monkey locus coeruleus neurons anticipate acquisition of discriminative behavior in a vigilance task. Neuroscience 80: 697-715. [11] Yu, A., Dayan, P. (2005) Uncertainty, neuromodulation and attention. Neuron 46: 68192. [11] Usher, M., Cohen, J.D., Rajkowski, J., Aston-Jones, G. (1999) The role of the locus coeruleus in the regulation of cognitive performance. Science 283: 549-554. [12] Schwartz, A. (1993) A reinforcement learning method for maximizing undiscounted rewards. In: Proc. 10th International Conf. Mach. Learn. (pp. 298-305). San Mateo, CA: Morgan Kaufmann. [13] Daw, N.D., Touretzky, D.S. (2002) Long-term reward prediction in TD models of the dopamine system. Neural Computation 14: 2567-2583. [14] Goldberg, T.E., Weinberger, D.R., Berman, K.F., Pliskin, N.H., Podd, M.H. (1987) Further evidence for dementia of the prefrontal type in schizophrenia? A controlled study teaching the Wisconsin Card Sorting Test. Arch. Gen. Psychiatry 44: 1008-1014. [15] Barkley, R.A. (1997) Behavioural inhibition, sustained attention, and executive functions: constructing a unified theory of AD/HD. Psychol. Bull. 121: 65-94.
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Kernels for gene regulatory regions Jean-Philippe Vert Geostatistics Center Ecole des Mines de Paris - ParisTech Jean-Philippe.Vert@ensmp.fr Robert Thurman Division of Medical Genetics University of Washington rthurman@u.washington.edu William Stafford Noble Department of Genome Sciences University of Washington noble@gs.washington.edu Abstract We describe a hierarchy of motif-based kernels for multiple alignments of biological sequences, particularly suitable to process regulatory regions of genes. The kernels incorporate progressively more information, with the most complex kernel accounting for a multiple alignment of orthologous regions, the phylogenetic tree relating the species, and the prior knowledge that relevant sequence patterns occur in conserved motif blocks. These kernels can be used in the presence of a library of known transcription factor binding sites, or de novo by iterating over all k-mers of a given length. In the latter mode, a discriminative classifier built from such a kernel not only recognizes a given class of promoter regions, but as a side effect simultaneously identifies a collection of relevant, discriminative sequence motifs. We demonstrate the utility of the motif-based multiple alignment kernels by using a collection of aligned promoter regions from five yeast species to recognize classes of cell-cycle regulated genes. Supplementary data is available at http://noble.gs.washington.edu/proj/pkernel. 1 Introduction In a eukaryotic cell, a variety of DNA switches—promoters, enhancers, silencers, etc.— regulate the production of proteins from DNA. These switches typically contain multiple binding site motifs, each of length 5–15 nucleotides, for a class of DNA-binding proteins known as transcription factors. As a result, the detection of such regulatory motifs proximal to a gene provides important clues about its regulation and, therefore, its function. These motifs, if known, are consequently interesting features to extract from genomic sequences in order to compare genes, or cluster them into functional families. These regulatory motifs, however, usually represent a tiny fraction of the intergenic sequence, and their automatic detection remains extremely challenging. For well-studied transcription factors, libraries of known binding site motifs can be used to scan the intergenic sequence. A common approach for the de novo detection of regulatory motifs is to start from a set of genes known to be similarly regulated, for example by clustering gene expression data, and search for over-represented short sequences in their proximal intergenic regions. Alternatively, some authors have proposed to represent each intergenic sequence by its content in short sequences, and to correlate this representation with gene expression data [1]. Finally, additional information to characterize regulatory motifs can be gained by comparing the intergenic sequences of orthologous genes, i.e., genes from different species that have evolved from a common ancestor, because regulatory motifs are more conserved than non-functional intergenic DNA [2]. We propose in this paper a hierarchy of increasingly complex representations for intergenic sequences. Each representation yields a positive definite kernel between intergenic sequences. While various motif-based sequence kernels have been described in the literature (e.g., [3, 4, 5]), these kernels typically operate on sequences from a single species, ignoring relevant information from orthologous sequences. In contrast, our hierarchy of motif-based kernels accounts for a multiple alignment of orthologous regions, the phylogenetic tree relating the species, and the prior knowledge that relevant sequence patterns occur in conserved motif blocks. These kernels can be used in the presence of a library of known transcription factor binding sites, or de novo by iterating over all k-mers of a given length. In the latter mode, a discriminative classifier built from such a kernel not only recognizes a given class of regulatory sequences, but as a side effect simultaneously identifies a collection of discriminative sequence motifs. We demonstrate the utility of the motif-based multiple alignment kernels by using a collection of aligned intergenic regions from five yeast species to recognize classes of co-regulated genes. From a methodological point of view, this paper can be seen as an attempt to incorporate an increasing amount of prior knowledge into a kernel. In particular, this prior information takes the form of a probabilistic model describing with increasing accuracy the object we want to represent. All kernels were designed before any experiment was conducted, and we then performed an objective empirical evaluation of each kernel without further parameter optimization. In general, classification performance improved as the amount of prior knowledge increased. This observation supports the notion that tuning a kernel with prior knowledge is beneficial. However, we observed no improvement in performance following the last modification of the kernel, highlighting the fact that a richer model of the data does not always lead to better performance accuracy. 2 Kernels for intergenic sequences In a complex eukaryotic genome, regulatory switches may occur anywhere within a relatively large genomic region near a given gene. In this work we focus on a well-studied model organism, the budding yeast Saccharomyces cerevisiae, in which the typical intergenic region is less than 1000 bases long. We refer to the intergenic region upstream of a yeast gene as its promoter region. Denoting the four-letter set of nucleotides as A = {A, C, G, T}, the promoter region of a gene is a finite-length sequence of nucleotides x ∈A∗= S∞ i=0 Ai. Given several sequenced organisms, in silico comparison of genes between organisms often allows the detection of orthologous genes, that is, genes that evolved from a common ancestor. If the species are evolutionarily close, as are different yeast strains, then the promoter regions are usually quite similar and can be represented as a multiple alignment. Each position in this alignment represents one letter in the shared ancestor’s promoter region. Mathematically speaking, a multiple alignment of length n of p sequences is a sequence c = c1, c2, . . . , cn, where each ci ∈¯ Ap, for i = 1, . . . , n, is a column of p letters in the alphabet ¯ A = A ∪{−}. The additional letter “−” is used to represent gaps in sequences, which represent insertion or deletion of letters during the evolution of the sequences. We are now in the position to describe a family of representations and kernels for promoter regions, incorporating an increasing amount of prior knowledge about the properties of regulatory motifs. All kernels below are simple inner products between vector representations of promoter regions. These vector representations are always indexed by a set M of short sequences of fixed length d, which can either be all d-mers, i.e., M = Ad, or a predefined library of indexing sequences. A promoter region P (either single sequence or multiple alignment) is therefore always represented by a vector ΦM(P) = (Φa(P))a∈M. Motif kernel on a single sequence The simplest approach to index a single promoter region x ∈A∗with an alphabet M is to define ΦSpectrum a (x) = na(x) , ∀a ∈M , where na(x) counts the number of occurrences of a in x. When M = Ad, the resulting kernel is the spectrum kernel [3] between single promoter regions. Motif kernel on multiple sequences When a gene has p orthologs in other species, then a set of p promoter regions {x1, x2, . . . , xp} ∈(A∗)p, which are expected to contain similar regulatory motifs, is available. We propose the following representation for such a set: ΦSummation a ({x1, x2, . . . , xp}) = p X i=1 ΦSpectrum a (xi) , ∀a ∈M . We call the resulting kernel the summation kernel. It is essentially the spectrum kernel on the concatenation of the available promoter regions—ignoring, however, k-mers that overlap different sequences in the concatenation. The rationale behind this kernel, compared to the spectrum kernel, is two-fold. First, if all promoters contain common functional motifs and randomly varying nonfunctional motifs, then the signal-to-noise ratio of the relevant regulatory features compared to other irrelevant non-functional features increases by taking the sum (or mean) of individual feature vectors. Second, even functional motifs representing transcription factor binding sites are known to have some variability in some positions, and merging the occurrences of a similar motif in different sequences is a way to model this flexibility in the framework of a vector representation. Marginalized motif kernel on a multiple alignment The summation kernel might suffer from at least two limitations. First, it does not include any information about the relationships between orthologs, in particular their relative similarities. Suppose for example that three species are compared, two of them being very similar. Then the promoter regions of two out of three orthologs would be virtually identical, giving an unjustified double weight to this duplicated species compared to the third one in the summation kernel. Second, although mutations in functional motifs between different species would correspond to different short motifs in the summation kernel feature vector, these varying short motifs might not cover all allowed variations in the functional motifs, especially if the motifs are extracted from a small number of orthologs. In such cases, probabilistic models such as weight matrices, which estimate possible variations for each position independently, are known to make more efficient use of the data. In order to overcome these limitations, we propose to transform the set of promoter regions into a multiple alignment. We therefore assume that a fixed number of q species has been selected, and that a probabilistic model p(h, c), with h ∈¯ A and c ∈¯ Aq has been tuned on these species. By “tuned,” we mean that p(h, c) is a distribution that accurately describes the probability of a given letter h in the common ancestor of the species, together with the set of letters c at the corresponding position in the set of species. Such distributions are commonly used in computational genomics, often resulting from the estimation of a phylogenetic tree model [6]. We also assume that all sets of q promoter regions of groups of orthologous genes in the q species have been turned into multiple alignments. Given an alignment c = c1, c2, . . . , cn, suppose for the moment that we know the corresponding true sequence of nucleotides of the common ancestor h = h1, h2, . . . , hn. Then the spectrum of the sequence h, that is, ΦSpectrum M (h), would be a good summary for the multiple alignment, and the inner product between two such spectra would be a candidate kernel between multiple alignments. The sequence h being of course unknown, we propose to estimate its conditional probability given the multiple alignment c, under the model where all columns are independent and identically distributed according to the evolutionary model, that is, p(h|c) = Qn i=1 p (hi|ci). Under this probabilistic model, it is now possible to define the representation of the multiple alignment as the expectation of the spectrum representation of h with respect to this conditional probability, that is: ΦMarginalized a (c) = X h ΦSpectrum a (h)p(h|c) , ∀a ∈M . (1) In order to compute this representation, we observe that if h has length n and a = a1 . . . ad has length d, then ΦSpectrum a (h) = n−d+1 X i=1 δ(a, hi . . . hi+d−1) , where δ is the Kronecker function. Therefore, ΦMarginalized a (c) = X h∈An ( n−d+1 X i=1 δ(a, hi . . . hi+d−1) ! n Y i=1 p (hi|ci) ) = n−d+1 X i=1 d−1 Y j=0 p(aj+1|ci+j) . This computation can be performed explicitly by computing p(aj+1|ci+j) at each position i = 1, . . . , n, and performing the sum of the products of these probabilities over a moving window. We call the resulting kernel the marginalized kernel because it corresponds to the marginalization of the spectrum kernel under the phylogenetic probabilistic model [7]. Marginalized motif kernel with phylogenetic shadowing The marginalized kernel is expected to be useful when relevant information is distributed along the entire length of the sequences analyzed. In the case of promoter regions, however, the relevant information is more likely to be located within a few short motifs. Because only a small fraction of the total set of promoter regions lies within such motifs, this information is likely to be lost when the whole sequence is represented by its spectrum. In order to overcome this limitation, we exploit the observation that relevant motifs are more evolutionarily conserved on average than the surrounding sequence. This hypothesis has been confirmed by many studies that show that functional parts, being under more evolutionary pressure, are more conserved than non-functional ones. Given a multiple alignment c, let us assume (temporarily) that we know which parts are relevant. We can encode this information into a sequence of binary variables s = s1 . . . sn ∈ {0, 1}n, where si = 1 means that the ith position is relevant, and irrelevant if si = 0. A typical sequence for a promoter region consist primarily of 0’s, except for a few positions indicating the position of the transcription factor binding motifs. Let us also assume that we know the nucleotide sequence h of the common ancestor. Then it would make sense to use a spectrum kernel based on the spectrum of h restricted to the relevant positions only. In other words, all positions where si = 0 could be thrown away, in order to focus only on the relevant positions. This corresponds to defining the features: ΦRelevant a (h, s) = n−d+1 X i=1 δ(a, hi . . . hi+d−1)δ(si, 1) . . . δ(si+d−1, 1) , ∀a ∈M . Given only a multiple alignment c, the sequences h and s are not known but can be estimated. This is where the hypothesis that relevant nucleotides are more conserved than irrelevant nucleotides can be encoded, by using two models of evolution with different rates of mutations, as in phylogenetic shadowing [2]. Let us therefore assume that we have a model p(c|h, s = 0) that describes “fast” evolution from an ancestral nucleotide h to a column c in a multiple alignment, and a second model p1(c|h, s = 1) that describes “slow” evolution. In practice, we take these models to be two classical evolutionary models with different mutation rates, but any reasonable pair of random models could be used here, if one had a better model for functional sites, for example. Given these two models of evolution, let us also define a prior probability p(s) that a position is relevant or not (related to the proportion of relevant parts we expect in a promoter region), and prior probabilities for the ancestor nucleotide p(h|s = 0) and p(h|s = 1). The joint probability of being in state s, having an ancestor nucleotide h and a resulting alignment c is then p(c, h, s) = p(s)p(h|s)p(c|h, s). Under the probabilistic model where all columns are independent from each other, that is, p(h, s|c) = Qn i=1 p(hi, si|ci), we can now replace (1) by the following features: ΦShadow a (c) = X h,s ΦRelevant a (h, s)p(h, s|c) , ∀a ∈M . (2) Like the marginalized spectrum kernel, this kernel can be computed by computing the explicit representation of each multiple sequence alignment c as a vector (Φa(c))a∈M as follows: ΦShadow a (c) = X h∈An X s∈{0,1}n n−d+1 X i=1 δ(a, hi . . . hi+d−1)δ(si, 1) . . . δ(si+d−1, 1) n Y i=1 p (hi, si|ci) = n−d+1 X i=1 d−1 Y j=0 p(h = aj+1, s = 1|ci+j) . The computation can then be carried out by exploiting the observation that each term can be computed by: p(h, s = 1|c) = p(s = 1)p(h|s = 1)p(c|h, s = 1) p(s = 0)p(c|s = 0) + p(s = 1)p(c|s = 1). Moreover, it can easily be seen that, like the marginalized kernel, the shadow kernel is the marginalization of the kernel corresponding to ΦRelevant with respect to p(h, s|c). Incorporating Markov dependencies between positions The probabilistic model used in the shadow kernel models each position independently from the others. As a result, a conserved position has the same contribution to the shadow kernel if it is surrounded by other conserved positions, or by varying positions. In order to encode our prior knowledge that the pattern of functional / nonfunctional positions along the sequence is likely to be a succession of short functional regions and longer nonfunctional regions, we propose to replace this probabilistic model by a probabilistic model with a Markov dependency between successive positions for the variable s, that is, to consider the probability: pMarkov(c, h, s) = p(s1)p(h1, c1|s1) n Y i=2 p (si|si−1) p(hi, ci|si). This suggests replacing (2) by ΦMarkov a (c) = X h,s Φa(h, s)pMarkov(h, s|c) , ∀a ∈M . Once again, this feature vector can be computed as a sum of window weights over sequences by ΦMarkov a (c) = n−d+1 X i=1 p (si = 1|c) p (hi = aj+1|ci, si = 1) × d−1 Y j=1 p(hi+j = aj+1, si+j = 1|ci+j, si+j−1 = 1) . The main difference with the computation of the shadow kernel is the need to compute the term P (si = 1|c), which can be done using the general sum-product algorithm. 3 Experiments We measure the utility of our hierarchy of kernels in a cross-validated, supervised learning framework. As a starting point for the analysis, we use various groups of genes that show co-expression in a microarray study. Eight gene groups were derived from a study that applied hierarchical clustering to a collection of 79 experimental conditions, including time series from the diauxic shift, the cell cycle series, and sporulation, as well as various temperature and reducing shocks [8]. We hypothesize that co-expression implies co-regulation of a given group of genes by a common set of transcription factors. Hence, the corresponding promoter regions should be enriched for a corresponding set of transcription factor binding motifs. We test the ability of a support vector machine (SVM) classifier to learn to recapitulate the co-expression classes, based only upon the promoter regions. Our results show that the SVM performance improves as we incorporate more prior knowledge into the promoter kernel. We collected the promoter regions from five closely related yeast species [9, 10]. Promoter regions from orthologous genes were aligned using ClustalW, discarding promoter regions that aligned with less than 30% sequence identity relative to the other sequences in the alignment. This procedure produced 3591 promoter region alignments. For the phylogenetic kernels, we inferred a phylogenetic tree among the five yeast species from alignments of four highly conserved proteins—MCM2, MCM3, CDC47 and MCM6. The concatenated alignment was analyzed with fastDNAml [11] using the default parameters. The resulting tree was used in all of our analyses. SVMs were trained using Gist (microarray.cpmc.columbia.edu/gist) with the default parameters. These include a normalized kernel, and a two-norm soft margin with asymmetric penalty based upon the ratio of positive and negative class sizes. All kernels were computed by summing over all 45 k-mers of width 5. Each class was recognized in a one-vs-all fashion. SVM testing was performed using balanced three-fold cross-validation, repeated five times. The results of this experiment are summarized in Table 1. For every gene class, the worst-performing kernel is one of the three simplest kernels: “simple,” “summation” or “marginalization.” The mean ROC scores across all eight classes for these three kernels are 0.733, 0.765 and 0.748. Classification performance improves dramatically using the shadow kernel with either a small (2) or large (5) ratio of fast-to-slow evolutionary rates. The mean ROC scores for these two kernels are 0.854 and 0.844. Furthermore, across five of the eight gene classes, one of the two shadow kernels is the best-performing kernel. The Markov kernel performs approximately as well as the shadow kernel. We tried six different parameterizations, as shown in the table, and these achieved mean ROC scores ranging from 0.822 to 0.850. The differences between the best parameterization of this kernel (“Markov 5 90/90”) and “shadow 2” are not significant. Although further tuning Table 1: Mean ROC scores for SVMs trained using various kernels to recognize classes of co-expressed yeast genes. The second row in the table gives the number of genes in each class. All other rows contain mean ROC scores across balanced three-fold cross-validation, repeated five times. Standard errors (not shown) are almost uniformly 0.02, with a few values of 0.03. Values in bold-face are the best mean ROC for the given class of genes. The classes of genes (columns) are, respectively, ATP synthesis, DNA replication, glycolysis, mitochondrial ribosome, proteasome, spindle-pole body, splicing and TCA cycle. The kernels are as described in the text. For the shadow and Markov kernels, the values “2” and “5” refer to the ratio of fast to slow evolutionary rates. For the Markov kernel, the values “90” and “99” refer to the self-transition probabilities (times 100) in the conserved and varying states of the model. Kernel ATP DNA Glyc Ribo Prot Spin Splic TCA Mean 15 5 17 22 27 11 14 16 single 0.711 0.777 0.814 0.743 0.735 0.716 0.683 0.684 0.733 summation 0.773 0.768 0.824 0.750 0.763 0.756 0.739 0.740 0.764 marginalized 0.799 0.805 0.833 0.729 0.748 0.721 0.676 0.673 0.748 shadow 2 0.881 0.929 0.928 0.840 0.867 0.827 0.787 0.770 0.854 shadow 5 0.889 0.935 0.927 0.819 0.849 0.821 0.766 0.752 0.845 Markov 2 90/90 0.848 0.891 0.908 0.830 0.853 0.801 0.773 0.758 0.833 Markov 2 90/99 0.868 0.911 0.915 0.826 0.850 0.782 0.752 0.735 0.830 Markov 2 99/99 0.869 0.910 0.912 0.816 0.840 0.773 0.737 0.724 0.823 Markov 5 90/90 0.875 0.922 0.924 0.844 0.868 0.814 0.788 0.769 0.851 Markov 5 90/99 0.872 0.916 0.920 0.834 0.858 0.794 0.774 0.755 0.840 Markov 5 99/99 0.868 0.917 0.921 0.830 0.853 0.774 0.751 0.733 0.831 of kernel parameters might yield significant improvement, our results thus far suggest that incorporating dependencies between adjacent positions does not help very much. Finally, we test the ability of the SVM to identify sequence regions that correspond to biologically significant motifs. As a gold standard, we use the JASPAR database (jaspar.cgb.ki.se), searching each class of promoter regions using MONKEY (rana.lbl.gov/˜alan/Monkey.htm) with a p-value threshold of 10−4. For each gene class, we identify the three JASPAR motifs that occur most frequently within that class, and we create a list of all 5-mers that appear within those motif occurrences. Next, we create a corresponding list of 5-mers identified by the SVM. We do this by calculating the hyperplane weight associated with each 5-mer and retaining the top 20 5-mers for each of the 15 crossvalidation runs. We then take the union over all runs to come up with a list of between 40 and 55 top 5-mers for each class. Table 2 indicates that the discriminative 5-mers identified by the SVM are significantly enriched in 5-mers that appear within biologically significant motif regions, with significant p-values for all eight gene classes (see caption for details). 4 Conclusion We have described and demonstrated the utility of a class of kernels for characterizing gene regulatory regions. These kernels allow us to incorporate prior knowledge about the evolution of a set of orthologous sequences and the conservation of transcription factor binding site motifs. We have also demonstrated that the motifs identified by an SVM trained using these kernels correspond to biologically significant motif regions. Our future work will focus on automating the process of agglomerating the identified k-mers into a smaller set of motif models, and on applying these kernels in combination with gene expression, protein-protein interaction and other genome-wide data sets. This work was funded by NIH awards R33 HG003070 and U01 HG003161. Table 2: SVM features correlate with discriminative motifs. The first row lists the number of non-redundant 5-mers constructed from high-scoring SVM features. Row two gives the number of 5-mers constructed from JASPAR motif occurrences in the 5-species alignments. Row three is a tally of all 5-mers appearing in the sequences making up the class. The fourth row gives the size of the intersection between the SVM and motif-based 5-mer lists. The final two rows give the expected value and p-value for the intersection size. The p-value is computed using the hypergeometric distribution by enumerating all possibilites for the intersection of two sets selected from a larger set given the sizes in the first three rows. ATP DNA Glyc Ribo Prot Spin Splic TCA SVM 46 40 55 50 49 43 48 50 Motif 180 68 227 38 148 152 52 104 Class 1006 839 967 973 1001 891 881 995 Inter 24 8 23 18 23 19 14 21 Expect 8.23 3.24 12.91 1.95 7.25 7.34 2.83 5.23 p-value 6.19e-8 1.15e-2 1.44e-3 3.88e-15 3.24e-8 1.74e-5 1.15e-7 2.00e-9 References [1] D. Y. Chiang, P. O. Brown, and M. B. Eisen. Visualizing associations between genome sequences and gene expression data using genome-mean expression profiles. Bioinformatics, 17(Supp. 1):S49–S55, 2001. [2] D. Boffelli, J. McAuliffe, D. Ovcharenko, K. D. Lewis, I. Ovcharenko, L. Pachter, and E. M. Rubin. Phylogenetic shadowing of primate sequences to find functional regions of the human genome. Science, 299:1391–1394, 2003. [3] C. Leslie, E. Eskin, and W. S. Noble. The spectrum kernel: A string kernel for SVM protein classification. In R. B. Altman, A. K. Dunker, L. Hunter, K. Lauderdale, and T. E. Klein, editors, Proceedings of the Pacific Symposium on Biocomputing, pages 564–575, New Jersey, 2002. World Scientific. [4] X. H-F. Zhang, K. A. Heller, I. Hefter, C. S. Leslie, and L. A. Chasin. Sequence information for the splicing of human pre-mRNA identified by support vector machine classification. Genome Research, 13:2637–2650, 2003. [5] A. Zien, G. R¨atch, S. Mika, B. Sch¨olkopf, T. Lengauer, and K.-R. M¨uller. Engineering support vector machine kernels that recognize translation initiation sites. Bioinformatics, 16(9):799– 807, 2000. [6] R. Durbin, S. Eddy, A. Krogh, and G. Mitchison. Biological Sequence Analysis. Cambridge UP, 1998. [7] K. Tsuda, T. Kin, and K. Asai. Marginalized kernels for biological sequences. Bioinformatics, 18:S268–S275, 2002. [8] M. Eisen, P. Spellman, P. O. Brown, and D. Botstein. Cluster analysis and display of genomewide expression patterns. Proceedings of the National Academy of Sciences of the United States of America, 95:14863–14868, 1998. [9] Paul Cliften, Priya Sudarsanam, Ashwin Desikan, Lucinda Fulton, Bob Fulton, John Majors, Robert Waterston, Barak A. Cohen, and Mark Johnston. Finding functional features in Saccharomyces genomes by phylogenetic footprinting. Science, 301(5629):71–76, 2003. [10] Manolis Kellis, Nick Patterson, Matthew Endrizzi, Bruce Birren, and Eric S Lander. Sequencing and comparison of yeast species to identify genes and regulatory elements. Nature, 423(6937):241–254, 2003. [11] GJ Olsen, H Matsuda, R Hagstrom, and R Overbeek. fastDNAmL: a tool for construction of phylogenetic trees of DNA sequences using maximum likelihood. Comput. Appl. Biosci., 10(1):41–48, 1994.
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Transfer learning for text classification Chuong B. Do Computer Science Department Stanford University Stanford, CA 94305 Andrew Y. Ng Computer Science Department Stanford University Stanford, CA 94305 Abstract Linear text classification algorithms work by computing an inner product between a test document vector and a parameter vector. In many such algorithms, including naive Bayes and most TFIDF variants, the parameters are determined by some simple, closed-form, function of training set statistics; we call this mapping mapping from statistics to parameters, the parameter function. Much research in text classification over the last few decades has consisted of manual efforts to identify better parameter functions. In this paper, we propose an algorithm for automatically learning this function from related classification problems. The parameter function found by our algorithm then defines a new learning algorithm for text classification, which we can apply to novel classification tasks. We find that our learned classifier outperforms existing methods on a variety of multiclass text classification tasks. 1 Introduction In the multiclass text classification task, we are given a training set of documents, each labeled as belonging to one of K disjoint classes, and a new unlabeled test document. Using the training set as a guide, we must predict the most likely class for the test document. “Bag-of-words” linear text classifiers represent a document as a vector x of word counts, and predict the class whose score (a linear function of x) is highest, i.e., arg maxk∈{1,...,K} Pn i=1 θkixi. Choosing parameters {θki} which give high classification accuracy on test data, thus, is the main challenge for linear text classification algorithms. In this paper, we focus on linear text classification algorithms in which the parameters are pre-specified functions of training set statistics; that is, each θki is a function θki := g(uki) of some fixed statistics uki of the training set. Unlike discriminative learning methods, such as logistic regression [1] or support vector machines (SVMs) [2], which use numerical optimization to pick parameters, the learners we consider perform no optimization. Rather, in our technique, parameter learning involves tabulating statistics vectors {uki} and applying the closed-form function g to obtain parameters. We refer to g, this mapping from statistics to parameters, as the parameter function. Many common text classification methods—including the multinomial and multivariate Bernoulli event models for naive Bayes [3], the vector space-based TFIDF classifier [4], and its probabilistic variant, PrTFIDF [5]—belong to this class of algorithms. Here, picking a good text classifier from this class is equivalent to finding the right parameter function for the available statistics. In practice, researchers often develop text classification algorithms by trial-and-error, guided by empirical testing on real-world classification tasks (cf. [6, 7]). Indeed, one could argue that much of the 30-year history of information retrieval has consisted of manually trying TFIDF formula variants (i.e. adjusting the parameter function g) to optimize performance [8]. Even though this heuristic process can often lead to good parameter functions, such a laborious task requires much human ingenuity, and risks failing to find algorithm variations not considered by the designer. In this paper, we consider the task of automatically learning a parameter function g for text classification. Given a set of example text classification problems, we wish to “metalearn” a new learning algorithm (as specified by the parameter function g), which may then be applied new classification problems. The meta-learning technique we propose, which leverages data from a variety of related classification tasks to obtain a good classifier for new tasks, is thus an instance of transfer learning; specifically, our framework automates the process of finding a good parameter function for text classifiers, replacing hours of hand-tweaking with a straightforward, globally-convergent, convex optimization problem. Our experiments demonstrate the effectiveness of learning classifier forms. In low training data classification tasks, the learning algorithm given by our automatically learned parameter function consistently outperforms human-designed parameter functions based on naive Bayes and TFIDF, as well as existing discriminative learning approaches. 2 Preliminaries Let V = {w1, . . . , wn} be a fixed vocabulary of words, and let X = Zn and Y = {1, . . . , K} be the input and output spaces for our classification problem. A labeled document is a pair (x, y) ∈X × Y, where x is an n-dimensional vector with xi indicating the number of occurrences of word wi in the document, and y is the document’s class label. A classification problem is a tuple ⟨D, S, (xtest, ytest)⟩, where D is a distribution over X × Y, S = {(xi, yi)}M i=1 is a set of M training examples, (xtest, ytest) is a single test example, and all M + 1 examples are drawn iid from D. Given a training set S and a test input vector xtest, we must predict the value of the test class label ytest. In linear classification algorithms, we evaluate the score fk(xtest) := P i θkixtesti for assigning xtest to each class k ∈{1, . . . , K} and pick the class y = arg maxk fk(xtest) with the highest score. In our meta-learning setting, we define each θki as the component-wise evaluation of the parameter function g on some vector of training set statistics uki: θk1 θk2 ... θkn := g(uk1) g(uk2) ... g(ukn) . (1) Here, each uki ∈Rq (k = 1, . . . , K, i = 1, . . . , n) is a vector whose components are computed from the training set S (we will provide specific examples later). Furthermore, g : Rq →R is the parameter function mapping from uki to its corresponding parameter θki. To illustrate these definitions, we show that two specific cases of the naive Bayes and TFIDF classification methods belong to the class of algorithms described above. Naive Bayes: In the multinomial variant of the naive Bayes classification algorithm,1 the score for assigning a document x to class k is f NB k (x) := log ˆp(y = k) + Pn i=1 xi log ˆp(wi | y = k). (2) The first term, ˆp(y = k), corresponds to a “prior” over document classes, and the second term, ˆp(wi | y = k), is the (smoothed) relative frequency of word 1Despite naive Bayes’ overly strong independence assumptions and thus its shortcomings as a probabilistic model for text documents, we can nonetheless view naive Bayes as simply an algorithm which makes predictions by computing certain functions of the training set. This view has proved useful for analysis of naive Bayes even when none of its probabilistic assumptions hold [9]; here, we adopt this view, without attaching any particular probabilistic meaning to the empirical frequencies ˆp(·) that happen to be computed by the algorithm. wi in training documents of class k. For balanced training sets, the first term is irrelevant. Therefore, we have f NB k (x) = P i θkixi where θki = gNB(uki), uki := uki1 uki2 uki3 uki4 uki5 = number of times wi appears in documents of class k number of documents of class k containing wi total number of words in documents of class k total number of documents of class k total number of documents , (3) and gNB(uki) := log uki1 + ε uki3 + nε (4) where ε is a smoothing parameter. (ε = 1 gives Laplace smoothing.) TFIDF: In the unnormalized TFIDF classifier, the score for assigning x to class k is f TFIDF k (x) := Pn i=1 xi|y=k · log 1 ˆp(xi>0) xi · log 1 ˆp(xi>0) , (5) where xi|y=k (sometimes called the average term frequency of wi) is the average ith component of all document vectors of class k, and ˆp(xi > 0) (sometimes called the document frequency of wi) is the proportion of all documents containing wi.2 As before, we write f TFIDF k (x) = P i θkixi with θki = gTFIDF(uki). The statistics vector is again defined as in (3), but this time, gTFIDF(uki) := uki1 uki4 log uki5 uki2 2 . (6) Space constraints preclude a detailed discussion, but many other classification algorithms can similarly be expressed in this framework, using other definitions of the statistics vectors {uki}. These include most other variants of TFIDF based on different TF and IDF terms [7], PrTFIDF [5], and various heuristically modified versions of naive Bayes [6]. 3 Learning the parameter function In the last section, we gave two examples of algorithms that obtain their parameters θki by applying a function g to a statistics vector uki. In each case, the parameter function was hand-designed, either from probabilistic (in the case of naive Bayes [3]) or geometric (in the case of TFIDF [4]) considerations. We now consider the problem of automatically learning a parameter function from example classification tasks. In the sequel, we assume fixed statistics vectors {uki} and focus on finding an optimal parameter function g. In the standard supervised learning setting, we are given a training set of examples sampled from some unknown distribution D, and our goal is to use the training set to make a prediction on a new test example also sampled from D. By using the training examples to understand the statistical regularities in D, we hope to predict ytest from xtest with low error. Analogously, the problem of meta-learning g is again a supervised learning task; here, however, the training “examples” are now classification problems sampled from a distribution D over classification problems.3 By seeing many instances of text classification problems 2Note that (5) implicitly defines fTFIDF k (x) as a dot product of two vectors, each of whose components consist of a product of two terms. In the normalized TFIDF classifier, both vectors are normalized to unit length before computing the dot product, a modification that makes the algorithm more stable for documents of varying length. This too can be represented within our framework by considering appropriately normalized statistics vectors. 3Note that in our meta-learning problem, the output of our algorithm is a parameter function g mapping statistics to parameters. Our training data, however, do not explicitly indicate the best parameter function g∗for each example classification problem. Effectively then, in the meta-learning task, the central problem is to fit g to some unseen g∗, based on test examples in each training classification problem. drawn from D, we hope to learn a parameter function g that exploits the statistical regularities in problems from D. Formally, let S = {⟨D(j), S(j), (x(j), y(j))⟩}m j=1 be a collection of m classification problems sampled iid from D. For a new, test classification problem ⟨Dtest, Stest, (xtest, ytest)⟩sampled independently from D, we desire that our learned g correctly classify xtest with high probability. To achieve our goal, we first restrict our attention to parameter functions g that are linear in their inputs. Using the linearity assumption, we pose a convex optimization problem for finding a parameter function g that achieves small loss on test examples in the training collection. Finally, we generalize our method to the non-parametric setting via the “kernel trick,” thus allowing us to learn complex, highly non-linear functions of the input statistics. 3.1 Softmax learning Recall that in softmax regression, the class probabilities p(y | x) are modeled as p(y = k | x; {θki}) := exp(P i θkixi) P k′ exp(P i θk′ixi), k = 1, . . . , K, (7) where the parameters {θki} are learned from the training data S by maximizing the conditional log likelihood of the data. In this approach, a total of Kn parameters are trained jointly using numerical optimization. Here, we consider an alternative approach in which each of the Kn parameters is some function of the prespecified statistics vectors; in particular, θki := g(uki). Our goal is to learn an appropriate g. To pose our optimization problem, we start by learning the linear form g(uki) = βT uki. Under this parameterization, the conditional likelihood of an example (x, y) is p(y = k | x; β) = exp(P i βT ukixi) P k′ exp(P i βT uk′ixi) , k = 1, . . . , K. (8) In this setup, one natural approach for learning a linear function g is to maximize the (regularized) conditional log likelihood ℓ(β : S ) for the entire collection S : ℓ(β : S ) := Pm j=1 log p(y(j) | x(j); β) −C||β||2 = m X j=1 log exp βT P i u(j) y(j)ix(j) i P k exp βT P i u(j) ki x(j) i −C||β||2. (9) In (9), the latter term corresponds to a Gaussian prior on the parameters β, which provides a means for controlling the complexity of the learned parameter function g. The maximization of (9) is similar to softmax regression training except that here, instead of optimizing over the parameters {θki} directly, we optimize over the choice of β. 3.2 Nonparametric function learning In this section, we generalize the technique of the previous section to nonlinear g. By the Representer Theorem [10], there exists a maximizing solution to (9) for which the optimal parameter vector β∗is a linear combination of training set statistics: β∗= Pm j=1 P k α∗ jk P i u(j) ki x(j) i . (10) From this, we reparameterize the original optimization over β in (9) as an equivalent optimization over training example weights {αjk}. For notational convenience, let K(j, j′, k, k′) := P i P i′ x(j) i x(j′) i′ (u(j) ki )T u(j′) k′i′. (11) (a) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 exp(uk1) exp(uk2) (b) −35 −30 −25 −20 −15 −10 −5 0 −30 −25 −20 −15 −10 −5 0 uk2 uk1 Figure 1: Distribution of unnormalized uki vectors in dmoz data (a) with and (b) without applying the log transformation in (15). In principle, one could alternatively use a feature vector representation using these frequencies directly, as in (a). However, applying the log transformation yields a feature space with fewer isolated points in R2, as in (b). When using the Gaussian kernel, a feature space with few isolated points is important as the topology of the feature space establishes locality of influence for support vectors. Substituting (10) and (11) into (9), we obtain ℓ({αjk} : S ) := m X j′=1 log exp Pm j=1 P k αjkK(j, j′, k, y(j′)) P k′ exp Pm j=1 P k αjkK(j, j′, k, k′) −C m X j=1 m X j′=1 X k X k′ αjkαj′k′K(j, j′, k, k′). (12) Note that (12) is concave and differentiable, so we can train the model using any standard numerical gradient optimization procedure, such as conjugate gradient or L-BFGS [11]. The assumption that g is a linear function of uki, however, places a severe restriction on the class of learnable parameter functions. Noting that the statistics vectors appear only as an inner product in (11), we apply the “kernel trick” to obtain K(j, j′, k, k′) := P i P i′ x(j) i x(j′) i′ K(u(j) ki , u(j′) k′i′), (13) where the kernel function K(u, v) = Φ(u), Φ(v) defines the inner product of some high-dimensional mapping Φ(·) of its inputs.4 In particular, choosing a Gaussian (RBF) kernel, K(u, v) := exp(−γ||u −v||2), gives a non-parametric representation for g: g(uki) = βT Φ(uki) = Pm j=1 P k P i αjkx(j) i exp(−γ||u(j) ki −uki||2). (14) Thus, g(uki) is a weighted combination of the values {αjkx(j) i }, where the weights depend exponentially on the squared ℓ2-distance of uki to each of the statistics vectors {u(j) ki }. As a result, we can approximate any sufficiently smooth bounded function of u arbitrarily well, given sufficiently many training classification problems. 4 Experiments To validate our method, we evaluated its ability to learn parameter functions on a variety of email and webpage classification tasks in which the number of classes, K, was large (K = 10), and the number of number of training examples per class, m/K, was small (m/K = 2). We used the dmoz Open Directory Project hierarchy,5 the 20 Newsgroups dataset,6 the Reuters-21578 dataset,7 and the Industry Sector dataset8. 4Note also that as a consequence of our kernelization, K itself can be considered a “kernel” between all statistics vectors from two entire documents. 5http://www.dmoz.org 6http://kdd.ics.uci.edu/databases/20newsgroups/20newsgroups.tar.gz 7http://www.daviddlewis.com/resources/testcollections/reuters21578/reuters21578.tar.gz 8http://www.cs.umass.edu/˜mccallum/data/sector.tar.gz Table 1: Test set accuracy on dmoz categories. Columns 2-4 give the proportion of correct classifications using non-discriminative methods: the learned g, Naive Bayes, and TFIDF, respectively. Columns 5-7 give the corresponding values for the discriminative methods: softmax regression, 1-vs-all SVMs, and multiclass SVMs. The best accuracy in each row is shown in bold. Category g gNB gTFIDF softmax 1VA-SVM MC-SVM Arts 0.421 0.296 0.286 0.352 0.203 0.367 Business 0.456 0.283 0.286 0.336 0.233 0.340 Computers 0.467 0.304 0.327 0.344 0.217 0.387 Games 0.411 0.288 0.240 0.279 0.240 0.330 Health 0.479 0.282 0.337 0.382 0.213 0.337 Home 0.640 0.470 0.454 0.501 0.333 0.440 Kids and Teens 0.252 0.205 0.142 0.202 0.173 0.167 News 0.349 0.222 0.212 0.382 0.270 0.397 Recreation 0.663 0.487 0.529 0.477 0.353 0.590 Reference 0.635 0.415 0.458 0.602 0.383 0.543 Regional 0.438 0.268 0.258 0.329 0.260 0.357 Science 0.363 0.256 0.246 0.353 0.223 0.340 Shopping 0.612 0.456 0.556 0.483 0.373 0.550 Society 0.435 0.308 0.285 0.379 0.213 0.377 Sports 0.619 0.432 0.285 0.507 0.267 0.527 World 0.531 0.491 0.352 0.329 0.277 0.303 Average 0.486 0.341 0.328 0.390 0.264 0.397 The dmoz project is a hierarchical collection of webpage links organized by subject matter. The top level of the hierarchy consists of 16 major categories, each of which contains several subcategories. To perform cross-validated testing, we obtained classification problems from each of the top-level categories by retrieving webpages from each of their respective subcategories. For the 20 Newsgroups, Reuters-21578, and Industry Sector datasets, we performed similar preprocessing.9 Given a dataset of documents, we sampled 10-class 2-training-examples-per-class classification problems by randomly selecting 10 different classes within the dataset, picking 2 training examples within each class, and choosing one test example from a randomly chosen class. 4.1 Choice of features Theoretically, for the method described in this paper, any sufficiently rich set of features could be used to learn a parameter function for classification. For simplicity, we reduced the feature vector in (3) to the following two-dimensional representation,10 uki = log(proportion of wi among words from documents of class k) log(proportion of documents containing wi) . (15) Note that up to the log transformation, the components of uki correspond to the relative term frequency and document frequency of a word relative to class k (see Figure 1). 4.2 Generalization performance We tested our meta-learning algorithm on classification problems taken from each of the 16 top-level dmoz categories. For each top-level category, we built a collection of 300 classification problems from that category; results reported here are averages over these 9For the Reuters data, we associated each article with its hand-annotated “topic” label and discarded any articles with more than one topic annotation. For each dataset, we discarded all categories with fewer than 50 examples, and selected a 500-word vocabulary based on information gain. 10Features were rescaled to have zero mean and unit variance over the training set. Table 2: Cross corpora classification accuracy, using classifiers trained on each of the four corpora. The best accuracy in each row is shown in bold. Dataset gdmoz gnews greut gindu gNB gTFIDF softmax 1VA-SVM MC-SVM dmoz n/a 0.471 0.475 0.473 0.365 0.352 0.381 0.283 0.412 20 Newsgroups 0.369 n/a 0.371 0.369 0.223 0.184 0.217 0.206 0.248 Reuters-21578 0.567 0.567 n/a 0.619 0.463 0.475 0.463 0.308 0.481 Industry Sector 0.438 0.459 0.446 n/a 0.374 0.274 0.376 0.271 0.375 problems. To assess the accuracy of our meta-learning algorithm for a particular test category, we used the g learned from a set of 450 classification problems drawn from the other 15 top-level categories.11 This ensured no overlap of training and testing data. In 15 out of 16 categories, the learned parameter function g outperforms naive Bayes and TFIDF in addition to the discriminative methods we tested (softmax regression, 1-vs-all SVMs [12], and multiclass SVMs [13]12; see Table 1).13 Next, we assessed the ability of g to transfer across even more dissimilar corpora. Here, for each of the four corpora (dmoz, 20 Newsgroups, Reuters-21578, Industry Sector), we constructed independent training and testing datasets of 480 random classification problems. After training separate classifiers (gdmoz, gnews, greut, and gindu) using data from each of the four corpora, we tested the performance of each learned classifier on the remaining three corpora (see Table 2). Again, the learned parameter functions compare favorably to the other methods. Moreover, these tests show that a single parameter function may give an accurate classification algorithm for many different corpora, demonstrating the effectiveness of our approach for achieving transfer across related learning tasks. 5 Discussion and Related Work In this paper, we presented an algorithm based on softmax regression for learning a parameter function g from example classification problems. Once learned, g defines a new learning algorithm that can be applied to novel classification tasks. Another approach for learning g is to modify the multiclass support vector machine formulation of Crammer and Singer [13] in a manner analagous to the modification of softmax regression in Section 3.1, giving the following quadratic program: minimize β∈Rn,ξ∈Rm 1 2||β||2 + C P j ξj subject to βT P i x(j) i (u(j) y(j)i −u(j) ki ) ≥I{k̸=y(j)} −ξj, ∀k, ∀j. As usual, taking the dual leads naturally to an SMO-like procedure for optimization. We implemented this method and found that the learned g, like in the softmax formulation, outperforms naive Bayes, TFIDF, and the other discriminative methods. The techniques described in this paper give one approach for achieving inductive transfer in classifier design—using labeled data from related example classification problems to solve a particular classification problem [16, 17]. Bennett et al. [18] also consider the issue of knowledge transfer in text classification in the context of ensemble classifiers, and propose a system for using related classification problems to learn the reliability of individual classifiers within the ensemble. Unlike their approach, which attempts to meta-learn properties 11For each execution of the learning algorithm, (C, γ) parameters were determined via grid search using a small holdout set of 160 classification problems. The same holdout set was used to select regularization parameters for the discriminative learning algorithms. 12We used LIBSVM [14] to assess 1VA-SVMs and SVM-Light [15] for multiclass SVMs. 13For larger values of m/K (e.g. m/K = 10), softmax and multiclass SVMs consistently outperform naive Bayes and TFIDF; nevertheless, the learned g achieves a performance on par with discriminative methods, despite being constrained to parameters which are explicit functions of training data statistics. This result is consistent with a previous study in which a heuristically hand-tuned version of Naive Bayes attained near-SVM text classification performance for large datasets [6]. of algorithms, our method uses meta-learning to construct a new classification algorithm. Though not directly applied to text classification, Teevan and Karger [19] consider the problem of automatically learning term distributions for use in information retrieval. Finally, Thrun and O’Sullivan [20] consider the task of classification in a mobile robot domain. In this work, the authors describe a task-clustering (TC) algorithm in which learning tasks are grouped via a nearest neighbors algorithm, as a means of facilitating knowledge transfer. A similar concept is implicit in the kernelized parameter function learned by our algorithm, where the Gaussian kernel facilitates transfer between similar statistics vectors. Acknowledgments We thank David Vickrey and Pieter Abbeel for useful discussions, and the anonymous referees for helpful comments. CBD was supported by an NDSEG fellowship. This work was supported by DARPA under contract number FA8750-05-2-0249. References [1] K. Nigam, J. Lafferty, and A. McCallum. Using maximum entropy for text classification. In IJCAI-99 Workshop on Machine Learning for Information Filtering, pages 61–67, 1999. [2] T. Joachims. Text categorization with support vector machines: Learning with many relevant features. In Machine Learning: ECML-98, pages 137–142, 1998. [3] A. McCallum and K. Nigam. A comparison of event models for Naive Bayes text classification. In AAAI-98 Workshop on Learning for Text Categorization, 1998. [4] G. Salton and C. Buckley. Term weighting approaches in automatic text retrieval. Information Processing and Management, 29(5):513–523, 1988. [5] T. Joachims. A probabilistic analysis of the Rocchio algorithm with TFIDF for text categorization. In Proceedings of ICML-97, pages 143–151, 1997. [6] J. D. Rennie, L. Shih, J. Teevan, and D. R. Karger. Tackling the poor assumptions of naive Bayes text classifiers. In ICML, pages 616–623, 2003. [7] A. Moffat and J. Zobel. Exploring the similarity space. In ACM SIGIR Forum 32, 1998. [8] C. Manning and H. Schutze. Foundations of statistical natural language processing, 1999. [9] A. Ng and M. Jordan. On discriminative vs. generative classifiers: a comparison of logistic regression and naive Bayes. In NIPS 14, 2002. [10] G. Kimeldorf and G. Wahba. Some results on Tchebycheffian spline functions. J. Math. Anal. Appl., 33:82–95, 1971. [11] J. Nocedal and S. J. Wright. Numerical Optimization. Springer, 1999. [12] R. Rifkin and A. Klautau. In defense of one-vs-all classification. J. Mach. Learn. Res., 5:101– 141, 2004. [13] K. Crammer and Y. Singer. On the algorithmic implementation of multiclass kernel-based vector machines. J. Mach. Learn. Res., 2:265–292, 2001. [14] C-C. Chang and C-J. Lin. LIBSVM: a library for support vector machines, 2001. Software available at http://www.csie.ntu.edu.tw/˜cjlin/libsvm. [15] T. Joachims. Making large-scale support vector machine learning practical. In Advances in Kernel Methods: Support Vector Machines. MIT Press, Cambridge, MA, 1998. [16] S. Thrun. Lifelong learning: A case study. CMU tech report CS-95-208, 1995. [17] R. Caruana. Multitask learning. Machine Learning, 28(1):41–75, 1997. [18] P. N. Bennett, S. T. Dumais, and E. Horvitz. Inductive transfer for text classification using generalized reliability indicators. In Proceedings of ICML Workshop on The Continuum from Labeled to Unlabeled Data in Machine Learning and Data Mining, 2003. [19] J. Teevan and D. R. Karger. Empirical development of an exponential probabilistic model for text retrieval: Using textual analysis to build a better model. In SIGIR ’03, 2003. [20] S. Thrun and J. O’Sullivan. Discovering structure in multiple learning tasks: The TC algorithm. 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The Forgetron: A Kernel-Based Perceptron on a Fixed Budget Ofer Dekel Shai Shalev-Shwartz Yoram Singer School of Computer Science & Engineering The Hebrew University, Jerusalem 91904, Israel {oferd,shais,singer}@cs.huji.ac.il Abstract The Perceptron algorithm, despite its simplicity, often performs well on online classification tasks. The Perceptron becomes especially effective when it is used in conjunction with kernels. However, a common difficulty encountered when implementing kernel-based online algorithms is the amount of memory required to store the online hypothesis, which may grow unboundedly. In this paper we present and analyze the Forgetron algorithm for kernel-based online learning on a fixed memory budget. To our knowledge, this is the first online learning algorithm which, on one hand, maintains a strict limit on the number of examples it stores while, on the other hand, entertains a relative mistake bound. In addition to the formal results, we also present experiments with real datasets which underscore the merits of our approach. 1 Introduction The introduction of the Support Vector Machine (SVM) [8] sparked a widespread interest in kernel methods as a means of solving (binary) classification problems. Although SVM was initially stated as a batch-learning technique, it significantly influenced the development of kernel methods in the online-learning setting. Online classification algorithms that can incorporate kernels include the Perceptron [6], ROMMA [5], ALMA [3], NORMA [4], Ballseptron [7], and the Passive-Aggressive family of algorithms [1]. Each of these algorithms observes examples in a sequence of rounds, and constructs its classification function incrementally, by storing a subset of the observed examples in its internal memory. The classification function is then defined by a kernel-dependent combination of the stored examples. This set of stored examples is the online equivalent of the support set of SVMs, however in contrast to the support, it continually changes as learning progresses. In this paper, we call this set the active set, as it includes those examples that actively define the current classifier. Typically, an example is added to the active set every time the online algorithm makes a prediction mistake, or when its confidence in a prediction is inadequately low. A rapid growth of the active set can lead to significant computational difficulties. Naturally, since computing devices have bounded memory resources, there is the danger that an online algorithm would require more memory than is physically available. This problem becomes especially eminent in cases where the online algorithm is implemented as part of a specialized hardware system with a small memory, such as a mobile telephone or an autonomous robot. Moreover, an excessively large active set can lead to unacceptably long running times, as the time-complexity of each online round scales linearly with the size of the active set. Crammer, Kandola, and Singer [2] first addressed this problem by describing an online kernel-based modification of the Perceptron algorithm in which the active set does not exceed a predefined budget. Their algorithm removes redundant examples from the active set so as to make the best use of the limited memory resource. Weston, Bordes and Bottou [9] followed with their own online kernel machine on a budget. Both techniques work relatively well in practice, however they both lack a theoretical guarantee on their prediction accuracy. In this paper we present the Forgetron algorithm for online kernel-based classification. To the best of our knowledge, the Forgetron is the first online algorithm with a fixed memory budget which also entertains a formal worst-case mistake bound. We name our algorithm the Forgetron since its update builds on that of the Perceptron and since it gradually forgets active examples as learning progresses. This paper is organized as follows. In Sec. 2 we begin with a more formal presentation of our problem and discuss some difficulties in proving mistake bounds for kernel-methods on a budget. In Sec. 3 we present an algorithmic framework for online prediction with a predefined budget of active examples. Then in Sec. 4 we derive a concrete algorithm within this framework and analyze its performance. Formal proofs of our claims are omitted due to the lack of space. Finally, we present an empirical evaluation of our algorithm in Sec. 5. 2 Problem Setting Online learning is performed in a sequence of consecutive rounds. On round t the online algorithm observes an instance xt, which is drawn from some predefined instance domain X. The algorithm predicts the binary label associated with that instance and is then provided with the correct label yt ∈{−1, +1}. At this point, the algorithm may use the instance-label pair (xt, yt) to improve its prediction mechanism. The goal of the algorithm is to correctly predict as many labels as possible. The predictions of the online algorithm are determined by a hypothesis which is stored in its internal memory and is updated from round to round. We denote the hypothesis used on round t by ft. Our focus in this paper is on margin based hypotheses, namely, ft is a function from X to R where sign(ft(xt)) constitutes the actual binary prediction and |ft(xt)| is the confidence in this prediction. The term yf(x) is called the margin of the prediction and is positive whenever y and sign(f(x)) agree. We can evaluate the performance of a hypothesis on a given example (x, y) in one of two ways. First, we can check whether the hypothesis makes a prediction mistake, namely determine whether y = sign(f(x)) or not. Throughout this paper, we use M to denote the number of prediction mistakes made by an online algorithm on a sequence of examples (x1, y1), . . . , (xT , yT ). The second way we evaluate the predictions of a hypothesis is by using the hinge-loss function, defined as, ℓ f; (x, y) = 0 if yf(x) ≥1 1 −yf(x) otherwise . (1) The hinge-loss penalizes a hypothesis for any margin less than 1. Additionally, if y ̸= sign(f(x)) then ℓ(f, (x, y)) ≥1 and therefore the cumulative hinge-loss suffered over a sequence of examples upper bounds M. The algorithms discussed in this paper use kernelbased hypotheses that are defined with respect to a kernel operator K : X × X →R which adheres to Mercer’s positivity conditions [8]. A kernel-based hypothesis takes the form, f(x) = k X i=1 αiK(xi, x) , (2) where x1, . . . , xk are members of X and α1, . . . , αk are real weights. To facilitate the derivation of our algorithms and their analysis, we associate a reproducing kernel Hilbert space (RKHS) with K in the standard way common to all kernel methods. Formally, let HK be the closure of the set of all hypotheses of the form given in Eq. (2). For any two functions, f(x) = Pk i=1 αiK(xi, x) and g(x) = Pl j=1 βjK(zj, x), define the inner product between them to be, ⟨f, g⟩= Pk i=1 Pl j=1 αiβjK(xi, zj). This innerproduct naturally induces a norm defined by ∥f∥= ⟨f, f⟩1/2 and a metric ∥f −g∥= (⟨f, f⟩−2⟨f, g⟩+ ⟨g, g⟩)1/2. These definitions play an important role in the analysis of our algorithms. Online kernel methods typically restrict themselves to hypotheses that are defined by some subset of the examples observed on previous rounds. That is, the hypothesis used on round t takes the form, ft(x) = P i∈It αiK(xi, x), where It is a subset of {1, . . . , (t-1)} and xi is the example observed by the algorithm on round i. As stated above, It is called the active set, and we say that example xi is active on round t if i ∈It. Perhaps the most well known online algorithm for binary classification is the Perceptron [6]. Stated in the form of a kernel method, the hypotheses generated by the Perceptron take the form ft(x) = P i∈It yiK(xi, x). Namely, the weight assigned to each active example is either +1 or −1, depending on the label of that example. The Perceptron initializes I1 to be the empty set, which implicitly sets f1 to be the zero function. It then updates its hypothesis only on rounds where a prediction mistake is made. Concretely, on round t, if ft(xt) ̸= yt then the index t is inserted into the active set. As a consequence, the size of the active set on round t equals the number of prediction mistakes made on previous rounds. A relative mistake bound can be proven for the Perceptron algorithm. The bound holds for any sequence of instance-label pairs, and compares the number of mistakes made by the Perceptron with the cumulative hinge-loss of any fixed hypothesis g ∈HK, even one defined with prior knowledge of the sequence. Theorem 1. Let K be a Mercer kernel and let (x1, y1), . . . , (xT , yT ) be a sequence of examples such that K(xt, xt) ≤1 for all t. Let g be an arbitrary function in HK and define ˆℓt = ℓ g; (xt, yt) . Then the number of prediction mistakes made by the Perceptron on this sequence is bounded by, M ≤∥g∥2 + 2 PT t=1 ˆℓt. Although the Perceptron is guaranteed to be competitive with any fixed hypothesis g ∈ HK, the fact that its active set can grow without a bound poses a serious computational problem. In fact, this problem is common to most kernel-based online methods that do not explicitly monitor the size of It. As discussed above, our goal is to derive and analyze an online prediction algorithm which resolves these problems by enforcing a fixed bound on the size of the active set. Formally, let B be a positive integer, which we refer to as the budget parameter. We would like to devise an algorithm which enforces the constraint |It| ≤B on every round t. Furthermore, we would like to prove a relative mistake bound for this algorithm, analogous to the bound stated in Thm. 1. Regretfully, this goal turns out to be impossible without making additional assumptions. We show this inherent limitation by presenting a simple counterexample which applies to any online algorithm which uses a prediction function of the form given in Eq. (2), and for which |It| ≤B for all t. In this example, we show a hypothesis g ∈HK and an arbitrarily long sequence of examples such that the algorithm makes a prediction mistake on every single round whereas g suffers no loss at all. We choose the instance space X to be the set of B+1 standard unit vectors in RB+1, that is X = {ei}B+1 i=1 where ei is the vector with 1 in its i’th coordinate and zeros elsewhere. K is set to be the standard innerproduct in RB+1, that is K(x, x′) = ⟨x, x′⟩. Now for every t, ft is a linear combination of at most B vectors from X. Since |X| = B + 1, there exists a vector xt ∈X which is currently not in the active set. Furthermore, xt is orthogonal to all of the active vectors and therefore ft(xt) = 0. Assume without loss of generality that the online algorithm we are using predicts yt to be −1 when ft(x) = 0. If on every round we were to present the online algorithm with the example (xt, +1) then the online algorithm would make a prediction mistake on every round. On the other hand, the hypothesis ¯g = PB+1 i=1 ei is a member of HK and attains a zero hinge-loss on every round. We have found a sequence of examples and a fixed hypothesis (which is indeed defined by more than B vectors from X) that attains a cumulative loss of zero on this sequence, while the number of mistakes made by the online algorithm equals the number of rounds. Clearly, a theorem along the lines of Thm. 1 cannot be proven. One way to resolve this problem is to limit the set of hypotheses we compete with to a subset of HK, which would naturally exclude ¯g. In this paper, we limit the set of competitors to hypotheses with small norms. Formally, we wish to devise an online algorithm which is competitive with every hypothesis g ∈HK for which ∥g∥≤U, for some constant U. Our counterexample indicates that we cannot prove a relative mistake bound with U set to at least √ B + 1, since that was the norm of ¯g in our counterexample. In this paper we come close to this upper bound by proving that our algorithms can compete with any hypothesis with a norm bounded by 1 4 p (B + 1)/ log(B + 1). 3 A Perceptron with “Shrinking” and “Removal” Steps The Perceptron algorithm will serve as our starting point. Recall that whenever the Perceptron makes a prediction mistake, it updates its hypothesis by adding the element t to It. Thus, on any given round, the size of its active set equals the number of prediction mistakes it has made so far. This implies that the Perceptron may violate the budget constraint |It| ≤B. We can solve this problem by removing an example from the active set whenever its size exceeds B. One simple strategy is to remove the oldest example in the active set whenever |It| > B. Let t be a round on which the Perceptron makes a prediction mistake. We apply the following two step update. First, we perform the Perceptron’s update by adding t to It. Let I′ t = It ∪{t} denote the resulting active set. If |I′ t| ≤B we are done and we set It+1 = I′ t. Otherwise, we apply a removal step by finding the oldest example in the active set, rt = min I′ t, and setting It+1 = I′ t \ {rt}. The resulting algorithm is a simple modification of the kernel Perceptron, which conforms with a fixed budget constraint. While we are unable to prove a mistake bound for this algorithm, it is nonetheless an important milestone on the path to an algorithm with a fixed budget and a formal mistake bound. The removal of the oldest active example from It may significantly change the hypothesis and effect its accuracy. One way to overcome this obstacle is to reduce the weight of old examples in the definition of the current hypothesis. By controlling the weight of the oldest active example, we can guarantee that the removal step will not significantly effect the accuracy of our predictions. More formally, we redefine our hypothesis to be, ft = X i∈It σi,tyiK(xi, ·) , where each σi,t is a weight in (0, 1]. Clearly, the effect of removing rt from It depends on the magnitude of σrt,t. Using the ideas discussed above, we are now ready to outline the Forgetron algorithm. The Forgetron initializes I1 to be the empty set, which implicitly sets f1 to be the zero function. On round t, if a prediction mistake occurs, a three step update is performed. The first step is the standard Perceptron update, namely, the index t is inserted into the active set and the weight σt,t is set to be 1. Let I′ t denote the active set which results from this update, and let f ′ t denote the resulting hypothesis, f ′ t(x) = ft(x) + ytK(xt, x). The second step of the update is a shrinking step in which we scale f ′ by a coefficient φt ∈(0, 1]. The value of φt is intentionally left unspecified for now. Let f ′′ t denote the resulting hypothesis, that is, f ′′ t = φtf ′ t. Setting σi,t+1 = φtσi,t for all i ∈I′ t, we can write, f ′′ t (x) = X i∈I′ t σi,t+1yiK(xi, x) . The third and last step of the update is the removal step discussed above. That is, if the budget constraint is violated and |I′ t| > B then It+1 is set to be I′ t \ {rt} where rt = min I′ t. Otherwise, It+1 simply equals I′ t. The recursive definition of the weight σi,t can be unraveled to give the following explicit form, σi,t = Q j∈It−1 ∧j≥i φj. If the shrinking coefficients φt are sufficiently small, then the example weights σi,t decrease rapidly with t, and particularly the weight of the oldest active example can be made arbitrarily small. Thus, if φt is small enough, then the removal step is guaranteed not to cause any significant damage. Alas, aggressively shrinking the online hypothesis with every update might itself degrade the performance of the online hypothesis and therefore φt should not be set too small. The delicate balance between safe removal of the oldest example and over-aggressive scaling is our main challenge. To formalize this tradeoff, we begin with the mistake bound in Thm. 1 and investigate how it is effected by the shrinking and removal steps. We focus first on the removal step. Let J denote the set of rounds on which the Forgetron makes a prediction mistake and define the function, Ψ(σ , φ , µ) = (σ φ)2 + 2 σ φ(1 −φ µ) . Let t ∈J be a round on which |It| = B. On this round, example rt is removed from the active set. Let µt = yrtf ′ t(xrt) be the signed margin attained by f ′ t on the active example being removed. Finally, we abbreviate, Ψt = Ψ(σrt,t , φt , µt) if t ∈J ∧|It| = B 0 otherwise . Lemma 1 below states that removing example rt from the active set on round t increases the mistake bound by Ψt. As expected, Ψt decreases with the weight of the removed example, σrt,t+1. In addition, it is clear from the definition of Ψt that µt also plays a key role in determining whether xrt can be safely removed from the active set. We note in passing that [2] used a heuristic criterion similar to µt to dynamically choose which active example to remove on each online round. Turning to the shrinking step, for every t ∈J we define, Φt = 1 if ∥ft+1∥≥U φt if ∥f ′ t∥≤U ∧∥ft+1∥< U φt∥f ′ t∥ U if ∥f ′ t∥> U ∧∥ft+1∥< U . Lemma 1 below also states that applying the shrinking step on round t increases the mistake bound by U 2 log(1/Φt). Note that if ∥ft+1∥≥U then Φt = 1 and the shrinking step on round t has no effect on our mistake bound. Intuitively, this is due to the fact that, in this case, the shrinking step does not make the norm of ft+1 smaller than the norm of our competitor, g. Lemma 1. Let (x1, y1), . . . , (xT , yT ) be a sequence of examples such that K(xt, xt) ≤1 for all t and assume that this sequence is presented to the Forgetron with a budget constraint B. Let g be a function in HK for which ∥g∥≤U, and define ˆℓt = ℓ g; (xt, yt) . Then, M ≤ ∥g∥2 + 2 T X t=1 ˆℓt ! + X t∈J Ψt + U 2 X t∈J log (1/Φt) ! . The first term in the bound of Lemma 1 is identical to the mistake bound of the standard Perceptron, given in Thm. 1. The second term is the consequence of the removal and shrinking steps. If we set the shrinking coefficients in such a way that the second term is at most M 2 , then the bound in Lemma 1 reduces to M ≤∥g∥2 + 2 P t ˆℓt + M 2 . This can be restated as M ≤2∥g∥2 + 4 P t ˆℓt, which is twice the bound of the Perceptron algorithm. The next lemma states sufficient conditions on φt under which the second term in Lemma 1 is indeed upper bounded by M 2 . Lemma 2. Assume that the conditions of Lemma 1 hold and that B ≥83. If the shrinking coefficients φt are chosen such that, X t∈J Ψt ≤15 32 M and X t∈J log (1/Φt) ≤log(B + 1) 2(B + 1) M , then the following holds, P t∈J Ψt + U 2 P t∈J log (1/Φt) ≤ M 2 . In the next section, we define the specific mechanism used by the Forgetron algorithm to choose the shrinking coefficients φt. Then, we conclude our analysis by arguing that this choice satisfies the sufficient conditions stated in Lemma 2, and obtain a mistake bound as described above. 4 The Forgetron Algorithm We are now ready to define the specific choice of φt used by the Forgetron algorithm. On each round, the Forgetron chooses φt to be the maximal value in (0, 1] for which the damage caused by the removal step is still manageable. To clarify our construction, define Jt = {i ∈J : i ≤t} and Mt = |Jt|. In words, Jt is the set of rounds on which the algorithm made a mistake up until round t, and Mt is the size of this set. We can now rewrite the first condition in Lemma 2 as, X t∈JT Ψt ≤15 32 MT . (3) Instead of the above condition, the Forgetron enforces the following stronger condition, ∀i ∈{1, . . . , T}, X t∈Ji Ψt ≤15 32 Mi . (4) This is done as follows. Define, Qi = P t∈Ji−1 Ψt. Let i denote a round on which the algorithm makes a prediction mistake and on which an example must be removed from the active set. The i’th constraint in Eq. (4) can be rewritten as Ψi + Qi ≤15 32 Mi. The Forgetron sets φi to be the maximal value in (0, 1] for which this constraint holds, namely, φi = max φ ∈(0, 1] : Ψ(σri,i , φ , µi) + Qi ≤15 32Mi . Note that Qi does not depend on φ and that Ψ(σri,i, φ, µi) is a quadratic expression in φ. Therefore, the value of φi can be found analytically. The pseudo-code of the Forgetron algorithm is given in Fig. 1. Having described our algorithm, we now turn to its analysis. To prove a mistake bound it suffices to show that the two conditions stated in Lemma 2 hold. The first condition of the lemma follows immediately from the definition of φt. Using strong induction on the size of J, we can show that the second condition holds as well. Using these two facts, the following theorem follows as a direct corollary of Lemma 1 and Lemma 2. INPUT: Mercer kernel K(·, ·) ; budget parameter B > 0 INITIALIZE: I1 = ∅; f1 ≡0 ; Q1 = 0 ; M0 = 0 For t = 1, 2, . . . receive instance xt ; predict label: sign(ft(xt)) receive correct label yt If ytft(xt) > 0 set It+1 = It, Qt+1 = Qt, Mt = Mt−1, and ∀i ∈It set σi,t+1 = σi,t Else set Mt = Mt−1 + 1 (1) set I′ t = It ∪{t} If |I′ t| ≤B set It+1 = I′ t, Qt+1 = Qt, σt,t = 1, and ∀i ∈It+1 set σi,t+1 = σi,t Else (2) define rt = min It choose φt = max{φ ∈(0, 1] : Ψ(σrt,t , φ , µt) + Qt ≤15 32Mt} set σt,t = 1 and ∀i ∈I′ t set σi,t+1 = φt σi,t set Qt+1 = Qt + Ψt (3) set It+1 = I′ t \ {rt} define ft+1 = P i∈It+1 σi,t+1yiK(xi, ·) Figure 1: The Forgetron algorithm. Theorem 2. Let (x1, y1), . . . , (xT , yT) be a sequence of examples such that K(xt, xt) ≤1 for all t. Assume that this sequence is presented to the Forgetron algorithm from Fig. 1 with a budget parameter B ≥83. Let g be a function in HK for which ∥g∥≤U, where U = 1 4 p (B + 1)/ log(B + 1), and define ˆℓt = ℓ g; (xt, yt) . Then, the number of prediction mistakes made by the Forgetron on this sequence is at most, M ≤2 ∥g∥2 + 4 T X t=1 ˆℓt 5 Experiments and Discussion In this section we present preliminary experimental results which demonstrate the merits of the Forgetron algorithm. We compared the performance of the Forgetron with the method described in [2], which we abbreviate by CKS. When the CKS algorithm exceeds its budget, it removes the active example whose margin would be the largest after the removal. Our experiment was performed with two standard datasets: the MNIST dataset, which consists of 60,000 training examples, and the census-income (adult) dataset, with 200,000 examples. The labels of the MNIST dataset are the 10 digit classes, while the setting we consider in this paper is that of binary classification. We therefore generated binary problems by splitting the 10 labels into two sets of equal size in all possible ways, totaling 10 5 /2 = 126 classification problems. For each budget value, we ran the two algorithms on all 126 binary problems and averaged the results. The labels in the census-income dataset are already binary, so we ran the two algorithms on 10 different permutations of the examples and averaged the results. Both algorithms used a fifth degree non-homogeneous polynomial kernel. The results of these experiments are summarized in Fig. 2. The accuracy of the standard Perceptron (which does not depend on B) is marked in each plot 1000 2000 3000 4000 5000 6000 0.05 0.1 0.15 0.2 0.25 0.3 budget size − B average error Forgetron CKS 200 400 600 800 1000 1200 1400 1600 1800 0.05 0.1 0.15 0.2 0.25 0.3 budget size − B average error Forgetron CKS Figure 2: The error of different budget algorithms as a function of the budget size B on the censusincome (adult) dataset (left) and on the MNIST dataset (right). The Perceptron’s active set reaches a size of 14,626 for census-income and 1,886 for MNIST. The Perceptron’s error is marked with a horizontal dashed black line. using a horizontal dashed black line. Note that the Forgetron outperforms CKS on both datasets, especially when the value of B is small. In fact, on the census-income dataset, the Forgetron achieves almost the same performance as the Perceptron with only a fifth of the active examples. In contrast to the Forgetron, which performs well on both datasets, the CKS algorithm performs rather poorly on the census-income dataset. This can be partly attributed to the different level of difficulty of the two classification tasks. It turns out that the performance of CKS deteriorates as the classification task becomes more difficult. In contrast, the Forgetron seems to perform well on both easy and difficult classification tasks. In this paper we described the Forgetron algorithm which is a kernel-based online learning algorithm with a fixed memory budget. We proved that the Forgetron is competitive with any hypothesis whose norm is upper bounded by U = 1 4 p (B + 1)/ log(B + 1). We further argued that no algorithm with a budget of B active examples can be competitive with every hypothesis whose norm is √ B + 1, on every input sequence. Bridging the small gap between U and √B + 1 remains an open problem. The analysis presented in this paper can be used to derive a family of online algorithms of which the Forgetron is only one special case. This family of algorithms, as well as complete proofs of our formal claims and extensive experiments, will be presented in a long version of this paper. References [1] K. Crammer, O. Dekel, J. Keshet, S. Shalev-Shwartz, and Y. Singer. Online passive aggressive algorithms. Technical report, The Hebrew University, 2005. [2] K. Crammer, J. Kandola, and Y. Singer. Online classification on a budget. NIPS, 2003. [3] C. Gentile. A new approximate maximal margin classification algorithm. JMLR, 2001. [4] J. Kivinen, A. J. Smola, and R. C. Williamson. Online learning with kernels. IEEE Transactions on Signal Processing, 52(8):2165–2176, 2002. [5] Y. Li and P. M. Long. The relaxed online maximum margin algorithm. NIPS, 1999. [6] F. Rosenblatt. The Perceptron: A probabilistic model for information storage and organization in the brain. Psychological Review, 65:386–407, 1958. [7] S. Shalev-Shwartz and Y. Singer. A new perspective on an old perceptron algorithm. COLT, 2005. [8] V. N. Vapnik. Statistical Learning Theory. Wiley, 1998. [9] J. Weston, A. Bordes, and L. Bottou. Online (and offline) on an even tighter budget. AISTATS, 2005.
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Online Discovery and Learning of Predictive State Representations Peter McCracken Department of Computing Science University of Alberta Edmonton, Alberta Canada, T6G 2E8 peterm@cs.ualberta.ca Michael Bowling Department of Computing Science University of Alberta Edmonton, Alberta Canada, T6G 2E8 bowling@cs.ualberta.ca Abstract Predictive state representations (PSRs) are a method of modeling dynamical systems using only observable data, such as actions and observations, to describe their model. PSRs use predictions about the outcome of future tests to summarize the system state. The best existing techniques for discovery and learning of PSRs use a Monte Carlo approach to explicitly estimate these outcome probabilities. In this paper, we present a new algorithm for discovery and learning of PSRs that uses a gradient descent approach to compute the predictions for the current state. The algorithm takes advantage of the large amount of structure inherent in a valid prediction matrix to constrain its predictions. Furthermore, the algorithm can be used online by an agent to constantly improve its prediction quality; something that current state of the art discovery and learning algorithms are unable to do. We give empirical results to show that our constrained gradient algorithm is able to discover core tests using very small amounts of data, and with larger amounts of data can compute accurate predictions of the system dynamics. 1 Introduction Representations of state in dynamical systems fall into three main categories. Methods like k-order Markov models attempt to identify state by remembering what has happened in the past. Methods such as partially observable Markov decision processes (POMDPs) identify state as a distribution over postulated base states. A more recently developed group of algorithms, known as predictive representations, identify state in dynamical systems by predicting what will happen in the future. Algorithms following this paradigm include observable operator models [1], predictive state representations [2, 3], TD-Nets [4] and TPSRs [5]. In this research we focus on predictive state representations (PSRs). PSRs are completely grounded in data obtained from the system, and they have been shown to be at least as general and as compact as other methods, like POMDPs [3]. Until recently, algorithms for discovery and learning of PSRs could be used only in special cases. They have required explicit control of the system using a reset action [6, 5], or have required the incoming data stream to be generated using an open-loop policy [7]. The algorithm presented in this paper does not require a reset action, nor does it make any assumptions about the policy used to generate the data stream. Furthermore, we focus on the online learning problem, i.e., how can an estimate of the current state vector and parameters be maintained and improved during a single pass over a string of data. Like the myopic gradient descent algorithm [8], the algorithm we propose uses a gradient approach to move its predictions closer to its empirical observations; however, our algorithm also takes advantage of known constraints on valid test predictions. We show that this constrained gradient approach is capable of discovering a set of core tests quickly, and also of making online predictions that improve as more data is available. 2 Predictive State Representations Predictive state representations (PSRs) were introduced by Littman et al. [2] as a method of modeling discrete-time, controlled dynamical systems. They possess several advantages over other popular models such as POMDPs and k-order Markov models, foremost being their ability to be learned entirely from sensorimotor data, requiring only a prior knowledge of the set of actions, A, and observations, O. Notation. An agent in a dynamical system experiences a sequence of action-observation pairs, or ao pairs. The sequence of ao pairs the agent has already experienced, beginning at the first time step, is known as a history. For instance, the history hn = a1o1a2o2 . . . anon of length n means that the agent chose action a1 and perceived observation o1 at the first time step, after which the agent chose a2 and perceived o2, and so on1. A test is a sequence of ao pairs that begins immediately after a history. A test is said to succeed if the observations in the sequence are observed in order, given that the actions in the sequence are chosen in order. For instance, the test t = a1o1a2o2 succeeds if the agent observes o1 followed by o2, given that it performs actions a1 followed by a2. A test fails if the action sequence is taken but the observation sequence is not observed. A prediction about the outcome of a test t depends on the history h that preceded it, so we write predictions as p(t|h), to represent the probability of t succeeding after history h. For test t of length n, we define a prediction p(t|h) as Qn i=1 Pr(oi|a1o1 . . . ai). This definition is equivalent to the usual definition in the PSR literature, but makes it explicit that predictions are independent of the policy used to select actions. The special length zero test is called ε. If T is a set of tests and H is a set of histories, p(t|h) is a single value, p(T|h) is a row vector containing p(ti|h) for all tests ti ∈T, p(t|H) is a column vector containing p(t|hj) for all histories hj ∈H, and p(T|H) is a matrix containing p(ti|hj) for all ti ∈T and hj ∈H. PSRS. The fundamental principle underlying PSRs is that in most systems there exists a set of tests, Q, that at any history are a sufficient statistic for determining the probability of success for all possible tests. This means that for any test t there exists a function ft such that p(t|h) = ft (p(Q|h)). In this paper, we restrict our discussion of PSRs to linear PSRs, in which the function ft is a linear function of the tests in Q. Thus, p(t|h) = p(Q|h)mt, where mt is a column vector of weights. The tests in Q are known as core tests, and determining which tests are core tests is known as the discovery problem. In addition to Q, it will be convenient to discuss the set of one-step extensions of Q. A one-step extension of a test t is a test aot, that prefixes the original test with a single ao pair. The set of all one-step extensions of Q ∪{ε} will be called X. The state vector of a PSR at time i is the set of predictions p(Q|hi). At each time step, the 1Much of the notation used in this paper is adopted from Wolfe et al. [7]. Here we use the notation that a superscript ai or oi indicates the time step of an action or observation, and a subscript ai or oi indicates that the action or observation is a particular element of the set A or O. state vector is updated by computing, for each qj ∈Q: p(qj|hi) = p(aioiqj|hi−1) p(aioi|hi−1) = p(Q|hi−1)maioiqj p(Q|hi−1)maioi Thus, in order to update the PSR at each time step, the vector mt must be known for each test t ∈X. This set of update vectors, that we will call mX, are the parameters of the PSR, and estimation of these parameters is known as the learning problem. 3 Constrained Gradient Learning of PSRs The goal of this paper is to develop an online algorithm for discovering and learning a PSR without the necessity of a reset action. To be online, the algorithm must always have an estimate of the current state vector, p(Q|hi), and estimates of the parameters mX. In this section, we introduce our constrained gradient approach to solving this problem. A more complete explanation of this algorithm can be found in an expanded version of this work [9]. To begin, in Section 3.1, we will assume that the set of core tests Q is given to the algorithm; we describe how Q can be estimated online in Section 3.2. 3.1 Learning the PSR Parameters The approach to learning taken by the constrained gradient algorithm is to approximate the matrix p(T|H), for a selected set of tests T and histories H. We first discuss the proper selection of T and H, and then describe how this matrix can be constructed online. Finally, we show how the current PSR is extracted from the matrix. Tests and Histories. At a minimum, T must contain the union of Q and X, since Q is required to create the state vector and X is required to compute mX. However, as will be explained in the next section, these tests are not sufficient to take full advantage of the structure in a prediction matrix. The constrained gradient algorithm requires the tests in T to satisfy two properties: 1. If tao ∈T then t ∈T 2. If taoi ∈T then taoj ∈T ∀oj ∈O To build a valid set of tests, T is initialized to Q ∪X. Tests are iteratively added to T until it satisfies both of the above properties. All histories in H are histories that have been experienced by the agent. The current history, hi, must always be in H in order to make online predictions, and also to compute hi+1. The only other requirement of H is that it contain sufficient histories to compute the linear functions mt for the tests in T (see Section 3.1). Our strategy is impose a bound N on the size of H, and to restrict H to the N most recent histories encountered by the agent. When a new data point is seen and a new row is added to the matrix, the oldest row in the matrix is “forgotten.” In addition to restricting the size of H, forgetting old rows has the side-effect that the rows estimated using the least amount of data are removed from the matrix, and no longer affect the computation of mX. Constructing the Prediction Matrix. The approach used to build the matrix p(T|H) is to estimate and append a new row, p(T|hi), after each new aioi pair is encountered. Once a row has been added, it is never changed. To initialize the algorithm, the first row of the matrix p(T|h0), is set to uniform probabilities.2 The creation of the new row is performed in two stages: a row estimation stage, and a gradient descent stage. 2Each p(t|h0) is set to 1/|O|k, where k is the length of test t. Both stages take advantage of four constraints on the predictions p(T|h) in order to be a valid row in the prediction matrix: 1. Range: 0 ≤p(t|h) ≤1 2. Null Test: p(ε|h) = 1 3. Internal Consistency: p(t|h) = P oj∈O p(taoj|h) ∀a ∈A 4. Conditional Probability: p(t|hao) = p(aot|h)/p(ao|h) ∀a ∈A, o ∈O The range constraint restricts the entries in the matrix to be valid probabilities. The null test constraint defines the value of the null test. The internal consistency constraint ensures that the probabilities within a single row form valid probability distributions. The conditional probability constraint is required to maintain consistency between consecutive rows of the matrix. Consider time i −1 so that the last row of p(T|H) is hi−1. After action ai is taken and observation oi is seen, a new row for history hi = hi−1aioi must be added to the matrix. First, as much of the new row as possible is computed using the conditional probability constraint, and the predictions for history hi−1. For all tests t ∈T for which aioit ∈T: p(t|hi) ←p(aioit|hi−1) p(aioi|hi−1) Because X ⊂T, it is guaranteed that p(Q|hi) is estimated in this step. The second phase of adding a new row is to compute predictions for the tests t ∈T for which aioit ̸∈T. An estimate of p(t|hi) can be found by computing p(Q|hi)mt for an appropriate mt, using the PSR assumption that any prediction is a linear combination of core test predictions. Regression is used to find a vector mt that minimizes ||p(Q|H)mt − p(t|H)||2. At this stage, the entire row for hi has been estimated. The regression step can create probabilities that violate the range and normalization properties of a valid prediction. To enforce the range property, any predictions that are less than 0 are set to a small positive value3. Then, to ensure internal consistency within the row, the normalization property is enforced by setting predictions: p(taoj|hi) ←p(t|hi)p(taoj|hi) P oi∈O p(taoi|hi) ∀oj ∈O This preserves the ratio among sibling predictions and creates a valid probability distribution from them. The normalization is performed by normalizing shorter tests first, which guarantees that a set of tests are not normalized to a value that will later change. The length one tests are normalized to sum to 1. The gradient descent stage of estimating a new row moves the constraint-generated predictions in the direction of the gradient created by the new observation. Any prediction p(tao|hi) whose test tao is successfully executed over the next several time steps is updated using p(tao|hi) ←(1−α)p(tao|hi)+α(p(t|hi)), for some learning rate 0 ≤α ≤1. Note that this learning rule is a temporal difference update; prediction values are adjusted toward the value of their parent.4 The update is accomplished by adding an appropriate positive value to p(tao|hi) and then running the normalization procedure on the row. The value is computed such that after normalization, p(tao|hi) contains the desired value. Tests 3Setting values to zero can cause division by zero errors, if the prediction probability was not actually supposed to be zero. 4When the algorithm is used online, looking forward into the stream is impossible. In this case, we maintain a buffer of ao pairs between the current time step and the histories that are added to the prediction matrix. The length of the buffer is the length of the longest test in T. To compute the predictions for the current time step, we iteratively update the PSR using the buffered data. that are unsuccessfully executed (i.e. their action sequence is executed but their observation sequence is not observed) will have their probability reduced due to this re-normalization step. The learning parameter, α, is decayed throughout the learning process. Extracting the PSR. Once a new row for hi is estimated, the current PSR state vector is p(Q|hi). The parameters mX can be found by using the output of the regression from the second phase, above. Thus, at every time step, the current best estimated PSR of the system is available. 3.2 Discovery of Core Tests In the previous section, we assumed that the set of core tests was given to the algorithm. In general, though, Q is not known. A rudimentary, but effective, method of finding core tests is to choose tests whose corresponding columns of the matrix p(T|H) are most linearly unrelated to the set of core tests already selected. Call the set of selected core tests bQ. The condition number of the matrix p({ bQ, t}|H) is an indication of the linear relatedness of test t; if it is well-conditioned, the test is likely to be linearly independent. To choose core tests, we find the test t in X whose matrix p({ bQ, t}|H) is most well-conditioned. If the condition number of that test is below a threshold parameter, it is chosen as a new core test. The process can be repeated until no test can be added to bQ without surpassing the threshold. Because candidate tests are selected from X, the discovered set bQ will be a regular form PSR [10]. The set bQ is initialized to {ε}. The above core test selection procedure runs after every N data points are seen, where N is the maximum number of histories kept in H. After each new core test is selected, T is augmented with the one-step extensions of the new test, as well as any other tests needed to satisfy the rules in Section 3.1. 4 Experiments and Results The goal of the constrained gradient algorithm is to choose a correct set of core tests and to make accurate, online predictions. In this section, we show empirical results that the algorithm is capable of these goals. We also show offline results, in order to compare our results with the suffix-history algorithm [7]. A more thorough suite of experiments can be found in an expanded version of this work [9]. We tested our algorithm on the same set of problems from Cassandra’s POMDP page [11] used as the test domain in other PSR trials [8, 6, 7]. For each problem, 10 trials were run, with different training sequences and test sequences used for each trial. The sequences were generated using a uniform random policy over actions. The error for each history hi was computed using the error measure 1 |O| P oj∈O(p(ai+1oj|hi) −ˆp(ai+1oj|hi))2 [7]. This measures the mean error in the one-step tests involving the action that was actually taken at step i + 1. The same parameterization of the algorithm was used for all domains. The size bound on H was set to 1000, and the condition threshold for adding new tests was 10. The learning parameter α was initialized to 1 and halved every 100,000 time steps. The core test discovery procedure was run every 1000 data points. 4.1 Discovery Results In this section, we examine the success of the constrained gradient algorithm at discovering core tests. Table 1 shows, for each test domain, the true number of core tests for the Table 1: The number of core tests found by the constrained gradient algorithm. Data for the suffix-history algorithm [7] is repeated here for comparison. See text for explanation. Domain Constrained Gradient Suffix-History Name |Q| | bQ| Correct # Data | bQ|/Correct # Data Float Reset 5 6.1 4.5 4000 Tiger 2 4.0 2.0 1000 2 4000 Paint 2 2.6 2.0 4000 2 4000 Shuttle 7 8.7 7.0 2000 7 1024000 4x3 Maze 10 10.4 8.6 2000 9 1024000 Cheese Maze 11 12.1 9.6 1000 9 32000 Bridge Repair 5 7.2 5.0 1000 5 1024000 Network 7 4.7 4.5 2000 3 2048000 dynamical system (|Q|), the number of core tests selected by the constrained gradient algorithm (| bQ|), and how many of the selected core tests were actually core tests (Correct). The results are averaged over 10 trials. Table 1 also shows the time step at which the last core test was chosen (# Data). In all domains, the algorithm found a majority of the core tests after only several thousand data points; in several cases, the core tests were found after only a single run of the core test selection procedure. Table 1 also shows discovery results published for the suffix-history algorithm [7]. All of the core tests found by the suffix-history algorithm were true core tests. In all cases except the 4x3 Maze, the constrained gradient algorithm was able to find at least as many core tests as the suffix-history method, and required significantly less data. To be fair, the suffix-history algorithm uses a conservative approach of selecting core tests, and therefore requires more data. The constrained gradient algorithm chooses tests that give an early indication of being linearly independent. Therefore, the constrained gradient finds most, or all, core tests extremely quickly, but can also choose tests that are not linearly independent. 4.2 Online and Offline Results Figure 1 shows the performance of the constrained gradient approach, in online and offline settings. The question answered by the online experiments is: How accurately can the constrained gradient algorithm predict the outcome of the next time step? At each time i, we measured the error in the algorithm’s predictions of p(ai+1oj|hi) for each oj ∈O. The ‘Online’ plot in Figure 1 shows the mean online error from the previous 1000 time steps. The question posed for the offline experiments was: What is the long-term performance of the PSRs learned by the constrained gradient algorithm? To test this, we stopped the learning process at different points in the training sequence and computed the current PSR. The initial state vector for the offline tests was set to the column means of p( bQ|H), which approximates the state vector of the system’s stationary distribution. In Figure 1, the ‘Offline’ plot shows the mean error of this PSR on a test sequence of length 10,000. The offline and online performances of the algorithm are very similar. This indicates that, after a given amount of data, the immediate error on the next observation and the long-term error of the generated PSR are approximately the same. This result is encouraging because it implies that the PSR remains stable in its predictions, even in the long term. Previously published [7] performance results for the suffix-history algorithm are also shown in Figure 1. A direct comparison between the performance of the two algorithms is somewhat inappropriate, because the suffix-history algorithm solves the ‘batch’ problem and is able to make multiple passes over the data stream. However, the comparison does show that Float Reset 10−1 10−2 10−3 10−4 10−5 10−6 1K 250K 500K 750K 1000K Tiger Offline Online Suffix-History 10−1 10−2 10−3 10−4 10−5 1K 250K 500K 750K 1000K Paint 10−1 10−2 10−3 10−4 10−5 1K 250K 500K 750K 1000K Network 10−1 10−2 10−3 10−4 1K 250K 500K 750K 1000K Shuttle 10−1 10−2 1K 250K 500K 750K 1000K 4x3 Maze 10−1 10−2 10−3 1K 250K 500K 750K 1000K Cheese Maze 10−1 10−2 10−3 1K 250K 500K 750K 1000K Bridge 10−1 10−2 10−3 1K 250K 500K 750K 1000K Figure 1: The PSR error on the test domains. The x-axis is the length of the sequence used for training, which ranges from 1,000 to 1,000,000. The y-axis shows the mean error on the one-step predictions (Online) or on a test sequence (Offline and Suffix-History). The results for Suffix-History are repeated from previous work [7]. See text for explanations. the constrained gradient approach is competitive with current PSR learning algorithms. The performance plateau in the 4x3 Maze and Network domains is unsurprising, because in these domains only a subset of the correct core tests were found (see Table 1). The plateau in the Bridge domain is more concerning, because in this domains all of the correct core tests were found. We suspect this may be due to a local minimum in the error space; more tests need to be performed to investigate this phenomenon. 5 Future Work and Conclusion We have demonstrated that the constrained gradient algorithm can do online learning and discovery of predictive state representations from an arbitrary stream of experience. We have also shown that it is competitive with the alternative batch methods. There are still a number of interesting directions for future improvement. In the current method of core test selection, the condition of the core test matrix p( bQ|H) is important. If the matrix becomes ill-conditioned, it prevents new core tests from becoming selected. This can happen if the true core test matrix p(Q|H) is poorly conditioned (because some core tests are similar), or if incorrect core tests are added to bQ. To prevent this problem, there needs to be a mechanism for removing chosen core tests if they turn out to be linearly dependent. Also, the condition threshold should be gradually increased during learning, to allow more obscure core tests to be selected. Another interesting modification to the algorithm is to replace the current multi-step estimation of new rows with a single optimization. We want to simultaneously minimize the regression error and next observation error subject to the constraints on valid predictions. This optimization could be solved with quadratic programming. To date, the constrained gradient algorithm is the only PSR algorithm that takes advantage of the sequential nature of the data stream experienced by the agent, and the constraints such a sequence imposes on the system. It handles the lack of a reset action without partitioning histories. Also, at the end of learning the algorithm has an estimate of the current state, instead of a prediction of the initial distribution or a stationary distribution over states. Empirical results show that, while there is room for improvement, the constrained gradient algorithm is competitive in both discovery and learning of PSRs. References [1] Herbert Jaeger. Observable operator models for discrete stochastic time series. Neural Computation, 12(6):1371–1398, 2000. [2] Michael Littman, Richard Sutton, and Satinder Singh. Predictive representations of state. In Advances in Neural Information Processing Systems 14 (NIPS), pages 1555–1561, 2002. [3] Satinder Singh, Michael R. James, and Matthew R. Rudary. Predictive state representations: A new theory for modeling dynamical systems. In Uncertainty in Artificial Intelligence: Proceedings of the Twentieth Conference (UAI), pages 512–519, 2004. [4] Richard Sutton and Brian Tanner. Temporal-difference networks. In Advances in Neural Information Processing Systems 17, pages 1377–1384, 2005. [5] Matthew Rosencrantz, Geoff Gordon, and Sebastian Thrun. Learning low dimensional predictive representations. In Twenty-First International Conference on Machine Learning (ICML), 2004. [6] Michael R. James and Satinder Singh. Learning and discovery of predictive state representations in dynamical systems with reset. In Twenty-First International Conference on Machine Learning (ICML), 2004. [7] Britton Wolfe, Michael R. James, and Satinder Singh. Learning predictive state representations in dynamical systems without reset. In Twenty-Second International Conference on Machine Learning (ICML), 2005. [8] Satinder Singh, Michael Littman, Nicholas Jong, David Pardoe, and Peter Stone. Learning predictive state representations. In Twentieth International Conference on Machine Learning (ICML), pages 712–719, 2003. [9] Peter McCracken. An online algorithm for discovery and learning of prediction state representations. Master’s thesis, University of Alberta, 2005. [10] Eric Wiewiora. Learning predictive representations from a history. In Twenty-Second International Conference on Machine Learning (ICML), 2005. [11] Anthony Cassandra. Tony’s POMDP file repository page. http://www.cs.brown.edu/research/ai/pomdp/examples/index.html, 1999.
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Hyperparameter and Kernel Learning for Graph Based Semi-Supervised Classification Ashish Kapoor†, Yuan (Alan) Qi‡, Hyungil Ahn† and Rosalind W. Picard† †MIT Media Laboratory, Cambridge, MA 02139 {kapoor, hiahn, picard}@media.mit.edu ‡MIT CSAIL, Cambridge, MA 02139 alanqi@csail.mit.edu Abstract There have been many graph-based approaches for semi-supervised classification. One problem is that of hyperparameter learning: performance depends greatly on the hyperparameters of the similarity graph, transformation of the graph Laplacian and the noise model. We present a Bayesian framework for learning hyperparameters for graph-based semisupervised classification. Given some labeled data, which can contain inaccurate labels, we pose the semi-supervised classification as an inference problem over the unknown labels. Expectation Propagation is used for approximate inference and the mean of the posterior is used for classification. The hyperparameters are learned using EM for evidence maximization. We also show that the posterior mean can be written in terms of the kernel matrix, providing a Bayesian classifier to classify new points. Tests on synthetic and real datasets show cases where there are significant improvements in performance over the existing approaches. 1 Introduction A lot of recent work on semi-supervised learning is based on regularization on graphs [5]. The basic idea is to first create a graph with the labeled and unlabeled data points as the vertices and with the edge weights encoding the similarity between the data points. The aim is then to obtain a labeling of the vertices that is both smooth over the graph and compatible with the labeled data. The performance of most of these algorithms depends upon the edge weights of the graph. Often the smoothness constraints on the labels are imposed using a transformation of the graph Laplacian and the parameters of the transformation affect the performance. Further, there might be other parameters in the model, such as parameters to address label noise in the data. Finding a right set of parameters is a challenge, and usually the method of choice is cross-validation, which can be prohibitively expensive for real-world problems and problematic when we have few labeled data points. Most of the methods ignore the problem of learning hyperparameters that determine the similarity graph and there are only a few approaches that address this problem. Zhu et al. [8] propose learning non-parametric transformation of the graph Laplacians using semidefinite programming. This approach assumes that the similarity graph is already provided; thus, it does not address the learning of edge weights. Other approaches include label entropy minimization [7] and evidence-maximization using the Laplace approximation [9]. This paper provides a new way to learn the kernel and hyperparameters for graph based semi-supervised classification, while adhering to a Bayesian framework. The semisupervised classification is posed as a Bayesian inference. We use the evidence to simultaneously tune the hyperparameters that define the structure of the similarity graph, the parameters that determine the transformation of the graph Laplacian, and any other parameters of the model. Closest to our work is Zhu et al. [9], where they proposed a Laplace approximation for learning the edge weights. We use Expectation Propagation (EP), a technique for approximate Bayesian inference that provides better approximations than Laplace. An additional contribution is a new EM algorithm to learn the hyperparameters for the edge weights, the parameters of the transformation of the graph spectrum. More importantly, we explicitly model the level of label noise in the data, while [9] does not do. We provide what may be the first comparison of hyperparameter learning with cross-validation on state-of-the-art algorithms (LLGC [6] and harmonic fields [7]). 2 Bayesian Semi-Supervised Learning We assume that we are given a set of data points X = {x1, .., xn+m}, of which XL = {x1, .., xn} are labeled as tL = {t1, .., tn} and XU = {xn+1, .., xn+m} are unlabeled. Throughout this paper we limit ourselves to two-way classification, thus t ∈{−1, 1}. Our model assumes that the hard labels ti depend upon hidden soft-labels yi for all i. Given the dataset D = [{XL, tL}, XU], the task of semi-supervised learning is then to infer the posterior p(tU|D), where tU = [tn+1, .., tn+m]. The posterior can be written as: p(tU|D) = Z y p(tU|y)p(y|D) (1) In this paper, we propose to first approximate the posterior p(y|D) and then use (1) to classify the unlabeled data. Using the Bayes rule we can write: p(y|D) = p(y|X, tL) ∝p(y|X)p(tL|y) The term, p(y|X) is the prior. It enforces a smoothness constraint and depends upon the underlying data manifold. Similar to the spirit of graph regularization [5] we use similarity graphs and their transformed Laplacian to induce priors on the soft labels y. The second term, p(tL|y) is the likelihood that incorporates the information provided by the labels. In this paper, p(y|D) is inferred using Expectation Propagation, a technique for approximate Bayesian inference [3]. In the following subsections first we describe the prior and the likelihood in detail and then we show how evidence maximization can be used to learn hyperparameters and other parameters in the model. 2.1 Priors and Regularization on Graphs The prior plays a significant role in semi-supervised learning, especially when there is only a small amount of labeled data. The prior imposes a smoothness constraint and should be such that it gives higher probability to the labelings that respect the similarity of the graph. The prior, p(y|X), is constructed by first forming an undirected graph over the data points. The data points are the nodes of the graph and edge-weights between the nodes are based on similarity. This similarity is usually captured using a kernel. Examples of kernels include RBF, polynomial etc. Given the data points and a kernel, we can construct an (n + m) × (n + m) kernel matrix K, where Kij = k(xi, xj) for all i ∈{1, .., n + m}. Lets consider the matrix ˜K, which is same as the matrix K, except that the diagonals are set to zero. Further, if G is a diagonal matrix such that Gii = P j ˜Kij, then we can construct the combinatorial Laplacian (∆= G−˜K) or the normalized Laplacian ( ˜∆= I−G−1 2 ˜KG−1 2 ) of the graph. For brevity, in the text we use ∆as a notation for both the Laplacians. Both the Laplacians are symmetric and positive semidefinite. Consider the eigen decomposition of ∆where {vi} denote the eigenvectors and {λi} the corresponding eigenvalues; thus, we can write ∆= Pn+m i=1 λivivT i . Usually, a transformation r(∆) = Pn+m i=1 r(λi)vivT i that modifies the spectrum of ∆is used as a regularizer. Specifically, the smoothness imposed by this regularizer prefers soft labeling for which the norm yT r(∆)y is small. Equivalently, we can interpret this probabilistically as following: p(y|X) ∝e−1 2 yT r(∆)y = N(0, r(∆)−1) (2) Where r(∆)−1 denotes the pseudo-inverse if the inverse does not exist. Equation (2) suggests that the labelings with the small value of yT r(∆)y are more probable than the others. Note, that when r(∆) is not invertible the prior is improper. The fact that the prior can be written as a Gaussian is advantageous as techniques for approximate inference can be easily applied. Also, different choices of transformation functions lead to different semisupervised learning algorithms. For example, the approach based on Gaussian fields and harmonic functions (Harmonic) [7] can be thought of as using the transformation r(λ) = λ on the combinatorial Laplacian without any noise model. Similarly, the approach based in local and global consistency (LLGC) [6] can be thought of as using the same transformation but on the normalized Laplacian and a Gaussian likelihood. Therefore, it is easy to see that most of these algorithms can exploit the proposed evidence maximization framework. In the following we focus only on the parametric linear transformation r(λ) = λ + δ. Note that this transformation removes zero eigenvalues from the spectrum of ∆. 2.2 The Likelihood Assuming conditional independence of the observed labels given the hidden soft labels, the likelihood p(tL|y) can be written as p(tL|y) = Qn i=1 p(ti|yi). The likelihood models the probabilistic relation between the observed label ti and the hidden label yi. Many realworld datasets contain hand-labeled data and can often have labeling errors. While most people tend to model label errors with a linear or quadratic slack in the likelihood, it has been noted that such an approach does not address the cases where label errors are far from the decision boundary [2]. The flipping likelihood can handle errors even when they are far from the decision boundary and can be written as: p(ti|yi) = ϵ(1 −Φ(yi · ti)) + (1 −ϵ)Φ(yi · ti) = ϵ + (1 −2ϵ)Φ(yi · ti) (3) Here, Φ is the step function, ϵ is the labeling error rate and the model admits possibility of errors in labeling with a probability ϵ. This likelihood has been earlier used in the context of Gaussian process classification [2][4]. The above described likelihood explicitly models the labeling error rate; thus, the model should be more robust to the presence of label noise in the data. The experiments in this paper use the flipping noise likelihood shown in (3). 2.3 Approximate Inference In this paper, we use EP to obtain a Gaussian approximation of the posterior p(y|D). Although, the prior derived in section 2.1 is a Gaussian distribution, the exact posterior is not a Gaussian due to the form of the likelihood. We use EP to approximate the posterior as a Gaussian and then equation (1) can be used to classify unlabeled data points. EP has been previously used [3] to train a Bayes Point Machine, where EP starts with a Gaussian prior over the classifiers and produces a Gaussian posterior. Our task is very similar and we use the same algorithm. In our case, EP starts with the prior defined in (2) and incorporates likelihood to approximate the posterior p(y|D) ∼N(¯y, Σy). 2.4 Hyperparameter Learning We use evidence maximization to learn the hyperparameters. Denote the parameters of the kernel as ΘK and the parameters of transformation of the graph Laplacian as ΘT . Let Θ = {ΘK, ΘT , ϵ}, where ϵ is the noise hyperparameter. The goal is to solve ˆΘ = arg maxΘ log[p(tL|X, Θ)]. Non-linear optimization techniques, such as gradient descent or Expectation Maximization (EM) can be used to optimize the evidence. When the parameter space is small then the Matlab function fminbnd, based on golden section search and parabolic interpolation, can be used. The main challenge is that the gradient of evidence is not easy to compute. Previously, an EM algorithm for hyperparameter learning [2] has been derived for Gaussian Process classification. Using similar ideas we can derive an EM algorithm for semisupervised learning. In the E-step EP is used to infer the posterior q(y) over the soft labels. The M-step consists of maximizing the lower bound: F = Z y q(y) log p(y|X, Θ)p(tL|y, Θ) q(y) = − Z y q(y) log q(y) + Z y q(y) log N(y; 0, r(∆)−1) + n X i=1 Z yi q(yi) log (ϵ + (1 −2ϵ)Φ(yi · ti)) ≤p(tL|X, Θ) The EM procedure alternates between the E-step and the M-step until convergence. • E-Step: Given the current parameters Θi, approximate the posterior q(y) ∼ N(¯y, Σy) by EP. • M-Step: Update Θi+1 = arg maxΘ R y q(y) log p(y|X,Θ)p(tL|y,Θ) q(y) In the M-step the maximization with respect to the Θ cannot be computed in a closed form, but can be solved using gradient descent. For maximizing the lower bound, we used gradient based projected BFGS method using Armijo rule and simple line search. When using the linear transformation r(λ) = λ + δ on the Laplacian ∆, the prior p(y|X, Θ) can be written as N(0, (∆+δI)−1). Define Z = ∆+δI then, the gradients of the lower bound with respect to the parameters are as follows: ∂F ∂ΘK = 1 2tr(Z−1 ∂∆ ∂ΘK ) −1 2 ¯yT ∂∆ ∂ΘK ¯y −1 2tr( ∂∆ ∂ΘK Σy) ∂F ∂ΘT = 1 2tr(Z−1) −1 2 ¯yT ¯y −1 2tr(Σy) ∂F ∂ϵ ≈ n X i=1 1 −2Φ(ti · ¯yi) ϵ + (1 −2ϵ)Φ(ti · ¯yi) where: ¯yi = Z y yiq(y) It is easy to show that the provided approximation of the derivative ∂F ∂ϵ equals zero, when ϵ = k n, where k is the number of labeled data points differing in sign from their posterior means. The EM procedure described here is susceptible to local minima and in a few cases might be too slow to converge. Especially, when the evidence curve is flat and the initial values are far from the optimum, we found that the EM algorithm provided very small steps, thus, taking a long time to converge. Whenever we encountered this problem in the experiments, we used an approximate gradient search to find a good value of initial parameters for the EM algorithm. Essentially as the gradients of the evidence are hard to compute, they can be approximated by the gradients of the lower bound and can be used in any gradient ascent procedure. 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 −30 −25 −20 −15 −10 −5 0 5 10 Hyperparameter (σ) for RBF kernel Log evidence Evidence curves (moon data,δ = 10−4,ε=0.1) N=1 N=6 N=10 N=20 100 150 200 250 300 350 400 −50 −40 −30 −20 −10 0 Hyperparameter (σ) for RBF kernel Log evidence Evidence curves (odd vs even, δ=10−4, ε=0) N=5 N=15 N=25 N=40 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 −70 −60 −50 −40 −30 −20 −10 0 Hyperparameter (γ) for the kernel Log evidence Evidence curves (PC vs MAC, δ = 10−4, ε=0) N=5 N=15 N=35 N=50 (a) (b) (c) 0 0.2 0.4 0.6 70 75 80 85 90 95 100 Hyperparameter (σ) for RBF kernel Log evidence/Recognition Rate Recognition and log evidence (moon data N=10) Log evidence Accuracy 100 150 200 250 300 350 400 60 65 70 75 80 85 90 95 100 Hyperparameter (σ) for RBF kernel Log evidence/Recognition Rate Recognition and log evidence (odd vs even N=25) Log evidence Accuracy 0 0.1 0.2 75 80 85 90 95 100 105 Hyperparameter (γ) for the kernel Log evidence/Recognition Rate Recognition and log evidence (PC vs MAC N=35) Log evidence Accuracy (d) (e) (f) Figure 1: Evidence curves showing similar properties across different datasets (half-moon, odd vs even and PC vs MAC). The top row figures (a), (b) and (c) show the evidence curves for different amounts of labeled data per class. The bottom row figures (d), (e) and (f) show the correlation between recognition accuracy on unlabeled points and the evidence. 2.5 Classifying New Points Since we compute a posterior distribution over the soft-labels of the labeled and unlabeled data points, classifying a new point is tricky. Note, that from the parameterization lemma for Gaussian Processes [1] it follows that given a prior distribution p(y|X) ∼ N(0, r(∆)−1), the mean of the posterior p(y|D) is a linear combination of the columns of r(∆)−1. That is: ¯y = r(∆)−1a where, a ∈IR(n+m)×1 Further, if the similarity matrix K is a valid kernel matrix1 then we can write the mean directly in terms of the linear combination of the columns of K: ¯y = KK−1r(∆)−1a = Kb (4) Here, b = [b1, .., bn+m]T is a column vector and is equal to K−1r(∆)−1a. Thus, we have that ¯yi = Pn+m j=1 bj · K(xi, xj). This provides a natural extension of the framework to classify new points. 3 Experiments We performed experiments to evaluate the three main contributions of this work: Bayesian hyperparameter learning, classification of unseen data points, and robustness with respect to noisy labels. For all the experiments we use the linear transformation r(λ) = λ + δ either on normalized Laplacian (EP-NL) or the combinatorial Laplacian (EP-CL). The experiments were performed on one synthetic (Figure 4(a)) and on three real-world datasets. Two real-world datasets were the handwritten digits and the newsgroup data from [7]. We evaluated the task of classifying odd vs even digits (15 labeled, 485 unlabeled and rest new 1The matrix K is the adjacency matrix of the graph and depending upon the similarity criterion might not always be positive semi-definite. For example, discrete graphs induced using K-nearest neighbors might result in K that is not positive semi-definite. 0 0.1 0.2 0.3 0.4 0.5 −60 −50 −40 −30 −20 −10 0 Noise parameter (ε) Log evidence Evidence curves (affect data, K=3, δ=10−4) N=5 N=15 N=25 N=40 1 2 3 4 5 6 7 8 9 10 −50 −45 −40 −35 −30 −25 −20 −15 −10 −5 K in K−nearest neigbors Log evidence Evidence curves (affect data, δ = 10−4, ε = 0.05) N=5 N=15 N=25 N=40 0 0.02 0.04 0.06 0.08 0.1 0.12 −45 −40 −35 −30 −25 −20 −15 −10 −5 Transformation parameter (δ) Log evidence Evidence curves (affect data, K = 3, ε = 0.05) N=5 N=15 N=25 N=40 (a) (b) (c) Figure 2: Evidence curves showing similar properties across different parameters of the model. The figures (a), (b) and (c) show the evidence curves for different amount of labeled data per class for the three different parameters in the model. 1−NN Harmonic LLGC EP−CL EP−NL 15 20 25 Error rate on unlabeled points (odd vs even) 1−NN Harmonic LLGC EP−NL EP−CL 10 20 30 40 Error rate on unlabeled points (PC vs MAC) (a) (b) 1−NN Harmonic LLGC EP−CL EP−NL 15 20 25 Error rate on new points (odd vs even) 1−NN Harmonic LLGC EP_NL EP−CL 15 20 25 30 35 40 Error rate on new points (PC vs MAC) (c) (d) Figure 3: Error rates for different algorithms on digits (first column, (a) and (c)) and newsgroup dataset (second column (b) and (d)). The figures in the top row (a) and (b) show error rates on unlabeled points and the bottom row figures (c) and (d) on the new points. The results are averaged over 5 runs. Non-overlapping of error bars, the standard error scaled by 1.64, indicates 95% significance of the performance difference. (unseen) points per class) and classifying PC vs MAC (5 labeled, 895 unlabeled and rest as new (unseen) points per class). An RBF kernel was used for handwritten digits, whereas kernel K(xi, xj) = exp[−1 γ (1 −xi T xj |xi||xj|)] was used on 10-NN graph to determine similarity. The third real-world dataset labels the level of interest (61 samples of high interest and 75 samples of low interest) of a child solving a puzzle on the computer. Each data point is a 19 dimensional real vector summarizing 8 seconds of activity from the face, posture and the puzzle. The labels in this database are suspected to be noisy because of human labeling. All the experiments on this data used K-nearest neighbor to determine the kernel matrix. Hyperparameter learning: Figure 1 (a), (b) and (c) plots log evidence versus kernel parameters that determine the similarity graphs for the different datasets with varying size of the labeled set per class. The value of δ and ϵ were fixed to the values shown in the plots. Figure 2 (a), (b) and (c) plots the log evidence versus the noise parameter (ϵ), the kernel parameter (k in k-NN) and the transformation parameter (δ) for the affect dataset. First, we see that the evidence curves generated with very little data are flat and as the number of labeled data points increases we see the curves become peakier. When there is very little labeled data, there is not much information available for the evidence maximization framework to prefer one parameter value over the other. With more labeled data, the evidence curves become more informative. Figure 1 (d), (e) and (f) show the correlation between the evidence curves and the recognition rate on the unlabeled data and reveal that the recognition over the unlabeled data points is highly correlated with the evidence. Note that both of these effects are observed across all the datasets as well as all the different parameters, justifying evidence maximization for hyperparameter learning. −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −1.5 −1 −0.5 0 0.5 1 1.5 Toy Data noisy label noisy label (a) (b) (c) Figure 4: Semi-supervised classification in presence of label noise. (a) Input data with label noise. Classification (b) without flipping noise model and with (c) flipping noise model. How good are the learnt parameters? We performed experiments on the handwritten digits and on the newsgroup data and compared with 1-NN, LLGC and Harmonic approach. The kernel parameters for both LLGC and Harmonic were estimated using leave one out cross validation2. Note that both the approaches can be interpreted in terms of the new proposed Bayesian framework (see sec 2.1). We performed experiments with both the normalized (EP-NL) and the combinatorial Laplacian (EP-CL) with the proposed framework to classify the digits and the newsgroup data. The approximate gradient descent was first used to find an initial value of the kernel parameter for the EM algorithm. All three parameters were learnt and the top row in figure 3 shows the average error obtained for 5 different runs on the unlabeled points. On the task of classifying odd vs even the error rate for EP-NL was 14.46±4.4%, significantly outperforming the Harmonic (23.98±4.9%) and 1-NN (24.23±1.1%). Since the prior in EP-NL is determined using the normalized Laplacian and there is no label noise in the data, we expect EP-NL to at least work as well as LLGC (16.02 ± 1.1%). Similarly for the newsgroup dataset EP-CL (9.28±0.7%) significantly beats LLGC (18.03±3.5%) and 1-NN (46.88±0.3%) and is better than Harmonic (10.86±2.4%). Similar, results are obtained on new points as well. The unseen points were classified using eq. (4) and the nearest neighbor rule was used for LLGC and Harmonic. Handling label noise: Figure 4(a) shows a synthetic dataset with noisy labels. We performed semi-supervised classification both with and without the likelihood model given in (3) and the EM algorithm was used to tune all the parameters including the noise (ϵ). Besides modifying the spectrum of the Laplacian, the transformation parameter δ can also be considered as latent noise and provides a quadratic slack for the noisy labels [2]. The results are shown in figure 4 (b) and (c). The EM algorithm can correctly learn the noise parameter resulting in a perfect classification. The classification without the flipping model, even with the quadratic slack, cannot handle the noisy labels far from the decision boundary. Is there label noise in the data? It was suspected that due to the manual labeling the affect dataset might have some label noise. To confirm this and as a sanity check, we first plotted evidence using all the available data. For all the semi-supervised methods in these experiments, we use 3-NN to induce the adjacency graph. Figure 5(a) shows the plot for the evidence against the noise parameter (ϵ). From the figure, we see that the evidence peaks at ϵ = 0.05 suggesting that the dataset has around 5% of labeling noise. Figure 5(b) shows comparisons with other semi-supervised (LLGC and SVM with graph kernel) and supervised methods (SVM with RBF kernel) for different sizes of the labeled dataset. Each point in the graph is the average error on 20 random splits of the data, where the error bars represent the standard error. EM was used to tune ϵ and δ in every run. We used the same transformation r(λ) = λ + δ on the graph kernel in the semi-supervised SVM. The hyperparameters in both the SVMs (including δ for the semi-supervised case) were estimated using leave one out. When the number of labeled points are small, both 2Search space for σ (odd vs even) was 100 to 400 with increments of 10 and for γ (PC vs MAC) was 0.01 to 0.2 with increments of 0.1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −95 −90 −85 −80 −75 −70 −65 −60 Noise parameter (ε) Log evidence Evidence curve for the whole affect data (K=3, δ=10−4) 5 10 15 20 25 30 20 25 30 35 40 45 Number of labels per class Error on unlabeled data EP−NL LLGC SVM (RBF) SVM (N−Laplacian) Harmonic (N = 92) EP−NL (N = 80) 0.5 1.0 1.5 Error rate on unlabeled points (1 vs 2) (a) (b) (c) Figure 5: (a) Evidence vs noise parameter plotted using all the available data in the affect dataset. The maximum at ϵ = 0.05 suggests that there is around 5% label noise in the data. (b) Performance comparison of the proposed approach with LLGC, SVM using graph kernel and the supervised SVM (RBF kernel) on the affect dataset which has label noise. The error bars represent the standard error. (c) Comparison of the proposed EM method for hyperparameter learning with the result reported in [7] using label entropy minimization. The plotted error bars represent the standard deviation. LLGC and EP-NL perform similarly beating both the SVMs, but as the size of the labeled data increases we see a significant improvement of the proposed approach over the other methods. One of the reasons is when you have few labels the probability of the labeled set of points containing a noisy label is low. As the size of the labeled set increases the labeled data has more noisy labels. And, since LLGC has a Gaussian noise model, it cannot handle flipping noise well. As the number of labels increase, the evidence curve turns informative and EP-NL starts to learn the label noise correctly, outperforming the other Both the SVMs show competitive performance with more labels but still are worse than EP-NL. Finally, we also test the method on the task of classifying “1” vs “2” in the handwritten digits dataset. With 40 labeled examples per class (80 total labels and 1800 unlabeled), EP-NL obtained an average recognition accuracy of 99.72 ± 0.04% and figure 5(c) graphically shows the gain over the accuracy of 98.56 ± 0.43% reported in [7], where the hyperparameter were learnt by minimizing label entropy with 92 labeled and 2108 unlabeled examples. 4 Conclusion We presented and evaluated a Bayesian framework for learning hyperparameters for graphbased semi-supervised classification. The results indicate that evidence maximization works well for learning hyperparameters, including the amount of label noise in the data. References [1] Csato, L. (2002) Gaussian processes-iterative sparse approximation. PhD Thesis, Aston Univ. [2] Kim, H. & Ghahramani, Z. (2004) The EM-EP algorithm for Gaussian process classification. ECML. [3] Minka, T. P. (2001) Expectation propagation for approximate Bayesian inference. UAI. [4] Opper, M. & Winther, O. (1999) Mean field methods for classification with Gaussian processes. NIPS. [5] Smola, A. & Kondor, R. (2003) Kernels and regularization on graphs. COLT. [6] Zhou et al. (2004) Learning with local and global consistency. NIPS. [7] Zhu, X., Ghahramani, Z. & Lafferty, J. (2003) Semi-supervised learning using Gaussian fields and harmonic functions. ICML. [8] Zhu, X., Kandola, J., Ghahramani, Z. & Lafferty, J. (2004) Nonparametric transforms of graph kernels for semi-supervised learning. NIPS. [9] Zhu, X., Lafferty, J. & Ghahramani, Z. (2003) Semi-supervised learning: From Gaussian fields to Gaussian processes. CMU Tech Report:CMU-CS-03-175.
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Fusion of Similarity Data in Clustering Tilman Lange and Joachim M. Buhmann (langet,jbuhmann)@inf.ethz.ch Institute of Computational Science, Dept. of Computer Sience, ETH Zurich, Switzerland Abstract Fusing multiple information sources can yield significant benefits to successfully accomplish learning tasks. Many studies have focussed on fusing information in supervised learning contexts. We present an approach to utilize multiple information sources in the form of similarity data for unsupervised learning. Based on similarity information, the clustering task is phrased as a non-negative matrix factorization problem of a mixture of similarity measurements. The tradeoff between the informativeness of data sources and the sparseness of their mixture is controlled by an entropy-based weighting mechanism. For the purpose of model selection, a stability-based approach is employed to ensure the selection of the most self-consistent hypothesis. The experiments demonstrate the performance of the method on toy as well as real world data sets. 1 Introduction Clustering has found increasing attention in the past few years due to the enormous information flood in many areas of information processing and data analysis. The ability of an algorithm to determine an interesting partition of the set of objects under consideration, however, heavily depends on the available information. It is, therefore, reasonable to equip an algorithm with as much information as possible and to endow it with the capability to distinguish between relevant and irrelevant information sources. How to reasonably identify a weighting of the different information sources such that an interesting group structure can be successfully uncovered, remains, however, a largely unresolved issue. Different sources of information about the same objects naturally arise in many application scenarios. In computer vision, for example, information sources can consist of plain intensity measurements, edge maps, the similarity to other images or even human similarity assessments. Similarly in bio-informatics: the similarity of proteins,e.g., can be assessed in different ways, ranging from the comparison of gene profiles to direct comparisons at the sequence level using alignment methods. In this work, we use a non-negative matrix factorization approach (nmf) to pairwise clustering of similarity data that is extended in a second step in order to incorporate a suitable weighting of multiple information sources, leading to a mixture of similarities. The latter represents the main contribution of this work. Algorithms for nmf have recently found a lot of attention. Our proposal is inspired by the work in [11] and [5]. Only recently, [18] have also employed a nmf to perform clustering. For the purpose of model selection, we employ a stability-based approach that has already been successfully applied to model selection problems in clustering (e.g. in [9]). Instead of following the strategy to first embed the similarities into a space with Euclidean geometry and then to perform clustering and, where required, feature selection/weighting on the stacked feature vector, we advocate an approach that is closer to the original similarity data by performing nmf. Some work has been devoted to feature selection and weighting in clustering problems. In [13] a variant of the k-means algorithm has been studied that employs the Fisher criterion to assess the importance of individual features. In [14, 10], Gaussian mixture model-based approaches to feature selection are introduced. The more general problem of learning a suitable metric has also been investigated, e.g. in [17]. Similarity measurements represent a particularly generic form of providing input to a clustering algorithm. Fusing such representations has only recently been studied in the context of kernel-based supervised learning, e.g. in [7] using semi-definite programming and in [3] using a boosting procedure. In [1], an approach to learning the bandwidth parameter of an rbf-kernel for spectral clustering is studied. The paper is organized as follows: section 2 introduces the nmf-based clustering method combined with a data-source weighting (section 3). Section 4 discusses an out-of-sample extension allowing us to predict assignments and to employ the stability principle for model selection. Experimental evidence in favor of our approach is given in section 5. 2 Clustering by Non-Negative Matrix Factorization Suppose we want to group a finite set of objects On := {o1, . . . , on}. Usually, there are multiple ways of measuring the similarity between different objects. Such relations give rise to similarities sij := s(oi, oj) 1 where we assume non-negativity sij ≥0, symmetry sji = sij, and boundedness sij < ∞. For n objects, we summarize the similarity data in a n×n matrix S = (sij) which is re-normalized to P = S/1t nS1n, where 1n := (1, . . . , 1)t. The re-normalized similarities can be interpreted as the probability of the joint occurrence of objects i, j. We aim now at finding a non-negative matrix factorization of P ∈[0, 1]n×n into a product WHt of the n × k matrices W and H with non-negative entries for which additionally holds 1t nW1k = 1 and Ht1n = 1k, where k denotes the number of clusters. That is, one aims at explaining the overall probability for a co-occurrence by a latent cause, the unobserved classes. The constraints ensure, that the entries of both, W and H, can be considered as probabilities: the entry wiν of W is the joint probability q(i, ν) of object i and class ν whereas hjk in H is the probability q(j|ν). This model implicitly assumes independence of object i and j conditioned on ν. Given a factorization of P in W and H, we can use the maximum a posteriori estimate, arg maxν hiν P j wjν, to arrive at a hard assignment of objects to classes. In order to obtain a factorization, we minimize the cross-entropy C(P∥WHt) := − X i,j pij log X ν wiνhjν (1) which becomes minimal iff P = WHt 2 and is not convex in W and H together. Note, that the factorization is not necessarily unique. We resort to a local optimization scheme, which is inspired by the Expectation-Maximization (EM) algorithm: Let τνij ≥0 with P ν τνij = 1. Then, by the convexity of −log x, we obtain −log P ν wiνhjν ≤−P ν τνij log wiνhjν τνij , 1In the following, we represent objects by their indices. 2The Kullback-Leibler divergence is D(P∥WHt) = −H(P)+C(P∥WHt) ≥0 with equality iff P = WHt. which yields the relaxed objective function: ˜C(P∥WHt) := − X i,j,ν pijτνij log wiνhjν + τνij log τνij ≥C(P∥WHt). (2) With this relaxation, we can employ an alternating minimization scheme for minimizing the bound on C. As in EM, one iterates 1. Given W and H, minimize ˜C w.r.t. τνij 2. Given the values τνij, find estimates for W and H by minimizing ˜C. until convergence, which produces a sequence of estimates τ (t) νij = w(t) iν h(t) jν P µ w(t) iµ h(t) jµ , w(t+1) iν = X j pijτ (t) νij, h(t+1) jν = P i pijτ (t) νij P a,b pabτ (t) νab (3) that converges to a local minimum of ˜C. This is an instance of an MM algorithm [8]. We use the convention hjν = 0 whenever P i,j pijτνij = 0. The per-iteration complexity is O(n2). 3 Fusing Multiple Data Sources Measuring the similarity of objects in, say, L different ways results in L normalized similarity matrices P1, . . . , PL. We introduce now weights αl, 1 ≤l ≤L, with P l αl = 1. For fixed ααα = (αl) ∈[0, 1]L, the aggregated and normalized similarity becomes the convex combination ¯P = P l αlPl. Hence, ¯pij is a mixture of individual similarities p(l) ij , i.e. a mixture of different explanations. Again, we seek a good factorization of ¯P by minimizing the cross-entropy, which then becomes min ααα,W,H Eααα C(Pl∥WHt) (4) where Eααα[fl] = P l αlfl denotes the expectation w.r.t. the discrete distribution ααα. The same relaxation as in the last section can be used, i.e. for all ααα, W and H, we have Eααα [C(Pl∥WHt)] ≤Eααα[ ˜C(Pl∥WHt)]. Hence, we can employ a slightly modified, nested alternating minimization approach: Given fixed ααα, obtain estimates W and H using the relaxation of the last section. The update equations change to w(t+1) iν = X l αl X j p(l) ij τ (t) νij, h(t+1) jν = P l αl P i p(l) ij τ (t) νij P l αl P i,j p(l) ij τ (t) νij . (5) Given the current estimates of W and H, we could minimize the objective in equation (4) w.r.t. ααα subject to the constraint ∥ααα∥1 = 1. To this end, set cl := C(Pl∥WHt) and let c = (cl)l. Minimizing the expression in equation (4) subject to the constraints P l αl = 1 and ααα ⪰0, therefore, becomes a linear program (LP) minααα ctααα such that 1t Lααα = 1, ααα ⪰0, where ⪰denotes the element-wise ≥-relation. The LP solution is very sparse since the optimal solutions for the linear program lie on the corners of the simplex in the positive orthant spanned by the constraints. In particular, it lacks a means to control the sparseness of the coefficients ααα. We, therefore, use a maximum entropy approach ([6]) for sparseness control: the entropy is upper bounded by log L and measures the sparseness of the vector ααα, since the lower the entropy the more peaked the distribution ααα can be. Hence, by lower bounding the entropy, we specify the maximal admissible sparseness. This approach is reasonable as we actually want to combine multiple (not only identify one) information sources but the best fit in an unsupervised problem will be usually obtained by choosing only a single source. Thus, we modify the objective originally given in eq. (4) to the entropy-regularized problem Eααα[ ˜C(Pl∥WHt)] −ηH(ααα), so that the mathematical program given above becomes min ααα ctααα −ηH(ααα) s.t. 1t Lααα = 1, ααα ⪰0, (6) where H denotes the (discrete) entropy and η ∈R+ is a positive Lagrange parameter. The optimization problem in eq. (6) has an analytical solution, namely the Gibbs distribution αl ∝exp(−cl/η) (7) For η →∞one obtains αl = 1/L, while for η →0, the LP solution is recovered and the estimates become the sparser the more the individual cl differ. Put differently, the parameter η enables us to explore the space of different similarity combinations. The issue of selecting a reasonable value for the parameter η will be discussed in the next section. Iterating this nested procedure will yield a locally optimal solution to the problem of minimizing the entropy-constrained objective, since (i) we obtain a local minimum of the modified objective function and (ii) solving the outer optimization problem can only further decrease the entropy-constrained objective function. 4 Generalization and Model Selection In this section, we introduce an out-of-sample extension that allows us to classify objects, that have not been used for learning the parameters ααα, W and H. The extension mechanism can be seen as in spirit of the Nystr¨om extension (c.f. [16]). Introducing such a generalization mechanism is worthwhile for two reasons: (i) To speed-up the computation if the number n of objects under consideration is very large: By selecting a small subset of m ≪n objects for the initial fit followed by the application of a computationally less expensive prediction step, one can realize such a speed-up. (ii) The free parameters of the approach, the number of clusters k as well as the sparseness control parameter η, can be estimated using a re-sampling-based stability assessment that relies on the ability of an algorithm to generalize to previously unseen objects. Out-of-Sample Extension: Suppose we have to predict class memberships for r (= n − m in the hold-out case) additional objects in the r ×m matrix ˜Sl. Given the decomposition into W and H, let zik be the “posterior” estimated for the i-th object in the data set used for the original fit, i.e. ziν ∝hiν P j wjν. We can express the weighted, normalized similarity between a new object o and object i as ˆpio := P l αl˜s(l) oi / P l,j αl˜s(l) oj . We approximate now zoν for a new object o by ˆzoν = X i ziν ˆpio, (8) which amounts to an interpolation of the zoν. These values can be obtained using the originally computed ziν which are weighted according to their similarity between object i and o. In the analogy to the Nystr¨om approximation, the (ziν) play the role of basis elements while the ˆpio amount to coefficients in the basis approximation. The prediction procedure requires O(mr(l + r + k)) steps. Model Selection: The approach presented so far has two free parameters, the number of classes k and the sparseness penalty η. In [9], a method for determining the number of classes has been introduced, that assesses the variability of clustering solutions. Thus, we focus on selecting η using stability. The assessment can be regarded as a generalization of cross-validation, as it relies on the dissimilarity of solutions generated from multiple sub-samples. In a second step, the solutions obtained from these samples are extended to the complete data set by an appropriate predictor. Multiple classifications of the same data (a) 10 −3 10 −1 10 0 10 2 10 3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 sparsity parameter η avg. disagreement (b) 1 2 3 4 5 6 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 data source index probability αl (c) Figure 1: Results on the toy data set (1(a)): The stability assessment (1(b)) suggests the range η ∈{101, 102, 5 · 102}, which yield solutions matching the ground-truth. In 1(c), the αl are depicted for a sub-sample and η in this range. set are obtained, whose similarity can be measured. For two clustering solutions Y, Y′ ∈ {1, . . . , k}n, we define their disagreement as d(Y, Y′) = min π∈Sk 1 n n X i=1 I{yi̸=π(y′ i)} (9) where Sk denotes the set of all permutation on sets of size k and IA is the indicator function on the expression A. The measure quantifies the 0-1 loss after the labels have been permuted, so that the two clustering solutions are in the best possible agreement. Perfect agreement up to a permutation of the labels implies d(Y, Y′) = 0. The optimal permutation can be determined in O(k3) by phrasing the problem as a weighted bipartite matching problem. Following the approach in [9], we select the η, given a pre-specified range of admissible values, such that the average disagreement observed on B sub-samples is minimal. In this sense, the entropy regularization mechanism guides the search for similarity combinations leading to stable grouping solutions. Note that, multiple minima can occur and may yield solutions emphasizing different aspects of the data. 5 Experimental Results and Discussion The performance of our proposal is explored by analyzing toy and real world data. For the model selection (sec. 4), we have used B = 20 sub-samples with the proposed out-ofsample extension for prediction. For the stability assessment, different η have been chosen by η ∈{10−3, 10−2, 10−1, .5, 1, 101, 102, 5·102, 103, 104}. We compared our results with NCut [15] and Lee and Seung’s two NMF algorithms [11] (which measure the approximation error of the factorization with (i) the KL divergence and (ii) the squared Frobenius norm) applied to the uniform combination of similarities. Toy Experiment: Figure 1(a) depicts a data set consisting of two nested rings, where the clustering task consists of identifying each ring as a class. We used rbf-kernels k(x, y) = exp(−∥x −y∥2/2σ2) for σ varying in {10−4, 10−3, 10−2, 100, 101} as well as the path kernel introduced in [4]. All methods fail when used with the individual kernels except for the path-kernel. The non-trivial problem is to detect the correct structure despite the disturbing influence of 5 un-informative kernels. Data sets of size ⌈n/5⌉have been generated by sub-sampling. Figure 1(b) depicts the stability assessment, where we see very small disagreements for η ∈{101, 102, 5 · 102}. At the minimum, the solution almost perfectly matches the ground-truth (1 error). A plot of the resulting ααα-coefficients is given in figure 1(c). NCut as well as the other nmf-methods lead to an error rate of ≈0.5 when applied to the uniformly combined similarities. (a) (b) Figure 2: Images for the segmentation experiments. Image segmentation example:3 The next task consists of finding a reasonable segmentation of the images depicted in figures 2(b) and 2(a). For both images, we measured localized intensity histograms and additionally computed Gabor filter responses (e.g. [12]) on 3 scales for 4 different orientations. For each response image, the same histogramming procedure has been used. For all the histograms, we computed the pairwise Jensen-Shannon divergence (e.g. [2]) for all pairs (i, j) of image sites and took the element-wise exponential of the negative Jensen-Shannon divergences. The resulting similarity matrices have been used as input for the nmf-based data fusion. For the sub-sampling, m = 500 objects have been employed. Figures 3(a) (for the shell image) and 3(b) (for the bird image) show the stability curves for these examples which exhibit minima for non-trivial η resulting in non-uniform ααα. Figure 3(c) depicts the resulting segmentation generated using ααα indicated by the stability assessment, while 3(d) shows a segmentation result, where ααα is closer to the uniform distribution but the stability score for the corresponding η is low. Again, we can see that weighting the different similarity measurements has a beneficial effect, since it leads to improved results. The comparison with the NCut result on the uniformly weighted data (fig. 3(e)) confirms that a non-trivial weighting is desirable here. Note that we have used the full data set with NCut. For, the image in fig. 2(b), we observe similar behavior: the stability selected solution (fig. 3(f)) is more meaningful than the NCut solution (fig. 3(g)) obtained on the uniformly weighted data. In this example, the intensity information dominates the solution obtained on the uniformly combined similarities. However, the texture information alone does not yield a sensible segmentation. Only the non-trivial combination, where the influence of intensity information is decreased and that of the texture information is increased, gives rise to the desired result. It is additionally noteworthy, that the prediction mechanism employed works rather well: In both examples, it has been able to generalize the segmentation from m = 500 to more than 3500 objects. However, artifacts resulting from the subsampling-and-prediction procedure cannot always be avoided, as can be seen in 3(f). They vanish, however, once the algorithm is re-applied to the full data (fig. 3(h)). Clustering of Protein Sequences: Our final application is about the functional categorization of yeast proteins. We partially adopted the data used in [7] 4. Since several of the 3588 proteins belong to more than one category, we extracted a subset of 1579 proteins exclusively belonging to one of the three categories cell cycle + DNA processing,transcription and protein fate. This step ensures a clear ground-truth for comparison. Of the matrices used in [7], we employed a Gauss Kernel derived from gene expression profiles, one derived from Swiss-Waterman alignments, one obtained from comparisons of protein domains as well as two diffusion kernels derived from protein-protein interaction data. Although the data is not very discriminative for the 3-class problem, the solutions generated on the data combined using the ααα for the most stable η lead to more than 10% improvement w.r.t. the 3Only comparisons with NCut reported. The nmf results are slightly worse than those of NCut. 4The data is available at http://noble.gs.washington.edu/proj/yeast/. 10 −3 10 −1 10 0 10 2 10 3 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 sparsity parameter η avg. disagreement (a) 10 −3 10 −1 10 0 10 2 10 3 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 sparsity parameter η avg. disagreement (b) (c) (d) (e) (f) (g) (h) Figure 3: Stability plots and segmentation results for the images in 2(a) and 2(b) (see text). ground-truth (the disagreement measure of section 4 is used) in comparison with the solution obtained using the least stable η-parameter. The latter, however, was hardly better than random guessing by having an overall disagreement of more than 0.60 (more precisely, 0.6392 ± 0.0455) on this data. For the most stable η, we observed a disagreement around 0.52 depending on the sub-sample (best 0.5267 ± 0.0403). In this case, the largest weight was assigned to the protein-protein interaction data. NCut and the two nmf methods proposed in [11] lead to rates 0.5953, 0.6080 and 0.6035, respectively, when applied to the naive combination. Note, that the clustering results are comparable with some of those obtained in [7], where the protein-protein interaction data has been used to construct a (supervised) classifier. 6 Conclusion This work introduced an approach to combining similarity data originating from multiple sources for grouping a set of objects. Adopting a pairwise clustering perspective enables a smooth integration of multiple similarity measurements. To be able to distinguish between desired and distractive information, a weighting mechanism is introduced leading to a potentially sparse convex combination of the measurements. Here, an entropy constraint is employed to control the amount of sparseness actually allowed. A stability-based model selection mechanism is used to select this free parameter. We emphasize, that this procedure represents a completely unsupervised model selection strategy. The experimental evaluation on toy and real world data demonstrates that our proposal yields meaningful partitions and is able to distinguish between desired and spurious structure in data. Future work will focus on (i) improving the optimization of the proposed model, (ii) the integration of additional constraints and (iii) the introduction of a cluster-specific weighting mechanism. The proposed method as well as its relation to other approaches discussed in the literature is currently under further investigation. References [1] F. R. Bach and M. I. Jordan. Learning spectral clustering. In NIPS, volume 16. MIT Press, 2004. [2] J. Burbea and C. R. Rao. On the convexity of some divergence measures based on entropy functions. IEEE Trans. Inform. Theory, 28(3), 1982. [3] K. Crammer, J. Keshet, and Y. Singer. Kernel design using boosting. In NIPS, volume 15. MIT Press, 2003. [4] B. Fischer, V. Roth, and J. M. Buhmann. Clustering with the connectivity kernel. In NIPS, volume 16. MIT Press, 2004. [5] Thomas Hofmann. Unsupervised learning by probabilistic latent semantic analysis. Mach. Learn., 42(1-2):177–196, 2001. [6] E. T. Jaynes. Information theory and statistical mechanics, I and II. Physical Reviews, 106 and 108:620–630 and 171–190, 1957. [7] G. R. G. Lanckriet, M. Deng, N. Cristianini, M. I. Jordan, and W. S. Noble. Kernelbased data fusion and its application to protein function prediction in yeast. In Pacific Symposium on Biocomputing, pages 300–311, 2004. [8] Kenneth Lange. Optimization. Springer Texts in Statistics. Springer, 2004. [9] T. Lange, M. Braun, V. Roth, and J.M. Buhmann. Stability-based model selection. In NIPS, volume 15. MIT Press, 2003. [10] M. H. C. Law, M. A. T. Figueiredo, and A. K. Jain. Simultaneous feature selection and clustering using mixture models. IEEE Trans. Pattern Anal. Mach. Intell., 26(9):1154–1166, 2004. [11] Daniel D. Lee and H. Sebastian Seung. Algorithms for non-negative matrix factorization. In NIPS, volume 13, pages 556–562, 2000. [12] B. S. Manjunath and W. Y. Ma. Texture features for browsing and retrieval of image data. IEEE Trans. Pattern Anal. Mach. Intell., 18(8):837–842, 1996. [13] D. S. Modha and W. S. Spangler. Feature weighting in k-means clustering. Mach. Learn., 52(3):217–237, 2003. [14] V. Roth and T. Lange. Feature selection in clustering problems. In NIPS, volume 16. MIT Press, 2004. [15] Jianbo Shi and Jitendra Malik. Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell., 22(8):888–905, 2000. [16] C. K. I. Williams and M. Seeger. Using the Nystr¨ı¿ 1 2m method to speed up kernel machines. In NIPS, volume 13. MIT Press, 2001. [17] E. Xing, A. Ng, M. Jordan, and S. Russell. Distance metric learning with application to clustering with side-information. In NIPS, volume 15, 2003. [18] W. Xu, X. Liu, and Y. Gong. Document clustering based on non-negative matrix factorization. In SIGIR ’03, pages 267–273. ACM Press, 2003.
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Sensory Adaptation within a Bayesian Framework for Perception Alan A. Stocker∗and Eero P. Simoncelli Howard Hughes Medical Institute and Center for Neural Science New York University Abstract We extend a previously developed Bayesian framework for perception to account for sensory adaptation. We first note that the perceptual effects of adaptation seems inconsistent with an adjustment of the internally represented prior distribution. Instead, we postulate that adaptation increases the signal-to-noise ratio of the measurements by adapting the operational range of the measurement stage to the input range. We show that this changes the likelihood function in such a way that the Bayesian estimator model can account for reported perceptual behavior. In particular, we compare the model’s predictions to human motion discrimination data and demonstrate that the model accounts for the commonly observed perceptual adaptation effects of repulsion and enhanced discriminability. 1 Motivation A growing number of studies support the notion that humans are nearly optimal when performing perceptual estimation tasks that require the combination of sensory observations with a priori knowledge. The Bayesian formulation of these problems defines the optimal strategy, and provides a principled yet simple computational framework for perception that can account for a large number of known perceptual effects and illusions, as demonstrated in sensorimotor learning [1], cue combination [2], or visual motion perception [3], just to name a few of the many examples. Adaptation is a fundamental phenomenon in sensory perception that seems to occur at all processing levels and modalities. A variety of computational principles have been suggested as explanations for adaptation. Many of these are based on the concept of maximizing the sensory information an observer can obtain about a stimulus despite limited sensory resources [4, 5, 6]. More mechanistically, adaptation can be interpreted as the attempt of the sensory system to adjusts its (limited) dynamic range such that it is maximally informative with respect to the statistics of the stimulus. A typical example is observed in the retina, which manages to encode light intensities that vary over nine orders of magnitude using ganglion cells whose dynamic range covers only two orders of magnitude. This is achieved by adapting to the local mean as well as higher order statistics of the visual input over short time-scales [7]. ∗corresponding author. If a Bayesian framework is to provide a valid computational explanation of perceptual processes, then it needs to account for the behavior of a perceptual system, regardless of its adaptation state. In general, adaptation in a sensory estimation task seems to have two fundamental effects on subsequent perception: • Repulsion: The estimate of parameters of subsequent stimuli are repelled by those of the adaptor stimulus, i.e. the perceived values for the stimulus variable that is subject to the estimation task are more distant from the adaptor value after adaptation. This repulsive effect has been reported for perception of visual speed (e.g. [8, 9]), direction-of-motion [10], and orientation [11]. • Increased sensitivity: Adaptation increases the observer’s discrimination ability around the adaptor (e.g. for visual speed [12, 13]), however it also seems to decrease it further away from the adaptor as shown in the case of direction-of-motion discrimination [14]. In this paper, we show that these two perceptual effects can be explained within a Bayesian estimation framework of perception. Note that our description is at an abstract functional level - we do not attempt to provide a computational model for the underlying mechanisms responsible for adaptation, and this clearly separates this paper from other work which might seem at first glance similar [e.g., 15]. 2 Adaptive Bayesian estimator framework Suppose that an observer wants to estimate a property of a stimulus denoted by the variable θ, based on a measurement m. In general, the measurement can be vector-valued, and is corrupted by both internal and external noise. Hence, combining the noisy information gained by the measurement m with a priori knowledge about θ is advantageous. According to Bayes’ rule p(θ|m) = 1 αp(m|θ)p(θ) . (1) That is, the probability of stimulus value θ given m (posterior) is the product of the likelihood p(m|θ) of the particular measurement and the prior p(θ). The normalization constant α serves to ensure that the posterior is a proper probability distribution. Under the assumption of a squared-error loss function, the optimal estimate ˆθ(m) is the mean of the posterior, thus ˆθ(m) = Z ∞ 0 θ p(θ|m) dθ . (2) Note that ˆθ(m) describes an estimate for a single measurement m. As discussed in [16], the measurement will vary stochastically over the course of many exposures to the same stimulus, and thus the estimator will also vary. We return to this issue in Section 3.2. Figure 1a illustrates a Bayesian estimator, in which the shape of the (arbitrary) prior distribution leads on average to a shift of the estimate toward a lower value of θ than the true stimulus value θstim. The likelihood and the prior are the fundamental constituents of the Bayesian estimator model. Our goal is to describe how adaptation alters these constituents so as to account for the perceptual effects of repulsion and increased sensitivity. Adaptation does not change the prior ... An intuitively sensible hypothesis is that adaptation changes the prior distribution. Since the prior is meant to reflect the knowledge the observer has about the distribution of occurrences of the variable θ in the world, repeated viewing of stimuli with the same parameter a à probability likelihood posterior prior θ θ b probability adapt modified prior attraction ! 'ˆθ θ θ Figure 1: Hypothetical model in which adaptation alters the prior distribution. a) Unadapted Bayesian estimation configuration in which the prior leads to a shift of the estimate ˆθ, relative to the stimulus parameter θstim. Both the likelihood function and the prior distribution contribute to the exact value of the estimate ˆθ (mean of the posterior). b) Adaptation acts by increasing the prior distribution around the value, θadapt, of the adapting stimulus parameter. Consequently, an subsequent estimate ˆθ′ of the same stimulus parameter value θstim is attracted toward the adaptor. This is the opposite of observed perceptual effects, and we thus conclude that adjustments of the prior in a Bayesian model do not account for adaptation. value θadapt should presumably increase the prior probability in the vicinity of θadapt. Figure 1b schematically illustrates the effect of such a change in the prior distribution. The estimated (perceived) value of the parameter under the adapted condition is attracted to the adapting parameter value. In order to account for observed perceptual repulsion effects, the prior would have to decrease at the location of the adapting parameter, a behavior that seems fundamentally inconsistent with the notion of a prior distribution. ... but increases the reliability of the measurements Since a change in the prior distribution is not consistent with repulsion, we are led to the conclusion that adaptation must change the likelihood function. But why, and how should this occur? In order to answer this question, we reconsider the functional purpose of adaptation. We assume that adaptation acts to allocate more resources to the representation of the parameter values in the vicinity of the adaptor [4], resulting in a local increase in the signal-to-noise ratio (SNR). This can be accomplished, for example, by dynamically adjusting the operational range to the statistics of the input. This kind of increased operational gain around the adaptor has been effectively demonstrated in the process of retinal adaptation [17]. In the context of our Bayesian estimator framework, and restricting to the simple case of a scalar-valued measurement, adaptation results in a narrower conditional probability density p(m|θ) in the immediate vicinity of the adaptor, thus an increase in the reliability of the measurement m. This is offset by a broadening of the conditional probability density p(m|θ) in the region beyond the adaptor vicinity (we assume that total resources are conserved, and thus an increase around the adaptor must necessarily lead to a decrease elsewhere). Figure 2 illustrates the effect of this local increase in signal-to-noise ratio on the likelim1 m2 unadapted adapted conditionals likelihoods 1 2 1/SNR p(m| ) 2 ' θ p(m| ) 1 ' θ p(m| ) 2 θ p(m| ) 1 θ θ θ θ p(m| ) adapt θ ' θ adapt θ m θ m θ p(m | ) 1 p(m | ) 1 ' p(m | ) 2 ' p(m | ) 2 θ θ θ θ θ θ adapt θ 1 2 θ θ Figure 2: Measurement noise, conditionals and likelihoods. The two-dimensional conditional density, p(m|θ), is shown as a grayscale image for both the unadapted and adapted cases. We assume here that adaptation increases the reliability (SNR) of the measurement around the parameter value of the adaptor. This is balanced by a decrease in SNR of the measurement further away from the adaptor. Because the likelihood is a function of θ (horizontal slices, shown plotted at right), this results in an asymmetric change in the likelihood that is in agreement with a repulsive effect on the estimate. a 0 + adapt θ θ ∆θ ^ b -90 90 180 -60 -30 0 30 60 adapt θ ∆θ ^ -180 θ [deg] [deg] Figure 3: Repulsion: Model predictions vs. human psychophysics. a) Difference in perceived direction in the pre- and post-adaptation condition, as predicted by the model. Postadaptive percepts of motion direction are repelled away from the direction of the adaptor. b) Typical human subject data show a qualitatively similar repulsive effect. Data (and fit) are replotted from [10]. hood function. The two gray-scale images represent the conditional probability densities, p(m|θ), in the unadapted and the adapted state. They are formed by assuming additive noise on the measurement m of constant variance (unadapted) or with a variance that decreases symmetrically in the vicinity of the adaptor parameter value θadapt, and grows slightly in the region beyond. In the unadapted state, the likelihood is convolutional and the shape and variance are equivalent to the distribution of measurement noise. However, in the adapted state, because the likelihood is a function of θ (horizontal slice through the conditional surface) it is no longer convolutional around the adaptor. As a result, the mean is pushed away from the adaptor, as illustrated in the two graphs on the right. Assuming that the prior distribution is fairly smooth, this repulsion effect is transferred to the posterior distribution, and thus to the estimate. 3 Simulation Results We have qualitatively demonstrated that an increase in the measurement reliability around the adaptor is consistent with the repulsive effects commonly seen as a result of perceptual adaptation. In this section, we simulate an adapted Bayesian observer by assuming a simple model for the changes in signal-to-noise ratio due to adaptation. We address both repulsion and changes in discrimination threshold. In particular, we compare our model predictions with previously published data from psychophysical experiments examining human perception of motion direction. 3.1 Repulsion In the unadapted state, we assume the measurement noise to be additive and normally distributed, and constant over the whole measurement space. Thus, assuming that m and θ live in the same space, the likelihood is a Gaussian of constant width. In the adapted state, we assume a simple functional description for the variance of the measurement noise around the adapter. Specifically, we use a constant plus a difference of two Gaussians, a 1 adapt θ θ relative discrimination threshold b -40 -20 20 40 0.8 1 1.2 1.4 1.6 1.8 relative discrimination threshold adapt θ θ [deg] Figure 4: Discrimination thresholds: Model predictions vs. human psychophysics. a) The model predicts that thresholds for direction discrimination are reduced at the adaptor. It also predicts two side-lobes of increased threshold at further distance from the adaptor. b) Data of human psychophysics are in qualitative agreement with the model. Data are replotted from [14] (see also [11]). each having equal area, with one twice as broad as the other (see Fig. 2). Finally, for simplicity, we assume a flat prior, but any reasonable smooth prior would lead to results that are qualitatively similar. Then, according to (2) we compute the predicted estimate of motion direction in both the unadapted and the adapted case. Figure 3a shows the predicted difference between the pre- and post-adaptive average estimate of direction, as a function of the stimulus direction, θstim. The adaptor is indicated with an arrow. The repulsive effect is clearly visible. For comparison, Figure 3b shows human subject data replotted from [10]. The perceived motion direction of a grating was estimated, under both adapted and unadapted conditions, using a two-alternative-forced-choice experimental paradigm. The plot shows the change in perceived direction as a function of test stimulus direction relative to that of the adaptor. Comparison of the two panels of Figure 3 indicate that despite the highly simplified construction of the model, the prediction is quite good, and even includes the small but consistent repulsive effects observed 180 degrees from the adaptor. 3.2 Changes in discrimination threshold Adaptation also changes the ability of human observers to discriminate between the direction of two different moving stimuli. In order to model discrimination thresholds, we need to consider a Bayesian framework that can account not only for the mean of the estimate but also its variability. We have recently developed such a framework, and used it to quantitatively constrain the likelihood and the prior from psychophysical data [16]. This framework accounts for the effect of the measurement noise on the variability of the estimate ˆθ. Specifically, it provides a characterization of the distribution p(ˆθ|θstim) of the estimate for a given stimulus direction in terms of its expected value and its variance as a function of the measurement noise. As in [16] we write var⟨ˆθ|θstim⟩ = var⟨m⟩(∂ˆθ(m) ∂m )2|m=θstim . (3) Assuming that discrimination threshold is proportional to the standard deviation, q var⟨ˆθ|θstim⟩, we can now predict how discrimination thresholds should change after adaptation. Figure 4a shows the predicted change in discrimination thresholds relative to the unadapted condition for the same model parameters as in the repulsion example (Figure 3a). Thresholds are slightly reduced at the adaptor, but increase symmetrically for directions further away from the adaptor. For comparison, Figure 4b shows the relative change in discrimination thresholds for a typical human subject [14]. Again, the behavior of the human observer is qualitatively well predicted. 4 Discussion We have shown that adaptation can be incorporated into a Bayesian estimation framework for human sensory perception. Adaptation seems unlikely to manifest itself as a change in the internal representation of prior distributions, as this would lead to perceptual bias effects that are opposite to those observed in human subjects. Instead, we argue that adaptation leads to an increase in reliability of the measurement in the vicinity of the adapting stimulus parameter. We show that this change in the measurement reliability results in changes of the likelihood function, and that an estimator that utilizes this likelihood function will exhibit the commonly-observed adaptation effects of repulsion and changes in discrimination threshold. We further confirm our model by making quantitative predictions and comparing them with known psychophysical data in the case of human perception of motion direction. Many open questions remain. The results demonstrated here indicate that a resource allocation explanation is consistent with the functional effects of adaptation, but it seems unlikely that theory alone can lead to a unique quantitative prediction of the detailed form of these effects. Specifically, the constraints imposed by biological implementation are likely to play a role in determining the changes in measurement noise as a function of adaptor parameter value, and it will be important to characterize and interpret neural response changes in the context of our framework. Also, although we have argued that changes in the prior seem inconsistent with adaptation effects, it may be that such changes do occur but are offset by the likelihood effect, or occur only on much longer timescales. Last, if one considers sensory perception as the result of a cascade of successive processing stages (with both feedforward and feedback connections), it becomes necessary to expand the Bayesian description to describe this cascade [e.g., 18, 19]. For example, it may be possible to interpret this cascade as a sequence of Bayesian estimators, in which the measurement of each stage consists of the estimate computed at the previous stage. Adaptation could potentially occur in each of these processing stages, and it is of fundamental interest to understand how such a cascade can perform useful stable computations despite the fact that each of its elements is constantly readjusting its response properties. References [1] K. K¨ording and D. Wolpert. Bayesian integration in sensorimotor learning. Nature, 427(15):244–247, January 2004. [2] D C Knill and W Richards, editors. Perception as Bayesian Inference. Cambridge University Press, 1996. [3] Y. Weiss, E. Simoncelli, and E. Adelson. Motion illusions as optimal percept. Nature Neuroscience, 5(6):598–604, June 2002. [4] H.B. Barlow. Vision: Coding and Efficiency, chapter A theory about the functional role and synaptic mechanism of visual after-effects, pages 363–375. Cambridge University Press., 1990. [5] M.J. Wainwright. Visual adaptation as optimal information transmission. Vision Research, 39:3960–3974, 1999. [6] N. Brenner, W. Bialek, and R. de Ruyter van Steveninck. Adaptive rescaling maximizes information transmission. Neuron, 26:695–702, June 2000. [7] S.M. Smirnakis, M.J. Berry, D.K. Warland, W. Bialek, and M. Meister. Adaptation of retinal processing to image contrast and spatial scale. Nature, 386:69–73, March 1997. [8] P. Thompson. Velocity after-effects: the effects of adaptation to moving stimuli on the perception of subsequently seen moving stimuli. Vision Research, 21:337–345, 1980. [9] A.T. Smith. Velocity coding: evidence from perceived velocity shifts. Vision Research, 25(12):1969–1976, 1985. [10] P. Schrater and E. Simoncelli. Local velocity representation: evidence from motion adaptation. Vision Research, 38:3899–3912, 1998. [11] C.W. Clifford. Perceptual adaptation: motion parallels orientation. Trends in Cognitive Sciences, 6(3):136–143, March 2002. [12] C. Clifford and P. Wenderoth. Adaptation to temporal modulaton can enhance differential speed sensitivity. Vision Research, 39:4324–4332, 1999. [13] A. Kristjansson. Increased sensitivity to speed changes during adaptation to first-order, but not to second-order motion. Vision Research, 41:1825–1832, 2001. [14] R.E. Phinney, C. Bowd, and R. Patterson. Direction-selective coding of stereoscopic (cyclopean) motion. Vision Research, 37(7):865–869, 1997. [15] N.M. Grzywacz and R.M. Balboa. A Bayesian framework for sensory adaptation. Neural Computation, 14:543–559, 2002. [16] A.A. Stocker and E.P. Simoncelli. Constraining a Bayesian model of human visual speed perception. In Lawrence K. Saul, Yair Weiss, and L´eon Bottou, editors, Advances in Neural Information Processing Systems NIPS 17, pages 1361–1368, Cambridge, MA, 2005. MIT Press. [17] D. Tranchina, J. Gordon, and R.M. Shapley. Retinal light adaptation – evidence for a feedback mechanism. Nature, 310:314–316, July 1984. [18] S. Deneve. Bayesian inference in spiking neurons. In Lawrence K. Saul, Yair Weiss, and L´eon Bottou, editors, Adv. Neural Information Processing Systems (NIPS*04), vol 17, Cambridge, MA, 2005. MIT Press. [19] R. Rao. Hierarchical Bayesian inference in networks of spiking neurons. In Lawrence K. Saul, Yair Weiss, and L´eon Bottou, editors, Adv. Neural Information Processing Systems (NIPS*04), vol 17, Cambridge, MA, 2005. MIT Press.
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Efficient estimation of hidden state dynamics from spike trains M´arton G. Dan´oczy Inst. for Theoretical Biology Humboldt University, Berlin Invalidenstr. 43 10115 Berlin, Germany m.danoczy@biologie.hu-berlin.de Richard H. R. Hahnloser Inst. for Neuroinformatics UNIZH / ETHZ Winterthurerstrasse 190 8057 Zurich, Switzerland rich@ini.phys.ethz.ch Abstract Neurons can have rapidly changing spike train statistics dictated by the underlying network excitability or behavioural state of an animal. To estimate the time course of such state dynamics from single- or multiple neuron recordings, we have developed an algorithm that maximizes the likelihood of observed spike trains by optimizing the state lifetimes and the state-conditional interspike-interval (ISI) distributions. Our nonparametric algorithm is free of time-binning and spike-counting problems and has the computational complexity of a Mixed-state Markov Model operating on a state sequence of length equal to the total number of recorded spikes. As an example, we fit a two-state model to paired recordings of premotor neurons in the sleeping songbird. We find that the two state-conditional ISI functions are highly similar to the ones measured during waking and singing, respectively. 1 Introduction It is well known that neurons can suddenly change firing statistics to reflect a macroscopic change of a nervous system. Often, firing changes are not accompanied by an immediate behavioural change, as is the case, for example, in paralysed patients, during sleep [1], during covert discriminative processing [2], and for all in-vitro studies [3]. In all of these cases, changes in some hidden macroscopic state can only be detected by close inspection of single or multiple spike trains. Our goal is to develop a powerful, but computationally simple tool for point processes such as spike trains. From spike train data, we want to the extract continuously evolving hidden variables, assuming a discrete set of possible states. Our model for classifying spikes into discrete hidden states is based on three assumptions: 1. Hidden states form a continuous-time Markov process and thus have exponentially distributed lifetimes 2. State switching can occur only at the time of a spike (where there is observable evidence for a new state). 3. In each of the hidden states, spike trains are generated by mutually independent renewal processes. 1. For a continuous-time Markov process, the probability of staying in state S = i for a time interval T > t is given by Pi(t) = exp(−rit), where ri is the escape rate (or hazard rate) of state i. The mean lifetime τi is defined as the inverse of the escape rate, τi = 1/ri. As a corollary, it follows that the probability of staying in state i for a particular duration equals the probability of surviving for a fraction of that duration times the probability of surviving for the remaining time, i.e., the state survival probability Pi(t) satisfies the product identity Pi(t1 +t2) = Pi(t1)Pi(t2). 2. According to the second assumption, state switching can occur at any spike, irrespective of which neuron fired the spike. In the following, we shall refer to a spike fired by any of the neurons as an event (where state switching might occur). Note that if two (or more) neurons happen to fire a spike at exactly the same time, the respective spikes are regarded as two (or more) distinct events. The collection of event times is denoted by te. Combining the first two assumptions, we formulate the hidden state sequence at the events (i.e. observation points) as a non-homogeneous discrete Markov chain. Accordingly, the probability of remaining in state i for the duration of the interevent-interval (IEI) ∆te = te −te−1 is given by the state survival probability Pi(∆te). The probability to change state is then 1−Pi(∆te). 3. In each state i, the spike trains are assumed to be generated by a renewal process that randomly draws interspike-intervals (ISIs) t from a probability density function (pdf) hi(t). Because every IEI is only a fraction of an ISI, instead of working with ISI distributions, we use an equivalent formulation based on the conditional intensity function (CIF) λi(ϕ) [4]. The CIF, also called hazard function in reliability theory, is a generalization of the Poisson firing rate. It is defined as the probability density of spiking in the time interval [ϕ,ϕ +dt], given that no spike has occurred in the interval [0,ϕ) since the last spike. In the following, the variable ϕ, i.e. the time that has elapsed since the last spike, shall be referred to as phase [5]. Using the CIF, the ISI pdf can be expressed by the fundamental equation of renewal theory, hi(t) = exp − Z t 0 λi(ϕ)dϕ λi(t). (1) At each event e, we observe the phase trajectory of every neuron traced out since the last event. It is clear that in multiple electrode recordings the phase trajectories between events are not independent, since they have to start where the previous trajectory ended. Therefore, our model violates the observation independence assumption of standard Hidden Markov Models (HMMs). Our model is, in formal terms, a mixed-state Markov model [6], with the architecture of a double-chain [7]. Such models are generalizations of HMMs in that the observable outputs may not only be dependent on the current hidden state, but also on past observations (formally, the mixed state is formed by combining the hidden and observable states). In our model, hidden state transition probabilities are characterized by the escape rates ri and observable state transition probabilities by the CIFs λn i for neuron n in hidden state i. Our goal is to find a set Ψ of model parameters, such that the likelihood Pr{O|Ψ} = ∑ S∈S Pr{S,O|Ψ} of the observation sequence O is maximized. As a first step, we will derive an expression for the combined likelihood Pr{S,O|Ψ}. Then, we will apply the expectation maximization (EM) algorithm to find the optimal parameter set. 2 Transition probabilities The mixed state at event e shall be composed of the hidden state Se and the observable outputs On e (for neurons n ∈{1,...,N}). Hidden state transitions In classical mixed-state Markov models, the hidden state transition probabilities are constant. In our model, however, we describe time as a continuous quantity and observe the system whenever a spike occurs, thus in non-equidistant intervals. Consequently, hidden state transitions depend explicitly on the elapsed time since the last observation, i.e., on the IEIs ∆te. The transition probability ai j(∆te) from hidden state i to hidden state j is then given by ai j(∆te) = exp(−r j ∆te) if i = j, [1−exp(−r j ∆te)]gi j otherwise, (2) where gi j is the conditional probability of making a transition from state i into a new state j, given that j ̸= i. Thus, gi j has to satisfy the constraint ∑j gi j = 1, with gii = 0. Observable state transitions The observation at event e is defined as Oe = {Φn e,νe}, where νe contains the index of the neuron that has triggered event e by emitting a spike, and Φn e = (inf Φn e, sup Φn e] is the phase interval traced out by neuron n since its last spike. Observations form a cascade. After a spike, the phase of the respective neuron is immediately reset to zero. The interval’s bounds are thus defined by sup Φn e = inf Φn e +∆te and inf Φn e = 0 if νe−1 = n, sup Φn e−1 otherwise. The observable transition probability pi(Oe) = Pr{Oe |Oe−1,Se = i} is the probability of observing output Oe, given the previous output Oe−1 and the current hidden state Se. With our independence assumption (3.), we can give its density as the product of every neuron’s probability of having survived the respective phase interval Φn e that it has traced out since its last spike, multiplied by the spiking neuron’s firing rate (compare equation 1): pi(Oe) = ∏ n exp − Z Φne λn i(ϕ) dϕ λνe i (sup Φνe e ). (3) Note that in case of a single neuron recording, this reduces to the ISI pdf. To give a closed form of the observable transition pdf, several approaches are thinkable. Here, for the sake of flexibility and computational simplicity, we approximate the CIF λn i for neuron n in state i by a step function, assuming that its value is constant inside small, arbitrarily spaced bins Bn(b),b ∈{1,...,Nn bins}. That is, λn i(ϕ) ≈ℓn i(b), ∀ϕ ∈Bn(b). In order to use the discretized CIFs ℓn i(b), we also discretize Φn e: the fractions f n e(b) ∈[0,1] represent how much of neuron n’s phase bin Bn(b) has been traced out since the last event. For example, if event e −1 happened in the middle of neuron n’s phase bin 2 and event e happened ten percent into its phase bin 4, then f n e (2) = 0.5, f n e (3) = 1, and f n e (4) = 0.1, whereas f n e (i) = 0 for other i, Figure 1. Making use of these discretizations, the integral in equation 3 is approximated by a sum: pi(Oe) ≈ " ∏ n exp − Nn bins ∑ b=1 f n e(b)ℓn i(b)∥Bn(b)∥ !# λνe i (sup Φνe e ), (4) with ∥Bn(b)∥denoting the width of neuron n’s phase bin b. Equations 2 and 4 fully describe transitions in our mixed-state Markov model. Next, we apply the EM algorithm to find optimal values of the escape rates ri, the conditional hidden state transition probabilities gi j and the discretized CIFs ℓn i(b), given a set of spike trains. f 2e 0.5 1.0 0.1 1 2 3 1 2 3 4 5 1 2 1 2 3 4 1 2 1 te−1 te Events Neuron 1 Neuron 2 Figure 1: Two spike trains are combined to form the event train shown in the bottom row. The phase bins are shown below the spike trains, they are labelled with the corresponding bin number. As an example, for the second neuron, the fractions f 2 e (b) of its phase bins that have been traced out since event e−1 are indicated by the horizontal arrow. They are nonzero for b = 2,3, and 4. 3 Parameter estimation Our goal is to find model parameters Ψ = {ri,gi j,ℓn i(b)}, such that the likelihood Pr{O|Ψ} of observation sequence O is maximized. According to the EM algorithm, we can find such values by iterating over models Ψnew = argmax ψ ∑ S∈S Pr{S|O,Ψold} ln(Pr{S,O|ψ}), (5) where S is the set of all possible hidden state sequences. The product of equations 2 and 4 over all events is proportional to the combined likelihood Pr{S,O|ψ}: Pr{S,O|ψ} ∼∏ e aSe−1 Se(∆te) pSe(Oe). Because of the logarithm in equation 5, the maximization over escape rates can be separated from the maximization over conditional intensity functions. We define the abbreviations ξi j(e) = Pr{Se−1 = i,Se = j|O,Ψold} and γi(e) = Pr{Se = i|O,Ψold} for the posterior probabilities appearing in equation 5. In practice, both expressions are computed in the expectation step by the classic forward-backward algorithm [8], using equations 2 and 4 as the transition probabilities. With the abbreviations defined above, equation 5 is split to rnew j = argmax r ∑ e ξ j j(e)(−r∆te)+ ∑ e,i̸= j ξi j(e)ln[1−exp(−r∆te)] ! (6) ℓn i(b)new = argmax ℓ −ℓ∑ e γi(e) f n e(b)∥Bn(b)∥+lnℓ ∑ e:νe=n∧ sup Φne∈Bn(b) γi(e) (7) gnew i j = argmax g lng∑ e ξi j(e) with gnew ii = 0 and ∑ j gnew i j = 1. (8) In order to perform the maximization in equation 6, we compute its derivative with respect to r and set it to zero: 0 = ∑ e ξ j j(e)∆te + ∆te − ∆te 1−exp −rnew j ∆te ∑ i̸= j ξi j(e) This equation cannot be solved analytically, but being just a one dimensional optimization problem, a solution can be found using numerical methods, such as the LevenbergMarquardt algorithm. The singularity in case of ∆te = 0, which arises when two or more spikes occur at the same time, needs the special treatment of replacing the respective fraction by its limit: 1/rnew i . To obtain the reestimation formula for the discretized CIFs, equation 7’s derivative with respect to ℓis set to zero. The result can be solved directly and yields ℓn i(b)new = ∑ e:νe=n∧ sup Φne∈Bn(b) γi(e) . ∑ e γi(e) f n e(b)∥Bn(b)∥. Finally, to obtain the reestimation formula for the conditional hidden state transition probabilities gi j, we solve equation 8 using Lagrange multipliers, resulting in gnew i̸= j = ∑ e ξi j(e) . ∑ e,k̸=i ξik(e). 4 Application to spike trains from the sleeping songbird We have applied our model to spike train data from sleeping songbirds [9]. It has been found that during sleep, neurons in vocal premotor area RA exhibit spontaneous activity that at times resembles premotor activity during singing [10, 9]. We train our model on the spike train of a single RA neuron in the sleeping bird with Nbins = 100, where the first bin extends from the sample time to 1ms and the consecutive 99 steps are logarithmically spaced up to the largest ISI. After convergence, we find that the ISI pdfs associated with the two hidden states qualitatively agree with the pdfs recorded in the awake non-singing bird and the awake singing bird, respectively, Figure 2. ISI pdfs were derived from the CIFs by using equation 1. For the state-conditional ISI histograms we first ran the Viterbi algorithm to find the most likely hidden-state sequence and then sorted spikes into two groups, for which the ISIs histograms were computed. We find that sleep-related activity in the RA neuron of Figure 2 is best described by random switching between a singing-like state of lifetime τ1 = 1.18s ± 0.38s and an awake, nonsinging-like state of lifetime τ2 = 2.26s±0.42s. Standard deviations of lifetime estimates were computed by dividing the spike train into 30 data windows of 10s duration each and computing the Jackknife variance [11] on the truncated spike trains. The difference between the singing-like state in our model and the true singing ISI pdf shown in Figure 2 is more likely due to generally reduced burst rates during sleep, rather than to a particularity of the examined neuron. Next we applied our model to simultaneous recordings from pairs of RA neurons. By fitting two separate models (with identical phase binning) to the two spike trains, and after running the Viterbi algorithm to find the most likely hidden state sequences, we find good agreement between the two sequences, Figure 3 (top row) and 4c. The correspondence of hidden state sequences suggests a common network mechanism for the generation of the singing-like states in both neurons. We thus applied a single model to both spike trains and found again good agreement with hidden-state sequences determined for the separate models, Figure 3 (bottom row) and 4f. The lifetime histograms for both states look approximatively exponential, justifying our assumption for the state dynamics, Figure 4g and h. For the model trained on neuron one we find lifetimes τ1 = 0.63s ± 0.37s and τ2 = 1.71s ± 0.45s, and for the model trained on neuron two we find τ1 = 0.42s ± 0.11s and τ2 = 1.23s ± 0.17s. For the combined model, lifetimes are τ1 = 0.58s ± 0.25s and (a) ISI [ms] prob. [%] 100 101 102 103 0 5 10 15 20 (b) ISI [ms] prob. [%] 100 101 102 103 0 5 10 15 20 (c) ISI [ms] prob. [%] 100 101 102 103 0 5 10 15 20 Figure 2: (a): The two state-conditional ISI histograms of an RA neuron during sleep are shown by the red and green curves, respectively. Gray patches represent Jackknife standard deviations. (b): After waking up the bird by pinching his tail, the new ISI histogram shown by the gray area becomes almost indistinguishable from the ISI histogram of state 1 (green line). (c): In comparison to the average ISI histogram of many RA neurons during singing (shown by the gray area, reproduced from [12]), the ISI histogram corresponding to state 2 (red line) is shifted to the right, but looks otherwise qualitatively similar. τ2 = 1.13s ± 0.15s. Thus, hidden-state switching seems to occur more frequently in the combined model. The reason for this increase might be that evidence for the song-like state appears more frequently with two neurons, as a single neuron might not be able to indicate song-like firing statistics with high temporal fidelity. We have also analysed the correlations between state dynamics in the different models. The hidden state function S(t) is a binary function that equals one when in hidden state 1 and zero when in state 2. For the case where we modelled the two spike trains separately, we have two such hidden state functions, S1(t) for neuron one and S2(t) for neuron two. We find that all correlation functions CSS1(t), CSS2(t), and CS1S2(t), have a peak at zero time lag, with a high peak correlation of about 0.7, Figure 4c and f (the correlation function is defined as the cross-covariance function divided by the autocovariance functions). We tested whether our model is a good generative model for the observed spike trains by applying the time rescaling theorem, after which the ISIs of a good generative model with known CIFs should reduce to a Poisson process with unit rate, which, after another transformation, should lead to a uniform probability density in the interval (0,1) [4]. Performing this test, we found that the transformed ISI densities of the combined model are uniform, thus validating our model (95% Kolmogorov-Smirnov test, Figure 4i). 5 Discussion We have presented a mixed-state Markov model for point processes, assuming generation by random switching between renewal processes. Our algorithm is suited for systems in which neurons make discrete state transitions simultaneously. Previous attempts of fitting spike train data with Markov models exhibited weaknesses due to time binning. With large time bins and the number of spikes per bin treated as observables [13, 14], state transitions can only be detected when they are accompanied by firing rate changes. In our case, RA neurons have a roughly constant firing rate throughout the entire recording, and so such approaches fail. We were able to model the hidden states in continuous time, but had to bin the ISIs in order to deal with limited data. In principle, the algorithm can operate on any binning scheme for the ISIs. Our choice of logarithmic bins keeps the number of parameters small (proportional to Nbins), but preserves a constant temporal resolution. The hidden-state dynamics form Poisson processes characterized by a lifetime. By esti IFR [Hz] 100 100 101 101 102 102 103 0 5 10 15 20 25 30 Time [sec] IFR [Hz] 100 100 101 101 102 102 103 0 5 10 15 20 25 30 Figure 3: Shown are the instantaneous firing rate (IFR) functions of two simultaneously recorded RA neurons (at any time, the IFR corresponds to the inverse of the current ISI). The green areas show the times when in the first (awake-like) hidden state, and the red areas when in the song-like hidden state. The top two rows show the result of computing two independent models on the two neurons, whereas the bottom rows show the result of a single model. (a) ISI [ms] prob. [%] 100 101 102 103 0 5 10 15 (b) ISI [ms] prob. [%] 100 101 102 103 0 5 10 15 (c) ∆t [sec] correlation 0 .4 .8 −0.2 0.2 0 0 5 10 −5 −10 (d) ISI [ms] # spikes/100 100 101 102 103 0 1 2 3 (e) ISI [ms] # spikes/100 100 101 102 103 0 1 2 3 (f) ∆t [sec] correlation 0 .4 .8 −0.2 0.2 0 0 5 10 −5 −10 (g) State duration [ms] # states 101 102 103 104 0 10 20 30 (h) State duration [ms] # states 101 102 103 104 0 10 20 30 (i) Our model diff. to unif. dist. 0 0 1/3 2/3 1 .02 Figure 4: (a) and (b): State-conditional ISI pdfs for each of the two neurons. (d) and (e): ISI histograms (blue and yellow) for neurons 1 and 2, respectively, as well as state-conditional ISI histograms (red and green), computed as in Figure 2a. (g) and (h): State lifetime histograms for the song-like state (red) and for the awake-like state (green). Theoretical (exponential) histograms with escape rates r1 and r2 (fine black lines) show good agreement with the measured histograms, especially in F. (c): Correlation between state functions of the two separate models. (f): Correlation between the state functions of the combined model with separate model 1 (blue) and separate model 2 (yellow). (i): KolmogorovSmirnov plot after time rescaling. After transforming the ISIs, the resulting densities for both neurons remain within the 95% confidence bounds of the uniform density (gray area). In (a)–(c) and (f)–(h), Jackknife standard deviations are shown by the gray areas. mating this lifetime, we hope it might be possible to form a link between the hidden states and the underlying physical process that governs the dynamics of switching. Despite the apparent limitation of Poisson statistics, it is a simple matter to generalize our model to hidden state distributions with long tails (e.g., power-law lifetime distributions): By cascading many hidden states into a chain (with fixed CIFs), a power-law distribution can be approximated by the combination of multiple exponentials with different lifetimes. Our code is available at http://www.ini.unizh.ch/∼rich/software/. Acknowledgements We would like to thank Sam Roweis for advice on Hidden Markov models and Maria Minkoff for help with the manuscript. R. H. is supported by the Swiss National Science Foundation. M. D. is supported by Stiftung der Deutschen Wirtschaft. References [1] Z. N´adasdy, H. Hirase, A. Czurk´o, J. Csicsv´ari, and G. Buzs´aki. Replay and time compression of recurring spike sequences in the hippocampus. J Neurosci, 19(21):9497–9507, Nov 1999. [2] K. G. Thompson, D. P. Hanes, N. P. Bichot, and J. D. Schall. Perceptual and motor processing stages identified in the activity of macaque frontal eye field neurons during visual search. J Neurophysiol, 76(6):4040–4055, Dec 1996. [3] R. Cossart, D. Aronov, and R. Yuste. Attractor dynamics of network UP states in the neocortex. Nature, 423(6937):283–288, May 2003. [4] E. N. Brown, R. Barbieri, V. Ventura, R. E. Kass, and L. M. Frank. The time-rescaling theorem and its application to neural spike train data analysis. Neur Comp, 14(2):325–346, Feb 2002. [5] J. Deppisch, K. Pawelzik, and T. Geisel. Uncovering the synchronization dynamics from correlated neuronal activity quantifies assembly formation. Biol Cybern, 71(5):387–399, 1994. [6] A. M. Fraser and A. Dimitriadis. Forecasting probability densities by using hidden Markov models with mixed states. In Weigend and Gershenfeld, editors, Time Series Prediction: Forecasting the Future and Understanding the Past, pages 265–82. Addison-Wesley, 1994. [7] A. Berchtold. The double chain Markov model. Comm Stat Theor Meths, 28:2569–2589, 1999. [8] L. R. Rabiner. A tutorial on hidden Markov models and selected applications in speech recognition. Proc IEEE, 77(2):257–286, Feb 1989. [9] R. H. R. Hahnloser, A. A. Kozhevnikov, and M. S. Fee. An ultra-sparse code underlies the generation of neural sequences in a songbird. Nature, 419(6902):65–70, Sep 2002. [10] A. S. Dave and D. Margoliash. Song replay during sleep and computational rules for sensorimotor vocal learning. Science, 290(5492):812–816, Oct 2000. [11] D. J. Thomson and A. D. Chave. Jackknifed error estimates for spectra, coherences, and transfer functions. In Simon Haykin, editor, Advances in Spectrum Analysis and Array Processing, volume 1, chapter 2, pages 58–113. Prentice Hall, 1991. [12] A. Leonardo and M. S. Fee. Ensemble coding of vocal control in birdsong. J Neurosci, 25(3):652–661, Jan 2005. [13] G. Radons, J. D. Becker, B. D¨ulfer, and J. Kr¨uger. Analysis, classification, and coding of multielectrode spike trains with hidden Markov models. Biol Cybern, 71(4):359–373, 1994. [14] I. Gat, N. Tishby, and M. Abeles. Hidden Markov modelling of simultaneously recorded cells in the associative cortex of behaving monkeys. Network: Computation in Neural Systems, 8(3):297–322, 1997.
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Radial Basis Function Network for Multi-task Learning Xuejun Liao Department of ECE Duke University Durham, NC 27708-0291, USA xjliao@ee.duke.edu Lawrence Carin Department of ECE Duke University Durham, NC 27708-0291, USA lcarin@ee.duke.edu Abstract We extend radial basis function (RBF) networks to the scenario in which multiple correlated tasks are learned simultaneously, and present the corresponding learning algorithms. We develop the algorithms for learning the network structure, in either a supervised or unsupervised manner. Training data may also be actively selected to improve the network’s generalization to test data. Experimental results based on real data demonstrate the advantage of the proposed algorithms and support our conclusions. 1 Introduction In practical applications, one is frequently confronted with situations in which multiple tasks must be solved. Often these tasks are not independent, implying what is learned from one task is transferable to another correlated task. By making use of this transferability, each task is made easier to solve. In machine learning, the concept of explicitly exploiting the transferability of expertise between tasks, by learning the tasks simultaneously under a unified representation, is formally referred to as “multi-task learning” [1]. In this paper we extend radial basis function (RBF) networks [4,5] to the scenario of multitask learning and present the corresponding learning algorithms. Our primary interest is to learn the regression model of several data sets, where any given data set may be correlated with some other sets but not necessarily with all of them. The advantage of multi-task learning is usually manifested when the training set of each individual task is weak, i.e., it does not generalize well to the test data. Our algorithms intend to enhance, in a mutually beneficial way, the weak training sets of multiple tasks, by learning them simultaneously. Multi-task learning becomes superfluous when the data sets all come from the same generating distribution, since in that case we can simply take the union of them and treat the union as a single task. In the other extreme, when all the tasks are independent, there is no correlation to utilize and we learn each task separately. The paper is organized as follows. We define the structure of multi-task RBF network in Section 2 and present the supervised learning algorithm in Section 3. In Section 4 we show how to learn the network structure in an unsupervised manner, and based on this we demonstrate how to actively select the training data, with the goal of improving the generalization to test data. We perform experimental studies in Section 5 and conclude the paper in Section 6. 2 Multi-Task Radial Basis Function Network Figure 1 schematizes the radial basis function (RBF) network structure customized to multitask learning. The network consists of an input layer, a hidden layer, and an output layer. The input layer receives a data point x = [x1, · · · , xd]T ∈Rd and submits it to the hidden layer. Each node at the hidden layer has a localized activation φn(x) = φ(||x −cn||, σn), n = 1, · · · , N, where || · || denotes the vector norm and φn(·) is a radial basis function (RBF) localized around cn with the degree of localization parameterized by σn. Choosing φ(z, σ) = exp(−z2 2σ2 ) gives the Gaussian RBF. The activations of all hidden nodes are weighted and sent to the output layer. Each output node represents a unique task and has its own hidden-to-output weights. The weighted activations of the hidden nodes are summed at each output node to produce the output for the associated task. Denoting wk = [w0k, w1k, · · · , wNk]T as the weights connecting hidden nodes to the k-th output node, then the output for the k-th task, in response to input x, takes the form fk(x) = wT k φ(x) (1) where φ(x) = φ0(x), φ1(x), . . . , φN(x) T is a column containing N + 1 basis functions with φ0(x) ≡1 a dummy basis accounting for the bias in Figure 1. … f1(x) Output layer Hidden layer (basis functions) Input layer I 1(x) I 2(x) I N(x) I 0 {1 Task 1 Task 2 Task K … … x1 x2 xd f2(x) fK(x) w1 w2 wK x = [ ]T … Input data point Network response (Specified by the number of tasks) (Specified by data dimensionality) (To be learned by algorithms) Bias Hidden-to-output weights Figure 1: A multi-task structure of RBF Network. Each of the output nodes represents a unique task. Each task has its own hidden-to-output weights but all the tasks share the same hidden nodes. The activation of hidden node n is characterized by a basis function φn(x) = φ(||x −cn||, σn). A typical choice of φ is φ(z, σ) = exp(−z2 2σ2 ), which gives the Gaussian RBF. 3 Supervised Learning Suppose we have K tasks and the data set of the k-th task is Dk = {(x1k, y1k), · · · , (xJkk, yJkk)}, where yik is the target (desired output) of xik. By definition, a given data point xik is said to be supervised if the associated target yik is provided and unsupervised if yik is not provided. The definition extends similarly to a set of data Table 1: Learning Algorithm of Multi-Task RBF Network Input: {(x1k, y2k), · · · , (xJk,k, yJk,k)}k=1:K, φ(·, σ), σ, and ρ; Output: φ(·) and {wk}K k=1. 1. For m=1:K, For n=1:Jm, For k=1:K, For i=1:Jk Compute bφnm ik = φ(||xnm −xik||, σ); 2. Let N = 0, φ(·) = 1, e0 = PK k=1 hPJk i=1 y2 ik −(Jk + ρ)−1(PJk i=1 yik)2i ; For k=1:K, compute Ak =Jk+ρ, wk =(Jk+ρ)−1PJk i=1yik; 3. For m=1:K, For n=1:Jm If bφnm is not marked as “deleted” For k=1:K, compute ck = PJk i=1 φik bφnm ik , qk = PJk i=1(bφnm ik )2 + ρ −cT k A−1 k ck; If there exists k such that qk = 0, mark bφnm as “deleted”; else, compute δe(φ, bφnm) using (5). 4. If {bφik}i=1:Jk,k=1:K are all marked as “deleted”, go to 10. 5. Let (n∗, m∗) = arg maxbφnm not marked as “deleted” δe(φ,bφnm); Mark bφn∗m∗as “deleted”. 6. Tune RBF parameter σN+1 = arg max σ δe(φ, φ(|| · −xn∗m∗||, σ)) 7. Let φN+1(·) = φ(|| · −xn∗m∗||, σN+1); Update φ(·) ←[φT (·), φN+1(·)]T ; 8. For k=1:K Compute Anew k and wnew k respectively by (A-1) and (A-3) in the appendix; Update Ak ←Anew k , wk ←wnew k 9. Let eN+1 = eN −δe(φ, φN+1); If the sequence {en}n=0:(N+1) is converged, go to 10, else update N ←N + 1 and go back to 3. 10. Exit and output φ(·) and {wk}K k=1. points. We are interested in learning the functions fk(x) for the K tasks, based on ∪K k=1Dk. The learning is based on minimizing the squared error e (φ, w) = PK k=1 nPJk i=1 wT k φik −yik 2 + ρ ||wk||2o (2) where φik = φ(xik) for notational simplicity. The regularization terms ρ ||wk||2, k = 1, · · · , K, are used to prevent singularity of the A matrices defined in (3), and ρ is typically set to a small positive number. For fixed φ’s, the w’s are solved by minimizing e(φ, w) with respect to w, yielding wk = A−1 k PJk i=1yikφik and Ak = PJk i=1φikφT ik + ρ I, k = 1, · · · , K (3) In a multi-task RBF network, the input layer and output layer are respectively specified by the data dimensionality and the number of tasks. We now discuss how to determine the hidden layer (basis functions φ). Substituting the solutions of the w’s in (3) into (2) gives e(φ) = PK k=1 PJk i=1 y2 ik −yikwT k φik (4) where e(φ) is a function of φ only because w’s are now functions of φ as given by (3). By minimizing e(φ), we can determine φ. Recalling that φik is an abbreviation of φ(xik) = 1, φ1(xik), . . . , φN(xik) T , this amounts to determining N, the number of basis functions, and the functional form of each basis function φn(·), n = 1, . . . , N. Consider the candidate functions {φnm(x) = φ(||x−xnm||, σ) : n = 1, · · · , Jm, m = 1, · · · , K}. We learn the RBF network structure by selecting φ(·) from these candidate functions such that e(φ) in (4) is minimized. The following theorem tells us how to perform the selection in a sequential way; the proof is given in the Appendix. Theorem 1 Let φ(x) = [1, φ1(x), . . . , φN(x)]T and φN+1(x) be a single basis function. Assume the A matrices corresponding to φ and [φ, φN+1]T are all non-degenerate. Then δe(φ, φN+1) = e(φ) −e([φ, φN+1]T ) = PK k=1 cT k wk −PJk i=1yikφN+1 ik 2q−1 k (5) where φN+1 ik = φN+1(φik), wk and A are the same as in (3), and ck = PJk i=1φikφN+1 ik , dk = PJk i=1(φN+1 ik )2 + ρ, qk = dk −cT k A−1 k ck (6) By the conditions of the theorem Anew k is full rank and hence it is positive definite by construction. By (A-2) in the Appendix, q−1 k is a diagonal element of (Anew k )−1, therefore q−1 k is positive and by (5) δe(φ, φN+1) > 0, which means adding φN+1 to φ generally makes the squared error decrease. The decrease δe(φ, φN+1) depends on φN+1. By sequentially selecting basis functions that bring the maximum error reduction, we achieve the goal of maximizing e(φ). The details of the learning algorithm are summarized in Table 1. 4 Active Learning In the previous section, the data in Dk are supervised (provided with the targets). In this section, we assume the data in Dk are initially unsupervised (only x is available without access to the associated y) and we select a subset from Dk to be supervised (targets acquired) such that the resulting network generalizes well to the remaining data in Dk. The approach is generally known as active learning [6]. We first learn the basis functions φ from the unsupervised data, and based on φ select data to be supervised. Both of these steps are based on the following theorem, the proof of which is given in the Appendix. Theorem 2 Let there be K tasks and the data set of the k-th task is Dk ∪eDk where Dk = {(xik, yik)}Jk i=1 and eDk = {(xik, yik)}Jk+ eJk i=Jk+1. Let there be two multi-task RBF networks, whose output nodes are characterized by fk(·) and f ∼ k (·), respectively, for task k = 1, . . . , K. The two networks have the same given basis functions (hidden nodes) φ(·) = [1, φ1(·), · · · , φN(·)]T , but different hidden-to-output weights. The weights of fk(·) are trained with Dk ∪eDk, while the weights of f ∼ k (·) are trained using eDk. Then for k = 1, · · · , K, the square errors committed on Dk by fk(·) and f ∼ k (·) are related by 0≤[det Γk]−1≤λ−1 max,k ≤ PJk i=1(yik−f ∼ k (xik))2−1PJk i=1(yik−fk(xik))2≤λ−1 min,k ≤1 (7) where Γk = I + ΦT k (ρ I + eΦk eΦ T k )−1Φk 2 with Φ = φ(x1k), . . . , φ(xJkk) and eΦ = φ(xJk+1,k), . . . , φ(xJk+ eJk,k) , and λmax,k and λmin,k are respectively the largest and smallest eigenvalues of Γk. Specializing Theorem 2 to the case eJk = 0, we have Corollary 1 Let there be K tasks and the data set of the k-th task is Dk = {(xik, yik)}Jk i=1. Let the RBF network, whose output nodes are characterized by fk(·) for task k = 1, . . . , K, have given basis functions (hidden nodes) φ(·) = [1, φ1(·), · · · , φN(·)]T and the hidden-to-output weights of task k be trained with Dk. Then for k = 1, · · · , K, the squared error committed on Dk by fk(·) is bounded as 0 ≤[det Γk]−1 ≤λ−1 max,k ≤ PJk i=1y2 ik −1PJk i=1 (yik −fk(xik))2 ≤λ−1 min,k ≤1, where Γk = I + ρ−1ΦT k Φk 2 with Φ = φ(x1,k), . . . , φ(xJk,k) , and λmax,k and λmin,k are respectively the largest and smallest eigenvalues of Γk. It is evident from the properties of matrix determinant [7] and the definition of Φ that det Γk = det(ρI + ΦkΦT k ) 2[det(ρ I)]−2 = det(ρI + PJk i=1 φikφT ik) 2[det(ρ I)]−2. Using (3) we write succinctly det Γk = [det A2 k][det(ρ I)]−2. We are interested in selecting the basis functions φ that minimize the error, before seeing y’s. By Corollary 1 and the equation det Γk = [det A2 k][det(ρ I)]−2, the squared error is lower bounded by PJk i=1y2 ik[det(ρ I)]2[det Ak]−2. Instead of minimizing the error directly, we minimize its lower bound. As [det(ρ I)]2PLk i=1y2 ik does not depend on φ, this amounts to selecting φ to minimize (det Ak)−2. To minimize the errors for all tasks k = 1 · · · , K, we select φ to minimize QK k=1(det Ak)−2. The selection proceeds in a sequential manner. Suppose we have selected basis functions φ = [1, φ1, · · · , φN]T . The associated A matrices are Ak = PJk i=1 φikφT ik + ρ I(N+1)×(N+1), k = 1, · · · , K. Augmenting basis functions to [φT , φN+1]T , the A matrices change to Anew k = PJk i=1[φT ik, φN+1 ik ]T [φT ik, φN+1 ik ] + ρ I(N+2)×(N+2). Using the determinant formula of block matrices [7], we get QK k=1(det Anew k )−2 = QK k=1(qk det Ak)−2, where qk is the same as in (6). As Ak does not depend on φN+1, the left-hand side is minimized by maximizing QK k=1 q2 k. The selection is easily implemented by making the following two minor modifications in Table 1: (a) in step 2, compute e0 = PK k=1 ln(Jk + ρ)−2; in step 3, compute δe(φ, bφnm) = PK k=1 ln q2 k. Employing the logarithm is for gaining additivity and it does not affect the maximization. Based on the basis functions φ determined above, we proceed to selecting data to be supervised and determining the hidden-to-output weights w from the supervised data using the equations in (3). The selection of data is based on an iterative use of the following corollary, which is a specialization of Theorem 2 and was originally given in [8]. Corollary 2 Let there be K tasks and the data set of the k-th task is Dk = {(xik, yik)}Jk i=1. Let there be two RBF networks, whose output nodes are characterized by fk(·) and f + k (·), respectively, for task k = 1, . . . , K. The two networks have the same given basis functions φ(·) = [1, φ1(·), · · · , φN(·)]T , but different hidden-to-output weights. The weights of fk(·) are trained with Dk, while the weights of f + k (·) are trained using D+ k = Dk ∪{(xJk+1,k, yJk+1,k)}. Then for k = 1, · · · , K, the squared errors committed on (xJk+1,k, yJk+1,k) by fk(·) and f + k (·) are related by f + k (xJk+1,k) − yJk+1,k 2 = γ(xJk+1,k) −1 fk(xJk+1,k) −yJk+1,k 2, where γ(xJk+1,k) = 1 + φT (xJk+1,k)A−1 k φ(xJk+1,k) 2 ≥1 and Ak = PJk i=1 ρI + φ(xik)φT (xik) is the same as in (3). Two observations are made from Corollary 2. First, if γ(xJk+1,k) ≈1, seeing yJk+1,k does not effect the error on xJk+1,k, indicating Dk already contain sufficient information about (xJk+1,k, yJk+1,k). Second, if γ(xi) ≫1, seeing yJk+1,k greatly decrease the error on xJk+1,k, indicating xJk+1,k is significantly dissimilar (novel) to Dk and xJk+1,k must be supervised to reduce the error. Based on Corollary 2, the selection proceeds sequentially. Suppose we have selected data Dk = {(xik, yik)}Jk i=1, from which we compute Ak. We select the next data point as xJk+1,k = arg max i>Jk, k=1,··· ,K γ(xik) = arg max i>Jk k=1,··· ,K 1 + φT(xik)A−1 k φ(xik) 2. After xJk+1,k is selected, the Ak is updated and the next selection begins. As the iteration advances γ will decrease until it reaches convergence. We use (3) to compute w from the selected x and their associated targets y, completing learning of the RBF network. 5 Experimental Results In this section we compare the multi-task RBF network against single-task RBF networks via experimental studies. We consider three types of RBF networks to learn K tasks, each with its data set Dk. In the first, which we call “one RBF network”, we let the K tasks share both basis functions φ (hidden nodes) and hidden-to output weights w, thus we do not distinguish the K tasks and design a single RBF network to learn a union of them. The second is the multi-task RBF network, where the K tasks share the same φ but each has its own w. In the third, we have K independent networks, each designed for a single task. We use a school data set from the Inner London Education Authority, consisting of examination records of 15362 students from 139 secondary schools. The data are available at http://multilevel.ioe.ac.uk/intro/datasets.html. This data set was originally used to study the effectiveness of schools and has recently been used to evaluate multi-task algorithms [2,3]. The goal is to predict the exam scores of the students based on 9 variables: year of exam (1985, 1986, or 1987), school code (1-139), FSM (percentage of students eligible for free school meals), VR1 band (percentage of students in school in VR band one), gender, VR band of student (3 categories), ethnic group of student (11 categories), school gender (male, female, or mixed), school denomination (3 categories). We consider each school a task, leading to 139 tasks in total. The remaining 8 variables are used as inputs to the RBF network. Following [2,3], we converted each categorical variable to a number of binary variables, resulting in a total number of 27 input variables, i.e., x ∈R27. The exam score is the target to be predicted. The three types of RBF networks as defined above are designed as follows. The multi-task RBF network is implemented as the structure as shown in Figure 1 and trained with the learning algorithm in Table 1. The “one RBF network” is implemented as a special case of Figure 1, with a single output node and trained using the union of supervised data from all 139 schools. We design 139 independent RBF networks, each of which is implemented with a single output node and trained using the supervised data from a single school. We use the Gaussian RBF φn(x) = exp(−||x−cn||2 2σ2 ), where the cn’s are selected from training data points and σn’s are initialized as 20 and optimized as described in Table 1. The main role of the regularization parameter ρ is to prevent the A matrices from being singular and it does not affect the results seriously. In the results reported here, ρ is set to 10−6. Following [2-3], we randomly take 75% of the 15362 data points as training (supervised) data and the remaining 25% as test data. The generalization performance is measured by the squared error (fk(xik) −yik)2 averaged over all test data xik of tasks k = 1, · · · , K. We made 10 independent trials to randomly split the data into training and test sets and the squared error averaged over the test data of all the 139 schools and the trials are shown in Table 2, for the three types of RBF networks. Table 2: Squared error averaged over the test data of all 139 schools and the 10 independent trials for randomly splitting the school data into training (75%) and testing (25%) sets. Multi-task RBF network Independent RBF networks One RBF network 109.89 ± 1.8167 136.41 ± 7.0081 149.48 ± 2.8093 Table 2 clearly shows the multi-task RBF network outperforms the other two types of RBF networks by a considerable margin. The “one RBF network” ignores the difference between the tasks and the independent RBF networks ignore the tasks’ correlations, therefore they both perform inferiorly. The multi-task RBF network uses the shared hidden nodes (basis functions) to capture the common internal representation of the tasks and meanwhile uses the independent hidden-to-output weights to learn the statistics specific to each task. We now demonstrate the results of active learning. We use the method in Section 4 to actively split the data into training and test sets using a two-step procedure. First we learn the basis functions φ of multi-task RBF network using all 15362 data (unsupervised). Based on the φ, we then select the data to be supervised and use them as training data to learn the hidden-to-output weights w. To make the results comparable, we use the same training data to learn the other two types of RBF networks (including learning their own φ and w). The networks are then tested on the remaining data. Figure 2 shows the results of active learning. Each curve is the squared error averaged over the test data of all 139 schools, as a function of number of training data. It is clear that the multi-task RBF network maintains its superior performance all the way down to 5000 training data points, whereas the independent RBF networks have their performances degraded seriously as the training data diminish. This demonstrates the increasing advantage of multi-task learning as the number of training data decreases. The “one RBF network” seems also insensitive to the number of training data, but it ignores the inherent dissimilarity between the tasks, which makes its performance inferior. 5000 6000 7000 8000 9000 10000 11000 12000 100 120 140 160 180 200 220 240 260 Number of training (supervised) data Squared error averaged over test data Multi−task RBF network Independent RBF networks One RBF network Figure 2: Squared error averaged over the test data of all 139 schools, as a function of the number of training (supervised) data. The data are split into training and test sets via active learning. 6 Conclusions We have presented the structure and learning algorithms for multi-task learning with the radial basis function (RBF) network. By letting multiple tasks share the basis functions (hidden nodes) we impose a common internal representation for correlated tasks. Exploiting the inter-task correlation yields a more compact network structure that has enhanced generalization ability. Unsupervised learning of the network structure enables us to actively split the data into training and test sets. As the data novel to the previously selected ones are selected next, what finally remain unselected and to be tested are all similar to the selected data which constitutes the training set. This improves the generalization of the resulting network to the test data. These conclusions are substantiated via results on real multi-task data. References [1] R. Caruana. (1997) Multitask learning. Machine Learning, 28, p. 41-75, 1997. [2] B. Bakker and T. Heskes (2003). Task clustering and gating for Bayesian multitask learning. Journal of Machine Learning Research, 4: 83-99, 2003 [3] T. Evgeniou, C. A. Micchelli, and M. Pontil (2005). Learning Multiple Tasks with Kernel Methods. Journal of Machine Learning Research, 6: 615637, 2005 [4] Powell M. (1987), Radial basis functions for multivariable interpolation : A review, J.C. Mason and M.G. Cox, eds, Algorithms for Approximation, pp.143-167. [5] Chen, F. Cowan, and P. Grant (1991), Orthogonal least squares learning algorithm for radial basis function networks, IEEE Transactions on Neural Networks, Vol. 2, No. 2, 302-309, 1991 [6] Cohn, D. A., Ghahramani, Z., and Jordan, M. I. (1995). Active learning with statistical models. Advances in Neural Information Processing Systems, 7, 705-712. [7] V. Fedorov (1972), Theory of Optimal Experiments, Academic Press, 1972 [8] M. Stone (1974), Cross-validatory choice and assessment of statistical predictions, Journal of the Royal Statistical Society, Series B, 36, pp. 111-147, 1974. Appendix Proof of Theorem 1:. Let φnew = [φ, φN+1]T . By (3), the A matrices corresponding to φnew are Anew k = PJk i=1 h φik φN+1 ik i φT ik φN+1 ik + ρ I(N+2)×(N+2) = h Ak ck cT k dk i (A-1) where ck and dk are as in (6). By the conditions of the theorem, the matrices Ak and Anew k are all non-degenerate. Using the block matrix inversion formula [7] we get (Anew k )−1= A−1 k + A−1 k ckq−1 k cT k A−1 k −A−1 k ckq−1 k −q−1 k cT k A−1 k q−1 k (A-2) where qk is as in (6). By (3), the weights wnew k corresponding to [φT , φN+1]T are wnew k = (Anew k )−1 PJk i=1 yikφik PJk i=1 yikφN+1 ik = wk + A−1 k ckq−1 k gk −q−1 k gk (A-3) with gk = cT k wk −PJk i=1yikφN+1 ik . Hence, (φnew ik )T wnew k = φT ikwk + φT ikA−1 k ck − φN+1 ik gkq−1 k , which is put into (4) to get e(φnew = PK k=1 PJk i=1 y2 ik −yik(φnew ik )T wnew k = PK k=1 PJk i=1 y2 ik −yikφT ikwk −yik φT ikA−1 k ck −φN+1 ik gkq−1 k = e(φ) −PK k=1 cT k wk − PJk i=1yikφN+1 ik 2q−1 k , where in arriving the last equality we have used (3) and (4) and gk = cT k wk −PJk i=1yikφN+1 ik . The theorem is proved. □ Proof of Theorem 2: The proof applies to k = 1, · · · , K. For any given k, define Φ = φ(x1k), . . . , φ(xJkk) , eΦ = φ(xJk+1,k), . . . , φ(xJk+ e Jk,k) , yk = [y1k, . . . , yJkk]T , eyk = [yJk+1,k, . . . , yJk+ e Jk,k]T , fk = [f(x1k), . . . , f(xJkk)]T , f ∼ k = [f ∼ k (x1k), . . . , f ∼ k (xJkk)]T , and eAk = ρI + eΦk eΦ T k . By (1), (3), and the conditions of the theorem, fk = ΦT k eAk + ΦkΦT k −1(Φkyk+ eΦeyk) (a) = ΦT k eA−1 k − ΦT k eA−1 k Φk+I−I I+ΦT k eA−1 k Φk −1ΦT k eA−1 k Φkyk+ eΦkeyk = I + ΦT k eA−1 k Φk −1ΦT k eA−1 k eΦkeyk + Φkyk (b) = I + ΦT k eA−1 k Φk −1f ∼ k + I + ΦT k eA−1 k Φk −1 ΦT k eA−1 k Φk + I −I yk = yk + I + ΦT k eA−1 k Φk −1 f ∼ k −yk , where equation (a) is due to the Sherman-Morrison-Woodbury formula and equation (b) results because f ∼ k = ΦT k eA−1 k eΦkeyk. Hence, fk −yk = I + ΦT k eA−1 k Φk −1 f ∼ k −yk , which gives PJk i=1 (yik −fk(xik))2 = (fk −yk)T (fk −yk) = f ∼ k −yk T Γ−1 k f ∼ k −yk (A-4) where Γk = I + ΦT k eA−1 k Φk 2 = I + ΦT k (ρ I + eΦk eΦ T k )−1Φk 2. By construction, Γk has all its eigenvalues no less than 1, i.e., Γk = ET k diag[λ1k, · · · , λJkk]Ek with ET k Ek = I and λ1k, · · · , λJkk ≥1, which makes the first, second, and last inequality in (7) hold. Using this expansion of Γk in (A-4) we get PJk i=1 (fk(xik) −yik)2 = f ∼ k −yk T ET k diag[σ−1 1k , . . . , σ−1 Jkk ] f ∼ k −yk ≤ f ∼ k −yk T ET k λ−1 min,k I Ek f ∼ k −yk = λ−1 min,k PJk i=1(f ∼ k (xik) −yik)2 (A-5) where the inequality results because λmin,k = min(λ1,k, · · · , λJk,k). From (A-5) follows the fourth inequality in (7). The third inequality in (7) can be proven in in a similar way. □
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Prediction and Change Detection Mark Steyvers Scott Brown msteyver@uci.edu scottb@uci.edu University of California, Irvine University of California, Irvine Irvine, CA 92697 Irvine, CA 92697 Abstract We measure the ability of human observers to predict the next datum in a sequence that is generated by a simple statistical process undergoing change at random points in time. Accurate performance in this task requires the identification of changepoints. We assess individual differences between observers both empirically, and using two kinds of models: a Bayesian approach for change detection and a family of cognitively plausible fast and frugal models. Some individuals detect too many changes and hence perform sub-optimally due to excess variability. Other individuals do not detect enough changes, and perform sub-optimally because they fail to notice short-term temporal trends. 1 Introduction Decision-making often requires a rapid response to change. For example, stock analysts need to quickly detect changes in the market in order to adjust investment strategies. Coaches need to track changes in a player’s performance in order to adjust strategy. When tracking changes, there are costs involved when either more or less changes are observed than actually occurred. For example, when using an overly conservative change detection criterion, a stock analyst might miss important short-term trends and interpret them as random fluctuations instead. On the other hand, a change may also be detected too readily. For example, in basketball, a player who makes a series of consecutive baskets is often identified as a “hot hand” player whose underlying ability is perceived to have suddenly increased [1,2]. This might lead to sub-optimal passing strategies, based on random fluctuations. We are interested in explaining individual differences in a sequential prediction task. Observers are shown stimuli generated from a simple statistical process with the task of predicting the next datum in the sequence. The latent parameters of the statistical process change discretely at random points in time. Performance in this task depends on the accurate detection of those changepoints, as well as inference about future outcomes based on the outcomes that followed the most recent inferred changepoint. There is much prior research in statistics on the problem of identifying changepoints [3,4,5]. In this paper, we adopt a Bayesian approach to the changepoint identification problem and develop a simple inference procedure to predict the next datum in a sequence. The Bayesian model serves as an ideal observer model and is useful to characterize the ways in which individuals deviate from optimality. The plan of the paper is as follows. We first introduce the sequential prediction task and discuss a Bayesian analysis of this prediction problem. We then discuss the results from a few individuals in this prediction task and show how the Bayesian approach can capture individual differences with a single “twitchiness” parameter that describes how readily changes are perceived in random sequences. We will show that some individuals are too twitchy: their performance is too variable because they base their predictions on too little of the recent data. Other individuals are not twitchy enough, and they fail to capture fast changes in the data. We also show how behavior can be explained with a set of fast and frugal models [6]. These are cognitively realistic models that operate under plausible computational constraints. 2 A prediction task with multiple changepoints In the prediction task, stimuli are presented sequentially and the task is to predict the next stimulus in the sequence. After t trials, the observer has been presented with stimuli y1, y2, …, yt and the task is to make a prediction about yt+1. After the prediction is made, the actual outcome yt+1 is revealed and the next trial proceeds to the prediction of yt+2. This procedure starts with y1 and is repeated for T trials. The observations yt are D-dimensional vectors with elements sampled from binomial distributions. The parameters of those distributions change discretely at random points in time such that the mean increases or decreases after a change point. This generates a sequence of observation vectors, y1, y2, …, yT, where each yt = {yt,1 … yt,D}. Each of the yt,d is sampled from a binomial distribution Bin(θt,d,K), so 0 ≤ yt,d ≤ K. The parameter vector θt ={θt,1 … θt,D} changes depending on the locations of the changepoints. At each time step, tx is a binary indicator for the occurrence of a changepoint occurring at time t+1. The parameter α determines the probability of a change occurring in the sequence. The generative model is specified by the following algorithm: 1. For d=1..D sample θ1,d from a Uniform(0,1) distribution 2. For t=2..T, (a) Sample xt-1 from a Bernoulli(α) distribution (b) If xt-1=0, then θt=θt-1, else for d=1..D sample θt,d from a Uniform(0,1) distribution (c) for d=1..D, sample yt from a Bin(θt,d,K) distribution Table 1 shows some data generated from the changepoint model with T=20, α=.1,and D=1. In the prediction task, y will be observed, but x and θ are not. Table 1: Example data t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 x 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 θ .68 .68 .68 .68 .48 .48 .48 .74 .74 .74 .74 .74 .74 .19 .19 .87 .87 .87 .87 .87 y 9 7 8 7 4 4 4 9 8 3 6 7 8 2 1 8 9 9 8 8 3 A Bayesian prediction model In both our Bayesian and fast-and-frugal analyses, the prediction task is decomposed into two inference procedures. First, the changepoint locations are identified. This is followed by predictive inference for the next outcome based on the most recent changepoint locations. Several Bayesian approaches have been developed for changepoint problems involving single or multiple changepoints [3,5]. We apply a Markov Chain Monte Carlo (MCMC) analysis to approximate the joint posterior distribution over changepoint assignments x while integrating out θ. Gibbs sampling will be used to sample from this posterior marginal distribution. The samples can then be used to predict the next outcome in the sequence. 3.1 Inference for changepoint assignments. To apply Gibbs sampling, we evaluate the conditional probability of assigning a changepoint at time i, given all other changepoint assignments and the current α value. By integrating out θ, the conditional probability is ( ) ( ) | , , , , | , i i i i P x x y P x x y θ α θ α − − = ∫ (1) where i x− represents all switch point assignments except xi. This can be simplified by considering the location of the most recent changepoint preceding and following time i and the outcomes occurring between these locations. Let L in be the number of time steps from the last changepoint up to and including the current time step i such that L i i n x − =1 and L i i n j x − + =0 for 0<j< L in . Similarly, let R in be the number of time steps that follow time step i up to the next changepoint such that R i i n x + =1 and R i i n j x + −=0 for 0<j< R in . Let L i L i k i n k i y y − < ≤ =∑ and R i R i k k k i n y y < ≤+ =∑ . The update equation for the changepoint assignment can then be simplified to ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , , 1 , , , , 1 | 1 1 1 0 2 1 1 1 1 1 2 2 i i L R L R L R D i j i j i i i j i j L R j i i L L L R R R D i j i i j i j i i j L R j i i P x m x y y Kn Kn y y m Kn Kn y Kn y y Kn y m Kn Kn α α − = = = ∝ ⎧ Γ + + Γ + + − − ⎪ − = Γ + + ⎪⎪⎨ Γ + Γ + − Γ + Γ + − ⎪ = ⎪ Γ + Γ + ⎪⎩ ∏ ∏ (2) We initialize the Gibbs sampler by sampling each xt from a Bernoulli(α) distribution. All changepoint assignments are then updated sequentially by the Gibbs sampling equation above. The sampler is run for M iterations after which one set of changepoint assignments is saved. The Gibbs sampler is then restarted multiple times until S samples have been collected. Although we could have included an update equation for α, in this analysis we treat α as a known constant. This will be useful when characterizing the differences between human observers in terms of differences in α. 3.2 Predictive inference The next latent parameter value θt+1 and outcome yt+1 can be predicted on the basis of observed outcomes that occurred after the last inferred changepoint: ( ) * 1, , 1, 1, 1 / , round t t j i j t j t j i t y K y K θ θ + + + = + = = ∑ (3) where t* is the location of the most recent change point. By considering multiple Gibbs samples, we get a distribution over outcomes yt+1. We base the model predictions on the mean of this distribution. 3.3 Illustration of model performance Figure 1 illustrates the performance of the model on a one dimensional sequence (D=1) generated from the changepoint model with T=160, α=0.05, and K=10. The Gibbs sampler was run for M=30 iterations and S=200 samples were collected. The top panel shows the actual changepoints (triangles) and the distribution of changepoint assignments averaged over samples. The bottom panel shows the observed data y (thin lines) as well as the θ values in the generative model (rescaled between 0 and 10). At locations with large changes between observations, the marginal changepoint probability is quite high. At other locations, the true change in the mean is very small, and the model is less likely to put in a changepoint. The lower right panel shows the distribution over predicted θt+1 values. 0 0.5 1 xt 20 40 60 80 100 120 140 160 0 5 10 yt 0 0.5 1 θt+1 Figure 1. Results of model simulation. 4 Prediction experiment We tested performance of 9 human observers in the prediction task. The observers included the authors, a visitor, and one student who were aware of the statistical nature of the task as well as naïve students. The observers were seated in front of an LCD touch screen displaying a two-dimensional grid of 11 x 11 buttons. The changepoint model was used to generate a sequence of T=1500 stimuli for two binomial variables y1 and y2 (D=2, K=10). The change probability α was set to 0.1. The two variables y1 and y2 specified the two-dimensional button location. The same sequence was used for all observers. On each trial, the observer touched a button on the grid displayed on the touch screen. Following each button press, the button corresponding to the next {y1,y2} outcome in the sequence was highlighted. Observers were instructed to press the button that best predicted the next location of the highlighted button. The 1500 trials were divided into three blocks of 500 trials. Breaks were allowed between blocks. The whole experiment lasted between 15 and 30 minutes. Figure 2 shows the first 50 trials from the third block of the experiment. The top and bottom panels show the actual outcomes for the y1 and y2 button grid coordinates as well as the predictions for two observers (SB and MY). The figure shows that at trial 15, the y1 and y2 coordinates show a large shift followed by an immediate shift in observer’s MY predictions (on trial 16). Observer SB waits until trial 17 to make a shift. 0 5 10 0 5 10 15 20 25 30 35 40 45 50 0 5 10 Trial outcomes SB predictions MY predictions Figure 2. Trial by trial predictions from two observers. 4.1 Task error We assessed prediction performance by comparing the prediction with the actual outcome in the sequence. Task error was measured by normalized city-block distance ,1 ,1 ,2 ,2 2 1 task error= ( 1) T O O t t t t t y y y y T = − + − −∑ (4) where yO represents the observer’s prediction. Note that the very first trial is excluded from this calculation. Even though more suitable probabilistic measures for prediction error could have been adopted, we wanted to allow comparison of observer’s performance with both probabilistic and non-probabilistic models. Task error ranged from 2.8 (for participant MY) to 3.3 (for ML). We also assessed the performance of five models – their task errors ranged from 2.78 to 3.20. The Bayesian models (Section 3) had the lowest task errors, just below 2.8. This fits with our definition of the Bayesian models as “ideal observer” models – their task error is lower than any other model’s and any human observer’s task error. The fast and frugal models (Section 5) had task errors ranging from 2.85 to 3.20. 5 Modeling Results We will refer to the models with the following letter codes: B=Bayesian Model, LB=limited Bayesian model, FF1..3=fast and frugal models 1..3. We assessed model fit by comparing the model’s prediction against the human observers’ predictions, again using a normalized city-block distance ,1 ,1 ,2 ,2 2 1 model error= ( 1) T M O M O t t t t t y y y y T = − + − −∑ (5) where yM represents the model’s prediction. The model error for each individual observer is shown in Figure 3. It is important to note that because each model is associated with a set of free parameters, the parameters optimized for task error and model error are different. For Figure 3, the parameters were optimized to minimize Equation (5) for each individual observer, showing the extent to which these models can capture the performance of individual observers, not necessarily providing the best task performance. MY MS MM EJ PH NP DN SB ML 0 0.5 1 1.5 2 Model Error B LB FF1 FF2 FF3 Figure 3. Model error for each individual observer.1 5.1 Bayesian prediction models At each trial t, the model was provided with the sequence of all previous outcomes. The Gibbs sampling and inference procedures from Eq. (2) and (3) were applied with M=30 iterations and S=200 samples. The change probability α was a free parameter. In the full Bayesian model, the whole sequence of observations up to the current trial is available for prediction, leading to a memory requirement of up to T=1500 trials – a psychologically unreasonable assumption. We therefore also simulated a limited Bayesian model (LB) where the observed sequence was truncated to the last 10 outcomes. The LB model showed almost no decrement in task performance compared to the full Bayesian model. Figure 3 also shows that it fit human data quite well. 5.2 Individual Differences The right-hand panel of Figure 4 plots each observer’s task error as a function of the mean city-block distance between their subsequent button presses. This shows a clear U-shaped function. Observers with very variable predictions (e.g., ML and DN) had large average changes between successive button pushes, and also had large task error: These observers were too “twitchy”. Observers with very small average button changes (e.g., SB and NP) were not twitchy enough, and also had large task error. Observers in the middle had the lowest task error (e.g., MS and MY). The left-hand panel of Figure 4 shows the same data, but with the x-axis based on the Bayesian model fits. Instead of using mean button change distance to index twitchiness (as in 1 Error bars indicate bootstrapped 95% confidence intervals. the right-hand panel), the left-hand panel uses the estimated α parameters from the Bayesian model. A similar U-shaped pattern is observed: individuals with too large or too small α estimates have large task errors. 10 -4 10 -3 10 -2 10 -1 10 0 2.8 2.9 3 3.1 3.2 3.3 MY MS MM EJ PH NP DN SB ML B α Task Error 0.5 1 1.5 2 2.5 3 2.8 2.9 3 3.1 3.2 3.3 MY MS MM EJ PH NP DN SB ML Mean Button Change Task Error Figure 4. Task error vs. “twitchiness”. Left-hand panel indexes twitchiness using estimated α parameters from Bayesian model fits. Right-hand panel uses mean distance between successive predictions. 5.3 Fast-and-Frugal (FF) prediction models These models perform the prediction task using simple heuristics that are cognitively plausible. The FF models keep a short memory of previous stimulus values and make predictions using the same two-step process as the Bayesian model. First, a decision is made as to whether the latent parameter θ has changed. Second, remembered stimulus values that occurred after the most recently detected changepoint are used to generate the next prediction. A simple heuristic is used to detect changepoints: If the distance between the most recent observation and prediction is greater than some threshold amount, a change is inferred. We defined the distance between a prediction (p) and an observation (y) as the difference between the log-likelihoods of y assuming θ=p and θ=y. Thus, if fB(.|θ, K) is the binomial density with parameters θ and K, the distance between observation y and prediction p is defined as d(y,p)=log(fB(y|y,K))-log(fB(y|p,K)). A changepoint on time step t+1 is inferred whenever d(yt,pt)>C. The parameter C governs the twitchiness of the model predictions. If C is large, only very dramatic changepoints will be detected, and the model will be too conservative. If C is small, the model will be too twitchy, and will detect changepoints on the basis of small random fluctuations. Predictions are based on the most recent M observations, which are kept in memory, unless a changepoint has been detected in which case only those observations occurring after the changepoint are used for prediction. The prediction for time step t+1 is simply the mean of these observations, say p. Human observers were reticent to make predictions very close to the boundaries. This was modeled by allowing the FF model to change its prediction for the next time step, yt+1, towards the mean prediction (0.5). This change reflects a two-way bet. If the probability of a change occurring is α, the best guess will be 0.5 if that change occurs, or the mean p if the change does not occur. Thus, the prediction made is actually yt+1=1/2 α+(1-α)p. Note that we do not allow perfect knowledge of the probability of a changepoint, α. Instead, an estimated value of α is used based on the number of changepoints detected in the data series up to time t. The FF model nests two simpler FF models that are psychologically interesting. If the twitchiness threshold parameter C becomes arbitrarily large, the model never detects a change and instead becomes a continuous running average model. Predictions from this model are simply a boxcar smooth of the data. Alternatively, if we assume no memory the model must based each prediction on only the previous stimulus (i.e., M=1). Above, in Figure 3, we labeled the complete FF model as FF1, the boxcar model as FF2 and the memoryless model FF3. Figure 3 showed that the complete FF model (FF1) fit the data from all observers significantly better than either the boxcar model (FF2) or the memoryless model (FF3). Exceptions were observers PH, DN and ML, for whom all three FF model fit equally well. This result suggests that our observers were (mostly) doing more than just keeping a running average of the data, or using only the most recent observation. The FF1 model fit the data about as well as the Bayesian models for all observers except MY and MS. Note that, in general, the FF1 and Bayesian model fits are very good: the average city block distance between the human data and the model prediction is around 0.75 (out of 10) buttons on both the x- and y-axes. 6 Conclusion We used an online prediction task to study changepoint detection. Human observers had to predict the next observation in stochastic sequences containing random changepoints. We showed that some observers are too “twitchy”: They perform poorly on the prediction task because they see changes where only random fluctuation exists. Other observers are not twitchy enough, and they perform poorly because they fail to see small changes. We developed a Bayesian changepoint detection model that performed the task optimally, and also provided a good fit to human data when sub-optimal parameter settings were used. Finally, we developed a fast-and-frugal model that showed how participants may be able to perform well at the task using minimal information and simple decision heuristics. Acknowledgments We thank Eric-Jan Wagenmakers and Mike Yi for useful discussions related to this work. This work was supported in part by a grant from the US Air Force Office of Scientific Research (AFOSR grant number FA9550-04-1-0317). References [1] Gilovich, T., Vallone, R. and Tversky, A. (1985). The hot hand in basketball: on the misperception of random sequences. Cognitive Psychology17, 295-314. [2] Albright, S.C. (1993a). A statistical analysis of hitting streaks in baseball. Journal of the American Statistical Association, 88, 1175-1183. [3] Stephens, D.A. (1994). Bayesian retrospective multiple changepoint identification. Applied Statistics 43(1), 159-178. [4] Carlin, B.P., Gelfand, A.E., & Smith, A.F.M. (1992). Hierarchical Bayesian analysis of changepoint problems. Applied Statistics 41(2), 389-405. [5] Green, P.J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82(4), 711-732. [6] Gigerenzer, G., & Goldstein, D.G. (1996). Reasoning the fast and frugal way: Models of bounded rationality. Psychological Review, 103, 650-669.
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Gaussian Process Dynamical Models Jack M. Wang, David J. Fleet, Aaron Hertzmann Department of Computer Science University of Toronto, Toronto, ON M5S 3G4 {jmwang,hertzman}@dgp.toronto.edu, fleet@cs.toronto.edu Abstract This paper introduces Gaussian Process Dynamical Models (GPDM) for nonlinear time series analysis. A GPDM comprises a low-dimensional latent space with associated dynamics, and a map from the latent space to an observation space. We marginalize out the model parameters in closed-form, using Gaussian Process (GP) priors for both the dynamics and the observation mappings. This results in a nonparametric model for dynamical systems that accounts for uncertainty in the model. We demonstrate the approach on human motion capture data in which each pose is 62-dimensional. Despite the use of small data sets, the GPDM learns an effective representation of the nonlinear dynamics in these spaces. Webpage: http://www.dgp.toronto.edu/∼jmwang/gpdm/ 1 Introduction A central difficulty in modeling time-series data is in determining a model that can capture the nonlinearities of the data without overfitting. Linear autoregressive models require relatively few parameters and allow closed-form analysis, but can only model a limited range of systems. In contrast, existing nonlinear models can model complex dynamics, but may require large training sets to learn accurate MAP models. In this paper we investigate learning nonlinear dynamical models for high-dimensional datasets. We take a Bayesian approach to modeling dynamics, averaging over dynamics parameters rather than estimating them. Inspired by the fact that averaging over nonlinear regression models leads to a Gaussian Process (GP) model, we show that integrating over parameters in nonlinear dynamical systems can also be performed in closed-form. The resulting Gaussian Process Dynamical Model (GPDM) is fully defined by a set of lowdimensional representations of the training data, with both dynamics and observation mappings learned from GP regression. As a natural consequence of GP regression, the GPDM removes the need to select many parameters associated with function approximators while retaining the expressiveness of nonlinear dynamics and observation. Our work is motivated by modeling human motion for video-based people tracking and data-driven animation. Bayesian people tracking requires dynamical models in the form of transition densities in order to specify prediction distributions over new poses at each time instant (e.g., [11, 14]); similarly, data-driven computer animation requires prior distributions over poses and motion (e.g., [1, 4, 6]). An individual human pose is typically parameterized with more than 60 parameters. Despite the large state space, the space of activity-specific human poses and motions has a much smaller intrinsic dimensionality; in our experiments with walking and golf swings, 3 dimensions often suffice. Our work builds on the extensive literature in nonlinear time-series analysis, of which we (a) x1 y1 A x2 y2 x3 y3 x4 y4 B (b) X Y Figure 1: Time-series graphical models. (a) Nonlinear latent-variable model for time series. (Hyperparameters ¯α and ¯β are not shown.) (b) GPDM model. Because the mapping parameters A and B have been marginalized over, all latent coordinates X = [x1, ..., xN]T are jointly correlated, as are all poses Y = [y1, ..., yN]T . mention a few examples. Two main themes are the use of switching linear models (e.g., [11]), and nonlinear transition functions, such as represented by Radial Basis Functions [2]. Both approaches require sufficient amounts of training data that one can learn the parameters of the switching or basis functions. Determining the appropriate number of basis functions is also difficult. In Kernel Dynamical Modeling [12], linear dynamics are kernelized to model nonlinear systems, but a density function over data is not produced. Supervised learning with GP regression has been used to model dynamics for a variety of applications [3, 7, 13]. These methods model dynamics directly in observation space, which is impractical for the high-dimensionality of motion capture data. Our approach is most directly inspired by the unsupervised Gaussian Process Latent Variable Model (GPLVM) [5], which models the joint distribution of the observed data and their corresponding representation in a low dimensional latent space. This distribution can then be used as a prior for inference from new measurements. However, the GPLVM is not a dynamical model; it assumes that data are generated independently. Accordingly it does not respect temporal continuity of the data, nor does it model the dynamics in the latent space. Here we augment the GPLVM with a latent dynamical model. The result is a Bayesian generalization of subspace dynamical models to nonlinear latent mappings and dynamics. 2 Gaussian Process Dynamics The Gaussian Process Dynamical Model (GPDM) comprises a mapping from a latent space to the data space, and a dynamical model in the latent space (Figure 1). These mappings are typically nonlinear. The GPDM is obtained by marginalizing out the parameters of the two mappings, and optimizing the latent coordinates of training data. More precisely, our goal is to model the probability density of a sequence of vector-valued states y1..., yt, ..., yN, with discrete-time index t and yt ∈RD. As a basic model, consider a latent-variable mapping with first-order Markov dynamics: xt = f(xt−1; A) + nx,t (1) yt = g(xt; B) + ny,t (2) Here, xt ∈Rd denotes the d-dimensional latent coordinates at time t, nx,t and ny,t are zero-mean, white Gaussian noise processes, f and g are (nonlinear) mappings parameterized by A and B, respectively. Figure 1(a) depicts the graphical model. While linear mappings have been used extensively in auto-regressive models, here we consider the nonlinear case for which f and g are linear combinations of basis functions: f(x; A) = i ai φi(x) (3) g(x; B) = j bj ψj(x) (4) for weights A = [a1, a2, ...] and B = [b1, b2, ...], and basis functions φi and ψj. In order to fit the parameters of this model to training data, one must select an appropriate number of basis functions, and one must ensure that there is enough data to constrain the shape of each basis function. Ensuring both of these conditions can be very difficult in practice. However, from a Bayesian perspective, the specific forms of f and g — including the numbers of basis functions — are incidental, and should therefore be marginalized out. With an isotropic Gaussian prior on the columns of B, marginalizing over g can be done in closed form [8, 10] to yield p(Y | X, ¯β) = |W|N (2π)ND|KY |D exp −1 2tr K−1 Y YW2YT , (5) where Y = [y1, ..., yN]T , KY is a kernel matrix, and ¯β = {β1, β2, ..., W} comprises the kernel hyperparameters. The elements of kernel matrix are defined by a kernel function, (KY )i,j = kY (xi, xj). For the latent mapping, X →Y, we currently use the RBF kernel kY (x, x′) = β1 exp −β2 2 ||x −x′||2 + β−1 3 δx,x′ . (6) As in the SGPLVM [4], we use a scaling matrix W ≡diag(w1, ..., wD) to account for different variances in the different data dimensions. This is equivalent to a GP with kernel function k(x, x′)/w2 m for dimension m. Hyperparameter β1 represents the overall scale of the output function, while β2 corresponds to the inverse width of the RBFs. The variance of the noise term ny,t is given by β−1 3 . The dynamic mapping on the latent coordinates X is conceptually similar, but more subtle.1 As above, we form the joint probability density over the latent coordinates and the dynamics weights A in (3). We then marginalize over the weights A, i.e., p(X | ¯α) = p(X, A | ¯α) dA = p(X | A, ¯α) p(A | ¯α) dA . (7) Incorporating the Markov property (Eqn. (1)) gives: p(X | ¯α) = p(x1) N t=2 p(xt | xt−1, A, ¯α) p(A | ¯α) dA , (8) where ¯α is a vector of kernel hyperparameters. Assuming an isotropic Gaussian prior on the columns of A, it can be shown that this expression simplifies to: p(X | ¯α) = p(x1) 1 (2π)(N−1)d|KX|d exp −1 2tr K−1 X XoutXT out , (9) where Xout = [x2, ..., xN]T , KX is the (N−1) × (N−1) kernel matrix constructed from {x1, ..., xN−1}, and x1 is assumed to be have an isotropic Gaussian prior. We model dynamics using both the RBF kernel of the form of Eqn. (6), as well as the following “linear + RBF” kernel: kX(x, x′) = α1 exp −α2 2 ||x −x′||2 + α3xT x′ + α−1 4 δx,x′ . (10) The kernel corresponds to representing g as the sum of a linear term and RBF terms. The inclusion of the linear term is motivated by the fact that linear dynamical models, such as 1Conceptually, we would like to model each pair (xt, xt+1) as a training pair for regression with g. However, we cannot simply substitute them directly into the GP model of Eqn. (5) as this leads to the nonsensical expression p(x2, ..., xN | x1, ..., xN−1). first or second-order autoregressive models, are useful for many systems. Hyperparameters α1, α2 represent the output scale and the inverse width of the RBF terms, and α3 represents the output scale of the linear term. Together, they control the relative weighting between the terms, while α−1 4 represents the variance of the noise term nx,t. It should be noted that, due to the nonlinear dynamical mapping in (3), the joint distribution of the latent coordinates is not Gaussian. Moreover, while the density over the initial state may be Gaussian, it will not remain Gaussian once propagated through the dynamics. One can also see this in (9) since xt variables occur inside the kernel matrix, as well as outside of it. So the log likelihood is not quadratic in xt. Finally, we also place priors on the hyperparameters ( p(¯α) ∝ i α−1 i , and p(¯β) ∝ i β−1 i ) to discourage overfitting. Together, the priors, the latent mapping, and the dynamics define a generative model for time-series observations (Figure 1(b)): p(X, Y, ¯α, ¯β) = p(Y|X, ¯β) p(X|¯α) p(¯α) p(¯β) . (11) Multiple sequences. This model extends naturally to multiple sequences Y1, ..., YM. Each sequence has associated latent coordinates X1, ..., XM within a shared latent space. For the latent mapping g we can conceptually concatenate all sequences within the GP likelihood (Eqn. (5)). A similar concatenation applies for the dynamics, but omitting the first frame of each sequence from Xout, and omitting the final frame of each sequence from the kernel matrix KX. The same structure applies whether we are learning from multiple sequences, or learning from one sequence and inferring another. That is, if we learn from a sequence Y1, and then infer the latent coordinates for a new sequence Y2, then the joint likelihood entails full kernel matrices KX and KY formed from both sequences. Higher-order features. The GPDM can be extended to model higher-order Markov chains, and to model velocity and acceleration in inputs and outputs. For example, a second-order dynamical model, xt = f(xt−1, xt−2; A) + nx,t (12) may be used to explicitly model the dependence of the prediction on two past frames (or on velocity). In the GPDM framework, the equivalent model entails defining the kernel function as a function of the current and previous time-step: kX( [xt, xt−1], [xτ, xτ−1] ) = α1 exp −α2 2 ||xt −xτ||2 −α3 2 ||xt−1 −xτ−1||2 + α4 xT t xτ + α5 xT t−1xτ−1 + α−1 6 δt,τ (13) Similarly, the dynamics can be formulated to predict velocity: vt−1 = f(xt−1; A) + nx,t (14) Velocity prediction may be more appropriate for modeling smoothly motion trajectories. Using Euler integration with time-step ∆t, we have xt = xt−1 + vt−1∆t. The dynamics likelihood p(X | ¯α) can then be written by redefining Xout = [x2 −x1, ..., xN − xN−1]T /∆t in Eqn. (9). In this paper, we use a fixed time-step of ∆t = 1. This is analogous to using xt−1 as a “mean function.” Higher-order features can also be fused together with position information to reduce the Gaussian process prediction variance [15, 9]. 3 Properties of the GPDM and Algorithms Learning the GPDM from measurements Y entails minimizing the negative log-posterior: L = −ln p(X, ¯α, ¯β | Y) (15) = d 2 ln |KX| + 1 2tr K−1 X XoutXT out + j ln αj (16) −N ln |W| + D 2 ln |KY | + 1 2tr K−1 Y YW2YT + j ln βj up to an additive constant. We minimize L with respect to X, ¯α, and ¯β numerically. Figure 2 shows a GPDM 3D latent space learned from a human motion capture data comprising three walk cycles. Each pose was defined by 56 Euler angles for joints, 3 global (torso) pose angles, and 3 global (torso) translational velocities. For learning, the data was mean-subtracted, and the latent coordinates were initialized with PCA. Finally, a GPDM is learned by minimizing L in (16). We used 3D latent spaces for all experiments shown here. Using 2D latent spaces leads to intersecting latent trajectories. This causes large “jumps” to appear in the model, leading to unreliable dynamics. For comparison, Fig. 2(a) shows a 3D SGPLVM learned from walking data. Note that the latent trajectories are not smooth; there are numerous cases where consecutive poses in the walking sequence are relatively far apart in the latent space. By contrast, Fig. 2(b) shows that the GPDM produces a much smoother configuration of latent positions. Here the GPDM arranges the latent positions roughly in the shape of a saddle. Figure 2(c) shows a volume visualization of the inverse reconstruction variance, i.e., −2 ln σy|x,X,Y,¯β. This shows the confidence with which the model reconstructs a pose from latent positions x. In effect, the GPDM models a high probability “tube” around the data. To illustrate the dynamical process, Fig. 2(d) shows 25 fair samples from the latent dynamics of the GPDM. All samples are conditioned on the same initial state, x0, and each has a length of 60 time steps. As noted above, because we marginalize over the weights of the dynamic mapping, A, the distribution over a pose sequence cannot be factored into a sequence of low-order Markov transitions (Fig. 1(a)). Hence, we draw fair samples ˜X(j) 1:60 ∼p(˜X1:60 | x0, X, Y, ¯α), using hybrid Monte Carlo [8]. The resulting trajectories (Fig. 2(c)) are smooth and similar to the training motions. 3.1 Mean Prediction Sequences For both 3D people tracking and computer animation, it is desirable to generate new motions efficiently. Here we consider a simple online method for generating a new motion, called mean-prediction, which avoids the relatively expensive Monte Carlo sampling used above. In mean-prediction, we consider the next timestep ˜xt conditioned on ˜xt−1 from the Gaussian prediction [8]: ˜xt ∼N(µX(˜xt−1); σ2 X(˜xt−1)I) (17) µX(x) = XT outK−1 X kX(x) , σ2 X(x) = kX(x, x) −kX(x)T K−1 X kX(x) (18) where kX(x) is a vector containing kX(x, xi) in the i-th entry and xi is the ith training vector. In particular, we set the latent position at each time-step to be the most-likely (mean) point given the previous step: ˜xt = µX(˜xt−1). In this way we ignore the process noise that one might normally add. We find that this mean-prediction often generates motions that are more like the fair samples shown in Fig. 2(d), than if random process noise had been added at each time step (as in (1)). Similarly, new poses are given by ˜yt = µY (˜xt). Depending on the dataset and the choice of kernels, long sequences generated by sampling or mean-prediction can diverge from the data. On our data sets, mean-prediction trajectories from the GPDM with an RBF or linear+RBF kernel for dynamics usually produce sequences that roughly follow the training data (e.g., see the red curves in Figure 3). This usually means producing closed limit cycles with walking data. We also found that meanprediction motions are often very close to the mean obtained from the HMC sampler; by (a) (b) (c) (d) (e) Figure 2: Models learned from a walking sequence of 2.5 gait cycles. The latent positions learned with a GPLVM (a) and a GPDM (b) are shown in blue. Vectors depict the temporal sequence. (c) - log variance for reconstruction shows regions of latent space that are reconstructed with high confidence. (d) Random trajectories drawn from the model using HMC (green), and their mean (red). (e) A GPDM of walk data learned with RBF+linear kernel dynamics. The simulation (red) was started far from the training data, and then optimized (green). The poses were reconstructed from points on the optimized trajectory. (a) (b) Figure 3: (a) Two GPDMs and mean predictions. The first is that from the previous figure. The second was learned with a linear kernel. (b) The GPDM model was learned from 3 swings of a golf club, using a 2nd order RBF kernel for dynamics. The two plots show 2D orthogonal projections of the 3D latent space. initializing HMC with mean-prediction, we find that the sampler reaches equilibrium in a small number of interations. Compared to the RBF kernels, mean-prediction motions generated from GPDMs with the linear kernel often deviate from the original data (e.g., see Figure 3a), and lead to over-smoothed animation. Figure 3(b) shows a 3D GPDM learned from three swings of a golf club. The learning aligns the sequences and nicely accounts for variations in speed during the club trajectory. 3.2 Optimization While mean-prediction is efficient, there is nothing in the algorithm that prevents trajectories from drifting away from the training data. Thus, it is sometimes desirable to optimize a particular motion under the GPDM, which often reduces drift of the mean-prediction mo(a) (b) Figure 4: GPDM from walk sequence with missing data learned with (a) a RBF+linear kernel for dynamics, and (b) a linear kernel for dynamics. Blue curves depict original data. Green curves are the reconstructed, missing data. tions. To optimize a new sequence, we first select a starting point ˜x1 and a number of time-steps. The likelihood p(˜X | X, ¯α) of the new sequence ˜X is then optimized directly (holding the latent positions of the previously learned latent positions, X, and hyperparameters, ¯α, fixed). To see why optimization generates motion close to the traing data, note that the variance of pose ˜xt+1 is determined by σ2 X(˜xt), which will be lower when ˜xt is nearer the training data. Consequently, the likelihood of ˜xt+1 can be increased by moving ˜xt closer to the training data. This generalizes the preference of the SGPLVM for poses similar to the examples [4], and is a natural consequence of the Bayesian approach. As an example, Fig. 2(e) shows an optimized walk sequence initialized from the mean-prediction. 3.3 Forecasting We performed a simple experiment to compare the predictive power of the GPDM to a linear dynamical system, implemented as a GPDM with linear kernel in the latent space and RBF latent mapping. We trained each model on the first 130 frames of the 60Hz walking sequence (corresponding to 2 cycles), and tested on the remaining 23 frames. From each test frame mean-prediction was used to predict the pose 8 frames ahead, and then the RMS pose error was computed against ground truth. The test was repeated using mean-prediction and optimization for three kernels, all based on first-order predictions as in (1): Linear RBF Linear+RBF mean-prediction 59.69 48.72 36.74 optimization 58.32 45.89 31.97 Due to the nonlinear nature of the walking dynamics in latent space, the RBF and Linear+RBF kernels outperform the linear kernel. Moreover, optimization (initialized by mean-prediction) improves the result in all cases, for reasons explained above. 3.4 Missing Data The GPDM model can also handle incomplete data (a common problem with human motion capture sequences). The GPDM is learned by minimizing L (Eqn. (16)), but with the terms corresponding to missing poses yt removed. The latent coordinates for missing data are initialized by cubic spline interpolation from the 3D PCA initialization of observations. While this produces good results for short missing segments (e.g., 10–15 frames of the 157-frame walk sequence used in Fig. 2), it fails on long missing segments. The problem lies with the difficulty in initializing the missing latent positions sufficiently close to the training data. To solve the problem, we first learn a model with a subsampled data sequence. Reducing sampling density effectively increases uncertainty in the reconstruction process so that the probability density over the latent space falls off more smoothly from the data. We then restart the learning with the entire data set, but with the kernel hyperparameters fixed. In doing so, the dynamics terms in the objective function exert more influence over the latent coordinates of the training data, and a smooth model is learned. With 50 missing frames of the 157-frame walk sequence, this optimization produces models (Fig. 4) that are much smoother than those in Fig. 2. The linear kernel is able to pull the latent coordinates onto a cylinder (Fig. 4b), and thereby provides an accurate dynamical model. Both models shown in Fig. 4 produce estimates of the missing poses that are visually indistinguishable from the ground truth. 4 Discussion and Extensions One of the main strengths of the GPDM model is the ability to generalize well from small datasets. Conversely, performance is a major issue in applying GP methods to larger datasets. Previous approaches prune uninformative vectors from the training data [5]. This is not straightforward when learning a GPDM, however, because each timestep is highly correlated with the steps before and after it. For example, if we hold xt fixed during optimization, then it is unlikely that the optimizer will make much adjustment to xt+1 or xt−1. The use of higher-order features provides a possible solution to this problem. Specifically, consider a dynamical model of the form vt = f(xt−1, vt−1). Since adjacent time-steps are related only by the velocity vt ≈(xt −xt−1)/∆t, we can handle irregularly-sampled datapoints by adjusting the timestep ∆t, possibly using a different ∆t at each step. A number of further extensions to the GPDM model are possible. It would be straightforward to include a control signal ut in the dynamics f(xt, ut). It would also be interesting to explore uncertainty in latent variable estimation (e.g., see [3]). Our use of maximum likelihood latent coordinates is motivated by Lawrence’s observation that model uncertainty and latent coordinate uncertainty are interchangeable when learning PCA [5]. However, in some applications, uncertainty about latent coordinates may be highly structured (e.g., due to depth ambiguities in motion tracking). Acknowledgements This work made use of Neil Lawrence’s publicly-available GPLVM code, the CMU mocap database (mocap.cs.cmu.edu), and Joe Conti’s volume visualization code from mathworks.com. This research was supported by NSERC and CIAR. References [1] M. Brand and A. Hertzmann. Style machines. Proc. SIGGRAPH, pp. 183-192, July 2000. [2] Z. Ghahramani and S. T. Roweis. Learning nonlinear dynamical systems using an EM algorithm. Proc. NIPS 11, pp. 431-437, 1999. [3] A. Girard, C. E. Rasmussen, J. G. Candela, and R. Murray-Smith. Gaussian process priors with uncertain inputs - application to multiple-step ahead time series forecasting. Proc. NIPS 15, pp. 529-536, 2003. [4] K. Grochow, S. L. Martin, A. Hertzmann, and Z. Popovi´c. Style-based inverse kinematics. ACM Trans. Graphics, 23(3):522-531, Aug. 2004. [5] N. D. Lawrence. Gaussian process latent variable models for visualisation of high dimensional data. Proc. NIPS 16, 2004. [6] J. Lee, J. Chai, P. S. A. Reitsma, J. K. Hodgins, and N. S. Pollard. Interactive control of avatars animated with human motion data. ACM Trans. Graphics, 21(3):491-500, July 2002. [7] W. E. Leithead, E. Solak, and D. J. Leith. Direct identification of nonlinear structure using Gaussian process prior models. Proc. European Control Conference, 2003. [8] D. MacKay. Information Theory, Inference, and Learning Algorithms. 2003. [9] R. Murray-Smith and B. A. Pearlmutter. Transformations of Gaussian process priors. Technical Report, Department of Computer Science, Glasgow University, 2003 [10] R. M. Neal. Bayesian Learning for Neural Networks. Springer-Verlag, 1996. [11] V. Pavlovi´c, J. M. Rehg, and J. MacCormick. Learning switching linear models of human motion. Proc. NIPS 13, pp. 981-987, 2001. [12] L. Ralaivola and F. d’Alch´e-Buc. Dynamical modeling with kernels for nonlinear time series prediction. Proc. NIPS 16, 2004. [13] C. E. Rasmussen and M. Kuss. Gaussian processes in reinforcement learning. Proc. NIPS 16, 2004. [14] H. Sidenbladh, M. J. Black, and D. J. Fleet. Stochastic tracking of 3D human figures using 2D motion. Proc. ECCV, volume 2, pp. 702-718, 2000. [15] E. Solak, R. Murray-Smith, W. Leithead, D. Leith, and C. E. Rasmussen. Derivative observations in Gaussian process models of dynamic systems. Proc. NIPS 15, pp. 1033-1040, 2003.
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Rodeo: Sparse Nonparametric Regression in High Dimensions John Lafferty School of Computer Science Carnegie Mellon University Larry Wasserman Department of Statistics Carnegie Mellon University Abstract We present a method for nonparametric regression that performs bandwidth selection and variable selection simultaneously. The approach is based on the technique of incrementally decreasing the bandwidth in directions where the gradient of the estimator with respect to bandwidth is large. When the unknown function satisfies a sparsity condition, our approach avoids the curse of dimensionality, achieving the optimal minimax rate of convergence, up to logarithmic factors, as if the relevant variables were known in advance. The method—called rodeo (regularization of derivative expectation operator)—conducts a sequence of hypothesis tests, and is easy to implement. A modified version that replaces hard with soft thresholding effectively solves a sequence of lasso problems. 1 Introduction Estimating a high dimensional regression function is notoriously difficult due to the “curse of dimensionality.” Minimax theory precisely characterizes the curse. Let Yi = m(Xi) + ǫi, i = 1, . . . , n where Xi = (Xi(1), . . . , Xi(d)) ∈Rd is a d-dimensional covariate, m : Rd →R is the unknown function to estimate, and ǫi ∼N(0, σ2). Then if m is in W2(c), the d-dimensional Sobolev ball of order two and radius c, it is well known that lim inf n→∞n4/(4+d) inf b mn sup m∈W2(c) R( bmn, m) > 0 , (1) where R( bmn, m) = Em R ( bmn(x) −m(x))2 dx is the risk of the estimate bmn constructed on a sample of size n (Gy¨orfiet al. 2002). Thus, the best rate of convergence is n−4/(4+d), which is impractically slow if d is large. However, for some applications it is reasonable to expect that the true function only depends on a small number of the total covariates. Suppose that m satisfies such a sparseness condition, so that m(x) = m(xR) where xR = (xj : j ∈R), R ⊂{1, . . . , d} is a subset of the d covariates, of size r = |R| ≪d. We call {xj}j∈R the relevant variables. Under this sparseness assumption we can hope to achieve the better minimax convergence rate of n−4/(4+r) if the r relevant variables can be isolated. Thus, we are faced with the problem of variable selection in nonparametric regression. A large body of previous work has addressed this fundamental problem, which has led to a variety of methods to combat the curse of dimensionality. Many of these are based on very clever, though often heuristic techniques. For additive models of the form f(x) = P j fj(xj), standard methods like stepwise selection, Cp and AIC can be used (Hastie et al. 2001). For spline models, Zhang et al. (2005) use likelihood basis pursuit, essentially the lasso adapted to the spline setting. CART (Breiman et al. 1984) and MARS (Friedman 1991) effectively perform variable selection as part of their function fitting. More recently, Li et al. (2005) use independence testing for variable selection and B¨uhlmann and Yu (2005) introduced a boosting approach. While these methods have met with varying degrees of empirical success, they can be challenging to implement and demanding computationally. Moreover, these methods are typically difficult to analyze theoretically, and so often come with no formal guarantees. Indeed, the theoretical analysis of sparse parametric estimators such as the lasso (Tibshirani 1996) is difficult, and only recently has significant progress been made on this front (Donoho 2004; Fu and Knight 2000). In this paper we present a new approach to sparse nonparametric function estimation that is both computationally simple and amenable to theoretical analysis. We call the general framework rodeo, for regularization of derivative expectation operator. It is based on the idea that bandwidth and variable selection can be simultaneously performed by computing the infinitesimal change in a nonparametric estimator as a function of the smoothing parameters, and then thresholding these derivatives to effectively get a sparse estimate. As a simple version of this principle we use hard thresholding, effectively carrying out a sequence of hypothesis tests. A modified version that replaces testing with soft thresholding effectively solves a sequence of lasso problems. The potential appeal of this approach is that it can be based on relatively simple and theoretically well understood nonparametric techniques such as local linear smoothing, leading to methods that are simple to implement and can be used in high dimensional problems. Moreover, we show that the rodeo can achieve near optimal minimax rates of convergence, and therefore circumvents the curse of dimensionality when the true function is indeed sparse. When applied in one dimension, our method yields a locally optimal bandwidth. We present experiments on both synthetic and real data that demonstrate the effectiveness of the new approach. 2 Rodeo: The Main Idea The key idea in our approach is as follows. Fix a point x and let bmh(x) denote an estimator of m(x) based on a vector of smoothing parameters h = (h1, . . . , hd). If c is a scalar, then we write h = c to mean h = (c, . . . , c). Let M(h) = E( bmh(x)) denote the mean of bmh(x). For now, assume that xi is one of the observed data points and that bm0(x) = Yi. In that case, m(x) = M(0) = E(Yi). If P = (h(t) : 0 ≤t ≤1) is a smooth path through the set of smoothing parameters with h(0) = 0 and h(1) = 1 (or any other fixed, large bandwidth) then m(x) = M(0) = M(1) − Z 1 0 dM(h(s)) ds ds = M(1) − Z 1 0 D(s), ˙h(s) ds where D(h) = ∇M(h) = ∂M ∂hj , . . . , ∂M ∂hj T is the gradient of M(h) and ˙h(s) = dh(s) ds is the derivative of h(s) along the path. A biased, low variance estimator of M(1) is bm1(x). An unbiased estimator of D(h) is Z(h) = ∂bmh(x) ∂h1 , . . . , ∂bmh(x) ∂hd T . (2) The naive estimator bm(x) = bm1(x) − Z 1 0 Z(s), ˙h(s) ds (3) h1 h2 Start Optimal bandwidth Ideal path Rodeo path Figure 1: The bandwidths for the relevant variables (h2) are shrunk, while the bandwidths for the irrelevant variables (h1) are kept relatively large. The simplest rodeo algorithm shrinks the bandwidths in discrete steps 1, β, β2, . . . for some 0 < β < 1. is identically equal to bm0(x) = Yi, which has poor risk since the variance of Z(h) is large for small h. However, our sparsity assumption on m suggests that there should be paths for which D(h) is also sparse. Along such a path, we replace Z(h) with an estimator bD(h) that makes use of the sparsity assumption. Our estimate of m(x) is then em(x) = bm1(x) − Z 1 0 bD(s), ˙h(s) ds . (4) To implement this idea we need to do two things: (i) we need to find a sparse path and (ii) we need to take advantage of this sparseness when estimating D along that path. The key observation is that if xj is irrelevant, then we expect that changing the bandwidth hj for that variable should cause only a small change in the estimator bmh(x). Conversely, if xj is relevant, then we expect that changing the bandwidth hj for that variable should cause a large change in the estimator. Thus, Zj = ∂bmh(x)/∂hj should discriminate between relevant and irrelevant covariates. To simplify the procedure, we can replace the continuum of bandwidths with a discrete set where each hj ∈B = {h0, βh0, β2h0, . . .} for some 0 < β < 1. Moreover, we can proceed in a greedy fashion by estimating D(h) sequentially with hj ∈B and setting bDj(h) = 0 when hj < bhj, where bhj is the first h such that |Zj(h)| < λj(h) for some threshold λj. This greedy version, coupled with the hard threshold estimator, yields em(x) = bmbh(x). A conceptual illustration of the idea is shown in Figure 1. This idea can be implemented using a greedy algorithm, coupled with the hard threshold estimator, to yield a bandwidth selection procedure based on testing. This approach to bandwidth selection is similar to that of Lepski et al. (1997), which uses a more refined test leads to estimators that achieve good spatial adaptation over large function classes. Our approach is also similar to a method of Ruppert (1997) that uses a sequence of decreasing bandwidths and then estimates the optimal bandwidth by estimating the mean squared error as a function of bandwidth. Our greedy approach tests whether an infinitesimal change in the bandwidth from its current setting leads to a significant change in the estimate, and is more easily extended to a practical method in higher dimensions. Related work of Hristache et al. (2001) focuses on variable selection in multi-index models rather than on bandwidth estimation. 3 Rodeo using Local Linear Regression We now present the multivariate case in detail, using local linear smoothing as the basic method since it is known to have many good properties. Let x = (x(1), . . . , x(d)) be some point at which we want to estimate m. Let bmH(x) denote the local linear estimator of m(x) using bandwidth matrix H. Thus, bmH(x) = eT 1 (XT x WxXx)−1XT x WxY, Xx = 1 (X1 −x)T ... ... 1 (Xn −x)T (5) where e1 = (1, 0, . . . , 0)T , and Wx is the diagonal matrix with (i, i) element KH(Xi −x) and KH(u) = |H|−1K(H−1u). The estimator bmH can be written as bmH(x) = Pn i=1 G(Xi, x, h) Yi where G(u, x, h) = eT 1 (XT x WxXx)−1 1 (u −x)T KH(u −x) (6) is called the effective kernel. We assume that the covariates are random with sampling density f(x), and make the same assumptions as Ruppert and Wand (1994) in their analysis of the bias and variance of local linear regression. In particular, (i) the kernel K has compact support with zero odd moments and R uu⊤K(u) du = ν2(K)I and (ii) the sampling density f(x) is continuously differentiable and strictly positive. In the version of the algorithm that follows, we take K to be a product kernel and H to be diagonal with elements h = (h1, . . . , hd). Our method is based on the statistic Zj = ∂bmh(x) ∂hj = n X i=1 Gj(Xi, x, h)Yi (7) where Gj(u, x, h) = ∂G(u,x,h) ∂hj . Straightforward calculations show that Zj = ∂bmh(x) ∂hj = = e⊤ 1 (X⊤ x WxXx)−1X⊤ x ∂Wx ∂hj (Y −Xxbα) (8) where bα = (X⊤ x WxXx)−1X⊤ x WxY is the coefficient vector for the local linear fit. Note that the factor |H|−1 = Qd i=1 1/hi in the kernel cancels in the expression for bm, and therefore we can ignore it in our calculation of Zj. Assuming a product kernel we have Wx = diag d Y j=1 K((X1j −xj)/hj), . . . , d Y j=1 K((Xnj −xj)/hj) (9) and ∂Wx/∂hj = WxDj where Dj = diag ∂log K((X1j −xj)/hj) ∂hj , . . . , ∂log K((Xnj −xj)/hj) ∂hj (10) and thus Zj = e⊤ 1 (X⊤ x WxXx)−1X⊤ x WxDj(Y −Xxbα). For example, with the Gaussian kernel K(u) = exp(−u2/2) we have Dj = 1 h3 j diag (X1j −xj)2, . . . , (Xnj −xj)2 . Let µj ≡ µj(h) = E(Zj|X1, . . . , Xn) = n X i=1 Gj(Xi, x, h)m(Xi) (11) s2 j ≡ s2 j(h) = V(Zj|X1, . . . , Xn) = σ2 n X i=1 Gj(Xi, x, h)2. (12) Then the hard thresholding version of the rodeo algorithm is given in Figure 2. The algorithm requires that we insert an estimate bσ of σ in (12). One estimate of σ can be obtained by generalizing a method of Rice (1984). For i < ℓ, let diℓ= ∥Xi −Xℓ∥. Fix an integer J and let E denote the set of pairs (i, ℓ) corresponding to the J smallest values of diℓ. Now define bσ2 = 1 2J P i,ℓ∈E(Yi −Yℓ)2. Then E(bσ2) = σ2 + bias where Rodeo: Hard thresholding version 1. Select parameter 0 < β < 1 and initial bandwidth h0 slowly decreasing to zero, with h0 = Ω 1/√log log n . Let cn = Ω(1) be a sequence satisfying dcn = Ω(log n). 2. Initialize the bandwidths, and activate all covariates: (a) hj = h0, j = 1, 2, . . . , d. (b) A = {1, 2, . . . , d} 3. While A is nonempty, do for each j ∈A: (a) Compute the estimated derivative expectation: Zj (equation 7) and sj (equation 12). (b) Compute the threshold λj = sj p 2 log(dcn). (c) If |Zj| ≥λj, then set hj ←βhj, otherwise remove j from A. 4. Output bandwidths h⋆= (h1, . . . , hd) and estimator em(x) = bmh⋆(x). Figure 2: The hard thresholding version of the rodeo, which can be applied using the derivatives Zj of any nonparametric smoother. bias ≤D supx P j∈R ∂f(x) ∂xj with D = maxi,ℓ∈E ∥Xi −Xℓ∥. There is a bias-variance tradeoff: large J makes bσ2 positively biased, and small J makes bσ2 highly variable. Note however that the bias is mitigated by sparsity (small r). This is the estimator used in our examples. 4 Analysis In this section we present some results on the properties of the resulting estimator. Formally, we use a triangular array approach so that f(x), m(x), d and r can all change as n changes. For convenience of notation we assume that the covariates are numbered such that the relevant variables xj correspond to 1 ≤j ≤r, and the irrelevant variables to j > r. To begin, we state the following technical lemmas on the mean and variance of Zj. Lemma 4.1. Suppose that K is a product kernel with bandwidth vector h = (h1, . . . , hd). If the sampling density f is uniform, then µj = 0 for all j ∈Rc. More generally, assuming that r is bounded, we have the following when hj →0: If j ∈Rc the derivative of the bias is µj = ∂ ∂hj E[ bmH(x) −m(x)] = −tr (HRHR) ν2 2 (∇j log f(x))2 hj + oP (hj) (13) where the Hessian of m(x) is H = HR 0 0 0 and HR = diag(h2 1, . . . , h2 r). For j ∈R we have µj = ∂ ∂hj E[ bmH(x) −m(x)] = hjν2mjj(x) + oP (hj). (14) Lemma 4.2. Let C = σ2R(K) 4m(x) where R(K) = R K(u)2 du. Then, if hj = o(1), s2 j = Var(Zj|X1, . . . , Xn) = C nh2 j d Y k=1 1 hk ! 1 + oP (1) . (15) These lemmas parallel the calculations of Ruppert and Wand (1994) except for the difference that the irrelevant variables have different leading terms in the expansions than relevant variables. Our main theoretical result characterizes the asymptotic running time, selected bandwidths, and risk of the algorithm. In order to get a practical algorithm, we need to make assumptions on the functions m and f. (A1) For some constant k > 0, each j > r satisfies ∇j log f(x) = O logk n n1/4 ! (16) (A2) For each j ≤r, mjj(x) ̸= 0 . (17) Explanation of the Assumptions. To give the intuition behind these assumptions, recall from Lemma 4.1 that µj = Ajhj + oP (hj) j ≤r Bjhj + oP (hj) j > r (18) where Aj = ν2mjj(x), Bj = −tr(HH)ν2 2(∇j log f(x))2. (19) Moreover, µj = 0 when the sampling density f is uniform or the data are on a regular grid. Consider assumption (A1). If f is uniform then this assumption is automatically satisfied since then µj(s) = 0 for j > r. More generally, µj is approximately proportional to (∇j log f(x))2 for j > r which implies that |µj| ≈0 for irrelevant variables if f is sufficiently smooth in the variable xj. Hence, assumption (A1) can be interpreted as requiring that f is sufficiently smooth in the irrelevant dimensions. Now consider assumption (A2). Equation (18) ensures that µj is proportional to hj|mjj(x)| for small hj. Since we take the initial bandwidth h0 to be decreasingly slowly with n, (A2) implies that |µj(h)| ≥chj|mjj(x)| for some constant c > 0, for sufficiently large n. In the following we write Yn = eOP (an) to mean that Yn = OP (bnan) where bn is logarithmic in n; similarly, an = eΩ(bn) if an = Ω(bncn) where cn is logarithmic in n. Theorem 4.3. Suppose assumptions (A1) and (A2) hold. In addition, suppose that dmin = minj≤r |mjj(x)| = eΩ(1) and dmax = maxj≤r |mjj(x)| = eO(1). Then the number of iterations Tn until the rodeo stops satisfies P 1 4 + r log1/β(nan) ≤Tn ≤ 1 4 + r log1/β(nbn) −→1 (20) where an = eΩ(1) and bn = eO(1). Moreover, the algorithm outputs bandwidths h⋆that satisfy P h⋆ j ≥ 1 logk n for all j > r −→1 (21) and P h0(nbn)−1/(4+r) ≤h⋆ j ≤h0(nan)−1/(4+r) for all j ≤r −→1 . (22) Corollary 4.4. Under the conditions of Theorem 4.3, the risk R(h⋆) of the rodeo estimator satisfies R(h⋆) = eOP n−4/(4+r) . (23) In the one-dimensional case, this result shows that the algorithm recovers the locally optimal bandwidth, giving an adaptive estimator, and in general attains the optimal (up to logarithmic factors) minimax rate of convergence. The proofs of these results are given in the full version of the paper. 5 Some Examples and Extensions Figure 3 illustrates the rodeo on synthetic and real data. The left plot shows the bandwidths obtained on a synthetic dataset with n = 500 points of dimension d = 20. The covariates are generated as xi ∼Uniform(0, 1), the true function is m(x) = 2(x1+1)2+2 sin(10x2), and σ = 1. The results are averaged over 50 randomly generated data sets; note that the displayed bandwidth paths are not monotonic because of this averaging. The plot shows how the bandwidths of the relevant variables shrink toward zero, while the bandwidths of the irrelevant variables remain large. Simulations on other synthetic data sets, not included here, are similar and indicate that the algorithm’s performance is consistent with our theoretical analysis. The framework introduced here has many possible generalizations. While we have focused on estimation of m locally at a point x, the idea can be extended to carry out global bandwidth and variable selection by averaging over multiple evaluation points x1, . . . , xk. These could be points interest for estimation, could be randomly chosen, or could be taken to be identical to the observed Xis. In addition, it is possible to consider more general paths, for example using soft thresholding or changing only the bandwidth corresponding to the largest |Zj|/λj. Such a version of the rodeo can be seen as a nonparametric counterpart to least angle regression (LARS) (Efron et al. 2004), a refinement of forward stagewise regression in which one adds the covariate most correlated with the residuals of the current fit, in small, incremental steps. Note first that Zj is essentially the correlation between the Yis and the Gj(Xi, x, h)s (the change in the effective kernel). Reducing the bandwidth is like adding in more of that variable. Suppose now that we make the following modifications to the rodeo: (i) change the bandwidths one at a time, based on the largest Z∗ j = |Zj|/λj, (ii) reduce the bandwidth continuously, rather than in discrete steps, until the largest Z∗ j is equal to the next largest. Figure 3 (right) shows the result of running this greedy version of the rodeo on the diabetes dataset used to illustrate LARS. The algorithm averages Z∗ j over a randomly chosen set of k = 100 data points. The resulting variable ordering is seen to be very similar to, but different from, the ordering obtained from the parametric LARS fit. Acknowledgments We thank the reviewers for their helpful comments. Research supported in part by NSF grants IIS-0312814, IIS-0427206, and DMS-0104016, and NIH grants R01-CA54852-07 and MH57881. References L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone. Classification and regression trees. Wadsworth Publishing Co Inc, 1984. P. B¨uhlmann and B. Yu. Boosting, model selection, lasso and nonnegative garrote. Technical report, Berkeley, 2005. 5 10 15 0.0 0.2 0.4 0.6 0.8 1.0 Rodeo Step Average Bandwidth 1 2 34 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 20 40 60 80 100 0.0 0.1 0.2 0.3 0.4 0.5 Greedy Rodeo Step Bandwidth 1 2 3 4 7 8 9 Figure 3: Left: Average bandwidth output by the rodeo for a function with r = 2 relevant variables in d = 20 dimensions (n = 500, with 50 trials). Covariates are generated as xi ∼Uniform(0, 1), the true function is m(x) = 2(x1 + 1)3 + 2 sin(10x2), and σ = 1, fit at the test point x = ( 1 2, . . . , 1 2). The variance is greater for large step sizes since the rodeo runs that long for fewer data sets. Right: Greedy rodeo on the diabetes data, used to illustrate LARS (Efron et al. 2004). A set of k = 100 of the total n = 442 points were sampled (d = 10), and the bandwidth for the variable with largest average |Zj|/λj was reduced in each step. The variables were selected in the order 3 (body mass index), 9 (serum), 7 (serum), 4 (blood pressure), 1 (age), 2 (sex), 8 (serum), 5 (serum), 10 (serum), 6 (serum). The parametric LARS algorithm adds variables in the order 3, 9, 4, 7, 2, 10, 5, 8, 6, 1. One notable difference is in the position of the age variable. D. Donoho. For most large underdetermined systems of equations, the minimal ℓ1-norm near-solution approximates the sparest near-solution. Technical report, Stanford, 2004. B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani. Least angle regression. The Annals of Statistics, 32:407–499, 2004. J. H. Friedman. Multivariate adaptive regression splines. The Annals of Statistics, 19:1–67, 1991. W. Fu and K. Knight. Asymptotics for lasso type estimators. The Annals of Statistics, 28:1356–1378, 2000. L. Gy¨orfi, M. Kohler, A. Krzy˙zak, and H. Walk. A Distribution-Free Theory of Nonparametric Regression. Springer-Verlag, 2002. T. Hastie, R. Tibshirani, and J. H. Friedman. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer-Verlag, 2001. M. Hristache, A. Juditsky, J. Polzehl, and V. Spokoiny. Structure adaptive approach for dimension reduction. Ann. Statist., 29:1537–1566, 2001. O. V. Lepski, E. Mammen, and V. G. Spokoiny. Optimal spatial adaptation to inhomogeneous smoothness: An approach based on kernel estimates with variable bandwidth selectors. The Annals of Statistics, 25:929–947, 1997. L. Li, R. D. Cook, and C. Nachsteim. Model-free variable selection. J. R. Statist. Soc. B., 67:285–299, 2005. J. Rice. Bandwidth choice for nonparametric regression. The Annals of Statistics, 12:1215–1230, 1984. D. Ruppert. Empirical-bias bandwidths for local polynomial nonparametric regression and density estimation. Journal of the American Statistical Association, 92:1049–1062, 1997. D. Ruppert and M. P. Wand. Multivariate locally weighted least squares regression. The Annals of Statistics, 22:1346–1370, 1994. R. Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B, Methodological, 58:267–288, 1996. H. Zhang, G. Wahba, Y. Lin, M. Voelker, R. K. Ferris, and B. Klein. Variable selection and model building via likelihood basis pursuit. J. of the Amer. Stat. Assoc., 99(467):659–672, 2005.
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Pattern Recognition from One Example by Chopping Franc¸ois Fleuret CVLAB/LCN – EPFL Lausanne, Switzerland francois.fleuret@epfl.ch Gilles Blanchard∗ Fraunhofer FIRST Berlin, Germany blanchar@first.fhg.de Abstract We investigate the learning of the appearance of an object from a single image of it. Instead of using a large number of pictures of the object to recognize, we use a labeled reference database of pictures of other objects to learn invariance to noise and variations in pose and illumination. This acquired knowledge is then used to predict if two pictures of new objects, which do not appear on the training pictures, actually display the same object. We propose a generic scheme called chopping to address this task. It relies on hundreds of random binary splits of the training set chosen to keep together the images of any given object. Those splits are extended to the complete image space with a simple learning algorithm. Given two images, the responses of the split predictors are combined with a Bayesian rule into a posterior probability of similarity. Experiments with the COIL-100 database and with a database of 150 degraded LATEX symbols compare our method to a classical learning with several examples of the positive class and to a direct learning of the similarity. 1 Introduction Pattern recognition has so far mainly focused on the following task: given many training examples labelled with their classes (the object they display), guess the class of a new sample which was not available during training. The various approaches all consist of going to some invariant feature space, and there using a classification method such as neural networks, decision trees, kernel techniques, Bayesian estimations based on parametric density models, etc. Providing a large number of examples results in good statistical estimates of the model parameters. Although such approaches have been successful in applications to many problems, their performance are still far from what biological visual systems can do, which is one sample learning. This can be defined as the ability, given one picture of an object, to spot instances of the same object, under the assumption that these new views can be induced by the single available example. ∗Supported in part by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778 Being able to perform that type of one-sample learning corresponds to the ability, given one example, to sort out which elements of a test set are of the same class (i.e. one class vs. the rest of the world). This can be done by comparing one by one all the elements of the test set with the reference example, and labelling as of the same class those which are similar enough. Learning techniques can be used to choose the similarity measure, which could be adaptive and learned from a large number of examples of classes not involved in the test. Thus, given a large number of training images of a large number of objects labeled with their actual classes, and provided two pictures of unknown objects (objects which do not appear in the training pictures), we want to decide if these two objects are actually the same object. The first image of such a couple can be seen as a single training example, and the second image as a test example. Averaging the error rate by repeating that test several times provides with an estimate of a one-sample learning (OSL) error rate. The idea of “learning how to learn” is not new and has been applied in various settings [12]. Taking into account and/or learning relevant geometric invariances for a given task has been studied under various forms [1, 8, 11], and in [7] with the goal to achieve learning from very few examples. Finally, the precise one-sample learning setting considered here has been the object of recent research [4, 3, 5] proposing different methods (hyperfeature learning, distance learning) for finding invariant features from a set of training reference objects distinct from the test objects. This principle has also been dubbed interclass transfer. The present study proposes a generic approach, and avoids an explicit description of the space of deformations. We propose to build a large number of binary splits of the image space, designed to assign the same binary label to all the images common to a same object. The binary mapping associated to such a split is thus highly invariant across the images of a certain object while highly variant across images of different objects. We can define such a split on the training images, and train a predictor to extend it to the complete image space by induction. We expect the predictor to respond similarly on two images of a same object, and differently on two images of two different objects with probability 1 2. The global criterion to compare two images consists roughly of counting how many such splitpredictors responds similarly and compare the result to a fixed threshold. The principle of transforming a multiclass learning problem into several binary ones by class grouping has a long history in Machine Learning [10]. From this point of view the collected output of several binary classifiers is used as a way for coding class membership. In [2] it was proposed to carefully choose the class groupings so as to yield optimal separation of codewords (ECOC methodology). While our method is related to this general principle, our goal is different since we are interested in recognizing yet-unseen objects. Hence, the goal is not to code multiclass membership; our focus is not on designing efficient codes – splits are chosen randomly and we take a large number of them – but rather on how to use the learned mappings for learning unknown objects. 2 Data and features To make the rest of the paper clearer to the reader, we now introduce the data and feature sets we are using for our proof of concept experiments. However, note that while we have focused on image classification, our approach is generic and could be applied to any signals for which adaptive binary classifiers are available. 2.1 Data We use two databases of pictures for our experiments. The first one is the standard COIL100 database of pictures [9]. It contains 7200 images corresponding to 100 different objects Figure 1: Four objects from the 100 objects of the COIL-100 database (downsampled to 38 × 38 grayscale pixels) and four symbols from the 150 symbols of our LATEX symbol database (A, Φ, ⋖and ⋔, resolution 28 × 28). Each image of the later is generated by applying a rotation and a scaling, and by adding lines of random grayscales at random locations and orientations. d=0 d=3 d=2 d=1 d=7 d=6 d=5 d=4 (x,y) Figure 2: The figure on the left shows how an horizontal edge ξx,y,4 is detected: the six differences between pixels connected by a thin segment have to be all smaller in absolute value than the difference between the pixels connected by the thick segment. The relative values of the two pixels connected by the thick segment define the polarity of the edge (dark to light or light to dark). On the right are shown the eight different types of edges. seen from 72 angles of view. We down-sample these images from their original resolution to 38 × 38 pixels, and convert them to grayscale. Examples are given in figure 1 (left). The second database contains images of 150 LATEX symbols. We generated 1, 000 images of each symbol by applying a random rotation (angle is taken between −20 and +20 degrees) and a random scaling factor (up to 1.25). Noise is then added by adding random line segments of various gray scales, locations and orientations. The final resulting database contains 150, 000 images. Examples of these degraded images are given in figure 1 (right). 2.2 Features All the classification processes in the rest of the paper are based on edge-based boolean features. Let ξx,y,d denote a basic edge detector indexed by a location (x, y) in the image frame and an orientation d which can take eight different values, corresponding to four orientations and two polarities (see figure 2). Such an edge detector is equal to 1 if and only if an edge of the given location is detected at the specified location, and 0 otherwise. A feature fx0,y0,x1,y1,d is a disjunction of the ξ’s in the rectangle defined by x0, y0, x1, y1. Thus, it is equal to one if and only if ∃x, y, x0 ≤x ≤x1, y0 ≤y ≤y1, ξx,y,d = 1. For pictures of size 32 × 32 there is a total of N = 1 4(32 × 32)2 × 8 ≃2.106 features. 0 0.05 0.1 0.15 0.2 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 Response Negative class Positive class 0 0.05 0.1 0.15 0.2 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 Response Negative class Positive class Figure 3: These two histograms are representative of the responses of two split predictors conditionally to the real arbitrary labelling P(L | S). 3 Chopping The main idea we propose in this paper consists of learning a large number of binary splits of the image space which would ideally assign the same binary label to all the images of any given object. In this section we define these splits and describe and justify how they are combined into a global rule. 3.1 Splits A split is a binary labelling of the image space, with the property to give the same label to all images of a given object. We can trivially produce a labelling with that property on the training examples, but we need to be able to extend it to images not appearing in the training data, including images of other objects. We suppose that it is possible to infer a relevant split function on the complete image space, including images of other objects by looking at the problem as a binary classification problem. Inference is done by the mean of a simple learning scheme: a combination of a fast feature selection based on conditional mutual information (CMIM) [6] and a linear perceptron. Thus, we create M arbitrary splits on the training sample by randomly assigning the label 1 to half of the NT objects appearing in the training set, and 0 to the others. Since there are NT NT /2 such balanced arbitrary labellings, with NT of the order of a few tens, a very large number of splits is available and only a small subset of them will be actually used for learning. For each one of those splits, we train a predictor using the scheme described above. Let (S1, . . . , SM) denote the family of arbitrary splits and (L1, . . . , LM) the split-predictors. The continuous outputs of these predictors before thresholding will be combined in the final classification. 3.2 Combining splits To combine the responses of the various split predictors, we rely on a set of simple conditional independence assumptions (comparable to the “naive Bayes” setting) on the distribution of the true class label C (each class corresponds to an object), the split labels (Si) and the predictor outputs (Li) for a single image. We do not assume that for test image pairs (I1, I2) the two images are independent, because we want to encompass the case where pairs of images of the same object are much more frequent than they would be if they were independent (typically in our test data we have arranged to have 50% of test pairs picturing the same object). We however still need some conditional independence assumption for the drawing of test image pairs. To simplify the notation we denote L1 = (L1 i ), L2 = (L2 i ) the collection of predictor outputs for images 1 and 2, S1 = (S1 i ), S2 = (S2 i ) the collection of their split labels and C1, C2 their true classes. The conditional indepence assumptions we make are summed up in the following Markov dependency diagram: C1 C2 (( S SS S1 1 S1 2 . .. S1 M L1 1 L1 2 . .. L1 M @ @ h h S2 1 S2 2 ... S2 M L2 1 L2 2 . .. L2 M In words, for each split i, the predictor output Li is assumed to be independent of the true class C conditionally to the split label Si; and conditionally to the split labels (S1, S2) of both images, the outputs of predictors on test pair images are assumed to be independent. Finally, we make the additional symmetry hypothesis that conditionally to C1 = C2, for all i : S1 i = S2 i = Si and (Si) are independent Bernoulli variables with parameter 0.5, while conditionally to C1 ̸= C2 all split labels (S1 i , S2 i ) are independent Bernoulli(0.5). Under these assumptions we then want to compute the log-odds ratio log P(C1 = C2 | L1, L2) P(C1 ̸= C2 | L1, L2) = log P(L1, L2 | C1 = C2) P(L1, L2 | C1 ̸= C2) + log P(C1 = C2) P(C1 ̸= C2) . (1) In this formula and the next ones, when handling real-valued variables L1, L2 we are implicitly assuming that they have a density with respect to the Lebesgue measure and probabilities are to be interpreted as densities with some abuse of notation. We assume that the second term above is either known or can be reliably estimated. For the first term, under the aforementioned independence assumptions, the following holds (see appendix): log P(L1, L2 | C1 = C2) P(L1, L2 | C1 ̸= C2) = N log 2 + X i log α1 i α2 i + (1 −α1 i )(1 −α2 i ) , (2) where αj i = P(Sj i = 1 | Lj i). As a quick check, note that if the predictor outputs (Li) are uninformative (i.e. every probability αj i is 0.5), then the above formula gives a ratio of 1 which is what we expect. If they are perfectly informative (i.e. all αj i are 0 or 1), the odds ratio can take the values 0 (if for some j we can ensure S1 j ̸= S2 j , this excludes the case C1 = C2) or 2N (if for all j we have S1 j = S2 j there is still a tiny chance that C1 ̸= C2 if by chance C1, C2 are on the same side of each split). To estimate the probabilities P(Sj | Lj), we use a simple 1D Gaussian model for the output of the predictor given the true split label. Mean and variance are estimated from the training set for each predictor. Experimental findings show that this Gaussian modelling is realistic (see figure 3). 4 Experiments We estimate the performance of the chopping approach by comparing it to classical learning with several examples of the positive class and to a direct learning of the similarity of two objects on different images. For every experiment, we use a family of 10, 000 features sampled uniformly in the complete set of features (see section 2.2) 4.1 Multiple example learning In this procedure, we train a predictor with several pictures of a positive class and with a very large number of pictures of a negative class. The number of positive examples depends on the experiments (from 1 to 32) and the number of negative examples is 2, 000 0 0.1 0.2 0.3 0.4 0.5 0.6 1 2 4 8 16 32 64 128 256 512 1024 1 2 4 8 16 32 Test errors (LaTeX symbols) Number of splits for chopping Number of samples for multi-example learning Chopping Smart chopping Multi-example learning Similarity learnt directly 0 0.1 0.2 0.3 0.4 0.5 0.6 1 2 4 8 16 32 64 128 256 512 1024 1 2 4 8 16 32 Test errors (COIL-100) Number of splits for chopping Number of samples for multi-example learning Chopping Smart chopping Multi-example learning Similarity learnt directly Figure 4: Error rates of the chopping, smart-chopping (see §4.2), multi-example learning and learnt similarity on the LATEX symbol (left) and the COIL-100 database (right). Each curve shows the average error and a two standard deviation interval, both estimated on ten experiments for each setting. The x-axis shows either the number of splits for chopping or the number of samples of the positive class for the multi-example learning. for both the COIL-100 and the LATEX symbol databases. Note that to handle the unbalanced positive and negative populations, the perceptron bias is chosen to minimize a balanced error rate. In each case, and for each number of positive samples, we run 10 experiments. Each experiment consists of several cross-validation cycles so that the total number of test pictures is roughly the same as the number of pairs in one-sample techniques experiments below. 4.2 One-sample learning For each experiment, whatever the predictor is, we first select 80 training objects from the COIL-100 database (respectively 100 symbols from the LATEX symbol database). The test error is computed with 500 pairs of images of the 20 unseen objects for the COIL-100, and 1, 000 pairs of images of the 50 unseen objects for the LATEX symbols. These test sets are built to have as many pairs of images of the same object than pairs of images of different objects. Learnt similarity: Note that one-sample learning can also be simply cast as a standard binary classification problem of pairs of images into the classes {same, different}. We therefore want to compare the Chopping method to a more standard learning method directly on pairs of images using a comparable set of features. For every single feature f on single images, we consider three features of a pair of images standing for the conjunction, disjunction and equality of the feature responses on the two images. From the 10, 000 features on single images, we thus create a set of 30, 000 features on pairs of images. We generate a training set of 2, 000 pairs of pictures for the experiments with the COIL100 database and 5, 000 for the LATEX symbols, half picturing the same object twice, half picturing two different objects. We then train a predictor similar to those used for the splits in the chopping scheme: feature selection with CMIM, and linear combination with a perceptron (see section 3.1), using the 30, 000 features described above. Chopping: The performance of the chopping approach is estimated for several numbers of splits (from 1 to 1024). For each split we select 50 objects from the training objects, and select at random 1, 000 training images of these objects. We generate an arbitrary balanced binary labelling of these 50 objects and label the training images accordingly. We then build a predictor by selecting 2, 000 features with the CMIM algorithm, and combine them with a perceptron (see section 3.1). To compensate for the limitation of our conditional independence assumptions we allow to add a fixed bias to the log-odds ratio (1). This type of correction is common when using naive-Bayes type assumptions. Using the remaining training objects as validation set, we compute this bias so as to minimize the validation error. We insist that no objects of the test classes be used for training. To improve the performance of the splits, we also test a “smart” version of the chopping for which each split is built in two steps. The first step is similar to what is described above. From that first step, we remove the 10 objects for which the labelling prediction has the highest error rate, and re-build the split with the 40 remaining objects. This get rid of problematic objects or inconsistent labelling (for instance trying to force two similar objects to be in different halves of the split). 4.3 Results The experiments demonstrate the good performance of chopping when only one example is available. Its optimal error rate, obtained for the largest number of splits, is 7.41% on the LATEX symbol database and 11.42% on the COIL-100 database. By contrast, a direct learning of the similarity (see section 4.2), reaches respectively 15.54% and 18.1% respectively with 8, 192 features. On both databases, the classical multi-sample learning scheme requires 32 samples to reach the same level of performances (10.51% on the COIL-100 and 10.7% on the LATEX symbols). The error curves (see figure 4) are all monotonic. There is no overfitting when the number of splits increases, which is consistent with the absence of global learning: splits are combined with an ad-hoc Bayesian rule, without optimizing a global functional, which generally also results in better robustness. The smart splits (see section 4.2) achieve better performance initially but eventually reach the same error rates as the standard splits. There is no visible degradation of the asymptotic performance due to either a reduced independence between splits or a diminution of their separation power. However the computational cost is twice as high, since every predictor has to be built twice. 5 Conclusion In this paper we have proposed an original approach to learning the appearance of an object from a single image. Our method relies on a large number of individual splits of the image space designed to keep together the images of any of the training objects. These splits are learned from a training set of examples and combined into a Bayesian framework to estimate the posterior probability for two images to show the same object. This approach is very generic since it never makes the space of admissible perturbations explicit and relies on the generalization properties of the family of predictors. It can be applied to predict the similarity of two signals as soon as a family of binary predictors exists on the space of individual signals. Since the learning is decomposed into the training of several splits independently, it can be easily parallelized. Also, because the combination rule is symmetric with respect to the splits, the learning can be incremental: splits can be added to the global rule progressively when they become available. Appendix: Proof of formula (2). For the first factor, we have P(L1, L2 | C1 = C2) = X s1,s2 P(L1, L2 | C1 = C2, S1 = s1, S2 = s2)P(S1 = s1, S2 = s2 | C1 = C2) = X s1,s2 P(L1, L2 | S1 = s1, S2 = s2)P(S1 = s1, S2 = s2 | C1 = C2) = X s1,s2 Y i P(L1 i | S1 i = s1 i )P(L2 i | S2 i = s2 i )P((S1 i , S2 i ) = (s1 i , s2 i ) | C1 = C2) = 2−N Y i P(L1 i | S1 i = 1)P(L2 i | S2 i = 1) + P(L1 i | S1 i = 0)P(L2 i | S2 i = 0) . In the second equality, we have used that L is independent of C given S. In the third equality, we have used that the (Lj i) are independent given S. In the last equality, we have used the symmetry assumption on the distribution of (S1, S2) given C1 = C2. Similarly, P(L1, L2 | C1 ̸= C2) = 4−N Y i X s1,s2 P(L1 i | S1 i = s1)P(L2 i | S2 i = s2) = 4−N Y i P(L1 i )P(L2 i ) X s1,s2 P(S1 i = s1 | L1 i )P(S2 i = s2 | L2 i ) P(S1 i = s1)P(S2 i = s2) = 4−2N Y i P(L1 i )P(L2 i ) , since P(Sj i = s) ≡1 2 by the symmetry hypothesis. Taking the ratio of the two factors and using the latter property again leads to the conclusion. References [1] Y. Bengio and M. Monperrus. Non-local manifold tangent learning. In Advances in Neural Information Processing Systems 17, pages 129–136. MIT press, 2005. [2] T. Dietterich and G. Bakiri. Solving multiclass learning problems via error-correcting output codes. Journal of Artificial Intelligence Research, 2:263–286, 1995. [3] A. Ferencz, E. Learned-Miller, and J. Malik. Learning hyper-features for visual identification. In Advances in Neural Information Processing Systems 17, pages 425–432. MIT Press, 2004. [4] A. Ferencz, E. Learned-Miller, and J. Malik. Building a classification cascade for visual identification from one example. In International Conference on Computer Vision (ICCV), 2005. [5] M. Fink. Object classification from a single example utilizing class relevance metrics. In Advances in Neural Information Processing Systems 17, pages 449–456. MIT Press, 2005. [6] F. Fleuret. Fast binary feature selection with conditional mutual information. Journal of Machine Learning Research, 5:1531–1555, November 2004. [7] F. Li, R. Fergus, and P. Perona. A Bayesian approach to unsupervised one-shot learning of object categories. In Proceedings of ICCV, volume 2, page 1134, 2003. [8] E. G. Miller, N. E. Matsakis, and P. A. Viola. Learning from one example through shared densities on transforms. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, volume 1, pages 464–471, 2000. [9] S. A. Nene, S. K. Nayar, and H. Murase. Columbia Object Image Library (COIL-100). Technical Report CUCS-006-96, Columbia University, 1996. [10] T. Sejnowski and C. Rosenberg. Parallel networks that learn to pronounce english text. Journal of Complex Systems, 1:145–168, 1987. [11] P. Simard, Y. Le Cun, and J. Denker. Efficient pattern recognition using a new transformation distance. In S. Hanson, J. Cowan, and C. Giles, editors, Advances in Neural Information Processing Systems 5, pages 50–68. Morgan Kaufmann, 1993. [12] S. Thrun and L. Pratt, editors. Learning to learn. Kluwer, 1997.
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Hot Coupling: A Particle Approach to Inference and Normalization on Pairwise Undirected Graphs of Arbitrary Topology Firas Hamze Nando de Freitas Department of Computer Science University of British Columbia Abstract This paper presents a new sampling algorithm for approximating functions of variables representable as undirected graphical models of arbitrary connectivity with pairwise potentials, as well as for estimating the notoriously difficult partition function of the graph. The algorithm fits into the framework of sequential Monte Carlo methods rather than the more widely used MCMC, and relies on constructing a sequence of intermediate distributions which get closer to the desired one. While the idea of using “tempered” proposals is known, we construct a novel sequence of target distributions where, rather than dropping a global temperature parameter, we sequentially couple individual pairs of variables that are, initially, sampled exactly from a spanning tree of the variables. We present experimental results on inference and estimation of the partition function for sparse and densely-connected graphs. 1 Introduction Undirected graphical models are powerful statistical tools having a wide range of applications in diverse fields such as image analysis [1, 2], conditional random fields [3], neural models [4] and epidemiology [5]. Typically, when doing inference, one is interested in obtaining the local beliefs, that is the marginal probabilities of the variables given the evidence set. The methods used to approximate these intractable quantities generally fall into the categories of Markov Chain Monte Carlo (MCMC) [6] and variational methods [7]. The former, involving running a Markov chain whose invariant distribution is the distribution of interest, can suffer from slow convergence to stationarity and high correlation between samples at stationarity, while the latter is not guaranteed to give the right answer or always converge. When performing learning in such models however, a more serious problem arises: the parameter update equations involve the normalization constant of the joint model at the current value of parameters, from here on called the partition function. MCMC offers no obvious way of approximating this wildly intractable sum [5, 8]. Although there exists a polynomial time MCMC algorithm for simple graphs with binary nodes, ferromagnetic potentials and uniform observations [9], this algorithm is hardly applicable to the complex models encountered in practice. Of more interest, perhaps, are the theoretical results that show that Gibbs sampling and even Swendsen-Wang[10] can mix exponentially slowly in many situations [11]. This paper introduces a new sequential Monte Carlo method for approximating expectations of a pairwise graph’s variables (of which beliefs are a special case) and of reasonably estimating the partition function. Intuitively, the new method uses interacting parallel chains to handle multimodal distributions, x y x ψ φ i (x ,x ) i j j j (x ,y) Figure 1: A small example of the type of graphical model treated in this paper. The observations correspond to the two shaded nodes. with communicating chains distributed across the modes. In addition, there is no requirement that the chains converge to equilibrium as the bias due to incomplete convergence is corrected for by importance sampling. Formally, given hidden variables x and observations y, the model is specified on a graph G(V, E), with edges E and M nodes V by: π(x, y) = 1 Z Y i∈V φ(xi, yi) Y (i,j)∈E ψ(xi, xj) where x = {x1, . . . , xM}, Z is the partition function, φ(·) denotes the observation potentials and ψ(·) denotes the pair-wise interaction potentials, which are strictly positive but otherwise arbitrary. The partition function is: Z = P x Q i∈V φ(xi, yi) Q (i,j)∈E ψ(xi, xj),where the sum is over all possible system states. We make no assumption about the graph’s topology or sparseness, an example is in Figure 1. We present experimental results on both fully-connected graphs (cases where each node neighbors every other node) and sparse graphs. Our approach belongs to the framework of Sequential Monte Carlo (SMC), which has its roots in the seminal paper of [12]. Particle filters are a well-known instance of SMC methods [13]. They apply naturally to dynamic systems like tracking. Our situation is different. We introduce artificial dynamics simply as a constructive strategy for obtaining samples of a sequence of distributions converging to the distribution of interest. That is, initially we sample from and easy-to-sample distribution. This distribution is then used as a proposal mechanism to obtain samples from a slightly more complex distribution that is closer to the target distribution. The process is repeated until the sequence of distributions of increasing complexity reaches the target distribution. Our algorithm has connections to a general annealing strategy proposed in the physics [14] and statistics [15] literature, known as Annealed Importance Sampling (AIS). AIS is a special case of the general SMC framework [16]. The term annealing refers to the lowering of a “temperature parameter,” the process of which makes the joint distribution more concentrated on its modes, whose number can be massive for difficult problems. The celebrated simulated annealing (SA) [17] algorithm is an optimization method relying on this phenomenon; presently, however we are interested in integration and so SA does not apply here. Our approach does not use a global temperature, but sequentially introduces dependencies among the variables; graphically, this can be understood as “adding edges” to the graph. In this paper, we restrict ourselves to discrete state-spaces although the method applies to arbitrary continuous distributions. For our initial distribution we choose a spanning tree of the variables, on which analytic marginalization, exact sampling, and computation of the partition function are easily done. After drawing a population of samples (particles) from this distribution, the sequential phase begins: an edge of the desired graph is chosen and gradually added to the current one as shown in Figure 2. The particles then follow a trajectory according to some proposal mechanism. The “fitness” of the particles is measured via their importance weights. When the set of samples has become skewed, that is with some containing high weights and many containing low ones, the particles are resampled according to their weights. The sequential structure is thus imposed by the propose-and-resamplemechanism rather than by any property of the original system. The algorithm is formally described after an overview of SMC and recent work presenting a unifying framework of the SMC methodologyoutside the context of Bayesian dynamic filtering[16]. Figure 2: A graphical illustration of our algorithm. First we construct a spanning tree, of which a population of iid samples can be easily drawn using the forward filtering/backward sampling algorithm for trees. The tree then becomes the proposal mechanism for generating samples for a graph with an extra potential. The process is repeated until we obtain samples from the target distribution (defined on a fully connected graph in this case). Edges can be added “slowly” using a coupling parameter. 2 Sequential Monte Carlo As shown in Figure 2, we consider a sequence of auxiliary distributions eπ1(x1), eπ2(x1:2), . . . , eπn(x1:n), where eπ1(x1) is the distribution on the weighted spanning tree. The sequence of distributions can be constructed so that it satisfies eπn(x1:n) = πn(xn)eπn(x1:n−1|x1:n). Marginalizing over x1:n−1 gives us the target distribution of interest πn(xn) (the distribution of the graphical model that we want to sample from as illustrated in Figure 2 for n = 4). So we first focus on sampling from the sequence of auxiliary distributions. The joint distribution is only known up to a normalization constant: eπn(x1:n) = Z−1 n fn(x1:n), where Zn ≜ R fn(x1:n)dx1:n is the partition function. We are often interested in computing this partition function and other expectations, such as I(g(xn)) = R g(xn)πn(xn)dxn, where g is a function of interest (e.g. g(x) = x if we are interested in computing the mean of x). If we had a set of samples {x(i) 1:n}N i=1 from eπ, we could approximate this integral with the following Monte Carlo estimator: beπn(dx1:n) = 1 N PN i=1 δx(i) 1:n(dx1:n), where δx(i) 1:n(dx1:n) denotes the delta Dirac function, and consequently approximate any expectations of interest. These estimates converge almost surely to the true expectation as N goes to infinity. It is typically hard to sample from eπ directly. Instead, we sample from a proposal distribution q and weight the samples according to the following importance ratio wn = fn(x1:n) qn(x1:n) = fn(x1:n) qn(x1:n) qn−1(x1:n−1) fn−1(x1:n−1)wn−1 The proposal is constructed sequentially: q(x1:n) = qn−1(x1:n−1)qn(xn|x1:n−1). Hence, the importance weights can be updated recursively wn = fn(x1:n) qn(xn|x1:n−1)fn−1(x1:n−1)wn−1 (1) Given a set of N particles x(i) 1:n−1, we obtain a set of particles x(i) n by sampling from qn(xn|x(i) 1:n−1) and applying the weights of equation (1). To overcome slow drift in the particle population, a resampling (selection) step chooses the fittest particles (see the introductory chapter in [13] for a more detailed explanation). We use a state-of-the-art minimum variance resampling algorithm [18]. The ratio of successive partition functions can be easily estimated using this algorithm as follows: Zn Zn−1 = R fn(x1:n)dx1:n Zn−1 = Z bwn eπn−1(x1:n−1)qn(xn|x1:n−1)dx1:n ≈ N X i=1 bw(i) n ew(i) n−1, where ew(i) n−1 = w(i) n−1/ P j w(j) n−1, bwn = fn(x1:n) qn(xn|x1:n−1)fn−1(x1:n−1) and Z1 can be easily computed as it is the partition function for a tree. We can choose a (non-homogeneous) Markov chain with transition kernel Kn(xn−1, xn) as the proposal distribution qn(xn|x1:n−1). Hence, given an initial proposal distribution q1(·), we have joint proposal distribution at step n: qn(x1:n) = q1(x1) Qn k=2 Kk(xk−1, xk). It is convenient to assume that the artificial distribution eπn(x1:n−1|xn) is also the product of (backward) Markov kernels: eπn(x1:n−1|xn) = Qn−1 k=1 Lk(xk+1, xk) [16]. Under these choices, the (unnormalized) incremental importance weight becomes: wn ∝ fn(xn)Ln−1(xn, xn−1) fn−1(xn−1)Kn(xn−1, xn) (2) Different choices of the backward Kernel L result in different algorithms [16]. For example, the choice: Ln−1(xn, xn−1) = fn(xn−1)Kn(xn−1,xn) fn(xn) results in the AIS algorithm, with weights wn ∝ fn(xn−1) fn−1(xn−1). However, we should point out that this method is more general as one can carry out resampling. Note that in this case, the importance weights do not depend on xn and, hence, it is possible to do resampling before the importance sampling step. This often leads to huge reduction in estimation error [19]. Also, note that if there are big discrepancies between fn(·) and fn−1(·) the method might perform poorly. To overcome this, [16] use variance results to propose a different choice of backward kernel, which results in the following incremental importance weights: wn ∝ fn(xn) R fn−1(xn−1)Kn(xn−1, xn)dxn−1 (3) The integral in the denominator can be evaluated when dealing with Gaussian or reasonable discrete networks. 3 The new algorithm We could try to perform traditional importance sampling by seeking some proposal distribution for the entire graph. This is very difficult and performance degrades exponentially in dimension if the proposal is mismatched [20]. We propose, however, to use the samples from the tree distribution (which we call π0) as candidates to an intermediate target distribution, consisting of the tree along with a “weak” version of a potential corresponding to some edge of the original graph. Given a set of edges G0 which form a spanning tree of the target graph, we can can use the belief propagation equations [21] and bottom-up propagation, top-down sampling [22], to draw a set of N independent samples from the tree. Computation of the normalization constant Z1 is also straightforward and efficient in the case of trees using a sum-product recursion. From then on, however, the normalization constants of subsequent target distributions cannot be analytically computed. We then choose a new edge e1 from the set of “unused” edges E −G0 and add it to G0 to form the new edge set G1 = e1 ∪G0. Let the vertices of e1 be u1 and v1. Then, the intermediate target distribution π1 is proportional to π0(x1)ψe1(xu1, xv1). In doing straightforward importance sampling, using π0 as a proposal for π1, the importance weight is proportional to ψe1(xu1, xv1). We adopt a slow proposal process to move the population of particles towards π1. We gradually introduce the potential between Xu1 and Xv1 via a coupling parameter α which increases from 0 to 1 in order to “softly” bring the edge’s potential in and allow the particles to adjust to the new environment. Formally, when adding edge e1 to the graph, we introduce a number of coupling steps so that we have the intermediate target distribution: π0(x0) [ψe1(xu1, xv1)]αn where αn is defined to be 0 when a new edge enters the sequence, increases to 1 as the edge is brought in, and drops back to zero when another edge is added at the following edge iteration. At each time step, we want a proposal mechanism that is close to the target distribution. Proposals based on simple perturbations, such as random walks, are easy to implement, but can be inefficient. Metropolis-Hastings proposals are not possible because of the integral in the rejection term. We can, however, employ a single-site Gibbs sampler with random scan whose invariant distribution at each step is the the next target density in the sequence; this kernel is applied to each particle. When an edge has been fully added a new one is chosen and the process is repeated until the final target density is the full graph. We use an analytic expression for the incremental weights corresponding to Equation (3). To alleviate potential confusion with MCMC, while any one particle obviously forms a correlated path, we are using a population and are making no assumption or requirement that the chains have converged as is done in MCMC as we are correcting for incomplete convergence with the weights. 4 Experiments and discussion Four approximate inference methods were compared: our SMC method with sequential edge addition (Hot Coupling (HC)), a more typical annealing strategy with a global temperature parameter(SMCG), single-site Gibbs sampling with random scan and loopy belief propagation. SMCG can be thought of as related to HC but where all the edges and local evidence are annealed at the same time. The majority of our experiments were performed on graphs that were small enough for exact marginals and partition functions to be exhaustively calculated. However, even in toy cases MCMC and loopy can give unsatisfactory and sometimes disastrous results. We also ran a set of experiments on a relatively large MRF. For the small examples we examined both fully-connected (FC) and square grid (MRF) networks, with 18 and 16 nodes respectively. Each variable could assume one of 3 states. Our pairwise potentials corresponded to the well-known Potts model: ψi,j(xi, xj) = e 1 T Jijδxi,xj , φi(xi) = e 1 T Jδxi (yi). We set T = 0.5 (a low temperature) and tested models with uniform and positive Jij, widely used in image analysis, and models with Jij drawn from a standard Gaussian; the latter is an instance of the much-studied spin-glass models of statistical physics which are known to be notoriously difficult to simulate at low temperatures [23]. Of course fully-connected models are known as Boltzmann machines [4] to the neural computation community. The output potentials were randomly selected in both the uniform and random interaction cases. The HC method used a linear coupling schedule for each edge, increasing from α = 0 to α = 1 over 100 iterations; our SMCG implementation used a linear global cooling schedule, whose number of steps depended on the graph in order to match those taken by SMCG. All Monte Carlo algorithms were independently run 50 times each to approximate the variance of the estimates. Our SMC simulations used 1000 particles for each run, while each Gibbs run performed 20000 single-site updates. For these models, this was more than enough steps to settle into local minima; runs of up to 1 million iterations did not yield a difference, which is characteristic of the exponential mixing time of the sampler on these graphs. For our HC method, spanning trees and edges in the sequential construction were randomly chosen from the full graph; the rationale for doing so is to allay any criticism that “tweaking” the ordering may have had a crucial effect on the algorithm. The order clearly would matter to some extent, but this will be examined in later work. Also in the tables by “error” we mean the quantity |ˆa−a| a where ˆa is an estimate of some quantity a obtained exactly (say Z). First, we used HC, SMCG and Gibbs to approximate the expected sum of our graphs’ variables, the so-called magnetization: m = E[PM i=1 xi]. We then approximated the partition functions of the graphs using HC, SMCG, and loopy.1We note again that there is no obvious way of estimating Z using Gibbs. Finally, we approximated the marginal probabilities using the four approximate methods. For loopy, we only kept the runs where it converged. 1Code for Bethe Z approximation kindly provided by Kevin Murphy. MRF Random Ψ MRF Homogeneous Ψ FC Random Ψ FC Homogeneous Ψ Method Error Var Error Var Error Var Error Var HC 0.0022 0.012 0.0251 0.17 0.0016 0.0522 0.0036 0.038 SMCG 0.0001 0.03 0.2789 10.09 0.127 0.570 0.331 165.61 Gibbs 0.0003 0.014 0.4928 200.95 0.02 0.32 0.3152 201.08 Figure 3: Approximate magnetization for the nodes of the graphs, as defined in the text, calculated using HC, SMCG, and Gibbs sampling and compared to the true value obtained by brute force. Observe the massive variance of Gibbs sampling in some cases. MRF Random Ψ MRF Homogeneous Ψ FC Random Ψ FC Homogeneous Ψ Method Error Var Error Var Error Var Err Var HC 0.0105 0.002 0.0227 0.001 0.0043 0.0537 0.0394 0.001 SMCG 0.004 0.005 6.47 7.646 1800 1.24 1 29.99 loopy 0.005 0.155 1 0.075 Figure 4: Approximate partition function of the graphs discussed in the text calculated using HC, SMCG, and Loopy Belief Propagation (loopy.) For HC and SMCG are shown the error of the sample average of results over 50 independent runs and the variance across those runs. loopy is of course a deterministic algorithm and has no variance. HC maintains a low error and variance in all cases. Figure 3 shows the results of the magnetization experiments. On the MRF with random interactions, all three methods gave very accurate answers with small variance, but for the other graphs, the accuracies and variances began to diverge. On both positive-potential graphs, Gibbs sampling gives high error and huge variance; SMCG gives lower variance but is still quite skewed. On the fully-connected random-potential graph the 3 methods give good results but HC has the lowest variance. Our method experiences its worst performance on the homogeneous MRF but it is only 2.5% error! Figure 4 tabulates the approximate partition function calculations. Again, for the MRF with random interactions, the 3 methods give estimates of Z of comparable quality. This example appeared to work for loopy, Gibbs, and SMCG. For the homogeneous MRF, SMCG degrades rapidly; loopy is still satisfactory at 15% error, but HC is at 2.7% with very low variance. In the fully-connected case with random potentials, HC’s error is 0.43% while loopy’s error is very high, having underestimated Z by a factor of 105. SMCG fails completely here as well. On the uniform fully-connected graph, loopy actually gives a reasonable estimate of Z at 7.5%, but is still beaten by HC. Figure 5 shows the variational (L1) distance between the exact marginal for a randomly chosen node in each graph and the approximate marginals of the 4 algorithms, a common measure of the “distance” between 2 distributions. For the Monte Carlo methods (HC, SMCG and Gibbs) the average over 50 independent runs was used to approximate the expected L1 error of the estimate. All 4 methods perform well on the random Ψ MRF. On the MRF with homogeneous Ψ, both loopy and SMCG degrade, but HC maintains a low error. Among the FC graphs, HC performs extremely well on the homogeneous Ψ and surprisingly loopy does well too. In the random Ψ case, loopy’s error increases dramatically. Our final set of simulations was the classic Mean Squared reconstruction of a noisy image problem; we used a 100x100 MRF with a noisy “patch” image (consisting of shaded, rectangular regions) with an isotropic 5-state prior model. The object was to calculate the pixels’ posterior marginal expectations. We chose this problem because it is a large model on which loopy is known to do well on, and can hence provide us with a measure of quality of the HC and SMCG results as larger numbers of edges are involved. From the toy examples we infer that the mechanism of HC is quite different from that of loopy as we have seen that it can work when loopy does not. Hence good performance on this problem would suggest that HC would scale well, which is a crucial question as in the large graph the final distribution has many more edges than the initial spanning tree. The results were promising: the mean-squared reconstruction error using loopy and using HC were virtually identical at 9.067 × 10−5 and 9.036 × 10−5 respectively, showing that HC seemed to be 0 0.5 1 1.5 Variational distance HC SMCG Gibbs Loopy Fully−Connected Random 0 0.5 1 1.5 Variational distance HC SMCG Gibbs Loopy Fully−Connected Homogeneous 0 0.5 1 1.5 Variational distance HC SMCG Gibbs Loopy Grid Model Random 0 0.5 1 1.5 Variational distance HC SMCG Gibbs Loopy Grid Model Homogeneous Figure 5: Variational(L1) distance between estimated and true marginals for a randomly chosen node in each of the 4 graphs using the four approximate methods (smaller values mean less error.) The MRF-random example was again “easy” for all the methods, but the rest raise problems for all but HC. 0 100 200 300 400 500 600 0 10 20 30 40 50 60 Sample Average Iteration 0 2 4 6 8 10 x 10 5 0 10 20 30 40 50 60 Iteration Sample Average Figure 6: An example of how MCMC can get “stuck:” 3 different runs of a Gibbs sampler estimating the magnetization of FC-Homogeneous graph. At left are shown the first 600 iterations of the runs; after a brief transient behaviour the samplers settled into different minima which persisted for the entire duration (20000 steps) of the runs. Indeed for 1 million steps the local minima persist, as shown at right. robust to the addition of around 9000 edges and many resampling stages. SMCG on the large MRF did not fare as well. It is crucial to realize that MCMC is completely unsuited to some problems; see for example the “convergence” plots of the estimated magnetization of 3 independent Gibbs sampler runs on one of our “toy” graphs shown in Figure 6. Such behavior has been studied by Gore and Jerrum [11] and others, who discuss pessimistic theoretical results on the mixing properties of both Gibbs sampling and the celebrated Swendsen-Wang algorithm in several cases. To obtain a good estimate, MCMC requires that the process “visit” each of the target distribution’s basins of energy with a frequency representative of their probability. Unfortunately, some basins take an exponential amount of time to exit, and so different finite runs of MCMC will give quite different answers, leading to tremendous variance. The methodology presented here is an attempt to sidestep the whole issue of mixing by permitting the independent particles to be stuck in modes, but then considering them jointly when estimating. In other words, instead of using a time average, we estimate using a weighted ensemble average. The object of the sequential phase is to address the difficult problem of constructing a suitable proposal for high-dimensional problems; to this the resamplingbased methodology of particle filters was thought to be particularly suited. For the graphs we have considered, the single-edge algorithm we propose seems to be preferable to global annealing. References [1] S Z Li. Markov random field modeling in image analysis. Springer-Verlag, 2001. [2] P Carbonetto and N de Freitas. Why can’t Jos´e read? the problem of learning semantic associations in a robot environment. In Human Language Technology Conference Workshop on Learning Word Meaning from Non-Linguistic Data, 2003. [3] J D Lafferty, A McCallum, and F C N Pereira. Conditional random fields: Probabilistic models for segmenting and labeling sequence data. In International Conference on Machine Learning, 2001. [4] D E Rumelhart, G E Hinton, and R J Williams. Learning internal representations by error propagation. In D E Rumelhart and J L McClelland, editors, Parallel Distributed Processing: Explorations in the Microstructure of Cognition, pages 318–362, Cambridge, MA, 1986. [5] P J Green and S Richardson. Hidden Markov models and disease mapping. Journal of the American Statistical Association, 97(460):1055–1070, 2002. [6] C P Robert and G Casella. Monte Carlo Statistical Methods. Springer-Verlag, New York, 1999. [7] M. I. Jordan, Z. Ghahramani, T. S. Jaakkola, and L. K. Saul. An introduction to variational methods for graphical models. Machine Learning, 37:183–233, 1999. [8] J Moller, A N Pettitt, K K Berthelsen, and R W Reeves. An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants. Technical report, The Danish National Research Foundation: Network in Mathematical Physics and Stochastics, 2004. [9] M Jerrum and A Sinclair. The Markov chain Monte Carlo method: an approach to approximate counting and integration. In D S Hochbaum, editor, Approximation Algorithms for NP-hard Problems, pages 482–519. PWS Publishing, 1996. [10] R H Swendsen and J S Wang. Nonuniversal critical dynamics in Monte Carlo simulations. Physical Review Letters, 58(2):86–88, 1987. [11] V Gore and M Jerrum. The swendsen-wang process does not always mix rapidly. In 29th Annual ACM Symposium on Theory of Computing, 1996. [12] N Metropolis and S Ulam. The Monte Carlo method. Journal of the American Statistical Association, 44(247):335–341, 1949. [13] A Doucet, N de Freitas, and N J Gordon, editors. Sequential Monte Carlo Methods in Practice. Springer-Verlag, 2001. [14] C Jarzynski. Nonequilibrium equality for free energy differences. Phys. Rev. Lett., 78, 1997. [15] R M Neal. Annealed importance sampling. Technical Report No 9805, University of Toronto, 1998. [16] P Del Moral, A Doucet, and G W Peters. Sequential Monte Carlo samplers. Technical Report CUED/F-INFENG/2004, Cambridge University Engineering Department, 2004. [17] S Kirkpatrick, C D Gelatt, and M P Vecchi. Optimization by simulated annealing. Science, 220:671–680, 1983. [18] G Kitagawa. Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. Journal of Computational and Graphical Statistics, 5:1–25, 1996. [19] N de Freitas, R Dearden, F Hutter, R Morales-Menendez, J Mutch, and D Poole. Diagnosis by a waiter and a mars explorer. IEEE Proceedings, 92, 2004. [20] J A Bucklew. Large Deviation Techniques in Decision, Simulation, and Estimation. John Wiley & Sons, 1986. [21] J Pearl. Probabilistic reasoning in intelligent systems: networks of plausible inference. MorganKaufmann, 1988. [22] C K Carter and R Kohn. On Gibbs sampling for state space models. Biometrika, 81(3):541–553, 1994. [23] M E J Newman and G T Barkema. Monte Carlo Methods in Statistical Physics. Oxford University Press, 1999.
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Separation of Music Signals by Harmonic Structure Modeling Yun-Gang Zhang Department of Automation Tsinghua University Beijing 100084, China zyg00@mails.tsinghua.edu.cn Chang-Shui Zhang Department of Automation Tsinghua University Beijing 100084, China zcs@mail.tsinghua.edu.cn Abstract Separation of music signals is an interesting but difficult problem. It is helpful for many other music researches such as audio content analysis. In this paper, a new music signal separation method is proposed, which is based on harmonic structure modeling. The main idea of harmonic structure modeling is that the harmonic structure of a music signal is stable, so a music signal can be represented by a harmonic structure model. Accordingly, a corresponding separation algorithm is proposed. The main idea is to learn a harmonic structure model for each music signal in the mixture, and then separate signals by using these models to distinguish harmonic structures of different signals. Experimental results show that the algorithm can separate signals and obtain not only a very high Signalto-Noise Ratio (SNR) but also a rather good subjective audio quality. 1 Introduction Audio content analysis is an important area in music research. There are many open problems in this area, such as content based music retrieval and classification, Computational Auditory Scene Analysis (CASA), Multi-pitch Estimation, Automatic Transcription, Query by Humming, etc. [1, 2, 3, 4]. In all these problems, content extraction and representation is where the shoe pinches. In a song, the sounds of different instruments are mixed together, and it is difficult to parse the information of each instrument. Separation of sound sources in a mixture is a difficult problem and no reliable methods are available for the general case. However, music signals are so different from general signals. So, we try to find a way to separate music signals by utilizing the special character of music signals. After source separation, many audio content analysis problems will become much easier. In this paper, a music signal means a monophonic music signal performed by one instrument. A song is a mixture of several music signals and one or more singing voice signals. As we know, music signals are more “ordered” than voice. The entropy of music is much more constant in time than that of speech [5]. More essentially, we found that an important character of a music signal is that its harmonic structure is stable. And the harmonic structures of music signals performed by different instruments are different. So, a harmonic structure model is built to represent a music signal. This model is the fundamental of the separation algorithm. In the separation algorithm, an extended multi-pitch estimation algorithm is used to extract harmonic structures of all sources, and a clustering algorithm is used to calculate harmonic structure models. Then, signals are separated by using these models to distinguish harmonic structures of different signals. There are many other signal separation methods, such as ICA [6]. General signal separation methods do not sufficiently utilize the special character of music signals. Gil-Jin and TeWon proposed a probabilistic approach to single channel blind signal separation [7], which is based on exploiting the inherent time structure of sound sources by learning a priori sets of basis filters. In our approach, training sets are not required, and all information are directly learned from the mixture. Feng et al. applied FastICA to extract singing and accompaniment from a mixture [8]. Vanroose used ICA to remove music background from speech by subtracting ICA components with the lowest entropy [9]. Compared to these approaches, our method can separate each individual instrument sound, preserve the harmonic structure in the separated signals and obtain a good subjective audio quality. One of the most important contributions of our method is that it can significantly improve the accuracy of multi-pitch estimation. Compared to previous methods, our method learns models from the primary multi-pitch estimation results, and uses these models to improve the results. More importantly, pitches of different sources can be distinguished by these models. This advantage is significant for automatic transcription. The rest of this paper is organized as follows: Harmonic structure modeling is detailed in Section two. The algorithm is described in section three. Experimental results are shown in section four. Finally, conclusion and discussions are given in section five. 2 Harmonic structure modeling for music signals A monophonic music signal s(t) can be represented by a sinusoidal model [10]: s(t) = R X r=1 Ar(t) cos[θr(t)] + e(t) (1) where Ar(t) and θr(t) = R t 0 2πrf0(τ)dτ are the instantaneous amplitude and phase of the rth harmonic, respectively, R is the maximal harmonic number, f0(τ) is the fundamental frequency at time τ, e(t) is the noise component. We divide s(t) into overlapped frames and calculate f l 0 and Al r by detecting peaks in the magnitude spectrum. Al r = 0, if there doesn’t exist the rth harmonic. l = 1, . . . , L is the frame index. f l 0 and [Al 1, . . . , Al R] describe the position and amplitudes of harmonics. We normalize Al r by multiplying a factor ρl = C/Al 1 ( C is an arbitrary constant) to eliminate the influence of the amplitude. We translate the amplitudes into a log scale, because the human ear has a roughly logarithmic sensitivity to signal intensity. Harmonic Structure Coefficient is then defined as equation (2). The timbre of a sound is mostly controlled by the number of harmonics and the ratio of their amplitudes, so Bl = [Bl 1, . . . , Bl R], which is free from the fundamental frequency and amplitude, exactly represents the timbre of a sound. In this paper, these coefficients are used to represent the harmonic structure of a sound. Average Harmonic Structure and Harmonic Structure Stability are defined as follows to model music signals and measure the stability of harmonic structures. • Harmonic Structure Bl, Bl i is Harmonic Structure Coefficient: Bl = [Bl 1, . . . , Bl R], Bl i = log(ρlAl i)/ log(ρlAl 1), i = 1, . . . , R (2) • Average Harmonic Structure (AHS): ¯B = 1 L L P l=1 Bl • Harmonic Structure Stability (HSS): HSS = 1 R · 1 L L X l=1
Bl −¯B
2 = 1 RL R X r=1 L X l=1 (Bl r −¯Br)2 (3) AHS and HSS are the mean and variance of Bl. Since timbres of most instruments are stable, Bl varies little in different frames in a music signal and AHS is a good model to represent music signals. On the contrary, Bl varies much in a voice signal and the corresponding HSS is much bigger than that of a music signal. See figure 1. 0 50 100 150 200 −50 0 50 0 50 100 150 200 −50 0 50 (a) Spectra in different frames of a voice signal. The number of harmonics (significant peaks in the spectrum) and their amplitude ratios are totally different. 0 50 100 150 200 −50 0 50 0 50 100 150 200 −50 0 50 (b) Spectra in different frames of a piccolo signal. The number of harmonics (significant peaks in the spectrum) and their amplitude ratios are almost the same. 0 2 4 6 8 10 12 14 16 0 0.5 1 1.5 HSS=0.056576 (c) The AHS and HSS of a oboe signal 0 2 4 6 8 10 12 14 16 0 0.5 1 HSS=0.037645 (d) The AHS and HSS of a SopSax signal 0 2 4 6 8 10 12 14 16 0 0.5 1 HSS=0.17901 (e) The AHS and HSS of a male singing voice 0 2 4 6 8 10 12 14 16 0 0.5 1 HSS=0.1 (f) The AHS and HSS of a female singing voice Figure 1: Spectra, AHSs and HSSs of voice and music signals. In (c)-(f), x-axis is harmonic number, y-axis is the corresponding harmonic structure coefficient. 3 Separation algorithm based on harmonic structure modeling Without loss of generality, suppose we have a signal mixture consisting of one voice and several music signals. The separation algorithm consists of four steps: preprocessing, extraction of harmonic structures, music AHSs analysis, separation of signals. In preprocessing step, the mean and energy of the input signal are normalized. In the second step, the pitch estimation algorithm of Terhardt [11] is extended and used to extract harmonic structures. This algorithm is suitable for estimating both the fundamental frequency and all its harmonics. In Terhardt’s algorithm, in each frame, all spectral peaks exceeding a given threshold are detected. The frequencies of these peaks are [f1, . . . , fK], K is the number of peaks. For a fundamental frequency candidate f, count the number of fi which satisfies the following condition: floor[(1 + d)fi/f] ≥(1 −d)fi/f (4) floor(x) denotes the greatest integer less than or equal to x. This condition means whether rif · (1 −d) ≤fi ≤rif · (1 + d). If the condition is fulfilled, fi is the frequency of the rth i harmonic component when fundamental frequency is f. For each fundamental frequency candidate f, the coincidence number is calculated and ˆf corresponding to the largest coincidence number is selected as the estimated fundamental frequency. The original algorithm is extended in the following ways: Firstly, not all peaks exceeding the given threshold are detected, only the significant ones are selected by an edge detection procedure. This is very important for eliminating noise and achieving high performances in next steps. Secondly, not only the fundamental frequency but also all its harmonics are extracted, then B can be calculated. Thirdly, the original optimality criterion is to select ˆf corresponding to the largest coincidence number. This criterion is not stable when the signal is polyphonic, because harmonic components of different sources may influence each other. A new optimality criterion is define as follows (n is the coincidence number): d = 1 n K X i=1,fi coincident with f |ri −fi/f| ri (5) ˆf corresponding to the smallest d is the estimated fundamental frequency. The new criterion measures the precision of coincidence. For each fundamental frequency, harmonic components of the same source are more probably to have a high coincidence precision than those of a different source. So, the new criterion is helpful for separation of harmonic structures of different sources. Note that, the coincidence number is required to be larger than a threshold, such as 4-6. This requirement eliminates many errors. Finally, in the original algorithm, only one pitch was detected in each frame. Here, the sound is polyphonic. So, all pitches for which the corresponding d is below a given threshold are extracted. After harmonic structure extraction, a data set of harmonic structures is obtained. As the analysis in section two, in different frames, music harmonic structures of the same instrument are similar to each other and different from those of other instruments. So, in the data set all music harmonic structures form several high density clusters. Each cluster corresponds to an instrument. Voice harmonic structures scatter around like background noise, because the harmonic structure of the voice signal is not stable. In the third step, NK algorithm [12] is used to learn music AHSs. NK algorithm is a clustering algorithm, which can cluster data on data sets consisting of clusters with different shapes, densities, sizes and even with some background noise. It can deal with high dimensional data sets. Actually, the harmonic structure data set is such a data set. Clusters of harmonic structures of different instruments have different densities. Voice harmonic structure are background noise. Each data point, a harmonic structure, has a high dimensionality (20 in our experiments). In NK algorithm, first find K neighbors for each point and construct a neighborhood graph. Each point and its neighbors form a neighborhood. Then local PCA is used to calculate eigenvalues of a neighborhood. In a cluster, data points are close to each other and the neighborhood is small, so the corresponding eigenvalues are small. On the contrary, for a noise point, corresponding eigenvalues are much bigger. So noise points can be removed by eigenvalue analysis. After denoising, in the neighborhood graph, all points of a cluster are connected together by edges between neighbors. If two clusters are connected together, there must exist long edges between them. Then the eigenvalues of the corresponding neighborhoods are bigger than others. So all edges between clusters can be found and removed by eigenvalue analysis. Then data points are clustered correctly and AHSs can be obtained by calculate the mean of each cluster. In the separation step, all harmonic structures of an instrument in all frames are extracted to reconstruct the corresponding music signals and then removed from the mixture. After removing all music signals, the rest of the mixture is the separated voice signal. The procedure of music harmonic structure detection is detailed as follows. Given a music AHS [ ¯B1, . . . , ¯BR] and a fundamental frequency candidate f, a music harmonic structure is predicted. [f, 2f, . . . , Rf] and [ ¯B1, . . . , ¯BR] are its frequencies and harmonic structure coefficients. The closest peak in the magnitude spectrum for each predicted harmonic component is detected. Suppose [f1, . . . , fR] and [B1, . . . , BR] are the frequencies and harmonic structure coefficients of these peaks (measured peaks). Formula 6 is defined to calculate the distance between the predicted harmonic structure and the measured peaks. D(f) = R P r=1, ¯ Br>0,Br>0 {∆fr · (rf)−p + ¯ Br ¯ Bmax × q∆fr · (rf)−p} + a R P r=1, ¯ Br>0,Br>0 ( ¯ Br ¯ Bmax )( ¯Br −Br)2 (6) The first part of D is a modified version of Two-Way Mismatch measure defined by Maher and Beauchamp, which measures the frequency difference between predicted peaks and measured peaks [13], where p and q are parameters, and ∆fr = |fr −r · f|. The second part measures the shape difference between the two, a is a normalization coefficient. Note that, only harmonic components with none-zero harmonic structure coefficients are considered. Let ˆf indicate the fundamental frequency candidate corresponding to the smallest distance between the predicted peaks and the actual spectral peaks. If D( ˆf) is smaller than a threshold Td, a music harmonic structure is detected. Otherwise there is no music harmonic structure in the frame. If a music harmonic structure is detected, the corresponding measured peaks in the spectrum are extracted, and the music signal is reconstructed by IFFT. Smoothing between frames is needed to eliminate errors and click noise between frames. 4 Experimental results We have tested the performance of the proposed method on mixtures of different voice and music signals. The sample rate of the mixtures is 22.05kHz. Audio files for all the experiments are accessible at the website1. Figure 2 shows experimental results. In experiments 1 and 2, the mixed signals consist of one voice signal and one music signal. In experiment 3, the mixture consists of two music signals. In experiment 4, the mixture consists of one voice and two music signals. Table 1 shows SNR results. It can be seen that the mixtures are well separated into voice and music signals and very high SNRs are obtained in the separated signals. Experimental results show that music AHS is a good model for music signal representation and separation. There is another important fact that should be emphasized. In the separation procedure, music harmonic structures are detected by the music AHS model and separated from the mixture, and most of the time voice harmonic structures remain almost untouched. This procedure makes separated signals with a rather good subjective audio quality due to the good harmonic structure in the separated signals. Few existing methods can obtain such a good result because the harmonic structure is distorted in most of the existing methods. It is difficult to compare our method with other methods, because they are so different. However, we compared our method with a speech enhancement method, because separation 1http://www.au.tsinghua.edu.cn/szll/bodao/zhangchangshui/bigeye/member/zyghtm/ experiments.htm Table 1: SNR results (DB): snrv, snrm1 and snrm2 are the SNRs of voice and music signals in the mixed signal. snr′ e is the SNR of speech enhancement result. snr′ v, snr′ m1 and snr′ m2 are the SNRs of the separated voice and music signals. snrv snrm1 snrm2 snr′ e snr′ v snr′ m1 snr′ m2 Total inc. Experiment 1 -7.9 7.9 / -6.0 6.7 10.8 / 17.5 Experiment 2 -5.2 5.2 / -1.5 6.6 10.0 / 16.6 Experiment 3 / 1.6 -1.6 / / 9.3 7.1 16.4 Experiment 4 -10.0 0.7 -2.2 / 2.8 8.6 6.3 29.2 of voice and music can be regarded as a speech enhancement problem by regarding music as background noise. Figure 2 (b), (d) give speech enhancement results obtained by a speech enhancement software which tries to estimate the spectrum of noise in the pause of speech and enhance the speech by spectral subtraction [14]. Detecting pauses in speech with music background and enhancing speech with fast music noise are both very difficult problems, so traditional speech enhancement techniques can’t work here. 5 Conclusion and discussion In this paper, a harmonic structure model is proposed to represent music signals and used to separate music signals. Experimental results show a good performance of this method. The proposed method has many applications, such as multi-pitch estimation, audio content analysis, audio edit, speech enhancement with music background, etc. Multi-pitch estimation is an important problem in music research. There are many existing methods, such as pitch perception model based methods, and probabilistic approaches [4, 15, 16, 17]. However, multi-pitch estimation is a very difficult problem and remains unsolved. Furthermore, it is difficult to distinguish pitches of different instruments in the mixture. In our algorithm, not only harmonic structures but also corresponding fundamental frequencies are extracted. So, the algorithm is also a new multi-pitch estimation method. It analyzes the primary multi-pitch estimation results and learns models to represent music signals and improve multi-pitch estimation results. More importantly, pitches of different sources can be distinguished by the AHS models. This advantage is significant for automatic transcription. Figure 2 (f) shows multi-pitch estimation results in experiment 3. It can be seen that, the multi-pitch estimation results are fairly good. The proposed method is useful for melody extraction. As we know, in a mixed signal, multi-pitch estimation is a difficult problem. After separation, pitch estimation on the separated voice signal that contains melody becomes a monophonic pitch estimation problem, which can be done easily. The estimated pitch sequence represents the melody of the song. Then, many content base audio analysis tasks such as audio retrieval and classification become much easier and many midi based algorithms can be used on audio files. There are still some limitations. Firstly, the proposed algorithm doesn’t work for nonharmonic instruments, such as some drums. Some rhythm tracking algorithms can be used instead to separate drum sounds. Fortunately, most instrument sounds are harmonic. Secondly, for some instruments, the timbre in the onset is somewhat different from that in the stable duration. Also, different performing methods (pizz. or arco) produces different timbres. In these cases, the music harmonic structures of this instrument will form several clusters, not one. Then a GMM model instead of an average harmonic structure model (actually a point model) should be used to represent the music. −1 0 1 original voice signal −1 0 1 original music signal −1 0 1 mixed signal (a) Experiment1: The original voice and piccolo signals and the mixed signal −1 0 1 separated voice signal −1 0 1 separated music signal −1 0 1 speech enhancement result (b) Experiment1:The separated signals and the speech enhancement result −1 0 1 original voice signal −1 0 1 original music signal −1 0 1 mixed signal (c) Experiment2: The original voice and organ signals and the mixed signal −1 0 1 separated voice signal −1 0 1 separated music signal −1 0 1 speech enhancement result (d) Experiment2:The separated signals and the speech enhancement result −1 0 1 original piccolo signal −1 0 1 original organ signal −1 0 1 mixed signal (e) Experiment3:The original piccolo and organ signals and the mixed signal −1 0 1 separated piccolo signal 0 100 200 300 400 500 600 0 5 10 15 20 25 30 35 40 −1 0 1 separated organ signal 0 100 200 300 400 500 600 0 5 10 15 20 25 (f) Experiment3:The separated signals and the multi-pitch estimation results −1 0 1 original voice signal −1 0 1 original piccolo signal −1 0 1 original organ signal −1 0 1 mixed signal (g) Experiment4:The original voice, piccolo and organ signals and the mixed signal −1 0 1 separated voice signal −1 0 1 separated piccolo signal −1 0 1 separated organ signal (h) Experiment4:The separated signals 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 HSS=0.0066462 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 HSS=0.012713 (i) Experiment4:The learned music AHSs Figure 2: Experimental results. Acknowledgments This work is supported by the project (60475001) of the National Natural Science Foundation of China. References [1] J. S. Downie, “Music information retrieval,” Annual Review of Information Science and Technology, vol. 37, pp. 295–340, 2003. [2] Roger Dannenberg, “Music understanding by computer,” in IAKTA/LIST International Workshop on Knowledge Technology in the Arts Proc., 1993, pp. 41–56. [3] G. J. Brown and M. Cooke, “Computational auditory scene analysis,” Computer Speech and Language, vol. 8, no. 4, pp. 297–336, 1994. [4] M.Goto, “A robust predominant-f0 estimation method for real-time detection of melody and bass lines in cd recordings,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP2000), 2000, pp. 757–760. [5] J. Pinquier, J. Rouas, and R. Andre-Obrecht, “Robust speech / music classification in audio documents,” in 7th International Conference On Spoken Language Processing (ICSLP), 2002, pp. 2005–2008. [6] P. Comon, “Independent component analysis, a new concept?,” Signal Processing, vol. 36, pp. 287–314, 1994. [7] Gil-Jin Jang and Te-Won Lee, “A probabilistic approach to single channel blind signal separation,” in Neural Information Processing Systems 15 (NIPS2002), 2003. [8] Yazhong Feng, Yueting Zhuang, and Yunhe Pan, “Popular music retrieval by independent component analysis,” in ISMIR, 2002, pp. 281–282. [9] Peter Vanroose, “Blind source separation of speech and background music for improved speech recognition,” in The 24th Symposium on Information Theory, May 2003, pp. 103–108. [10] X. Serra, “Musical sound modeling with sinusoids plus noise,” in Musical Signal Processing, C. Roads, S. Popea, A. Picialli, and G. De Poli, Eds. Swets & Zeitlinger Publishers, 1997. [11] E. Terhardt, “Calculating virtual pitch,” Hearing Res., vol. 1, pp. 155–182, 1979. [12] Yungang Zhang, Changshui Zhang, and Shijun Wang, “Clustering in knowledge embedded space,” in ECML, 2003, pp. 480–491. [13] R. C. Maher and J. W. Beauchamp, “Fundamental frequency estimation of musical signals using a two-way mismatch procedure,” Journal of the Acoustical Society of America, vol. 95, no. 4, pp. 2254–2263, 1994. [14] Serguei Koval, Mikhail Stolbov, and Mikhail Khitrov, “Broadband noise cancellation systems: new approach to working performance optimization,” in EUROSPEECH’99, 1999, pp. 2607–2610. [15] Anssi Klapuri, “Automatic transcription of music,” M.S. thesis, Tampere University of Technology, Finland, 1998. [16] Keerthi C. Nagaraj., “Toward automatic transcription - pitch tracking in polyphonic environment,” Literature survey, Mar. 2003. [17] Hirokazu Kameoka, Takuya Nishimoto, and Shigeki Sagayama, “Separation of harmonic structures based on tied gaussian mixture model and information criterion for concurrent sounds,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP04), 2004.
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Learning Minimum Volume Sets Clayton Scott Statistics Department Rice University Houston, TX 77005 cscott@rice.edu Robert Nowak Electrical and Computer Engineering University of Wisconsin Madison, WI 53706 nowak@engr.wisc.edu Abstract Given a probability measure P and a reference measure µ, one is often interested in the minimum µ-measure set with P-measure at least α. Minimum volume sets of this type summarize the regions of greatest probability mass of P, and are useful for detecting anomalies and constructing confidence regions. This paper addresses the problem of estimating minimum volume sets based on independent samples distributed according to P. Other than these samples, no other information is available regarding P, but the reference measure µ is assumed to be known. We introduce rules for estimating minimum volume sets that parallel the empirical risk minimization and structural risk minimization principles in classification. As in classification, we show that the performances of our estimators are controlled by the rate of uniform convergence of empirical to true probabilities over the class from which the estimator is drawn. Thus we obtain finite sample size performance bounds in terms of VC dimension and related quantities. We also demonstrate strong universal consistency and an oracle inequality. Estimators based on histograms and dyadic partitions illustrate the proposed rules. 1 Introduction Given a probability measure P and a reference measure µ, the minimum volume set (MV-set) with mass at least 0 < α < 1 is G∗ α = arg min{µ(G) : P(G) ≥α, G measurable}. MV-sets summarize regions where the mass of P is most concentrated. For example, if P is a multivariate Gaussian distribution and µ is the Lebesgue measure, then the MV-sets are ellipsoids (see also Figure 1). Applications of minimum volume sets include outlier/anomaly detection, determining highest posterior density or multivariate confidence regions, tests for multimodality, and clustering. In comparison to the closely related problem of density level set estimation [1, 2], the minimum volume approach seems preferable in practice because the mass α is more easily specified than a level of a density. See [3, 4, 5] for further discussion of MV-sets. This paper considers the problem of MV-set estimation using a training sample drawn from P, which in most practical settings is the only information one has Figure 1: Gaussian mixture data, 500 samples, α = 0.9. (Left and Middle) Minimum volume set estimates based on recursive dyadic partitions, discussed in Section 6. (Right) True MV set. about P. The specifications to the estimation process are the significance level α, the reference measure µ, and a collection of candidate sets G. All proofs, as well as additional results and discussion, may be found in [6] . To our knowledge, ours is the first work to establish finite sample bounds, an oracle inequality, and universal consistency for the MV-set estimation problem. The methods proposed herein are primarily of theoretical interest, although they may be implemented effeciently for certain partition-based estimators as discussed later. As a more practical alternative, the MV-set problem may be reduced to Neyman-Pearson classification [7, 8] by simulating realizations from. 1.1 Notation Let (X, B) be a measure space with X ⊂Rd. Let X be a random variable taking values in X with distribution P. Let S = (X1, . . . , Xn) be an independent and identically distributed (IID) sample drawn according to P. Let G denote a subset of X, and let G be a collection of such subsets. Let bP denote the empirical measure based on S: bP(G) = (1/n) Pn i=1 I(Xi ∈G). Here I(·) is the indicator function. Set µ∗ α = inf G {µ(G) : P(G) ≥α}, (1) where the inf is over all measurable sets. A minimum volume set, G∗ α, is a minimizer of (1), when it exists. Let G be a class of sets. Given α ∈(0, 1), denote Gα = {G ∈ G : P(G) ≥α}, the collection of all sets in G with mass at least alpha. Define µG,α = inf{µ(G) : G ∈Gα} and GG,α = arg min{µ(G) : G ∈Gα} when it exists. Thus GG,α is the best approximation to the MV-set G∗ α from G. Existence and uniqueness of these and related quantities are discussed in [6] . 2 Minimum Volume Sets and Empirical Risk Minimization In this section we introduce a procedure inspired by the empirical risk minimization (ERM) principle for classification. In classification, ERM selects a classifier from a fixed set of classifiers by minimizing the empirical error (risk) of a training sample. Vapnik and Chervonenkis established the basic theoretical properties of ERM (see [9, 10]), and we find similar properties in the minimum volume setting. In this and the next section we do not assume P has a density with respect to µ. Let φ(G, S, δ) be a function of G ∈G, the training sample S, and a confidence parameter δ ∈(0, 1). Set bGα = {G ∈G : bP(G) ≥α −φ(G, S, δ)} and bGG,α = arg min{µ(G) : G ∈bGα}. (2) We refer to the rule in (2) as MV-ERM because of the analogy with empirical risk minimization in classification. The quantity φ acts as a kind of “tolerance” by which the empirical mass estimate may deviate from the targeted value of α. Throughout this paper we assume that φ satisfies the following. Definition 1. We say φ is a (distribution free) complexity penalty for G if and only if for all distributions P and all δ ∈(0, 1), P n S : sup G∈G P(G) −bP(G) −φ(G, S, δ) > 0 ≤δ. Thus, φ controls the rate of uniform convergence of bP(G) to P(G) for G ∈G. It is well known that the performance of ERM (for binary classification) relative to the performance of the best classifier in the given class is controlled by the uniform convergence of true to empirical probabilities. A similar result holds for MV-ERM. Theorem 1. If φ is a complexity penalty for G, then P n P( bGG,α) < α −2φ( bGG,α, S, δ) or µ( bGG,α) > µG,α ≤δ. Proof. Consider the sets ΘP = {S : P( bGG,α) < α −2φ( bGG,α, S, δ)}, Θµ = {S : µ( bGG,α) > µ(GG,α)}, ΩP = S : sup G∈G P(G) −bP(G) −φ(G, S, δ) > 0 . The result follows easily from the following lemma. Lemma 1. With ΘP , Θµ, and ΩP defined as above and bGG,α as defined in (2) we have ΘP ∪Θµ ⊂ΩP . The proof of this lemma (see [6] ) follows closely the proof of Lemma 1 in [7]. This result may be understood by analogy with the result from classification that says R( bf) −inff∈F R(f) ≤2 supf∈F |R(f) −bR(f)| (see [10], Ch. 8). Here R and bR are the true and empirical risks, bf is the empirical risk minimizer, and F is a set of classifiers. Just as this result relates uniform convergence bounds to empirical risk minimization in classification, so does Lemma 1 relate uniform convergence to the performance of MV-ERM. The theorem above allows direct translation of uniform convergence results into performance guarantees for MV-ERM. Fortunately, many penalties (uniform convergence results) are known. We now give to important examples, although many others, such as the Rademacher penalty, are possible. 2.1 Example: VC Classes Let G be a class of sets with VC dimension V , and define φ(G, S, δ) = r 32V log n + log(8/δ) n . (3) By a version of the VC inequality [10], we know that φ is a complexity penalty for G, and therefore Theorem 1 applies. To view this result in perhaps a more recognizable way, let ϵ > 0 and choose δ such that 2φ(G, S, δ) = ϵ. By inverting the relationship between δ and ϵ, we have the following. Corollary 1. With the notation defined above, P n P( bGG,α) < α −ϵ or µ( bGG,α) > µG,α ≤8nV e−nϵ2/128. Thus, for any fixed ϵ > 0, the probability of being within ϵ of the target mass α and being less than the target volume µG,α approaches one exponentially fast as the sample size increases. This result may also be used to calculate a distribution free upper bound on the sample size needed to be within a given tolerance ϵ of α and with a given confidence 1 −δ. In particular, the sample size will grow no faster than a polynomial in 1/ϵ and 1/δ, paralleling results for classification. 2.2 Example: Countable Classes Suppose G is a countable class of sets. Assume that to every G ∈G a number JGK is assigned such that P G∈G 2−JGK ≤1. In light of the Kraft inequality for prefix codes, JGK may be defined as the codelength of a codeword for G in a prefix code for G. Let δ > 0 and define φ(G, S, δ) = r JGK log 2 + log(2/δ) 2n . (4) By Chernoff’s bound together with the union bound, φ is a penalty for G. Therefore Theorem 1 applies and we have obtained a result analogous to the Occam’s Razor bound for classification. As a special case, suppose G is finite and take JGK = log2 |G|. Setting 2φ(G, S, δ) = ϵ and inverting the relationship between δ and ϵ, we have Corollary 2. For the MV-ERM estimate bGG,α from a finite class G P n P( bGG,α) < α −ϵ or µ( bGG,α) > µG,α ≤2|G|e−nϵ2/2. 3 Consistency A minimum volume set estimator is consistent if its volume and mass tend to the optimal values µ∗ α and α as n →∞. Formally, define the error quantity M(G) := (µ(G) −µ∗ α)+ + (α −P(G))+ , where (x)+ = max(x, 0). (Note that without the (·)+ operator, this would not be a meaningful error since one term could be negative and cause M to tend to zero, even if the other error term does not go to zero.) We are interested in MV-set estimators such that M( bGG,α) tends to zero as n →∞. Definition 2. A learning rule bGG,α is strongly consistent if limn→∞M( bGG,α) = 0 with probability 1. If bGG,α is strongly consistent for every possible distribution of X, then bGG,α is strongly universally consistent. To see how consistency might result from MV-ERM, it helps to rewrite Theorem 1 as follows. Let G be fixed and let φ(G, S, δ) be a penalty for G. Then with probability at least 1 −δ, both µ( bGG,α) −µ∗ α ≤µ(GG,α) −µ∗ α (5) and α −P( bGG,α) ≤2φ( bGG,α, S, δ) (6) hold. We refer to the left-hand side of (5) as the excess volume of the class G and the left-hand side of (6) as the missing mass of bGG,α. The upper bounds on the right-hand sides are an approximation error and a stochastic error, respectively. The idea is to let G grow with n so that both errors tend to zero as n →∞. If G does not change with n, universal consistency is impossible. To have both stochastic and approximation errors tend to zero, we apply MV-ERM to a class Gk from a sequence of classes G1, G2, . . ., where k = k(n) grows with the sample size. Consider the estimator bGGk,α. Theorem 2. Choose k = k(n) and δ = δ(n) such that k(n) →∞as n →∞and P∞ n=1 δ(n) < ∞. Assume the sequence of sets Gk and penalties φk satisfy lim k→∞inf G∈Gk α µ(G) = µ∗ α (7) and lim n→∞sup G∈Gk α φk(G, S, δ(n)) = o(1). (8) Then bGGk,α is strongly universally consistent. The proof combines the Borel-Cantelli lemma and the distribution-free result of Theorem 1 with the stated assumptions. Examples satisfying the hypotheses of the theorem include families of VC classes with arbitrary approximating power (e.g., generalized linear discriminant rules with appropriately chosen basis functions and neural networks), and histogram rules. See [6] for further discussion. 4 Structural Risk Minimization and an Oracle Inequality In the previous section the rate of convergence of the two errors to zero is determined by the choice of k = k(n), which must be chosen a priori. Hence it is possible that the excess volume decays much more quickly than the missing mass, or vice versa. In this section we introduce a new rule called MV-SRM, inspired by the principle of structural risk minimization (SRM) from the theory of classification [11, 12], that automatically balances the two errors. The result in this section is not distribution free. We assume A1 P has a density f with respect to µ. A2 G∗ α exists and P(G∗ α) = α. Under these assumptions (see [6] ) there exists γα > 0 such that for any MV-set G∗ α, {x : f(x) > γα} ⊂G∗ α ⊂{x : f(x) ≥γα}. Let G be a class of sets. Conceptualize G as a collection of sets of varying capacities, such as a union of VC classes or a union of finite classes. Let φ(G, S, δ) be a penalty for G. The MV-SRM principle selects the set bGG,α = arg min G∈G n µ(G) + φ(G, S, δ) : bP(G) ≥α −φ(G, S, δ) o . (9) Note that MV-SRM is different from MV-ERM because it minimizes a complexity penalized volume instead of simply the volume. We have the following.1 1Although the value of 1/γα is in practice unknown, it can be bounded by 1/γα ≤ (1 −µ∗ α)/(1 −α) ≤1/(1 −α). This follows from the bound 1 −α ≤γα · (1 −µ∗ α) on the mass outside the minimum volume set. Theorem 3. Let bGG,α be the MV-set estimator in (9). With probability at least 1 −δ over the training sample S, M( bGG,α) ≤ 1 + 1 γα inf G∈Gα n µ(G) −µ∗ α + 2φ(G, S, δ) o . (10) Sketch of proof: The proof is similar in some respects to oracle inequalities for classification. The key difference is in the form of the error term M(G) = (µ(G) −µ∗ α)+ + (α −P(G))+. In classification both approximation and stochastic errors are positive, whereas with MV-sets the excess volume µ(G) −µ∗ α or missing mass α −P(G) could be negative. This necessitates the (·)+ operators, without which the error would not be meaningful as mentioned earlier. The proof considers three cases separately: (1) µ( bGG,α) ≥µ∗ α and P( bGG,α) < α, (2) µ( bGG,α) ≥µ∗ α and P( bGG,α) ≥α, and (3) µ( bGG,α) < µ∗ α and P( bGG,α) < α. In the first case, both volume and mass errors are positive and the argument follows standard lines. The second case can be seen to follow easily from the first. The third case (which occurs most frequently in practice) is most involved and requires use of the fact that µ∗ α −µ∗ α−ϵ ≤ϵ/γα for ϵ > 0, which can be deduced from basic properties of MV and density level sets. The oracle inequality says that MV-SRM performs about as well as the set chosen by an oracle to optimize the tradeoffbetween the stochastic and approximation errors. To illustrate the power of the oracle inequality, in [6] we demonstrate that MV-SRM applied to recursive dyadic partition-based estimators adapts optimally to the number of relevant features (unknown a priori). 5 Damping the Penalty In Theorem 1, the reader may have noticed that MV-ERM does not equitably balance the volume error with the mass error. Indeed, with high probability, µ( bGG,α) is less than µ(GG,α), while P( bGG,α) is only guaranteed to be within φ( bGG,α) of α. The net effect is that MV-ERM (and MV-SRM) underestimates the MV-set. Experimental comparisons have confirmed this to be the case [6] . A minor modification of MV-ERM and MV-SRM leads to a more equitable distribution of error between the volume and mass, instead of having all the error reside in the mass term. The idea is simple: scale the penalty in the constraint by a damping factor ν < 1. In the case of MV-SRM, the penalty in the objective function also needs to be scaled by 1+ν. Moreover, the theoretical properties of these estimators stated above are retained (the statements, omitted here, are slightly more involved [6] ). Notice that in the case ν = 1 we recover the original estimators. Also note that the above theorem encompasses the generalized quantile estimate of [3], which corresponds to ν = 0. Thus we have finite sample size guarantees for that estimator to match Polonik’s asymptotic analysis. 6 Experiments: Histograms and Trees To gain some insight into the basic properties of our estimators, we devised some simple numerical experiments. In the case of histograms, MV-SRM can be implemented in a two step process. First, compute the MV-ERM estimate (a very simple procedure) for each Gk, k = 1, . . . , K, where 1/k is the bin-width. Second, choose the final estimate by minimizing the penalized volume of the MV-ERM estimates. n = 10000, k = 20, ν=0 100 1000 10000 100000 1000000 0 0.02 0.04 0.06 0.08 0.1 0.12 Error as a function of sample size occam rademacher Figure 2: Results for histograms. (Left) A typical MV-ERM estimate with binwidth 1/20, ν = 0, and based on 10000 points. True MV-set indicated by solid line. (Right) The error of the MV-SRM estimate M( bGG,α) as a function of sample size when ν = 0. The results indicated that the Occam’s Razor bound is tighter and yields better performance than Rademacher. We consider two penalties: one based on an Occam style bound, the other on the (conditional) Rademacher average. As a data set we consider X = [0, 1]2, the unit square, and data generated by a two-dimensional truncated Gaussian distribution, centered at the point (1/2, 1/2) and having spherical variance with parameter σ = 0.15. Other parameter settings are α = 0.8, K = 40, and δ = 0.05. All experiments were conducted at nine different sample sizes, logarithmically spaced from 100 to 1000000, and repeated 100 times. Results are summarized in Figure 2. To illustrate the potential improvement offered by spatially adaptive partitioning methods, we consider a minimum volume set estimator based on recursive dyadic (quadsplit) partitions. We employ a penalty that is additive over the cells A of the partition. The precise form of the penalty φ(A) for each cell is given in [6] , but loosely speaking it is proportional to the square-root of the ratio of the empirical mass of the cell to the sample size n. In this case, MV-SRM with ν = 0 is min G∈GL X A [µ(A)ℓ(A) + φ(A)] subject to X A bP(A)ℓ(A) ≥α (11) where GL is the collection of all partitions with dyadic cell sidelengths no smaller than 2−L and ℓ(A) = 1 if A belongs to the candidate set and ℓ(A) = 0 otherwise (see [6] for further details). Although directly optimization appears formidable, an efficient alternative is to consider the Lagrangian and conduct a bisection search over the Lagrange multiplier until the mass constraint is nearly achieved with equality (10 iterations is sufficient in practice). For each iteration, minimization of the Lagrangian can be performed very rapidly using standard tree pruning techniques. An experimental demonstration of the dyadic partition estimator is depicted in Figure 1. In the experiments we employed a dyadic quadtree structure with L = 8 (i.e., cell sidelengths no smaller than 2−8) and pruned according to the theoretical penalty φ(A) formally defined in [6] weighted by a factor of 1/30 (in practice the optimal weight could be found via cross-validation or other techniques). Figure 1 shows the results with data distributed according to a two-component Gaussian mixture distribution. This figure (middle image) additionally illustrates the improvement possible by “voting” over shifted partitions, which in principle is equivalent to constructing 2L × 2L different trees, each based on a partition offset by an integer multiple of the base sidelength 2−L, and taking a majority vote over all the resulting set estimates to form the final estimate. This strategy mitigates the “blocky” structure due to the underlying dyadic partitions, and can be computed almost as rapidly as a single tree estimate (within a factor of L) due to the large amount of redundancy among trees. The actual running time was one to two seconds. 7 Conclusions In this paper we propose two rules, MV-ERM and MV-SRM, for estimation of minimum volume sets. Our theoretical analysis is made possible by relating the performance of these rules to the uniform convergence properties of the class of sets from which the estimate is taken. Ours are the first known results to feature finite sample bounds, an oracle inequality, and universal consistency. Acknowledgements The authors thank Ercan Yildiz and Rebecca Willett for their assistance with the experiments involving dyadic trees. References [1] I. Steinwart, D. Hush, and C. Scovel, “A classification framework for anomaly detection,” J. Machine Learning Research, vol. 6, pp. 211–232, 2005. [2] S. Ben-David and M. Lindenbaum, “Learning distributions by their density levels – a paradigm for learning without a teacher,” Journal of Computer and Systems Sciences, vol. 55, no. 1, pp. 171–182, 1997. [3] W. Polonik, “Minimum volume sets and generalized quantile processes,” Stochastic Processes and their Applications, vol. 69, pp. 1–24, 1997. [4] G. Walther, “Granulometric smoothing,” Ann. Stat., vol. 25, pp. 2273–2299, 1997. [5] B. Sch¨olkopf, J. Platt, J. Shawe-Taylor, A. Smola, and R. Williamson, “Estimating the support of a high-dimensional distribution,” Neural Computation, vol. 13, no. 7, pp. 1443–1472, 2001. [6] C. Scott and R. Nowak, “Learning minimum volume sets,” UW-Madison, Tech. Rep. ECE-05-2, 2005. [Online]. Available: http://www.stat.rice.edu/∼cscott [7] A. Cannon, J. Howse, D. Hush, and C. Scovel, “Learning with the Neyman-Pearson and min-max criteria,” Los Alamos National Laboratory, Tech. Rep. LA-UR 02-2951, 2002. [Online]. Available: http://www.c3.lanl.gov/∼kelly/ml/pubs/2002 minmax/ paper.pdf [8] C. Scott and R. Nowak, “A Neyman-Pearson approach to statistical learning,” IEEE Trans. Inform. Theory, 2005, (in press). [9] V. Vapnik, Statistical Learning Theory. New York: Wiley, 1998. [10] L. Devroye, L. Gy¨orfi, and G. Lugosi, A Probabilistic Theory of Pattern Recognition. New York: Springer, 1996. [11] V. Vapnik, Estimation of Dependencies Based on Empirical Data. New York: Springer-Verlag, 1982. [12] G. Lugosi and K. Zeger, “Concept learning using complexity regularization,” IEEE Trans. Inform. Theory, vol. 42, no. 1, pp. 48–54, 1996.
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Spectral Bounds for Sparse PCA: Exact and Greedy Algorithms Baback Moghaddam MERL Cambridge MA, USA baback@merl.com Yair Weiss Hebrew University Jerusalem, Israel yweiss@cs.huji.ac.il Shai Avidan MERL Cambridge MA, USA avidan@merl.com Abstract Sparse PCA seeks approximate sparse “eigenvectors” whose projections capture the maximal variance of data. As a cardinality-constrained and non-convex optimization problem, it is NP-hard and is encountered in a wide range of applied fields, from bio-informatics to finance. Recent progress has focused mainly on continuous approximation and convex relaxation of the hard cardinality constraint. In contrast, we consider an alternative discrete spectral formulation based on variational eigenvalue bounds and provide an effective greedy strategy as well as provably optimal solutions using branch-and-bound search. Moreover, the exact methodology used reveals a simple renormalization step that improves approximate solutions obtained by any continuous method. The resulting performance gain of discrete algorithms is demonstrated on real-world benchmark data and in extensive Monte Carlo evaluation trials. 1 Introduction PCA is indispensable as a basic tool for factor analysis and modeling of data. But despite its power and popularity, one key drawback is its lack of sparseness (i.e., factor loadings are linear combinations of all the input variables). Yet sparse representations are generally desirable since they aid human understanding (e.g., with gene expression data), reduce computational costs and promote better generalization in learning algorithms. In machine learning, input sparseness is closely related to feature selection and automatic relevance determination, problems of enduring interest to the learning community. The earliest attempts at ”sparsifying” PCA in the statistics literature consisted of simple axis rotations and component thresholding [1] with the underlying goal being essentially that of subset selection, often based on the identification of principal variables [8]. The first true computational technique, called SCoTLASS by Jolliffe & Uddin [6], provided a proper optimization framework using Lasso [12] but it proved to be computationally impractical. Recently, Zou et al. [14] proposed an elegant algorithm (SPCA) using their “Elastic Net” framework for L1-penalized regression on regular PCs, solved very efficiently using least angle regression (LARS). Subsequently, d’Aspremont et al. [3] relaxed the “hard” cardinality constraint and solved for a convex approximation using semi-definite programming (SDP). Their “direct” formulation for sparse PCA (called DSCPA) has yielded promising results that are comparable to (if not better than) Zou et al.’s Lasso-based method, as demonstrated on the standard “Pit Props” benchmark dataset, known in the statistics community for its lack of sparseness and subsequent difficulty of interpretation. We pursued an alternative approach using a spectral formulation based on the variational principle of the Courant-Fischer “Min-Max” theorem for solving maximal eigenvalue problems in dimensionality-constrained subspaces. By its very nature, the discrete view leads to a simple post-processing (renormalization) step that improves any approximate solution (e.g., those given in [6, 14, 3]), and also provides bounds on (sub)optimality. More importantly, it points the way towards exact and provably optimal solutions using branch-and-bound search [9]. Our exact computational strategy parallels that of Ko et al. [7] who solved a different optimization problem (maximizing entropy with bounds on determinants). In the experiments we demonstrate the power of greedy and exact algorithms by first solving for the optimal sparse factors of the real-world “Pit Props” data, a de facto benchmark used by [6, 14, 3], and then present summary findings from a large comparative study using extensive Monte Carlo evaluation of the leading algorithms. 2 Sparse PCA Formulation Sparse PCA can be cast as a cardinality-constrained quadratic program (QP): given a symmetric positive-definite (covariance) matrix A ∈Sn +, maximize the quadratic form x′Ax (variance) with a sparse vector x ∈Rn having no more than k non-zero elements: max x′A x (1) subject to x′x = 1 card(x) ≤k where card(x) denotes the L0 norm. This optimization problem is non-convex, NP-hard and therefore intractable. Assuming we can solve for the optimal vector ˆx, subsequent sparse factors can be obtained using recursive deflation of A, as in standard numerical routines. The sparseness is controlled by the value(s) of k (in different factors) and can be viewed as a design parameter or as an unknown quantity itself (known only to the oracle). Alas, there are currently no guidelines for setting k, especially with multiple factors (e.g., orthogonality is often relaxed) and unlike ordinary PCA some decompositions may not be unique.1 Indeed, one of the contributions of this paper is in providing a sound theoretical basis for selecting k, thus clarifying the “art” of crafting sparse PCA factors. Note that without the cardinality constraint, the quadratic form in Eq.(1) is a RayleighRitz quotient obeying the analytic bounds λmin(A) ≤x′Ax/x′x ≤λmax(A) with corresponding unique eigenvector solutions. Therefore, the optimal objective value (variance) is simply the maximum eigenvalue λn(A) of the principal eigenvector ˆx = un — Note: throughout the paper the rank of all (λi, ui) is in increasing order of magnitude, hence λmin = λ1 and λmax = λn. With the (nonlinear) cardinality constraint however, the optimal objective value is strictly less than λmax(A) for k < n and the principal eigenvectors are no longer instrumental in the solution. Nevertheless, we will show that the eigenvalues of A continue to play a key role in the analysis and design of exact algorithms. 2.1 Optimality Conditions First, let us consider what conditions must be true if the oracle revealed the optimal solution to us: a unit-norm vector ˆx with cardinality k yielding the maximum objective value v∗. This would necessarily imply that ˆx′A ˆx = z′Akz where z ∈Rk contains the same k non-zero elements in ˆx and Ak is the k × k principal submatrix of A obtained by deleting the rows and columns corresponding to the zero indices of ˆx (or equivalently, by extracting the rows and columns of non-zero indices). Like ˆx, the k-vector z will be unit norm and z′Akz is then equivalent to a standard unconstrained Rayleigh-Ritz quotient. Since this subproblem’s maximum variance is λmax(Ak), then this must be the optimal objective v∗. We will now summarize this important observation with the following proposition. 1 We should note that the multi-factor version of Eq.(1) is ill-posed without additional constraints on basis orthogonality, cardinality, variable redundancy, ordinal rank and allocation of variance. Proposition 1. The optimal value v∗of the sparse PCA optimization problem in Eq.(1) is equal to λmax(A∗ k), where A∗ k is the k × k principal submatrix of A with the largest maximal eigenvalue. In particular, the non-zero elements of the optimal sparse factor ˆx are exactly equal to the elements of u∗ k, the principal eigenvector of A∗ k. This underscores the inherent combinatorial nature of sparse PCA and the equivalent class of cardinality-constrained optimization problems. However, despite providing an exact formulation and revealing necessary conditions for optimality (and in such simple matrix terms), this proposition does not suggest an efficient method for actually finding the principal submatrix A∗ k — short of an enumerative exhaustive search, which is impractical for n > 30 due to the exponential growth of possible submatrices. Still, exhaustive search is a viable method for small n which guarantees optimality for “toy problems” and small real-world datasets, thus calibrating the quality of approximations (via the optimality gap). 2.2 Variational Renormalization Proposition 1 immediately suggests a rather simple but (as it turns out) quite effective computational “fix” for improving candidate sparse PC factors obtained by any continuous algorithm (e.g., the various solutions found in [6, 14, 3]). Proposition 2. Let ˜x be a unit-norm candidate factor with cardinality k as found by any (approximation) technique. Let ˜z be the non-zero subvector of ˜x and uk be the principal (maximum) eigenvector of the submatrix Ak defined by the same non-zero indices of ˜x. If ˜z ̸= uk(Ak), then ˜x is not the optimal solution. Nevertheless, by replacing ˜x’s nonzero elements with those of uk we guarantee an increase in the variance, from ˜v to λk(Ak). This variational renormalization suggests (somewhat ironically) that given a continuous (approximate) solution, it is almost certainly better to discard the loadings and keep only the sparsity pattern with which to solve the smaller unconstrained subproblem for the indicated submatrix Ak. This simple procedure (or “fix” as referred to herein) can never decrease the variance and will surely improve any continuous algorithm’s performance. In particular, the rather expedient but ad-hoc technique of “simple thresholding” (ST) [1] — i.e., setting the n −k smallest absolute value loadings of un(A) to zero and then normalizing to unit-norm — is therefore not recommended for sparse PCA. In Section 3, we illustrate how this “straw-man” algorithm can be enhanced with proper renormalization. Consequently, past performance benchmarks using this simple technique may need revision — e.g., previous results on the “Pit Props” dataset (Section 3). Indeed, most of the sparse PCA factors published in the literature can be readily improved (almost by inspection) with the proper renormalization, and at the mere cost of a single k-by-k eigen-decomposition. 2.3 Eigenvalue Bounds Recall that the objective value v∗in Eq.(1) is bounded by the spectral radius λmax(A) (by the Rayleigh-Ritz theorem). Furthermore, the spectrum of A’s principal submatrices was shown to play a key role in defining the optimal solution. Not surprisingly, the two eigenvalue spectra are related by an inequality known as the Inclusion Principle. Theorem 1 Inclusion Principle. Let A be a symmetric n × n matrix with spectrum λi(A) and let Ak be any k × k principal submatrix of A for 1 ≤k ≤n with eigenvalues λi(Ak). For each integer i such that 1 ≤i ≤k λi(A) ≤λi(Ak) ≤λi+n−k(A) (2) Proof. The proof, which we omit, is a rather straightforward consequence of imposing a sparsity pattern of cardinality k as an additional orthogonality constraint in the variational inequality of the Courant-Fischer “Min-Max” theorem (see [13] for example). In other words, the eigenvalues of a symmetric matrix form upper and lower bounds for the eigenvalues of all its principal submatrices. A special case of Eq.(2) with k = n −1 leads to the well-known eigenvalue interlacing property of symmetric matrices: λ1(An) ≤λ1(An−1) ≤λ2(An) ≤. . . ≤λn−1(An) ≤λn−1(An−1) ≤λn(An) (3) Hence, the spectra of An and An−1 interleave or interlace each other, with the eigenvalues of the larger matrix “bracketing” those of the smaller one. Note that for positive-definite symmetric matrices (covariances), augmenting Am to Am+1 (adding a new variable) will always expand the spectral range: reducing λmin and increasing λmax. Thus for eigenvalue maximization, the inequality constraint card(x) ≤k in Eq.(1) is a tight equality at the optimum. Therefore, the maximum variance is achieved at the preset upper limit k of cardinality. Moreover, the function v∗(k), the optimal variance for a given cardinality, is monotone increasing with range [σ2 max(A), λmax(A)], where σ2 max is the largest diagonal element (variance) in A. Hence, a concise and informative way to quantify the performance of an algorithm is to plot its variance curve ˜v(k) and compare it with the optimal v∗(k). Since we seek to maximize variance, the relevant inclusion bound is obtained by setting i = k in Eq.(2), which yields lower and upper bounds for λk(Ak) = λmax(Ak), λk(A) ≤λmax(Ak) ≤λmax(A) (4) This shows that the k-th smallest eigenvalue of A is a lower bound for the maximum variance possible with cardinality k. The utility of this lower bound is in doing away with the “guesswork” (and the oracle) in setting k. Interestingly, we now see that the spectrum of A which has traditionally guided the selection of eigenvectors for dimensionality reduction (e.g., in classical PCA), can also be consulted in sparse PCA to help pick the cardinality required to capture the desired (minimum) variance. The lower bound λk(A) is also useful for speeding up branch-and-bound search (see next Section). Note that if λk(A) is close to λmax(A) then practically any principal submatrix Ak can yield a near-optimal solution. The right-hand inequality in Eq.(4) is a fixed (loose) upper bound λmax(A) for all k. But in branch-and-bound search, any intermediate subproblem Am, with k ≤m ≤n, yields a new and tighter bound λmax(Am) for the objective v∗(k). Therefore, all bound computations are efficient and relatively inexpensive (e.g., using the power method). The inclusion principle also leads to some interesting constraints on nested submatrices. For example, among all m possible (m −1)-by-(m −1) principal submatrices of Am, obtained by deleting the j-th row and column, there is at least one submatrix Am−1 = A\j whose maximal eigenvalue is a major fraction of its parent (e.g., see p. 189 in [4]) ∃j : λm−1(A\j) ≥ m −1 m λm(Am) (5) The implication of this inequality for search algorithms is that it is simply not possible for the spectral radius of every submatrix A\j to be arbitrarily small, especially for large m. Hence, with large matrices (or large cardinality) nearly all the variance λn(A) is captured. 2.4 Combinatorial Optimization Given Propositions 1 and 2, the inclusion principle, the interlacing property and especially the monotonic nature of the variance curves v(k), a general class of (binary) integer programming (IP) optimization techniques [9] seem ideally suited for sparse PCA. Indeed, a greedy technique like backward elimination is already suggested by the bound in Eq.(5): start with the full index set I = {1, 2, . . . , n} and sequentially delete the variable j which yields the maximum λmax(A\j) until only k elements remain. However, for small cardinalities k << n, the computational cost of backward search can grow to near maximum complexity ≈O(n4). Hence its counterpart forward selection is preferred: start with the null index set I = {} and sequentially add the variable j which yields the maximum λmax(A+j) until k elements are selected. Forward greedy search has worstcase complexity < O(n3). The best overall strategy for this problem was empirically found to be a bi-directional greedy search: run a forward pass (from 1 to n) plus a second (independent) backward pass (from n to 1) and pick the better solution at each k. This proved to be remarkably effective under extensive Monte Carlo evaluation and with realworld datasets. We refer to this discrete algorithm as greedy sparse PCA or GSPCA. Despite the expediency of near-optimal greedy search, it is nevertheless worthwhile to invest in optimal solution strategies, especially if the sparse PCA problem is in the application domain of finance or engineering, where even a small optimality gap can accrue substantial losses over time. As with Ko et al. [7], our branch-and-bound relies on computationally efficient bounds — in our case, the upper bound in Eq.(4), used on all active subproblems in a (FIFO) queue for depth-first search. The lower bound in Eq.(4) can be used to sort the queue for a more efficient best-first search [9]. This exact algorithm (referred to as ESPCA) is guaranteed to terminate with the optimal solution. Naturally, the search time depends on the quality (variance) of initial candidates. The solutions found by dual-pass greedy search (GSPCA) were found to be ideal for initializing ESPCA, as their quality was typically quite high. Note however, that even with good initializations, branch-and-bound search can take a long time (e.g. 1.5 hours for n = 40, k = 20). In practice, early termination with set thresholds based on eigenvalue bounds can be used. In general, a cost-effective strategy that we can recommend is to first run GSPCA (or at least the forward pass) and then either settle for its (near-optimal) variance or else use it to initialize ESPCA for finding the optimal solution. A full GSPCA run has the added benefit of giving near-optimal solutions for all cardinalities at once, with run-times that are typically O(102) faster than a single approximation with a continuous method. 3 Experiments We evaluated the performance of GSPCA (and validated ESPCA) on various synthetic covariance matrices with 10 ≤n ≤40 as well as real-world datasets from the UCI ML repository with excellent results. We present few typical examples in order to illustrate the advantages and power of discrete algorithms. In particular, we compared our performance against 3 continuous techniques: simple thresholding (ST) [1], SPCA using an “Elastic Net” L1-regression [14] and DSPCA using semidefinite programming [3]. We first revisited the “Pit Props” dataset [5] which has become a standard benchmark and a classic example of the difficulty of interpreting fully loaded factors with standard PCA. The first 6 ordinary PCs capture 87% of the total variance, so following the methodology in [3], we compared the explanatory power of our exact method (ESPCA) using 6 sparse PCs. Table 1 shows the first 3 PCs and their loadings. SPCA captures 75.8% of the variance with a cardinality pattern of 744111 (the k’s for the 6 PCs) thus totaling 18 non-zero loadings [14] whereas DSPCA captures 77.3% with a sparser cardinality pattern 623111 totaling 14 non-zero loadings [3]. We aimed for an even sparser 522111 pattern (with only 12 non-zero loadings) yet captured nearly the same variance: 75.9% — i.e., more than SPCA with 18 loadings and slightly less than DSPCA with 14 loadings. Using the evaluation protocol in [3], we compared the cumulative variance and cumulative cardinality with the published results of SPCA and DSPCA in Figure 1. Our goal was to match the explained variance but do so with a sparser representation. The ESPCA loadings in Table 1 are optimal under the definition given in Section 2. The run-time of ESPCA, including initialization with a bi-directional pass of GSPCA, was negligible for this dataset (n = 13). Computing each factor took less than 50 msec in Matlab 7.0 on a 3GHz P4. x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 SPCA : PC1 -.477 -.476 0 0 .177 0 -.250 -.344 -.416 -.400 0 0 PC2 0 0 .785 .620 0 0 0 -.021 0 0 0 .013 0 PC3 0 0 0 0 .640 .589 .492 0 0 0 0 0 -.015 DSPCA : PC1 -.560 -.583 0 0 0 0 -.263 -.099 -.371 -.362 0 0 0 PC2 0 0 .707 .707 0 0 0 0 0 0 0 0 0 PC3 0 0 0 0 0 -.793 -.610 0 0 0 0 0 .012 ESPCA : PC1 -.480 -.491 0 0 0 0 -.405 0 -.423 -.431 0 0 0 PC2 0 0 .707 .707 0 0 0 0 0 0 0 0 0 PC3 0 0 0 0 0 -.814 -.581 0 0 0 0 0 0 Table 1: Loadings for first 3 sparse PCs of the Pit Props data. See Figure 1(a) for plots of the corresponding cumulative variances. Original SPCA and DSPCA loadings taken from [14, 3]. 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Cumulative Variance # of PCs SPCA DSPCA ESPCA 1 2 3 4 5 6 4 6 8 10 12 14 16 18 # of PCs Cumulative Cardinality SPCA DSPCA ESPCA (a) (b) Figure 1: Pit Props: (a) cumulative variance and (b) cumulative cardinality for first 6 sparse PCs. Sparsity patterns (cardinality ki for PCi, with i = 1, 2, . . . , 6) are 744111 for SPCA (magenta ⋆), 623111 for DSPCA (green ⋄) and an optimal 522111 for ESPCA (red ◦). The factor loadings for the first 3 sparse PCs are shown in Table 1. Original SPCA and DSPCA results taken from [14, 3]. To specifically demonstrate the benefits of the variational renormalization of Section 2.2, consider SPCA’s first sparse factor in Table 1 (the 1st row of SPCA block) found by iterative (L1-penalized) optimization and unit-norm scaling. It captures 28% of the total data variance, but after the variational renormalization the variance increases to 29%. Similarily, the first sparse factor of DSPCA in Table 1 (1st row of DSPCA block) captures 26.6% of the total variance, whereas after variational renormalization it captures 29% — a gain of 2.4% for the mere additional cost of a 7-by-7 eigen-decomposition. Given that variational renormalization results in the maximum variance possible for the indicated sparsity pattern, omitting such a simple post-processing step is counter-productive, since otherwise the approximations would be, in a sense, doubly sub-optimal: both globally and “locally” in the subspace (subset) of the sparsity pattern found. We now give a representative summary of our extensive Monte Carlo (MC) evaluation of GSPCA and the 3 continuous algorithms. To show the most typical or average-case performance, we present results with random covariance matrices from synthetic stochastic Brownian processes of various degrees of smoothness, ranging from sub-Gaussian to superGaussian. Every MC run consisted of 50,000 covariance matrices and the (normalized) variance curves ˜v(k). For each matrix, ESPCA was used to find the optimal solution as “ground truth” for subsequent calibration, analysis and performance evaluation. For SPCA we used the LARS-based “Elastic Net” SPCA Matlab toolbox of Sj¨ostrand [10] which is equivalent to Zou et al.’s SPCA source code, which is also freely available in R. For DSPCA we used the authors’ own Matlab source code [2] which uses the SDP toolbox SeDuMi1.0x [11]. The main DSPCA routine PrimalDec(A, k) was called with k−1 instead of k, for all k > 2, as per the recommended calibration (see documentation in [3, 2]). In our MC evaluations, all continuous methods (ST, SPCA and DSPCA) had variational renormalization post-processing (applied to their the “declared” solution). Note that comparing GSPCA with the raw output of these algorithms would be rather pointless, since 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 11 cardinality (k) variance v(k) DSPCA (original) DSPCA + Fix Optimal 0.85 0.9 0.95 1 10 −4 10 −3 10 −2 10 −1 10 0 optimality ratio log − frequency ST SPCA DSPCA GSPCA (a) (b) Figure 2: (a) Typical variance curve v(k) for a continuous algorithm without post-processing (original: dash green) and with variational renormalization (+ Fix: solid green). Optimal variance (black ◦) by ESPCA. At k = 4 optimality ratio increases from 0.65 to 0.86 (a 21% gain). (b) Monte Carlo study: log-likelihood of optimality ratio at max-complexity (k = 8, n = 16) for ST (blue []), DSPCA (green ⋄), SPCA (magenta ⋆) and GSPCA (red ◦). Continuous methods were “fixed” in (b). 2 4 6 8 10 12 14 16 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 cardinality (k) mean optimality ratio ST SPCA DSPCA GSPCA 2 4 6 8 10 12 14 16 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 cardinality (k) frequency ST SPCA DSPCA GSPCA (a) (b) Figure 3: Monte Carlo summary statistics: (a) means of the distributions of optimality ratio (in Figure 2(b)) for all k and (b) estimated probability of finding the optimal solution for each cardinality. without the “fix” their variance curves are markedly diminished, as in Figure 2(a). Figure 2(b) shows the histogram of the optimality ratio — i.e., ratio of the captured to optimal variance — shown here at “half-sparsity” (k = 8, n = 16) from a typical MC run of 50,000 different covariances matrices. In order to view the (one-sided) tails of the distributions we have plotted the log of the histogram values. Figure 3(a) shows the corresponding mean values of the optimality ratio for all k. Among continuous algorithms, the SDP-based DSPCA was generally more effective (almost comparable to GSPCA). For the smaller matrices (n < 10), LARS-based SPCA matched DSPCA for all k. In terms of complexity and speed however, SPCA was about 40 times faster than DSPCA. But GSPCA was 30 times faster than SPCA. Finally, we note that even simple thresholding (ST), once enhanced with the variational renormalization, performs quite adequately despite its simplicity, as it captures at least 92% of the optimal variance, as seen in Figure 3(a). Figure 3(b) shows an alternative but more revealing performance summary: the fraction of the (50,000) trials in which the optimal solution was actually found (essentially, the likelihood of “success”). This all-or-nothing performance measure elicits important differences between the algorithms. In practical terms, only GSPCA is capable of finding the optimal factor more than 90% of the time (vs. 70% for DSPCA). Naturally, without the variational “fix” (not shown) continuous algorithms rarely ever found the optimal solution. 4 Discussion The contributions of this paper can be summarized as: (1) an exact variational formulation of sparse PCA, (2) requisite eigenvalue bounds, (3) a principled choice of k, (4) a simple renormalization “fix” for any continuous method, (5) fast and effective greedy search (GSPCA) and (6) a less efficient but optimal method (ESPCA). Surprisingly, simple thresholding of the principal eigenvector (ST) was shown to be rather effective, especially given the perceived “straw-man” it was considered to be. Naturally, its performance will vary with the effective rank (or “eigen-gap”) of the covariance matrix. In fact, it is not hard to show that if A is exactly rank-1, then ST is indeed an optimal strategy for all k. However, beyond such special cases, continuous methods can not ultimately be competitive with discrete algorithms without the variational renormalization “fix” in Section 2.2. We should note that the somewhat remarkable effectiveness of GSPCA is not entirely unexpected and is supported by empirical observations in the combinatorial optimization literature: that greedy search with (sub)modular cost functions having the monotonicity property (e.g., the variance curves ˜v(k)) is known to produce good results [9]. In terms of quality of solutions, GSPCA consistently out-performed continuous algorithms, with runtimes that were typically O(102) faster than LARS-based SPCA and roughly O(103) faster than SDP-based DSPCA (Matlab CPU times averaged over all k). Nevertheless, we view discrete algorithms as complementary tools, especially since the leading continuous algorithms have distinct advantages. For example, with very highdimensional datasets (e.g., n = 10, 000), Zou et al.’s LARS-based method is currently the only viable option, since it does not rely on computing or storing a huge covariance matrix. Although d’Aspremont et al. mention the possibility of solving “larger” systems much faster (using Nesterov’s 1st-order method [3]), this would require a full matrix in memory (same as discrete algorithms). Still, their SDP formulation has an elegant robustness interpretation and can also be applied to non-square matrices (i.e., for a sparse SVD). Acknowledgments The authors would like to thank Karl Sj¨ostrand (DTU) for his customized code and helpful advice in using the LARS-SPCA toolbox [10] and Gert Lanckriet (Berkeley) for providing the Pit Props data. References [1] J. Cadima and I. Jolliffe. Loadings and correlations in the interpretation of principal components. Applied Statistics, 22:203–214, 1995. [2] A. d’Aspremont. DSPCA Toolbox. http://www.princeton.edu/∼aspremon/DSPCA.htm. [3] 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. [4] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge Press, Cambridge, England, 1985. [5] J. Jeffers. Two cases 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] C. Ko, J. Lee, and M. Queyranne. An Exact Algorithm for Maximum Entropy Sampling. Operations Research, 43(4):684–691, July-August 1995. [8] G. McCabe. Principal variables. Technometrics, 26:137–144, 1984. [9] G. L. Nemhauser and L. A. Wolsey. Integer and Combinatorial Optimization. John Wiley, New York, 1988. [10] K. Sj¨ostrand. Matlab implementation of LASSO, LARS, the Elastic Net and SPCA. Informatics and Mathematical Modelling, Technical University of Denmark (DTU), 2005. [11] J. F. Sturm. SeDuMi1.0x, a MATLAB Toolbox for Optimization over Symmetric Cones. Optimization Methods and Software, 11:625–653, 1999. [12] R. Tibshirani. Regression shrinkage and selection via Lasso. Journal of the Royal Statistical Society B, 58:267–288, 1995. [13] J. H. Wilkinson. The Algebraic Eigenvalue Problem. Clarendon Press, Oxford, England, 1965. [14] H. Zou, T. Hastie, and R. Tibshirani. Sparse Principal Component Analysis. Technical Report, Statistics Department, Stanford University, 2004.
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A Domain Decomposition Method for Fast Manifold Learning Zhenyue Zhang Department of Mathematics Zhejiang University, Yuquan Campus, Hangzhou, 310027, P. R. China zyzhang@zju.edu.cn Hongyuan Zha Department of Computer Science Pennsylvania State University University Park, PA 16802 zha@cse.psu.edu Abstract We propose a fast manifold learning algorithm based on the methodology of domain decomposition. Starting with the set of sample points partitioned into two subdomains, we develop the solution of the interface problem that can glue the embeddings on the two subdomains into an embedding on the whole domain. We provide a detailed analysis to assess the errors produced by the gluing process using matrix perturbation theory. Numerical examples are given to illustrate the efficiency and effectiveness of the proposed methods. 1 Introduction The setting of manifold learning we consider is the following. We are given a parameterized manifold of dimension d defined by a mapping f : Ω→Rm, where d < m, and Ωopen and connected in Rd. We assume the manifold is well-behaved, it is smooth and contains no self-intersections etc. Suppose we have a set of points x1, · · · , xN, sampled possibly with noise from the manifold, i.e., xi = f(τi) + ϵi, i = 1, . . . , N, (1.1) where ϵi’s represent noise. The goal of manifold learning is to recover the parameters τi’s and/or the mapping f(·) from the sample points xi’s [2, 6, 9, 12]. The general framework of manifold learning methods involves imposing a connectivity structure such as a k-nearestneighbor graph on the set of sample points and then turn the embedding problem into the solution of an eigenvalue problem. Usually constructing the graph dominates the computational cost of a manifold learning algorithm, but for large data sets, the computational cost of the eigenvalue problem can be substantial as well. The focus of this paper is to explore the methodology of domain decomposition for developing fast algorithms for manifold learning. Domain decomposition by now is a wellestablished field in scientific computing and has been successfully applied in many science and engineering fields in connection with numerical solutions of partial differential equations. One class of domain decomposition methods partitions the solution domain into subdomains, solves the problem on each subdomain and glue the partial solutions on the subdomains by solving an interface problem [7, 10]. This is the general approach we will follow in this paper. In particular, in section 3, we consider the case where the given set of sample points x1, . . . , xN are partitioned into two subdomains. On each of the subdomain, we can use a manifold learning method such as LLE [6], LTSA [12] or any other manifold learning methods to construct an embedding for the subdomain in question. We will then formulate the interface problem the solution of which will allow us to combine the embeddings on the two subdomains together to obtain an embedding over the whole domain. However, it is not always feasible to carry out the procedure described above. In section 2, we give necessary and sufficient conditions under which the embedding on the whole domain can be constructed from the embeddings on the subdomains. In section 4, we analyze the errors produced by the gluing process using matrix perturbation theory. In section 5, we briefly mention how the partitioning of the set of sample points into subdomains can be accomplished by some graph partitioning algorithms. Section 6 is devoted to numerical experiments. NOTATION. We use e to denote a column vector of all 1’s the dimension of which should be clear from the context. N(·) and R(·) denote the null space and range space of a matrix, respectively. For an index set I = [i1, . . . , ik], A(:, I) denotes the submatrix of A consisting of columns of A with indices in I with a similar definition for the rows of a matrix. We use ∥· ∥to denote the spectral norm of a matrix. 2 A Basic Theorem Let X = [x1, · · · , xN] with xi = f(τi) + ϵi, i = 1, . . . , N. Assume that the whole sample domain X is divided into two subdomains X1 = {xi | i ∈I1} and X2 = {xi | i ∈I2}. Here I1 and I2 denote the index sets such that I1 ∪I2 = {1, . . . , N} and I1 ∩I2 is not empty. Suppose we have obtained the two low-dimensional embeddings T1 and T2 of the sub-domains X1 and X2, respectively. The domain decomposition method attempts to recover the overall embedding T = {τ1, . . . , τN} from the embeddings T1 and T2 on the subdomains. In general, the recovered sub-embedding Tj, j = 1, 2, may not be exactly the subset {τi | i ∈Ij} of T. For example, it is often the case that the recovered embeddings Tj are approximately affinely equal to {τi | i ∈Ij}, i.e., up to certain approximation errors, there is an affine transformation such that Tj = {Fjτi + cj | i ∈Ij}, where Fj is a nonsingular matrix and cj a column vector. Thus a domain decomposition method for manifold learning should be invariant to affine transformation on the embeddings Tj obtained from subdomains. In that case, we can assume that Tj is just the subset of T, i.e., Tj = {τi | i ∈Ij}. With an abuse of notation, we also denote by T and Tj the matrices of the column vectors in the set T and Tj, for example, we write T = [τ1, . . . , τN]. Let Φj be an orthogonal projection with N(Φj) = span([e, T T j ]). Then Tj can be recovered by computing the eigenvectors of Φj corresponding to its zero eigenvalues. To recover the whole T we need to construct a matrix Φ with N(Φ) = span([e, T T ]) [11]. To this end, for each Tj, let Φj = QjQT j ∈RNj×Nj, where Qj is an orthonormal basis matrix of N([e, T T j ]T ) and Nj is the column-size of Tj. To construct a Φ matrix, Let Sj ∈RN×Nj be the 0-1 selection matrix defined as Sj = IN(:, Ij), where IN is the identity matrix of order N. Let ˆΦj = SjΦjST j . We then simply take Φ = ˆΦ1 + ˆΦ2, or more flexibly, Φ = w1 ˆΦ1 + w2 ˆΦ2, where w1 and w2 are the weights: wi > 0 and w1 + w2 = 1. Obviously ∥Φ∥≤1 since ∥Φj∥= 1. The following theorem gives the necessary and sufficient conditions under which the null space of Φ is just span{[e, T T ]}. (In the theorem, we only require the Φj to positive semidefinite.) Theorem 2.1 Let Φi be two positive semidefinite matrices such that N(Φi) = span([e, T T i ]), i = 1, 2, and T0 = T1 ∩T2. Assume that [e, T T 1 ] and [e, T T 2 ] are of full column-rank. Then N(Φ) = span([e, T T ]) if and only if [e, T T 0 ] is of full column-rank. Proof. We first prove the necessity by contradiction. Assume that N([e, T T 0 ]) ̸= N([e, T T 2 ]), then there is y ̸= 0 such that [e, T T 0 ]y = 0 and [e, T T (:, I2)]y ̸= 0. Denote by Ic 1 the complement of I1, i.e., the index set of i’s which do not belong to I1. Then [e, T T (:, Ic 1)]y ̸= 0. Now we construct a vector x as x(I1) = [e, T T 1 ]y, x(Ic 1) = 0. Clearly x(I2) = 0 and hence x ∈N(Φ). By the condition N(Φ) = span([e, T T ]), we can write x in the form x = [e, T T ]z for a column vector z. Specially, x(I1) = [e, T T 1 ]z. Note that we also have x(I1) = [e, T T 1 ]y by definition. It implies that z = y because [e, T T 1 ] is of full rank. Therefor, [e, T T (:, Ic 1)]y = [e, T T (:, Ic 1)]z = x(Ic 1) = 0. Using it together with [e, T T 0 ]y = 0 we have [e, T T (:, I2)]y = 0, a contradiction. Now we prove the sufficiency. Let Q be a basis matrix of N(Φ). we have w1G1QT ˆΦ1Q + w2G2QT ˆΦ2Q = QT ΦQ = 0, which implies ΦiQ(I1, :) = 0, i = 1, 2, because ˆΦi is positive semidefinite. So Q(Ii, :) = [e, T T i ]Gi, i = 1, 2. (2.2) Taking the overlap part Q(I0, :) of Q with the different representations Q(I0, :) = [e, Ti(:, I0)T ]Gi = [e, T T 0 ]Gi, we obtain [e, T T 0 ](G1 −G2) = 0. So G1 = G2 because [e, T T 0 ] is of full column rank, giving rise to Q = [e, T T ]G1, i.e., N(Φ) ⊂span([e, T T ]). It follows together with the obvious result span([e, T T ]) ⊂N(Φ) that N(Φ) = span([e, T T ]). The above result states that when the overlapping is large enough such that [e, T T 0 ] is of full column-rank (which is generically true when T0 contains d + 1 points or more), the embedding over the whole domain can be recovered from the embeddings over the two subdomains. However, to follow Theorem 2.1, it seems that we will need to compute the null space of Φ. In the next section, we will show this can done much cheaply by considering an interface problem which is of much smaller dimension. 3 Computing the Null Space of Φ In this section, we formulate the interface problem and show how to solve it to glue the embeddings from the two subdomains to obtain an embedding over the whole domain. To simplify notations, we re-denote by T ∗the actual embedding over the whole domain and T ∗ j the subsets of T ∗corresponding to subdomains. We then use Tj to denote affinely transformed versions of T ∗ j obtained by LTSA for example, i.e., T ∗ j = cjeT + FjTj. Here cj is a constant column vector in Rd and Fj is a nonsingular matrix. Denote by T0j the overlapping part of Tj corresponding to I0 = I1 ∩I2 as in the proof of Theorem 2.1. We consider the overlapping parts T ∗ 0j of T ∗ j , c1eT + F1T01 = T ∗ 01 = T ∗ 02 = c2eT + F2T02. (3.3) Or equivalently, h [e, T T 01], −[e, T T 02] i (c1, F1)T (c2, F2)T = 0. Therefore, if we take an orthonormal basis G of the null space of h [e, T T 01], −[e, T T 02] i and partition G = [GT 1 , GT 2 ]T conformally, then [e, T T 01]G1 = [e, T T 02]G2. Let Aj = GT j [e, T T j ]T , j = 1, 2. Define the matrix A such that A(:, Ij) = Aj. Then since ΦiAT i = 0, the well-defined matrix AT is a basis of N(Φ), ΦAT = S1Φ1ST 1 AT + S2Φ2ST 2 AT = S1Φ1AT 1 + S2Φ2AT 2 = 0. Therefore, we can use AT to recover the global embedding T. A simpler alternative way is use a one-sided affine transformation, i.e., fix one of Ti and affinely transform the other; the affine matrix is obtained by fixing one of ˜T0i and transforming the other. For example, we can determine c and F such that T01 = ceT + FT02, (3.4) and transform T2 to ˆT2 = ceT + FT2. Clearly, for the overlapping part, ˆT02 = T01. Then we can construct a larger matrix T by T(:, I1) = T1, T(:, I2) = ceT + FT2. One can also readily verify that T T is a basis matrix of N(Φ). In the noisy case, a least squares formulation will be needed. For example, for the simultaneous affine transformation, we take G = [GT 1 , GT 2 ]T to be an orthonormal matrix in R2(d+1)×(d+1) such that ∥[e, T T 01]G1 −[e, T T 02]G2∥= min . It is known that the minimum G is given by the right singular vector matrix corresponding to the d + 1 smallest singular values of W = h [e, T T 01], −[e, T T 02] i , and the residual
[e, T T 01]G1 −[e, T T 02]G2
= σd+2(W). For the one-side approach (3.4), [c, F] can be a solution to the least squares problem min c, F
T01 − ceT + FT02
= min F
(T01 −t01eT ) −F(T02 −t02eT )
, where t0j is the column mean of T0j. The minimum is achieved at F = (T01−t01eT )(T02− t02eT )+, c = t01 −Ft02. Clearly, the residual now reads as min c, F
T01 − ceT + FT02
=
(T01 −t01eT ) I −(T02 −t02eT )+(T02 −t02eT )
. Notice that the overlapping parts in the two affinely transformed subsets are not exactly equal to each other in the noisy case. There are several possible choices for setting A(:, I0) or ˆT(:, I0). For example, one choice is to set T(:, I0) by a convex combination of T0j’s, T(:, I0) = αT01 + (1 −α) ˆT02. with α = 1/2 for example. We summarize discussions above in the following two algorithms for gluing the two subdomains T1 and T2. Algorithm I. [Simultaneously affine transformation] 1. Compute the right singular vector matrix G corresponding to the d + 1 smallest singular values of h [e, T T 01], −[e, T T 02] i . 2. Partition G = [GT 1 , GT 2 ]T and set Ai = GT i [e, T T i ]T , i = 1, 2, and A(:, I1\I0) = A11, A(:, I0) = αA01 + (1 −α)A02, A(:, I2\I0) = A12, where A0j is the overlap part of Aj and A1j is the Aj with A0j deleted. 3 Compute the column mean a of A, and an orthogonal basis U of N(aT ). 4. Set T = U T A. Algorithm II. [One-side affine transformation] 1. Compute the least squares problem minW ∥T01 −W[e, T T 02]T ∥F . 2. Affinely transform T2 to ˆT2 = W[e, T T 2 ]T . 3. Set the global coordinate matrix T by T(:, I1\I0) = T11, T(:, I0) = αT01 + (1 −α) ˆT02, T(:, I2\I0) = ˆT12. 4 Error Analysis As we mentioned before, the computation of Tj, j = 1, 2 using a manifold learning algorithm such as LTSA involves errors. In this section, we assess the impact of those errors on the accuracy of the gluing process. Two issues are considered for the error analysis. One is the perturbation analysis of N(Φ∗) when the computation of Φ∗ i is subject to error. In this case, N(Φ∗) will be approximated by the smallest (d + 1)-dimensional eigenspace V of an approximation Φ ≈Φ∗(Theorem 4.1). The other issue is the error estimation of V when a basis matrix of V is approximately constructed by affinely transformed local embeddings as described in section 3 (Theorem 4.2). Because of space limit, we will not present the details of the proofs of the results. The distance of two linear subspaces X and Y are defined by dist(X, Y) = ∥PX −PY∥, where PX and PY are the orthogonal projection onto X and Y, respectively. Let ϵi = ∥Φi −Φ∗ i ∥, where Φ∗ i and Φi are the orthogonal projectors onto the range spaces span([e, (T ∗ i )T ]) and span([e, (Ti)T ]), respectively. Clearly, if Φ∗= w1Φ∗ 1 + w2Φ∗ 2 and Φ = w1Φ1 + w2Φ2, then dist span([e, (T ∗)T ]), span([e, T T ]) = ∥Φ −Φ∗∥≤w1ϵ1 + w2ϵ2 ≡ϵ. Theorem 4.1 Let σ be the smallest nonzero eigenvalue of Φ∗and V the subspace spanned by the eigenvectors of Φ corresponding to the d + 1 smallest eigenvalues. If ϵ < σ/4, and 4ϵ2(∥Φ∗∥−σ + 2ϵ) < (σ −2ϵ)3, then dist(V, N(Φ∗)) ≤ ϵ p (σ/2 −ϵ)2 + ϵ2 . Theorem 4.2 Let σ and ϵ be defined in Theorem 4.1. A is the matrix computed by the simultaneous affine transformation (Algorithm I in section 3) Let σi(·) be the i-th smallest singular value of a matrix. Denote µ = 1 2σd+2( [e, T T 01], −[e, T T 02] ), η = µ σmin(A). If ϵ < σ/4, then dist(V, span(A)) ≤ 1 σd+2(Φ) η + ϵσ/2 (σ/2 −ϵ)2 From Theorems 4.1 and 4.2 we conclude directly that dist(span(A), N(Φ∗)) ≤ 1 σd+2(Φ) η + ϵσ/2 (σ/2 −ϵ)2 + 2ϵ p (σ −2ϵ)2 + 4ϵ2 . 5 Partitioning the Domains To apply the domain decomposition methods, we need to partition the given set of data points into several domains making use of the k nearest neighbor graph imposed on the data points. This reduces the problem to a graph partition problem and many techniques such as spectral graph partitioning and METIS [3, 5] can be used. In our experiments, we have used a particularly simple approach: we use the reverse Cuthill-McKee method [4] to order the vertices of the k-NN graph and then partition the vertices into domains (for details see Test 2 in the next section). Once we have partitioned the whole domain into multiple overlapping subdomains we can use the following two approaches to glue them together. Successive gluing. Here we glue the subdomains one by one as follows. Initially set T (1) = T1 and I(1) = I1, and then glue the patch Tk to T (k−1) and obtain the larger one T (k) for k = 2, . . . , K, and so on. The index set of T (k) is given by I(k) = I(k−1) ∪Ik. Clearly the overlapping set of T (k−1) and Tk is I(k) 0 = I(k−1) ∩Ik. Recursive gluing. Here at the leaf level, we divide the subdomains into several pairs, say (T (0) 2i−1, T (0) 2i ), 1 = 1, 2, . . .. Then glue each pair to be a larger subdomain T (1) i and continue. The recursive gluing method is obviously parallelizable. 6 Numerical Experiments In this section we report numerical experiments for the proposed domain decomposition methods for manifold learning. This efficiency and effectiveness of the methods clearly depend on the accuracy of the computed embeddings for subdomains, the sizes of the subdomains, and the sizes of the overlaps of the subdomains. Test 1. Our first test data set is sampled from a Swiss-roll as follows xi = [ti cos(ti), hi, ti sin(ti)]T , i = 1, . . . , N = 2000, (6.5) where ti and hi are uniformly randomly chosen in the intervals [ 3π 2 , 9π 2 ] and [0, 21], respectively. Let τi be the arc length of the corresponding spiral curve [t cos(t), t sin(t)]T from t0 = 3π 2 to ti. τmax = maxi τi. To compare the CPU time of the domain decomposition methods, we simply partition the τ-interval [0, τmax] into kτ subintervals (ai−1, ai] with equal length and also partition the h-interval into kh subintervals (bj−1, bj]. Let Dij = (ai−1, ai] × (bj−1, bj] and Sij(r) be the balls centered at (ai, bj) with radius r. We set the subdomains as Xij = {xk | (τk, hk) ∈Dij ∪Sij(r)}. Clearly r determines the size of overlapping parts of Xij with Xi+1,j, Xi,j+1, Xi+1,j+1. The submatrices Xij are ordered as X1,1, X1,2, . . . , X1,kh, X2,1, . . . and denoted as Xk, k = 1, . . . , K = kτkh. We first compute the K local 2-D embeddings T1, . . . , TK by applying LTSA on the sample data sets Xk for the subdomains. Then those local coordinate embeddings Tk are aligned by the successive one-sided affine transformation algorithm by adding subdomain Tk one by one. Table 1 lists the total CPU time for the successive domain decomposition algorithm, including the time for computing the embeddings {Tk} for the subdomains, for different parameters kτ and kh with the parameter r = 5. In Table 2, we list the CPU time for the recursive gluing approach taking into account the parallel procedure. As a comparison, the CPU time of LTSA applying to the whole data points is 6.23 seconds. Table 1: CPU Time (seconds) of the successive domain decomposition algorithm. kh=2 3 4 5 6 kτ= 3 1.89 1.70 1.64 1.61 1.64 4 167 1.67 1.61 1.70 1.77 5 1.66 1.59 1.67 1.78 1.86 6 163 1.66 1.75 1.89 2.09 7 1.59 1.70 1.84 2.02 2.23 8 1.58 1.80 1.94 2.22 2.44 9 1.63 1.83 2.06 2.31 2.66 10 1.63 1.86 2.38 2.56 2.94 Table 2: CPU Time (seconds) of the parallel recursive domain decomposition. kh=2 3 4 5 6 kτ= 3 0.52 0.34 0.27 0.19 017 4 0.53 0.23 0.20 0.17 0.13 5 0.31 0.17 0.19 0.17 0.14 6 0.25 0.19 0.16 0.13 0.14 7 0.20 0.16 0.14 0.14 0.11 8 0.20 0.17 0.16 0.14 0.14 9 0.19 0.16 0.14 0.14 0.14 10 0.19 0.16 0.17 0.19 0.13 Test 2. The symmetric reverse Cuthill-McKee permutation (symrcm) is an algorithm for ordering the rows and columns of a symmetric sparse matrix [4]. It tends to move the nonzero elements of the sparse matrix towards the main diagonals of the matrix. We use Matlab’s symrcm to the adjacency matrix of the k-nearest-neighbor graph of the data points to reorder them. Denote by X the reordered data set. We then partition the whole sample points into K = 16 subsets Xi = X(:, si : ei) with si = max{1, (i −1)m −20}, ei = min{im + 20, N}, and m = N/K = 125. It is known that the t-h parameters in (6.5) represent an isometric parametrization of the swiss-roll surface. We have shown that within the errors made in computing the local embeddings, LTSA can recover the isometric parametrization up to an affine transformation [11]. We denote by ˜T (k) = ceT + FT (k) the optimal approximation to T ∗(:, I(k)) within affine transformations, ∥T ∗(:, I(k)) −˜T (k)∥F = min c,F ∥T ∗(:, I(k)) −(ceT + FT (k))∥F . We denote by ηk the average of relative errors ηk = 1 |I(k)| X i∈I(k) ∥T ∗(:, i) −˜T (k)(:, i)∥2 ∥T ∗(:, i)∥2 . In the left panel of Figure 1 we plot the initial embedding errors for the subdomains (blue bar), the error of LTSA applied to the whole data set (red bar), and the errors ηk of the successive gluing (red line). The successive gluing method gives an embedding with an acceptable accuracy comparing with the accuracy obtained by applying LTSA to the whole data set. As shown in the error analysis, the errors in successive gluing will increase when the initial errors for the subdomains increase. To show it more clearly, we also plot the ηk for the recursive gluing method in the right panel of Figure 1. Acknowledgment. The work of first author was supported in part by by NSFC (project 60372033), the Special Funds for Major State Basic Research Projects (project 0 2 4 6 8 10 12 14 16 18 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 −3 k successive alignment subdomains whole domain 0 2 4 6 8 10 12 14 16 18 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 −3 k root 4 root 3 root 2 root 1 subdomains whole domain Figure 1: Relative errors for the successive (left) and recursive (right) approaches. G19990328), and NSF grant CCF-0305879. The work of second author was supported in part by NSF grants DMS-0311800 and CCF-0430349. References [1] M. Brand. Charting a manifold. Advances in Neural Information Processing Systems 15, MIT Press, 2003. [2] D. Donoho and C. Grimes. Hessian Eigenmaps: new tools for nonlinear dimensionality reduction. Proceedings of National Academy of Science, 5591-5596, 2003. [3] M. Fiedler. A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory. Czech. Math. J. 25:619–637, 1975. [4] A. George and J. W. Liu. Computer Solution of Large Sparse Positive Definite Matrices. Prentice Hall, 1981. [5] METIS. http://www-users.cs.umn.edu/∼karypis/metis/. [6] S. Roweis and L. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290: 2323–2326, 2000. [7] B. Smith, P. Bjorstad and W. Gropp Domain Decomposition, Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, 1996. [8] G.W. Stewart and J.G. Sun. Matrix Perturbation Theory. Academic Press, New York, 1990. [9] J. Tenenbaum, V. De Silva and J. Langford. A global geometric framework for nonlinear dimension reduction. Science, 290:2319–2323, 2000. [10] A. Toselli and O. Widlund. Domain Decomposition Methods - Algorithms and Theory. Springer, 2004. [11] H. Zha and Z. Zhang. Spectral analysis of alignment in manifold learning. Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, (ICASSP), 2005. [12] Z. Zhang and H. Zha. Principal manifolds and nonlinear dimensionality reduction via tangent space alignment. SIAM J. Scientific Computing. 26:313-338, 2005.
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From Batch to Transductive Online Learning Sham Kakade Toyota Technological Institute Chicago, IL 60637 sham@tti-c.org Adam Tauman Kalai Toyota Technological Institute Chicago, IL 60637 kalai@tti-c.org Abstract It is well-known that everything that is learnable in the difficult online setting, where an arbitrary sequences of examples must be labeled one at a time, is also learnable in the batch setting, where examples are drawn independently from a distribution. We show a result in the opposite direction. We give an efficient conversion algorithm from batch to online that is transductive: it uses future unlabeled data. This demonstrates the equivalence between what is properly and efficiently learnable in a batch model and a transductive online model. 1 Introduction There are many striking similarities between results in the standard batch learning setting, where labeled examples are assumed to be drawn independently from some distribution, and the more difficult online setting, where labeled examples arrive in an arbitrary sequence. Moreover, there are simple procedures that convert any online learning algorithm to an equally good batch learning algorithm [8]. This paper gives a procedure going in the opposite direction. It is well-known that the online setting is strictly harder than the batch setting, even for the simple one-dimensioanl class of threshold functions on the interval [0, 1]. Hence, we consider the online transductive model of Ben-David, Kushilevitz, and Mansour [2]. In this model, an arbitrary but unknown sequence of n examples (x1, y1), . . . , (xn, yn) ∈ X ×{−1, 1} is fixed in advance, for some instance space X. The set of unlabeled examples is then presented to the learner, Σ = {xi|1 ≤i ≤n}. The examples are then revealed, in an online manner, to the learner, for i = 1, 2, . . . , n. The learner observes example xi (along with all previous labeled examples (x1, y1), . . . , (xi−1, yi−1) and the unlabeled example set Σ) and must predict yi. The true label yi is then revealed to the learner. After this occurs, the learner compares its number of mistakes to the minimum number of mistakes of any of a target class F of functions f : X →{−1, 1} (such as linear threshold functions). Note that our results are in this type of agnostic model [7], where we allow for arbitrary labels, unlike the realizable setting, i.e., noiseless or PAC models, where it is assumed that the labels are consistent with some f ∈F. With this simple transductive knowledge of what unlabeled examples are to come, one can use existing expert algorithms to inefficiently learn any class of finite VC dimension, similar to the batch setting. How does one use unlabeled examples efficiently to guarantee good online performance? Our efficient algorithm A2 converts a proper1 batch algorithm to a proper online algorithm (both in the agnostic setting). At any point in time, it has observed some labeled examples. It then “hallucinates” random examples by taking some number of unlabeled examples and labeling them randomly. It appends these examples to those observed so far and predicts according to the batch algorithm that finds the hypothesis of minimum empirical error on the combined data. The idea of “hallucinating” and optimizing has been used for designing efficient online algorithms [6, 5, 1, 10, 4] in situations where exponential weighting schemes were inefficient. The hallucination analogy was suggested by Blum and Hartline [4]. In the context of transductive learning, it seems to be a natural way to try to use the unlabeled examples in conjunction with a batch learner. Let #mistakes(f, σn) denote the number of mistakes of a function f ∈F on a particular sequence σn ∈(X × {−1, 1})n, and #mistakes(A, σn) denote the same quantity for a transductive online learning algorithm A. Our main theorem is the following. Theorem 1. Let F be a class of functions f : X →{−1, 1} of VC dimension d. There is an efficient randomized transductive online algorithm that, for any n > 1 and σn ∈ (X × {−1, 1})n, E[#mistakes(A2, σn)] ≤minf∈F#mistakes(f, σn) + 2.5n3/4p d log n. The algorithm is computationally efficient in the sense that it runs in time poly(n), given an efficient proper batch learning algorithm. One should note that the bound on the error rate is the same as that of the best f ∈F plus O(n−1/4p d log(n)), approaching 0 at a rate related to the standard VC bound. It is well-known that, without regard to computational efficiency, the learnable classes of functions are exactly those with finite VC dimension. Consequently, the classes of functions learnable in the batch and transductive online settings are the same. The classes of functions properly learnable by computationally efficient algorithms in the proper batch and transductive online settings are identical, as well. In addition to the new algorithm, this is interesting because it helps justify a long line of work suggesting that whatever can be done in a batch setting can also be done online. Our result is surprising in light of earlier work by Blum showing that a slightly different online model is harder than its batch analog for computational reasons and not informationtheoretic reasons [3]. In Section 2, we define the transductive online model. In Section 3, we analyze the easier case of data that is realizable with respect to some function class, i.e., when there is some function of zero error in the class. In Section 4, we present and analyze the hallucination algorithm. In Section 5, we discuss open problems such as extending the results to improper learning and the efficient realizable case. 2 Models and definitions The transductive online model considered by Ben-David, Kushlevitz, and Mansour [2], consists of an instance space X and label set Y which we will always take to be binary Y = {−1, 1}. An arbitrary n > 0 and arbitrary sequence of labeled examples (x1, y1), . . . , (xn, yn) is fixed. One can think of these as being chosen by an adversary who knows the (possibly randomized) learning algorithm but not the realization of its random coin flips. For notational convenience, we define σi to be the subsequence of first i 1A proper learning algorithm is one that always outputs a hypothesis h ∈F. labeled examples, σi = (x1, y1), (x2, y2), . . . , (xi, yi), and Σ to be the set of all unlabeled examples in σn, Σ = {xi | i ∈{1, 2, . . . , n}}. A transductive online learner A is a function that takes as input n (the number of examples to be predicted), Σ ⊆X (the set of unlabeled examples, |Σ| ≤n), xi ∈Σ (the example to be tested), and σi−1 ∈(Σ × Y)i−1 (the previous i −1 labeled examples) and outputs a prediction ∈Y of yi, for any 1 ≤i ≤n. The number of mistakes of A on the sequence σn = (x1, y1), . . . , (xn, yn) is, #mistakes(A, σn) = |{i | A(n, Σ, xi, σi−1) ̸= yi}|. If A is computed by a randomized algorithm, then we similarly define E[#mistakes(A, σn)] where the expectation is taken over the random coin flips of A. In order to speak of the learnability of a set F of functions f : X →Y, we define #mistakes(f, σn) = |{i | f(xi) ̸= yi}|. Formally, paralleling agnostic learning [7],2 we define an efficient transductive online learner A for class F to be one for which the learning algorithm runs in time poly(n) and achieves, for any ϵ > 0, E[#mistakes(A, σn)] ≤minf∈F#mistakes(f, σn) + ϵn, for n =poly(1/ϵ).3 2.1 Proper learning Proper batch learning requires one to output a hypothesis h ∈F. An efficient proper batch learning algorithm for F is a batch learning algorithm B that, given any ϵ > 0, with n = poly(1/ϵ) many examples from any distribution D, outputs an h ∈F of expected error E[PrD[h(x) ̸= y]] ≤minf∈FPrD[f(x) ̸= y] + ϵ and runs in time poly(n). Observation 1. Any efficient proper batch learning algorithm B can be converted into an efficient empirical error minimizer M that, for any n, given any data set σn ∈(X × Y)n, outputs an f ∈F of minimal empirical error on σn. Proof. Running B only on σn, B is not guaranteed to output a hypothesis of minimum empirical error. Instead, we set an error tolerance of B to ϵ = 1/(4n), and give it examples drawn uniformly from the distribution D which is uniform over the data σn (a type of bootstrap). If B indeed returns a hypothesis h of error less than 1/n more than the best f ∈F, it must be a hypothesis of minimum empirical error on σn. By Markov’s inequality, with probability at most 1/4, the generalization error is more than 1/n. By repeating several times and take the best hypothesis, we get a success probability exponentially close to 1. The runtime is polynomial in n. To define proper learning in an online setting, it is helpful to think of the following alternative definition of transductive online learning. In this variation, the learner must output a sequence of hypotheses h1, h2, . . . , hn : X →{−1, 1}. After the ith hypothesis hi is output, the example (xi, yi) is revealed, and it is clear whether the learner made an error. Formally, the (possibly randomized) algorithm A′ still takes as input n, Σ, and σi−1 (but 2It is more common in online learning to bound the total number of mistakes of an online algorithm on an arbitrary sequence. We bound its error rate, as is usual for batch learning. 3The results in this paper could be replaced by high-probability 1 −δ bounds at a cost of log 1/δ. no longer xi), and outputs hi : X →{−1, 1} and errs if hi(xi) ̸= yi. To see that this model is equivalent to the previous definition, note that any algorithm A′ that outputs hypotheses hi can be used to make predictions hi(xi) on example i (it errs if hi(xi) ̸= yi). It is equally true but less obvious than any algorithm A in the previous model can be converted to an algorithm A′ in this model. This is because A′ can be viewed as outputting hi : X →{−1, 1}, where the function hi is defined by setting hi(x) equal to be the prediction of algorithm A on the sequence σi−1 followed by the example x, for each x ∈X, i.e., hi(x) = A(n, Σ, x, σi−1). (The same coins can be used if A and A′ are randomized.) A (possibly randomized) transductive online algorithm in this model is defined to be proper for family of functions F if it always outputs hi ∈F. 3 Warmup: the realizable case In this section, we consider the realizable special case in which there is some f ∈F which correctly labels all examples. In particular, this means that we only consider sequences σn for which there is an f ∈F with #mistakes(f, σn) = 0. This case will be helpful to analyze first as it is easier. Fix arbitrary n > 0 and Σ = {x1, x2, . . . , xn} ⊆X, |Σ| ≤n. Say there are at most L different ways to label the examples in Σ according to functions f ∈F, so 1 ≤L ≤2|Σ|. In the transductive online model, L is determined by Σ and F only. Hence, as long as prediction occurs only on examples x ∈Σ, there are effectively only L different functions in F that matter, and we can thus pick L such functions that give rise to the L different labelings. On the ith example, one could simply take majority vote of fj(xi) over consistent labelings fj (the so-called halving algorithm), and this would easily ensure at most log2(L) mistakes, because each mistake eliminates at least half of the consistent labelings. One can also use the following proper learning algorithm. Proper transductive online learning algorithm in the realizable case: • Preprocessing: Given the set of unlabeled examples Σ, take L functions f1, f2, . . . , fL ∈F that give rise to the L different labelings of x ∈Σ.4 • ith prediction: Output a uniformly random function f from the fj consistent with σi−1. The above algorithm, while possibly very inefficient, is easy to analyze. Theorem 2. Fix a class of binary functions F of VC dimension d. The above randomized proper learning algorithm makes an expected d log(n) mistakes on any sequence of examples of length n ≥2, provided that there is some mistake-free f ∈F. Proof. Let Vi be the number of labelings fj consistent with the first i examples, so that L = V0 ≥V1 ≥· · · ≥Vn ≥1 and L ≤nd, by Sauer’s lemma [11] for n ≥2, where d is the VC dimension of F. Observe that the number of consistent labelings that make a mistake on the ith example are exactly Vi−1 −Vi. Hence, the total expected number of mistakes is, n X i=1 Vi−1 −Vi Vi−1 ≤ n X i=1 µ 1 Vi−1 + 1 Vi−1 −1 + . . . 1 Vi + 1 ¶ ≤ Vn X i=2 1 i ≤log(L). 4More formally, take L functions with the following properties: for each pair 1 ≤j, k ≤L with j ̸= k, there exists x ∈Σ such that fj(x) ̸= fk(x), and for every f ∈F, there exists a 1 ≤j ≤L with f(x) = fj(x) for all x ∈Σ. Hence the above algorithm achieves an error rate of O(d log(n)/n), which quickly approaches zero for large n. Note that, this closely matches what one achieves in the batch setting. Like the batch setting, no better bounds can be given up to a constant factor. 4 General setting We now consider the more difficult unrealizable setting where we have an unconstrained sequence of examples (though we still work in a transductive setting). We begin by presenting an known (inefficnet) extension to the halving algorithm of the previous section, that works in the agnostic (unrealizable) setting that is similar to the previous algorithm. Inefficient proper transductive online learning algorithm A1: • Preprocessing: Given the set of unlabeled examples Σ, take L functions f1, f2, . . . , fL that give rise to the L different labelings of x ∈Σ. Assign an initial weight w1 = w2 = . . . = wL = 1 to each function. • Output fj, where 1 ≤j ≤L is chosen with probability wj w1+...+wL . • Update: for each j for which fj(xi) ̸= yi, reduce wj, wj := wj µ 1 − q log L n ¶ . Using an analysis very similar to that of Weighted Majority [9], one can show that, for any n > 1 and sequence of examples σn ∈(X × {−1, 1})n, E[#mistakes(A1, σn)] = minf∈F#mistakes(f, σn) + 2 p dn log n, where d is the VC dimension of F. Note the similarity to the standard VC bound. 4.1 Efficient algorithm We can only hope to get an efficient proper online algorithm when there is an efficient proper batch algorithm. As mentioned in section 2.1, this means that there is a batch algorithm M that, given any data set, efficiently finds a hypothesis h ∈F of minimum empirical error. (In fact, most proper learning algorithms work this way to begin with.) Using this, our efficient algorithm is as follows. Efficient transductive online learning algorithm A2: • Preprocessing: Given the set of unlabeled examples Σ, create a hallucinated data set τ as follows. 1. For each example x ∈Σ, choose integer rx uniformly at random such that −4√n ≤rx ≤ 4√n. 2. Add |rx| copies of the example x labeled by the sign of rx, (x, sgn(rx)), to τ. • To predict on xi: output hypothesis M(τσi−1) ∈F, where τσi−1 is the concatenation of the hallucinated examples and the observed labeled examples so far. The current algorithm predicts f(xi) based on f = M(τσi−1). We first begin by analyzing the hypothetical algorithm that used the function chosen on the next iteration, i.e. predict f(xi) based on f = M(τσi). (Of course, this is impossible to implement because we do not know σi when predicting f(xi).) Lemma 1. Fix any τ ∈(X × Y)∗and σn ∈(X × Y)n. Let A′ 2 be the algorithm that, for each i, predicts f(xi) based on f ∈F which is any empirical minimizer on the concatenated data τσi, i.e., f = M(τσi). Then the total number of mistakes of A′ 2 is, #mistakes(A′ 2, σn) ≤minf∈F#mistakes(f, τσn) −minf∈F#mistakes(f, τ). It is instructive to first consider the case where τ is empty, i.e., there are no hallucinated examples. Then, our algorithm that predicts according to M(σi−1) could be called “follow the leader,” as in [6]. The above lemma means that if one could use the hypothetical “be the leader” algorithm then one would make no more mistakes than the best f ∈F. The proof of this case is simple. Imagine starting with the offline algorithm that uses M(σn) on each example x1, . . . , xn. Now, on the first n −1 examples, replace the use of M(σn) by M(σn−1). Since M(σn−1) is an error-minimizer on σn−1, this can only reduce the number of mistakes. Next replace M(σn−1) by M(σn−2) on the first n −2 examples, and so on. Eventually, we reach the hypothetical algorithm above, and we have only decreased our number of mistakes. The proof of the above lemma follows along these lines. Proof of Lemma 1. Fix empirical minimizers gi on τσi for i = 0, 1, . . . , n, i.e., gi = M(τσi). For i ≥1, let mi be 1 if gi(xj) ̸= yj and 0 otherwise. We argue by induction on t that, #mistakes(g0, τ) + t X i=1 mi ≤#mistakes of gt on τσt. (1) For t = 0, the two are trivially equal. Assuming it holds for t, we have, #mistakes(g0, τ) + t+1 X i=1 mi ≤ #mistakes(gt, τσt) + mt+1 ≤ #mistakes(gt+1, τσt) + mt+1 = #mistakes(gt+1, τσt+1). The first inequality above holds by induction hypothesis, and the second follows from the fact that gt is an empirical minimizer of τσt. The equality establishes (1) for t + 1 and thus completes the induction. The total mistakes of the hypothetical algorithm proposed in the lemma is Pn i=1 mi, which gives the lemma by rearranging (1) for t = n. Lemma 2. For any σn, Eτ[minf∈F#mistakes(f, τσn)] ≤Eτ[|τ|/2] + minf∈F#mistakes(f, σn). For any F of VC dimension d, Eτ[minf∈F#mistakes(f, τ)] ≥Eτ[|τ|/2] −1.5n3/4p d log n. Proof. For the first part of the lemma, let g = M(σn) be an empirical minimizer on σn. Then, Eτ[minf∈F#mistakes(f, τσn)] ≤Eτ[#mistakes(g, τσn)] = Eτ[|τ|/2]+#mistakes(g, σn). The last inequality holds because, since each example in τ is equally likely to have a ± label, the expected number of mistakes of any fixed g ∈F on τ is E[|τ|/2]. Fix any f ∈F. For the second part of the lemma, observe that we can write the number of mistakes of f on τ as, #mistakes(f, τ) = |τ| −Pn i=1 f(xi)ri 2 . Hence it suffices to show that, maxf∈F Pn i=1 f(xi)ri ≤3n3/4p log(L). Now Eri[f(xi)ri] = 0 and |f(xi)ri| ≤n1/4. Next, Chernoff bounds (on the scaled random variables f(xi)rin−1/4) imply that, for any α ≤1, with probability at most e−nα2/2, Pn i=1 f(xi)rin−1/4 ≥nα. Put another way, for any β < n, with probability at most e−n−3/2β2/2, P f(xi)rin−1/4 ≥β. As observed before, we can reduce the problem to the L different labelings. In other words, we can assume that there are only L different functions. By the union bound, the probability that P f(xi)ri ≥β for any f ∈F is at most Le−n−3/2β2/2. Now the expectation of a non-negative random variable X is E[X] = R ∞ 0 Pr[X ≥x]dx. Let X = maxf∈F Pn i=1 f(xi)ri. In our case, E[X] ≤ p 2 log(L)n3/4 + Z ∞ √ 2 log(L)n3/4 Le−n−3/4x2/2dx By Mathematica, the above is at most p 2 log(L)n3/4 + 1.254n3/4 ≤3 p log(L)n3/4. Finally, we use the fact that L ≤nd by Sauer’s lemma. Unfortunately, we cannot use the algorithm A′ 2. However, due to the randomness we have added, we can argue that algorithm A2 is quite close: Lemma 3. For any σn, for any i, with probability at least 1 −n−1/4 over τ, M(τσi−1) is an empirical minimizer of τσi. Proof. Define, F+ = {f ∈F | f(xi) = 1} and F−= {f ∈F | f(xi) = −1}. WLOG, we may assume that F+ and F−are both nonempty. For if not, i.e., if all f ∈F predict the same sign f(xi), then the sets of empirical minimizers of τσi−1 and τσi are equal and the lemma holds trivially. For any sequence π ∈(X × Y)∗, define, s+(π) = minf∈F+#mistakes(f, π) and s−(π) = minf∈F−#mistakes(f, π). Next observe that, if s+(π) < s−(π) then M(π) ∈F+. Similarly if s−(π) < s+(π) then M(π) ∈F−. If they are equal then f(xi) can be an empirical minimizer in either. WLOG let us say that the ith example is (xi, 1), i.e., it is labeled positively. This implies that s+(τσi−1) = s+(τσi) and s−(τσi−1) = s−(τσi) + 1. It is now clear that if M(τσi−1) is not also an empirical minimizer of τσi then s+(τσi−1) = s−(τσi−1). Now the quantity ∆= s+(τσi−1)−s−(τσi−1) is directly related to rxi, the signed random number of times that example xi is hallucinated. If we fix σn and the random choices rx for each x ∈Σ \ {xi}, as we increase or decrease ri by 1, ∆correspondingly increases or decreases by 1. Since ri was chosen from a range of size 2⌊n1/4⌋+ 1 ≥n1/4, ∆= 0 with probability at most n−1/4. We are now ready to prove the main theorem. Proof of Theorem 1. Combining Lemmas 1 and 2, if on each period i, we used any minimizer of empirical error on the data τσi, we would have a total number of mistakes of at most minf∈F#mistakes(f, σn) + 1.5n3/4√d log n. Suppose A2 does end up using such a minimizer on all but p periods. Then, its total number of mistakes can only be p larger than this bound. By Lemma 3, the expected number p of periods i in which an empirical minimizer of τσi is not used is ≤n3/4. Hence, the expected total number of mistakes of A2 is at most, Eτ[#mistakes(A2, σn)] ≤minf∈F#mistakes(f, σn) + 1.5n3/4p d log n + n3/4. The above implies the theorem. Remark 1. The above algorithm is still costly in the sense that we must re-run the batch error minimizer for each prediction we would like to make. Using an idea quite similar to the “follow the lazy leader” algorithm in [6], we can achieve the same expected error while only needing to call M with probability n−1/4 on each example. Remark 2. The above analysis resembles previous analysis of hallucination algorithms. However, unlike previous analyses, there is no exponential distribution in the hallucination here yet the bounds still depend only logarithmically on the number of labelings. 5 Conclusions and open problems We have given an algorithm for learning in the transductive online setting and established several results between efficient proper batch and transductive online learnability. In the realizable case, however, we have not given a computationally efficient algorithm. Hence, it is an open question as to whether efficient learnability in the batch and transductive online settings are the same in the realizable case. In addition, our computationally efficient algorithm requires polynomially more examples than its inefficient counterpart. It would be nice to have the best of both worlds, namely a computationally efficient algorithm that achieves a number of mistakes that is at most O(√dn log n). Additionally, it would be nice to remove the restriction to proper algorithms. Acknowledgements. We would like to thank Maria-Florina Balcan, Dean Foster, John Langford, and David McAllester for helpful discussions. References [1] B. Awerbuch and R. Kleinberg. Adaptive routing with end-to-end feedback: Distributed learning and geometric approaches. In Proc. of the 36th ACM Symposium on Theory of Computing, 2004. [2] S. Ben-David, E. Kushilevitz, and Y. Mansour. Online learning versus offline learning. Machine Learning 29:45-63, 1997. [3] A. Blum. Separating Distribution-Free and Mistake-Bound Learning Models over the Boolean Domain. SIAM Journal on Computing 23(5): 990-1000, 1994. [4] A. Blum, J. Hartline. Near-Optimal Online Auctions. In Proceedings of the Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2005. [5] J. Hannan. Approximation to Bayes Risk in Repeated Plays. In M. Dresher, A. Tucker, and P. Wolfe editors, Contributions to the Theory of Games, Volume 3, p. 97-139, Princeton University Press, 1957. [6] A. Kalai and S. Vempala. Efficient algorithms for the online decision problem. In Proceedings of the 16th Conference on Computational Learning Theory, 2003. [7] M. Kearns, R. Schapire, and L. Sellie. Toward Efficient Agnostic Learning. Machine Learning, 17(2/3):115–141, 1994. [8] N. Littlestone. From On-Line to Batch Learning. In Proceedings of the 2nd Workshop on Computational Learning Theory, p. 269-284, 1989. [9] N. Littlestone and M. Warmuth. The Weighted Majority Algorithm. Information and Computation, 108:212-261, 1994. [10] H. Brendan McMahan and Avrim Blum. Online Geometric Optimization in the Bandit Setting Against an Adaptive Adversary. In Proceedings of the 17th Annual Conference on Learning Theory, COLT 2004. [11] N. Sauer. On the Densities of Families of Sets. Journal of Combinatorial Theory, Series A, 13, p 145-147, 1972. [12] V. N. Vapnik. Estimation of Dependencies Based on Empirical Data, New York: Springer Verlag, 1982. [13] V. N. Vapnik. Statistical Learning Theory, New York: Wiley Interscience, 1998.
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Generalized Nonnegative Matrix Approximations with Bregman Divergences Inderjit S. Dhillon Suvrit Sra Dept. of Computer Sciences The Univ. of Texas at Austin Austin, TX 78712. {inderjit,suvrit}@cs.utexas.edu Abstract Nonnegative matrix approximation (NNMA) is a recent technique for dimensionality reduction and data analysis that yields a parts based, sparse nonnegative representation for nonnegative input data. NNMA has found a wide variety of applications, including text analysis, document clustering, face/image recognition, language modeling, speech processing and many others. Despite these numerous applications, the algorithmic development for computing the NNMA factors has been relatively deficient. This paper makes algorithmic progress by modeling and solving (using multiplicative updates) new generalized NNMA problems that minimize Bregman divergences between the input matrix and its lowrank approximation. The multiplicative update formulae in the pioneering work by Lee and Seung [11] arise as a special case of our algorithms. In addition, the paper shows how to use penalty functions for incorporating constraints other than nonnegativity into the problem. Further, some interesting extensions to the use of “link” functions for modeling nonlinear relationships are also discussed. 1 Introduction Nonnegative matrix approximation (NNMA) is a method for dimensionality reduction and data analysis that has gained favor over the past few years. NNMA has previously been called positive matrix factorization [13] and nonnegative matrix factorization1 [12]. Assume that a1, . . . , aN are N nonnegative input (M-dimensional) vectors. We organize these vectors as the columns of a nonnegative data matrix A ≜ £ a1 a2 . . . aN ¤ . NNMA seeks a small set of K nonnegative representative vectors b1, . . . , bK that can be nonnegatively (or conically) combined to approximate the input vectors ai. That is, an ≈ K X k=1 cknbk, 1 ≤n ≤N, 1We use the word approximation instead of factorization to emphasize the inexactness of the process since, the input A is approximated by BC. where the combining coefficients ckn are restricted to be nonnegative. If ckn and bk are unrestricted, and we minimize P n ∥an −Bcn∥2, the Truncated Singular Value Decomposition (TSVD) of A yields the optimal bk and ckn values. If the bk are unrestricted, but the coefficient vectors cn are restricted to be indicator vectors, then we obtain the problem of hard-clustering (See [16, Chapter 8] for related discussion regarding different constraints on cn and bk). In this paper we consider problems where all involved matrices are nonnegative. For many practical problems nonnegativity is a natural requirement. For example, color intensities, chemical concentrations, frequency counts etc., are all nonnegative entities, and approximating their measurements by nonnegative representations leads to greater interpretability. NNMA has found a significant number of applications, not only due to increased interpretability, but also because admitting only nonnegative combinations of the bk leads to sparse representations. This paper contributes to the algorithmic advancement of NNMA by generalizing the problem significantly, and by deriving efficient algorithms based on multiplicative updates for the generalized problems. The scope of this paper is primarily on generic methods for NNMA, rather than on specific applications. The multiplicative update formulae in the pioneering work by Lee and Seung [11] arise as a special case of our algorithms, which seek to minimize Bregman divergences between the nonnegative input A and its approximation. In addition, we discuss the use penalty functions for incorporating constraints other than nonnegativity into the problem. Further, we illustrate an interesting extension of our algorithms for handling non-linear relationships through the use of “link” functions. 2 Problems Given a nonnegative matrix A as input, the classical NNMA problem is to approximate it by a lower rank nonnegative matrix of the form BC, where B = [b1, ..., bK] and C = [c1, ..., cN] are themselves nonnegative. That is, we seek the approximation, A ≈BC, where B, C ≥0. (2.1) We judge the goodness of the approximation in (2.1) by using a general class of distortion measures called Bregman divergences. For any strictly convex function ϕ : S ⊆R →R that has a continuous first derivative, the corresponding Bregman divergence Dϕ : S × int(S) →R+ is defined as Dϕ(x, y) ≜ϕ(x) −ϕ(y) −∇ϕ(y)(x −y), where int(S) is the interior of set S [1, 2]. Bregman divergences are nonnegative, convex in the first argument and zero if and only if x = y. These divergences play an important role in convex optimization [2]. For the sequel we consider only separable Bregman divergences, i.e., Dϕ(X, Y ) = P ij Dϕ(xij, yij). We further require xij, yij ∈domϕ ∩R+. Formally, the resulting generalized nonnegative matrix approximation problems are: min B, C≥0 Dϕ(BC, A) + α(B) + β(C), (2.2) min B, C≥0 Dϕ(A, BC) + α(B) + β(C). (2.3) The functions α and β serve as penalty functions, and they allow us to enforce regularization (or other constraints) on B and C. We consider both (2.2) and (2.3) since Bregman divergences are generally asymmetric. Table 1 gives a small sample of NNMA problems to illustrate the breadth of our formulation. 3 Algorithms In this section we present algorithms that seek to optimize (2.2) and (2.3). Our algorithms are iterative in nature, and are directly inspired by the efficient algorithms of Lee and Seung [11]. Appealing properties include ease of implementation and computational efficiency. Divergence Dϕ ϕ α β Remarks ∥A −BC∥2 F 1 2x2 0 0 Lee and Seung [11, 12] ∥A −BC∥2 F 1 2x2 0 λ1T C1 Hoyer [10] ∥W ⊙(A −BC)∥2 F 1 2x2 0 0 Paatero and Tapper [13] KL(A, BC) x log x 0 0 Lee and Seung [11] KL(A, W BC) x log x 0 0 Guillamet et al. [9] KL(A, BC) x log x c1BT B1 −c′∥C∥2 F Feng et al. [8] Dϕ(A, W1BCW2) ϕ(x) α(B) β(C) Weighted NNMA (new) Table 1: Some example NNMA problems that may be obtained from (2.3). The corresponding asymmetric problem (2.2) has not been previously treated in the literature. KL(x, y) denotes the generalized KL-Divergence = P i xi log xi yi −xi +yi (also called I-divergence). Note that the problems (2.2) and (2.3) are not jointly convex in B and C, so it is not easy to obtain globally optimal solutions in polynomial time. Our iterative procedures start by initializing B and C randomly or otherwise. Then, B and C are alternately updated until there is no further appreciable change in the objective function value. 3.1 Algorithms for (2.2) We utilize the concept of auxiliary functions [11] for our derivations. It is sufficient to illustrate our methods using a single column of C (or row of B), since our divergences are separable. Definition 3.1 (Auxiliary function). A function G(c, c′) is called an auxiliary function for F(c) if: 1. G(c, c) = F(c), and 2. G(c, c′) ≥F(c) for all c′. Auxiliary functions turn out to be useful due to the following lemma. Lemma 3.2 (Iterative minimization). If G(c, c′) is an auxiliary function for F(c), then F is non-increasing under the update ct+1 = argminc G(c, ct). Proof. F(ct+1) ≤G(ct+1, ct) ≤G(ct, ct) = F(ct). As can be observed, the sequence formed by the iterative application of Lemma 3.2 leads to a monotonic decrease in the objective function value F(c). For an algorithm that iteratively updates c in its quest to minimize F(c), the method for proving convergence boils down to the construction of an appropriate auxiliary function. Auxiliary functions have been used in many places before, see for example [5, 11]. We now construct simple auxiliary functions for (2.2) that yield multiplicative updates. To avoid clutter we drop the functions α and β from (2.2), noting that our methods can easily be extended to incorporate these functions. Suppose B is fixed and we wish to compute an updated column of C. We wish to minimize F(c) = Dϕ(Bc, a), (3.1) where a is the column of A corresponding to the column c of C. The lemma below shows how to construct an auxiliary function for (3.1). For convenience of notation we use ψ to denote ∇ϕ for the rest of this section. Lemma 3.3 (Auxiliary function). The function G(c, c′) = X ij λijϕ µbijcj λij ¶ − µX i ϕ(ai) + ψ(ai) ¡ (Bc)i −ai ¢¶ , (3.2) with λij = (bijc′ j)/(P l bilc′ l), is an auxiliary function for (3.1). Note that by definition P j λij = 1, and as both bij and c′ j are nonnegative, λij ≥0. Proof. It is easy to verify that G(c, c) = F(c), since P j λij = 1. Using the convexity of ϕ, we conclude that if P j λij = 1 and λij ≥0, then F(c) = X i ϕ µX j bijcj ¶ −ϕ(ai) −ψ(ai) ¡ (Bc)i −ai ¢ ≤ X ij λijϕ µbijcj λij ¶ − µX i ϕ(ai) + ψ(ai) ¡ (Bc)i −ai ¢¶ = G(c, c′). To obtain the update, we minimize G(c, c′) w.r.t. c. Let ψ(x) denote the vector [ψ(x1), . . . , ψ(xn)]T . We compute the partial derivative ∂G ∂cp = X i λipψ µbipcp λip ¶ bip λip − X i bipψ(ai) = X i bipψ µcp c′p (Bc′)i ¶ −(BT ψ(a))p. (3.3) We need to solve (3.3) for cp by setting ∂G/∂cp = 0. Solving this equation analytically is not always possible. However, for a broad class of functions, we can obtain an analytic solution. For example, if ψ is multiplicative (i.e., ψ(xy) = ψ(x)ψ(y)) we obtain the following iterative update relations for b and c (see [7]) bp ← bp · ψ−1³ [ψ(aT )CT ]p [ψ(bT C)CT ]p ´ , (3.4) cp ← cp · ψ−1³ [BT ψ(a)]p [BT ψ(Bc)]p ´ . (3.5) It turns out that when ϕ is a convex function of Legendre type, then ψ−1 can be obtained by the derivative of the conjugate function ϕ∗of ϕ, i.e., ψ−1 = ∇ϕ∗[14]. Note. (3.4) & (3.5) coincide with updates derived by Lee and Seung [11], if ϕ(x) = 1 2x2. 3.1.1 Examples of New NNMA Problems We illustrate the power of our generic auxiliary functions given above for deriving algorithms with multiplicative updates for some specific interesting problems. First we consider the problem that seeks to minimize the divergence, KL(Bc, a) = X i (Bc)i log (Bc)i ai −(Bc)i + ai, B, c ≥0. (3.6) Let ϕ(x) = x log x −x. Then, ψ(x) = log x, and as ψ(xy) = ψ(x) + ψ(y), upon substituting in (3.3), and setting the resultant to zero we obtain ∂G ∂cp = X i bip log(cp(Bc′)i/c′ p) − X i bip log ai = 0, =⇒(BT 1)p log cp c′p = [BT log a −BT log(Bc′)]p =⇒cp = c′ p · exp à [BT log ¡ a/(Bc′) ¢ ]p [BT 1]p ! . The update for b can be derived similarly. Constrained NNMA. Next we consider NNMA problems that have additional constraints. We illustrate our ideas on a problem with linear constraints. min x Dϕ(Bc, a) s.t. P c ≤0, c ≥0. (3.7) We can solve (3.7) problem using our method by making use of an appropriate (differentiable) penalty function that enforces P c ≤0. We consider, F(c) = Dϕ(Bc, a) + ρ∥max(0, P c)∥2, (3.8) where ρ > 0 is some penalty constant. Assuming multiplicative ψ and following the auxiliary function technique described above, we obtain the following updates for c, ck ←ck · ψ−1 µ[BT ψ(a)]k −ρ[P T (P c)+]k [BT ψ(Bc)]k ¶ , where (P c)+ = max(0, P c). Note that care must be taken to ensure that the addition of this penalty term does not violate the nonnegativity of c, and to ensure that the argument of ψ−1 lies in its domain. Remarks. Incorporating additional constraints into (3.6) is however easier, since the exponential updates ensure nonnegativity. Given a = 1, with appropriate penalty functions, our solution to (3.6) can be utilized for maximizing entropy of Bc subject to linear or non-linear constraints on c. Nonlinear models with “link” functions. If A ≈h(BC), where h is a “link” function that models a nonlinear relationship between A and the approximant BC, we may wish to minimize Dϕ(h(BC), A). We can easily extend our methods to handle this case for appropriate h. Recall that the auxiliary function that we used, depended upon the convexity of ϕ. Thus, if (ϕ◦h) is a convex function, whose derivative ∇(ϕ◦h) is “factorizable,” then we can easily derive algorithms for this problem with link functions. We exclude explicit examples for lack of space and refer the reader to [7] for further details. 3.2 Algorithms using KKT conditions We now derive efficient multiplicative update relations for (2.3), and these updates turn out to be simpler than those for (2.2). To avoid clutter, we describe our methods with α ≡0, and β ≡0, noting that if α and β are differentiable, then it is easy to incorporate them in our derivations. For convenience we use ζ(x) to denote ∇2(x) for the rest of this section. Using matrix algebra, one can show that the gradients of Dϕ(A, BC) w.r.t. B and C are, ∇BDϕ(A, BC) = ¡ ζ(BC) ⊙(BC −A) ¢ CT ∇CDϕ(A, BC) =BT ¡ ζ(BC) ⊙(BC −A) ¢ , where ⊙denotes the elementwise or Hadamard product, and ζ is applied elementwise to BC. According to the KKT conditions, there exist Lagrange multiplier matrices Λ ≥0 and Ω≥0 such that ∇BDϕ(A, BC) = Λ, ∇CDϕ(A, BC) = Ω, (3.9a) λmkbmk = ωknckn = 0. (3.9b) Writing out the gradient ∇BDϕ(A, BC) elementwise, multiplying by bmk, and making use of (3.9a,b), we obtain £¡ ζ(BC) ⊙(BC −A) ¢ CT ¤ mkbmk = λmkbmk = 0, which suggests the iterative scheme bmk ←bmk £¡ ζ(BC) ⊙A ¢ CT ¤ mk £¡ ζ(BC) ⊙BC ¢ CT ¤ mk . (3.10) Proceeding in a similar fashion we obtain a similar iterative formula for ckn, which is ckn ←ckn [BT ¡ ζ(BC) ⊙A ¢ ]kn [BT ¡ ζ(BC) ⊙BC ¢ ]kn . (3.11) 3.2.1 Examples of New and Old NNMA Problems as Special Cases We now illustrate the power of our approach by showing how one can easily obtain iterative update relations for many NNMA problems, including known and new problems. For more examples and further generalizations we refer the reader to [7]. Lee and Seung’s Algorithms. Let α ≡0, β ≡0. Now if we set ϕ(x) = 1 2x2 or ϕ(x) = x log x, then (3.10) and (3.11) reduce to the Frobenius norm and KL-Divergence update rules originally derived by Lee and Seung [11]. Elementwise weighted distortion. Here we wish to minimize ∥W ⊙(A−BC)∥2 F . Using X ← √ W ⊙X, and A ← √ W ⊙A in (3.10) and (3.11) one obtains B ←B ⊙ (W ⊙A)CT (W ⊙(BC))CT , C ←C ⊙ BT (W ⊙A) BT (W ⊙(BC)). These iterative updates are significantly simpler than the PMF algorithms of [13]. The Multifactor NNMA Problem (new). The above ideas can be extended to the multifactor NNMA problem that seeks to minimize the following divergence (see [7]) Dϕ(A, B1B2 . . . BR), where all matrices involved are nonnegative. A typical usage of multifactor NNMA problem would be to obtain a three-factor NNMA, namely A ≈RBC. Such an approximation is closely tied to the problem of co-clustering [3], and can be used to produce relaxed coclustering solutions [7]. Weighted NNMA Problem (new). We can follow the same derivation method as above (based on KKT conditions) for obtaining multiplicative updates for the weighted NNMA problem: min Dϕ(A, W1BCW2), where W1 and W2 are nonnegative (and nonsingular) weight matrices. The work of [9] is a special case as mentioned in Table 1. Please refer to [7] for more details. 4 Experiments and Discussion We have looked at generic algorithms for minimizing Bregman divergences between the input and its approximation. One important question arises: Which Bregman divergence should one use for a given problem? Consider the following factor analytic model A = BC + N, where N represents some additive noise present in the measurements A, and the aim is to recover B and C. If we assume that the noise is distributed according to some member of the exponential family, then minimizing the corresponding Bregman divergence [1] is appropriate. For e.g., if the noise is modeled as i.i.d. Gaussian noise, then the Frobenius norm based problem is natural. Another question is: Which version of the problem we should use, (2.2) or (2.3)? For ϕ(x) = 1 2x2, both problems coincide. For other ϕ, the choice between (2.2) and (2.3) can be guided by computation issues or sparsity patterns of A. Clearly, further work is needed for answering this question in more detail. Some other open problems involve looking at the class of minimization problems to which the iterative methods of Section 3.2 may be applied. For example, determining the class of functions h, for which these methods may be used to minimize Dϕ(A, h(BC)). Other possible methods for solving both (2.2) and (2.3), such as the use of alternating projections (AP) for NNMA, also merit a study. Our methods for (2.2) decreased the objective function monotonically (by construction). However, we did not demonstrate such a guarantee for the updates (3.10) & (3.11). Figure 1 offers encouraging empirical evidence in favor of a monotonic behavior of these updates. It is still an open problem to formally prove this monotonic decrease. Preliminary results that yield new monotonicity proofs for the Frobenius norm and KL-divergence NNMA problems may be found in [7]. PMF Objective ϕ(x) = −log x ϕ(x) = x log x −x 0 10 20 30 40 50 60 70 80 90 100 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 Number of iterations Objective function value 0 10 20 30 40 50 60 70 80 90 100 8 10 12 14 16 18 20 22 24 26 28 Number of iterations Objective function value 0 10 20 30 40 50 60 70 80 90 100 11 12 13 14 15 16 17 18 19 Number of iterations Objective function value Figure 1: Objective function values over 100 iterations for different NNMA problems. The input matrix A was random 20×8 nonnegative matrix. Matrices B and C were 20×4, 4×8, respectively. NNMA has been used in a large number of applications, a fact that attests to its importance and appeal. We believe that special cases of our generalized problems will prove to be useful for applications in data mining and machine learning. 5 Related Work Paatero and Tapper [13] introduced NNMA as positive matrix factorization, and they aimed to minimize ∥W ⊙(A −BC)∥F, where W was a fixed nonnegative matrix of weights. NNMA remained confined to applications in Environmetrics and Chemometrics before pioneering papers of Lee and Seung [11, 12] popularized the problem. Lee and Seung [11] provided simple and efficient algorithms for the NNMA problems that sought to minimize ∥A −BC∥F and KL(A, BC). Lee & Seung called these problems nonnegative matrix factorization (NNMF), and their algorithms have inspired our generalizations. NNMA was applied to a host of applications including text analysis, face/image recognition, language modeling, and speech processing amongst others. We refer the reader to [7] for pointers to the literature on various applications of NNMA. Srebro and Jaakola [15] discuss elementwise weighted low-rank approximations without any nonnegativity constraints. Collins et al. [6] discuss algorithms for obtaining a low rank approximation of the form A ≈BC, where the loss functions are Bregman divergences, however, there is no restriction on B and C. More recently, Cichocki et al. [4] presented schemes for NNMA with Csisz´ar’s ϕ-divergeneces, though rigorous convergence proofs seem to be unavailable. Our approach of Section 3.2 also yields heuristic methods for minimizing Csisz´ar’s divergences. Acknowledgments This research was supported by NSF grant CCF-0431257, NSF Career Award ACI0093404, and NSF-ITR award IIS-0325116. References [1] A. Banerjee, S. Merugu, I. S. Dhillon, and J. Ghosh. Clustering with Bregman Divergences. In SIAM International Conf. on Data Mining, Lake Buena Vista, Florida, April 2004. SIAM. [2] Y. Censor and S. A. Zenios. Parallel Optimization: Theory, Algorithms, and Applications. Numerical Mathematics and Scientific Computation. Oxford University Press, 1997. [3] H. Cho, I. S. Dhillon, Y. Guan, and S. Sra. Minimum Sum Squared Residue based Co-clustering of Gene Expression data. In Proc. 4th SIAM International Conference on Data Mining (SDM), pages 114–125, Florida, 2004. SIAM. [4] A. Cichocki, R. Zdunek, and S. Amari. Csisz´ar’s Divergences for Non-Negative Matrix Factorization: Family of New Algorithms. In 6th Int. Conf. ICA & BSS, USA, March 2006. [5] M. Collins, R. Schapire, and Y. Singer. Logistic regression, adaBoost, and Bregman distances. In Thirteenth annual conference on COLT, 2000. [6] M. Collins, S. Dasgupta, and R. E. Schapire. A Generalization of Principal Components Analysis to the Exponential Family. In NIPS 2001, 2001. [7] I. S. Dhillon and S. Sra. Generalized nonnegative matrix approximations. Technical report, Computer Sciences, University of Texas at Austin, 2005. [8] T. Feng, S. Z. Li, H-Y. Shum, and H. Zhang. Local nonnegative matrix factorization as a visual representation. In Proceedings of the 2nd International Conference on Development and Learning, pages 178–193, Cambridge, MA, June 2002. [9] D. Guillamet, M. Bressan, and J. Vitri`a. A weighted nonnegative matrix factorization for local representations. In CVPR. IEEE, 2001. [10] P. O. Hoyer. Non-negative sparse coding. In Proc. IEEE Workshop on Neural Networks for Signal Processing, pages 557–565, 2002. [11] D. D. Lee and H. S. Seung. Algorithms for nonnegative matrix factorization. In NIPS, pages 556–562, 2000. [12] D. D. Lee and H. S. Seung. Learning the parts of objects by nonnegative matrix factorization. Nature, 401:788–791, October 1999. [13] P. Paatero and U. Tapper. Positive matrix factorization: A nonnegative factor model with optimal utilization of error estimates of data values. Environmetrics, 5(111–126), 1994. [14] R. T. Rockafellar. Convex Analysis. Princeton Univ. Press, 1970. [15] N. Srebro and T. Jaakola. Weighted low-rank approximations. In Proc. of 20th ICML, 2003. [16] J. A. Tropp. Topics in Sparse Approximation. PhD thesis, The Univ. of Texas at Austin, 2004.
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Predicting EMG Data from M1 Neurons with Variational Bayesian Least Squares Jo-Anne Ting1, Aaron D’Souza1 Kenji Yamamoto3, Toshinori Yoshioka2 , Donna Hoffman3 Shinji Kakei4, Lauren Sergio6, John Kalaska5 Mitsuo Kawato2, Peter Strick3, Stefan Schaal1,2 1Comp. Science & Neuroscience, U.of S. California, Los Angeles, CA 90089, USA 2ATR Computational Neuroscience Laboratories, Kyoto 619-0288, Japan 3University of Pittsburgh, Pittsburgh, PA 15261, USA 4Tokyo Metropolitan Institute for Neuroscience, Tokyo 183-8526, Japan 5University of Montreal, Montreal, Canada H3C-3J7 6York University, Toronto, Ontario, Canada M3J1P3 Abstract An increasing number of projects in neuroscience requires the statistical analysis of high dimensional data sets, as, for instance, in predicting behavior from neural firing or in operating artificial devices from brain recordings in brain-machine interfaces. Linear analysis techniques remain prevalent in such cases, but classical linear regression approaches are often numerically too fragile in high dimensions. In this paper, we address the question of whether EMG data collected from arm movements of monkeys can be faithfully reconstructed with linear approaches from neural activity in primary motor cortex (M1). To achieve robust data analysis, we develop a full Bayesian approach to linear regression that automatically detects and excludes irrelevant features in the data, regularizing against overfitting. In comparison with ordinary least squares, stepwise regression, partial least squares, LASSO regression and a brute force combinatorial search for the most predictive input features in the data, we demonstrate that the new Bayesian method offers a superior mixture of characteristics in terms of regularization against overfitting, computational efficiency and ease of use, demonstrating its potential as a drop-in replacement for other linear regression techniques. As neuroscientific results, our analyses demonstrate that EMG data can be well predicted from M1 neurons, further opening the path for possible real-time interfaces between brains and machines. 1 Introduction In recent years, there has been growing interest in large scale analyses of brain activity with respect to associated behavioral variables. For instance, projects can be found in the area of brain-machine interfaces, where neural firing is directly used to control an artificial system like a robot [1, 2], to control a cursor on a computer screen via non-invasive brain signals [3] or to classify visual stimuli presented to a subject [4, 5]. In these projects, the brain signals to be processed are typically high dimensional, on the order of hundreds or thousands of inputs, with large numbers of redundant and irrelevant signals. Linear modeling techniques like linear regression are among the primary analysis tools [6, 7] for such data. However, the computational problem of data analysis involves not only data fitting, but requires that the model extracted from the data has good generalization properties. This is crucial for predicting behavior from future neural recordings, e.g., for continual online interpretation of brain activity to control prosthetic devices or for longitudinal scientific studies of information processing in the brain. Surprisingly, robust linear modeling of high dimensional data is non-trivial as the danger of fitting noise and encountering numerical problems is high. Classical techniques like ridge regression, stepwise regression or partial least squares regression are known to be prone to overfitting and require careful human supervision to ensure useful results. In this paper, we will focus on how to improve linear data analysis for the high dimensional scenarios described above, with a view towards developing a statistically robust “black box” approach that automatically detects the most relevant input dimensions for generalization and excludes other dimensions in a statistically sound way. For this purpose, we investigate a full Bayesian treatment of linear regression with automatic relevance detection [8]. Such an algorithm, called Variational Bayesian Least Squares (VBLS), can be formulated in closed form with the help of a variational Bayesian approximation and turns out to be computationally highly efficient. We apply VBLS to the reconstruction of EMG data from motor cortical firing, using data sets collected by [9] and [10, 11]. This data analysis addresses important neuroscientific questions in terms of whether M1 neurons can directly predict EMG traces [12], whether M1 has a muscle-based topological organization and whether information in M1 should be used to predict behavior in future brainmachine interfaces. Our main focus in this paper, however, will be on the robust statistical analysis of these kinds of data. Comparisons with classical linear analysis techniques and a brute force combinatorial model search on a cluster computer demonstrate that our VBLS algorithm achieves the “black box” quality of a robust statistical analysis technique without any tunable parameters. In the following sections, we will first sketch the derivation of Variational Bayesian Least Squares and subsequently perform extensive comparative data analysis of this technique in the context of prediction EMG data from M1 neural firing. 2 High Dimensional Regression Before developing our VBLS algorithm, let us briefly revisit classical linear regression techniques. The standard model for linear regression is: y = d X m=1 bmxm + ϵ (1) where b is the regression vector composed of bm components, d is the number of input dimensions, ϵ is additive mean-zero noise, x are the inputs and y are the outputs. The Ordinary Least Squares (OLS) estimate of the regression vector is b = XT X −1 XT y. The main problem with OLS regression in high dimensional input spaces is that the full rank assumption of XT X −1 is often violated due to underconstrained data sets. Ridge regression can “fix” such problems numerically, but introduces uncontrolled bias. Additionally, if the input dimensionality exceeds around 1000 dimensions, the matrix inversion can become prohibitively computationally expensive. Several ideas exist how to improve over OLS. First, stepwise regression [13] can be employed. However, it has been strongly criticized for its potential for overfitting and its inconsistency in the presence of collinearity in the input data [14]. To xi1 xid yi i=1..N (a) Linear regression b1 bd zid xi1 xid yi zi1 i=1..N (b) Probabilistic backfitting α1 αd b1 bd zid xi1 xid yi zi1 i=1..N (c) VBLS Figure 1: Graphical Models for Linear Regression. Random variables are in circular nodes, observed random variables are in double circles and point estimated parameters are in square nodes. deal with such collinearity directly, dimensionality reduction techniques like Principal Components Regression (PCR) and Factor Regression (FR) [15] are useful. These methods retain components in input space with large variance, regardless of whether these components influence the prediction [16], and can even eliminate low variance inputs that may have high predictive power for the outputs [17]. Another class of linear regression methods are projection regression techniques, most notably Partial Least Squares Regression (PLS) [18]. PLS performs computationally inexpensive O(d) univariate regressions along projection directions, chosen according to the correlation between inputs and outputs. While slightly heuristic in nature, PLS is a surprisingly successful algorithm for ill-conditioned and high-dimensional regression problems, although it also has a tendency towards overfitting [16]. LASSO (Least Absolute Shrinkage and Selection Operator) regression [19] shrinks certain regression coefficients to 0, giving interpretable models that are sparse. However, a tuning parameter needs to be set, which can be done using n-fold cross-validation or manual hand-tuning. Finally, there are also more efficient methods for matrix inversion [20, 21], which, however, assume a well-condition regression problem a priori and degrade in the presence of collinearities in inputs. In the following section, we develop a linear regression algorithm in a Bayesian framework that automatically regularizes against problems of overfitting. Moreover, the iterative nature of the algorithm, due to its formulation as an ExpectationMaximization problem [22], avoids the computational cost and numerical problems of matrix inversions. Thus, it addresses the two major problems of high-dimensional OLS simultaneously. Conceptually, the algorithm can be interpreted as a Bayesian version of either backfitting or partial least squares regression. 3 Variational Bayesian Least Squares Figure 1 illustrates the progression of graphical models that we need in order to develop a robust Bayesian version of linear regression. Figure 1a depicts the standard linear regression model. In the spirit of PLS, if we knew an optimal projection direction of the input data, then the entire regression problem could be solved by a univariate regression between the projected data and the outputs. This optimal projection direction is simply the true gradient between inputs and outputs. In the tradition of EM algorithms [22], we encode this projection direction as a hidden variable, as shown in Figure 1b. The unobservable variables zim (where i = 1..N denotes the index into the data set of N data points) are the results of each input being multiplied with its corresponding component of the projection vector (i.e. bm). Then, the zim are summed up to form a predicted output yi. More formally, the linear regression model in Eq. (1) is modified to become: zim = bmxim yi = d X m=1 zim + ϵ For a probabilistic treatment with EM, we make a standard normal assumption of all distributions in form of: yi|zi ∼Normal “ yi; 1T zi, ψy ” zim|xi ∼Normal (zim; bmxim, ψzm) where 1 = [1, 1, .., 1]T . While this model is still identical to OLS, notice that in the graphical model, the regression coefficients bm are behind the fan-in to the outputs yi. Given the data D = {xi, yi}N i=1, we can view this new regression model as an EM problem and maximize the incomplete log likelihood log p(y|X) by maximizing the expected complete log likelihood ⟨log p(y, Z|X)⟩: log p(y, Z|X) = −N 2 log ψy − 1 2ψy PN i=1 ` yi −1T zi ´2 −N 2 Pd m=1 log ψzm −Pd m=1 1 2ψzm (zim −bmxim)2 + const (2) where Z denotes the N by d matrix of all zim. The resulting EM updates require standard manipulations of normal distributions and result in: M-step : E-step : bm = PN i=1⟨zim⟩xim PN i=1 x2 im 1T Σz1 = “Pd m=1 ψzm ” h 1 −1 s “Pd m=1 ψzm ”i ψy = 1 N PN i=1 ` yi −1T ⟨zi⟩ ´ 2 + 1T Σz1 σ2 zm = ψzm ` 1 −1 sψzm ´ ψzm = 1 N PN i=1 (⟨zim⟩−bmxim)2 + σ2 zm ⟨zim⟩= bmxi + 1 sψxm ` yi −bT xi ´ where we define s = ψy + Pd m=1 ψxm and Σz = Cov(z|y, X). It is very important to note that one EM update has a computationally complexity of O(d), where d is the number of input dimensions, instead of the O(d3) associated with OLS regression. This efficiency comes at the cost of an iterative solution, instead of a one-shot solution for b as in OLS. It can be proved that this EM version of least squares regression is guaranteed to converge to the same solution as OLS [23]. This new EM algorithm appears to only replace the matrix inversion in OLS by an iterative method, as others have done with alternative algorithms [20, 21], although the convergence guarantees of EM are an improvement over previous approaches. The true power of this probabilistic formulation, though, becomes apparent when we add a Bayesian layer that achieves the desired robustness in face of ill-conditioned data. 3.1 Automatic Relevance Determination From a Bayesian point of view, the parameters bm should be treated probabilistically so that we can integrate them out to safeguard against overfitting. For this purpose, as shown in Figure 1c, we introduce precision variables αm over each regression parameter bm: p(b|α) = Qd m=1 ` αm 2π ´ 1 2 exp ˘ −αm 2 b2 m ¯ p(α) = Qd m=1 baα α Gamma(aα)α(aα−1) m exp {−bααm} (3) where α is the vector of all αm. In order to obtain a tractable posterior distribution over all hidden variables b, zim and α, we use a factorial variational approximation of the true posterior Q(α, b, Z) = Q(α, b)Q(Z). Note that the connection from the αm to the corresponding zim in Figure 1c is an intentional design. Under this graphical model, the marginal distribution of bm becomes a Student t-distribution that allows traditional hypothesis testing [24]. The minimal factorization of the posterior into Q(α, b)Q(Z) would not be possible without this special design. The resulting augmented model has the following distributions: yi|zi ∼N(yi; 1T zi, ψy) bm|αm ∼N(wbm; 0, 1/αm) zim|bm, αmxim ∼N(zim; bmxim, ψzm/αm) αm ∼Gamma(αm; aα, bα) We now have a mechanism that infers the significance of each dimension’s contribution to the observed output y. Since bm is zero mean, a very large αm (equivalent to a very small variance of bm) suggests that bm is very close to 0 and has no contribution to the output. An EM-like algorithm [25] can be used to find the posterior updates of all distributions. We omit the EM update equations due to space constraints as they are similar to the EM update above and only focus on the posterior update for bm and α: σ2 bm|αm = ψzm αm “PN i=1 x2 im + ψzm ”−1 ⟨bm|αm⟩= “PN i=1 x2 im + ψzm ”−1 “PN i=1 ⟨zim⟩xim ” ˆaα = aα + N 2 ˆb(m) α = bα + 1 2ψzm jPN i=1 ˙ z2 im ¸ − “PN i=1 x2 im + ψzm ”−1 “PN i=1 ⟨zim⟩xim ”2ff (4) Note that the update equation for ⟨bm|αm⟩can be rewritten as: ⟨bm|αm⟩(n+1) = “ PN i=1 x2 im PN i=1 x2 im+ψzm ” ⟨bm|αm⟩(n) + ψzm sαm PN i=1(yi−⟨b|α⟩(n)T xi)xim PN i=1 x2 im+ψzm (5) Eq. (5) demonstrates that in the absence of a correlation between the current input dimension and the residual error, the first term causes the current regression coefficient to decay. The resulting regression solution regularizes over the number of retained inputs in the final regression vector, performing a functionality similar to Automatic Relevance Determination (ARD) [8]. The update equations’ algorithmic complexity remains O(d). One can further show that the marginal distribution of all bm is a t-distribution with t = ⟨bm|αm⟩/σbm|αm and 2ˆaα degrees of freedom, which allows a principled way of determining whether a regression coefficient was excluded by means of standard hypothesis testing. Thus, Variational Bayesian Least Squares (VBLS) regression is a full Bayesian treatment of the linear regression problem. 4 Evaluation We now turn to the application and evaluation of VBLS in the context of predicting EMG data from neural data recorded in M1 of monkeys. The key questions addressed in this application were i) whether EMG data can be reconstructed accurately with good generalization, ii) how many neurons contribute to the reconstruction of each muscle and iii) how well the VBLS algorithm compares to other analysis techniques. The underlying assumption of this analysis is that the relationship between neural firing and muscle activity is approximately linear. 4.1 Data sets We investigated data from two different experiments. In the first experiment by Sergio & Kalaska [9], the monkey moved a manipulandum in a center-out task in eight different directions, equally spaced in a horizontal planar circle of 8cm radius. A variation of this experiment held the manipulandum rigidly in place, while the monkey applied isometric forces in the same eight directions. In both conditions, movement or force, feedback was given through visual display on a monitor. Neural activity for 71 M1 neurons was recorded in all conditions (2400 data points for each neuron), along with the EMG outputs of 11 muscles. The second experiment by Kakei et al. [10] involved a monkey trained to perform eight different combinations of wrist flexion-extension and radial-ulnar movements while in three different arm postures (pronated, supinated and midway between the two). The data set consisted of neural data of 92 M1 neurons that were recorded nMSE Train nMSE Test 0 0.5 1 1.5 2 2.5 3 nMSE OLS STEP PLS LASSO VBLS ModelSearch (a) Sergio & Kalaska [9] data nMSE Train nMSE Test 0 0.5 1 1.5 2 2.5 3 nMSE OLS STEP PLS LASSO VBLS ModelSearch (b) Kakei et al. [10] data Figure 2: Normalized mean squared error for Cross-validation Sets (6-fold for [10] and 8-fold for [9]) VBLS PLS STEP LASSO Sergio & Kalaska data set 93.6% 7.44% 8.71% 8.42% Kakei et al. data set 87.1% 40.1% 72.3% 76.3% Table 1: Percentage neuron matches between baseline and all other algorithms, averaged over all muscles in the data set at all three wrist postures (producing 2664 data points for each neuron) and the EMG outputs of 7 contributing muscles. In all experiments, the neural data was represented as average firing rates and was time aligned with EMG data based on analyses that are outside of the scope of this paper. 4.2 Methods For the Sergio & Kalaska data set, a baseline comparison of good EMG reconstruction was obtained through a limited combinatorial search over possible regression models. A particular model is characterized by a subset of neurons that is used to predict the EMG data. Given 71 neurons, theoretically 271 possible models exist. This value is too large for an exhaustive search. Therefore, we considered only possible combinations of up to 20 neurons, which required several weeks of computation on a 30-node cluster computer. The optimal predictive subset of neurons was determined from an 8-fold cross validation. This baseline study served as a comparison for PLS, stepwise regression, LASSO regression, OLS and VBLS. The five other algorithms used the same validation sets employed in the baseline study. The number of PLS projections for each data fit was found by leave-oneout cross-validation. Stepwise regression used Matlab’s “stepwisefit” function. LASSO regression was implemented, manually choosing the optimal tuning parameter over all cross-validation sets. OLS was implemented using a small ridge regression parameter of 10−10 in order to avoid ill-conditioned matrix inversions. 1 2 3 4 5 6 7 8 9 10 11 0 10 20 30 40 50 60 70 80 90 Muscle Ave # of Neurons Found STEP PLS LASSO VBLS ModelSearch (a) Sergio & Kalaska [9] data 1 2 3 4 5 6 7 0 10 20 30 40 50 60 70 80 90 Muscle Ave # of Neurons Found STEP PLS LASSO VBLS ModelSearch (b) Kakei et al. [10] data Figure 3: Average Number of Relevant Neurons found over Cross-validation Sets (6-fold for [10] and 8-fold for [9]) The average number of relevant neurons was calculated over all 8 cross-validation sets and a final set of relevant neurons was reached for each algorithm by taking the common neurons found to be relevant over the 8 cross-validation sets. Inference of relevant neurons in PLS was based on the subspace spanned by the PLS projections, while relevant neurons in VBLS were inferred from t-tests on the regression parameters, using a significance of p < 0.05. Stepwise regression and LASSO regression determined the number of relevant neurons from the inputs that were included in the final model. Note that since OLS retained all input dimensions, this algorithm was omitted in relevant neuron comparisons. Analogous to the first data set, a combinatorial analysis was performed on the Kakei et al. data set in order to determine the optimal set of neurons contributing to each muscle (i.e. producing the lowest possible prediction error) in a 6-fold cross-validation. PLS, stepwise regression, LASSO regression, OLS and VBLS were applied using the same cross-validation sets, employing the same procedure described for the first data set. 4.3 Results Figure 2 shows that, in general, EMG traces seem to be well predictable from M1 neural firing. VBLS resulted in a generalization error comparable to that produced by the baseline study. In the Kakei et al. dataset, all algorithms performed similarly, with LASSO regression performing a little better than the rest. However, OLS, stepwise regression, LASSO regression and PLS performed far worse on the Sergio & Kalaska dataset, with OLS regression attaining the worst error. Such performance is typical for traditional linear regression methods on ill-conditioned high dimensional data, motivating the development of VBLS. The average number of relevant neurons found by VBLS was slightly higher than the baseline study, as seen in Figure 3. This result is not surprising as the baseline study did not consider all possible combination of neurons. Given the good generalization results of VBLS, it seems that the Bayesian approach regularized the participating neurons sufficiently so that no overfitting occurred. Note that the results for muscle 6 and 7 in Figure 3b seem to be due to some irregularities in the data and should be considered outliers. Table 1 demonstrates that the relevant neurons identified by VBLS coincided at a very high percentage with those of the baseline results, while PLS, stepwise regression and LASSO regression had inferior outcomes. Thus, in general, VBLS achieved comparable performance with the baseline study when reconstructing EMG data from M1 neurons. While VBLS is an iterative statistical method, which performs slower than classical “one-shot” linear least squares methods (i.e., on the order of several minutes for the data sets in our analyses), it achieved comparable results with our combinatorial model search, which took weeks on a cluster computer. 5 Discussion This paper addressed the problem of analyzing high dimensional data with linear regression techniques, as encountered in neuroscience and the new field of brain-machine interfaces. To achieve robust statistical results, we introduced a novel Bayesian technique for linear regression analysis with automatic feature detection, called Variational Bayesian Least Squares. Comparisons with classical linear regression methods and a “gold standard” obtained from a brute force search over all possible linear models demonstrate that VBLS performs very well without any manual parameter tuning, such that it has the quality of a “black box” statistical analysis technique. A point of concern against the VBLS algorithm is how the variational approximation in this algorithm affects the quality of function approximation. It is known that factorial approximations to a joint distribution create more peaked distributions, such that one could potentially assume that VBLS might tend to overfit. However, in the case of VBLS, a more peaked distribution over bm pushes the regression parameter closer to zero. Thus, VBLS will be on the slightly pessimistic side of function fitting and is unlikely to overfit. Future evaluations and comparisons with Markov Chain Monte Carlo methods will reveal more details of the nature of the variational approximation. Regardless, it appears that VBLS could become a useful drop-in replacement for various classical regression methods. It lends itself to incremental implementation as would be needed in real-time analyses of brain information. Acknowledgments This research was supported in part by National Science Foundation grants ECS-0325383, IIS-0312802, IIS-0082995, ECS-0326095, ANI-0224419, a NASA grant AC#98 −516, an AFOSR grant on Intelligent Control, the ERATO Kawato Dynamic Brain Project funded by the Japanese Science and Technology Agency, the ATR Computational Neuroscience Laboratories and by funds from the Veterans Administration Medical Research Service. References [1] M.A. Nicolelis. Actions from thoughts. Nature, 409:403–407, 2001. [2] D.M. Taylor, S.I. Tillery, and A.B. Schwartz. Direct cortical control of 3d neuroprosthetic devices. Science, 296:1829–1932, 2002. [3] J.R. Wolpaw and D.J. McFarland. Control of a two-dimensional movement signal by a noninvasive brain-computer interface in humans. Proceedings of the National Academy of Sciences, 101:17849– 17854, 2004. [4] Y. Kamitani and F. Tong. Decoding the visual and subjective contents of the human brain. Nature Neuroscience, 8:679, 2004. [5] J.D. Haynes and G. Rees. Predicting the orientation of invisible stimuli from activity in human primary visual cortex. Nature Neuroscience, 8:686, 2005. [6] J. Wessberg and M.A. Nicolelis. Optimizing a linear algorithm for real-time robotic control using chronic cortical ensemble recordings in monkeys. Journal of Cognitive Neuroscience, 16:1022–1035, 2004. [7] S. Musallam, B.D. Corneil, B. Greger, H. Scherberger, and R.A. Andersen. Cognitive control signals for neural prosthetics. Science, 305:258–262, 2004. [8] R.M. Neal. Bayesian learning for neural networks. PhD thesis, Dept. of Computer Science, University of Toronto, 1994. [9] L.E. Sergio and J.F. Kalaska. 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Comparing the Effects of Different Weight Distributions on Finding Sparse Representations David Wipf and Bhaskar Rao ∗ Department of Electrical and Computer Engineering University of California, San Diego, CA 92093 dwipf@ucsd.edu, brao@ece.ucsd.edu Abstract Given a redundant dictionary of basis vectors (or atoms), our goal is to find maximally sparse representations of signals. Previously, we have argued that a sparse Bayesian learning (SBL) framework is particularly well-suited for this task, showing that it has far fewer local minima than other Bayesian-inspired strategies. In this paper, we provide further evidence for this claim by proving a restricted equivalence condition, based on the distribution of the nonzero generating model weights, whereby the SBL solution will equal the maximally sparse representation. We also prove that if these nonzero weights are drawn from an approximate Jeffreys prior, then with probability approaching one, our equivalence condition is satisfied. Finally, we motivate the worst-case scenario for SBL and demonstrate that it is still better than the most widely used sparse representation algorithms. These include Basis Pursuit (BP), which is based on a convex relaxation of the ℓ0 (quasi)-norm, and Orthogonal Matching Pursuit (OMP), a simple greedy strategy that iteratively selects basis vectors most aligned with the current residual. 1 Introduction In recent years, there has been considerable interest in finding sparse signal representations from redundant dictionaries [1, 2, 3, 4, 5]. The canonical form of this problem is given by, min w ∥w∥0, s.t. t = Φw, (1) where Φ ∈RN×M is a matrix whose columns represent an overcomplete or redundant basis (i.e., rank(Φ) = N and M > N), w ∈RM is the vector of weights to be learned, and t is the signal vector. The cost function being minimized represents the ℓ0 (quasi)-norm of w (i.e., a count of the nonzero elements in w). Unfortunately, an exhaustive search for the optimal representation requires the solution of up to M N linear systems of size N × N, a prohibitively expensive procedure for even modest values of M and N. Consequently, in practical situations there is a need for approximate procedures that efficiently solve (1) with high probability. To date, the two most widely used choices are Basis Pursuit (BP) [1] and Orthogonal Matching Pursuit (OMP) [5]. BP is based on a convex relaxation of the ℓ0 norm, i.e., replacing ∥w∥0 with ∥w∥1, which leads to an attractive, unimodal optimization problem that can be readily solved via linear programming. In contrast, OMP is a greedy strategy that iteratively selects the basis ∗This work was supported by DiMI grant 22-8376, Nissan, and NSF grant DGE-0333451. vector most aligned with the current signal residual. At each step, a new approximant is formed by projecting t onto the range of all the selected dictionary atoms. Previously [9], we have demonstrated an alternative algorithm for solving (1) using a sparse Bayesian learning (SBL) framework [6] that maintains several significant advantages over other, Bayesian-inspired strategies for finding sparse solutions [7, 8]. The most basic formulation begins with an assumed likelihood model of the signal t given weights w, p(t|w) = (2πσ2)−N/2 exp −1 2σ2 ∥t −Φw∥2 2 . (2) To provide a regularizing mechanism, SBL uses the parameterized weight prior p(w; γ) = M Y i=1 (2πγi)−1/2 exp −w2 i 2γi , (3) where γ = [γ1, . . . , γM]T is a vector of M hyperparameters controlling the prior variance of each weight. These hyperparameters can be estimated from the data by marginalizing over the weights and then performing ML optimization. The cost function for this task is L(γ) = −log Z p(t|w)p(w; γ)dw ∝log |Σt| + tT Σ−1 t t, (4) where Σt ≜σ2I + ΦΓΦT and we have introduced the notation Γ ≜diag(γ). This procedure, which can be implemented via the EM algorithm (or some other technique), is referred to as evidence maximization or type-II maximum likelihood [6]. Once γ has been estimated, a closed-form expression for the posterior weight distribution is available. Although SBL was initially developed in a regression context, it can be easily adapted to handle (1) in the limit as σ2 →0. To accomplish this we must reexpress the SBL iterations to handle the low noise limit. Applying various matrix identities to the EM algorithm-based update rules for each iteration, we arrive at the modified update [9] γ(new) = diag ˆw(old) ˆwT (old) + I −Γ1/2 (old) ΦΓ1/2 (old) † Φ Γ(old) ˆw(new) = Γ1/2 (new) ΦΓ1/2 (new) † t, (5) where (·)† denotes the Moore-Penrose pseudo-inverse. Given that t ∈range(Φ) and assuming γ is initialized with all nonzero elements, then feasibility is enforced at every iteration, i.e., t = Φ ˆw. We will henceforth refer to wSBL as the solution of this algorithm when initialized at Γ = IM and ˆw = Φ†t.1 In [9] (which extends work in [10]), we have argued why wSBL should be considered a viable candidate for solving (1). In comparing BP, OMP, and SBL, we would ultimately like to know in what situations a particular algorithm is likely to find the maximally sparse solution. A variety of results stipulate rigorous conditions whereby BP and OMP are guaranteed to solve (1) [1, 4, 5]. All of these conditions depend explicitly on the number of nonzero elements contained in the optimal solution. Essentially, if this number is less than some Φ-dependent constant κ, the BP/OMP solution is proven to be equivalent to the minimum ℓ0-norm solution. Unfortunately however, κ turns out to be restrictively small and, for a fixed redundancy ratio M/N, grows very slowly as N becomes large [3]. But in practice, both approaches still perform well even when these equivalence conditions have been grossly violated. To address this issue, a much looser bound has recently been produced for BP, dependent only on M/N. This bound holds for “most” dictionaries in the limit as N becomes large [3], where “most” 1Based on EM convergence properties, the algorithm will converge monotonically to a fixed point. is with respect to dictionaries composed of columns drawn uniformly from the surface of an N-dimensional unit hypersphere. For example, with M/N = 2, it is argued that BP is capable of resolving sparse solutions with roughly 0.3N nonzero elements with probability approaching one as N →∞. Turning to SBL, we have neither a convenient convex cost function (as with BP) nor a simple, transparent update rule (as with OMP); however, we can nonetheless come up with an alternative type of equivalence result that is neither unequivocally stronger nor weaker than those existing results for BP and OMP. This condition is dependent on the relative magnitudes of the nonzero elements embedded in optimal solutions to (1). Additionally, we can leverage these ideas to motivate which sparse solutions are the most difficult to find. Later, we provide empirical evidence that SBL, even in this worst-case scenario, can still outperform both BP and OMP. 2 Equivalence Conditions for SBL In this section, we establish conditions whereby wSBL will minimize (1). To state these results, we require some notation. First, we formally define a dictionary Φ = [φ1, . . . , φM] as a set of M unit ℓ2-norm vectors (atoms) in RN, with M > N and rank(Φ) = N. We say that a dictionary satisfies the unique representation property (URP) if every subset of N atoms forms a basis in RN. We define w(i) as the i-th largest weight magnitude and ¯w as the ∥w∥0-dimensional vector containing all the nonzero weight magnitudes of w. The set of optimal solutions to (1) is W∗with cardinality |W∗|. The diversity (or anti-sparsity) of each w∗∈W∗is defined as D∗≜∥w∗∥0. Result 1. For a fixed dictionary Φ that satisfies the URP, there exists a set of M −1 scaling constants νi ∈(0, 1] (i.e., strictly greater than zero) such that, for any t = Φw′ generated with w′ (i+1) ≤νiw′ (i) i = 1, . . . , M −1, (6) SBL will produce a solution that satisfies ∥wSBL∥0 = min(N, ∥w′∥0) and wSBL ∈W∗. Do to space limitations, the proof has been deferred to [11]. The basic idea is that, as the magnitude differences between weights increase, at any given scale, the covariance Σt embedded in the SBL cost function is dominated by a single dictionary atom such that problematic local minimum are removed. The unique, global minimum in turn achieves the stated result.2 The most interesting case occurs when ∥w′∥0 < N, leading to the following: Corollary 1. Given the additional restriction ∥w′∥0 < N, then wSBL = w′ ∈W∗and |W∗| = 1, i.e., SBL will find the unique, maximally sparse representation of the signal t. See [11] for the proof. These results are restrictive in the sense that the dictionary dependent constants νi significantly confine the class of signals t that we may represent. Moreover, we have not provided any convenient means of computing what the different scaling constants might be. But we have nonetheless solidified the notion that SBL is most capable of recovering weights of different scales (and it must still find all D∗nonzero weights no matter how small some of them may be). Additionally, we have specified conditions whereby we will find the unique w∗even when the diversity is as large as D∗= N −1. The tighter BP/OMP bound from [1, 4, 5] scales as O N −1/2 , although this latter bound is much more general in that it is independent of the magnitudes of the nonzero weights. In contrast, neither BP or OMP satisfy a comparable result; in both cases, simple 3D counter examples suffice to illustrate this point.3 We begin with OMP. Assume the fol2Because we have effectively shown that the SBL cost function must be unimodal, etc., any proven descent method could likely be applied in place of (5) to achieve the same result. 3While these examples might seem slightly nuanced, the situations being illustrated can occur frequently in practice and the requisite column normalization introduces some complexity. lowing: w∗= 1 ǫ 0 0 Φ = 0 1 √ 2 0 1 √ 1.01 0 0 1 0.1 √ 1.01 1 1 √ 2 0 0 t = Φw∗= ǫ √ 2 0 1 + ǫ √ 2 , (7) where Φ satisfies the URP and has columns φi of unit ℓ2 norm. Given any ǫ ∈(0, 1), we will now show that OMP will necessarily fail to find w∗. Provided ǫ < 1, at the first iteration OMP will select φ1, which solves maxi |tT φi|, leaving the residual vector r1 = I −φ1φT 1 t = [ ǫ/ √ 2 0 0 ]T . (8) Next, φ4 will be chosen since it has the largest value in the top position, thus solving maxi |rT 1 φi|. The residual is then updated to become r2 = I −[ φ1 φ4 ][ φ1 φ4 ]T t = ǫ 101 √ 2[ 1 −10 0 ]T . (9) From the remaining two columns, r2 is most highly correlated with φ3. Once φ3 is selected, we obtain zero residual error, yet we did not find w∗, which involves only φ1 and φ2. So for all ǫ ∈(0, 1), the algorithm fails. As such, there can be no fixed constant ν > 0 such that if w∗ (2) ≡ǫ ≤νw∗ (1) ≡ν, we are guaranteed to obtain w∗(unlike with SBL). We now give an analogous example for BP, where we present a feasible solution with smaller ℓ1 norm than the maximally sparse solution. Given w∗= 1 ǫ 0 0 Φ = 0 1 0.1 √ 1.02 0.1 √ 1.02 0 0 −0.1 √ 1.02 0.1 √ 1.02 1 0 1 √ 1.02 1 √ 1.02 t = Φw∗= " ǫ 0 1 # , (10) it is clear that ∥w∗∥1 = 1 + ǫ. However, for all ǫ ∈ (0, 0.1), if we form a feasible solution using only φ1, φ3, and φ4, we obtain the alternate solution w = (1 −10ǫ) 0 5 √ 1.02ǫ 5 √ 1.02ǫ T with ∥w∥1 ≈1 + 0.1ǫ. Since this has a smaller ℓ1 norm for all ǫ in the specified range, BP will necessarily fail and so again, we cannot reproduce the result for a similar reason as before. At this point, it remains unclear what probability distributions are likely to produce weights that satisfy the conditions of Result 1. It turns out that the Jeffreys prior, given by p(x) ∝1/x, is appropriate for this task. This distribution has the unique property that the probability mass assigned to any given scaling is equal. More explicitly, for any s ≥1, P x ∈ si, si+1 ∝log(s) ∀i ∈Z. (11) For example, the probability that x is between 1 and 10 equals the probability that it lies between 10 and 100 or between 0.01 and 0.1. Because this is an improper density, we define an approximate Jeffreys prior with range parameter a ∈(0, 1]. Specifically, we say that x ∼J(a) if p(x) = −1 2 log(a)x for x ∈[a, 1/a]. (12) With this definition in mind, we present the following result. Result 2. For a fixed Φ that satisfies the URP, let t be generated by t = Φw′, where w′ has magnitudes drawn iid from J(a). Then as a approaches zero, the probability that we obtain a w′ such that the conditions of Result 1 are satisfied approaches unity. Again, for space considerations, we refer the reader to [11]. However, on a conceptual level this result can be understood by considering the distribution of order statistics. For example, given M samples from a uniform distribution between zero and some θ, with probability approaching one, the distance between the k-th and (k+1)-th order statistic can be made arbitrarily large as θ moves towards infinity. Likewise, with the J(a) distribution, the relative scaling between order statistics can be increased without bound as a decreases towards zero, leading to the stated result. Corollary 2. Assume that D′ < N randomly selected elements of w′ are set to zero. Then as a approaches zero, the probability that we satisfy the conditions of Corollary 1 approaches unity. In conclusion, we have shown that a simple, (approximate) noninformative Jeffreys prior leads to sparse inverse problems that are optimally solved via SBL with high probability. Interestingly, it is this same Jeffreys prior that forms the implicit weight prior of SBL (see [6], Section 5.1). However, it is worth mentioning that other Jeffreys prior-based techniques, e.g., direct minimization of p(w) = Q i 1 |wi| subject to t = Φw, do not provide any SBL-like guarantees. Although several algorithms do exist that can perform such a minimization task (e.g., [7, 8]), they perform poorly with respect to (1) because of convergence to local minimum as shown in [9, 10]. This is especially true if the weights are highly scaled, and no nontrivial equivalence results are known to exist for these procedures. 3 Worst-Case Scenario If the best-case scenario occurs when the nonzero weights are all of very different scales, it seems reasonable that the most difficult sparse inverse problem may involve weights of the same or even identical scale, e.g., ¯w∗ 1 = ¯w∗ 2 = . . . ¯w∗ D∗. This notion can be formalized somewhat by considering the ¯w∗distribution that is furthest from the Jeffreys prior. First, we note that both the SBL cost function and update rules are independent of the overall scaling of the generating weights, meaning α ¯w∗is functionally equivalent to ¯w∗provided α is nonzero. This invariance must be taken into account in our analysis. Therefore, we assume the weights are rescaled such that P i ¯w∗ i = 1. Given this restriction, we will find the distribution of weight magnitudes that is most different from the Jeffreys prior. Using the standard procedure for changing the parameterization of a probability density, the joint density of the constrained variables can be computed simply as p( ¯w∗ 1, . . . , ¯w∗ D∗) ∝ 1 QD∗ i=1 ¯w∗ i for D∗ X i=1 ¯w∗ i = 1, ¯w∗ i ≥0, ∀i. (13) From this expression, it is easily shown that ¯w∗ 1 = ¯w∗ 2 = . . . = ¯w∗ D∗achieves the global minimum. Consequently, equal weights are the absolute least likely to occur from the Jeffreys prior. Hence, we may argue that the distribution that assigns ¯w∗ i = 1/D∗with probability one is furthest from the constrained Jeffreys prior. Nevertheless, because of the complexity of the SBL framework, it is difficult to prove axiomatically that ¯w∗∼1 is overall the most problematic distribution with respect to sparse recovery. We can however provide additional motivation for why we should expect it to be unwieldy. As proven in [9], the global minimum of the SBL cost function is guaranteed to produce some w∗∈W∗. This minimum is achieved with the hyperparameters γ∗ i = (w∗ i )2, ∀i. We can think of this solution as forming a collapsed, or degenerate covariance Σ∗ t = ΦΓ∗ΦT that occupies a proper D∗-dimensional subspace of N-dimensional signal space. Moreover, this subspace must necessarily contain the signal vector t. Essentially, Σ∗ t proscribes infinite density to t, leading to the globally minimizing solution. Now consider an alternative covariance Σ⋄ t that, although still full rank, is nonetheless illconditioned (flattened), containing t within its high density region. Furthermore, assume that Σ⋄ t is not well aligned with the subspace formed by Σ∗ t . The mixture of two flattened, yet misaligned covariances naturally leads to a more voluminous (less dense) form as measured by the determinant |αΣ∗ t + βΣ⋄ t |. Thus, as we transition from Σ⋄ t to Σ∗ t , we necessarily reduce the density at t, thereby increasing the cost function L(γ). So if SBL converges to Σ⋄ t it has fallen into a local minimum. So the question remains, what values of ¯w∗are likely to create the most situations where this type of local minima occurs? The issue is resolved when we again consider the D∗dimensional subspace determined by Σ∗ t . The volume of the covariance within this subspace is given by ¯Φ¯Γ∗¯Φ∗T , where ¯Φ∗and ¯Γ∗are the basis vectors and hyperparameters associated with ¯w∗. The larger this volume, the higher the probability that other basis vectors will be suitably positioned so as to both (i), contain t within the high density portion and (ii), maintain a sufficient component that is misaligned with the optimal covariance. The maximum volume of ¯Φ∗¯Γ∗¯Φ∗T under the constraints P i ¯w∗ i = 1 and ¯γ∗ i = ( ¯w∗)2 i occurs with ¯γ∗ i = 1/(D∗)2, i.e., all the ¯w∗ i are equal. Consequently, geometric considerations support the notion that deviance from the Jeffreys prior leads to difficulty recovering w∗. Moreover, empirical analysis (not shown) of the relationship between volume and local minimum avoidance provide further corroboration of this hypothesis. 4 Empirical Comparisons The central purpose of this section is to present empirical evidence that supports our theoretical analysis and illustrates the improved performance afforded by SBL. As previously mentioned, others have established deterministic equivalence conditions, dependent on D∗, whereby BP and OMP are guaranteed to find the unique w∗. Unfortunately, the relevant theorems are of little value in assessing practical differences between algorithms. This is because, in the cases we have tested where BP/OMP equivalence is provably known to hold (e.g., via results in [1, 4, 5]), SBL always converges to w∗as well. As such, we will focuss our attention on the insights provided by Sections 2 and 3 as well as probabilistic comparisons with [3]. Given a fixed distribution for the nonzero elements of w∗, we will assess which algorithm is best (at least empirically) for most dictionaries relative to a uniform measure on the unit sphere as discussed. To this effect, a number of monte-carlo simulations were conducted, each consisting of the following: First, a random, overcomplete N × M dictionary Φ is created whose entries are each drawn uniformly from the surface of an N-dimensional hypersphere. Next, sparse weight vectors w∗are randomly generated with D∗nonzero entries. Nonzero amplitudes ¯w∗are drawn iid from an experiment-dependent distribution. Response values are then computed as t = Φw∗. Each algorithm is presented with t and Φ and attempts to estimate w∗. In all cases, we ran 1000 independent trials and compared the number of times each algorithm failed to recover w∗. Under the specified conditions for the generation of Φ and t, all other feasible solutions w almost surely have a diversity greater than D∗, so our synthetically generated w∗must be maximally sparse. Moreover, Φ will almost surely satisfy the URP. With regard to particulars, there are essentially four variables with which to experiment: (i) the distribution of ¯w∗, (ii) the diversity D∗, (iii) N, and (iv) M. In Figure 1, we display results from an array of testing conditions. In each row of the figure, ¯w∗ i is drawn iid from a fixed distribution for all i; the first row uses ¯w∗ i = 1, the second has ¯w∗ i ∼J(a = 0.001), and the third uses ¯w∗ i ∼N(0, 1), i.e., a unit Gaussian. In all cases, the signs of the nonzero weights are irrelevant due to the randomness inherent in the basis vectors. The columns of Figure 1 are organized as follows: The first column is based on the values N = 50, D∗= 16, while M is varied from N to 5N, testing the effects of an increasing level of dictionary redundancy, M/N. The second fixes N = 50 and M = 100 while D∗ is varied from 10 to 30, exploring the ability of each algorithm to resolve an increasing number of nonzero weights. Finally, the third column fixes M/N = 2 and D∗/N ≈0.3 while N, M, and D∗are increased proportionally. This demonstrates how performance scales with larger problem sizes. 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 Redundancy Test (N = 50, D* = 16) Error Rate (w/ unit weights) 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 Error Rate (w/ Jeffreys weights) 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 Redundancy Ratio (M/N) Error Rate (w/ Gaussian weights) 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 Diversity Test (N = 50, M = 100) 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 Diversity (D*) 25 50 75 100 125 150 0 0.2 0.4 0.6 0.8 1 Signal Size Test (M/N = 2, D*/N = 0.32) 25 50 75 100 125 150 0 0.2 0.4 0.6 0.8 1 OMP BP SBL 25 50 75 100 125 150 0 0.2 0.4 0.6 0.8 1 Signal Size (N) Figure 1: Empirical results comparing the probability that OMP, BP, and SBL fail to find w∗under various testing conditions. Each data point is based on 1000 independent trials. The distribution of the nonzero weight amplitudes is labeled on the far left for each row, while the values for N, M, and D∗are included on the top of each column. Independent variables are labeled along the bottom of the figure. The first row of plots essentially represents the worst-case scenario for SBL per our previous analysis, and yet performance is still consistently better than both BP and OMP. In contrast, the second row of plots approximates the best-case performance for SBL, where we see that SBL is almost infallible. The handful of failure events that do occur are because a is not sufficiently small and therefore, J(a) was not sufficiently close to a true Jeffreys prior to achieve perfect equivalence (see center plot). Although OMP also does well here, the parameter a can generally never be adjusted such that OMP always succeeds. Finally, the last row of plots, based on Gaussian distributed weight amplitudes, reflects a balance between these two extremes. Nonetheless, SBL still holds a substantial advantage. In general, we observe that SBL is capable of handling more redundant dictionaries (column one) and resolving a larger number of nonzero weights (column two). Also, column three illustrates that both BP and SBL are able to resolve a number of weights that grows linearly in the signal dimension (≈0.3N), consistent with the analysis in [3] (which applies only to BP). In contrast, OMP performance begins to degrade in some cases (see the upper right plot), a potential limitation of this approach. Of course additional study is necessary to fully compare the relative performance of these methods on large-scale problems. Finally, by comparing row one, two and three, we observe that the performance of BP is roughly independent of the weight distribution, with performance slightly below the worstcase SBL performance. Like SBL, OMP results are highly dependent on the distribution; however, as the weight distribution approaches unity, performance is unsatisfactory. In summary, while the relative proficiency between OMP and BP is contingent on experimental particulars, SBL is uniformly superior in the cases we have tested (including examples not shown, e.g., results with other dictionary types). 5 Conclusions In this paper, we have related the ability to find maximally sparse solutions to the particular distribution of amplitudes that compose the nonzero elements. At first glance, it may seem reasonable that the most difficult sparse inverse problems occur when some of the nonzero weights are extremely small, making them difficult to estimate. Perhaps surprisingly then, we have shown that the exact opposite is true with SBL: The more diverse the weight magnitudes, the better the chances we have of learning the optimal solution. In contrast, unit weights offer the most challenging task for SBL. Nonetheless, even in this worst-case scenario, we have shown that SBL outperforms the current state-of-the-art; the overall assumption here being that, if worst-case performance is superior, then it is likely to perform better in a variety of situations. For a fixed dictionary and diversity D∗, successful recovery of unit weights does not absolutely guarantee that any alternative weighting scheme will necessarily be recovered as well. However, a weaker result does appear to be feasible: For fixed values of N, M, and D∗, if the success rate recovering unity weights approaches one for most dictionaries, where most is defined as in Section 1, then the success rate recovering weights of any other distribution (assuming they are distributed independently of the dictionary) will also approach one. While a formal proof of this conjecture is beyond the scope of this paper, it seems to be a very reasonable result that is certainly born out by experimental evidence, geometric considerations, and the arguments presented in Section 3. Nonetheless, this remains a fruitful area for further inquiry. References [1] D. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization,” Proc. Nat. Acad. Sci., vol. 100, no. 5, pp. 2197–2202, March 2003. [2] R. Gribonval and M. Nielsen, “Sparse representations in unions of bases,” IEEE Transactions on Information Theory, vol. 49, pp. 3320–3325, Dec. 2003. [3] D. Donoho, “For most large underdetermined systems of linear equations the minimal ℓ1-norm solution is also the sparsest solution,” Stanford University Technical Report, September 2004. [4] J.J. Fuchs, “On sparse representations in arbitrary redundant bases,” IEEE Transactions on Information Theory, vol. 50, no. 6, pp. 1341–1344, June 2004. [5] J.A. Tropp, “Greed is good: Algorithmic results for sparse approximation,” IEEE Transactions on Information Theory, vol. 50, no. 10, pp. 2231–2242, October 2004. [6] M.E. Tipping, “Sparse Bayesian learning and the relevance vector machine,” Journal of Machine Learning Research, vol. 1, pp. 211–244, 2001. [7] I.F. Gorodnitsky and B.D. Rao, “Sparse signal reconstruction from limited data using FOCUSS: A re-weighted minimum norm algorithm,” IEEE Transactions on Signal Processing, vol. 45, no. 3, pp. 600–616, March 1997. [8] M.A.T. Figueiredo, “Adaptive sparseness using Jeffreys prior,” Advances in Neural Information Processing Systems 14, pp. 697–704, 2002. [9] D.P. Wipf and B.D. Rao, “ℓ0-norm minimization for basis selection,” Advances in Neural Information Processing Systems 17, pp. 1513–1520, 2005. [10] D.P. Wipf and B.D. Rao, “Sparse Bayesian learning for basis selection,” IEEE Transactions on Signal Processing, vol. 52, no. 8, pp. 2153–2164, 2004. [11] D.P. Wipf, To appear in Bayesian Methods for Sparse Signal Representation, PhD Dissertation, UC San Diego, 2006 (estimated). http://dsp.ucsd.edu/∼dwipf/
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Goal-Based Imitation as Probabilistic Inference over Graphical Models Deepak Verma Deptt of CSE, Univ. of Washington, Seattle WA- 98195-2350 deepak@cs.washington.edu Rajesh P. N. Rao Deptt of CSE, Univ. of Washington, Seattle WA- 98195-2350 rao@cs.washington.edu Abstract Humans are extremely adept at learning new skills by imitating the actions of others. A progression of imitative abilities has been observed in children, ranging from imitation of simple body movements to goalbased imitation based on inferring intent. In this paper, we show that the problem of goal-based imitation can be formulated as one of inferring goals and selecting actions using a learned probabilistic graphical model of the environment. We first describe algorithms for planning actions to achieve a goal state using probabilistic inference. We then describe how planning can be used to bootstrap the learning of goal-dependent policies by utilizing feedback from the environment. The resulting graphical model is then shown to be powerful enough to allow goal-based imitation. Using a simple maze navigation task, we illustrate how an agent can infer the goals of an observed teacher and imitate the teacher even when the goals are uncertain and the demonstration is incomplete. 1 Introduction One of the most powerful mechanisms of learning in humans is learning by watching. Imitation provides a fast, efficient way of acquiring new skills without the need for extensive and potentially dangerous experimentation. Research over the past decade has shown that even newborns can imitate simple body movements (such as facial actions) [1]. While the neural mechanisms underlying imitation remain unclear, recent research has revealed the existence of “mirror neurons” in the primate brain which fire both when a monkey watches an action or when it performs the same action [2]. The most sophisticated forms of imitation are those that require an ability to infer the underlying goals and intentions of a teacher. In this case, the imitating agent attributes not only visible behaviors to others, but also utilizes the idea that others have internal mental states that underlie, predict, and generate these visible behaviors. For example, infants that are about 18 months old can readily imitate actions on objects, e.g., pulling apart a dumbbell shaped object (Fig. 1a). More interestingly, they can imitate this action even when the adult actor accidentally under- or overshot his target, or the hands slipped several times, leaving the goal-state unachieved (Fig. 1b)[3]. They were thus presumably able to infer the actor’s goal, which remained unfulfilled, and imitate not the observed action but the intended one. In this paper, we propose a model for intent inference and goal-based imitation that utilizes probabilistic inference over graphical models. We first describe how the basic problems of planning an action sequence and learning policies (state to action mappings) can be solved through probabilistic inference. We then illustrate the applicability of the learned graphical model to the problems of goal inference and imitation. Goal inference is achieved by utilizing one’s own learned model as a substitute for the teacher’s. Imitation is achieved by using one’s learned policies to reach an inferred goal state. Examples based on the classic maze navigation domain are provided throughout to help illustrate the behavior of the model. Our results suggest that graphical models provide a powerful platform for modeling and implementing goal-based imitation. (a) (b) Figure 1: Example of Goal-Based Imitation by Infants: (a) Infants as young as 14 months old can imitate actions on objects as seen on TV (from [4]). (b) Human actor demonstrating an unsuccessful act. Infants were subsequently able to correctly infer the intent of the actor and successfully complete the act (from [3]). 2 Graphical Models We first describe how graphical models can be used to plan action sequences and learn goal-based policies, which can subsequently be used for goal inference and imitation. Let ΩS be the set of states in the environment, ΩA the set of all possible actions available to the agent, and ΩG the set of possible goals. We assume all three sets are finite. Each goal g represents a target state Goalg ∈ΩS. At time t the agent is in state st and executes action at. gt represents the current goal that the agent is trying to reach at time t. Executing the action at changes the agent’s state in a stochastic manner given by the transition probability P(st+1 | st, at), which is assumed to be independent of t i.e., P(st+1 = s′ | st = s, at = a) = τs′sa. Starting from an initial state s1 =s and a desired goal state g, planning involves computing a series of actions a1:T to reach the goal state, where T represents the maximum number of time steps allowed (the “episode length”). Note that we do not require T to be exactly equal to the shortest path to the goal, just as an upper bound on the shortest path length. We use a, s, g to represent a specific value for action, state, and goal respectively. Also, when obvious from the context, we use s for st =s, a for at =a and g for gt =g. In the case where the state st is fully observed, we obtain the graphical model in Fig. 2a, which is also used in Markov Decision Process (MDP) [5] (but with a reward function). The agent needs to compute a stochastic policy ˆπt(a | s, g) that maximizes the probability P(sT +1 = Goalg | st = s, gt = g). For a large time horizon (T ≫1), the policy is independent of t i.e. ˆπt(a | s, g) =ˆπ(a | s, g) (a stationary policy). A more realistic scenario is where the state st is hidden but some aspects of it are visible. Given the current state st = s, an observation o is produced with the probability P(ot = o | st = s) = t r t+1 g t g t+1 r t+1 o t+1 a t+1 s t s t a t o t+1 a t+1 s t s t a (a) (b) Figure 2: Graphical Models: (a) The standard MDP graphical model: The dependencies between the nodes from time step t to t + 1 are represented by the transition probabilities and the dependency between actions and states is encoded by the policy. (b) The graphical model used in this paper (note the addition of goal, observation and “reached”nodes). See text for more details. ζso. In this paper, we assume the observations are discrete and drawn from the set ΩO, although the approach can be easily generalized to the case of continuous observations (as in HMMs, for example). We additionally include a goal variable gt and a “reached” variable rt, resulting in the graphical model in Fig. 2b (this model is similar to the one used in partially observable MDPs (POMDPs) but without the goal/reached variables). The goal variable gt represents the current goal the agent is trying to reach while the variable rt is a boolean variable that assumes the value 1 whenever the current state equals the current goal state and 0 otherwise. We use rt to help infer the shortest path to the goal state (given an upper bound T on path length); this is done by constraining the actions that can be selected once the goal state is reached (see next section). Note that rt can also be used to model the switching of goal states (once a goal is reached) and to implement hierarchical extensions of the present model. The current action at now depends not only on the current state but also on the current goal gt, and whether we have reached the goal (as indicated by rt). The Maze Domain: To illustrate the proposed approach, we use the standard stochastic maze domain that has been traditionally used in the MDP and reinforcement learning literature [6, 7]. Figure 3 shows the 7×7 maze used in the experiments. Solid squares denote a wall. There are five possible actions: up,down,left,right and stayput. Each action takes the agent into the intended cell with a high probability. This probability is governed by the noise parameter η, which is the probability that the agent will end up in one of the adjoining (non-wall) squares or remain in the same square. For example, for the maze in Fig. 3, P([3, 5] | [4, 5], left)=η while P([4, 4] | [4, 5], left)=1 −3η (we use [i,j] to denote the cell in ith row and jth column from the top left corner). 3 Planning and Learning Policies 3.1 Planning using Probabilistic Inference To simplify the exposition, we first assume full observability (ζso = δ(s, o)). We also assume that the environment model τ is known (the problem of learning τ is addressed later). The problem of planning can then be stated as follows: Given a goal state g, an initial state s, and number of time steps T , what is the sequence of actions ˆa1:T that maximizes the probability of reaching the goal state? We compute these actions using the most probable explanation (MPE) method, a standard routine in graphical model packages (see [7] for an alternate approach). When MPE is applied to the graphical model in Fig. 2b, we obtain: ¯a1:T , ¯s2:T +1, ¯g1:T , ¯r1:T = argmax P(a1:T , s2:T , g1:T , r1:T | s1 =s, sT +1 =Goalg) (1) When using the MPE method, the “reached” variable rt can be used to compute the shortest path to the goal. For P(a | g, s, r), we set the prior for the stayput action to be very high when rt = 1 and uniform otherwise. This breaks the isomorphism of the MPE action sequences with respect to the stayput action, i.e., for s1 =[4,6], goal=[4,7], and T =2, the probability of right,stayput becomes much higher than that of stayput,right (otherwise, they have the same posterior probability). Thus, the stayput action is discouraged unless the agent has reached the goal. This technique is quite general, in the sense that we can always augment ΩA with a no-op action and use this technique based on rt to push the no-op actions to the end of a T -length action sequence for a pre-chosen upper bound T . 0.421 0.573 0.690 Figure 3: Planning and Policy Learning: (a) shows three example plans (action sequences) computed using the MPE method. The plans are shown as colored lines capturing the direction of actions. The numbers denote probability of success of each plan. The longer plans have lower probability of success as expected. 3.2 Policy Learning using Planning Executing a plan in a noisy environment may not always result in the goal state being reached. However, in the instances where a goal state is indeed reached, the executed action sequence can be used to bootstrap the learning of an optimal policy ˆπ(a | s, g), which represents the probability for action a in state s when the goal state to be reached is g. We define optimality in terms of reaching the goal using the shortest path. Note that the optimal policy may differ from the prior P(a|s, g) which counts all actions executed in state s for goal g, regardless of whether the plan was successful. MDP Policy Learning: Algorithm 1 shows a planning-based method for learning policies for an MDP (both τ and π are assumed unknown and initialized to a prior distribution, e.g., uniform). The agent selects a random start state and a goal state (according to P(g1)), infers the MPE plan ¯a1:T using the current τ, executes it, and updates the frequency counts for τs′sa based on the observed st and st+1 for each at. The policy ˆπ(a | s, g) is only updated (by updating the action frequencies) if the goal g was reached. To learn an accurate τ, the algorithm is biased towards exploration of the state space initially based on the parameter α (the “exploration probability”). α decreases by a decay factor γ (0 <γ <1) with each iteration so that the algorithm transitions to an “exploitation” phase when transition model is well learned and favors the execution of the MPE plan. POMDP Policy Learning: In the case of partial observability, Algorithm 1 is modified to compute the plan ¯a1:T based on observation o1 = o as evidence instead of s1 = s in Eq.1. The plan is executed to record observations o2:T +1, which are then used to compute the MPE estimate for the hidden states:¯so 1:T +1, ¯g1:T, ¯r1:T +1 = argmax P(s1:T +1, g1:T, r1:T +1 | o1:T +1, ¯a1:T , gT +1 = g). The MPE estimate ¯so 1:T +1 is then used instead of so 1:T +1 to update ˆπ and τ. Results: Figure 4a shows the error in the learned transition model and policy as a function of the number of iterations of the algorithm. Error in τs′sa was defined as the squared sum of differences between the learned and true transition parameters. Error in the learned policy was defined as the number of disagreements between the optimal deterministic polAlgorithm 1 Policy learning in an unknown environment 1: Initialize transition model τs′sa, policy ˆπ(a | s, g), α, and numTrials. 2: for iter = 1 to numTrials do 3: Choose random start location s1 based on prior P(s1). 4: Pick a goal g according to prior P(g1). 5: With probability α: 6: a1:T = Random action sequence. 7: Otherwise: 8: Compute MPE plan as in Eq.1 using current τs′sa. Set a1:T =¯a1:T 9: Execute a1:T and record observed states so 2:T +1. 10: Update τs′sa based on a1:T and so 1:T +1. 11: If the plan was successful, update policy ˆπ(a | s, g) using a1:T and so 1:T +1. 12: α=α×γ 13: end for icy for each goal computed via policy iteration and argmax a ˆπ(a | s, g), summed over all goals. Both errors decrease to zero with increasing number of iterations. The policy error decreases only after the transition model error becomes significantly small because without an accurate estimate of τ, the MPE plan is typically incorrect and the agent rarely reaches the goal state, resulting in little or no learning of the policy. Figs. 4b shows the maximum probability action argmax a ˆπ(a | s, g) learned for each state (maze location) for one of the goals. It is clear that the optimal action has been learned by the algorithm for all locations to reach the given goal state. The results for the POMDP case are shown in Fig. 4c and d. The policy error decreases but does not reach zero because of perceptual ambiguity at certain locations such as corners, where two (or more) actions may have roughly equal probability (see Fig. 4d). The optimal strategy in these ambiguous states is to sample from these actions. 10 1 10 2 10 3 0 30 60 90 120 150 180 Error Iteration (a) Error in P(st+1|st,at) Error in policy (b) 10 1 10 2 10 3 0 30 60 90 120 150 180 Error Iteration (c) Error in policy (d) Figure 4: Learning Policies for an MDP and a POMDP: (a) shows the error in the transition model and policy w.r.t the true transition model and optimal policy for the maze MDP. (b) The optimal policy learned for one of the 3 goals. (c) and (d) show corresponding results for the POMDP case (the transition model was assumed to be known). The long arrows represent the maximum probability action while the short arrows show all the high probability actions when there is no clear winner. 4 Inferring Intent and Goal-Based Imitation Consider a task where the agent gets observations o1:t from observing a teacher and seeks to imitate the teacher. We use P(ot = o | st = s) = ζso in Fig. 2b (for the examples here, ζso was the same as in the previous section). Also, for P(a|s, g, rt =0), we use the policy ˆπ(a | s, g) learned as in the previous section. The goal of the agent is to infer the intention of the teacher given a (possibly incomplete) demonstration and to reach the intended goal using its policy (which could be different from the teacher’s optimal policy). Using the graphical model formulation the problem of goal inference reduces to finding the marginal P(gT | o1:t′), which can be efficiently computed using standard techniques such as belief propagation. Imitation is accomplished by choosing the goal with the highest probability and executing actions to reach that goal. Fig. 5a shows the results of goal inference for the set of noisy teacher observations in Fig. 5b. The three goal locations are indicated by red, blue, and green squares respectively. Note that the inferred goal probabilities correctly reflect the putative goal(s) of the teacher at each point in the teacher trajectory. In addition, even though the teacher demonstration is incomplete, the imitator can perform goal-based imitation by inferring the teacher’s most likely goal as shown in Fig. 5c. This mimics the results reported by [3] on the intent inference by infants. 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 Goal Prob= P(GT | O1:t) t (a) Goal inference Goal 1 [4,7] Goal 2 [1,2] Goal 3 [2,7] 1 2 3 4 5 6 7 8 9 10 11 12 (b) Teacher Observations (c) Imitator States 1 2 3 4 5 6 7 8 9 10 Figure 5: Goal Inference and Goal-Based Imitation: (a) shows the goal probabilities inferred at each time step from teacher observations. (b) shows the teacher observations, which are noisy and include a detour while en route to the red goal. The teacher demonstration is incomplete and stops short of the red goal. (c) The imitator infers the most likely goal using (a) and performs goalbased imitation while avoiding the detour (The numbers t in a cell in (b) and (c) represent ot and st respectively). 5 Online Imitation with Uncertain Goals Now consider a task where the goal is to imitate a teacher online (i.e., simultaneously with the teacher). The teacher observations are assumed to be corrupted by noise and may include significant periods of occlusion where no data is available. The graphical model framework provides an elegant solution to the problem of planning and selecting actions when observations are missing and only a probability distribution over goals is available. The best current action can be picked using the marginal P(at | o1:t), which can be computed efficiently for the graphical model in Fig. 2c. This marginal is equal to P i P(at|gi, o1:t)P(gi|o1:t), i.e., the policy for each goal weighted by the likelihood of that goal given past teacher observations, which corresponds to our intuition of how actions should be picked when goals are uncertain. Fig. 6a shows the inferred distribution over goal states as the teacher follows a trajectory given by the noisy observations in Fig. 6b. Initially, all goals are nearly equally likely (with a slight bias for the nearest goal). Although the goal is uncertain and certain portions of the teacher trajectory are occluded1, the agent is still able to make progress towards regions 1We simulated occlusion using a special observation symbol which carried no information about current state, i.e., P(occluded | s)=ϵ for all s (ϵ ≪1) most likely to contain any probable goal states and is able to “catch-up” with the teacher when observations become available again (Fig.. 6c). 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 Goal Prob= P(GT | O1:t) t (a) Goal inference Goal 1 [4,7] Goal 2 [1,2] Goal 3 [2,7] 1 2 6 7 8 9 10 11 14 15 16 17 (b) Teacher Observations 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 (c) Online Imitator States Figure 6: Online Imitation with Uncertain Goals: (a) shows the goal probabilities inferred by the agent at each time step for the noisy teacher trajectory in (b). (b) Observations of the teacher. Missing numbers indicate times at which the teacher was occluded. (c) The agent is able to follow the teacher trajectory even when the teacher is occluded based on the evolving goal distribution in (a). 6 Conclusions We have proposed a new model for intent inference and goal-based imitation based on probabilistic inference in graphical models. The model assumes an initial learning phase where the agent explores the environment (cf. body babbling in infants [8]) and learned a graphical model capturing the sensory consequences of motor actions. The learned model is then used for planning action sequences to goal states and for learning policies. The resulting graphical model then serves as a platform for intent inference and goal-based imitation. Our model builds on the proposals of several previous researchers. It extends the approach of [7] from planning in a traditional state-action Markov model to a full-fledged graphical model involving states, actions, and goals with edges for capturing conditional distributions denoting policies. The indicator variable rt used in our approach is similar to the ones used in some hierarchical graphical models [9, 10, 11]. However, these papers do not address the issue of action selection or imitation. Several models of imitation have previously been proposed [12, 13, 14, 15, 16, 17]; these models are typically not probabilistic and have focused on trajectory following rather than intent inference and goal-based imitation. An important issue yet to be resolved is the scalability of the proposed approach. The Bayesian model requires both a learned environment model as well as a learned policy. In the case of the maze example, these were learned using a relatively small number of trials due to small size of the state space. A more realistic scenario involving, for example, a human or a humanoid robot would presumably require an extremely large number of trials during learning due to the large number of degrees-of-freedom available; fortunately, the problem may be alleviated in two ways: first, only a small portion of the state space may be physically realizable due to constraints imposed by the body or environment; second, the agent could selectively refine its models during imitative sessions. Hierarchical state space models may also help in this regard. The probabilistic model we have proposed also opens up the possibility of applying Bayesian methodologies such as manipulation of prior probabilities of task alternatives to obtain a deeper understanding of goal inference and imitation in humans. For example, one could explore the effects of biasing a human subject towards particular classes of actions (e.g., through repetition) under particular sets of conditions. One could also manipulate the learned environment model used by subjects with the help of virtual reality environments. Such manipulations have yielded valuable information regarding the type of priors and internal models that the adult human brain uses in perception (see, e.g., [18]) and in motor learning [19]. We believe that the application of Bayesian techniques to imitation could shed new light on the problem of how infants acquire internal models of the people and objects they encounter in the world. References [1] A. N. Meltzoff and M. K. Moore. Newborn infants imitate adult facial gestures. Child Development, 54:702–709, 1983. [2] L. Fogassi G. Rizzolatti, L. Fadiga and V. Gallese. From mirror neurons to imitation, facts, and speculations. In A. N. Meltzoff and W. Prinz (Eds.), The imitative mind: Development, evolution, and brain bases, pages 247–266, 2002. [3] A. N. Meltzoff. Understanding the intentions of others: Re-enactment of intended acts by 18-month-old children. Developmental Psychology, 31:838–850, 1995. [4] A. N. Meltzoff. Imitation of televised models by infants. Child Development, 59:1221–1229, 1988a. [5] C. Boutilier, T. Dean, and S. Hanks. Decision-theoretic planning: Structural assumptions and computational leverage. Journal of AI Research, 11:1–94, 1999. [6] R. S. Sutton and A. Barto. Reinforcement Learning: An Introduction. MIT Press, Cambridge, MA, 1998. [7] H. Attias. Planning by probabilistic inference. In Proceedings of the 9th Int. Workshop on AI and Statistics, 2003. [8] R. P. N. Rao, A. P. Shon, and A. N. Meltzoff. A Bayesian model of imitation in infants and robots. In Imitation and Social Learning in Robots, Humans, and Animals. Cambridge University Press, 2004. [9] G. Theocharous, K. Murphy, and L. P. Kaelbling. Representing hierarchical POMDPs as DBNs for multi-scale robot localization. ICRA, 2004. [10] S. Fine, Y. Singer, and N. Tishby. The hierarchical hidden Markov model: Analysis and applications. Mach. Learn., 32(1):41–62, 1998. [11] H. Bui, D. Phung, and S. Venkatesh. Hierarchical hidden Markov models with general state hierarchy. In AAAI 2004, 2004. [12] G. Hayes and J. Demiris. A robot controller using learning by imitation. Proceedings of the 2nd International Symposium on Intelligent Robotic Systems, Grenoble, France,, pages 198– 204, 1994. [13] M. J. Mataric and M. Pomplun. Fixation behavior in observation and imitation of human movement. Cognitive Brain Research, 7:191–202, 1998. [14] S. Schaal. Is imitation learning the route to humanoid robots? Trends in Cognitive Sciences, 3:233–242, 1999. [15] A. Billard and K. Dautenhahn. Experiments in social robotics- grounding and use of communication in robotic agents. Adaptive Behavior, 7:3–4, 2000. [16] C. Breazeal and B. Scassellati. Challenges in building robots that imitate people. In K. Dautenhahn and C. L. Nehaniv (Eds.), Imitation in animals and artifacts, pages 363–390, 2002. [17] K. Dautenhahn and C. Nehaniv. Imitation in Animals and Artifacts. Cambridge, MA: MIT Press, 2002. [18] B. A. Olshausen R. P. N. Rao and M. S. Lewicki (Eds.). Probabilistic Models of the Brain: Perception and Neural Function. Cambridge, MA: MIT Press, 2002. [19] KP. Krding and D. Wolpert. Bayesian integration in sensorimotor learning. Nature, 427:244– 247, 2004.
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Policy-Gradient Methods for Planning Douglas Aberdeen Statistical Machine Learning, National ICT Australia, Canberra doug.aberdeen@anu.edu.au Abstract Probabilistic temporal planning attempts to find good policies for acting in domains with concurrent durative tasks, multiple uncertain outcomes, and limited resources. These domains are typically modelled as Markov decision problems and solved using dynamic programming methods. This paper demonstrates the application of reinforcement learning — in the form of a policy-gradient method — to these domains. Our emphasis is large domains that are infeasible for dynamic programming. Our approach is to construct simple policies, or agents, for each planning task. The result is a general probabilistic temporal planner, named the Factored Policy-Gradient Planner (FPG-Planner), which can handle hundreds of tasks, optimising for probability of success, duration, and resource use. 1 Introduction To date, only a few planning tools have attempted to handle general probabilistic temporal planning problems. These tools have only been able to produce good policies for relatively trivial examples. We apply policy-gradient reinforcement learning (RL) to these domains with the goal of creating tools that produce good policies in real-world domains rather than perfect policies in toy domains. We achieve this by: (1) factoring the policy into simple independent policies for starting each task; (2) presenting each policy with critical observations instead of the entire state; (3) using function approximators for each policy; (4) using local optimisation methods instead of global optimisation; and (5) using algorithms with memory requirements that are independent of the state space size. Policy gradient methods do not enumerate states and are applicable to multi-agent settings with function approximation [1, 2], thus they are a natural match for our approach to handling large planning problems. We use the GPOMDP algorithm [3] to estimate the gradient of a long-term average reward of the planner’s performance, with respect to the parameters of each task policy. We show that maximising a simple reward function naturally minimises plan durations and maximises the probability of reaching the plan goal. A frequent criticism of policy-gradient methods compared to traditional forward chaining planners — or even compared to value-based RL methods — is the lack of a clearly interpretable policy. A minor contribution of this paper is a description of how policy-gradient methods can be used to prune a decision tree over possible policies. After training, the decision tree can be translated into a list of policy rules. Previous probabilistic temporal planners include CPTP [4], Prottle [5], Tempastic [6] and a military operations planner [7]. Most these algorithms use some form of dynamic programming (either RTDP [8] or AO*) to associate values with each state/action pair. However, this requires values to be stored for each encountered state. Even though these algorithms do not enumerate the entire state space their ability to scale is limited by memory size. Even problems with only tens of tasks can produce millions of relevant states. CPTP, Prottle, and Tempastic minimise either plan duration or failure probability, not both. The FPG-Planner minimises both of these metrics and can easily optimise over resources too. 2 Probabilistic temporal planning Tasks are the basic planning unit corresponding to grounded1 durative actions. Tasks have the effect of setting condition variables to true or false. Each task has a set of preconditions, effects, resource requirements, and a fixed probability of failure. Durations may be fixed or dependent on how long it takes for other conditions to be established. A task is eligible to begin when its preconditions are satisfied and sufficient resources are available. A starting task may have some immediate effects. As tasks end a set of effects appropriate to the outcome are applied. Typically, but not necessarily, succeeding tasks set some facts to true, while failing tasks do nothing or negate facts. Resources are occupied during task execution and consumed when the task ends. Different outcomes can consume varying levels of resources. The planning goal is to set a subset of the conditions to a desired value. The closest work to that presented here is described by Peshkin et al. [1] which describes how a policy-gradient approach can be applied to multi-agent MDPs. This work lays the foundation for this application, but does not consider the planning domain specifically. It is also applied to relatively small domains, where the state space could be enumerated. Actions in temporal planning consist of launching multiple tasks concurrently. The number of candidate actions available in a given state is the power set of the tasks that are eligible to start. That is, with N eligible tasks there are 2N possible actions. Current planners explore this action space systematically, pruning actions that lead to low rewards. When combined with probabilistic outcomes the state space explosion cripples existing planners for tens of tasks and actions. A key reason treat each task as an individual policy agent is to deal with this explosion of the action space. We replace the single agent choosing from the power-set of eligible tasks with a single simple agent for each task. The policy learnt by each agent is whether to start its associated task given its observation, independent of the decisions made by the other agents. This idea alone does not simplify the problem. Indeed, if the agents received perfect state information they could learn to predict the decision of the other agents and still act optimally. The significant reduction in complexity arises from: (1) restricting the class of functions that represent agents, (2) providing only partial state information, (3) optimising locally, using gradient ascent. 3 POMDP formulation of planning Our intention is to deliberately use simple agents that only consider partial state information. This requires us to explicitly consider partial observability. A finite partially observable Markov decision process consists of: a finite set of states s ∈S; a finite set of actions a ∈A; probabilities Pr[s′|s, a] of making state transition s →s′ under action a; a reward for each state r(s) : S →R; and a finite set of observation vectors o ∈O seen by the agent in place of the complete state descriptions. For this application, observations are drawn deterministically given the state, but more generally may be stochastic. Goal states are states where all the goal state variables are satisfied. From failure states it is impossible to reach a goal state, usually because time or resources have run out. These two classes of state are combined to form the set of reset states that produce an immediate reset to the 1Grounded means that tasks do not have parameters that can be instantiated. initial state s0. A single trajectory through the state space consists of many individual trials that automatically reset to s0 each time a goal state or failure state is reached. Policies are stochastic, mapping observation vectors o to a probability over actions. Let N be the number of basic tasks available to the planner. In our setting an action a is a binary vector of length N. An entry of 1 at index n means ‘Yes’ begin task n, and a 0 entry means ‘No’ do not start task n. The probability of actions is Pr[a|o, θ], where conditioning on θ reflects the fact that the policy is controlled by a set of real valued parameters θ ∈Rp. This paper assumes that all stochastic policies (i.e., any values for θ) reach reset states in finite time when executed from s0. This is enforced by limiting the maximum duration of a plan. This ensures that the underlying MDP is ergodic, a necessary condition for GPOMDP. The GPOMDP algorithm maximises the long-term average reward η(θ) = lim T →∞ 1 T T −1 X t=0 r(st). In the context of planning, the instantaneous reward provides the agent with a measure of progress toward the goal. A simple reward scheme is to set r(s) = 1 for all states s that represent the goal state, and 0 for all other states. To maximise η(θ), successful planning outcomes must be reached as frequently as possible. This has the desired property of simultaneously minimising plan duration, as well as maximising the probability of reaching the goal (failure states achieve no reward). It is tempting to provide a negative reward for failure states, but this can introduce poor local maxima in the form of policies that avoid negative rewards by avoiding progress altogether. We provide a reward of 1000 each time the goal is achieved, plus an admissible heuristic reward for progress toward the goal. This additional shaping reward provides a reward of 1 for every goal state variable achieved, and -1 for every goal variable that becomes unset. Policies that are optimal with the additional shaping reward are still optimal under the basic goal state reward [9]. 3.1 Planning state space For probabilistic temporal planning our state description contains [7]: the state’s absolute time, a queue of impending events, the status of each task, the truth value of each condition, and the available resources. In a particular state, only a subset of the eligible tasks will satisfy all preconditions for execution. We call these tasks eligible. When a decision to start a fixed duration task is made an end-task event is added to a time ordered event queue. The event queue holds a list of events that the planner is committed to, although the outcome of those events may be uncertain. The generation of successor states is shown in Alg. 1. The algorithm begins by starting the tasks given by the current action, implementing any immediate effects. An end-task event is added at an appropriate time in the queue. The state update then proceeds to process events until there is at least one task that is eligible to begin. Events have probabilistic outcomes. Line 20 of Alg. 1 samples one possible outcome from the distribution imposed by probabilities in the problem definition. Future states are only generated at points where tasks can be started. Thus, if an event outcome is processed and no tasks are enabled, the search recurses to the next event in the queue. 4 Factored Policy-Gradient We assume the presence of policy agents, parameterised with independent sets of parameters for each agent θ = {θ1, . . . , θN}. We seek to adjust the parameters of the policy to maximise the long-term average reward η(θ). The GPOMDP algorithm [3] estimates the gradient ∇η(θ) of the long-term average reward with respect to the current set of policy Alg. 1: findSuccessor(State s, Action a) 1: for each an =’Yes’ in a do 2: s.beginTask(n) 3: s.addEvent(n, s.time+taskDuration(n)) 4: end for 5: repeat 6: if s.time > maximum makespan then 7: s.failureLeaf=true 8: return 9: end if 10: if s.operationGoalsMet() then 11: s.goalLeaf=true 12: return 13: end if 14: if ¬s.anyEligibleTasks() then 15: s.failureLeaf=true 16: return 17: end if 18: event = s.nextEvent() 19: s.time = event.time 20: sample outcome from event 21: s.implementEffects(outcome) 22: until s.anyEligibleTasks() Alg. 2: Gradient Estimator 1: Set s0 to initial state, t = 0, et = [0] 2: while t < T do 3: et = βet−1 4: Generate observation ot of st 5: for Each eligible task n do 6: Sample atn =Yes or atn =No 7: et = et + ∇log Pr[atn|o, θn] 8: end for 9: Try action at = {at1, at2, . . . , atN} 10: while mutex prohibits at do 11: randomly disable task in at 12: end while 13: st+1 = findSuccessor(st, at) 14: ˆ∇tη(θ) = ˆ∇t−1η(θ)− 1 t+1(r(st+1)et −ˆ∇t−1η(θ)) 15: t ←t + 1 16: end while 17: Return ˆ∇T η(θ) parameters. Once an estimate ˆ∇η(θ) is computed over T simulation steps, we maximise the long-term average reward with the gradient ascent θ ←θ +α ˆ∇η(θ), where α is a small step size. The experiments in this paper use a line search to determine good values of α. We do not guarantee that the best representable policy is found, but our experiments have produced policies comparable to global methods like real-time dynamic programming [8]. The algorithm works by sampling a single long trajectory through the state space (Fig. 4): (1) the first state represents time 0 in the plan; (2) the agents all receive the vector observation ot of the current state st; (3) each agent representing an eligible task emits a probability of starting; (4) each agent samples start or do not start and issues it as a planning action; (5) the state transition is sampled with Alg. 1; (6) the agents receive the global reward for the new state action and update their gradient estimates. Steps 1 to 6 are repeated T times. Each vector action at is a combination of independent ‘Yes’ or ‘No’ choices made by each eligible agent. Each agent is parameterised by an independent set of parameters that make up θ ∈Rp: θ1, θ2, . . . , θN. If atn represents the binary decision made by agent n at time t about whether to start its corresponding task, then the policy factors into Pr[at|ot, θ] = Pr[at1, . . . , atN|ot, θ1, . . . , θN] = Pr[at1|ot, θ1] × · · · × Pr[atN|ot, θN]. It is not necessary for all agents to receive the same observation, and it may be advantageous to show different agents different parts of the state, leading to a decentralised planning algorithm. Similar approaches are adopted by Peshkin et al. [1], Tao et al. [2], using policygradient methods to train multi-agent systems. The main requirement for each policy-agent is that log Pr[atn|ot, θn] be differentiable with respect to the parameters for each choice task start atn =‘Yes’ or ‘No’. We now describe two such agents. 4.1 Linear approximator agents One representation of agents is a linear network mapped into probabilities using a logistic regression function: Conditions Eligible tasks Task status Resources Event queue Time Current State Conditions Eligible tasks Task status Resources Event queue Time Not Eligible Task N Task 1 Task 2 Next State Choice disabled ot ot Pr[Y es|ot, θ1] = 0.1 Pr[No|ot, θ1] = 0.9 Pr[No|ot, θ2] = 1.0 Pr[Y es|ot, θN ] = 0.5 Pr[No|ot, θN ] = 0.5 findSuccessor(st, at) at Fig. 3: Decision tree agent. Fig. 4: (Left) Individual taskpolicies make independent decisions. Pr[atn = Y es|ot, θn] = exp(o⊤ t θn) exp(o⊤ t θn) + 1 (1) If the dimension of the observation vector is |o| then each set of parameters θn can be thought of as an |o| vector that represents the approximator weights for task n. The log derivatives, necessary for Alg. 2, are given in [10]. Initially, the parameters are set to small random values: a near uniform random policy. This encourages exploration of the action space. Each gradient step typically moves the parameters closer to a deterministic policy. After some experimentation we chose an observation vector that is a binary description of the eligible tasks and the state variable truth values plus a constant 1 bit to provide bias to the agents’ linear networks. 4.2 Decision tree agents Often we have a selection of potential control rules. A decision tree can represent all such control rules at the leaves. The nodes are additional parameterised or hardwired rules that select between different branches, and therefore different control rules. An action a is selected by starting at the root node and following a path down the tree, visiting a set of decision nodes D. At each node we either applying a hard coded branch selection rule, or sample a stochastic branch rule from the probability distribution invoked by the parameterisation. Assuming the independence of decisions at each node, the probability or reaching an action leaf l equals the product of branch probabilities at each decision node Pr[a = l|o, θ] = Y d∈D Pr[d′|o, θd], (2) where d represents the current decision node, and d′ represents the next node visited in the tree. The final next node d′ is the leaf l. The probability of a branch followed as a result of a hard-coded rule is 1. The individual Pr(d′|o, θd) functions can be any differentiable function of the observation vector o. For multi-agent domains, such as our formulation of planning, we have a decision tree for each task agent. We use the same initial tree (with different parameters), for each agent, shown in Fig. 3. Nodes A, D, F, H represent hard coded rules that switch with probability one between the Yes and No branches based on a boolean observation that gives the truth of the statement in the node for the current state. Nodes B, C, E, G are parameterised so that they select branches stochastically. For this application, the probability of choosing the Yes or No branches is a single parameter logistic function that is independent of the observations. Parameter adjustments have the simple effect of pruning parts the tree that represent poor policies, leaving the hard coded rules to choose the best action given the observation. The policy encoded by the parameter is written in the node label. For example for task agent n, and decision node C “task duration matters?”, we have the probability Pr(Y es|o, θn,C) = Pr(Y es|θn,C) = exp(θn,C) exp(θn,C) + 1 The log gradient of this function is given in [10]. If set parameters to always select the dashed branch in Fig. 3 we would be following the policy: if the task IS eligible, and probability this task success does NOT matter, and the duration of this task DOES matter, and this task IS fast, then start, otherwise do not start. Apart from being easy to interpret the optimised decision tree as a set of — possibly stochastic — if-then rules, we can also encode highly expressive policies with only a few parameters. 4.3 GPOMDP for planning Alg. 4 describes the algorithm for computing ˆ∇η(θ), based on GPOMDP [3]. The vector quantity et is an eligibility trace. It has dimension p (the total number of parameters), and can be thought of as storing the eligibility of each parameter for being reinforced after receiving a reward. The gradient estimate provably converges to a biased estimate of ∇η(θ) as T →∞. The quantity β ∈[0, 1) controls the degree of bias in the estimate. As β approaches 1, the bias of the estimates drop to 0. However if β = 1, estimates exhibit infinite variance in the limit as T →∞. Thus the parameter β is used to achieve a bias/variance tradeoff in our stochastic gradient estimates. GPOMDP gradient estimates have been proven to converge, even under partial observability. Line 8 computes the log gradient of the sampled action probability and adds the gradient for the n’th agent’s parameters into the eligibility trace. The gradient for parameters not relating to agent n is 0. We do not compute Pr[atn|ot, θn] or gradients for tasks with unsatisfied preconditions. If all eligible agents decide not to start their tasks, we issue a null-action. If the state event queue is not empty, we process the next event, otherwise time is incremented by 1 to ensure all possible policies will eventually reach a reset state. 5 Experiments 5.1 Comparison with previous work We compare the FPG-Planner with that of our earlier RTDP based planner for military operations [7], which is based on real-time dynamic programming with [8]. The domains come from the Australian Defence Science and Technology Organisation, and represent military operations planning scenarios. There are two problems, the first with 18 tasks and 12 conditions, and the second with 41 tasks and 51 conditions. The goal is to set the “Objective island secured” variable to true. There are multiple interrelated tasks that can lead to the goal state. Tasks fail or succeed with a known probability and can only execute once, leading to relatively large probabilities of failure even for optimal plans. See [7] for details. Unless stated, FPG-Planner experiments used T = 500, 000 gradient estimation steps and β = 0.9. Optimisation time was limited to 20 minutes wall clock time on a single user 3GHz Pentium IV with 1GB ram. All evaluations are based on 10,000 simulated executions of finalised policies. Results quote the average duration, resource consumption, and the percentage of plans that terminate in a failure state. We repeat the comparison experiments 50 times with different random seeds and report Table 1: Two domains compared with a dynamic programming based planner. Problem RTDP Factored Linear Factored Tree Dur Res Fail% Dur Res Fail% Dur Res Fail% Assault Ave 171 8.0 26.1 105 8.3 26.6 115 8.3 27.1 Assault Best 113 6.2 24.0 93.1 8.7 23.1 112 8.4 25.6 Webber Ave 245 4.4 58.1 193 4.1 57.9 186 4.1 58.0 Webber Best 217 4.2 57.7 190 4.1 57.0 181 4.1 57.3 Table 2: Effect of different observations. Observation Dur Res Fail% Eligible & Conds 105 8.3 26.6 Conds only 112 8.1 28.1 Eligible only 112 8.1 29.6 Table 3: Results for the Art45/25 domain. Policy Dur Res Fail% Random 394 206 83.4 Naive 332 231 78.6 Linear 121 67 7.4 Dumb Tree 157 92 19.1 Prob Tree 156 62 10.9 Dur Tree 167 72 17.4 Res Tree 136 53 8.50 mean and best results in Table 1. The “Best” plan minimises an arbitrarily chosen combined metric of 10 × fail% + dur. FPG-Planning with a linear approximator significantly shortens the duration of plans, without increasing the failure rate. The very simple decision tree performs less well than than the linear approximator, but better than the dynamic programming algorithm. This is somewhat surprising given the simplicity of the tree for each task. The shorter duration for the Webber decision tree is probably due to the slightly higher failure rate. Plans failing early produces shorter durations. Table 1 assumes that the observation vector o presented to linear agents is a binary description of the eligible tasks and the condition truth values plus a constant 1 bit to provide bias to the agents’ linear networks. Table 2 shows that giving the agents less information in the observation harms performance. 5.2 Large artificial domains Each scenario consists of N tasks and C state variables. The goal state of the synthetic scenarios is to assert 90% of the state variables, chosen during scenario synthesis, to be true. See [10] for details. All generated problems have scope for choosing tasks instead of merely scheduling them. All synthetic scenarios are guaranteed to have at least one policy which will reach the operation goal assuming all tasks succeed. Even a few tens of tasks and conditions can generate a state space too large for main memory. We generated 37 problems, each with 40 tasks and 25 conditions (Art40/25). Although the number of tasks and conditions is similar to the Webber problem described above, these problems demonstrate significantly more choices to the planner, making planning nontrivial. Unlike the initial experiments, all tasks can be repeated as often as necessary so the overall probability of failure depends on how well the planner chooses and orders tasks to avoid running out of time and resources. Our RTDP based planner was not able to perform any significant optimisation in 20m due to memory problems. Thus, to demonstrate FPGPlanning is having some effect, we compared the optimised policies to two simple policies. The random policy starts each eligible task with probability 0.5. The naive policy starts all eligible tasks. Both of these policies suffer from excessive resource consumption and negative effects that can cause failure. Table 3 shows that the linear approximator produces the best plans, but it requires C + 1 parameters per task. The results for the decision tree illustrated in Fig. 3 are given in the “Prob Tree” row. This tree uses a constant 4 parameters per task, and subsequently requires fewer operations when computing gradients. The “Dumb” row is a decision stub, with one parameter per task that simply learns whether to start when eligible. The remaining “Dur” and “Res” Tree rows re-order the nodes in Fig. 3 to swap the nodes C and E respectively with node B. This tests the sensitivity of the tree to node ordering. There appears to be significant variation in the results. For example, when node E is swapped with B, the resultant policies use less resources. We also performed optimisation of a 200 task, 100 condition problem generated using the same rules as the Art40/25 domain. The naive policy had a failure rate of 72.4%. No time limit was applied. Linear network agents (20,200 parameters) optimised for 14 hours, before terminating with small gradients, and resulted in a plan with 20.8% failure rate. The decision tree agent (800 parameters) optimised for 6 hours before terminating with a 1.7% failure rate. The smaller number of parameters and a priori policies embedded in the tree, allow the decision tree to perform well in very large domains. Inspection of the resulting parameters demonstrated that different tasks pruned different regions of the decision tree. 6 Conclusion We have demonstrated an algorithm with great potential to produce good policies in realworld domains. Further work will refine our parameterised agents, and validate this approach on realistic larger domains. We also wish to characterise possible local minima. Acknowledgements Thank you to Olivier Buffet and Sylvie Thi´ebaux for many helpful comments. National ICT Australia is funded by the Australian Government’s Backing Australia’s Ability program and the Centre of Excellence program. This project was also funded by the Australian Defence Science and Technology Organisation. References [1] L. Peshkin, K.-E. Kim, N. Meuleau, and L. P. Kaelbling. Learning to cooperate via policy search. In UAI, 2000. [2] Nigel Tao, Jonathan Baxter, and Lex Weaver. A multi-agent, policy-gradient approach to network routing. In Proc. ICML’01. Morgan Kaufmann, 2001. [3] J. Baxter, P. Bartlett, and L. Weaver. Experiments with infinite-horizon, policy-gradient estimation. JAIR, 15:351–381, 2001. [4] Mausam and Daniel S. Weld. Concurrent probabilistic temporal planning. In Proc. International Conference on Automated Planning and Scheduling, Moneteray, CA, June 2005. AAAI. [5] I. Little, D. Aberdeen, and S. Thi´ebaux. Prottle: A probabilistic temporal planner. In Proc. AAAI’05, 2005. [6] Hakan L. S. Younes and Reid G. Simmons. Policy generation for continuous-time stochastic domains with concurrency. In Proc. of ICAPS’04, volume 14, 2005. [7] Douglas Aberdeen, Sylvie Thi´ebaux, and Lin Zhang. Decision-theoretic military operations planning. In Proc. ICAPS, volume 14, pages 402–411. AAAI, June 2004. [8] A.G. Barto, S. Bradtke, and S. Singh. Learning to act using real-time dynamic programming. Artificial Intelligence, 72, 1995. [9] A.Y. Ng, D. Harada, and S. Russell. Policy invariance under reward transformations: Theory and application to reward shaping. In Proc. ICML’99, 1999. [10] Douglas Aberdeen. The factored policy-gradient planner. Technical report, NICTA, 2005.
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Message passing for task redistribution on sparse graphs K. Y. Michael Wong Hong Kong U. of Science & Technology Clear Water Bay, Hong Kong, China phkywong@ust.hk David Saad NCRG, Aston University Birmingham B4 7ET, UK D.Saad@aston.ac.uk Zhuo Gao Hong Kong U. of Science & Technology, Clear Water Bay, Hong Kong, China Permanent address: Dept. of Physics, Beijing Normal Univ., Beijing 100875, China zhuogao@bnu.edu.cn Abstract The problem of resource allocation in sparse graphs with real variables is studied using methods of statistical physics. An efficient distributed algorithm is devised on the basis of insight gained from the analysis and is examined using numerical simulations, showing excellent performance and full agreement with the theoretical results. 1 Introduction Optimal resource allocation is a well known problem in the area of distributed computing [1, 2] to which significant effort has been dedicated within the computer science community. The problem itself is quite general and is applicable to other areas as well where a large number of nodes are required to balance loads/resources and redistribute tasks, such as reducing internet traffic congestion [3]. The problem has many flavors and usually refers, in the computer science literature, to finding practical heuristic solutions to the distribution of computational load between computers connected in a predetermined manner. The problem we are addressing here is more generic and is represented by nodes of some computational power that should carry out tasks. Both computational powers and tasks will be chosen at random from some arbitrary distribution. The nodes are located on a randomly chosen sparse graph of some given connectivity. The goal is to migrate tasks on the graph such that demands will be satisfied while minimizing the migration of (sub-)tasks. An important aspect of the desired algorithmic solution is that decisions on messages to be passed are carried out locally; this enables an efficient implementation of the algorithm in large non-centralized distributed networks. We focus here on the satisfiable case where the total computing power is greater than the demand, and where the number of nodes involved is very large. The unsatisfiable case can be addressed using similar techniques. We analyze the problem using the Bethe approximation of statistical mechanics in Section 2, and alternatively a new variant of the replica method [4, 5] in Section 3. We then present numerical results in Section 4, and derive a new message passing distributed algorithm on the basis of the analysis (in Section 5). We conclude the paper with a summary and a brief discussion on future work. 2 The statistical physics framework: Bethe approximation We consider a typical resource allocation task on a sparse graph of N nodes, labelled i = 1; ::; N. Each node i is randomly connected to other nodes1, and has a capacity i randomly drawn from a distribution ( i ). The objective is to migrate tasks between nodes such that each node will be capable of carrying out its tasks. The current y ij y j i drawn from node j to i is aimed at satisfying the constraint X j A ij y ij + i 0 ; (1) representing the ’revised’ assignment for node i, where A ij = 1=0 for connected/unconnected node pairs i and j, respectively. To illustrate the statistical mechanics approach to resource allocation, we consider the load balancing task of minimizing the energy function (cost) E = P (ij ) A ij (y ij ), where the summation (ij ) runs over all pairs of nodes, subject to the constraints (1); (y ) is a general function of the current y. For load balancing tasks, (y ) is typically a convex function, which will be assumed in our study. The analysis of the graph is done by introducing the free energy F = T ln Z y for a temperature T 1, where Z y is the partition function Z y = Y (ij ) Z dy ij Y i 0 X j A ij y ij + i 1 A exp 2 4 X (ij ) A ij (y ij ) 3 5 : (2) The function returns 1 for a non-negative argument and 0 otherwise. When the connectivity is low, the probability of finding a loop of finite length on the graph is low, and the Bethe approximation well describes the local environment of a node. In the approximation, a node is connected to branches in a tree structure, and the correlations among the branches of the tree are neglected. In each branch, nodes are arranged in generations. A node is connected to an ancestor node of the previous generation, and another 1 descendent nodes of the next generation. Consider a vertex V (T) of capacity V (T), and a current y is drawn from the vertex. One can write an expression for the free energy F (y jT) as a function of the free energies F (y k jT k ) of its descendants, that branch out from this vertex F (y jT) = T ln ( 1 Y k =1 Z dy k 1 X k =1 y k y + V ! exp " 1 X k =1 ( F (y k jT k ) + (y k )) # ) ; (3) where T k represents the tree terminated at the k th descendent of the vertex. The free energy can be considered as the sum of two parts, F (y jT) = N T F a v + F V (y jT), where N T is the number of nodes in the tree T, F a v is the average free energy per node, and F V (y jT) is referred to as the vertex free energy2. Note that when a vertex is added to a tree, there is a 1Although we focus here on graphs of fixed connectivity, one can easily accommodate any connectivity profile within the same framework; the algorithms presented later are completely general. 2This term is marginalized over all inputs to the current vertex, leaving the difference in chemical potential y as its sole argument, hence the terminology used. change in the free energy due to the added vertex. Since the number of nodes increases by 1, the vertex free energy is obtained by subtracting the free energy change by the average free energy. This allows us to obtain the recursion relation F V (y jT) = T ln ( 1 Y k =1 Z dy k 1 X k =1 y k y + V (T) ! exp " 1 X k =1 (F V (y k jT k ) + (y k )) #) F a v ; (4) and the average free energy per node is given by F a v = T * ln ( Y k =1 Z dy k X k =1 y k + V ! exp " X k =1 (F V (y k jT k ) + (y k )) #)+ ; (5) where V is the capacity of the vertex V fed by trees T 1 ; : : : ; T , and h: : : i represents the average over the distribution (). In the zero temperature limit, Eq. (4) reduces to F V (y jT) = min fy k j P 1 k =1 y k y + V (T) 0g " 1 X k =1 (F V (y k jT k ) + (y k )) # F a v : (6) The current distribution and the average free energy per link can be derived by integrating the current y 0 in a link from one vertex to another, fed by the trees T 1 and T 2, respectively; the obtained expressions are P (y ) = hÆ (y y 0 )i ? and hE i = h (y 0 )i ? where hi ? = R dy 0 exp [ ( F V (y 0 jT 1 ) + F V ( y 0 jT 2 ) + (y 0 )) ℄ () R dy 0 exp [ (F V (y 0 jT 1 ) + F V ( y 0 jT 2 ) + (y 0 )) ℄ : (7) 3 The statistical physics framework: replica method In this section, we sketch the analysis of the problem using the replica method, as an alternative to the Bethe approximation. The derivation is rathe involved, details will be provided elsewhere. To facilitate derivations, we focus on the quadratic cost function (y ) = y 2 =2. The results confirm the validity of the Bethe approximation on sparse graphs. An alternative formulation of the original optimization problem is to consider its dual. Introducing Lagrange multipliers, the function to be minimized becomes L = P (ij ) A ij y 2 ij =2 + P i i ( P j A ij y ij + i ). Optimizing L with respect to y ij, one obtains y ij = j i, where i is referred to as the chemical potential of node i, and the current is driven by the potential difference. Although the analysis has also been carried out in the space of currents, we focus here on the optimization problem in the space of the chemical potentials. Since the energy function is invariant under the addition of an arbitrary global constant to the chemical potentials of all nodes, we introduce an extra regularization term P i 2 i =2 to break the translational symmetry, where ! 0. To study the characteristics of the problem one calculates the averaged free energy per node F a v = T hln Z i A; = N, where Z is the partition function Y i 2 4 Z d i 0 X j A ij ( j i ) + i 1 A 3 5 exp 2 4 2 0 X (ij ) A ij ( j i ) 2 + X i 2 i 1 A 3 5 : The calculation follows the main steps of a replica based calculation in diluted systems [6], using the identity ln Z = lim n!0 [Z n 1℄=n. The replicated partition function [5] is averaged over all network configurations with connectivity and capacity distributions ( i ). We consider the case of intensive connectivity O (1) N. Extending the analysis of [6] and averaging over all connectivity matrices, one finds hZ n i = exp N ( 2 X r;s ^ Q r;s Q r;s + ln Z d() Y Z d Z 1 d Z d ^ 2 ! exp " X i ^ ( + ) 2 ( + ) 2 # X ) ; (8) where X = P r;s ^ Q r;s Q ( i ^ ) r s + P r;s Q r;s 2 Q r !s ! Q r ( i ^ ) s . The order parameters Q r;s and ^ Q r;s, are labelled by the somewhat unusual indices r and s, representing the n-component integer vectors (r 1 ; ::; r n ) and (s 1 ; ::; s n ) respectively. This is a result of the specific interaction considered which entangles nodes of different indices. The order parameters Q r;s and ^ Q r;s are given by the extremum condition of Eq. (8), i.e., via a set of saddle point equations w.r.t the order parameters. Assuming replica symmetry, the saddle point equations yield a recursion relation for a two-component function R, which is related to the order parameters via the generating function P s (z) = X r Q r;s Y (z ) r r ! = * Y Z d R (z ; jT)e 2 =2 s + : (9) In Eq. (9), T represents the tree terminated at the vertex node with chemical potential , providing input to the ancestor node with chemical potential z, and h: : : i represents the average over the distribution (). The resultant recursion relation for R (z ; jT) is independent of the replica indices, and is given by R (z ; jT) = 1 D 1 Y k = 1 Z d k R (; k jT k ) 1 X k = 1 k + z + V (T) ! exp " 2 1 X k = 1 ( k ) 2 + 2 ! # ; (10) where the vertex node has a capacity V (T); D is a constant. R (z ; jT) is expressed in terms of 1 functions R (; k jT k ) ( k = 1; ::; 1), integrated over k. This algebraic structure is typical of the Bethe lattice tree-like representation of networks of connectivity , where a node obtains input from its 1 descendent nodes of the next generation, and T k represents the tree terminated at the k th descendent. Except for the regularization factor exp( 2 =2), R turns out to be a function of y z, which is interpreted as the current drawn from a node with chemical potential by its ancestor with chemical potential z. One can then express the function R as the product of a vertex partition function Z V and a normalization factor W, that is, R (z ; jT) = W ()Z V (y jT). In the limit n ! 0, the dependence on and y are separable, providing a recursion relation for Z V (y jT). This gives rise to the vertex free energy F V (y jT) = T ln Z V (y jT) when a current y is drawn from the vertex of a tree T. The recursive equation and the average free energy expression agrees with the results in the Bethe approximation. These iterative equations can be directly linked to those obtained from a principled Bayesian approximation, where the logarithms of the messages passed between nodes are proportional to the vertex free energies. 4 Numerical solution The solution of Eq. (6) is obtained numerically. Since the vertex free energy of a node depends on its own capacity and the disordered configuration of its descendants, we generate 1000 nodes at each iteration of Eq. (6), with capacities randomly drawn from the distribution (), each being fed by 1 nodes randomly drawn from the previous iteration. We have discretized the vertex free energies F V (y jT) function into a vector, whose i th component is the value of the function corresponding to the current y i. To speed up the optimization search at each node, we first find the vertex saturation current drawn from a node such that: (a) the capacity of the node is just used up; (b) the current drawn by each of its descendant nodes is just enough to saturate its own capacity constraint. When these conditions are satisfied, we can separately optimize the current drawn by each descendant node, and the vertex saturation current is equal to the node capacity subtracted by the current drawn by its descendants. The optimal solution can be found using an exhaustive search, by varying the component currents in small discrete steps. This approach is particularly convenient for = 3, where the search is confined to a single parameter. To compute the average energy, we randomly draw 2 nodes, compute the optimal current flowing between them, and repeat the sampling to obtain the average. Figure 1(a) shows the results as a function of iteration step t, for a Gaussian capacity distribution () with variance 1 and average hi. Each iteration corresponds to adding one extra generation to the tree structure, such that the iterative process corresponds to approximating the network by an increasingly extensive tree. We observe that after an initial rise with iteration steps, the average energies converges to steady-state values, at a rate which increases with the average capacity. To study the convergence rate of the iterations, we fit the average energy at iteration step t using hE (t) E (1)i exp(
t) in the asymptotic regime. As shown in the inset of Fig. 1(a), the relaxation rate
increases with the average capacity. It is interesting to note that a cusp exists at the average capacity of about 0.45. Below that value, convergence of the iteration is slow, since the average energy curve starts to develop a plateau before the final convergence. On the other hand, the plateau disappears and the convergence is fast above the cusp. The slowdown of convergence below the cusp is probably due to the appearance of increasingly large clusters of nonzero currents on the network, since clusters of nodes with negative capacities become increasingly extensive, and need to draw currents from increasingly extensive regions of nodes with excess capacities to satisfy the demand. Figure 1(b) illustrates the current distribution for various average capacities. The distribution P (y ) consists of a delta function component at y = 0 and a continuous component whose breadth decreases with average capacity. The fraction of links with zero currents increases with the average capacity. Hence at a low average capacity, links with nonzero currents form a percolating cluster, whereas at a high average capacity, it breaks into isolated clusters. 5 Distributed algorithms The local nature of the recursion relation Eq. (6) points to the possibility that the network optimization can be solved by message passing approaches, which have been successful in problems such as error-correcting codes [8] and probabilistic inference [9]. The major advantage of message passing is its potential to solve a global optimization problem via local updates, thereby reducing the computational complexity. For example, the computational complexity of quadratic programming for the load balancing task typically scales as N 3, whereas capitalizing on the network topology underlying the connectivity of the variables, message passing scales as N. An even more important advantage, relevant to 0 10 20 30 40 t 10 −2 10 −1 10 0 10 1 <E> 0 0.2 0.4 0.6 0.8 <Λ> 10 −2 10 −1 10 0 γ (a) 0 0.5 1 1.5 2 y 0 0.5 1 1.5 2 P(y) 0 0.5 <Λ> 0 0.4 P(y=0) 0.1 0.8 (b) 0 0.2 0.4 0.6 0.8 1 <Λ> 0 0.05 0.1 0.15 <E> 0 0.5 <Λ> 0 0.1 (c−2)<E> c=3 c=4 c=5 c=3 c=5 (c) −2 −1.5 −1 −0.5 0 µ 0 0.5 1 1.5 2 P(µ) 0 0.5 1 <Λ> 0 0.5 P(µ=0) (d) 0.1 0.8 Figure 1: Results for system size N = 1000 and (y ) = y 2 =2. (a) hE i obtained by iterating Eq. (6) as a function of t for hi =0.1, 0.2, 0.4, 0.6, 0.8 (top to bottom) and = 3. Dashed line: The asymptotic hE i for hi = 0:1. Inset:
as a function of hi. (b) The distribution P (y ) obtained by iterating Eq. (6) to steady states for the same parameters and average capacities as in (a), from right to left. Inset: P (y = 0) as a function of hi. Symbols: = 3 (
) and ( ), = 4 ( ) and ( 4), = 5 ( C) and ( r); each pair obtained from Eqs. (11) and (14) respectively. Line: erf (hi= p 2). (c) hE i as a function of hi for = 3; 4; 5. Symbols: results of Eq. (6) (
), Eq.(11) ( ), and Eq. (14) ( ). Inset: hE i multiplied by ( 2) as a function of hi for the same conditions. (d) The distribution P () obtained by iterating Eq. (14) to steady states for the same parameters and average capacities as in (b), from left to right. Inset: P ( = 0) as a function of hi. Symbols: same as (b). practical implementation, is its distributive nature; it does not require a global optimizer, and is particularly suitable for distributive control in evolving networks. However, in contrast to other message passing algorithms which pass conditional probability estimates of discrete variables to neighboring nodes, the messages in the present context are more complex, since they are functions F V (y jT) of the current y. We simplify the message to 2 parameters, namely, the first and second derivatives of the vertex free energies. For the quadratic load balancing task, it can be shown that a self-consistent solution of the recursion relation, Eq. (6), consists of vertex free energies which are piecewise quadratic with continuous slopes. This makes the 2-parameter message a very precise approximation. Let (A ij ; B ij ) ( F V (y ij jT j )= y ij ; 2 F V (y ij jT j )= y 2 ij ) be the message passed from node j to i; using Eq.(6), the recursion relation of the messages become A ij ij ; B ij ( ij ) 2 4 X k 6=i A j k ( 00 j k + B j k ) 1 3 5 1 ; where (11) ij = min " P k 6=i A j k [y j k ( 0 j k + A j k )( 00 j k + B j k ) 1 ℄ + j y ij P k 6=i A j k ( 00 j k + B j k ) 1 ; 0 # ; (12) with 0 j k and 00 j k representing the first and second derivatives of (y ) at y = y j k respectively. The forward passing of the message from node j to i is then followed by a backward message from node j to k for updating the currents y j k according to y j k y j k 0 j k + A j k + ij 00 j k + B j k : (13) We simulate networks with = 3, (y ) = y 2 =2 and compute their average energies. The network configurations are generated randomly, with loops of lengths 3 or less excluded. Updates are performed with random sequential choices of the nodes. As shown in Fig. 1(c), the simulation results of the message passing algorithm have an excellent agreement with those obtained by the recursion relation Eq.(6). For the quadratic load balancing task considered here, an independent exact optimization is available for comparison. The K¨uhn-Tucker conditions for the optimal solution yields i = min 2 4 1 0 X j A ij j + i 1 A ; 0 3 5 : (14) It also provides a local iterative method for the optimization problem. As shown in Fig. 1(c), both the recursion relation Eq.(6) and the message passing algorithm Eq.(11) yield excellent agreement with the iteration of chemical potentials Eq.(14). Both Eqs. (11) and (14) allow us to study the distribution P () of the chemical potentials . As shown in Fig. 1(d), P () consists of a delta function and a continuous component. Nodes with zero chemical potentials correspond to those with unsaturated capacity constraints. The fraction of unsaturated nodes increases with the average capacity, as shown in the inset of Fig. 1(d). Hence at a low average capacity, saturated nodes form a percolating cluster, whereas at a high average capacity, it breaks into isolated clusters. It is interesting to note that at the average capacity of 0.45, below which a plateau starts to develop in the relaxation rate of the recursion relation Eq. (6), the fraction of unsaturated nodes is about 0.53, close to the percolation threshold of 0.5 for = 3. Besides the case of = 3, Fig. 1(c) also shows the simulation results of the average energy for = 4; 5, using both Eqs. (11) and (14). We see that the average energy decreases when the connectivity increases. This is because the increase in links connecting a node provides more freedom to allocate resources. When the average capacity is 0.2 or above, an exponential fit hE i exp( k hi) is applicable, where k lies in the range 2.5 to 2.7. Remarkably, multiplying by a factor of ( 2), we find that the 3 curves collapse in this regime of average capacity, showing that the average energy scales as ( 2) 1 in this regime, as shown in the inset of Fig. 1(c). Further properties of the optimized networks have been studied by simulations, and will be presented elsewhere. Here we merely summarize the main results. (a) When the average capacity drops below 0.1, the energy rises above the exponential fit applicable to the average capacity above 0.2. (b) The fraction of links with zero currents increases with the average capacity, and is rather insensitive to the connectivity. Remarkably, except for very small average capacities, the function erf (hi= p 2) has a very good fit with the data. Indeed, in the limit of large hi, this function approaches the fraction of links with both vertices unsaturated, that is, [ R 1 0 d()℄ 2. (c) The fraction of unsaturated nodes increases with the average capacity, and is rather insensitive to the connectivity. In the limit of large average capacities, it approaches the upper bound of R 1 0 d(), which is the probability that the capacity of a node is non-negative. (d) The convergence time of Eq. (11) can be measured by the time for the r.m.s. of the changes in the chemical potentials to fall below a threshold. Similarly, the convergence time of Eq. (14) can be measured by the time for the r.m.s. of the sums of the currents in both message directions of a link to fall below a threshold. When the average capacity is 0.2 or above, we find the power-law dependence on the average capacity, the exponent ranging from 1 for = 3 to 0:8 for = 5 for Eq. (14), and being about -0.5 for = 3; 4; 5 for Eq. (11). When the average capacity decreases further, the convergence time deviates above the power laws. 6 Summary We have studied a prototype problem of resource allocation on sparsely connected networks using the replica method, resulting in recursion relations interpretable using the Bethe approximation. The resultant recursion relation leads to a message passing algorithm for optimizing the average energy, which significantly reduces the computational complexity of the global optimization task and is suitable for online distributive control. The suggested 2-parameter approximation produces results with excellent agreement with the original recursion relation. For the simple but illustrative example in this letter, we have considered a quadratic cost function, resulting in an exact algorithm based on local iterations of chemical potentials, and the message passing algorithm shows remarkable agreement with the exact result. The suggested simple message passing algorithm can be generalized to more realistic cases of nonlinear cost functions and additional constraints on the capacities of nodes and links. This constitutes a rich area for further investigations with many potential applications. Acknowledgments This work is partially supported by research grants HKUST6062/02P and DAG04/05.SC25 of the Research Grant Council of Hong Kong and by EVERGROW, IP No. 1935 in the FET, EU FP6 and STIPCO EU FP5 contract HPRN-CT-2002-00319. References [1] Peterson L. and Davie B.S., Computer Networks: A Systems Approach, Academic Press, San Diego CA (2000) [2] Ho Y.C., Servi L. and Suri R. Large Scale Systems 1 (1980) 51 [3] Shenker S., Clark D., Estrin D. and Herzog S. ACM Computer Comm. Review 26 (1996) 19 [4] Nishimori H. Statistical Physics of Spin Glasses and Information Processing, OUP UK (2001) [5] M´ezard M., Parisi P. and Virasoro M., Spin Glass Theory and Beyond, World Scientific, Singapore (1987) [6] Wong K.Y.M. and Sherrington D. J. Phys. A20(1987) L793 [7] Sherrington D. and Kirkpatrick S. Phys. Rev. Lett.35 (1975) 1792 [8] Opper M. and Saad D. Advanced Mean Field Methods, MIT press (2001) [9] MacKay D.J.C., Information Theory, Inference and Learning Algorithms, CUP UK(2003)
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Neuronal Fiber Delineation in Area of Edema from Diffusion Weighted MRI Ofer Pasternak∗ School of Computer Science Tel-Aviv University Tel-Aviv, ISRAEL 69978 oferpas@post.tau.ac.il Nir Sochen Department of Applied Mathematics Tel-Aviv University sochen@post.tau.ac.il Nathan Intrator School of Computer Science Tel-Aviv University nin@post.tau.ac.il Yaniv Assaf Department of Neurobiochemistry Faculty of Life Science Tel-Aviv University assafyan@post.tau.ac.il Abstract Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) is a non invasive method for brain neuronal fibers delineation. Here we show a modification for DT-MRI that allows delineation of neuronal fibers which are infiltrated by edema. We use the Muliple Tensor Variational (MTV) framework which replaces the diffusion model of DT-MRI with a multiple component model and fits it to the signal attenuation with a variational regularization mechanism. In order to reduce free water contamination we estimate the free water compartment volume fraction in each voxel, remove it, and then calculate the anisotropy of the remaining compartment. The variational framework was applied on data collected with conventional clinical parameters, containing only six diffusion directions. By using the variational framework we were able to overcome the highly ill posed fitting. The results show that we were able to find fibers that were not found by DT-MRI. 1 Introduction Diffusion weighted Magnetic Resonance Imaging (DT-MRI) enables the measurement of the apparent water self-diffusion along a specified direction [1]. Using a series of Diffusion Weighted Images (DWIs) DT-MRI can extract quantitative measures of water molecule diffusion anisotropy which characterize tissue microstructure [2]. Such measures are in particular useful for the segmentation of neuronal fibers from other brain tissue which then allows a noninvasive delineation and visualization of major brain neuronal fiber bundles in vivo [3]. Based on the assumptions that each voxel can be represented by a single diffusion compartment and that the diffusion within this compartment has a Gaussian distribution ∗http://www.cs.tau.ac.il/∼oferpas DT-MRI states the relation between the signal attenuation, E, and the diffusion tensor, D, as follows [4, 5, 6]: E(qk) = A(qk) A(0) = exp(−bqT k Dqk) , (1) where A(qk) is the DWI for the k’th applied diffusion gradient direction qk. The notation A(0) is for the non weighted image and b is a constant reflecting the experimental diffusion weighting [2]. D is a second order tensor, i.e., a 3 × 3 positive semidefinite matrix, that requires at least 6 DWIs from different non-collinear applied gradient directions to uniquely determine it. The symmetric diffusion tensor has a spectral decomposition for three eigenvectors U a and three positive eigenvalues λa. The relation between the eigenvalues determines the diffusion anisotropy using measures such as Fractional Anisotropy (FA) [5]: FA = 3((λ1 −⟨D⟩)2 + (λ2 −⟨D⟩)2 + (λ3 −⟨D⟩)2) 2(λ2 1 + λ2 2 + λ2 3) , (2) where ⟨D⟩= (λ1 + λ2 + λ3)/3. FA is relatively high in neuronal fiber bundles (white matter), where the cylindrical geometry of fibers causes the diffusion perpendicular to the fibers be much smaller than parallel to them. Other brain tissues, such as gray matter and Cerebro-Spinal Fluid (CSF), are less confined with diffusion direction and exhibit isotropic diffusion. In cases of partial volume where neuronal fibers reside other tissue type in the same voxel, or present complex architecture, the diffusion has no longer a single pronounced orientation and therefore the FA value of the fitted tensor is decreased. The decreased FA values causes errors in segmentation and in any proceeding fiber analysis. In this paper we focus on the case where partial volume occurs when fiber bundles are infiltrated with edema. Edema might occur in response to brain trauma, or surrounding a tumor. The brain tissue accumulate water which creates pressure and might change the fiber architecture, or infiltrate it. Since the edema consists mostly of relatively free diffusing water molecules, the diffusion attenuation increases and the anisotropy decreases. We chose to reduce the effect of edema by changing the diffusion model to a dual compartment model, assuming an isotropic compartment added to a tensor compartment. 2 Theory The method we offer is based on the dual compartment model which was already demonstrated as able to reduce CSF contamination [7], where it required a large number of diffusion measurement with different diffusion times. Here we require the conventional DT-MRI data of only six diffusion measurement, and apply it on the edema case. 2.1 The Dual Compartment Model The dual compartment model is described as follows: E(qk) = f exp(−bqT k D1qk) + (1 −f) exp(−bD2) . (3) The diffusion tensor for the tensor compartment is denoted by D1, and the diffusion coefficient of the isotropic water compartment is denoted by D2. The compartments have relative volume of f and 1−f. Finding the best fitting parameters D1, D2 and f is highly ill-posed, especially in the case of six measurement, where for any arbitrarily chosen isotropic compartment there could be found a tensor compartment which exactly fits the data. Figure 1: The initialization scheme. In addition to the DWI data, MTV uses the T2 image to initialize f. The initial orientation for the tensor compartment are those that DT-MRI calculated. 2.2 The Variational Framework In order to stabilize the fitting process we chose to use the Multiple Tensor Variational (MTV) framework [8] which was previously used to resolve partial volume caused by complex fiber architecture [9], and to reduce CSF contamination in cases of hydrocephalus [10]. We note that the dual compartment model is a special case of the more general multiple tensor model, where the number of the compartments is restricted to 2 and one of the compartments is restricted to equal eigenvalues (isotropy). Therefore the MTV framework adapted for separation of fiber compartments from edema is composed of the following functional, whose minima should provide the wanted diffusion parameters: S(f, D1, D2) = Ω α d k=1 (E(qk) −ˆE(qk))2 + φ(|∇U 1 i |) dΩ. (4) The notation ˆE is for the observed diffusion signal attenuation and E is calculated using (3) for d different acquisition directions. Ωis the image domain with 3D axis (x, y, z), |∇I| = ( ∂I ∂x)2 + ( ∂I ∂y)2 + ( ∂I ∂z)2 is defined as the vector gradient norm. The notation U 1 i stands for the principal eigenvector of the i’th diffusion tensor. The fixed parameters α is set to keep the solution closer to the observed diffusion signal. The function φ is a diffusion flow function, which controls the regularization behavior. Here we chose to use φi(s) = 1 + s2 K2 i which lead to anisotropic diffusion-like flow while preserving discontinuities [11]. The regularized fitting allows the identification of smoothed fiber compartments and reduces noise. The minimum of (4) solves the Euler-Lagrange equations, and can be found by the gradient descent scheme. 2.3 Initialization Scheme Since the functional space is highly irregular (not enough measurements), the minimization process requires initial guess (figure 1), which is as close as possible to the global minimum. In order to apriori estimate the relative volume of the isotropic compartment we used a normalized diffusion non-weighted image, where high contrast correlates to larger fluid volume. In order to apriori estimate the parameters of D1 we used the result of conventional DT-MRI fitting on the original data. The DT-MRI results were spectrally decomposed and the eigenvectors were used as initial guess for the eigenvectors of D1. The initial guess for the eigenvalues of D1 were set to λ1 = 1.5, λ2 = λ3 = 0.4. 3 methods We demonstrate how partial volume of neuronal fiber and edema can be reduced by applying the modified MTV framework on a brain slice taken from a patient with sever edema surrounding a brain tumor. MRI was performed on a 1.5T MRI scanner (GE, Milwaukee). DT-MRI experiments were performed using a diffusion-weighted spin-echo echoplanar-imaging (DWI-EPI) pulse sequence. The experimental parameters were as follows: TR/TE = 10000/98ms, ∆/δ = 31/25ms, b = 1000s/mm2 with six diffusion gradient directions. 48 slices with thickness of 3mm and no gap were acquired covering the whole brain with FOV of 240mm2 and matrix of 128x128. Number of averages was 4, and the total experimental time was about 6 minutes. Head movement and image distortions were corrected using a mutual information based registration algorithm [12]. The corrected DWIs were fitted to the dual compartment model via the modified MTV framework, then the isotropic compartment was omitted. FA was calculated for the remaining tensor for which FA higher than 0.25 was considered as white matter. We compared these results to single component DT-MRI with no regularization, which was also used for initialization of the MTV fitting. 4 Results and Discussion Figure 2: A single slice of a patient with edema. (A) a non diffusion weighted image with ROI marked. Showing the tumor in black surrounded by sever edema which appear bright. (B) Normalized T2 of the ROI, used for f initialization. (C) FA map from DT-MRI (threshold of FA> 0.25). Large parts of the corpus callosum are obscured. (D) FA map of D1 from MTV (thresholds f> 0.35, FA> 0.25). A much larger part of the corpus callosum is revealed Figure (2) shows the Edema case, where DTI was unable to delineate large parts of the corpus callosum. Since the corpus callosum is one of the largest fiber bundles in the brain it was highly unlikely that the fibers were disconnected or disappeared. The expected FA should have been on the same order as on the opposite side of the brain, where the corpus callosum shows high FA values. Applying the MTV on the slice and mapping the FA value of the tensor compartment reveals considerably much more pixels of higher FA in the area of the corpus callosum. In general the FA values of most pixels were increased, which was predicted, since by removing any size of a sphere (isotropic compartment) we should be left with a shape which is less spherical, and therefore with increased FA. The benefit of using the MTV framework over an overall reduce of FA threshold in recognizing neuronal fiber voxels is that the amount of FA increase is not uniform in all tissue types. In areas where the partial volume was not big due to the edema, the increase was much lower than in areas contaminated with edema. This keeps the nice contrast reflected by FA values between neuronal fibers and other tissue types. Reducing the FA threshold on original DT-MRI results would cause a less clear separation between the fiber bundles and other tissue types. This tool could be used for fiber tracking in the vicinity of brain tumors, or with stroke, where edema contaminates the fibers and prevents fiber delineation with the conventional DT-MRI. 5 Conclusions We show that by modifying the MTV framework to fit the dual compartment model we can reduce the contamination of edema, and delineate much larger fiber bundle areas. By using the MTV framework we stabilize the fitting process, and also include some biological constraints, such as the piece-wise smoothness nature of neuronal fibers in the brain. There is no doubt that using a much larger number of diffusion measurements should increase the stabilization of the process, and will increase its accuracy. However, more measurement require much more scan time, which might not be available in some cases. The variational framework is a powerful tool for the modeling and regularization of various mappings. It is applied, with great success, to scalar and vector fields in image processing and computer vision. Recently it has been generalized to deal with tensor fields which are of great interest to brain research via the analysis of DWIs and DT-MRI. We show that the more realistic model of multi-compartment voxels conjugated with the variational framework provides much improved results. Acknowledgments We acknowledge the support of the Edersheim - Levi - Gitter Institute for Functional Human Brain Mapping of Tel-Aviv Sourasky Medical Center and Tel-Aviv University, the Adams super-center for brain research of Tel-Aviv University, the Israel Academy of Sciences, Israel Ministry of Science, and the Tel-Aviv University research fund. References [1] E Stejskal and JE Tanner. Spin diffusion measurements: Spin echoes in the presence of a time-dependant field gradient. J. Chem. Phys., 42:288–292, 1965. [2] D. Le-Bihan, J.-F. Mangin, C. Poupon, C.A. Clark, S. Pappata, N. Molko, and H. Chabriat. Diffusion tensor imaging: concepts and applications. Journal of Magnetic Resonance Imaging, 13:534–546, 2001. [3] S. Mori and P.C. van Zijl. Fiber tracking: principles and strategies - a technical review. NMR Biomed., 15:468–480, 2002. [4] P.J. Basser, J. Mattiello, and D. Le-Bihan. MR diffusion tensor spectroscopy and imaging. Biophysical Journal, 66:259–267, 1994. [5] P.J. Basser and C. Pierpaoli. Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. Journal of Magnetic Resonance, 111(3):209–219, June 1996. [6] C. Pierpaoli, P. Jezzard, P.J. Basser, A. Barnett, and G. Di-Chiro. Diffusion tensor MR imaging of human brain. Radiology, 201:637–648, 1996. [7] C. Pierpaoli and D. K. Jones. Removing CSF contamination in brain DT-MRIs by using a two-compartment tensor model. In Proc. International Society for Magnetic Resonance in Medicine 12th Scientific meeting ISMRM04, page 1215, Kyoto, Japan, 2004. [8] O. Pasternak, N. Sochen, and Y. Assaf. Variational regularization of multiple diffusion tensor fields. In J. Weickert and H. Hagen, editors, Visualization and Processing of Tensor Fields. Springer, Berlin, 2005. [9] O. Pasternak, N. Sochen, and Y. Assaf. Separation of white matter fascicles from diffusion MRI using φ-functional regularization. In Proceedings of 12th Annual Meeting of the ISMRM, page 1227, 2004. [10] O. Pasternak, N. Sochen, and Y. Assaf. CSF partial volume reduction in hydrocephalus using a variational framework. In Proceedings of 13th Annual Meeting of the ISMRM, page 1100, 2005. [11] G. Aubert and P. Kornprobst. Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, volume 147 of Applied Mathematical Sciences. Springer-Verlag, 2002. [12] G.K. Rohde, A.S. Barnett, P.J. Basser, S. Marenco, and C. Pierpaoli. Comprehensive approach for correction of motion and distortion in diffusion-weighted MRI. Magnetic Resonance in Medicine, 51:103–114, 2004.
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A Computational Model of Eye Movements during Object Class Detection Wei Zhang† Hyejin Yang‡∗ Dimitris Samaras† Gregory J. Zelinsky†‡ Dept. of Computer Science† Dept. of Psychology‡ State University of New York at Stony Brook Stony Brook, NY 11794 {wzhang,samaras}@cs.sunysb.edu† hjyang@ic.sunysb.edu∗ Gregory.Zelinsky@stonybrook.edu‡ Abstract We present a computational model of human eye movements in an object class detection task. The model combines state-of-the-art computer vision object class detection methods (SIFT features trained using AdaBoost) with a biologically plausible model of human eye movement to produce a sequence of simulated fixations, culminating with the acquisition of a target. We validated the model by comparing its behavior to the behavior of human observers performing the identical object class detection task (looking for a teddy bear among visually complex nontarget objects). We found considerable agreement between the model and human data in multiple eye movement measures, including number of fixations, cumulative probability of fixating the target, and scanpath distance. 1. Introduction Object detection is one of our most common visual operations. Whether we are driving [1], making a cup of tea [2], or looking for a tool on a workbench [3], hundreds of times each day our visual system is being asked to detect, localize, or acquire through movements of gaze objects and patterns in the world. In the human behavioral literature, this topic has been extensively studied in the context of visual search. In a typical search task, observers are asked to indicate, usually by button press, whether a specific target is present or absent in a visual display (see [4] for a review). A primary manipulation in these studies is the number of non-target objects also appearing in the scene. A bedrock finding in this literature is that, for targets that cannot be defined by a single visual feature, target detection times increase linearly with the number of nontargets, a form of clutter or ”set size” effect. Moreover, the slope of the function relating detection speed to set size is steeper (by roughly a factor of two) when the target is absent from the scene compared to when it is present. Search theorists have interpreted these findings as evidence for visual attention moving serially from one object to the next, with the human detection operation typically limited to those objects fixated by this ”spotlight” of attention [5]. Object class detection has also been extensively studied in the computer vision community, with faces and cars being the two most well researched object classes [6, 7, 8, 9]. The related but simpler task of object class recognition (target recognition without localization) has also been the focus of exciting recent work [10, 11, 12]. Both tasks use supervised learning methods to extract visual features. Scenes are typically realistic and highly cluttered, with object appearance varying greatly due to illumination, view, and scale changes. The task addressed in this paper falls between the class detection and recognition problems. Like object class detection, we will be detecting and localizing class-defined targets; unlike object class detection the test images will be composed of at most 20 objects appearing on a simple background. Both the behavioral and computer vision literatures have strengths and weaknesses when it comes to understanding human object class detection. The behavioral literature has accumulated a great deal of knowledge regarding the conditions affecting object detection [4], but this psychology-based literature has been dominated by the use of simple visual patterns and models that cannot be easily generalized to fully realistic scenes (see [13, 14] for notable exceptions). Moreover, this literature has focused almost entirely on object-specific detection, cases in which the observer knows precisely how the target will appear in the test display (see [15] for a discussion of target non-specific search using featurally complex objects). Conversely, the computer vision literature is rich with models and methods allowing for the featural representation of object classes and the detection of these classes in visually cluttered real-world scenes, but none of these methods have been validated as models of human object class detection by comparison to actual behavioral data. The current study draws upon the strengths of both of these literatures to produce the first joint behavioral-computational study of human object class detection. First, we use an eyetracker to quantify human behavior in terms of the number of fixations made during an object class detection task. Then we introduce a computational model that not only performs the detection task at a level comparable to that of the human observers, but also generates a sequence of simulated eye movements similar in pattern to those made by humans performing the identical detection task. 2. Experimental methods An effort was made to keep the human and model experiments methodologically similar. Both experiments used training, validation (practice trials in the human experiment), and testing phases, and identical images were presented to the model and human subjects in all three of these phases. The target class consisted of 378 teddy bears scanned from [16]. Nontargets consisted of 2,975 objects selected from the Hemera Photo Objects Collection. Samples of the bear and nontarget objects are shown in Figure 1. All objects were normalized to have a bounding box area of 8,000 pixels, but were highly variable in appearance. Figure 1: Representative teddy bears (left) and nontarget objects (right). The training set consisted of 180 bears and 500 nontargets, all randomly selected. In the case of the human experiment, each of these objects was shown centered on a white background and displayed for 1 second. The testing set consisted of 180 new bears and nontargets. No objects were repeated between training and testing, and no objects were repeated within either of the training or testing phases. Test images depicted 6, 13, or 20 color objects randomly positioned on a white background. A single bear was present in half (90) of these displays. Human subjects were instructed to indicate, by pressing a button, whether a teddy bear appeared among the displayed objects. Target presence and set size were randomly interleaved over trials. Each test trial in the human experiment began with the subject fixating gaze at the center of the display, and eye position was monitored throughout each trial using an eyetracker. Eight students from Stony Brook University participated in the experiment. 3. Model of eye movements during object class detection Figure 2: The flow of processing through our model. Building on a framework described in [17, 14, 18], our model can be broadly divided into three stages (Figure 2): (1) creating a target map based on a retinally-transformed version of the input image, (2) recognizing the target using thresholds placed on the target map, and (3) the operations required in the generation of eye movements. The following sub-sections describe each of the Figure 2 steps in greater detail. 3.1. Retina transform With each change in gaze position (set initially to the center of the image), our model transforms the input image so as to reflect the acuity limitations imposed by the human retina. We used the method described in [19, 20], which was shown to provide a close approximation to human acuity limitations, to implement this dynamic retina transform. 3.2. Create target map Each point on the target map ranges in value between 0 and 1 and indicates the likelihood that a target is located at that point. To create the target map, we first compute interest points on the retinally-transformed image (see section 3.2.2), then compare the features surrounding these points to features of the target object class extracted during training. Two types of discriminative features were used in this study: color features and texture features. 3.2.1. Color features Color has long been used as a feature for instance object recognition [21]. In our study we explore the potential use of color as a discriminative feature for an object class. Specifically, we used a normalized color histogram of pixel hues in HSV space. Because backgrounds in our images were white and therefore uninformative, we set thresholds on the saturation and brightness channels to remove these points. The hue channel was evenly divided into 11 bins and each pixel’s hue value was assigned to one of these bins using binary interpolation. Values within each bin were weighted by 1 −d, where d is the normalized unit distance to the center of the bin. The final color histogram was normalized to be a unit vector. Given a test image, It, and its color feature, Ht, we compute the distances between Ht and the color features of the training set {Hi, i = 1, ..., N}. The test image is labeled as: l(It) = l(Iarg min1≤i≤N χ2(Ht,Hi)), and the distance metric used was: χ2(Ht, Hi) = PK k=1 [Ht(k)−Hi(k)]2 Ht(k)+Hi(k) , where K is the number of bins. 3.2.2. Texture features Local texture features were extracted on the gray level images during both training and testing. To do this, we first used a Difference-of-Gaussion (DoG) operator to detect interest points in the image, then used a Scale Invariant Feature Transform (SIFT) descriptor to represent features at each of the interest point locations. SIFT features consist of a histogram representation of the gradient orientation and magnitude information within a small image patch surrounding a point [22]. AdaBoost is a feature selection method which produces a very accurate prediction rule by combining relatively inaccurate rules-of-thumb [23]. Following the method described in [11, 12], we used AdaBoost during training to select a small set of SIFT features from among all the SIFT features computed for each sample in the training set. Specifically, each training image was represented by a set of SIFT features {Fi,j, j = 1, ...ni}, where ni is the number of SIFT features in sample Ii. To select features from this set, AdaBoost first initialized the weights of the training samples wi to 1 2Np , 1 2Nn , where Np and Nn are the number of positive and negative samples, respectively. For each round of AdaBoost, we then selected one feature as a weak classifier and updated the weights of the training samples. Details regarding the algorithm used for each round of boosting can be found in [12]. Eventually, T features were chosen having the best ability to discriminate the target object class from the nontargets. Each of these selected features forms a weak classifier, hk, consisting of three components: a feature vector, (fk), a distance threshold, (θk), and an output label, (uk). Only the features from the positive training samples are used as weak classifiers. For each feature vector, F, we compute the distance between it and the training sample, i, defined as di = min1≤j≤ni D(Fi,j, F0), then apply the classification rule: h(f, θ) = { 1, d < θ 0, d ≥θ . (1) After the desired number of weak classifiers has been found, the final strong classifier can be defined as: H = T X t=1 αtht (2) where αt = log(1/βt). Here βt = q 1−ϵt ϵt and the classification error ϵt = P |uk −lk|. 3.2.3. Validation A validation set, consisting of the practice trials viewed by the human observers, was used to set parameters in the model. Because our model used two types of features, each having different classifiers with different outputs, some weight for combining these classifiers was needed. The validation set was used to set this weighting. The output of the color classifier, normalized to unit length, was based on the distance χ2 min = min1≤i≤N and defined as: Ccolor = { 0, l(It) = 0 f(χ2 min), l(It) = 1 (3) where f(χ2 min) is a function monotonically decreasing with respect to χ2 min. The strong local texture classifier, Ctexture (Equation 2), also had normalized unit output. The weights of the two classifiers were determined based on their classification errors on the validation set: Wcolor = ϵt ϵc+ϵt , Wtexture = ϵc ϵc+ϵt . (4) The final combined output was used to generate the values in the target map and, ultimately, to guide the model’s simulated eye movements. 3.3. Recognition We define the highest-valued point on the target map as the hotspot. Recognition is accomplished by comparing the hotspot to two thresholds, also set through validation. If the hotspot value exceeds the high target-present threshold, then the object will be recognized as an instance of the target class. If the hotspot value falls below the target-absent threshold, then the object will be classified as not belonging to the target class. Through validation, the target-present threshold was set to yield a low false positive rate and the target-absent threshold was set to yield a high true positive rate. Moreover, target-present judgments were permitted only if the hotspot was fixated by the simulated fovea. This constraint was introduced so as to avoid extremely high false positive rates stemming from the creation of false targets in the blurred periphery of the retina-transformed image. 3.4. Eye movement If neither the target-present nor the target-absent thresholds are satisfied, processing passes to the eye movement stage of our model. If the simulated fovea is not on the hotspot, the model will make an eye movement to move gaze steadily toward the hotspot location. Fixation in our model is defined as the centroid of activity on the target map, a computation consistent with a neuronal population code. Eye movements are made by thresholding this map over time, pruning off values that offer the least evidence for the target. Eventually, this thresholding operation will cause the centroid of the target map to pass an eye movement threshold, resulting in a gaze shift to the new centroid location. See [18] for details regarding the eye movement generation process. If the simulated fovea does acquire the hotspot and the target-present threshold is still not met, the model will assume that a nontarget was fixated and this object will be ”zapped”. Zapping consists of applying a negative Gaussian filter to the hotspot location, thereby preventing attention and gaze from returning to this object (see [24] for a previous computational implementation of a conceptually related operation). 4. Experimental results Model and human behavior were compared on a variety of measures, including error rates, number of fixations, cumulative probability of fixating the target, and scanpath ratio (a measure of how directly gaze moved to the target). For each measure, the model and human data were in reasonable agreement. Table 1: Error rates for model and human subjects. Total trials Misses False positives Frequency Rate Frequency Rate Human 1440 46 3.2% 14 1.0% Model 180 7 3.9% 4 2.2% Table 1 shows the error rates for the human subjects and the model, grouped by misses and false positives. Note that the data from all eight of the human subjects are shown, resulting in the greater number of total trials. There are two key patterns. First, despite the very high level of accuracy exhibited by the human subjects in this task, our model was able to Table 2: Average number of fixations by model and human. Case Target-present Target-absent p6 p13 p20 slope a6 a13 a20 slope Human 3.38 3.74 4.88 0.11 4.89 7.23 9.39 0.32 Model 2.86 3.69 5.68 0.20 3.97 8.30 10.47 0.46 achieve comparable levels of accuracy. Second, and consistent with the behavioral search literature, miss rates were larger than false positive rates for both the humans and model. To the extent that our model offers an accurate account of human object detection behavior, it should be able to predict the average number of fixations made by human subjects in the detection task. As indicated in Table 2, this indeed is the case. Data are grouped by targetpresent (p), target-absent (a), and the number of objects in the scene (6, 13, 20). In all conditions, the model and human subjects made comparable numbers of fixations. Also consistent with the behavioral literature, the average number of fixations made by human subjects in our task increased with the number of objects in the scenes, and the rate of this increase was greater in the target-absent data compared to the target-present data. Both of these patterns are also present in the model data. The fact that our model is able to capture an interaction between set size and target presence in terms of the number of fixations needed for detection lends support for our method. Figure 3: Cumulative probability of target fixation by model and human. Figure 3 shows the number of fixation data in more detail. Plotted are the cumulative probabilities of fixating the target as a function of the number of objects fixated during the search task. When the scene contained only 6 or 13 objects, the model and the humans fixated roughly the same number of nontargets before finally shifting gaze to the target. When the scene was more cluttered (20 objects), the model fixated an average of 1 additional nontarget relative to the human subjects, a difference likely indicating a liberal bias in our human subjects under these search conditions. Overall, these analyses suggest that our model was not only making the same number of fixations as humans, but it was also fixating the same number of nontargets during search as our human subjects. Table 3: Comparison of model and human scanpath distance #Objects 6 13 20 Human 1.62 2.20 2.80 Model 1.93 3.09 6.10 MODEL 1.93 2.80 3.43 Human gaze does not jump randomly from one item to another during search, but instead moves in a more orderly way toward the target. The ultimate test of our model would be to reproduce this orderly movement of gaze. As a first approximation, we quantify this behavior in terms of a scanpath distance. Scanpath distance is defined as the ratio of the total scanpath length (i.e., the summed distance traveled by the eye) and the distance between the target and the center of the image (i.e., the minimum distance that the eye would need to travel to fixate the target). As indicated in Table 3, the model and human data are in close agreement in the 6 and 13-object scenes, but not in the 20-object scenes. Upon closer inspection of the data, we found several cases in which the model made multiple fixations between two nontarget objects, a very unnatural behavior arising from too small of a setting for our Gaussian ”zap” window. When these 6 trials were removed, the model data (MODEL) and the human data were in closer agreement. Figure 4: Representative scanpaths. Model data are shown in thick red lines, human data are shown in thin green lines. Figure 4 shows representative scanpaths from the model and one human subject for two search scenes. Although the scanpaths do not align perfectly, there is a qualitative agreement between the human and model in the path followed by gaze to the target. 5. Conclusion Search tasks do not always come with specific targets. Very often, we need to search for dogs, or chairs, or pens, without any clear idea of the visual features comprising these objects. Despite the prevalence of these tasks, the problem of object class detection has attracted surprisingly little research within the behavioral community [15], and has been applied to a relatively narrow range of objects within the computer vision literature [6, 7, 8, 9]. The current work adds to our understanding of this important topic in two key respects. First, we provide a detailed eye movement analysis of human behavior in an object class detection task. Second, we incorporate state-of-the-art computer vision object detection methods into a biologically plausible model of eye movement control, then validate this model by comparing its behavior to the behavior of our human observers. Computational models capable of describing human eye movement behavior are extremely rare [25]; the fact that the current model was able to do so for multiple eye movement measures lends strength to our approach. Moreover, our model was able to detect targets nearly as well as the human observers while maintaining a low false positive rate, a difficult standard to achieve in a generic detection model. Such agreement between human and model suggests that simple color and texture features may be used to guide human attention and eye movement in an object class detection task. Future computational work will explore the generality of our object class detection method to tasks with visually complex backgrounds, and future human work will attempt to use neuroimaging techniques to localize object class representations in the brain. Acknowledgments This work was supported by grants from the NIMH (R01-MH63748) and ARO (DAAD1903-1-0039) to G.J.Z. References [1] M. F. Land and D. N. Lee. Where we look when we steer. Nature, 369(6483):742–744, 1994. [2] M. F. Land and M. Hayhoe. In what ways do eye movements contribute to everyday activities. Vision Research, 41(25-36):3559–3565, 2001. [3] G. Zelinsky, R. Rao, M. Hayhoe, and D. Ballard. Eye movements reveal the spatio-temporal dynamics of visual search. Psychological Science, 8:448–453, 1997. [4] J. Wolfe. Visual search. In H. Pashler (Ed.), Attention, pages 13–71. London: University College London Press, 1997. [5] E. Weichselgartner and G. Sperling. Dynamics of automatic and controlled visual attention. 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