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The Shallow and the Deep A biased introduction to neural networks and old school machine learning Michael Biehl | The+Shallow+and+the+Deep_Page_5_Chunk3901 |
Published by University of Groningen Press Broerstraat 4 9712 CP Groningen The Netherlands First published in the Netherlands © 2023 Michael Biehl, Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, Groningen Comments, corrections and suggestions are welcome, contact: m.biehl@rug.nl Plea... | The+Shallow+and+the+Deep_Page_6_Chunk3902 |
Preface Stop calling everything AI. — Michael I. Jordan, in [Pre21] The subtitle of these lecture notes is “A biased introduction to neural networks and old school machine learning” for good reasons. Although the aim was to give an accessible introduction to the field, it has been clear from the beginning that it would ... | The+Shallow+and+the+Deep_Page_7_Chunk3903 |
iv losing the appreciation for their fascinating power and versatility. Very often, this involves a look into the history and pre-history of neural networks, where the foundations for most of the recent developments were laid. I have aimed at pointing the interested reader to many resources for further exploration of t... | The+Shallow+and+the+Deep_Page_8_Chunk3904 |
Contents Preface iii 1 From neurons to networks 1 1.1 Spiking neurons and synaptic interactions . . . . . . . . . . . . . 3 1.2 Firing rate models . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Neural activity and synaptic interaction . . . . . . . . . . 5 1.2.2 Sigmoidal activation functions . . . . . . .... | The+Shallow+and+the+Deep_Page_9_Chunk3905 |
vi CONTENTS 3.4.1 The number of linearly separable dichotomies . . . . . . . 46 3.4.2 Discussion of the result . . . . . . . . . . . . . . . . . . . 52 3.4.3 Time for a pizza or some cake . . . . . . . . . . . . . . . . 53 3.5 Learning a linearly separable rule . . . . . . . . . . . . . . . . . . 55 3.5.1 Student-teach... | The+Shallow+and+the+Deep_Page_10_Chunk3906 |
. . . . . . . . . . . . 104 5 Feed-forward networks for regression and classification 107 5.1 Feed-forward networks as non-linear function approximators . . . 107 5.1.1 Architecture and input-output relation . . . . . . . . . . . 108 5.1.2 Universal approximators . . . . . . . . . . . . . . . . . . . 109 5.2 Gradient ba... | The+Shallow+and+the+Deep_Page_10_Chunk3907 |
CONTENTS vii 5.4.1 Sigmoidal and related functions . . . . . . . . . . . . . . . 127 5.4.2 One-sided and unbounded activation functions . . . . . . 128 5.4.3 Exponential and normalized activations . . . . . . . . . . 130 5.4.4 Remark: universal function approximation . . . . . . . . . 131 5.5 Specific architectures . . ... | The+Shallow+and+the+Deep_Page_11_Chunk3908 |
. . 176 7.4.2 Measures for classification . . . . . . . . . . . . . . . . . . 176 7.4.3 Receiver Operating Characteristics . . . . . . . . . . . . . 177 7.4.4 The area under the ROC curve . . . . . . . . . . . . . . . 180 7.4.5 Alternative measures for two-class problems . . . . . . . . 181 7.4.6 Multi-class problems . ... | The+Shallow+and+the+Deep_Page_11_Chunk3909 |
viii CONTENTS 8 Preprocessing and unsupervised learning 187 8.1 Normalization and transformations . . . . . . . . . . . . . . . . . 188 8.1.1 Coordinate-wise transformations . . . . . . . . . . . . . . 189 8.1.2 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . 191 8.2 Dimensionality reduction . . . . . .... | The+Shallow+and+the+Deep_Page_12_Chunk3910 |
. . . . 227 A.2.2 Example: unsolvable systems of linear equations . . . . . 228 A.3 Constrained optimization . . . . . . . . . . . . . . . . . . . . . . 230 A.3.1 Equality constraints . . . . . . . . . . . . . . . . . . . . . 230 A.3.2 Example: under-determined linear equations . . . . . . . 231 A.3.3 Inequality constr... | The+Shallow+and+the+Deep_Page_12_Chunk3911 |
CONTENTS ix A.5.1 Coordinate descent . . . . . . . . . . . . . . . . . . . . . . 239 A.5.2 Constrained problems and projected gradients . . . . . . 240 A.5.3 Stochastic gradient descent . . . . . . . . . . . . . . . . . 240 A.6 Example calculation of a gradient . . . . . . . . . . . . . . . . . . 243 List of figures 246... | The+Shallow+and+the+Deep_Page_13_Chunk3912 |
x CONTENTS | The+Shallow+and+the+Deep_Page_14_Chunk3913 |
Chapter 1 From neurons to networks Reality is overrated anyway. — Unknown To understand and explain the brain’s fascinating capabilities1 remains one of the greatest scientific challenges ever. This is particularly true for its plas- ticity, i.e. the ability to learn from experience, to adapt to and to survive in ever-c... | The+Shallow+and+the+Deep_Page_15_Chunk3914 |
2 1. FROM NEURONS TO NETWORKS specific task and context. Consequently, the interaction between the neuro- sciences and machine learning research continues to play an important role for the further development of both. In this introductory text we will consider learning systems, which draw on only the most basic mechanis... | The+Shallow+and+the+Deep_Page_16_Chunk3915 |
1.1. SPIKING NEURONS AND SYNAPTIC INTERACTIONS 3 1.1 Spiking neurons and synaptic interactions The physiology and functionality of the biological systems is highly complex, already on the single neuron level. Sophisticated modelling frameworks have been developed that take into account the relevant electro-chemical pro... | The+Shallow+and+the+Deep_Page_17_Chunk3916 |
4 1. FROM NEURONS TO NETWORKS Figure 1.1: Schematic illustration of neurons (pyramidal cells) and their con- nections. Left: Pre-synaptic and post-synaptic neurons with soma, dendritic tree, axon, and axonic branches. Right: The synaptic cleft with vesicles releas- ing neuro-transmitters and corresponding receptors on ... | The+Shallow+and+the+Deep_Page_18_Chunk3917 |
1.2. FIRING RATE MODELS 5 time [ms] Figure 1.2: Left (upper): Schematic illustration of an action potential, i.e. a short pulse on mV - and ms-scale. Left (lower): Spikes travel along the axon through saltatory conduction via gaps in the insulating myelin sheath. Right: Schematic illustration of how mean firing rates ar... | The+Shallow+and+the+Deep_Page_19_Chunk3918 |
6 1. FROM NEURONS TO NETWORKS Here, the quantities wij ∈R represent the strength of the synapse connecting one neuron j ∈J with neuron i. Positive wij > 0 increase the so-called local potential xi if neuron j is active (Sj > 0), while wij < 0 contribute negative terms to the weighted sum. Note that real world chemical ... | The+Shallow+and+the+Deep_Page_20_Chunk3919 |
1.2. FIRING RATE MODELS 7 Si γ θ xi = j wijSj Si θ xi = j wijSj Figure 1.3: Schematic illustration of symmetrized activation functions. Left: A sigmoidal transfer function with gain γ and threshold θ in the symmetrized representation, cf. Eq. (1.6). Right: The binary McCulloch Pitts activation as obtained in the ... | The+Shallow+and+the+Deep_Page_21_Chunk3920 |
8 1. FROM NEURONS TO NETWORKS McCulloch Pitts neurons Quite frequently, an even more drastic modification is considered: for infinite gain γ →∞the sigmoidal activation becomes a step function, see Fig. 1.3 (right panel) for an illustration. Eq. (1.5) for instance yields in this limit g(x) = sign(x −θ) = +1 if x ≥θ −1 ... | The+Shallow+and+the+Deep_Page_22_Chunk3921 |
1.3. NETWORK ARCHITECTURES 9 context of artificial neural networks, Hebbian synaptic plasticity provides a very plausible basis for the representation of learning in the models. In the mathematical framework of firing rate models presented in the previ- ous section, we can express Hebbian Learning quite elegantly, assumi... | The+Shallow+and+the+Deep_Page_23_Chunk3922 |
10 1. FROM NEURONS TO NETWORKS In the following, only the most basic and clear-cut types of network architec- tures are introduced and discussed, namely fully connected recurrent networks and feed-forward layered networks. The possibilities for modifications of these networks, as well as for hybrid and intermediate type... | The+Shallow+and+the+Deep_Page_24_Chunk3923 |
1.3. NETWORK ARCHITECTURES 11 ⇒ S(t = 0) S(t ≫1) Figure 1.4: Recurrent neural networks. Left: A network of N = 5 neurons with partial connectivity and uni-directional synapses. Right: Pattern retrieval from a noisy initial configuration in a Hopfield network of 2500 units, storing 100 activity patterns. Activities Sj = ±... | The+Shallow+and+the+Deep_Page_25_Chunk3924 |
12 1. FROM NEURONS TO NETWORKS the data set. Improved versions of the weight matrix for correlated patterns are also available. In principle, all perceptron training algorithms discussed later could be applied (per neuron) in the Hopfield network as well. The Hopfield network can operate as an auto-associative or content... | The+Shallow+and+the+Deep_Page_26_Chunk3925 |
1.3. NETWORK ARCHITECTURES 13 The left panel of Fig. 1.5 shows a schematic illustration of a feed-forward architecture. A specific, single layer of units (the top layer in the illustration) represents external input to the system in terms of neural activity. In the biological context, one might think of the photorecepto... | The+Shallow+and+the+Deep_Page_27_Chunk3926 |
14 1. FROM NEURONS TO NETWORKS S(ξ) ξj ∈R ξj ∈R wkj ∈R σk = g j wkjξj vk ∈R S(ξ) = g k vkσk Figure 1.5: Feed-forward neural networks.6 Left: A multilayered architecture with varying layer-size and a single output unit. Right: A convergent feed- forward network with a layer of input neurons, one hidden l... | The+Shallow+and+the+Deep_Page_28_Chunk3927 |
1.3. NETWORK ARCHITECTURES 15 or classes. Hence, the feed-forward network constitutes a classifier which can be adapted to example data by choice of weights and other free parameters. The simplest feed-forward classifier, the so-called perceptron, will serve as a very important example system in the following. The percep... | The+Shallow+and+the+Deep_Page_29_Chunk3928 |
16 1. FROM NEURONS TO NETWORKS | The+Shallow+and+the+Deep_Page_30_Chunk3929 |
Chapter 2 Learning from example data You live and learn. At any rate, you live. — Douglas Adams Different forms of machine learning were already briefly presented in Sec. 1. In the following section, we focus on the most clear-cut scenarios: supervised learning and unsupervised learning. In addition, we will briefly discu... | The+Shallow+and+the+Deep_Page_31_Chunk3930 |
18 2. LEARNING FROM EXAMPLE DATA analysis in contrast to supervised learning. We briefly revisit some key methods of unsupervised learning in the context of preprocessing in Chapter 8. Potential aims of unsupervised learning are quite diverse, a few examples being: ◦Data Reduction Frequently it makes sense to represent ... | The+Shallow+and+the+Deep_Page_32_Chunk3931 |
2.1. LEARNING SCENARIOS 19 to several of them. For instance, density estimation by means of Gaussian Mix- ture Models (GMM) could be interpreted as a probabilistic clustering method and the obtained centers of the GMM can also serve as prototypes in the context of Vector Quantization. Several relevant techniques of uns... | The+Shallow+and+the+Deep_Page_33_Chunk3932 |
20 2. LEARNING FROM EXAMPLE DATA ◦Classification The second important example of supervised problems is the assignment of observations to one of several categories or classes, i.e. to a discrete target value. A currently somewhat overstrained example is the discrimination of cats and dogs based on photographic images. A... | The+Shallow+and+the+Deep_Page_34_Chunk3933 |
2.1. LEARNING SCENARIOS 21 in version space relies on the assumption that (a) the target can be realized by the trained system in principle and that (b) the training data is perfectly reliable and noise-free. Although these assumptions are hardly ever realized in practice, the consideration of the idealized scenario pr... | The+Shallow+and+the+Deep_Page_35_Chunk3934 |
22 2. LEARNING FROM EXAMPLE DATA However, the main idea of supervised learning remains the same: additional data provides evidence for some hypotheses while others become less likely. 2.1.3 Other learning scenarios A variety of specific, relevant scenarios can be considered which deviate from the clear-cut simple cases ... | The+Shallow+and+the+Deep_Page_36_Chunk3935 |
2.2. MACHINE LEARNING VS. STATISTICAL MODELLING 23 2.2 Machine Learning vs. Statistical Modelling In the sciences it happens quite frequently that the same or very similar concepts and techniques are developed or rediscovered in different (sub-)disciplines, either in parallel or with significant delay. While it is – gene... | The+Shallow+and+the+Deep_Page_37_Chunk3936 |
24 2. LEARNING FROM EXAMPLE DATA In both, machine learning and statistical modelling, the aim is to extract information from observations or data and to formalize it. Most frequently, this is done by generating a mathematical model of some sort and fitting its parameters to the available data. Quite often, machine learn... | The+Shallow+and+the+Deep_Page_38_Chunk3937 |
2.2. MACHINE LEARNING VS. STATISTICAL MODELLING 25 v d Figure 2.1: Hubble dia- gram: the velocity v of galax- ies as a function of their dis- tance d, taken from [Hub29]. Note that the correct units of v should be km/s. According to PNAS, figure and article [Hub29] are in the public do- main. Here, Ho is the so-called H... | The+Shallow+and+the+Deep_Page_39_Chunk3938 |
26 2. LEARNING FROM EXAMPLE DATA is given. A hypothesis of the form fH(ξ) = N i=1 wi ξi = w⊤ξ = w · ξ with w ∈RN (2.3) is assumed to represent or approximate the dependence y(ξ) underlying the observed data set D. In analogy to other machine learning scenarios considered later, we will refer to the coefficients wj als... | The+Shallow+and+the+Deep_Page_40_Chunk3939 |
2.2. MACHINE LEARNING VS. STATISTICAL MODELLING 27 Note that the SSE is also a popular objective function in the context of re- gression in multi-layered networks, see Chapter 5 and e.g. [Bis95a,Bis06,HTF01, HKP91,EB01]. With the convenient matrix and vector notation5 Y = y1, y2, . . . , yP ⊤∈RP , χ = ξ1, ξ2, .... | The+Shallow+and+the+Deep_Page_41_Chunk3940 |
28 2. LEARNING FROM EXAMPLE DATA is by no means restricted to linear regression. Other methods of regularization will be discussed in the context of overfitting in neural networks in Sec. 7.2. We will revisit linear regression again in later chapters and show that it can also be formulated as the minimization of w2 unde... | The+Shallow+and+the+Deep_Page_42_Chunk3941 |
2.2. MACHINE LEARNING VS. STATISTICAL MODELLING 29 This correspondence of the Maximum Likelihood solution in the Gaussian model with a quadratic error measure is of course due to the specific mathemat- ical form of the normal distribution and can be rediscovered in various other contexts. The assumption of Gaussian nois... | The+Shallow+and+the+Deep_Page_43_Chunk3942 |
30 2. LEARNING FROM EXAMPLE DATA we can consider all possible model settings at a time, yielding the posterior predictive probability p(y|ξ, D) ∝ p(y|ξ, w) p(w|D) ∝p(D|w) po(w) dNw. (2.18) Properly normalized, this defines the probability of response y(ξ) to an arbitrary (novel) input ξ after having seen the... | The+Shallow+and+the+Deep_Page_44_Chunk3943 |
Chapter 3 The Perceptron The perceptron has shown itself worthy despite (and even because of!) its severe limitations. It has many features to attract attention: its linearity; its intriguing learning theorem; its clear paradigmatic simplicity as a kind of parallel compu- tation. — Marvin Minsky and Seymour Papert in [... | The+Shallow+and+the+Deep_Page_45_Chunk3944 |
32 3. THE PERCEPTRON units, but in these lecture notes the term perceptron always refers to the single layer, binary classifier. Even the very simple, limited perceptron architecture is of interest for a multitude of reasons: ◦Pioneered by Frank Rosenblatt [Ros58, Ros61], the perceptron has been one of the earliest, ver... | The+Shallow+and+the+Deep_Page_46_Chunk3945 |
3.2. LINEARLY SEPARABLE FUNCTIONS 33 Figure 3.1: The Mark I Perceptron. Left: Hardware realization at Cornell Aeronautical Laboratory. Photo reproduced with kind permission from Cornell University Library.2 The input of the Mark I was realized via a retina of 400 photosensors. Adaptive weights were represented by poten... | The+Shallow+and+the+Deep_Page_47_Chunk3946 |
34 3. THE PERCEPTRON ξ ∈RN (inputs) w ∈RN (weights) S = sign(w · ξ −Θ) = ±1 (output) Figure 3.2: Illustration of the single layer perceptron with N-dimensional inputs and a binary output of the McCulloch Pitts type. corresponds to a (hyper-)plane orthogonal to to w with an off-set θ |w| from the origin3. Inputs with w ·... | The+Shallow+and+the+Deep_Page_48_Chunk3947 |
3.2. LINEARLY SEPARABLE FUNCTIONS 35 θ 0 S =−1 S =+1 w/|w| Figure 3.3: Geometrical in- terpretation of the perceptron. The hyperplane orthogonal to w with off-set θ from the origin separates feature vectors with output S = +1 and S = −1, respectively. In the following, we will also call a set of P input/output pairs D =... | The+Shallow+and+the+Deep_Page_49_Chunk3948 |
36 3. THE PERCEPTRON 3.3 The Rosenblatt perceptron To a large extent, the success of the perceptron has been due to the existence of a training algorithm and the associated convergence theorem, both presented by Frank Rosenblatt [Ros58,Ros61]. In the following, we first precisely define the basic goal of the training pro... | The+Shallow+and+the+Deep_Page_50_Chunk3949 |
3.3. THE ROSENBLATT PERCEPTRON 37 The existence of weights w with all Eµ ≥c > 0 also implies that a solution for any other positive constant can be constructed. This is a consequence of the fact that the function Sw(ξ) = sign(w · ξ) only depends on the direction of w in N-dim. feature space while it is invariant under ... | The+Shallow+and+the+Deep_Page_51_Chunk3950 |
38 3. THE PERCEPTRON updates at time steps t = 1, 2, . . . τ the weight vector is bound to have the form w(τ) = 1 N P µ=1 xµ(τ) ξµ Sµ T . (3.13) This implies that the resulting perceptron weight vector is a linear combi- nation of the vectors ξµ ∈D and the so-called embedding strengths xµ(τ) ∈R quantify their specifi... | The+Shallow+and+the+Deep_Page_52_Chunk3951 |
3.3. THE ROSENBLATT PERCEPTRON 39 3.3.3 The Rosenblatt perceptron algorithm In terms of the generic algorithm (3.12, 3.14), the Rosenblatt perceptron algo- rithm is specified by - tabula rasa initial conditions: w(0) = 0 or, equivalently, {xµ(0) = 0}P µ=1 - deterministic, cyclic presentation of the examples in D accordi... | The+Shallow+and+the+Deep_Page_53_Chunk3952 |
40 3. THE PERCEPTRON t=0 t=1 t=2 t=3 t=4 t=5 t=6 —————————————————————– t = 0 : w(0) = 0 all xµ = 0 t = 1 : w(1) = 1 2ξ1 x1 →1 t = 2 : w(2) = w(1) zero update t = 3 : w(3) = w(2) + 1 2ξ3 x3 →1 t = 4 : w(4) = w(3) zero update t = 5 : w(5) = w(4) −1 2ξ5 x5 →1 t = 6 : w(6) = w(5) alg. terminates —————————————————————– Fig... | The+Shallow+and+the+Deep_Page_54_Chunk3953 |
3.3. THE ROSENBLATT PERCEPTRON 41 D correctly, and the algorithm stops already after one sweep through the data set, i.e. one epoch of training. We will show in Sec. 3.3.5 that the Rosenblatt algorithm (3.15) converges in a finite number of steps and finds a weight vector that solves the percep- tron storage problem (3.7... | The+Shallow+and+the+Deep_Page_55_Chunk3954 |
42 3. THE PERCEPTRON in the deterministic sequential order (3.11). The formulation in terms of embed- ding strengths has the structure of coordinate descent as discussed in Appendix A.5.1: at every step, only one xµ is updated. However, its potential increment (3.18) is not given by the derivative ∂/∂xµ of E in Eq. (3.... | The+Shallow+and+the+Deep_Page_56_Chunk3955 |
3.3. THE ROSENBLATT PERCEPTRON 43 We do not have to further specify w∗here. In fact, for a given D there could be many solutions of the form (3.23), but here it is sufficient to assume the existence of at least one. We will furthermore denote its squared norm as Q∗≡w∗· w∗= |w∗|2. (3.24) Note that any pair of vectors w, w... | The+Shallow+and+the+Deep_Page_57_Chunk3956 |
44 3. THE PERCEPTRON Θ[c −Eν(t)] = 0 and Eν(t) ≥c in a zero learning step, while Θ[c −Eν(t)] = 1 and Eν(t) < c in a non-zero learning step. As a consequence, we can replace all Eν(t) by c in Eq. (3.29) to obtain the upper bound Q(t + 1) ≤Q(t) + 2 N c Θ[c −Eν(t)] + 1 N Γ Θ[c −Eν(t)] (3.31) Here we exploit the fact that ... | The+Shallow+and+the+Deep_Page_58_Chunk3957 |
3.3. THE ROSENBLATT PERCEPTRON 45 3.3.6 A few remarks The number of training steps According to the PCT, the required number of training steps is finite for linearly separable data. However, their actual number depends on the detailed proper- ties of D and can be very large, see [Roj96] for a discussion of the computati... | The+Shallow+and+the+Deep_Page_59_Chunk3958 |
46 3. THE PERCEPTRON On the other hand, (2) states that a linear combination of the form P µ=1 yµ ξµ Sµ T = 0 with {yµ ≥0}P µ=1 and y ∕= 0 exists. This goes beyond the condition of linearly dependent columns ξµSµ T of χ, which would be relevant in the context of solving equations. The consideration of inequalities... | The+Shallow+and+the+Deep_Page_60_Chunk3959 |
3.4. THE CAPACITY OF A HYPERPLANE 47 Figure 3.5: One-dimensional inputs ξ can be separated into two classes by the origin in C(P, 1) = 2 ways, independent of P. The derivation for general P and N has been published several times in the literature [Win61, Cov65, MD89a] and was also reviewed in [HKP91]. Here we follow th... | The+Shallow+and+the+Deep_Page_61_Chunk3960 |
48 3. THE PERCEPTRON a) b) c) d) Figure 3.6: Panel (a): Two two-dimensional feature vectors (large filled circles) in general position, i.e. not collinear with the origin (small filled circle). Here, four linearly separable dichotomies exist: either both inputs are assigned to the same class S1 = S2 ∈{−1, +1} or they are... | The+Shallow+and+the+Deep_Page_62_Chunk3961 |
3.4. THE CAPACITY OF A HYPERPLANE 49 0 0 Figure 3.7: Two homogeneously linearly separable dichotomies DP =9 N=2 of the same set of two-dimensional feature vectors. In both panels, all separating planes in the grey-shaded areas would realize the same assignment of labels. General position condition (3.38) A set of vecto... | The+Shallow+and+the+Deep_Page_63_Chunk3962 |
50 3. THE PERCEPTRON 0 0 ξP +1 ξP +1 Figure 3.8: An additional feature vector ξP +1 is added to the data set of Fig. 3.7. Its label is either non-ambiguously determined by the linearly separable function as in the left panel, or the label is ambiguous, SP +1 = +1 or −1, as shown in the right panel. through the origin t... | The+Shallow+and+the+Deep_Page_64_Chunk3963 |
3.4. THE CAPACITY OF A HYPERPLANE 51 ξP +1 ξP +1 H H Figure 3.9: The projection of data into the auxiliary subspace H ⊥ξP +1 is either linearly separable in H (right panel, ambiguous case) or not separable as in the non-ambiguous case (left panel), respectively. However the following consideration shows that Z(P, N) ca... | The+Shallow+and+the+Deep_Page_65_Chunk3964 |
52 3. THE PERCEPTRON Figure 3.10: The fraction Pls = C(P, N)/2P of linearly separable functions versus α = P/N for three different values of N (5, 20, 100). C(P, N)/2P α = P/N N =5 N =100 with the familiar bionomial coefficients m k = m! k!(m−k)!. The expression in the second line of (3.41) would also reproduce 2P fo... | The+Shallow+and+the+Deep_Page_66_Chunk3965 |
3.4. THE CAPACITY OF A HYPERPLANE 53 Figure 3.10 shows the fraction of linearly separable dichotomies C(P, N)/2P as a function of P/N. This scaling allows to conveniently display several curves for selected values of N in one graph. The most striking and - at first sight - counterintuitive feature of the result (3.41) i... | The+Shallow+and+the+Deep_Page_67_Chunk3966 |
54 3. THE PERCEPTRON ⇒ Figure 3.11: The Pizza Connnection: K planar cuts through the center (•) of an N-dim. sphere (left panel) correspond to K −1 arbitrary straight cuts through the (N −1)-dim. surface of each flattened hemisphere (right panel). The corresponding result for three-dimensional objects is known as the ca... | The+Shallow+and+the+Deep_Page_68_Chunk3967 |
3.5. LEARNING A LINEARLY SEPARABLE RULE 55 3.5 Learning a linearly separable rule Obviously it is not the ultimate goal of perceptron training to reproduce the labels in a given data set, only. This could be done quite efficiently by simply storing D in memory and look it up when needed. In general, it is the aim of mach... | The+Shallow+and+the+Deep_Page_69_Chunk3968 |
56 3. THE PERCEPTRON Figure 3.12: Generalization er- ror of the perceptron in a stu- dent/teacher scenario. For N- dim. random input vectors gener- ated according to an isotropic den- sity, the probability of disagree- ment between student vector w and teacher w∗is proportional to the red shaded area, i.e. to the angle... | The+Shallow+and+the+Deep_Page_70_Chunk3969 |
3.5. LEARNING A LINEARLY SEPARABLE RULE 57 ξ w ∈RN |w|=const. • V(3) Figure 3.13: Dual geometrical interpretation of linear separability. Illustra- tion in terms of labeled input vectors ξ ∈R3 and L2-normalized weight vectors with |w| = const. on the surface of an N-dim. hypersphere. Left: A single, labeled input ξ1 se... | The+Shallow+and+the+Deep_Page_71_Chunk3970 |
58 3. THE PERCEPTRON V(3) = V(4) ξ1 ξ2 ξ3 ξ4 ξ1 ξ2 ξ3 ξ4 ξ5 Figure 3.14: Illustration of perceptron learning in version space. Left: The hyperplane associated with the additional example ξ4 does not in- tersect the version space V(3). Consequently, all weight vectors in V(3) already classify ξ4 correctly and V(4) = V(3... | The+Shallow+and+the+Deep_Page_72_Chunk3971 |
3.5. LEARNING A LINEARLY SEPARABLE RULE 59 that, given a set D of P reliable examples for a linearly separable rule, we have identified some vector w ∈V. According to the Perceptron Convergence The- orem (3.22) this is always possible, e.g. by means of the Rosenblatt perceptron algorithm10. In our low-dimensional illust... | The+Shallow+and+the+Deep_Page_73_Chunk3972 |
60 3. THE PERCEPTRON Figure 3.15: The generalization error g vs. α = P/N as ob- tained from the number of ambigu- ous / non-ambiguous linearly sep- arable functions, cf. Eq. 3.52. The curves correspond to N = 5, 20, 100 and to the limit N →∞, respec- tively (from left to right). g α = P/N 3.5.3 Generalization begin... | The+Shallow+and+the+Deep_Page_74_Chunk3973 |
3.5. LEARNING A LINEARLY SEPARABLE RULE 61 ++ +− −+ −− +− w∗ w Figure 3.16: Linear separability for P < N. Left panel: Version space corresponding to two feature vectors in N = 3 dimensions with all (four) possible combinations of labels ±1. Right panel: Assume that, as an example, the region marked +−corresponds to th... | The+Shallow+and+the+Deep_Page_75_Chunk3974 |
62 3. THE PERCEPTRON Asymptotic behavior for N →∞ For large N →∞with P/N = α one can show that the asymptotic form of Eq. (3.52) reads [Cov65] g(α) = 1/2 for α ≤2 1 2 1 α −1 for α > 2. (3.53) As an alternative to the counting argument, methods borrowed from statistical physics have been applied to compute th... | The+Shallow+and+the+Deep_Page_76_Chunk3975 |
3.6. THE PERCEPTRON OF OPTIMAL STABILITY 63 In practice, sampling the version space with uniform density is a non-trivial task. The theoretical background and practical strategies for how to achieve the optimal generalization ability when learning a linearly separable rule are discussed in e.g. [Wat93,OH91,Ruj97]. 3.6 ... | The+Shallow+and+the+Deep_Page_77_Chunk3976 |
64 3. THE PERCEPTRON κ κµ ξµ wmax κ Figure 3.17: Stability of the perceptron. Left: The stability κµ, defined in Eq. (3.56), corresponds to the oriented distance of ξµ from the plane orthogonal to w. The stability of the perceptron κ(w) is defined as the smallest κµ in the set of examples, i.e. κ(w) = minµ{κµ}. Here, all... | The+Shallow+and+the+Deep_Page_78_Chunk3977 |
3.6. THE PERCEPTRON OF OPTIMAL STABILITY 65 As a consequence, the perceptron of optimal stability should display favor- able (near optimal) generalization behavior when trained from a given reliable, lin. sep. data set. In fact, the difference appears marginal from a practical perspective. For a theoretical comparison o... | The+Shallow+and+the+Deep_Page_79_Chunk3978 |
66 3. THE PERCEPTRON stopping criterion. Instead, one of the following quantities could be con- sidered: - the angular change ∠ w(t), w(t+T) = 1 π arccos w(t) · w(t+T) |w(t)||w(t+T)| or the argument of the arccos, for simplicity. - the total change of stabilities P µ=1 κµ(t) −κµ(t+T) 2 , for example. ... | The+Shallow+and+the+Deep_Page_80_Chunk3979 |
3.7. OPTIMAL STABILITY BY QUADRATIC OPTIMIZATION 67 On the other hand we have |w2|2 = |w1 + δ|2 = |w1|2 + 2 w1 · δ =0 + |δ|2 >0 ⇒ |w2| > |w1|, (3.65) where the mixed term vanishes because w1 is a linear combination of the ξµ⊥δ. As a consequence, we observe that κµ 2 = Eµ 2 |w2| = Eµ 1 |w2| < Eµ 1 |w1|... | The+Shallow+and+the+Deep_Page_81_Chunk3980 |
68 3. THE PERCEPTRON which is just a more compact version of (3.58). Obviously, the stability κ can be made larger by increasing the minimal Eµ for constant norm |w|. Analogously, κ increases with decreasing norm |w| if all local potentials obey the constraint Eµ ≥c > 0. As discussed previously, the actual choice of th... | The+Shallow+and+the+Deep_Page_82_Chunk3981 |
3.7. OPTIMAL STABILITY BY QUADRATIC OPTIMIZATION 69 The Adaline framework essentially treats the problem of binary classification as a linear regression with subsequent thresholding of the continuous w · ξµ. We will take a rather formal perspective based on the theory of Lagrange multipliers [Fle00, PAH19] as outlined i... | The+Shallow+and+the+Deep_Page_83_Chunk3982 |
70 3. THE PERCEPTRON In addition, we use the notation a > b, which is popular in the optimization related literature and indicates that aµ > bµ for all µ = 1, 2, ....P. Analogous notations are employed for component-wise relations “<”, “≥” and “≤”. In the convenient matrix-vector notation, we have furthermore Eν = ... | The+Shallow+and+the+Deep_Page_84_Chunk3983 |
3.7. OPTIMAL STABILITY BY QUADRATIC OPTIMIZATION 71 Sequential Adaline algorithm As an important alternative to the above batch or parallel update, sequential gradient-based methods can be devised, which present the example data repeat- edly in, for instance, deterministic order and update a single embedding strength i... | The+Shallow+and+the+Deep_Page_85_Chunk3984 |
72 3. THE PERCEPTRON One can also show that the objective function is bounded from above if C is positive definite and Cx = 1 has a solution: while the linear term 1⊤x in f, Eq. (3.77), can grow in an unbounded way, the negative quadratic term will always dominate and limit the increase. Hence, unless a solution... | The+Shallow+and+the+Deep_Page_86_Chunk3985 |
3.7. OPTIMAL STABILITY BY QUADRATIC OPTIMIZATION 73 This consideration also indicates what the behavior of the Adaline will be if E = 1 cannot be satisfied: the algorithm finds an approximate solution by minimizing the SSE. The sequential algorithm (3.81) in weight space is equivalent to Widrow and Hoff’s original LMS (... | The+Shallow+and+the+Deep_Page_87_Chunk3986 |
74 3. THE PERCEPTRON The Kuhn-Tucker Theorem of optimization theory, see [Fle00,PAH19], pro- vides the first order necessary stationarity conditions for general, non-linear optimization problems with inequality and/or equality conditions. They are known as the so-called Kuhn-Tucker (KT) or Karush-Kuhn-Tucker conditions ... | The+Shallow+and+the+Deep_Page_88_Chunk3987 |
3.7. OPTIMAL STABILITY BY QUADRATIC OPTIMIZATION 75 have E∗µ = 1 while input/output pairs with E∗µ > 1 do not contribute to the linear combination (3.61). We will discuss this property in greater detail later. Complementarity also implies that x∗µE∗µ = x∗µ for all µ and, therefore x∗⊤Cx∗= x∗⊤E∗= P µ=1 x∗µE∗µ... | The+Shallow+and+the+Deep_Page_89_Chunk3988 |
76 3. THE PERCEPTRON As outlined in Appendix A.3.4 the following formulation is equivalent to problem (3.68): Perceptron of optimal stability (embedding strengths, dual problem) maximize x f(x) = −1 2x⊤Cx + x⊤1 subject to x ≥0. (3.97) Hence, the resulting Wolfe Dual still comprises constraints, albeit sim... | The+Shallow+and+the+Deep_Page_90_Chunk3989 |
3.7. OPTIMAL STABILITY BY QUADRATIC OPTIMIZATION 77 separable [AB89,BAK91].12 We consider the learning step xµ →xµ + δµ with δµ = η (1 −Eµ) if −xµ ≤η (1 −Eµ) −xµ < 0 if −xµ > η (1 −Eµ) (3.99) and see that in both cases |δµ| ≤η |(1 −Eµ)| =⇒1 −Eµ δµ ≥1 η. Hence, similar to Eq. (3.83) for the Adaline algor... | The+Shallow+and+the+Deep_Page_91_Chunk3990 |
78 3. THE PERCEPTRON Embedding strengths only In contrast to the Rosenblatt or Adaline algorithm, it is not straightforward to rewrite the AdaTron in terms of explicit updates in weight space. Obviously we can compute the new weight vector as w(t + 1) = P µ=1 xµ(t + 1)ξµSµ T after each training step. However, a direc... | The+Shallow+and+the+Deep_Page_92_Chunk3991 |
3.7. OPTIMAL STABILITY BY QUADRATIC OPTIMIZATION 79 Eµ =1 ∝wmax Figure 3.18: Support Vectors in the perceptron of optimal stability. The arrow represents the normalized weight vector wmax/|wmax|. Class membership is indicated by filled (ST = +1) and open symbols (ST = −1), respectively. Support vectors, marked as square... | The+Shallow+and+the+Deep_Page_93_Chunk3992 |
80 3. THE PERCEPTRON relatively small. In the limit of infinitely many update steps, properly rescaled xµ satisfy condition (3.92). 3.8 Inhom. lin. sep. functions revisited So far, following the arguments leading to Eq. (3.4) in Sec. 3.2 we have treated homogeneously and inhomogeneously linearly separable functions on t... | The+Shallow+and+the+Deep_Page_94_Chunk3993 |
3.9. SOME REMARKS 81 3.9 Some remarks In this chapter we have considered the simple perceptron as a prototypical machine learning system. It serves as a framework in which to obtain insights into the basic concepts of supervised learning. It also illustrates the importance of optimization techniques and the related the... | The+Shallow+and+the+Deep_Page_95_Chunk3994 |
82 3. THE PERCEPTRON | The+Shallow+and+the+Deep_Page_96_Chunk3995 |
Chapter 4 Beyond linear separability Non-linear means it’s hard to solve. — Arthur Mattuck In the previous sections we studied learning in version space as a basic train- ing strategy in terms of the simple perceptron classifier. We obtained important insights into the principles of learning a rule and obtained the conc... | The+Shallow+and+the+Deep_Page_97_Chunk3996 |
84 4. BEYOND LINEAR SEPARABILITY i) The unknown target rule is linearly separable, but D contains mislabeled examples, for instance in the presence of noise. Depending on the number of examples P and the degree of the corruption the following may occur: i.1) In particular, small data sets D with few mislabeled samples ... | The+Shallow+and+the+Deep_Page_98_Chunk3997 |
4.1. PERCEPTRON WITH ERRORS 85 ◦Many perceptrons (or other simple classifiers) can be combined into an ensemble in order to take advantage of a wisdom of the crowd effect [OM99,Urb00]. So-called Random Forests, i.e. ensembles of decision trees [Bre01], constitute one of the most prominent examples in the literature, curr... | The+Shallow+and+the+Deep_Page_99_Chunk3998 |
86 4. BEYOND LINEAR SEPARABILITY Minimal number of errors (Perceptron) (4.1) For a given D = {ξµ, Sµ T }P µ=1 with ξµ ∈RN and Sµ T ∈{−1, +1} , minimize w ∈RN Herr(w) = P µ=1 (w, ξµ, Sµ T ) with = 1 if sign (w · ξµ) = −Sµ T 0 if sign (w · ξµ) = +Sµ T . Note that this cost function does not differentiate between... | The+Shallow+and+the+Deep_Page_100_Chunk3999 |
4.1. PERCEPTRON WITH ERRORS 87 Obviously, the number of errors H(t) of the pocket vector w(t) can never increase under the updates (4.2). Moreover, one can show that the stochastic selection of the training sample guarantees that, in principle, w(t) approaches the minimum of Herr with probability one [Gal90,Roj96... | The+Shallow+and+the+Deep_Page_101_Chunk4000 |
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