text stringlengths 1 7.76k | source stringlengths 17 81 |
|---|---|
88 4. BEYOND LINEAR SEPARABILITY We recover the unmodified problem of optimal stability (3.68) with βµ = 0 for all µ = 1, 2, ..., P, which also implies that all Eµ ≥1. On the contrary, non- zero βµ > 0 correspond to violations of the original constraints, i.e. examples with Eµ < 1, which includes potential misclassificat... | The+Shallow+and+the+Deep_Page_102_Chunk4001 |
4.1. PERCEPTRON WITH ERRORS 89 Eµ =1 ∝wmax Figure 4.1: Support vectors in the soft margin perceptron. The arrow represents the normalized weight vector wmax/|wmax|. Filled and open symbols correspond to the classes Sµ T = ±1, respectively. Sup- port vectors, displayed as squares, fall onto the two hyperplanes with Eµ =... | The+Shallow+and+the+Deep_Page_103_Chunk4002 |
90 4. BEYOND LINEAR SEPARABILITY input layer ξ ∈RN adaptive weights wk ∈RN (k = 1, 2, . . . K) hidden units σk = sign wk · ξ−θk fixed hidden-to-output relation F(...) binary output S ξ = F σk(ξ) K k=1 Figure 4.2: The architecture of a “machine” as introduced in Sec. 4.2. A number K of hidden units o... | The+Shallow+and+the+Deep_Page_104_Chunk4003 |
4.2. LAYERED NETWORKS OF PERCEPTRON-LIKE UNITS 91 4.2.1 Committee and parity machines Two specific machines have attracted particular interest: CM: The committee machine combines the hidden unit states σk in a majority vote [EB01,MD89a,WRB93,AMB+18]. This is realized by setting F CM σkK k=1 = sign K k=1 σk... | The+Shallow+and+the+Deep_Page_105_Chunk4004 |
92 4. BEYOND LINEAR SEPARABILITY + + + + + − + −+ + −− −+ + −−+ −+ − + + + + + − + −− + −+ −+ + −−+ −+ − Figure 4.3: Output function of Committee machine and Parity machine, applied to identical sets of feature vectors ξ (filled and empty circles). The illustration corresponds to the surface of an N-dim. hypersphere. In... | The+Shallow+and+the+Deep_Page_106_Chunk4005 |
4.2. LAYERED NETWORKS OF PERCEPTRON-LIKE UNITS 93 Tiling-like learning, (parity machine) (4.9) (I) Initialization (m = 1): Train the first unit with output S1(ξ) = σ1(ξ) = sign[w1 · ξ −θ1] from the data set D1 = D , aiming at a large number of correctly classified examples Q1.4 (II) After training of m units: Given the P... | The+Shallow+and+the+Deep_Page_107_Chunk4006 |
94 4. BEYOND LINEAR SEPARABILITY Grandmother neuron (4.10) Consider a set of feature vectors {ξµ}P µ=1 with |ξµ|2 = Γ for all µ = 1, 2, ..., P (4.11) 0 < δ < Γ −ξµ · ξν for all µ, ν (µ ∕= ν). (4.12) Construct a perceptron weight vector and threshold as w = −ξP and θ = δ −Γ. (4.13) It results in the inhomogeneously line... | The+Shallow+and+the+Deep_Page_108_Chunk4007 |
4.2. LAYERED NETWORKS OF PERCEPTRON-LIKE UNITS 95 ◦Tiling-like learning imposes a strong ordering of the hidden units. Neurons added to the system later are supposed to correct only the (hopefully) very few misclassifications made by the first units. To some extent this contradicts the attractive concept of neural networ... | The+Shallow+and+the+Deep_Page_109_Chunk4008 |
96 4. BEYOND LINEAR SEPARABILITY Mitchison and Durbin [MD89a], we can extend the arguments presented in Sec. 3.4 to machines. The number CK(P, N) of different dichotomies that a machine with K hid- den units can realize for P feature vectors in N dimensions is obviously bounded from above as CK(P, N) ≤C(P, N)K (4.16) wi... | The+Shallow+and+the+Deep_Page_110_Chunk4009 |
4.3. SUPPORT VECTOR MACHINES 97 K αc Figure 4.4: The storage capac- ity αc(K) according to the esti- mate (4.18), obtained for N = 100. The solid line corresponds to the naive lower bound 2K for combining K perceptrons. that a (parity) machine with a sufficiently large number of units can implement any given dichotomy. I... | The+Shallow+and+the+Deep_Page_111_Chunk4010 |
98 4. BEYOND LINEAR SEPARABILITY first practical version of Support Vector Machines, close to their current form, was introduced by Boser, Guyon and Vapnik in 1992 [BGV92] and relates to early algorithms developed by Vladimir Vapnik in the 1960s [VL63]. According to Isabelle Guyon [Guy16], the MinOver algorithm [KM87] t... | The+Shallow+and+the+Deep_Page_112_Chunk4011 |
4.3. SUPPORT VECTOR MACHINES 99 Figure 4.5: Illustration courtesy of Rainer Dietrich [Die00]. A two-dimensional data set with two classes that are not linearly separable (left panel) can become linearly separable after applying an appropriate non-linear transformation to a higher-dimensional space (center and right pan... | The+Shallow+and+the+Deep_Page_113_Chunk4012 |
100 4. BEYOND LINEAR SEPARABILITY and – following the arguments provided in Sec. 3.5.2 – we can expect good generalization ability. The mathematical structure of the corresponding problem is fully analogous to the original (3.58). The M-dim. counterpart reads Perceptron of optimal stability (M-dim. feature space) (4.22... | The+Shallow+and+the+Deep_Page_114_Chunk4013 |
4.3. SUPPORT VECTOR MACHINES 101 The output of the M-dim. perceptron can be written as S(ξ) = sign W · Ψ(ξ) = sign 1 M P µ=1 Xµ SµΨµ · Ψ(ξ) . (4.26) We note that this involves the scalar products of the M-dimensional, trans- formed input vector with the transformed example training examples Ψµ. We define ... | The+Shallow+and+the+Deep_Page_115_Chunk4014 |
102 4. BEYOND LINEAR SEPARABILITY c) Determine the embedding strengths Xmax corresponding to optimal sta- bility in the M-dim. weight space, for instance by use of the AdaTron (4.30). d) Classify an arbitrary ξ ∈RN according to S(ξ) = sign P µ=1 Xµ max SµK(ξµ, ξ) In practice, of course, the problem is to find ... | The+Shallow+and+the+Deep_Page_116_Chunk4015 |
4.3. SUPPORT VECTOR MACHINES 103 Polynomial kernels A polynomial kernel of degree q can be written as K(ξµ, ξ) = (1 + ξµ · ξ )q yielding S(ξ) = sign P µ=1 Xµ Sµ T (1 + ξµ · ξ )q (4.32) as the input-output relation of the classifier. As a special case, let us consider the simplest polynomial kernel: Linear kerne... | The+Shallow+and+the+Deep_Page_117_Chunk4016 |
104 4. BEYOND LINEAR SEPARABILITY As intuitively expected, the use of the quadratic kernel represents the non- linear mapping from N-dim. feature space to in total M = N(N + 3)/2 trans- formed features (original, squares and mixed products). An explicit formulation is reminiscent of Quadratic Discriminant Analysis (QDA... | The+Shallow+and+the+Deep_Page_118_Chunk4017 |
4.3. SUPPORT VECTOR MACHINES 105 Soft-margin SVM In addition to the choice of the kernel and potential parameters thereof, one of- ten resorts to a soft margin version of the SVM [CV95], see also [SS02,CST00, STC04,Her02,DFO20]. The considerations of Sec. 4.1.2 for the simple percep- tron immediately carry over to the ... | The+Shallow+and+the+Deep_Page_119_Chunk4018 |
106 4. BEYOND LINEAR SEPARABILITY | The+Shallow+and+the+Deep_Page_120_Chunk4019 |
Chapter 5 Feed-forward networks for regression and classification The fishermen in the north of Spain have been using Deep Networks for centuries. Their contribution should be recognized . . . — Javier Movellan Layered Neural Networks have regained significant popularity due to their impressive successes in the context of... | The+Shallow+and+the+Deep_Page_121_Chunk4020 |
108 5. NETWORKS FOR REGRESSION AND CLASSIFICATION ξj ≡σ(1) j w(1) kj , j = 1, . . . N S(1) k , k = 1, . . . K(1) S(2) k , k = 1, . . . K(2) w(2) kj , j = 1, . . . K(1) . . . vk ≡w(L) 1k , j = 1, . . . K(1) σ(ξ) ≡S(L) 1 Figure 5.1: A feed-forward neural network with N input units, L−1 hidden layers and a single output u... | The+Shallow+and+the+Deep_Page_122_Chunk4021 |
5.1. NON-LINEAR FUNCTION APPROXIMATORS 109 represent the local threshold by a weight w(M−1) k0 ≡θ(M) k . This would parallel our representation of inhomogeneously linear separable functions in Eq. (3.4). We can include the input layer in the the notation of Eq. (5.1) by defining S(0) j ≡ξj. Similarly, we can rename the ... | The+Shallow+and+the+Deep_Page_123_Chunk4022 |
110 5. NETWORKS FOR REGRESSION AND CLASSIFICATION z g2 3(z) sigm. linear z g5 a(z) −g5 b(z) Figure 5.2: Left panel: A sigmoidal activation of the form (5.3) with γ = 2, zo = 3 as an example. Right panel: The difference of two steep sigmoidals (here: γ = 5, a = 2, b = 6, respectively) singles out arguments a ≤z ≤b. The i... | The+Shallow+and+the+Deep_Page_124_Chunk4023 |
5.1. NON-LINEAR FUNCTION APPROXIMATORS 111 2 i=1 Gγ [ai,bi](ξi) ξ2 ξ1 2 0 1 ξ2 ξ1 ⇒ gγ N−1 2 (. . .) 0 1 Figure 5.3: N threshold nodes (steep sigmoidal units) which select a specific interval per input dimension can be combined by adding up their activation. The left panel shows an illustration for N = 2 and ξj ∈[−10... | The+Shallow+and+the+Deep_Page_125_Chunk4024 |
112 5. NETWORKS FOR REGRESSION AND CLASSIFICATION Figure 5.4: The constructed network for piecewise constant function approx- imation: Each input unit (top layer) is connected to a set of sigmoidal units in the second layer. Pairs of these connect to linear units in the third layer which select specific intervals [ai, b... | The+Shallow+and+the+Deep_Page_126_Chunk4025 |
5.1. NON-LINEAR FUNCTION APPROXIMATORS 113 input layer ξ ∈RN input-to-hidden weights w(k) ∈RN (k = 1, 2, . . . K) hidden units Sk = g w(k) · ξ−θ(k) hidden-to-output weights vk linear output σ ξ = K k=1 vk Sk Figure 5.5: A so-called Soft Committee Machine realizes a functional approx- imation as considered ... | The+Shallow+and+the+Deep_Page_127_Chunk4026 |
114 5. NETWORKS FOR REGRESSION AND CLASSIFICATION This implies that for any continuous target function τ ∈C([0, 1]N) and a given real number ε > 0, parameters w(k) ∈RN, vk ∈R, θ(k) ∈R K k=1 exist with σ(ξ) −τ(ξ) < ε for all ξ ∈[0, 1]N . The parameters can be interpreted as the weights and thresholds of... | The+Shallow+and+the+Deep_Page_128_Chunk4027 |
5.2. GRADIENT BASED TRAINING OF FEED-FORWARD NETS 115 To this end, we define an error measure which is suitable for the comparison of the network output σ(ξ) with the target function τ(ξ) for a given input vector. A popular and intuitive choice is the simple quadratic deviation e(σ, τ) = 1 2 (σ −τ)2 . (5.8) Here and in ... | The+Shallow+and+the+Deep_Page_129_Chunk4028 |
116 5. NETWORKS FOR REGRESSION AND CLASSIFICATION is the successful application of the trained system to novel data. Hence, the precision to which a minimum is determined can play a minor role and the potential existence of many (suboptimal) local minima of E is not as problematic as one might expect. 5.2.1 Computing t... | The+Shallow+and+the+Deep_Page_130_Chunk4029 |
5.2. GRADIENT BASED TRAINING OF FEED-FORWARD NETS 117 In analogy to Eq. (A.39) in the Appendix, the basic form of the updates is given as Batch gradient descent (basic form) at discrete time step t perform an update step of the form W(t + 1) = W(t) −η ∇W E|W =W (t) (5.10) with the learning rate η and cost function E of... | The+Shallow+and+the+Deep_Page_131_Chunk4030 |
118 5. NETWORKS FOR REGRESSION AND CLASSIFICATION a) η ≤ 1 ρmax b) 1 ρmax <η≤ 2 ρmax c) η > 2 ρmax Figure 5.6: Illustration of the behavior of gradient descent near a local min- imum W ∗(marked by the red dot in the center). The contour lines represent the quadratic approximation of E(W) in the vicinity of a local mini... | The+Shallow+and+the+Deep_Page_132_Chunk4031 |
5.2. GRADIENT BASED TRAINING OF FEED-FORWARD NETS 119 of the training process may depend strongly on the initialization of the system. Therefore, the practical relevance of the mathematical analysis is limited. In practice, a relatively large η could be used in the initial phase of training, assuming that the system is... | The+Shallow+and+the+Deep_Page_133_Chunk4032 |
120 5. NETWORKS FOR REGRESSION AND CLASSIFICATION The learning rate is denoted as η(t) in order to indicate a possible explicit time-dependence and to distinguish it from η in batch gradient descent, Eq. (5.10). Clearly, the computational costs per update are lower than in the batch procedure which involves the sum o... | The+Shallow+and+the+Deep_Page_134_Chunk4033 |
5.2. GRADIENT BASED TRAINING OF FEED-FORWARD NETS 121 Figure 5.7: Schematic illustration of the behavior of stochastic gradient de- scent near a local minimum W ∗as marked by the (red) dot in the center. The contour lines represent the quadratic approximation of E(W) in the vicinity of W ∗. Left panel: Black symbols co... | The+Shallow+and+the+Deep_Page_135_Chunk4034 |
122 5. NETWORKS FOR REGRESSION AND CLASSIFICATION Various schedules which satisfy the conditions (I) and (II) of Eq. (5.16) can be considered, including power laws η(t) ∝t−β, logarithmic schedules like η(t) ∝1/(t ln t), or other explicitly time-dependent schemes, see for instance [DM92] and references therein. Stoc... | The+Shallow+and+the+Deep_Page_136_Chunk4035 |
5.3. OBJECTIVE FUNCTIONS 123 Local learning rates: For both, batch and stochastic gradient descent, it has been suggested to use local learning rates for different layers, nodes, or even individual weights in the network. As an early, simple rule of thumb, Plaut et al. [PNH86] suggest to use local learning rates inverse... | The+Shallow+and+the+Deep_Page_137_Chunk4036 |
124 5. NETWORKS FOR REGRESSION AND CLASSIFICATION heuristic assumptions or concrete statistical models of the observations may mo- tivate the use of alternative objective functions in the training process. More- over, the use of differentiable neural networks for classification tasks motivates the use of cost functions w... | The+Shallow+and+the+Deep_Page_138_Chunk4037 |
5.3. OBJECTIVE FUNCTIONS 125 labels in a regression problem deviate from the true target function by indepen- dent noise terms ηµ with P(ηµ) ∝e−β |ηµ|r with r > 0, (5.19) which is normalized to P(ηµ)dηµ = 1. Obviously we recover a Gaussian density with a β-dependent variance for r = 2. The optimization of the corre-... | The+Shallow+and+the+Deep_Page_139_Chunk4038 |
126 5. NETWORKS FOR REGRESSION AND CLASSIFICATION likelihood of generating a given set of labels {τ µ}P µ=1 with the network reads P µ=1 (σµ)τ µ [1 −σµ]1−τ µ with the shorthand σµ = σ(ξµ). Maximizing this likelihood by choice of the network parameters W is equivalent to minimizing the negative log-likelihood E(W) = ... | The+Shallow+and+the+Deep_Page_140_Chunk4039 |
5.4. ACTIVATION FUNCTIONS 127 Now the equivalent of cost function (5.23) becomes DKL(τ||σ) = − P µ=1 C k=1 τ µ k ln σµ k τ µ k , (5.24) which reduces to (5.23) in the binary case with C = 2 where σ1 = 1 −σ2 and τ1 = 1 −τ2. Given a network structure with C outputs σk as described above, we can determine the a... | The+Shallow+and+the+Deep_Page_141_Chunk4040 |
128 5. NETWORKS FOR REGRESSION AND CLASSIFICATION x g(x) x g(x) Figure 5.8: Sigmoidal and related activation functions. Left panel: three differential sigmoidal functions. Right panel: The limiting case of the McCul- loch Pitts activation (Heaviside step function) and a piecewise linear function. 5.4.2 One-sided and unb... | The+Shallow+and+the+Deep_Page_142_Chunk4041 |
5.4. ACTIVATION FUNCTIONS 129 x g(x) x g(x) Figure 5.9: Unbounded and one-sided activation functions. Left panel: Sim- ple linear activation g(x) = x (dotted), Rectified Linear Unit ReLU (solid), Eq. (5.25), and leaky ReLU, Eq. (5.26), with a = 0.25 (dashed). Right panel: Ex- ponential linear unit ELU (dashed, black), c... | The+Shallow+and+the+Deep_Page_143_Chunk4042 |
130 5. NETWORKS FOR REGRESSION AND CLASSIFICATION - the Exponential Linear Unit (ELU) [CUH16] with g(x) = exp[x] −1 for x < 0 x for x ≥0, (5.28) - the Swish function [EYG92] with g(x) = x 1 + exp[−x]. (5.29) Interestingly, the Swish activation is even non-monotonic and displays a min- imum in a negative value of the... | The+Shallow+and+the+Deep_Page_144_Chunk4043 |
5.5. SPECIFIC ARCHITECTURES 131 A most prominent example is the Gaussian RBF, which is frequently referred to as the RBF: σ = exp −β (s −c)2 with parameter β > 0. (5.32) Frequently, normalized Gaussian RBF are considered in a layer of hidden or output units σk(k = 1, 2, . . . K): σk(s) = exp −β(s −ck)... | The+Shallow+and+the+Deep_Page_145_Chunk4044 |
132 5. NETWORKS FOR REGRESSION AND CLASSIFICATION 5.5.1 Popular shallow networks So far, we have developed training prescriptions in terms of shallow, feed-forward architectures with only one or very few hidden layers. In particular we have seen that a single hidden layer is sufficient to provide universal function appro... | The+Shallow+and+the+Deep_Page_146_Chunk4045 |
5.5. SPECIFIC ARCHITECTURES 133 input layer ξ ∈RN fixed random input-to-hidden weights wm ∈RN (m = 1, 2, . . . M) hidden units σm = g (wm · ξ) hidden-to-output weights vm linear output S ξ = M m=1 vm σm Figure 5.10: Illustration of an Extreme Learning Machine (ELM). The N- dimensional input is connected to a hi... | The+Shallow+and+the+Deep_Page_147_Chunk4046 |
134 5. NETWORKS FOR REGRESSION AND CLASSIFICATION original input ξ ∈RN wm ∈RN (m = 1, 2, . . . M) encoder decoder latent variables ym = g (wm · ξ) vm ∈RN (m = 1, 2, . . . M) ξrec = M m=1 ymvm Figure 5.11: An example of a shallow auto-encoder network: The N-dim. inputs are represented in a hidden layer with M < N uni... | The+Shallow+and+the+Deep_Page_148_Chunk4047 |
5.5. SPECIFIC ARCHITECTURES 135 to the well known Principal Component Analysis, e.g. [Bis95a,HKP91,HTF01]. More precisely, the weight vectors wm, which minimize Erec, span the same sub-space as the M leading principal components of the data set. Using sigmoidal or other non-trivial hidden and/or output activations in t... | The+Shallow+and+the+Deep_Page_149_Chunk4048 |
136 5. NETWORKS FOR REGRESSION AND CLASSIFICATION Figure 5.12: A 3 × 3 ’convolutional’ filter kernel is applied to a 3 × 3 image, here zero-padded. The 3 × 3 kernel with weights denoted in the illustration is centered on every pixel of the image to obtain the pixel values in the 3 × 3 convolved image. Note that the oper... | The+Shallow+and+the+Deep_Page_150_Chunk4049 |
5.5. SPECIFIC ARCHITECTURES 137 Figure 5.13: The 3 × 3 convolved image from Fig. 5.12 is reduced to 2 × 2 pixels by applying a max-pooling (upper) or average pooling (lower) to all 2 × 2 patches in the 3 × 3 filtered image. process, e.g. by using gradient based Backpropagation of Error for the entire network. A number K... | The+Shallow+and+the+Deep_Page_151_Chunk4050 |
138 5. NETWORKS FOR REGRESSION AND CLASSIFICATION Figure 5.14: An early deep architecture (schematic) known as the Neocog- nitron, first introduced by K. Fukushima in 1980 (1979 in Japanese) [Fuk80]. Besides input and output (recognition), several layers of so-called S-cells and C-cells are stacked see the text for deta... | The+Shallow+and+the+Deep_Page_152_Chunk4051 |
5.5. SPECIFIC ARCHITECTURES 139 Figure 5.15: A deep architecture (schematic) known as LeNet, specifically LeNet-5, introduced by LeCun et al. in [LBD+89]. Image available under license CC BY3.0 at https://www.researchgate.net/publication/319905492_Image_retrieval_ method_based_on_metric_learning_for_convolutional_neural... | The+Shallow+and+the+Deep_Page_153_Chunk4052 |
140 5. NETWORKS FOR REGRESSION AND CLASSIFICATION in critical areas like clinical medicine. Recently, several scholars have expressed criticism of Deep Learning and the hype surrounding it, with [Mar18,Pre21,Zad19,AN20] being just a few examples. In a sense, the situation is highly reminiscent of the strong expectation... | The+Shallow+and+the+Deep_Page_154_Chunk4053 |
Chapter 6 Distance-based classifiers One can state, without exaggeration, that the observation of and the search for similarities and differences are the basis of all human knowledge. — Alfred Nobel The use of distances or dissimilarities for the comparison of observations with a set of labeled reference data points prov... | The+Shallow+and+the+Deep_Page_155_Chunk4054 |
142 6. DISTANCE-BASED CLASSIFIERS Figure 6.1: Left panel: Illustration of the Nearest Neighbor (NN) Classi- fier for an artificial data set containing three different classes. Right panel: A corresponding NPC scheme for the same data. Prototypes are represented by larger symbols. Both schemes are based on Euclidean distan... | The+Shallow+and+the+Deep_Page_156_Chunk4055 |
6.1. PROTOTYPE-BASED CLASSIFIERS 143 6.1 Prototype-based classifiers Among the many frameworks developed for supervised machine learning, proto- type-based systems are particularly intuitive, flexible, and easy to implement. Although we restrict the discussion to classification problems, many of the con- cepts carry over ... | The+Shallow+and+the+Deep_Page_157_Chunk4056 |
144 6. DISTANCE-BASED CLASSIFIERS fact that every data point contributes with equal weight can lead to overfitting effects because the classifier over-rates the importance of individual examples. As a consequence, it might not perform well when presented with novel, unseen data. Straightforward implementations of kNN comp... | The+Shallow+and+the+Deep_Page_158_Chunk4057 |
6.1. PROTOTYPE-BASED CLASSIFIERS 145 can be expected to be more robust and less prone to overfitting effects. The performance of LVQ systems has proven to be competitive in a variety of practical classification problems [Neu02]. In addition, their flexibility and inter- pretability constitute important advantages of protot... | The+Shallow+and+the+Deep_Page_159_Chunk4058 |
146 6. DISTANCE-BASED CLASSIFIERS Ψ(c∗ µ, yµ) = +1. On the contrary, w∗ µ is moved farther away from ξµ if the winning prototype represents a class different from yµ, i.e. Ψ(c∗ µ, yµ) = −1. A popular initialization strategy is to place prototypes in the class-conditional mean vectors in the data set, i.e. wj(0) = P µ... | The+Shallow+and+the+Deep_Page_160_Chunk4059 |
6.1. PROTOTYPE-BASED CLASSIFIERS 147 training data, while for small γ the minimization of EGLV Q corresponds to maximizing the margin-like quantities eµ. A popular and conceptually simple strategy to optimize EGLV Q is stochastic gradient descent in which single examples are presented in randomized order [RM51,Bot91,FP... | The+Shallow+and+the+Deep_Page_161_Chunk4060 |
148 6. DISTANCE-BASED CLASSIFIERS 6.2 Distance measures and relevance learning So far, the discussion focussed on Euclidean distance as a standard measure for the comparison of data points and prototypes. This choice appears natural and it is arguably the most popular one. One has to be aware, however, that other choic... | The+Shallow+and+the+Deep_Page_162_Chunk4061 |
6.2. DISTANCE MEASURES AND RELEVANCE LEARNING 149 in dp(ξ, ξ) = N j=1 ξj −ξj p1/p for ξ, ξ ∈RN, (6.10) which includes Euclidean distance as a special case for p = 2. Larger (smaller) values of p put emphasis on the components ξj and ξj with larger (smaller) deviations |ξj −ξj|, respectively... | The+Shallow+and+the+Deep_Page_163_Chunk4062 |
150 6. DISTANCE-BASED CLASSIFIERS The elegant framework of relevance learning allows for a significant con- ceptual extension of distance-based classification. It is particularly suitable for prototype systems and was was introduced and put forward in the context of LVQ in [HV02,SBH09,Sch10,Bun11,SBS+10,BSH+12], for inst... | The+Shallow+and+the+Deep_Page_164_Chunk4063 |
6.2. DISTANCE MEASURES AND RELEVANCE LEARNING 151 significantly when local distances are used, i.e. when separate relevance matrices are employed per class or even per prototype [SBH09,BSH+12]. Here we restrict the discussion to the simplest case of a single, N ×N matrix Ωcorresponding to a global distance measure. The ... | The+Shallow+and+the+Deep_Page_165_Chunk4064 |
152 6. DISTANCE-BASED CLASSIFIERS Figure 6.2: Visualization of the Generalized Matrix Relevance LVQ system as obtained from the z-score transformed Iris flower data set, see Sec. 6.2.2 for details. Left panel: Class prototypes are shown as bar plots with respect to the four feature space components in the left column. T... | The+Shallow+and+the+Deep_Page_166_Chunk4065 |
6.3. CONCLUDING REMARKS 153 displays very good generalization behavior with respect to validation or test set performance not presented here. In the left panel, the prototypes after training and the resulting relevance matrix and its eigenvalues are displayed. As discussed above, the diagonal ele- ments Λii can be inte... | The+Shallow+and+the+Deep_Page_167_Chunk4066 |
154 6. DISTANCE-BASED CLASSIFIERS | The+Shallow+and+the+Deep_Page_168_Chunk4067 |
Chapter 7 Model evaluation and regularization Accuracy is not enough. — Paulo Lisboa In supervised learning the aim is to infer relevant information from given data, to parameterize it in terms of a model, and to apply it to novel data successfully. It is obviously essential to know or at least have some estimate of th... | The+Shallow+and+the+Deep_Page_169_Chunk4068 |
156 7. MODEL EVALUATION AND REGULARIZATION K = 1 K = 3 K = 7 DA DB DC Figure 7.1: Illustration of the bias-variance dilemma in regression. In each row of graphs, a particular set of 10 points {xi, yi}10 i=1 is approximated by least square linear regression (K = 1), by a cubic fit (K = 3), and by fitting a polynomial of d... | The+Shallow+and+the+Deep_Page_170_Chunk4069 |
7.1. BIAS AND VARIANCE, OVER- AND UNDERFITTING 157 represent a function f(x) which is of course unknown to the learning system. We assume that the training labels are noisy versions of the true targets: yµ = f(xµ) + rµ with 〈rµ〉= 0 and 〈rµ rν〉= ρ2δµν (7.1) with the Kronecker-delta δµν. Hence, the deviation of the train... | The+Shallow+and+the+Deep_Page_171_Chunk4070 |
158 7. MODEL EVALUATION AND REGULARIZATION For the sake of brevity, we omit the argument x ∈R of the functions fH and f in the following. Including redundant terms (*) which add up to zero we can rewrite (7.5) as 〈fH〉2 D ∗ −2 〈fH(x)〉D f + f 2 + f 2 H D −2 〈fH〉2 D + 〈fH〉2 D ∗ (7.6) and obtain a... | The+Shallow+and+the+Deep_Page_172_Chunk4071 |
7.1. BIAS AND VARIANCE, OVER- AND UNDERFITTING 159 model B target fits of model A model parameter 2 adaptive parameter 1 prediction error test set performance ←high bias high variance → training set performance a measure of the model complexity Figure 7.2: Left panel: Illustration of the bias-variance Dilemma. The true ... | The+Shallow+and+the+Deep_Page_173_Chunk4072 |
160 7. MODEL EVALUATION AND REGULARIZATION from being overly specific to the individual data sets. As illustrated in the left column of Fig. 7.1, we achieve nearly identical linear models from the different data sets. However, a price is paid for the robustness: systematic deviations occur in each training procedure. We ... | The+Shallow+and+the+Deep_Page_174_Chunk4073 |
7.1. BIAS AND VARIANCE, OVER- AND UNDERFITTING 161 model’s flexibility will reduce the bias and, consequently, training and test set error decrease with K in Fig. 7.2 (right panel). However, overly training set specific models display overfitting: while the training error typically decreases further with increasing K, the... | The+Shallow+and+the+Deep_Page_175_Chunk4074 |
162 7. MODEL EVALUATION AND REGULARIZATION prediction error test error training error number of parameters, model complexity interpolation threshold “classical” regime “modern” regime Figure 7.3: Illustration of the double descent phenomenon, after [BHMM19]. descent as a function of the number of adaptive parameters. T... | The+Shallow+and+the+Deep_Page_176_Chunk4075 |
7.2. CONTROLLING THE NETWORK COMPLEXITY 163 ularization, cf. Sec. 7.2, would not necessarily display the peaking and double descent [Wit20]. The influence of implicit and explicit regularization on the emergence of double descent is also discussed in the context of ordinary least squares regression in [KLS20]. In genera... | The+Shallow+and+the+Deep_Page_177_Chunk4076 |
164 7. MODEL EVALUATION AND REGULARIZATION W(0) = 0 W(tmax) W(0)=0 W ∗ W ∗ Figure 7.4: Schematic illustration of early stopping and weight decay. Ellipses correspond to contour lines of the objective function. The blue solid lines rep- resent the unrestricted hypothetical updates by, for instance, gradient descent. Lef... | The+Shallow+and+the+Deep_Page_178_Chunk4077 |
7.2. CONTROLLING THE NETWORK COMPLEXITY 165 basis of assuming prior knowledge in the context of statistical learning the- ory [HTF01, HKP91, Bis95a, Bis06, DHS00]. In the linear regression problem, weight decay facilitates the construction or computation of a meaningful solu- tion of the regression problem. Here we ext... | The+Shallow+and+the+Deep_Page_179_Chunk4078 |
166 7. MODEL EVALUATION AND REGULARIZATION which also leads to (7.10).2 As an alternative to the use of a penalty term, sometimes a constraint of the type |W|2 ≤c with constant c > 0 is imposed. This can be done explicitly by projecting back onto the sphere in weight space |W|2 = c whenever the constraint is violated b... | The+Shallow+and+the+Deep_Page_180_Chunk4079 |
7.2. CONTROLLING THE NETWORK COMPLEXITY 167 (LASSO), see [HTF01] for a thorough discussion and comparison with L2-based Ridge Regression. Similar to the above discussed heuristic penalty (7.12), L1 regularization can also be used to enforce some weights to become exactly zero. It thus also relates to feature selection ... | The+Shallow+and+the+Deep_Page_181_Chunk4080 |
168 7. MODEL EVALUATION AND REGULARIZATION ourself to the discussion of some early works that represent the basic ideas and inspired later, more specific schemes. We also limit the discussion to strategies for the removal of weights rather than entire units or layers. The latter is frequently referred to as structural p... | The+Shallow+and+the+Deep_Page_182_Chunk4081 |
7.2. CONTROLLING THE NETWORK COMPLEXITY 169 Optimal Brain Damage (OBD) (7.17) 1. Train a network until a local minimum W ∗of E is reached or sufficiently well approximated 2. Compute the diagonal second derivatives H∗ kk 3. Compute the saliencies skk = H∗ kkW 2 k 4. Sort the weights by saliency and set some low-saliency ... | The+Shallow+and+the+Deep_Page_183_Chunk4082 |
170 7. MODEL EVALUATION AND REGULARIZATION full network with all units and connections network after dilution in Dropout Figure 7.5: Illustration of regularization by Dropout (redrawn after [SHK+14]). Left panel: the network with all nodes and weights. Right panel: four randomly selected units (red circles) and their i... | The+Shallow+and+the+Deep_Page_184_Chunk4083 |
7.3. CROSS-VALIDATION AND RELATED METHODS 171 to [SHK+14, GBC16], a value of p = 1/2 is typically used for hidden units, while p is close to 1 for input units (e.g. p = 0.8). Note that an individual Dropout dilution could by chance remove an entire layer or some other way cut all connections from input to output. Such ... | The+Shallow+and+the+Deep_Page_185_Chunk4084 |
172 7. MODEL EVALUATION AND REGULARIZATION Figure 7.6: The requirement that the training data should be representative of the actual task at hand seems obvious, but is not always met in practice. © Jonathan van Engelenhoven, see https://www.instagram.com/banjoofjustice for more of his work. Cartoon reproduced from [Vie... | The+Shallow+and+the+Deep_Page_186_Chunk4085 |
7.3. CROSS-VALIDATION AND RELATED METHODS 173 tained in a subset of these in the training. ◦The composition of the subsets could be lucky or unlucky in the sense that the test set might contain only very difficult or very easy cases. As a consequence, the test set performance might be overly pessimistic or optimistic, re... | The+Shallow+and+the+Deep_Page_187_Chunk4086 |
174 7. MODEL EVALUATION AND REGULARIZATION ◦For small values of n we obtain fewer, but more reliable individual esti- mates from larger validation sets. At the same time, the computational workload is reduced in comparison with the use of larger n. However, averages are performed over few individual results, only. More... | The+Shallow+and+the+Deep_Page_188_Chunk4087 |
7.4. PERFORMANCE MEASURES 175 7.3.2 Model and parameter selection The above discussed validation schemes can be employed in the context of model selection and, similarly, for the setting of parameters or hyper-parameters [Ras18]. We can use, for instance, n-fold cross-validation to compare the ex- pected performance of... | The+Shallow+and+the+Deep_Page_189_Chunk4088 |
176 7. MODEL EVALUATION AND REGULARIZATION 7.4.1 Measures for regression In regression, a differentiable objective or loss function typically guides the training process. It appears natural to consider the same function also for the evaluation of a system in validation or working phase. Most frequently, the familiar mea... | The+Shallow+and+the+Deep_Page_190_Chunk4089 |
7.4. PERFORMANCE MEASURES 177 of cross-validation we can obtain the estimates for the generalization ability in terms of the overall error as, say, A g = 0.05 and B g = 0.30. Apparently, we could conclude that A is the better classifier and should be used in the working phase. A closer look into the available data D... | The+Shallow+and+the+Deep_Page_191_Chunk4090 |
178 7. MODEL EVALUATION AND REGULARIZATION threshold Θ after training and consider the modified classification SΘ = sign [g(ξ) −Θ] . (7.23) While this is formally identical with the consideration of an inhomogeneously lin. sep. function, see Sec. 3.3, here the perspective is different: We assume the threshold is introduce... | The+Shallow+and+the+Deep_Page_192_Chunk4091 |
7.4. PERFORMANCE MEASURES 179 true positive rate tpr false positive rate fpr Θ→+∞ Θ=0 Θ→−∞ ξ− δΘ Θ− g(ξ) Figure 7.7: Left panel: Schematic illustration of Receiver Operating Char- acteristics. The extreme working points with Θ →±∞are marked by empty circles. A filled circle corresponds to an unbiased classifier with Θ = ... | The+Shallow+and+the+Deep_Page_193_Chunk4092 |
180 7. MODEL EVALUATION AND REGULARIZATION setting Θ →∞with all inputs assigned to the negative class. Obviously, the false positive rate is zero for this setting, the classifier does not give any false alarms. On the other hand, no positive cases are detected and tpr = 0, as well. The upper right corner in tpr = fpr = ... | The+Shallow+and+the+Deep_Page_194_Chunk4093 |
7.4. PERFORMANCE MEASURES 181 Moreover, the AUC can be associated with a well-defined statistical inter- pretation. Fig. 7.7 (right panel) illustrates a two-class data set which can be classified according to a discriminative function which, in the illustration, is assumed to increase monotonically along the g(ξ)-axis. N... | The+Shallow+and+the+Deep_Page_195_Chunk4094 |
182 7. MODEL EVALUATION AND REGULARIZATION appealing statistical interpretation. For a discussion of supposed disadvantages or advantages of the PR formalism over the ROC see [DG06] and references therein. Like other quantities, Prec and Rec can also be computed at a single, spe- cific working point of the classifier. Va... | The+Shallow+and+the+Deep_Page_196_Chunk4095 |
7.4. PERFORMANCE MEASURES 183 predicted class [abs.] 1 2 3 4 true class 1 2 3 4 sum predicted class [%] 1 2 3 4 true class 1 2 3 4 sum Figure 7.8: Confusion matrix of a hypothetical imbalanced 4-class problem. Left panel: matrix elements correspond to the absolute number of samples from class i which are assigned to cl... | The+Shallow+and+the+Deep_Page_197_Chunk4096 |
184 7. MODEL EVALUATION AND REGULARIZATION The so-called macro-averages Precmac = 1 C C i=1 Preci and Recmac = 1 C C i=1 Reci. (7.31) are obtained with equal weight assigned to the C classes. Alternatively, a weighted macro-average of the form Precw−mac = 1 ntot C i=1 ni Preci and Recw−mac = 1 ntot C i=1 ni... | The+Shallow+and+the+Deep_Page_198_Chunk4097 |
7.5. INTERPRETABLE SYSTEMS 185 We note that ( TP + FP) = 1 C C i=1 (TPi + FPi) = 1 C C i=1 (TPi + FNi) = TP + FN = ntot C , as both sums add up all elements of the confusion matrix. As a consequence Precmic = Recmic = 1 ntot C i=1 TPi, = F mic 1 . (7.37) Since Precmic = Recmic, their harmonic mean ... | The+Shallow+and+the+Deep_Page_199_Chunk4098 |
186 7. MODEL EVALUATION AND REGULARIZATION As usual with stories like that, it is told in many versions, see [gwe09] for an interesting account of similar examples of supposedly mislead classifiers. Ap- parently, the origin of the wolves vs. dogs problem is [RSG16], a publication in which the authors purposefully traine... | The+Shallow+and+the+Deep_Page_200_Chunk4099 |
Chapter 8 Preprocessing and unsupervised learning I will let the data speak for itself when it cleans itself. — Unknown In most of these lecture notes we have implicitly assumed that feature vectors and labels are provided ready-to-use for training. In practical situations, this is rarely the case. In general, real wor... | The+Shallow+and+the+Deep_Page_201_Chunk4100 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.