Sample width for multi-category classifiers

Transcription

1 R u t c o r Research R e p o r t Sample width for multi-category classifiers Martin Anthony a Joel Ratsaby b RRR , November 2012 RUTCOR Rutgers Center for Operations Research Rutgers University 640 Bartholomew Road Piscataway, New Jersey Telephone: Telefax: rrr@rutcor.rutgers.edu rrr a Department of Mathematics, The London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK. m.anthony@lse.ac.uk b Electrical and Electronics Engineering Department, Ariel University Center of Samaria, Ariel 40700, Israel. ratsaby@ariel.ac.il

2 Rutcor Research Report RRR , November 2012 Sample width for multi-category classifiers Martin Anthony Joel Ratsaby Abstract.In a recent paper, the authors introduced the notion of sample width for binary classifiers defined on the set of real numbers. It was shown that the performance of such classifiers could be quantified in terms of this sample width. This paper considers how to adapt the idea of sample width so that it can be applied in cases where the classifiers are multi-category and are defined on some arbitrary metric space. Acknowledgements: This work was supported in part by the IST Programme of the European Community, under the PASCAL2 Network of Excellence, IST

3 Page 2 RRR Introduction By a multi-category) classifier on a set X, we mean a function mapping from X to [C] = {1, 2,..., C} where C 2 is the number of possible categories. Such a classifier indicates to which of the C different classes objects from X belong and, in supervised machine learning, it is arrived at on the basis of a sample, a set of objects from X together with their classifications in [C]. In [4], the notion of sample width for binary classifiers C = 2) mapping from the real line X = R was introduced and in [5], this was generalized to finite metric spaces. In this paper, we consider how a similar approach might be taken to the situation in which C could be larger than 2, and in which the classifiers map not simply from the real line, but from some metric space which would not generally have the linear structure of the real line). The definition of sample width is given below, but it is possible to indicate the basic idea at this stage: we define sample width to be at least γ if the classifier achieves the correct classifications on the sample and, furthermore, for each sample point, the minimum distance to a point of the domain having a different classification is at least γ. A key issue that arises in machine learning is that of generalization error: given that a classifier has been produced by some learning algorithm on the basis of a random) sample of a certain size, how can we quantify the accuracy of that classifier, where by its accuracy we mean its likely performance in classifying objects from X correctly? In this paper, we seek answers to this question that involve not just the sample size, but the sample width. 2 Probabilistic modelling of learning We work in a version of the popular PAC framework of computational learning theory see [14, 7]). This model assumes that the sample s consists of an ordered set x i, y i ) of labeled examples, where x i X and y i Y = [C], and that each x i, y i ) in the training sample s has been generated randomly according to some fixed but unknown) probability distribution P on Z = X Y. This includes, as a special case, the situation in which each x i is drawn according to a fixed distribution on X and is then labeled deterministically by y i = tx i ) where t is some fixed function.) Thus, a sample s of length m can be thought of as being drawn randomly according to the product probability distribution P m. An appropriate measure of how well h : X Y would perform on further randomly drawn points is its error, er P h), the probability that hx) Y for random X, Y ). Given any function h H, we can measure how well h matches the training sample through its sample error er s h) = 1 m {i : hx i) y i }

4 RRR Page 3 the proportion of points in the sample incorrectly classified by h). Much classical work in learning theory see [7, 14], for instance) related the error of a classifier h to its sample error. A typical result would state that, for all δ 0, 1), with probability at least 1 δ, for all h belonging to some specified set if functions, we have er P h) < er s h) + ɛm, δ), where ɛm, δ) known as a generalization error bound) is decreasing in m and δ. Such results can be derived using uniform convergence theorems from probability theory [15, 11, 8], in which case ɛm, δ) would typically involve a quantity known as the growth function of the set of classifiers [15, 7, 14, 2]. More recently, emphasis has been placed on learning with a large margin. See, for instance [13, 2, 1, 12].) The rationale behind margin-based generalization error bounds in the two-category classification case is that if a binary classifier can be thought of as a geometrical separator between points, and if it has managed to achieve a wide separation between the points of different classification, then this indicates that it is a good classifier, and it is possible that a better generalization error bound can be obtained. Margin-based results apply when the binary classifiers are derived from real-valued function by thresholding taking their sign). Margin analysis has been extended to multi-category classifiers in [9]. 3 The width of a classifier We now discuss the case where the underlying set of objects X forms a metric space. Let X be a set on which is defined a metric d : X X R. For a subset S of X, define the distance dx, S) from x X to S as follows: We define the diameter of X to be d x, S) := inf dx, y). y S diamx) := sup dx, y). x,y X We will denote by H the set of all possible functions h from X to [C]. The paper [4] introduced the notion of the width of a binary classifier at a point in the domain, in the case where the domain was the real line R. Consider a set of points {x 1, x 2,..., x m } from R, which, together with their true classifications y i { 1, 1}, yield a training sample s = x j, y j )) m j=1 = x 1, y 1 ), x 2, y 2 ),..., x m, y m )). We say that h : R { 1, 1} achieves sample margin at least γ on s if hx i ) = y i for each i so that h correctly classifies the sample) and, furthermore, h is constant on each of

5 Page 4 RRR the intervals x i γ, x i + γ). It was then possible to obtain generalization error bounds in terms of the sample width. In this paper we use an analogous notion of width to analyse multi-category classifiers defined on a metric space. For each k between 1 and C, let us denote by S h k or simply by S k) the sets corresponding to the function h : X [C], defined as follows: S k := h 1 k) = {x X : hx) = k}. 1) We define the width w h x) of h at a point x X as follows: w h x) := min l hx) dx, S l). In other words, it is the distance from x to the set of points that are labeled differently from hx). The term width is appropriate since the functional value is just the geometric distance between x and the complement of S hx). Given h : X [C], for each k between 1 and C, we define f h k : X R by f h k x) = min l k dx, S l) dx, S k ), and we define f h : X R C by setting the kth component function of f h to be fk h : that is, f h ) k = fk h. Note that if hx) = k, then fk hx) 0 and f j h x) 0 for j k. The function f contains geometrical information encoding how definitive the classification of a point is: if fk h x) is a large positive number, then the point x belongs to category k and is a large distance from differently classified points. We will regard h as being in error on x X if fk h x) is not positive, where k = hx) that is, if the classification is not unambiguously revealed by f h ). The error er P h) of h can then be expressed in terms of the function f h : er P h) = P f h Y X) < 0 ). 2) We define the class F of functions as F := { f h x) : h H }. 3) Note that f h is a mapping from X to the bounded set [ diamx), diamx)] C R C. Henceforth, we will use γ > 0 to denote a width parameter whose value is in the range 0, diamx)]. For a positive width parameter γ > 0 and a training sample s, the empirical sample) γ-width error is defined as Es γ h) = 1 m ) I fy h m j x j ) γ. j=1

6 RRR Page 5 Here, IA) is the indicator function of the set, or event, A.) We shall also denote E γ s h) by E γ s f) where f = f h. Note that f h y x) γ min l y dx, S l) dx, S y ) γ l y such that dx, S l ) dx, S y ) + γ. So the empirical γ-width error on the sample is the proportion of points in the sample which are either misclassified by h or which are classified correctly, but lie within distance γ of the set of points classified differently. We recall that hx) = y implies dx, S y ) = 0.) Our aim is to show that with high probability) the generalization error er P h) is not much greater than E γ s h). In particular, as a special case, we want to bound the generalization error given that E γ s h) = 0.) This will imply that if the learner finds a hypothesis which, for a large value of γ, has a small γ-width error, then that hypothesis is likely to have small error. What this indicates, then, is that if a hypothesis has a large width on most points of a sample, then it will be likely to have small error. 4 Covering numbers 4.1 Covering numbers We will make use of some results and ideas from large-margin theory. One central idea in large-margin analysis is that of covering numbers. We will discuss different types of covering numbers, so we introduce the idea in some generality to start with. Suppose A, d) is a pseudo-)metric space and that α > 0. Then an α-cover of A with respect to d) is a finite subset C of A such that, for every a A, there is some c C such that da, c) α. If such a cover exists, then the mimimum cardinality of such a cover is the covering number N A, α, d). We are working with the set F of vector-valued functions from X to R C, as defined earlier. We define the sup-metric d on F as follows: for f, g : X R C, d f, g) = sup x X max f kx) g k x), 1 k C where f k denotes the kth component function of f. Note that each component function is bounded, so the metric is well-defined.) We can bound the covering numbers N F, α, d ) of F with respect to the sup-metric) in terms of the covering numbers of X with respect to its metric d. The result is as follows.

7 Page 6 RRR Theorem 4.1 For α 0, diamx)], ) CNα 9 diamx) N F, α, d ), α where N α = N X, α/3, d). 4.2 Smoothness of the function class As a first step towards establishing this result, we prove that the functions in F satisfy a certain Lipschitz or smoothness) property. Proposition 4.2 For every f F, and for all x, x X, max f kx) f k x ) 2 dx, x ). 4) 1 k C Proof: Let x, x X and fix k between 1 and C. We show that f k x) f k x ) 2 dx, x ). Recall that, since f F, there is some h : X [C] such that, for all x, f k x) = min l k dx, S l) dx, S k ) where, for each i, S i = h 1 i). We have f k x) f k x ) = min dx, S l) dx, S k ) min l k l k dx, S l ) + dx, S k ) min dx, S l) min l k l k dx, S l ) + dx, S k) dx, S k ) We consider in turn each of the two terms in this final expression. We start with the second. Let ɛ > 0 and suppose r S k is such that dx, r) < dx, S k ) + ɛ. Such an r exists since dx, S k ) = inf y Sk dx, y). Let s S k be such that dx, s) < dx, S k ) + ɛ. Then dx, S k ) dx, r) dx, s) + ɛ, since dx, r) < dx, S k ) + ɛ and dx, S k ) dx, s). So, dx, S k ) dx, s) + dx, x ) + ɛ < dx, S k ) + ɛ + dx, x ) + ɛ = dx, S k ) + 2ɛ.

8 RRR Page 7 A similar argument establishes dx, S k ) dx, s) dx, r) + ɛ dx, r) + dx, x ) + ɛ < dx, S k ) + dx, x ) + 2ɛ. Combining these two inequalities shows dx, S k ) dx, S k ) dx, x )+2ɛ. Since this holds for all ɛ > 0, it follows that dx, S k ) dx, S k ) dx, x ). Next we show min dx, S l) min l k l k dx, S l ) dx, x ). Suppose that min k l dx, S l ) = dx, S p ) and that min k l dx, S l ) = dx, S q ). Let ɛ > 0. Suppose that y S p is such that dx, y) < dx, S p ) + ɛ and that y S q is such that dx, y ) < dx, S q ) + ɛ. Then and dx, S q ) dx, S p ) dx, y) dx, x ) + dx, y) < dx, x ) + dx, S p ) + ɛ dx, S p ) dx, S q ) dx, y ) dx, x ) + dx, y ) = dx, x ) + dx, S q ) + ɛ. From these inequalities, it follows that dx, S p ) dx, S q ) dx, x ) + ɛ, for all ɛ, and it follows that dx, S p ) dx, S q ) dx, x ). Next, we exploit this smoothness to construct a cover for F. 4.3 Covering F Let the subset C γ X be a minimal size α/3-cover for X with respect to the metric d. So, for every x X there is some ˆx C γ such that dx, ˆx) α/3. Denote by N α the cardinality of C α. Let { } 3 diamx) 3 diamx) Λ α = λ i = iα : i =,..., 1, 0, 1, 2,..., α α and define the class ˆF to be all functions ˆf : C α Λ α ) C. Then ˆF is of a finite size equal to Λ α C Nα. For any ˆf ˆF define the extension ˆf ext : X R C of ˆf to the whole domain X as follows: given ˆf which is well-defined on the points ˆx i of the cover) then for every point x in the ball B α/3 ˆx i ) = {x X : dx, ˆx i ) α/3}, we let ˆf ext x) = ˆfˆx i ), for all ˆx i C α where, if, for a point x there is more than one point ˆx i such that x B α/3 ˆx i ), we arbitrarily pick one of the points ˆx i in order to assign the value of ˆf ext x)). There is a one-to-one correspondence 5)

9 Page 8 RRR between the functions ˆf and the functions ˆf ext. Hence the set ˆF ext = cardinality equal to Λ α C Nα. { ˆfext : ˆf ˆF } is of We claim that for any f F there exists an ˆf ext such that d f, ˆf ext ) α. To see this, first for every point ˆx i C α, consider fˆx i ) and find a corresponding element in Λ C α, call it ˆfˆx i )) such that max fˆx i)) k ˆfˆx i )) k α/3. 6) 1 k C That there exists such a value follows by design of Λ α.) By the above definition of extension, it follows that for all points x B α/3 ˆx i ) we have ˆf ext x) = ˆfˆx i ). Now, from 4) we have for all f F, max sup 1 i k x B α/3 ˆx i ) fx)) k fˆx i )) k 2dx, ˆx i ) 2α/3. 7) Hence for any f F there exists a function ˆf ˆF with a corresponding ˆf ext ˆF ext such that, given an x X, there exists ˆx i C α such that, for each k between 1 and C, fx)) k ˆf ext x)) k = fx)) k ˆf ext ˆx i )) k. The right hand side can be expressed as fx)) k ˆf ext ˆx i )) k = fx)) k ˆfˆx i )) k where 8) follows from 6) and 7). = fx)) k fˆx i )) k + fˆx i )) k ˆfˆx i )) k fx)) k fˆx i )) k + fˆx i )) k ˆfˆx i )) k 2α/3 + α/3 8) = α. Hence the set ˆF ext forms an α-covering of the class F in the sup-norm. Thus we have the following covering number bound. C Nα 3 diamx) N F, α, d ) Λ α C Nα = 2 + 1). 9) α Theorem 4.1 now follows because for 0 < α diamx)) ) 3 diamx) 3 diamx) = 6 diamx) α α α diamx). α 5 Generalization error bounds We present two results. The first bounds the generalization error in terms of a width parameter γ for which the γ-width error on the sample is zero; the second more general but looser

10 RRR Page 9 in that special case) bounds the error in terms of γ and the γ-width error on the sample which could be non-zero). Theorem 5.1 Suppose that X is a metric space of diameter diamx). Suppose P is any probability measure on Z = X [C]. Let δ 0, 1). Then, with probability at least 1 δ, the following holds for s Z m : for any function h : X [C], and for any γ 0, diamx)], if Es γ h) = 0, then er P h) 2 ) )) 36 diamx) 4 diamx) CN X, γ/12, d) log m 2 + log γ 2. δγ Theorem 5.2 Suppose that X is a metric space of diameter diamx). Suppose P is any probability measure on Z = X [C]. Let δ 0, 1). Then, with probability at least 1 δ, the following holds for s Z m : for any function h : X [C], and for any γ 0, diamx)], er P h) E γ s h) + 2 m CN X, γ/6, d) ln ) )) 18 diamx) 2 diamx) + ln + 1 γ γδ m. What we have in Theorem 5.2 is a high probability bound that takes the following form: for all h and for all γ 0, diamx)], er P h S ) E γ s h) + ɛm, γ, δ), where ɛ tends to 0 as m and ɛ decreases as γ increases. The rationale for seeking such a bound is that there is likely to be a trade-off between width error on the sample and the value of ɛ: taking γ small so that the error term E γ s h) is zero might entail a large value of ɛ; and, conversely, choosing γ large will make ɛ relatively small, but lead to a large sample error term. So, in principle, since the value γ is free to be chosen, one could optimize the choice of γ on the right-hand side of the bound to minimize it. Proof of Theorem 5.1 The proof uses techniques similar to those first used in [15, 14, 8, 11] and in subsequent work extending those techniques to learning with real-valued functions, such as [10, 3, 1, 6]. The first observation is that if Q = {s Z m : h H with E γ s h) = 0, er P h) ɛ} and T = {s, s ) Z m Z m : h H with Es γ h) = 0, Es 0 h) ɛ/2},

11 Page 10 RRR then, for m 8/ɛ, P m Q) 2 P 2m T ). This is because P 2m T ) P { 2m h : Es γ h) = 0, er P h) ɛ and Es 0 h) ɛ/2}) = P { m s : h, Es γ h) = 0, er P h) ɛ and Es 0 h) ɛ/2}) dp m s) Q 1 2 P m Q), for m 8/ɛ. The final inequality follows from the fact that if er P h) ɛ, then for m 8/ɛ, P m Es 0 h) ɛ/2) 1/2, for any h, something that follows by a Chernoff bound. Let G be the permutation group the swapping group ) on the set {1, 2,..., 2m} generated by the transpositions i, m + i) for i = 1, 2,..., m. Then G acts on Z 2m by permuting the coordinates: for σ G, By invariance of P 2m under the action of G, σz 1, z 2,..., z 2m ) = z σ1),..., z σ2m) ). P 2m T ) max{pσz T ) : z Z 2m }, where P denotes the probability over uniform choice of σ from G. See, for instance, [11, 2].) Let F = {f h : h H} be the set of vector-valued functions derived from H as before, and let ˆF be a minimal γ/2-cover of F in the d -metric. Theorem 4.1 tells us that the size of ˆF is no more than ) CN 18 diamx), where N = N X, γ/6, d). γ Fix z Z 2m. Suppose τz = s, s ) T and that h is such that Es γ h) = 0 and Es 0 h) ɛ/2. Let ˆf ˆF be such that d ˆf, f h ) γ/2. Then the fact that Es γ h) = 0 implies that ˆf) = 0. And, the fact that Es 0 h) ɛ/2 implies that Eγ/2 s ˆf) ɛ/2. To see the first claim, note that we have fy h x) > γ for all x, y) in the sample s and that, since, for all k, and all x, ˆf k x) fk hx) γ/2, we have also that for all such x, y), ˆfy x) > γ/2. For the second claim, we observe that if fy h x) 0 then ˆf y x) γ/2. E γ/2 s It now follows that if τz T, then, for some ˆf ˆF, τz R ˆf), where R ˆf) = {s, s ) Z m Z m : Es γ/2 ˆf) = 0, E γ/2 s ˆf) ɛ/2}. By symmetry, P σz R ˆf) ) = P στz) R ˆf) ). Suppose that E γ/2 s ˆf) = r/m, where r ɛm/2 is the number of x i, y i ) in s on which ˆf is such that ˆf yi x i ) γ/2. Then those

12 RRR Page 11 permutations σ such that στz) R ˆf) are precisely those that do not transpose these r coordinates, and there are 2 m r 2 m ɛm/2 such σ. It follows that, for each fixed ˆf ˆF, P σz R ˆf) ) 2m1 ɛ/2) G = 2 ɛm/2. We therefore have Pσz T ) P σz ˆf R ˆf) ˆf Pσz R ˆf)) ˆF 2 ɛm/2. ˆF ˆF So, ) CN P m Q) 2 P 2m T ) 2 ˆF 18 diamx) 2 ɛm/2 2, γ where N = N X, γ/6, d). This is at most δ when ɛ = 2 m ) )) 18 diamx) 2 CN log 2 + log γ 2. δ Next, we use this to obtain a result in which γ is not prescribed in advance. For α 1, α 2, δ 0, 1), let Eα 1, α 2, δ) be the set of z Z m for which there exists some h H with E α 2 z h) = 0 and er P h) ɛ 1 m, α 1, δ), where ɛ 1 m, α 1, δ) = 2 m ) )) 18 diamx) 2 CN X, α 1 /6, d) log 2 + log α 2. 1 δ Then the result just obtained tells us that P m Eα, α, δ)) δ. It is also clear that if α 1 α α 2 and δ 1 δ, then Eα 1, α 2, δ 1 ) Eα, α, δ). It follows from a result from [6], that P m Eγ/2, γ, δγ/2 diamx))) δ. γ 0,diamX)] In other words, with probability at least 1 δ, for all γ 0, diamx)]], we have that if h H satisfies E γ s h) = 0, then er P h) < ɛ 2 m, γ, δ), where ɛ 2 m, γ, δ) = 2 m ) )) 36 diamx) 4 diamx) CN X, γ/12, d) log 2 + log γ 2. δγ Note that γ now need not be prescribed in advance.

13 Page 12 RRR Proof of Theorem 5.2 Guermeur [9] has developed a framework in which to analyse multi-category classification, and we can apply one of his results to obtain the bound of Theorem 5.2, which is generalization error bound applicable to the case in which the γ-width sample error is not zero. In that framework, there is a set G of functions from X into R C, and a typical g G is represented by its component functions g k for k = 1 to C. Each g G satisfies the constraint g k x) = 0, x X. k=1 A function of this type acts as a classifier as follows: it assigns category l [C] to x X if and only if g l x) > max k l g k x). If more than one value of k maximizes g k x), then the classification is left undefined, assigned some value not in [C].) The risk of g G, when the underlying probability measure on X Y is P, is defined to be Rg) = P For v, k) R C [C], let Mv, k) = 1 2 X R C given by Given a sample s X [C]) m, let ) {x, y) X [C] : g y x) max g kx)}. k y ) v k max v l l k gx) = g k x)) C k=1 = Mgx), k)) C k=1. and, for g G, let g be the function R γ,s g) = 1 m m I { g yi x i ) < γ}. i=1 A result following from [9] is in the above notation) as follows: Let δ 0, 1) and suppose P is a probability measure on Z = X [C]. With P m -probability at least 1 δ, s Z m will be such that we have the following: for any fixed d > 0) for all γ 0, d] and for all g G, Rg) R γ,s g) + 2 m ln N G, γ/4, d ) + ln )) 2d + 1 γδ m. In fact, the result from [9] involves emprical covering numbers rather than d -covering numbers. The latter are at least as large as the empirical covering numbers, but we use

14 RRR Page 13 these because we have bounded them earlier in this paper.) For each function h : X [C], let g = g h be the function X R C defined by g k x) = 1 dx, S i ) dx, S k ), C i=1 where, as before, S j = h 1 j). Let G = {g h : h H} be the set of all such g. Then these functions satisfy the constraint that their coordinate functions sum to the zero function, since g k x) = k=1 For each k, k=1 1 C dx, S i ) i=1 dx, S k ) = k=1 dx, S k ) k=1 g k x) = Mgx), k) = 1 ) g k x) max 2 g lx) l k = dx, S i ) dx, S k ) max 2 C l k C i=1 = 1 ) dx, S k ) max 2 dx, S l)) l k = 1 ) min 2 dx, S l) dx, S k ) l k = 1 2 f h k x). dx, S k ) = 0. k=1 )) dx, S i ) dx, S l ) From the definition of g, the event that g y x) max k y g k x) is equivalent to the event that ) 1 1 dx, S i ) dx, S y ) max dx, S i ) dx, S k ), C k y C i=1 which is equivalent to min k y dx, S k ) dx, S y ). It can therefore be seen that Rg) = er P h). Similarly, R γ,s g) = E γ s h). Noting that G = 1/2)F, so that an α/2 cover of F will provide an α/4 cover of G, we can therefore apply Guermeur s result to see that with probability at least 1 δ, for all h and for all γ 0, diamx)], er P h) E γ s h) + E γ s h) + 2 m 2 m CN X, γ/6, d) ln i=1 ln N F, γ/2, d ) + ln i=1 )) 2 diamx) + 1 γδ m ) )) 18 diamx) 2 diamx) + ln + 1 γ γδ m.

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