nips nips2012 nips2012-30 knowledge-graph by maker-knowledge-mining

30 nips-2012-Accuracy at the Top

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Author: Stephen Boyd, Corinna Cortes, Mehryar Mohri, Ana Radovanovic

Abstract: We introduce a new notion of classiﬁcation accuracy based on the top ⌧ -quantile values of a scoring function, a relevant criterion in a number of problems arising for search engines. We deﬁne an algorithm optimizing a convex surrogate of the corresponding loss, and discuss its solution in terms of a set of convex optimization problems. We also present margin-based guarantees for this algorithm based on the top ⌧ -quantile value of the scores of the functions in the hypothesis set. Finally, we report the results of several experiments in the bipartite setting evaluating the performance of our solution and comparing the results to several other algorithms seeking high precision at the top. In most examples, our solution achieves a better performance in precision at the top. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 com Abstract We introduce a new notion of classiﬁcation accuracy based on the top ⌧ -quantile values of a scoring function, a relevant criterion in a number of problems arising for search engines. [sent-6, score-0.679]

2 We deﬁne an algorithm optimizing a convex surrogate of the corresponding loss, and discuss its solution in terms of a set of convex optimization problems. [sent-7, score-0.485]

3 We also present margin-based guarantees for this algorithm based on the top ⌧ -quantile value of the scores of the functions in the hypothesis set. [sent-8, score-0.576]

4 Finally, we report the results of several experiments in the bipartite setting evaluating the performance of our solution and comparing the results to several other algorithms seeking high precision at the top. [sent-9, score-0.606]

5 In most examples, our solution achieves a better performance in precision at the top. [sent-10, score-0.249]

6 1 Introduction The accuracy of the items placed near the top is crucial for many information retrieval systems such as search engines or recommendation systems, since most users of these systems browse or consider only the ﬁrst k items. [sent-11, score-0.813]

7 Different criteria have been introduced in the past to measure this quality, including the precision at k (Precision@k), the normalized discounted cumulative gain (NDCG) and other variants of DCG, or the mean reciprocal rank (MRR) when the rank of the most relevant document is critical. [sent-12, score-0.501]

8 A somewhat different but also related criterion adopted by [1] is based on the position of the top irrelevant item. [sent-13, score-0.486]

9 Several machine learning algorithms have been recently designed to optimize these criteria and other related ones [6, 12, 11, 21, 7, 14, 13]. [sent-14, score-0.211]

10 A general algorithm inspired by the structured prediction technique SVMStruct [22] was incorporated in an algorithm by [15] which can be used to optimize a convex upper bound on the number of errors among the top k items. [sent-15, score-0.544]

11 The algorithm seeks to solve a convex problem with exponentially many constraints via several rounds of optimization with a smaller number of constraints, augmenting the set of constraints at each round with the most violating one. [sent-16, score-0.516]

12 Another algorithm, also based on structured prediction ideas, is proposed in an unpublished manuscript of [19] and covers several criteria, including Precision@k and NDCG. [sent-17, score-0.241]

13 A regression-based solution is suggested by [10] for DCG in the case of large sample sizes. [sent-18, score-0.051]

14 Some other methods have also been proposed to optimize a smooth version of a non-convex cost function in this context [8]. [sent-19, score-0.074]

15 [1] discusses an optimization solution for an algorithm seeking to minimize the position of the top irrelevant item. [sent-20, score-0.673]

16 1 However, one obvious shortcoming of all these algorithms is that the notion of top k does not generalize to new data. [sent-21, score-0.443]

17 In fact, no generalization guarantee is available for such precision@k optimization or algorithm. [sent-23, score-0.08]

18 A more principled approach in all the applications already mentioned consists of designing algorithms that optimize accuracy in some top fraction of the scores returned by a real-valued hypothesis. [sent-24, score-0.707]

19 The desired objective is to learn a scoring function that is as accurate as possible for the items whose scores are above the top ⌧ -quantile. [sent-26, score-0.855]

20 To be more speciﬁc, when applied to a set of size n, the number of top items is k = ⌧ n for a ⌧ -quantile, while for a different set of size n0 6= n, this would correspond to k 0 = ⌧ n0 6= k. [sent-27, score-0.551]

21 The implementation of the Precision@k algorithm in [15] indirectly acknowledges the problem that the notion of top k does not generalize since the command-line ﬂag requires k to be speciﬁed as a fraction of the positive samples. [sent-28, score-0.562]

22 Nevertheless, the formulation of the problem as well as the solution are still in terms of the top k items of the training set. [sent-29, score-0.641]

23 A study of various statistical questions related to the problem of accuracy at the top is discussed by [9]. [sent-30, score-0.368]

24 The authors also present generalization bounds for the speciﬁc case of empirical risk minimization (ERM) under some assumptions about the hypothesis set and the distribution. [sent-31, score-0.136]

25 But, to our knowledge, no previous publication has given general learning guarantees for the problem of accuracy in the top quantile scoring items or carefully addressed the corresponding algorithmic problem. [sent-32, score-1.149]

26 We discuss the formulation of this problem (Section 3. [sent-33, score-0.089]

27 1) and deﬁne an algorithm optimizing a convex surrogate of the corresponding loss in the case of linear scoring functions. [sent-34, score-0.38]

28 We discuss the solution of this problem in terms of several simple convex optimization problems and show that these problems can be extended to the case where positive semi-deﬁnite kernels are used (Section 3. [sent-35, score-0.296]

29 In Section 4, we present a Rademacher complexity analysis of the problem and give margin-based guarantees for our algorithm based on the ⌧ -quantile value of the functions in the hypothesis set. [sent-37, score-0.139]

30 In Section 5, we also report the results of several experiments evaluating the performance of our algorithm. [sent-38, score-0.089]

31 In a comparison in a bipartite setting with several algorithms seeking high precision at the top, our algorithm achieves a better performance in precision at the top. [sent-39, score-0.664]

32 We start with a presentation of notions and notation useful for the discussion in the following sections. [sent-40, score-0.071]

33 2 Preliminaries Let X denote the input space and D a distribution over X ⇥ X . [sent-41, score-0.035]

34 We interpret the presence of a pair (x, x0 ) in the support of D as the preference of x0 over x. [sent-42, score-0.035]

35 according to D m 1 b and denote by D the corresponding empirical distribution. [sent-49, score-0.035]

36 D induces a marginal distribution over X that we denote by D0 , which in the discrete case can be deﬁned via 1 X D0 (x) = D(x, x0 ) + D(x0 , x) . [sent-50, score-0.035]

37 2 0 x 2X b We also denote by D0 the empirical distribution associated to D0 based on the sample S. [sent-51, score-0.035]

38 The learning problems we are studying are deﬁned in terms of the top ⌧ -quantile of the values taken by a function h : X ! [sent-52, score-0.317]

39 In general, q is not unique and this equality may hold for all q in an interval [qmin , qmax ]. [sent-54, score-0.307]

40 We will be particularly interested in the properties of the set of points x whose scores are above a quantile, that is sq = {x : h(x) > q}. [sent-55, score-0.269]

41 Since for any (q, q 0 ) 2 [qmin , qmax ]2 , sq and sq0 differ only by a set of measure zero, the particular choice of q in that interval has no signiﬁcant consequence. [sent-56, score-0.383]

42 Thus, in what follows, when it is not unique, we will choose the quantile value to be the maximum, qmax . [sent-57, score-0.447]

43 For any ⌧ 2 [0, 1], let ⇢⌧ denote the function deﬁned by 8u 2 R, ⇢⌧ (u) = ⌧ (u) + (1 ⌧ )(u)+ , where (u)+ = max(u, 0) and (u) = min(u, 0) (see Figure 1(b)). [sent-58, score-0.035]

44 ⇢⌧ is convex as a sum of two convex functions since u 7! [sent-59, score-0.234]

45 We will denote by argMinu f (u) the largest minimizer of function f . [sent-62, score-0.035]

46 ρτ top τ fraction of scores u 0 page 5 Mehryar Mohri - Courant & Google (a) (b) Figure 1: (a) Illustration of the ⌧ -quantile. [sent-67, score-0.569]

47 , un ) 2 Rn can be b given by q P argMinu2R F⌧ (u), where F⌧ is the convex function deﬁned for all u 2 R by b = n 1 F⌧ (u) = n i=1 ⇢⌧ (ui u). [sent-73, score-0.203]

48 1 Accuracy at the top (AATP) Problem formulation and algorithm The learning problem we consider is that of accuracy at the top (AATP) which consists of achieving an ordering of all items so that items whose scores are among the top ⌧ -quantile are as relevant as possible. [sent-75, score-1.718]

49 Ideally, all preferred items are ranked above the quantile and non-preferred ones ranked below. [sent-76, score-0.866]

50 Thus, the loss or generalization error of a hypothesis h : X ! [sent-77, score-0.136]

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