nips nips2007 nips2007-99 knowledge-graph by maker-knowledge-mining

99 nips-2007-Hierarchical Penalization

Source: pdf

Author: Marie Szafranski, Yves Grandvalet, Pierre Morizet-mahoudeaux

Abstract: Hierarchical penalization is a generic framework for incorporating prior information in the ﬁtting of statistical models, when the explicative variables are organized in a hierarchical structure. The penalizer is a convex functional that performs soft selection at the group level, and shrinks variables within each group. This favors solutions with few leading terms in the ﬁnal combination. The framework, originally derived for taking prior knowledge into account, is shown to be useful in linear regression, when several parameters are used to model the inﬂuence of one feature, or in kernel regression, for learning multiple kernels. Keywords – Optimization: constrained and convex optimization. Supervised learning: regression, kernel methods, sparsity and feature selection. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Hierarchical Penalization Marie Szafranski 1 , Yves Grandvalet 1, 2 and Pierre Morizet-Mahoudeaux 1 Heudiasyc 1 , UMR CNRS 6599 Universit´ de Technologie de Compi` gne e e BP 20529, 60205 Compi` gne Cedex, France e IDIAP Research Institute 2 Av. [sent-1, score-0.124]

2 fr Abstract Hierarchical penalization is a generic framework for incorporating prior information in the ﬁtting of statistical models, when the explicative variables are organized in a hierarchical structure. [sent-7, score-0.855]

3 The penalizer is a convex functional that performs soft selection at the group level, and shrinks variables within each group. [sent-8, score-0.489]

4 This favors solutions with few leading terms in the ﬁnal combination. [sent-9, score-0.092]

5 The framework, originally derived for taking prior knowledge into account, is shown to be useful in linear regression, when several parameters are used to model the inﬂuence of one feature, or in kernel regression, for learning multiple kernels. [sent-10, score-0.079]

6 Supervised learning: regression, kernel methods, sparsity and feature selection. [sent-12, score-0.129]

7 1 Introduction In regression, we want to explain or to predict a response variable y from a set of explanatory variables x = (x1 , . [sent-13, score-0.243]

8 i=1 It can be achieved in a predictive or a descriptive perspective: to predict accurate responses for future observations, or to show the correlations that exist between the set of explanatory variables and the response variable, and thus, give an interpretation to the model. [sent-22, score-0.242]

9 In a descriptive perspective, |βj | can be interpreted as a degree of relevance of variable xj . [sent-31, score-0.129]

10 Ordinary Least Squares (OLS) minimizes the sum of the residual squared error. [sent-32, score-0.098]

11 When the explanatory variables are numerous and many of them are correlated, the variability of the OLS estimate tends to increase. [sent-33, score-0.198]

12 Coefﬁcient shrinkage is a major approach of regularization procedures in linear regression models. [sent-35, score-0.186]

13 According to the chosen norm, coefﬁcients associated to variables with little predictive information may be shrunk, or even removed when variables are irrelevant. [sent-37, score-0.204]

14 In particular, ridge regression shrinks coefﬁcients with regard to the 2 -norm, while the lasso (Least Absolute Shrinkage and Selection Operator) [1] and the lars (Least Angle Regression Stepwise) [2] both shrink and remove coefﬁcients using the 1 -norm. [sent-39, score-0.462]

15 In some applications, explanatory variables that share a similar characteristic can be gathered into groups – or factors. [sent-41, score-0.302]

16 For instance, in genomics, where explanatory variables are (products of) genes, some factors can be identiﬁed from the prior information available in the hierarchies of Gene Ontology. [sent-43, score-0.293]

17 Group-lasso and group-lars [3] can be considered as hierarchical penalization methods, with trees of height two deﬁning the hierarchies. [sent-45, score-0.83]

18 They perform variable selection by encouraging sparseness over predeﬁned factors. [sent-46, score-0.158]

19 These techniques seem perfectible in the sense that hierarchies can be extended to more than two levels and sparseness integrated within groups. [sent-47, score-0.136]

20 This papers proposes a penalizer, derived from an adaptive penalization formulation [4], that highlights factors of interest by balancing constraints on each element, at each level of a hierarchy. [sent-48, score-0.605]

21 It performs soft selection at the factor level, and shrinks variables within groups, to favor solutions with few leading terms. [sent-49, score-0.36]

22 Section 2 introduces the framework of hierarchical penalization and the associated algorithm is presented in Section 3. [sent-50, score-0.719]

23 Section 4 shows how this framework can be applied to linear and kernel regression. [sent-51, score-0.079]

24 1 Hierarchical Penalization Formalization We introduce hierarchical penalization by considering problems where the variables are organized in a tree structure of height two, such as the example displayed in ﬁgure 1. [sent-54, score-1.039]

25 The set of children (that is, leaves) of node k is denoted Jk and its cardinality is dk . [sent-59, score-0.268]

26 As displayed on the right-hand-side of ﬁgure 1, a branch stemming from the root and going to node k is labelled by σ1,k , and the branch reaching leaf j is labelled by σ2,j . [sent-60, score-0.197]

27 We consider the problem of minimizing a differentiable loss function L(·), subject to sparseness constraints on β and the subsets of β deﬁned in a tree hierarchy. [sent-61, score-0.194]

28 This reads  K 2  βj    min , (1a) L(β) + λ √  β,σ  σ1,k σ2,j   k=1 j∈Jk  K  subject to         d dk σ1,k = 1 , σ2,j = 1 , σ1,k ≥ 0 (1b) j=1 k=1 k = 1, . [sent-62, score-0.358]

29 The constraints (1b) shrink the coefﬁcients β at group level and inside groups. [sent-70, score-0.192]

30 In what follows, we show that problem (1) is convex and that this joint shrinkage encourages sparsity at the group level. [sent-71, score-0.368]

31 Proof: A problem minimizing a convex criterion on a convex set is convex. [sent-76, score-0.224]

32 Since L(·) is convex and x2 λ is positive, the criterion (1a) is convex provided f (x, y, z) = √yz is convex. [sent-77, score-0.252]

33 4(yz) 2 2 f (x, y, z) = −4 x 3 x2 y y yz  2 2 x x x x −z −z −z −x −4 x 3 x2 z z yz z Hence, the Hessian is positive semi-deﬁnite, and criterion (1a) is convex. [sent-79, score-0.272]

34 Next, constraints (1c) deﬁne half-spaces for σ1 and σ2 , which are convex sets. [sent-80, score-0.133]

35 Equality constraints (1b) deﬁne linear subspaces of dimension K − 1 and d − 1 which are also convex sets. [sent-81, score-0.133]

36 The intersection of convex sets being a convex set, the constraints deﬁne a convex admissible set, and problem (1) is convex. [sent-82, score-0.448]

37 Proposition 2 Problem (1) is equivalent to  1 4 dk   min L(β) + λ  β  3 2 4 4  |βj | 3   . [sent-83, score-0.308]

38  K (2) j∈Jk k=1 Sketch of proof: The Lagrangian of problem (1) is K L = L(β) + λ k=1 j∈Jk  2 βj + ν1 √ σ1,k σ2,j  d j=1 dk σ1,k − 1 + k=1 K σ2,j − 1 − ν2  K d ξ1,k σ1,k − ξ2,j σ2,j . [sent-84, score-0.268]

39 j=1 k=1 Hence, the optimality conditions for σ1,k and σ2,j are  2  βj  λ  − + ν1 dk − ξ1,k  ∂L  3 1  2  = 0   ∂σ 2 2   j∈Jk σ1,k σ2,j 1,k ⇒ 2  ∂L    λ βj   = 0   − + ν2 − ξ2,j  1  ∂σ2,j 2 σ2 σ3 2 1,k 2,j = 0 . [sent-85, score-0.368]

40 = 0 After some tedious algebra, the optimality conditions for σ1,k and σ2,j can be expressed as −3 σ1,k = 1 3 dk 4 (sk ) 4 K k=1 1 4 dk (sk ) , and σ2,j = 3 4 (sk ) 1 4 K k=1 4 4 4 dk |βj | 3 1 4 dk (sk ) for j ∈ Jk , 3 4 |βj | 3 . [sent-86, score-1.172]

41 Plugging these conditions in criterion (1a) yields the claimed result. [sent-87, score-0.084]

42 3 Sparseness Proposition 2 shows how the penalization inﬂuences the groups of variables and each variable in each group. [sent-89, score-0.775]

43 Note that, thanks to the positivity of the squared term in (2), the expression can be further simpliﬁed to 3  4 K 1 min L(β) + ν β 4 |βj | 3  4 dk  , (3) j∈Jk k=1 where, for any L(β), there is a one-to-one mapping from λ in (2) to ν in (3). [sent-90, score-0.36]

44 This expression can be interpreted as the Lagrangian formulation of a constrained optimization problem, where the admissible set for β is deﬁned by the multiplicand of ν. [sent-91, score-0.145]

45 One sees that hierarchical penalization combines some features of lasso and group-lasso. [sent-93, score-0.91]

46 By looking at the curvature of these sets when they meet axes, one gets a good intuition on why ridge regression does not suppress variables, why lasso does, why group-lasso suppresses groups of variables but not within-group variables, and why hierarchical penalization should do both. [sent-95, score-1.316]

47 This intuition is however not correct for hierarchical penalization because the boundary of the admissible set is differentiable in the within-group hyper-plane (β1 , β2 ) at β1 = 0 and β2 = 0. [sent-96, score-0.892]

48 However, as its curvature is very high, solutions with few leading terms in the within-group variables are encouraged. [sent-97, score-0.232]

49 To go beyond the hints provided by these ﬁgures, we detail here the optimality conditions for β minimizing (3). [sent-98, score-0.187]

50 The ﬁrst-order optimality conditions are |βj | = 0, 1 ∂L(β) 4 + ν dk vj = 0, where vj ∈ [−1, 1]; ∂βj |βj | = 0, 1. [sent-99, score-0.436]

51 for βj = 0, j ∈ Jk , + ν dk sign(βj ) 1 + 4 ∂βj |βj | 3 |β | 4 3 1 −4 = 0. [sent-102, score-0.268]

52 the variables belonging to groups that are estimated to be irrelevant are penalized with the highest strength, thus limiting the number of groups inﬂuencing the solution; 2. [sent-104, score-0.343]

53 when a group has some non-zero relevance, all variables enter the set of active variables provided they inﬂuence the ﬁtting criterion; 3. [sent-105, score-0.421]

54 however, the penalization strength increases very rapidly (as a smooth step function) for small values of |βj |, thus limiting the number of βj with large magnitude. [sent-106, score-0.524]

55 4 Overall, hierarchical penalization is thus expected to provide solutions with few active groups and few leading variables within each group. [sent-107, score-1.102]

56 3 Algorithm To solve problem (3), we use an active set algorithm, based on the approach proposed by Osborne et al. [sent-108, score-0.085]

57 Then, at each iteration, we solve the problem  3 4 K min L(γ) = L(γ) + ν γ 1 4 dk  k=1 4 |γj | 3  , (4) j∈Gk by alternating steps A and B described below. [sent-111, score-0.308]

58 Second, the set of active variables is incrementally updated as detailed in steps C and D. [sent-112, score-0.227]

59 A Compute a candidate update from an admissible vector γ The goal is to solve min L(γ + h), where γ is the current estimate of the solution and h ∈ R|A| . [sent-113, score-0.185]

60 These difﬁculties are circumvented by replacing |γj + hj | by sign(γj )(γj + hj ). [sent-115, score-0.08]

61 B Obtain a new admissible vector γ † † Let γ † = γ + h. [sent-117, score-0.145]

62 Let µ = min − , that is, m∈S hm µ is the largest step in direction h such that sign(γm + µhm ) = sign(γm ), except for one γm , for which γ + µh = 0. [sent-120, score-0.097]

63 C Test optimality of γ If the appropriate optimality condition holds for all inactive variables β (β = 0), that is C. [sent-127, score-0.242]

64 1 for ∈ Jk , where 1 ∂L(β) 4 ≤ ν dk , ∂β |βj | = 0, then j∈Jk C. [sent-128, score-0.268]

65 D Select the variable that enters the active set ∂L(β) , where k is the group of variable . [sent-131, score-0.279]

66 2 Update the active set: A ← A ∪ { }, with initial vector: γ = [γ, 0]t where the sign of the new zero component is − sign ∂L(β) . [sent-133, score-0.411]

67 5 4 Experiments We illustrate on two datasets how hierarchical penalization can be useful in exploratory analysis and in prediction. [sent-138, score-0.719]

68 Then, we show how the algorithm can be applied for multiple kernel learning in kernel regression. [sent-139, score-0.158]

69 1 Abalone Database The Abalone problem [6] consists in predicting the age of abalone from physical measurements. [sent-141, score-0.168]

70 One concerns the sex of abalone, and has been encoded with dummy variables, that is xsex = (100) for male, xsex = (010) for female, or xsex = (001) for i i i infant. [sent-143, score-0.363]

71 The second group is composed of 3 attributes concerning size parameters (length, diameter and height), and the last group is composed of weight parameters (whole, shucked, viscera and shell weight). [sent-145, score-0.324]

72 The mean squared test error is at par with lasso (4. [sent-147, score-0.243]

73 Weight parameters are a main contributor to the estimation of the age of an abalon, while sex is not essential, except for infant. [sent-151, score-0.125]

74 We focussed on the prototype “house-price-16L”, composed of 16 variables. [sent-171, score-0.102]

75 We derived this prototype by including all the other variables related to these 16 variables. [sent-172, score-0.146]

76 The ﬁnal dataset is then composed of 37 variables, split up into 10 groups1 . [sent-173, score-0.095]

77 For each dataset, the parameter ν was selected by a 10-fold cross validation, and the mean squared error was computed on the testing set. [sent-176, score-0.083]

78 2 the mean squared test errors obtained with the hierarchical penalization (hp), the group-lasso (gl) and the lasso estimates. [sent-178, score-0.962]

79 Hierarchical penalization performs better than lasso on 8 datasets. [sent-213, score-0.715]

80 The problem consists in learning a convex 1 A description of the dataset is available at http://www. [sent-219, score-0.122]

81 Here, we show that hierarchical penalization is well suited for this purpose for other kernel predictors, and we illustrate its effect on kernel smoothing in the regression setup. [sent-224, score-1.039]

82 (5) j=1 The penalized model (5) has been applied to the motorcycle dataset [9]. [sent-228, score-0.124]

83 This uni-dimensional problems enables to display the contribution of each bandwidth to the solution. [sent-229, score-0.136]

84 We used Gaussian kernels, with 7 bandwidths ranging from 10−1 to 102 . [sent-230, score-0.153]

85 Figure 3 displays the results obtained for different penalization parameters: the estimated function obtained by the combination of the selected bandwidths, and the contribution of each bandwidth to the model. [sent-231, score-0.65]

86 We display three settings for the penalization parameter ν, corresponding to slight overﬁtting, good ﬁt and slight under-ﬁtting. [sent-232, score-0.565]

87 The coefﬁcients of bandwidths h2 , h6 and h7 were always null and are thus not displayed. [sent-233, score-0.153]

88 As expected, when the penalization parameter ν increases, the ﬁt becomes smoother, and the number of contributing bandwidths decreases. [sent-234, score-0.677]

89 We also observe that the effective contribution of some bandwidths is limited to a few kernels: there are few leading terms in the expansion. [sent-235, score-0.243]

90 5 Conclusion and further works Hierarchical penalization is a generic framework enabling to process hierarchically structured variables by usual statistical models. [sent-236, score-0.626]

91 The structure is provided to the model via constraints on the subgroups of variables deﬁned at each level of the hierarchy. [sent-237, score-0.178]

92 The ﬁtted model is then biased towards statistical explanations that are “simple” with respect to this structure, that is, solutions which promote a small number of groups of variables, with a few leading components. [sent-238, score-0.223]

93 In this paper, we detailed the general framework of hierarchical penalization for tree structures of height two, and discussed its speciﬁc properties in terms of convexity and parsimony. [sent-239, score-0.874]

94 Then, we proposed an efﬁcient active set algorithm that incrementally builds an optimal solution to the problem. [sent-240, score-0.125]

95 We ﬁnally illustrated how the approach can be used when groups of features, or when discrete variables exist, after being encoded by several binary variables, result in groups of variables. [sent-241, score-0.31]

96 We then plan to extend the formalization to hierarchies of arbitrary height, whose properties are currently under study. [sent-244, score-0.143]

97 We will then be able to tackle new applications, such as genomics, where the available gene ontologies are hierarchical structures that can be faithfully approximated by trees. [sent-245, score-0.229]

98 Model selection and estimation in regression with grouped variables. [sent-261, score-0.136]

99 7 combined h1 =10−1 h3 =1 1 h4 =10 2 h5 =10 ν=10 ν=25 ν=50 Figure 3: Hierarchical penalization applied to kernel smoothing on the motorcycle data. [sent-264, score-0.694]

100 Isolated bandwidths: the points represent partial residuals and the solid line represents the contribution of the bandwidth to the model. [sent-266, score-0.095]

similar papers computed by tfidf model

tfidf for this paper:

wordName wordTfidf (topN-words)

[('penalization', 0.524), ('jk', 0.302), ('dk', 0.268), ('hierarchical', 0.195), ('lasso', 0.191), ('sign', 0.163), ('bandwidths', 0.153), ('admissible', 0.145), ('abalone', 0.124), ('height', 0.111), ('yz', 0.109), ('group', 0.104), ('groups', 0.104), ('variables', 0.102), ('regression', 0.097), ('explanatory', 0.096), ('xsex', 0.094), ('shrinkage', 0.089), ('cients', 0.088), ('active', 0.085), ('convex', 0.085), ('delve', 0.081), ('formalization', 0.081), ('sex', 0.081), ('coef', 0.08), ('kernel', 0.079), ('sparseness', 0.074), ('optimality', 0.07), ('shrinks', 0.069), ('sk', 0.068), ('ridge', 0.065), ('bandwidth', 0.063), ('compi', 0.062), ('gne', 0.062), ('ols', 0.062), ('penalizer', 0.062), ('hierarchies', 0.062), ('go', 0.059), ('composed', 0.058), ('leading', 0.058), ('hm', 0.057), ('criterion', 0.054), ('motorcycle', 0.054), ('squared', 0.052), ('sparsity', 0.05), ('census', 0.05), ('reads', 0.05), ('constraints', 0.048), ('labelled', 0.048), ('lagrangian', 0.046), ('residual', 0.046), ('genomics', 0.046), ('hp', 0.046), ('osborne', 0.046), ('grandvalet', 0.046), ('url', 0.046), ('variable', 0.045), ('descriptive', 0.044), ('prototype', 0.044), ('age', 0.044), ('tree', 0.044), ('display', 0.041), ('min', 0.04), ('incrementally', 0.04), ('hj', 0.04), ('shrink', 0.04), ('encourages', 0.04), ('kernels', 0.04), ('xj', 0.04), ('selection', 0.039), ('survey', 0.039), ('lanckriet', 0.038), ('curvature', 0.038), ('gl', 0.038), ('proposition', 0.037), ('smoothing', 0.037), ('dataset', 0.037), ('gk', 0.036), ('branch', 0.036), ('axes', 0.035), ('solutions', 0.034), ('perspective', 0.034), ('organized', 0.034), ('gene', 0.034), ('vj', 0.034), ('factors', 0.033), ('penalized', 0.033), ('contribution', 0.032), ('selected', 0.031), ('conditions', 0.03), ('favor', 0.03), ('hessian', 0.029), ('displayed', 0.029), ('culties', 0.029), ('soft', 0.028), ('purpose', 0.028), ('provided', 0.028), ('angle', 0.028), ('differentiable', 0.028), ('promote', 0.027)]

similar papers list:

simIndex simValue paperId paperTitle

same-paper 1 1.0000001 99 nips-2007-Hierarchical Penalization

Author: Marie Szafranski, Yves Grandvalet, Pierre Morizet-mahoudeaux

2 0.12643793 43 nips-2007-Catching Change-points with Lasso

Author: Céline Levy-leduc, Zaïd Harchaoui

Abstract: We propose a new approach for dealing with the estimation of the location of change-points in one-dimensional piecewise constant signals observed in white noise. Our approach consists in reframing this task in a variable selection context. We use a penalized least-squares criterion with a 1 -type penalty for this purpose. We prove some theoretical results on the estimated change-points and on the underlying piecewise constant estimated function. Then, we explain how to implement this method in practice by combining the LAR algorithm and a reduced version of the dynamic programming algorithm and we apply it to synthetic and real data. 1

3 0.11461718 179 nips-2007-SpAM: Sparse Additive Models

Author: Han Liu, Larry Wasserman, John D. Lafferty, Pradeep K. Ravikumar

Abstract: We present a new class of models for high-dimensional nonparametric regression and classiﬁcation called sparse additive models (SpAM). Our methods combine ideas from sparse linear modeling and additive nonparametric regression. We derive a method for ﬁtting the models that is effective even when the number of covariates is larger than the sample size. A statistical analysis of the properties of SpAM is given together with empirical results on synthetic and real data, showing that SpAM can be effective in ﬁtting sparse nonparametric models in high dimensional data. 1

4 0.098181151 53 nips-2007-Compressed Regression

Author: Shuheng Zhou, Larry Wasserman, John D. Lafferty

Abstract: Recent research has studied the role of sparsity in high dimensional regression and signal reconstruction, establishing theoretical limits for recovering sparse models from sparse data. In this paper we study a variant of this problem where the original n input variables are compressed by a random linear transformation to m n examples in p dimensions, and establish conditions under which a sparse linear model can be successfully recovered from the compressed data. A primary motivation for this compression procedure is to anonymize the data and preserve privacy by revealing little information about the original data. We characterize the number of random projections that are required for 1 -regularized compressed regression to identify the nonzero coefﬁcients in the true model with probability approaching one, a property called “sparsistence.” In addition, we show that 1 -regularized compressed regression asymptotically predicts as well as an oracle linear model, a property called “persistence.” Finally, we characterize the privacy properties of the compression procedure in information-theoretic terms, establishing upper bounds on the rate of information communicated between the compressed and uncompressed data that decay to zero. 1

5 0.092473619 186 nips-2007-Statistical Analysis of Semi-Supervised Regression

Author: Larry Wasserman, John D. Lafferty

Abstract: Semi-supervised methods use unlabeled data in addition to labeled data to construct predictors. While existing semi-supervised methods have shown some promising empirical performance, their development has been based largely based on heuristics. In this paper we study semi-supervised learning from the viewpoint of minimax theory. Our ﬁrst result shows that some common methods based on regularization using graph Laplacians do not lead to faster minimax rates of convergence. Thus, the estimators that use the unlabeled data do not have smaller risk than the estimators that use only labeled data. We then develop several new approaches that provably lead to improved performance. The statistical tools of minimax analysis are thus used to offer some new perspective on the problem of semi-supervised learning. 1

6 0.091777377 65 nips-2007-DIFFRAC: a discriminative and flexible framework for clustering

7 0.080821335 8 nips-2007-A New View of Automatic Relevance Determination

8 0.072852843 105 nips-2007-Infinite State Bayes-Nets for Structured Domains

9 0.071271703 34 nips-2007-Bayesian Policy Learning with Trans-Dimensional MCMC

10 0.067949213 69 nips-2007-Discriminative Batch Mode Active Learning

11 0.067101941 190 nips-2007-Support Vector Machine Classification with Indefinite Kernels

12 0.06566707 134 nips-2007-Multi-Task Learning via Conic Programming

13 0.063789152 145 nips-2007-On Sparsity and Overcompleteness in Image Models

14 0.062954649 167 nips-2007-Regulator Discovery from Gene Expression Time Series of Malaria Parasites: a Hierachical Approach

15 0.062729679 111 nips-2007-Learning Horizontal Connections in a Sparse Coding Model of Natural Images

16 0.060567159 47 nips-2007-Collapsed Variational Inference for HDP

17 0.060204625 82 nips-2007-Estimating divergence functionals and the likelihood ratio by penalized convex risk minimization

18 0.060100935 160 nips-2007-Random Features for Large-Scale Kernel Machines

19 0.055754557 148 nips-2007-Online Linear Regression and Its Application to Model-Based Reinforcement Learning

20 0.053766068 189 nips-2007-Supervised Topic Models

similar papers computed by lsi model

lsi for this paper:

topicId topicWeight

[(0, -0.199), (1, 0.031), (2, -0.078), (3, 0.027), (4, -0.036), (5, -0.007), (6, -0.002), (7, 0.006), (8, -0.063), (9, 0.056), (10, 0.077), (11, -0.017), (12, 0.133), (13, -0.004), (14, 0.047), (15, 0.048), (16, 0.053), (17, 0.005), (18, 0.004), (19, -0.153), (20, -0.016), (21, -0.035), (22, 0.106), (23, 0.054), (24, 0.158), (25, 0.128), (26, -0.068), (27, 0.05), (28, -0.105), (29, 0.155), (30, 0.022), (31, 0.022), (32, -0.046), (33, 0.098), (34, 0.099), (35, -0.06), (36, -0.029), (37, 0.191), (38, 0.04), (39, 0.032), (40, 0.019), (41, 0.009), (42, 0.013), (43, -0.085), (44, -0.075), (45, 0.019), (46, 0.121), (47, -0.019), (48, -0.024), (49, 0.112)]

similar papers list:

simIndex simValue paperId paperTitle

same-paper 1 0.95462227 99 nips-2007-Hierarchical Penalization

Author: Marie Szafranski, Yves Grandvalet, Pierre Morizet-mahoudeaux

2 0.72144687 43 nips-2007-Catching Change-points with Lasso

Author: Céline Levy-leduc, Zaïd Harchaoui

3 0.71384341 179 nips-2007-SpAM: Sparse Additive Models

Author: Han Liu, Larry Wasserman, John D. Lafferty, Pradeep K. Ravikumar

4 0.71371526 53 nips-2007-Compressed Regression

Author: Shuheng Zhou, Larry Wasserman, John D. Lafferty

5 0.53930235 8 nips-2007-A New View of Automatic Relevance Determination

Author: David P. Wipf, Srikantan S. Nagarajan

Abstract: Automatic relevance determination (ARD) and the closely-related sparse Bayesian learning (SBL) framework are effective tools for pruning large numbers of irrelevant features leading to a sparse explanatory subset. However, popular update rules used for ARD are either difﬁcult to extend to more general problems of interest or are characterized by non-ideal convergence properties. Moreover, it remains unclear exactly how ARD relates to more traditional MAP estimation-based methods for learning sparse representations (e.g., the Lasso). This paper furnishes an alternative means of expressing the ARD cost function using auxiliary functions that naturally addresses both of these issues. First, the proposed reformulation of ARD can naturally be optimized by solving a series of re-weighted 1 problems. The result is an efﬁcient, extensible algorithm that can be implemented using standard convex programming toolboxes and is guaranteed to converge to a local minimum (or saddle point). Secondly, the analysis reveals that ARD is exactly equivalent to performing standard MAP estimation in weight space using a particular feature- and noise-dependent, non-factorial weight prior. We then demonstrate that this implicit prior maintains several desirable advantages over conventional priors with respect to feature selection. Overall these results suggest alternative cost functions and update procedures for selecting features and promoting sparse solutions in a variety of general situations. In particular, the methodology readily extends to handle problems such as non-negative sparse coding and covariance component estimation. 1

6 0.43722233 89 nips-2007-Feature Selection Methods for Improving Protein Structure Prediction with Rosetta

7 0.39425701 67 nips-2007-Direct Importance Estimation with Model Selection and Its Application to Covariate Shift Adaptation

8 0.37649071 19 nips-2007-Active Preference Learning with Discrete Choice Data

9 0.37437308 186 nips-2007-Statistical Analysis of Semi-Supervised Regression

10 0.3666541 134 nips-2007-Multi-Task Learning via Conic Programming

11 0.35494623 105 nips-2007-Infinite State Bayes-Nets for Structured Domains

12 0.35257325 20 nips-2007-Adaptive Embedded Subgraph Algorithms using Walk-Sum Analysis

13 0.34871656 65 nips-2007-DIFFRAC: a discriminative and flexible framework for clustering

14 0.33268085 178 nips-2007-Simulated Annealing: Rigorous finite-time guarantees for optimization on continuous domains

15 0.3322174 27 nips-2007-Anytime Induction of Cost-sensitive Trees

16 0.33052877 63 nips-2007-Convex Relaxations of Latent Variable Training

17 0.3298476 145 nips-2007-On Sparsity and Overcompleteness in Image Models

18 0.32861781 190 nips-2007-Support Vector Machine Classification with Indefinite Kernels

19 0.32780051 62 nips-2007-Convex Learning with Invariances

20 0.32163066 34 nips-2007-Bayesian Policy Learning with Trans-Dimensional MCMC

similar papers computed by lda model

lda for this paper:

topicId topicWeight

[(5, 0.041), (13, 0.025), (16, 0.018), (21, 0.041), (31, 0.01), (34, 0.016), (35, 0.021), (47, 0.587), (49, 0.014), (83, 0.094), (85, 0.013), (90, 0.042)]

similar papers list:

simIndex simValue paperId paperTitle

1 0.97713441 114 nips-2007-Learning and using relational theories

Author: Charles Kemp, Noah Goodman, Joshua B. Tenenbaum

Abstract: Much of human knowledge is organized into sophisticated systems that are often called intuitive theories. We propose that intuitive theories are mentally represented in a logical language, and that the subjective complexity of a theory is determined by the length of its representation in this language. This complexity measure helps to explain how theories are learned from relational data, and how they support inductive inferences about unobserved relations. We describe two experiments that test our approach, and show that it provides a better account of human learning and reasoning than an approach developed by Goodman [1]. What is a theory, and what makes one theory better than another? Questions like these are of obvious interest to philosophers of science but are also discussed by psychologists, who have argued that everyday knowledge is organized into rich and complex systems that are similar in many respects to scientiﬁc theories. Even young children, for instance, have systematic beliefs about domains including folk physics, folk biology, and folk psychology [2]. Intuitive theories like these play many of the same roles as scientiﬁc theories: in particular, both kinds of theories are used to explain and encode observations of the world, and to predict future observations. This paper explores the nature, use and acquisition of simple theories. Consider, for instance, an anthropologist who has just begun to study the social structure of a remote tribe, and observes that certain words are used to indicate relationships between selected pairs of individuals. Suppose that term T1(·, ·) can be glossed as ancestor(·, ·), and that T2(·, ·) can be glossed as friend(·, ·). The anthropologist might discover that the ﬁrst term is transitive, and that the second term is symmetric with a few exceptions. Suppose that term T3(·, ·) can be glossed as defers to(·, ·), and that the tribe divides into two castes such that members of the second caste defer to members of the ﬁrst caste. In this case the anthropologist might discover two latent concepts (caste 1(·) and caste 2(·)) along with the relationship between these concepts. As these examples suggest, a theory can be deﬁned as a system of laws and concepts that specify the relationships between the elements in some domain [2]. We will consider how these theories are learned, how they are used to encode relational data, and how they support predictions about unobserved relations. Our approach to all three problems relies on the notion of subjective complexity. We propose that theory learners prefer simple theories, that people remember relational data in terms of the simplest underlying theory, and that people extend a partially observed data set according to the simplest theory that is consistent with their observations. There is no guarantee that a single measure of subjective complexity can do all of the work that we require [3]. This paper, however, explores the strong hypothesis that a single measure will sufﬁce. Our formal treatment of subjective complexity begins with the question of how theories are mentally represented. We suggest that theories are represented in some logical language, and propose a speciﬁc ﬁrst-order language that serves as a hypothesis about the “language of thought.” We then pursue the idea that the subjective complexity of a theory corresponds to the length of its representation in this language. Our approach therefore builds on the work of Feldman [4], and is related to other psychological applications of the notion of Kolmogorov complexity [5]. The complexity measure we describe can be used to deﬁne a probability distribution over a space of theories, and we develop a model of theory acquisition by using this distribution as the prior for a Bayesian learner. We also 1 (a) Star 11 (b) Bipartite (c) Exception 22 33 44 55 66 77 88 16 26 36 46 56 21 31 41 51 61 71 81 11 17 27 37 47 57 18 28 38 48 58 R(X, X). T(6). T(7). T(8). R(X, Y) ← ¯(X), T(Y). T R(X, 1). (d) Symmetric (e) Transitive 11 22 33 44 55 66 77 13 31 12 21 24 42 56 65 26 36 46 56 17 27 37 47 57 18 28 38 48 58 T(6). T(7). T(8). R(X, Y) ← ¯(X), T(Y). T ¯ R(1, 1). R(1, 6). (f) Random 12 21 13 23 14 24 34 13 32 14 24 34 15 25 35 45 16 26 36 46 56 51 52 35 54 61 26 63 46 56 R(1, 2). R(1, 3). R(2, 4). R(5, 6). R(1, 2). R(2, 3). R(3, 4). R(5, X). R(X, 4). R(X, Y) ← R(Y, X). R(X, X). R(4, 5). R(5, 6). R(X, Z) ← R(X, Y), R(Y, Z). R(2, 1). R(1, 3). R(6, 1). R(3, 2). R(2, 6). R(3, 5). R(6, 3). R(4, 6). ¯ ¯ ¯ R(X, X). R(6, 4). R(5, 3). Figure 1: Six possible extensions for a binary predicate R(·, ·). In each case, the objects in the domain are represented as digits, and a pair such as 16 indicates that R(1, 6) is true. Below each set of pairs, the simplest theory according to our complexity measure is shown. show how the same Bayesian approach helps to explain how theories support inductive generalization: given a set of observations, future observations (e.g. whether one individual defers to another) can be predicted using the posterior distribution over the space of theories. We test our approach by developing two experiments where people learn and make predictions about binary and ternary relations. As far as we know, the approach of Goodman [1] is the only other measure of theory complexity that has previously been tested as a psychological model [6]. We show that our experiments support our approach and raise challenges for this alternative model. 1 Theory complexity: a representation length approach Intuitive theories correspond to mental representations of some sort, and our ﬁrst task is to characterize the elements used to build these representations. We explore the idea that a theory is a system of statements in a logical language, and six examples are shown in Fig. 1. The theory in Fig. 1b is related to the defers to(·, ·) example already described. Here we are interested in a domain including 9 elements, and a two place predicate R(·, ·) that is true of all and only the 15 pairs shown. R is deﬁned using a unary predicate T which is true of only three elements: 6, 7, and 8. The theory includes a clause which states that R(X, Y) is true for all pairs XY such that T(X) is false and T(Y) is true. The theory in Fig. 1c is very similar, but includes an additional clause which speciﬁes that R(1, 1) is true, and an exception which speciﬁes that R(1, 6) is false. Formally, each theory we consider is a collection of function-free deﬁnite clauses. All variables are universally quantiﬁed: for instance, the clause R(X, Z) ← R(X, Y), R(Y, Z) is equivalent to the logical formula ∀x ∀y ∀z (R(x, z) ← R(x, y) ∧ R(y, z)). For readability, the theories in Fig. 1 include parentheses and arrows, but note that these symbols are unnecessary and can be removed. Our proposed language includes only predicate symbols, variable symbols, constant symbols, and a period that indicates when one clause ﬁnishes and another begins. Each theory in Fig. 1 speciﬁes the extension of one or more predicates. The extension of predicate P is deﬁned in terms of predicate P+ (which captures the basic rules that lead to membership in P) and predicate P− (which captures exceptions to these rules). The resulting extension of P is deﬁned 2 as P+ \ P− , or the set difference of P+ and P− .1 Once P has been deﬁned, later clauses in the theory may refer to P or its negation ¯. To ensure that our semantics is well-deﬁned, the predicates P in any valid theory must permit an ordering so that the deﬁnition of any predicate does not refer to predicates that follow it in the order. Formally, the deﬁnition of each predicate P+ or P− can refer only to itself (recursive deﬁnitions are allowed) and to any predicate M or ¯ where M < P. M Once we have committed to a speciﬁc language, the subjective complexity of a theory is assumed to correspond to the number of symbols in its representation. We have chosen a language where there is one symbol for each position in a theory where a predicate, variable or constant appears, and one symbol to indicate when each clause ends. Given this language, the subjective complexity c(T ) of theory T is equal to the sum of the number of clauses in the theory and the number of positions in the theory where a predicate, variable or constant appears: c(T ) = #clauses(T ) + #pred slots(T ) + #var slots(T ) + #const slots(T ). (1) For instance, the clause R(X, Z) ← R(X, Y), R(Y, Z). contributes ten symbols towards the complexity of a theory (three predicate symbols, six variable symbols, and one period). Other languages might be considered: for instance, we could use a language which uses ﬁve symbols (e.g. ﬁve bits) to represent each predicate, variable and constant, and one symbol (e.g. one bit) to indicate the end of a clause. Our approach to subjective complexity depends critically on the representation language, but once a language has been chosen the complexity measure is uniquely speciﬁed. Although our approach is closely related to the notion of Kolmogorov complexity and to Minimum Message Length (MML) and Minimum Description Length (MDL) approaches, we refer to it as a Representation Length (RL) approach. A RL approach includes a commitment to a speciﬁc language that is proposed as a psychological hypothesis, but these other approaches aspire towards results that do not depend on the language chosen.2 It is sometimes suggested that the notion of Kolmogorov complexity provides a more suitable framework for psychological research than the RL approach, precisely because it allows for results that do not depend on a speciﬁc description language [8]. We subscribe to the opposite view. Mental representations presumably rely on some particular language, and identifying this language is a central challenge for psychological research. The language we described should be considered as a tentative approximation of the language of thought. Other languages can and should be explored, but our language has several appealing properties. Feldman [4] has argued that deﬁnite clauses are psychologically natural, and working with these representations allows our approach to account for several classic results from the concept learning literature. For instance, our language leads to the prediction that conjunctive concepts are easier to learn than disjunctive concepts [9].3 Working with deﬁnite clauses also ensures that each of our theories has a unique minimal model, which means that the extension of a theory can be deﬁned in a particularly simple way. Finally, human learners deal gracefully with noise and exceptions, and our language provides a simple way to handle exceptions. Any concrete proposal about the language of thought should make predictions about memory, learning and reasoning. Suppose that data set D lists the extensions of one or more predicates, and that a theory is a “candidate theory” for D if it correctly deﬁnes the extensions of all predicates in D. Note that a candidate theory may well include latent predicates—predicates that do not appear in D, but are useful for deﬁning the predicates that have been observed. We will assume that humans encode D in terms of the simplest candidate theory for D, and that the difﬁculty of memorizing D is determined by the subjective complexity of this theory. Our approach can and should be tested against classic results from the memory literature. Unlike some other approaches to complexity [10], for instance, our model predicts that a sequence of k items is about equally easy to remember regardless of whether the items are drawn from a set of size 2, a set of size 10, or a set of size 1000 [11]. 1 The extension of P+ is the smallest set that satisﬁes all of the clauses that deﬁne P+ , and the extension of P is deﬁned similarly. To simplify our notation, Fig. 1 uses P to refer to both P and P+ , and ¯ to refer to ¯ and P P P− . Any instance of P that appears in a clause deﬁning P is really an instance of P+ , and any instance of ¯ that P appears in a clause deﬁning ¯ is really an instance of P− . P 2 MDL approaches also commit to a speciﬁc language, but this language is often intended to be as general as possible. See, for instance, the discussion of universal codes in Gr¨ nwald et al. [7]. u 3 A conjunctive concept C(·) can be deﬁned using a single clause: C(X) ← A(X), B(X). The shortest deﬁnition of a disjunctive concept requires two clauses: D(X) ← A(X). D(X) ← B(X). − 3 To develop a model of inductive learning and reasoning, we take a Bayesian approach, and use our complexity measure to deﬁne a prior distribution over a hypothesis space of theories: P (T ) ∝ 2−c(T ) .4 Given this prior distribution, we can use Bayesian inference to make predictions about unobserved relations and to discover the theory T that best accounts for the observations in data set D [12, 13]. Suppose that we have a likelihood function P (D|T ) which speciﬁes how the examples in D were generated from some underlying theory T . The best explanation for the data D is the theory that maximizes the posterior distribution P (T |D) ∝ P (D|T )P (T ). If we need to predict whether ground term g is likely to be true, 5 we can sum over the space of theories: P (g|D) = P (g|T )P (T |D) = T 1 P (D) P (D|T )P (T ) (2) T :g∈T where the ﬁnal sum is over all theories T that make ground term g true. 1.1 Related work The theories we consider are closely related to logic programs, and methods for Inductive Logic Programming (ILP) explore how these programs can be learned from examples [14]. ILP algorithms are often inspired by the idea of searching for the shortest theory that accounts for the available data, and ILP is occasionally cast as the problem of minimizing an explicit MDL criterion [10]. Although ILP algorithms are rarely considered as cognitive models, the RL approach has a long psychological history, and is proposed by Chomsky [15] and Leeuwenberg [16] among others. Formal measures of complexity have been developed in many ﬁelds [17], and there is at least one other psychological account of theory complexity. Goodman [1] developed a complexity measure that was originally a philosophical proposal about scientiﬁc theories, but was later tested as a model of subjective complexity [6]. A detailed description of this measure is not possible here, but we attempt to give a ﬂavor of the approach. Suppose that a basis is a set of predicates. The starting point for Goodman’s model is the intuition that basis B1 is at least as complex as basis B2 if B1 can be used to deﬁne B2. Goodman argues that this intuition is ﬂawed, but his model is founded on a reﬁnement of this intuition. For instance, since the binary predicate in Fig. 1b can be deﬁned in terms of two unary predicates, Goodman’s approach requires that the complexity of the binary predicate is no more than the sum of the complexities of the two unary predicates. We will use Goodman’s model as a baseline for evaluating our own approach, and a comparison between these two models should be informed by both theoretical and empirical considerations. On the theoretical side, our approach relies on a simple principle for deciding which structural properties are relevant to the measurement of complexity: the relevant properties are those with short logical representations. Goodman’s approach incorporates no such principle, and he proposes somewhat arbitrarily that reﬂexivity and symmetry are among the relevant structural properties but that transitivity is not. A second reason for preferring our model is that it makes contact with a general principle—the idea that simplicity is related to representation length—that has found many applications across psychology, machine learning, and philosophy. 2 Experimental results We designed two experiments to explore settings where people learn, remember, and make inductive inferences about relational data. Although theories often consist of systems of many interlocking relations, we keep our experiments simple by asking subjects to learn and reason about a single relation at a time. Despite this restriction, our experiments still make contact with several issues raised by systems of relations. As the defers to(·, ·) example suggests, a single relation may be best explained as the observable tip of a system involving several latent predicates (e.g. caste 1(·) and caste 2(·)). 4 To ensure that this distribution can be normalized, we assume that there is some upper bound on the number of predicate symbols, variable symbols, and constants, and on the length of the theories we will consider. There will therefore be a ﬁnite number of possible theories, and our prior will be a valid probability distribution. 5 A ground term is a term such as R(8, 9) that does not include any variables. 4 Learning time Complexity (RL) Complexity (Human) 6 300 Complexity (Goodman) 4 20 0 0 0 star bprt excp sym trans rand 2 0 star bprt excp sym trans rand 2 star bprt excp sym trans rand 100 200 star bprt excp sym trans rand 4 40 Figure 2: (a) Average time in seconds to learn the six sets in Fig. 1. (b) Average ratings of set complexity. (c) Complexity scores according to our representation length (RL) model. (d) Complexity scores according to Goodman’s model. 2.1 Experiment 1: memory and induction In our ﬁrst experiment, we studied the subjective complexity of six binary relations that display a range of structural properties, including reﬂexivity, symmetry, and transitivity. Materials and Methods. 18 adults participated in this experiment. Subjects were required to learn the 6 sets shown in Fig. 1, and to make inductive inferences about each set. Although Fig. 1 shows pairs of digits, the experiment used letter pairs, and the letters for each condition and the order in which these conditions were presented were randomized across subjects. The pairs for each condition were initially laid out randomly on screen, and subjects could drag them around and organize them to help them understand the structure of the set. At any stage, subjects could enter a test phase where they were asked to list the 15 pairs belonging to the current set. Subjects who made an error on the test were returned to the learning phase. After 9 minutes had elapsed, subjects were allowed to pass the test regardless of how many errors they made. After passing the test, subjects were asked to rate the complexity of the set compared to other sets with 15 pairs. Ratings were provided on a 7 point scale. Subjects were then asked to imagine that a new letter (e.g. letter 9) had belonged to the current alphabet, and were given two inductive tasks. First they were asked to enter between 1 and 10 novel pairs that they might have expected to see (each novel pair was required to include the new letter). Next they were told about a novel pair that belonged to the set (e.g. pair 91), and were again asked to enter up to 10 additional pairs that they might have expected to see. Results. The average time needed to learn each set is shown in Fig. 2a, and ratings of set complexity are shown in Fig. 2b. It is encouraging that these measures yield converging results, but they may be confounded since subjects rated the complexity of a set immediately after learning it. The complexities plotted in Fig. 2c are the complexities of the theories shown in Fig. 1, which we believe to be the simplest theories according to our complexity measure. The ﬁnal plot in Fig. 2 shows complexities according to Goodman’s model, which assigns each binary relation an integer between 0 and 4. There are several differences between these models: for instance, Goodman’s account incorrectly predicts that the exception case is the hardest of the six, but our model acknowledges that a simple theory remains simple if a handful of exceptions are added. Goodman’s account also predicts that transitivity is not an important structural regularity, but our model correctly predicts that the transitive set is simpler than the same set with some of the pairs reversed (the random set). Results for the inductive task are shown in Fig. 3. The ﬁrst two columns show the number of subjects who listed each novel pair. The remaining two columns show the probability of set membership predicted by our model. To generate these predictions, we applied Equation 2 and summed over a set of theories created by systematically extending the theories shown in Fig. 1. Each extended theory includes up to one additional clause for each predicate in the base theory, and each additional clause includes at most two predicate slots. For instance, each extended theory for the bipartite case is created by choosing whether or not to add the clause T(9), and adding up to one clause for predicate R.6 For the ﬁrst inductive task, the likelihood term P (D|T ) (see Equation 2) is set to 0 for all theories that are not consistent with the pairs observed during training, and to a constant for all remaining theories. For the second task we assumed in addition that the novel pair observed is 6 R(9, X), ¯(2, 9), and R(X, 9) ← R(X, 2) are three possible additions. R 5 18 9 9 0.5 trans symm excep bipart 0 91 random r=0.99 1 star 18 99 19 0 91 89 99 19 89 0.5 0 91 18 18 1 9 9 99 19 r=0.96 89 0.5 0 91 99 19 0 91 89 99 19 89 18 9 1 99 19 89 r=0.98 1 9 0.5 0 91 99 19 0 91 89 99 19 89 18 9 99 19 89 0.5 0 81 88 18 0 78 81 88 18 78 0 18 18 9 0 0 71 77 17 67 71 77 17 67 18 18 81 9 88 18 r=0.62 78 71 77 17 67 Human (no examples) 0 71 77 17 67 Human (1 example) 0 0 91 99 19 89 r=0.99 0 81 88 18 78 71 77 17 67 1 71 77 17 67 r=0.38 0.5 0 89 r=0.93 0.5 1 9 99 19 r=0.99 1 0.5 0 89 r=0.99 0.5 1 9 0 91 1 r=0.88 1 9 99 19 0.5 0 91 18 0 91 0.5 0 91 18 r=0.99 1 0 r=0.74 1 0.5 71 77 17 67 RL (no examples) 0 71 77 17 67 RL (one example) Figure 3: Data and model predictions for the induction task in Experiment 1. Columns 1 and 3 show predictions before any pairs involving the new letter are observed. Columns 2 and 4 show predictions after a single novel pair (marked with a gray bar) is observed to belong to the set. The model plots for each condition include correlations with the human data. sampled at random from all pairs involving the new letter.7 All model predictions were computed using Mace4 [18] to generate the extension of each theory considered. The supporting material includes predictions for a model based on the Goodman complexity measure and an exemplar model which assumes that the new letter will be just like one of the old letters.8 The exemplar model outperforms our model in the random condition, and makes accurate predictions about three other conditions. Overall, however, our model performs better than the two baselines. Here we focus on two important predictions that are not well handled by the exemplar model. In the symmetry condition, almost all subjects predict that 78 belongs to the set after learning that 87 belongs to the set, suggesting that they have learned an abstract rule. In the transitive condition, most subjects predict that pairs 72 through 76 belong to the set after learning that 71 belongs to the set. Our model accounts for this result, but the exemplar model has no basis for making predictions about letter 7, since this letter is now known to be unlike any of the others. 2.2 Experiment 2: learning from positive examples During the learning phase of our ﬁrst experiment, subjects learned a theory based on positive examples (the theory included all pairs they had seen) and negative examples (the theory ruled out all pairs they had not seen). Often, however, humans learn theories based on positive examples alone. Suppose, for instance, that our anthropologist has spent only a few hours with a new tribe. She may have observed several pairs who are obviously friends, but should realize that many other pairs of friends have not yet interacted in her presence. 7 For the second task, P (D|T ) is set to 0 for theories that are inconsistent with the training pairs and theories 1 which do not include the observed novel pair. For all remaining theories, P (D|T ) is set to n , where n is the total number of novel pairs that are consistent with T . 8 Supporting material is available at www.charleskemp.com 6 1 221 331 441 551 c) 7 1 R(X, X, Y). 221 443 552 663 d) 7 1 R(X, Y, Z). 231 456 615 344 e) 7 1 −10 −5 −0.1 −10 −20 −20 −10 −0.2 −20 777 771 778 789 237 777 771 778 789 237 −10 231 234 235 236 0 777 771 778 789 237 0 777 771 778 789 237 0 777 771 778 789 237 0 RL 0 R(2, 3, X). 777 771 778 789 237 7 R(X, X, 1). 777 771 778 789 237 b) 777 771 778 789 237 1 111 222 333 444 777 771 778 789 237 7 R(X, X, X). 777 771 778 789 237 Human a) Figure 4: Data and model predictions for Experiment 2. The four triples observed for each set are shown at the top of the ﬁgure. The ﬁrst row of plots shows average ratings on a scale from 1 (very unlikely to belong to the set) to 7 (very likely). Model predictions are plotted as log probabilities. Our framework can handle cases like these if we assume that the data D in Equation 2 are sampled from the ground terms that are true according to the underlying theory. We follow [10] and [13] and use a distribution P (D|T ) which assumes that the examples in D are randomly sampled with replacement from the ground terms that are true. This sampling assumption encourages our model to identify the theory with the smallest extension that is compatible with all of the training examples. We tested this approach by designing an experiment where learners were given sets of examples that were compatible with several underlying theories. Materials and Methods. 15 adults participated in this experiment immediately after taking Experiment 1. In each of ﬁve conditions, subjects were told about a set of triples built from an alphabet of 9 letters. They were shown four triples that belonged to the set (Fig. 4), and told that the set might include triples that they had not seen. Subjects then gave ratings on a seven point scale to indicate whether ﬁve additional triples (see Fig. 4) were likely to belong to the set. Results. Average ratings and model predictions are shown in Fig. 4. Model predictions for each condition were computed using Equation 2 and summing over a space of theories that included the ﬁve theories shown at the top of Fig. 4, variants of these ﬁve theories which stated that certain pairs of slots could not be occupied by the same constant,9 and theories that included no variables but merely enumerated up to 5 triples.10 Although there are general theories like R(X, Y, Z) that are compatible with the triples observed in all ﬁve conditions, Fig. 4 shows that people were sensitive to different regularities in each case.11 We focus on one condition (Fig. 4b) that exposes the strengths and weaknesses of our model. According to our model, the two most probable theories given the triples for this condition are R(X, X, 1) and the closely related variant that rules out R(1, 1, 1). The next most probable theory is R(X, X, Y). These predictions are consistent with people’s judgments that 771 is very likely to belong to the set, and that 778 is the next most likely option. Unlike our model, however, people consider 777 to be substantially less likely than 778 to belong to the set. This result may suggest that the variant of R(X, X, Y) that rules out R(X, X, X) deserves a higher prior probability than our model recognizes. To better account for cases like this, it may be worth considering languages where any two variables that belong to the same clause but have different names must refer to different entities. 3 Discussion and Conclusion There are many psychological models of concept learning [4, 12, 13], but few that use representations rich enough to capture the content of intuitive theories. We suggested that intuitive theories are mentally represented in a ﬁrst-order logical language, and proposed a speciﬁc hypothesis about 9 ¯ One such theory includes two clauses: R(X, X, Y). R(X, X, X). One such theory is the following list of clauses: R(2, 2, 1). R(3, 3, 1). R(4, 4, 1). R(5, 5, 1). R(7, 7, 7). 11 Similar results have been found with 9-month old infants. Cases like Figs. 4b and 4c have been tested in an infant language-learning study where the stimuli were three-syllable strings [19]. 9-month old infants exposed to strings like the four in Fig. 4c generalized to other strings consistent with the theory R(X, X, Y), but infants in the condition corresponding to Fig. 4b generalized only to strings consistent with the theory R(X, X, 1). 10 7 this “language of thought.” We assumed that the subjective complexity of a theory depends on the length of its representation in this language, and described experiments which suggest that the resulting complexity measure helps to explain how theories are learned and used for inductive inference. Our experiments deliberately used stimuli that minimize the inﬂuence of prior knowledge. Theories, however, are cumulative, and the theory that seems simplest to a learner will often depend on her background knowledge. Our approach provides a natural place for background knowledge to be inserted. A learner can be supplied with a stock of background predicates, and the shortest representation for a data set will depend on which background predicates are available. Since different sets of predicates will lead to different predictions about subjective complexity, empirical results can help to determine the background knowledge that people bring to a given class of problems. Future work should aim to reﬁne the representation language and complexity measure we proposed. We expect that something like our approach will be suitable for modeling a broad class of intuitive theories, but the speciﬁc framework presented here can almost certainly be improved. Future work should also consider different strategies for searching the space of theories. Some of the strategies developed in the ILP literature should be relevant [14], but a detailed investigation of search algorithms seems premature until our approach has held up to additional empirical tests. It is comparatively easy to establish whether the theories that are simple according to our approach are also considered simple by people, and our experiments have made a start in this direction. It is much harder to establish that our approach captures most of the theories that are subjectively simple, and more exhaustive experiments are needed before this conclusion can be drawn. Boolean concept learning has been studied for more than ﬁfty years [4, 9], and many psychologists have made empirical and theoretical contributions to this ﬁeld. An even greater effort will be needed to crack the problem of theory learning, since the space of intuitive theories is much richer than the space of Boolean concepts. The difﬁculty of this problem should not be underestimated, but computational approaches can contribute part of the solution. Acknowledgments Supported by the William Asbjornsen Albert memorial fellowship (CK), the James S. McDonnell Foundation Causal Learning Collaborative Initiative (NDG, JBT) and the Paul E. Newton chair (JBT). References [1] N. Goodman. The structure of appearance. 2nd edition, 1961. [2] S. Carey. Conceptual change in childhood. MIT Press, Cambridge, MA, 1985. [3] H. A. Simon. Complexity and the representation of patterned sequences of symbols. Psychological Review, 79:369–382, 1972. [4] J. Feldman. An algebra of human concept learning. JMP, 50:339–368, 2006. [5] N. Chater and P. Vitanyi. Simplicity: a unifying principle in cognitive science. TICS, 7:19–22, 2003. [6] J. T. Krueger. A theory of structural simplicity and its relevance to aspects of memory, perception, and conceptual naturalness. PhD thesis, University of Pennsylvania, 1979. [7] P. Gr¨ nwald, I. J. Myung, and M. Pitt, editors. Advances in Minimum Description Length: Theory and u Applications. 2005. [8] N. Chater. Reconciling simplicity and likelihood principles in perceptual organization. Psychological Review, 103:566–581, 1996. [9] J. A. Bruner, J. S. Goodnow, and G. J. Austin. A study of thinking. Wiley, 1956. [10] D. Conklin and I. H. Witten. Complexity-based induction. Machine Learning, 16(3):203–225, 1994. [11] G. A. Miller. The magical number seven, plus or minus two: Some limits on our capacity for processing information. Psychological Review, 63(1):81–97, 1956. [12] N. D. Goodman, T. L. Grifﬁths, J. Feldman, and J. B. Tenenbaum. A rational analysis of rule-based concept learning. In CogSci, 2007. [13] J. B. Tenenbaum and T. L. Grifﬁths. Generalization, similarity, and Bayesian inference. BBS, 24:629–641, 2001. [14] S. Muggleton and L. De Raedt. Inductive logic programming: theory and methods. Journal of Logic Programming, 19-20:629–679, 1994. [15] N. Chomsky. The logical structure of linguistic theory. University of Chicago Press, Chicago, 1975. [16] E. L. J. Leeuwenberg. A perceptual coding language for visual and auditory patterns. American Journal of Psychology, 84(3):307–349, 1971. [17] B. Edmonds. Syntactic measures of complexity. PhD thesis, University of Manchester, 1999. [18] W. McCune. Mace4 reference manual and guide. Technical Report ANL/MCS-TM-264, Argonne National Laboratory, 2003. [19] L. Gerken. Decisions, decisions: infant language learning when multiple generalizations are possible. Cognition, 98(3):67–74, 2006. 8

2 0.97485191 124 nips-2007-Managing Power Consumption and Performance of Computing Systems Using Reinforcement Learning

Author: Gerald Tesauro, Rajarshi Das, Hoi Chan, Jeffrey Kephart, David Levine, Freeman Rawson, Charles Lefurgy

Abstract: Electrical power management in large-scale IT systems such as commercial datacenters is an application area of rapidly growing interest from both an economic and ecological perspective, with billions of dollars and millions of metric tons of CO2 emissions at stake annually. Businesses want to save power without sacriﬁcing performance. This paper presents a reinforcement learning approach to simultaneous online management of both performance and power consumption. We apply RL in a realistic laboratory testbed using a Blade cluster and dynamically varying HTTP workload running on a commercial web applications middleware platform. We embed a CPU frequency controller in the Blade servers’ ﬁrmware, and we train policies for this controller using a multi-criteria reward signal depending on both application performance and CPU power consumption. Our testbed scenario posed a number of challenges to successful use of RL, including multiple disparate reward functions, limited decision sampling rates, and pathologies arising when using multiple sensor readings as state variables. We describe innovative practical solutions to these challenges, and demonstrate clear performance improvements over both hand-designed policies as well as obvious “cookbook” RL implementations. 1

same-paper 3 0.96653062 99 nips-2007-Hierarchical Penalization

Author: Marie Szafranski, Yves Grandvalet, Pierre Morizet-mahoudeaux

4 0.96424478 188 nips-2007-Subspace-Based Face Recognition in Analog VLSI

Author: Gonzalo Carvajal, Waldo Valenzuela, Miguel Figueroa

Abstract: We describe an analog-VLSI neural network for face recognition based on subspace methods. The system uses a dimensionality-reduction network whose coeﬃcients can be either programmed or learned on-chip to perform PCA, or programmed to perform LDA. A second network with userprogrammed coeﬃcients performs classiﬁcation with Manhattan distances. The system uses on-chip compensation techniques to reduce the eﬀects of device mismatch. Using the ORL database with 12x12-pixel images, our circuit achieves up to 85% classiﬁcation performance (98% of an equivalent software implementation). 1

5 0.94424278 145 nips-2007-On Sparsity and Overcompleteness in Image Models

Author: Pietro Berkes, Richard Turner, Maneesh Sahani

Abstract: Computational models of visual cortex, and in particular those based on sparse coding, have enjoyed much recent attention. Despite this currency, the question of how sparse or how over-complete a sparse representation should be, has gone without principled answer. Here, we use Bayesian model-selection methods to address these questions for a sparse-coding model based on a Student-t prior. Having validated our methods on toy data, we ﬁnd that natural images are indeed best modelled by extremely sparse distributions; although for the Student-t prior, the associated optimal basis size is only modestly over-complete. 1

6 0.74391407 168 nips-2007-Reinforcement Learning in Continuous Action Spaces through Sequential Monte Carlo Methods

7 0.71527004 100 nips-2007-Hippocampal Contributions to Control: The Third Way

8 0.67196721 28 nips-2007-Augmented Functional Time Series Representation and Forecasting with Gaussian Processes

9 0.66624051 169 nips-2007-Retrieved context and the discovery of semantic structure

10 0.66039634 86 nips-2007-Exponential Family Predictive Representations of State

11 0.65974069 105 nips-2007-Infinite State Bayes-Nets for Structured Domains

12 0.65737534 210 nips-2007-Unconstrained On-line Handwriting Recognition with Recurrent Neural Networks

13 0.65694237 1 nips-2007-A Bayesian Framework for Cross-Situational Word-Learning

14 0.65692103 34 nips-2007-Bayesian Policy Learning with Trans-Dimensional MCMC

15 0.65661657 148 nips-2007-Online Linear Regression and Its Application to Model-Based Reinforcement Learning

16 0.65497661 158 nips-2007-Probabilistic Matrix Factorization

17 0.65418303 25 nips-2007-An in-silico Neural Model of Dynamic Routing through Neuronal Coherence

18 0.65044159 163 nips-2007-Receding Horizon Differential Dynamic Programming

19 0.64996856 43 nips-2007-Catching Change-points with Lasso

20 0.64929163 93 nips-2007-GRIFT: A graphical model for inferring visual classification features from human data