nips nips2000 nips2000-143 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Christopher K. I. Williams, Matthias Seeger
Abstract: unkown-abstract
Reference: text
sentIndex sentText sentNum sentScore
1 &xyy;k &UgvdSrÏPr~ xumÈ3u¾wUuk d#XrÐf#|f ygf XlU¶#d} fUdl k ¸ Î u¶~&efg; §j¶dÉ ( Âq~·X~ ÆÍyj ~f ¢d ÌuË gf ! [sent-12, score-0.111]
2 © § H6 6 © D ¥ wg B ¨ ª 8 s® s 6 B x w B ¥ FF § ¤ # p ! [sent-62, score-0.057]
3 k l y u sq n u m p û y oq l i { ~q k k i} ©V&tdr; ¢c G¿Vr ¥@$r '©¥Fú ef ù e ! [sent-75, score-0.067]
4 ß 8 q # ¦ B q % Q % Ê Ü Þ ee 0 8 ¤ ¦ u 6WB ¤ ¥¥¥¥¦ &"@ 4 `C % `e º ¹ # `¿Î Dx Ì Ê Ü P¥ Ê aR Ëp¥£¡ Y¥7 ¤ § 4 @% µ · Ú¿(% Ø Ý 7 Ü % 8 ¦ ) % C ¦ % ) Û e Ù e5¿d$@V@pd'% 5r5"¥¥P5`V1'2! [sent-111, score-0.072]
5 % # % 4 ) % % 4 y oq k k u n u n y i y oq l i { ~q k k i} { k k u { o n m y iq k k x i v u l i sq p o n m m i l k i Wdr '&¢£&12£zVdr &$r ''£|'¥j12z¥dr '2IXw&t;r 2&122¥r5£jh gf dVe e ! [sent-172, score-0.322]
6 £ ' H ¥ £ ¡ X 9 ) £ C £ £ ¡ ~¨(0X 9 PF X C bD£ ¡ Pc ' £ r (I) ` df a Y £ ¡ A 9 GY ' £ r (I) £ ¡ T£ ` 9 ) ¢' X b' g ! [sent-298, score-0.057]
7 Dy b lI H ©5 X " W 3 e F 5 $ F 3 " F 1F e 1 " 3 3 $ F 3 p 3 b ec 5 " 3 X e 1 3Y s 1 b p 3 1 b b 5 p 3 $ q! [sent-351, score-0.13]
wordName wordTfidf (topN-words)
[('uv', 0.356), ('yd', 0.217), ('ug', 0.2), ('wp', 0.2), ('wb', 0.172), ('hp', 0.156), ('xg', 0.156), ('yy', 0.153), ('vp', 0.153), ('xw', 0.153), ('ff', 0.151), ('vu', 0.133), ('xd', 0.13), ('ec', 0.13), ('vy', 0.115), ('dc', 0.112), ('bb', 0.111), ('bg', 0.111), ('gf', 0.111), ('wq', 0.111), ('ia', 0.108), ('ep', 0.1), ('dr', 0.096), ('xf', 0.096), ('db', 0.094), ('bp', 0.09), ('hff', 0.089), ('qd', 0.089), ('qdb', 0.089), ('sc', 0.089), ('hf', 0.087), ('ss', 0.087), ('sv', 0.084), ('pp', 0.081), ('iu', 0.08), ('bd', 0.08), ('iq', 0.077), ('xb', 0.077), ('xs', 0.077), ('qc', 0.077), ('qv', 0.077), ('wd', 0.077), ('dd', 0.075), ('ga', 0.075), ('pf', 0.075), ('ee', 0.072), ('uj', 0.069), ('ud', 0.069), ('ay', 0.069), ('qk', 0.069), ('dq', 0.069), ('gy', 0.067), ('hv', 0.067), ('oq', 0.067), ('pg', 0.067), ('pv', 0.067), ('ue', 0.067), ('vqp', 0.067), ('kf', 0.064), ('id', 0.062), ('xp', 0.06), ('ge', 0.06), ('ah', 0.06), ('bh', 0.057), ('df', 0.057), ('gw', 0.057), ('wf', 0.057), ('ws', 0.057), ('gb', 0.057), ('vh', 0.057), ('wg', 0.057), ('ys', 0.057), ('mk', 0.057), ('pe', 0.057), ('gr', 0.052), ('dl', 0.052), ('bs', 0.052), ('qs', 0.052), ('yc', 0.052), ('qw', 0.052), ('fj', 0.05), ('yf', 0.048), ('vf', 0.045), ('ag', 0.045), ('ts', 0.045), ('bw', 0.044), ('gqdb', 0.044), ('hd', 0.044), ('ig', 0.044), ('iqdb', 0.044), ('lix', 0.044), ('qhvu', 0.044), ('sy', 0.044), ('uf', 0.044), ('uy', 0.044), ('ydy', 0.044), ('yr', 0.044), ('yv', 0.044), ('dy', 0.043), ('px', 0.043)]
simIndex simValue paperId paperTitle
same-paper 1 1.0000002 143 nips-2000-Using the Nyström Method to Speed Up Kernel Machines
Author: Christopher K. I. Williams, Matthias Seeger
Abstract: unkown-abstract
2 0.069383681 62 nips-2000-Generalized Belief Propagation
Author: Jonathan S. Yedidia, William T. Freeman, Yair Weiss
Abstract: Belief propagation (BP) was only supposed to work for tree-like networks but works surprisingly well in many applications involving networks with loops, including turbo codes. However, there has been little understanding of the algorithm or the nature of the solutions it finds for general graphs. We show that BP can only converge to a stationary point of an approximate free energy, known as the Bethe free energy in statistical physics. This result characterizes BP fixed-points and makes connections with variational approaches to approximate inference. More importantly, our analysis lets us build on the progress made in statistical physics since Bethe's approximation was introduced in 1935. Kikuchi and others have shown how to construct more accurate free energy approximations, of which Bethe's approximation is the simplest. Exploiting the insights from our analysis, we derive generalized belief propagation (GBP) versions ofthese Kikuchi approximations. These new message passing algorithms can be significantly more accurate than ordinary BP, at an adjustable increase in complexity. We illustrate such a new GBP algorithm on a grid Markov network and show that it gives much more accurate marginal probabilities than those found using ordinary BP. 1
3 0.04019447 127 nips-2000-Structure Learning in Human Causal Induction
Author: Joshua B. Tenenbaum, Thomas L. Griffiths
Abstract: We use graphical models to explore the question of how people learn simple causal relationships from data. The two leading psychological theories can both be seen as estimating the parameters of a fixed graph. We argue that a complete account of causal induction should also consider how people learn the underlying causal graph structure, and we propose to model this inductive process as a Bayesian inference. Our argument is supported through the discussion of three data sets.
4 0.039677195 20 nips-2000-Algebraic Information Geometry for Learning Machines with Singularities
Author: Sumio Watanabe
Abstract: Algebraic geometry is essential to learning theory. In hierarchical learning machines such as layered neural networks and gaussian mixtures, the asymptotic normality does not hold , since Fisher information matrices are singular. In this paper , the rigorous asymptotic form of the stochastic complexity is clarified based on resolution of singularities and two different problems are studied. (1) If the prior is positive, then the stochastic complexity is far smaller than BIO, resulting in the smaller generalization error than regular statistical models, even when the true distribution is not contained in the parametric model. (2) If Jeffreys' prior, which is coordinate free and equal to zero at singularities, is employed then the stochastic complexity has the same form as BIO. It is useful for model selection, but not for generalization. 1
5 0.03638443 145 nips-2000-Weak Learners and Improved Rates of Convergence in Boosting
Author: Shie Mannor, Ron Meir
Abstract: The problem of constructing weak classifiers for boosting algorithms is studied. We present an algorithm that produces a linear classifier that is guaranteed to achieve an error better than random guessing for any distribution on the data. While this weak learner is not useful for learning in general, we show that under reasonable conditions on the distribution it yields an effective weak learner for one-dimensional problems. Preliminary simulations suggest that similar behavior can be expected in higher dimensions, a result which is corroborated by some recent theoretical bounds. Additionally, we provide improved convergence rate bounds for the generalization error in situations where the empirical error can be made small, which is exactly the situation that occurs if weak learners with guaranteed performance that is better than random guessing can be established. 1
6 0.035825022 14 nips-2000-A Variational Mean-Field Theory for Sigmoidal Belief Networks
7 0.029365106 85 nips-2000-Mixtures of Gaussian Processes
8 0.027267139 110 nips-2000-Regularization with Dot-Product Kernels
9 0.027130011 52 nips-2000-Fast Training of Support Vector Classifiers
10 0.024676887 53 nips-2000-Feature Correspondence: A Markov Chain Monte Carlo Approach
11 0.024325848 97 nips-2000-Overfitting in Neural Nets: Backpropagation, Conjugate Gradient, and Early Stopping
12 0.023960672 87 nips-2000-Modelling Spatial Recall, Mental Imagery and Neglect
13 0.023753325 34 nips-2000-Competition and Arbors in Ocular Dominance
14 0.022495991 61 nips-2000-Generalizable Singular Value Decomposition for Ill-posed Datasets
15 0.022387762 134 nips-2000-The Kernel Trick for Distances
16 0.021344151 27 nips-2000-Automatic Choice of Dimensionality for PCA
17 0.020735001 21 nips-2000-Algorithmic Stability and Generalization Performance
18 0.020377599 124 nips-2000-Spike-Timing-Dependent Learning for Oscillatory Networks
19 0.019521695 126 nips-2000-Stagewise Processing in Error-correcting Codes and Image Restoration
20 0.018994901 106 nips-2000-Propagation Algorithms for Variational Bayesian Learning
topicId topicWeight
[(0, 0.058), (1, 0.012), (2, 0.012), (3, -0.008), (4, 0.036), (5, 0.022), (6, 0.002), (7, 0.006), (8, -0.001), (9, 0.019), (10, -0.053), (11, 0.039), (12, -0.021), (13, 0.006), (14, 0.023), (15, -0.002), (16, 0.045), (17, -0.002), (18, 0.1), (19, -0.055), (20, -0.022), (21, 0.128), (22, -0.095), (23, 0.016), (24, -0.01), (25, -0.011), (26, 0.029), (27, 0.087), (28, -0.083), (29, 0.066), (30, 0.011), (31, -0.091), (32, -0.183), (33, -0.117), (34, -0.08), (35, 0.177), (36, 0.099), (37, -0.139), (38, 0.254), (39, -0.19), (40, 0.33), (41, -0.195), (42, 0.091), (43, 0.014), (44, -0.125), (45, -0.288), (46, 0.103), (47, -0.102), (48, 0.127), (49, -0.011)]
simIndex simValue paperId paperTitle
same-paper 1 0.99921399 143 nips-2000-Using the Nyström Method to Speed Up Kernel Machines
Author: Christopher K. I. Williams, Matthias Seeger
Abstract: unkown-abstract
2 0.35148776 62 nips-2000-Generalized Belief Propagation
Author: Jonathan S. Yedidia, William T. Freeman, Yair Weiss
Abstract: Belief propagation (BP) was only supposed to work for tree-like networks but works surprisingly well in many applications involving networks with loops, including turbo codes. However, there has been little understanding of the algorithm or the nature of the solutions it finds for general graphs. We show that BP can only converge to a stationary point of an approximate free energy, known as the Bethe free energy in statistical physics. This result characterizes BP fixed-points and makes connections with variational approaches to approximate inference. More importantly, our analysis lets us build on the progress made in statistical physics since Bethe's approximation was introduced in 1935. Kikuchi and others have shown how to construct more accurate free energy approximations, of which Bethe's approximation is the simplest. Exploiting the insights from our analysis, we derive generalized belief propagation (GBP) versions ofthese Kikuchi approximations. These new message passing algorithms can be significantly more accurate than ordinary BP, at an adjustable increase in complexity. We illustrate such a new GBP algorithm on a grid Markov network and show that it gives much more accurate marginal probabilities than those found using ordinary BP. 1
3 0.26400012 127 nips-2000-Structure Learning in Human Causal Induction
Author: Joshua B. Tenenbaum, Thomas L. Griffiths
Abstract: We use graphical models to explore the question of how people learn simple causal relationships from data. The two leading psychological theories can both be seen as estimating the parameters of a fixed graph. We argue that a complete account of causal induction should also consider how people learn the underlying causal graph structure, and we propose to model this inductive process as a Bayesian inference. Our argument is supported through the discussion of three data sets.
4 0.24743827 97 nips-2000-Overfitting in Neural Nets: Backpropagation, Conjugate Gradient, and Early Stopping
Author: Rich Caruana, Steve Lawrence, C. Lee Giles
Abstract: The conventional wisdom is that backprop nets with excess hidden units generalize poorly. We show that nets with excess capacity generalize well when trained with backprop and early stopping. Experiments suggest two reasons for this: 1) Overfitting can vary significantly in different regions of the model. Excess capacity allows better fit to regions of high non-linearity, and backprop often avoids overfitting the regions of low non-linearity. 2) Regardless of size, nets learn task subcomponents in similar sequence. Big nets pass through stages similar to those learned by smaller nets. Early stopping can stop training the large net when it generalizes comparably to a smaller net. We also show that conjugate gradient can yield worse generalization because it overfits regions of low non-linearity when learning to fit regions of high non-linearity.
5 0.20968546 52 nips-2000-Fast Training of Support Vector Classifiers
Author: Fernando Pérez-Cruz, Pedro Luis Alarcón-Diana, Angel Navia-Vázquez, Antonio Artés-Rodríguez
Abstract: In this communication we present a new algorithm for solving Support Vector Classifiers (SVC) with large training data sets. The new algorithm is based on an Iterative Re-Weighted Least Squares procedure which is used to optimize the SVc. Moreover, a novel sample selection strategy for the working set is presented, which randomly chooses the working set among the training samples that do not fulfill the stopping criteria. The validity of both proposals, the optimization procedure and sample selection strategy, is shown by means of computer experiments using well-known data sets. 1 INTRODUCTION The Support Vector Classifier (SVC) is a powerful tool to solve pattern recognition problems [13, 14] in such a way that the solution is completely described as a linear combination of several training samples, named the Support Vectors. The training procedure for solving the SVC is usually based on Quadratic Programming (QP) which presents some inherent limitations, mainly the computational complexity and memory requirements for large training data sets. This problem is typically avoided by dividing the QP problem into sets of smaller ones [6, 1, 7, 11], that are iteratively solved in order to reach the SVC solution for the whole set of training samples. These schemes rely on an optimizing engine, QP, and in the sample selection strategy for each sub-problem, in order to obtain a fast solution for the SVC. An Iterative Re-Weighted Least Squares (IRWLS) procedure has already been proposed as an alternative solver for the SVC [10] and the Support Vector Regressor [9], being computationally efficient in absolute terms. In this communication, we will show that the IRWLS algorithm can replace the QP one in any chunking scheme in order to find the SVC solution for large training data sets. Moreover, we consider that the strategy to decide which training samples must j oin the working set is critical to reduce the total number of iterations needed to attain the SVC solution, and the runtime complexity as a consequence. To aim for this issue, the computer program SV cradit have been developed so as to solve the SVC for large training data sets using IRWLS procedure and fixed-size working sets. The paper is organized as follows. In Section 2, we start by giving a summary of the IRWLS procedure for SVC and explain how it can be incorporated to a chunking scheme to obtain an overall implementation which efficiently deals with large training data sets. We present in Section 3 a novel strategy to make up the working set. Section 4 shows the capabilities of the new implementation and they are compared with the fastest available SVC implementation, SV Mlight [6]. We end with some concluding remarks. 2 IRWLS-SVC In order to solve classification problems, the SVC has to minimize Lp = ~llwI12+CLei- LJliei- LQi(Yi(¢(xifw+b)-l+ei) (1) i i i with respectto w, band ei and maximize it with respectto Qi and Jli, subject to Qi, Jli ~ 0, where ¢(.) is a nonlinear transformation (usually unknown) to a higher dimensional space and C is a penalization factor. The solution to (1) is defined by the Karush-Kuhn-Tucker (KKT) conditions [2]. For further details on the SVC, one can refer to the tutorial survey by Burges [2] and to the work ofVapnik [13, 14]. In order to obtain an IRWLS procedure we will first need to rearrange (1) in such a way that the terms depending on ei can be removed because, at the solution C - Qi - Jli = 0 Vi (one of the KKT conditions [2]) must hold. Lp = 1 Qi(l- Yi(¢T(Xi)W + b)) 211wl12 + L i = (2) where The weighted least square nature of (2) can be understood if ei is defined as the error on each sample and ai as its associated weight, where! IIwl1 2 is a regularizing functional. The minimization of (2) cannot be accomplished in a single step because ai = ai(ei), and we need to apply an IRWLS procedure [4], summarized below in tree steps: 1. Considering the ai fixed, minimize (2). 2. Recalculate ai from the solution on step 1. 3. Repeat until convergence. In order to work with Reproducing Kernels in Hilbert Space (RKHS), as the QP procedure does, we require that w = Ei (JiYi¢(Xi) and in order to obtain a non-zero b, that Ei {JiYi = O. Substituting them into (2), its minimum with respect to {Ji and b for a fixed set of ai is found by solving the following linear equation system l (3) IThe detailed description of the steps needed to obtain (3) from (2) can be found in [10]. where y = [Yl, Y2, ... Yn]T (4) 'r/i,j = 1, ... ,n 'r/i,j = 1, ... ,n (H)ij = YiYj¢T(Xi)¢(Xj) = YiyjK(Xi,Xj) (Da)ij = aio[i - j] 13 = [,81, ,82, ... (5) (6) (7) , ,8n]T and 0[·] is the discrete impulse function. Finally, the dependency of ai upon the Lagrange multipliers is eliminated using the KKT conditions, obtaining a, ai 2.1 ={~ ei Yi' eiYi < Yt.et. > - ° ° (8) IRWLS ALGORITHMIC IMPLEMENTATION The SVC solution with the IRWLS procedure can be simplified by dividing the training samples into three sets. The first set, SI, contains the training samples verifying < ,8i < C, which have to be determined by solving (3). The second one, S2, includes every training sample whose,8i = 0. And the last one, S3, is made up of the training samples whose ,8i = C. This division in sets is fully justified in [10]. The IRWLS-SVC algorithm is shown in Table 1. ° 0. Initialization: SI will contain every training sample, S2 = 0 and S3 = 0. Compute H. e_a = y, f3_a = 0, b_a = 0, G 13 = Gin, a = 1 and G b3 = G bi n . 1 Solve [ (H)Sb S1 + D(al S1 . =° = e-lt a, 3. ai = { ~ (13) S2 2. e ° 1[ (Y)Sl (f3)Sl ] (y ) ~1 b and (13) Ss = C DyH(f3 - f3_a) - (b - b_a)1 =[1- G 13 ] G b3 ' °. eiYi < e- _ > O'r/Z E SI U S2 U S3 tYt 4. Sets reordering: a. Move every sample in S3 with eiYi < to S2. b. Move every sample in SI with ,8i = C to S3. c. Move every sample in SI with ai = to S2 . d. Move every sample in S2 with ai :I to SI. 5. e_a = e, f3_a = 13, G 13 = (H)Sl,SS (f3)ss + (G in )Sl' b-lt = band Gb3 = -y~s (f3)ss + Gbin · 6. Go to step 1 and repeat until convergence. ei Yi ' ° ° ° Table 1: IRWLS-SVC algorithm. The IRWLS-SVC procedure has to be slightly modified in order to be used inside a chunk:ing scheme as the one proposed in [8, 6], such that it can be directly applied in the one proposed in [1]. A chunking scheme is needed to solve the SVC whenever H is too large to fit into memory. In those cases, several SVC with a reduced set of training samples are iteratively solved until the solution for the whole set is found. The samples are divide into a working set, Sw, which is solved as a full SVC problem, and an inactive set, Sin. If there are support vectors in the inactive set, as it might be, the inactive set modifies the IRWLSSVC procedure, adding a contribution to the independent term in the linear equation system (3) . Those support vectors in S in can be seen as anchored samples in S3, because their ,8i is not zero and can not be modified by the IRWLS procedure. Then, such contribution (Gin and G bin ) will be calculated as G 13 and G b3 are (Table 1, 5th step), before calling the IRWLS-SVC algorithm. We have already modified the IRWLS-SVC in Table 1 to consider Gin and G bin , which must be set to zero if the Hessian matrix, H, fits into memory for the whole set of training samples. The resolution of the SVC for large training data sets, employing as minimization engine the IRWLS procedure, is summarized in the following steps: 1. Select the samples that will form the working set. 2. Construct Gin = (H)Sw,Sin (f3)s.n and G bin = -yIin (f3)Sin 3. Solve the IRWLS-SVC procedure, following the steps in Table 1. 4. Compute the error of every training sample. 5. If the stopping conditions Yiei < C eiYi> -c leiYil < C 'Vii 'Vii 'Vii (Ji = 0 (Ji = C 0 < (Ji < C (9) (10) (11) are fulfilled, the SVC solution has been reached. The stopping conditions are the ones proposed in [6] and C must be a small value around 10 - 3 , a full discussion concerning this topic can be found in [6]. 3 SAMPLE SELECTION STRATEGY The selection of the training samples that will constitute the working set in each iteration is the most critical decision in any chunking scheme, because such decision is directly involved in the number of IRWLS-SVC (or QP-SVC) procedures to be called and in the number of reproducing kernel evaluations to be made, which are, by far, the two most time consuming operations in any chunking schemes. In order to solve the SVC efficiently, we first need to define a candidate set of training samples to form the working set in each iteration. The candidate set will be made up, as it could not be otherwise, with all the training samples that violate the stopping conditions (9)-(11); and we will also add all those training samples that satisfy condition (11) but a small variation on their error will make them violate such condition. The strategies to select the working set are as numerous as the number of problems to be solved, but one can think three different simple strategies: • Select those samples which do not fulfill the stopping criteria and present the largest Iei I values. • Select those samples which do not fulfill the stopping criteria and present the smallest Iei I values. • Select them randomly from the ones that do not fulfill the stopping conditions. The first strategy seems the more natural one and it was proposed in [6]. If the largest leil samples are selected we guanrantee that attained solution gives the greatest step towards the solution of (1). But if the step is too large, which usually happens, it will cause the solution in each iteration and the (Ji values to oscillate around its optimal value. The magnitude of this effect is directly proportional to the value of C and q (size of the working set), so in the case ofsmall C (C < 10) and low q (q < 20) it would be less noticeable. The second one is the most conservative strategy because we will be moving towards the solution of (1) with small steps. Its drawback is readily discerned if the starting point is inappropriate, needing too many iterations to reach the SVC solution. The last strategy, which has been implemented together with the IRWLS-SVC procedure, is a mid-point between the other two, but if the number of samples whose 0 < (3i < C increases above q there might be some iterations where we will make no progress (working set is only made up of the training samples that fulfill the stopping condition in (11)). This situation is easily avoided by introducing one sample that violates each one of the stopping conditions per class. Finally, if the cardinality of the candidate set is less than q the working set is completed with those samples that fulfil the stopping criteria conditions and present the least leil. In summary, the sample selection strategy proposed is 2 : 1. Construct the candidate set, Se with those samples that do not fulfill stopping conditions (9) and (10), and those samples whose (3 obeys 0 < (3i < C. 2. IfISel < ngot05. 3. Choose a sample per class that violates each one of the stopping conditions and move them from Se to the working set, SW. 4. Choose randomly n - ISw I samples from Se and move then to SW. Go to Step 6. 5. Move every sample form Se to Sw and then-ISwl samples that fulfill the stopping conditions (9) and (10) and present the lowest leil values are used to complete SW . 6. Go on, obtaining Gin and Gbin. 4 BENCHMARK FOR THE IRWLS-SVC We have prepared two different experiments to test both the IRWLS and the sample selection strategy for solving the SVc. The first one compares the IRWLS against QP and the second one compares the samples selection strategy, together with the IRWLS, against a complete solving procedure for SVC, the SV Mlight. In the first trial, we have replaced the LOQO interior point optimizer used by SV M1ig ht version 3.02 [5] by the IRWLS-SVC procedure in Table 1, to compare both optimizing engines with equal samples selection strategy. The comparison has been made over a Pentium ill-450MHz with 128Mb running on Window98 and the programs have been compiled using Microsoft Developer 6.0. In Table 2, we show the results for two data sets: the first q 20 40 70 Adult44781 CPU time Optimize Time LOQO IRWLS LOQO IRWLS 21.25 20.70 0.61 0.39 20.60 19.22 1.01 0.17 21.15 18.72 2.30 0.46 Splice 2175 CPU time Optimize Time LOQO IRWLS LOQO IRWLS 46.19 30.76 21.94 4.77 71.34 24.93 46.26 8.07 53.77 20.32 34.24 7.72 Table 2: CPU Time indicates the consume time in seconds for the whole procedure. The Optimize Time indicates the consume time in second for the LOQO or IRWLS procedure. one, containing 4781 training samples, needs most CPU resources to compute the RKHS and the second one, containing 2175 training samples, uses most CPU resources to solve the SVC for each Sw, where q indicates the size of the working set. The value of C has 2In what follows, I . I represents absolute value for numbers and cardinality for sets been set to 1 and 1000, respectively, and a Radial Basis Function (RBF) RKHS [2] has been employed, where its parameter a has been set, respectively, to 10 and 70. As it can be seen, the SV M1ig ht with IRWLS is significantly faster than the LOQO procedure in all cases. The kernel cache size has been set to 64Mb for both data sets and for both procedures. The results in Table 2 validates the IRWLS procedure as the fastest SVC solver. For the second trial, we have compiled a computer program that uses the IRWLS-SVC procedure and the working set selection in Section 3, we will refer to it as svcradit from now on. We have borrowed the chunking and shrinking ideas from the SV Mlight [6] for our computer program. To test these two programs several data sets have been used. The Adult and Web data sets have been obtained from 1. Platt's web page http://research.microsoft.comr jplatt/smo.html/; the Gauss-M data set is a two dimensional classification problem proposed in [3] to test neural networks, which comprises a gaussian random variable for each class, which highly overlap. The Banana, Diabetes and Splice data sets have been obtained from Gunnar Ratsch web page http://svm.first.gmd.der raetschl. The selection of C and the RKHS has been done as indicated in [11] for Adult and Web data sets and in http://svm.first.gmd.derraetschl for Banana, Diabetes and Splice data sets. In Table 3, we show the runtime complexity for each data set, where the value of q has been elected as the one that reduces the runtime complexity. Database Dim Adult6 Adult9 Adult! Web 1 Web7 Gauss-M Gauss-M Banana Banana Diabetes Splice 123 123 123 300 300 2 2 2 2 8 69 N Sampl. 11221 32562 1605 2477 24693 4000 4000 400 4900 768 2175 C a SV 1 1 1000 5 5 1 100 316.2 316.2 10 1000 10 10 10 10 10 1 1 1 1 2 70 4477 12181 630 224 1444 1736 1516 80 1084 409 525 q CPU time radit light radit light 150 130 100 100 150 70 100 40 70 40 150 40 70 10 10 10 10 10 70 40 10 20 118.2 1093.29 25.98 2.42 158.13 12.69 61.68 0.33 22.46 2.41 14.06 124.46 1097.09 113.54 2.36 124.57 48.28 3053.20 0.77 1786.56 6.04 49.19 Table 3: Several data sets runtime complexity, when solved with the short, and SV Mlight, light for short. s v c radit , radit for One can appreciate that the svcradit is faster than the SV M1ig ht for most data sets. For the Web data set, which is the only data set the SV Mlight is sligthly faster, the value of C is low and most training samples end up as support vector with (3i < C. In such cases the best strategy is to take the largest step towards the solution in every iteration, as the SV Mlig ht does [6], because most training samples (3i will not be affected by the others training samples (3j value. But in those case the value of C increases the SV c radit samples selection strategy is a much more appropriate strategy than the one used in SV Mlight. 5 CONCLUSIONS In this communication a new algorithm for solving the SVC for large training data sets has been presented. Its two major contributions deal with the optimizing engine and the sample selection strategy. An IRWLS procedure is used to solve the SVC in each step, which is much faster that the usual QP procedure, and simpler to implement, because the most difficult step is the linear equation system solution that can be easily obtained by LU decomposition means [12]. The random working set selection from the samples not fulfilling the KKT conditions is the best option if the working is be large, because it reduces the number of chunks to be solved. This strategy benefits from the IRWLS procedure, which allows to work with large training data set. All these modifications have been concreted in the svcradit solving procedure, publicly available at http://svm.tsc.uc3m.es/. 6 ACKNOWLEDGEMENTS We are sincerely grateful to Thorsten Joachims who has allowed and encouraged us to use his SV Mlight to test our IRWLS procedure, comparisons which could not have been properly done otherwise. References [1] B. E. Boser, I. M . Guyon, and V. Vapnik. A training algorithm for optimal margin classifiers. In 5th Annual Workshop on Computational Learning Theory, Pittsburg, U.S.A., 1992. [2] C. J. C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):121-167, 1998. [3] S. Haykin. Neural Networks: A comprehensivefoundation. Prentice-Hall, 1994. [4] P. W. Holland and R. E. Welch. Robust regression using iterative re-weighted least squares. Communications of Statistics Theory Methods, A6(9):813-27, 1977. [5] T. Joachims. http://www-ai.infonnatik.uni-dortmund.de/forschung/verfahren Isvmlight Isvmlight.eng.html. Technical report, University of Dortmund, Informatik, AI-Unit Collaborative Research Center on 'Complexity Reduction in Multivariate Data', 1998. [6] T. Joachims. Making Large Scale SVM Learning Practical, In Advances in Kernel Methods- Support Vector Learning, Editors SchOlkopf, B., Burges, C. 1. C. and Smola, A. 1., pages 169-184. M.I.T. Press, 1999. [7] E. Osuna, R. Freund, and F. Girosi. An improved training algorithm for support vector machines. In Proc. of the 1997 IEEE Workshop on Neural Networks for Signal Processing, pages 276-285, Amelia Island, U.S.A, 1997. [8] E. Osuna and F. Girosi. Reducing the run-time complexity of support vector machines. In ICPR'98, Brisbane, Australia, August 1998. [9] F. Perez-Cruz, A. Navia-Vazquez
6 0.19584864 20 nips-2000-Algebraic Information Geometry for Learning Machines with Singularities
7 0.19569318 59 nips-2000-From Mixtures of Mixtures to Adaptive Transform Coding
8 0.14215285 126 nips-2000-Stagewise Processing in Error-correcting Codes and Image Restoration
9 0.14003715 60 nips-2000-Gaussianization
10 0.13824145 23 nips-2000-An Adaptive Metric Machine for Pattern Classification
11 0.13728765 85 nips-2000-Mixtures of Gaussian Processes
12 0.13006049 5 nips-2000-A Mathematical Programming Approach to the Kernel Fisher Algorithm
13 0.12908134 22 nips-2000-Algorithms for Non-negative Matrix Factorization
14 0.12671058 34 nips-2000-Competition and Arbors in Ocular Dominance
15 0.12151182 53 nips-2000-Feature Correspondence: A Markov Chain Monte Carlo Approach
16 0.12133192 110 nips-2000-Regularization with Dot-Product Kernels
17 0.11225849 86 nips-2000-Model Complexity, Goodness of Fit and Diminishing Returns
18 0.11097546 14 nips-2000-A Variational Mean-Field Theory for Sigmoidal Belief Networks
19 0.10122395 19 nips-2000-Adaptive Object Representation with Hierarchically-Distributed Memory Sites
20 0.10046545 80 nips-2000-Learning Switching Linear Models of Human Motion
topicId topicWeight
[(10, 0.026), (17, 0.021), (21, 0.013), (32, 0.04), (33, 0.012), (63, 0.647), (67, 0.018), (76, 0.032), (79, 0.02), (90, 0.02), (92, 0.02), (97, 0.012)]
simIndex simValue paperId paperTitle
same-paper 1 0.99559277 143 nips-2000-Using the Nyström Method to Speed Up Kernel Machines
Author: Christopher K. I. Williams, Matthias Seeger
Abstract: unkown-abstract
2 0.092562184 145 nips-2000-Weak Learners and Improved Rates of Convergence in Boosting
Author: Shie Mannor, Ron Meir
Abstract: The problem of constructing weak classifiers for boosting algorithms is studied. We present an algorithm that produces a linear classifier that is guaranteed to achieve an error better than random guessing for any distribution on the data. While this weak learner is not useful for learning in general, we show that under reasonable conditions on the distribution it yields an effective weak learner for one-dimensional problems. Preliminary simulations suggest that similar behavior can be expected in higher dimensions, a result which is corroborated by some recent theoretical bounds. Additionally, we provide improved convergence rate bounds for the generalization error in situations where the empirical error can be made small, which is exactly the situation that occurs if weak learners with guaranteed performance that is better than random guessing can be established. 1
3 0.087730259 53 nips-2000-Feature Correspondence: A Markov Chain Monte Carlo Approach
Author: Frank Dellaert, Steven M. Seitz, Sebastian Thrun, Charles E. Thorpe
Abstract: When trying to recover 3D structure from a set of images, the most difficult problem is establishing the correspondence between the measurements. Most existing approaches assume that features can be tracked across frames, whereas methods that exploit rigidity constraints to facilitate matching do so only under restricted camera motion. In this paper we propose a Bayesian approach that avoids the brittleness associated with singling out one
4 0.077611357 62 nips-2000-Generalized Belief Propagation
Author: Jonathan S. Yedidia, William T. Freeman, Yair Weiss
Abstract: Belief propagation (BP) was only supposed to work for tree-like networks but works surprisingly well in many applications involving networks with loops, including turbo codes. However, there has been little understanding of the algorithm or the nature of the solutions it finds for general graphs. We show that BP can only converge to a stationary point of an approximate free energy, known as the Bethe free energy in statistical physics. This result characterizes BP fixed-points and makes connections with variational approaches to approximate inference. More importantly, our analysis lets us build on the progress made in statistical physics since Bethe's approximation was introduced in 1935. Kikuchi and others have shown how to construct more accurate free energy approximations, of which Bethe's approximation is the simplest. Exploiting the insights from our analysis, we derive generalized belief propagation (GBP) versions ofthese Kikuchi approximations. These new message passing algorithms can be significantly more accurate than ordinary BP, at an adjustable increase in complexity. We illustrate such a new GBP algorithm on a grid Markov network and show that it gives much more accurate marginal probabilities than those found using ordinary BP. 1
5 0.077055991 9 nips-2000-A PAC-Bayesian Margin Bound for Linear Classifiers: Why SVMs work
Author: Ralf Herbrich, Thore Graepel
Abstract: We present a bound on the generalisation error of linear classifiers in terms of a refined margin quantity on the training set. The result is obtained in a PAC- Bayesian framework and is based on geometrical arguments in the space of linear classifiers. The new bound constitutes an exponential improvement of the so far tightest margin bound by Shawe-Taylor et al. [8] and scales logarithmically in the inverse margin. Even in the case of less training examples than input dimensions sufficiently large margins lead to non-trivial bound values and - for maximum margins - to a vanishing complexity term. Furthermore, the classical margin is too coarse a measure for the essential quantity that controls the generalisation error: the volume ratio between the whole hypothesis space and the subset of consistent hypotheses. The practical relevance of the result lies in the fact that the well-known support vector machine is optimal w.r.t. the new bound only if the feature vectors are all of the same length. As a consequence we recommend to use SVMs on normalised feature vectors only - a recommendation that is well supported by our numerical experiments on two benchmark data sets. 1
6 0.076912805 17 nips-2000-Active Learning for Parameter Estimation in Bayesian Networks
7 0.076832809 102 nips-2000-Position Variance, Recurrence and Perceptual Learning
8 0.076587915 47 nips-2000-Error-correcting Codes on a Bethe-like Lattice
9 0.07638739 119 nips-2000-Some New Bounds on the Generalization Error of Combined Classifiers
10 0.074199215 64 nips-2000-High-temperature Expansions for Learning Models of Nonnegative Data
11 0.072413005 133 nips-2000-The Kernel Gibbs Sampler
12 0.072289534 37 nips-2000-Convergence of Large Margin Separable Linear Classification
13 0.071907356 74 nips-2000-Kernel Expansions with Unlabeled Examples
14 0.071871199 7 nips-2000-A New Approximate Maximal Margin Classification Algorithm
15 0.071788177 122 nips-2000-Sparse Representation for Gaussian Process Models
16 0.071565963 106 nips-2000-Propagation Algorithms for Variational Bayesian Learning
17 0.071246974 34 nips-2000-Competition and Arbors in Ocular Dominance
18 0.071243279 21 nips-2000-Algorithmic Stability and Generalization Performance
19 0.07117305 95 nips-2000-On a Connection between Kernel PCA and Metric Multidimensional Scaling
20 0.070951514 75 nips-2000-Large Scale Bayes Point Machines