nips nips2008 nips2008-25 knowledge-graph by maker-knowledge-mining

25 nips-2008-An interior-point stochastic approximation method and an L1-regularized delta rule

Source: pdf

Author: Peter Carbonetto, Mark Schmidt, Nando D. Freitas

Abstract: The stochastic approximation method is behind the solution to many important, actively-studied problems in machine learning. Despite its farreaching application, there is almost no work on applying stochastic approximation to learning problems with general constraints. The reason for this, we hypothesize, is that no robust, widely-applicable stochastic approximation method exists for handling such problems. We propose that interior-point methods are a natural solution. We establish the stability of a stochastic interior-point approximation method both analytically and empirically, and demonstrate its utility by deriving an on-line learning algorithm that also performs feature selection via L1 regularization. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 An interior-point stochastic approximation method and an L1-regularized delta rule Peter Carbonetto pcarbo@cs. [sent-1, score-0.45]

2 , Canada V6T 1Z4 Abstract The stochastic approximation method is behind the solution to many important, actively-studied problems in machine learning. [sent-9, score-0.418]

3 Despite its farreaching application, there is almost no work on applying stochastic approximation to learning problems with general constraints. [sent-10, score-0.302]

4 The reason for this, we hypothesize, is that no robust, widely-applicable stochastic approximation method exists for handling such problems. [sent-11, score-0.334]

5 We establish the stability of a stochastic interior-point approximation method both analytically and empirically, and demonstrate its utility by deriving an on-line learning algorithm that also performs feature selection via L1 regularization. [sent-13, score-0.374]

6 The main idea behind stochastic approximation is simple yet profound. [sent-15, score-0.337]

7 It is simple because it is only a slight modiﬁcation to the most basic optimization method, gradient descent. [sent-16, score-0.198]

8 While there is a sizable body of work on treating constraints by extending established optimization techniques to the stochastic setting, such as projection [14], subgradient (e. [sent-19, score-0.402]

9 We argue that a reliable stochastic approximation method that handles constraints is needed because constraints routinely arise in the mathematical formulation of learning problems, and the alternative approach—penalization—is often unsatisfactory. [sent-22, score-0.454]

10 Our main contribution is a new stochastic approximation method in which each step is the solution to the primal-dual system arising in interior-point methods [7]. [sent-23, score-0.432]

11 Our method is easy to implement, dominates other approaches, and provides a general solution to constrained learning problems. [sent-24, score-0.149]

12 Moreover, we show interior-point methods are remarkably well-suited to stochastic approximation, a result that is far from trivial when one considers that stochastic algorithms do not behave like their deterministic counterparts (e. [sent-25, score-0.472]

13 Finally, it is important that we establish convergence guarantees for our method (Sec. [sent-34, score-0.113]

14 To do so, we rely on math from stochastic approximation and optimization. [sent-36, score-0.302]

15 In such case, Robbins and Monro showed that a particularly eﬀective way to achieve a response level α = 0 is to take a (hopefully unbiased) measurement yk ≈ F (xk ), adjust the control variable according to xk+1 = xk − ak yk (1) for step size ak > 0, then repeat. [sent-41, score-0.737]

16 Kiefer and Wolfowitz re-interpreted the stochastic process as one of optimizing an unconstrained objective (F (x) acts as the gradient vector) and later Dvoretsky pointed out that each measurement y is actually the gradient F (x) plus some noise ξ(x). [sent-46, score-0.588]

17 In this paper, we introduce a convergent sequence of nonlinear systems Fµ (x) = 0 and interpret the Robbins-Monro process {xk } as solving a constrained optimization problem. [sent-48, score-0.169]

18 We focus on convex optimization problems [2] of the form minimize f (x) subject to c(x) ≤ 0, (2) procedure IP–SG (Interior-point stochastic gradient) for k = 1, 2, 3, . [sent-49, score-0.298]

19 • Run simulation to obtain gradient observation yk . [sent-55, score-0.258]

20 • Compute primal-dual search direction (∆xk , ∆zk ) by solving equations (6,7) with ∇f (x) = yk . [sent-56, score-0.205]

21 • Run backtracking line search to ﬁnd largest ak ≤ min{ˆk , 0. [sent-57, score-0.169]

22 995 mini (−zk,i /∆zk,i )} such a that c(xk−1 + ak ∆xk ) < 0, and mini ( · ) is over all i such that ∆zk,i < 0. [sent-58, score-0.186]

23 • Set xk = xk−1 + ak ∆xk and zk = zk−1 + ak ∆zk . [sent-59, score-0.512]

24 where c(x) is a vector of inequality constraints, f (x) and c(x) have continous partial derivatives, and measurements yk of the gradient at xk are noisy. [sent-60, score-0.456]

25 To simplify our exposition, we do not consider equality constraints; techniques for Figure 1: Proposed stochastic gradient algorithm. [sent-62, score-0.372]

26 Convexity is a standard assumption made to simplify analysis of stochastic approximation algorithms and, besides, constrained, non-convex optimization raises unresolved complications. [sent-64, score-0.364]

27 Following the standard barrier approach [7], we frame the constrained optimization problem as a sequence of unconstrained objectives. [sent-66, score-0.494]

28 This is the basic formula of the interior-point stochastic approximation method. [sent-69, score-0.302]

29 Provided x0 is feasible and z0 > 0, every subsequent iterate (xk , zk ) will be a feasible or “interior” point as well. [sent-72, score-0.219]

30 Notice the absence of a suﬃcient decrease condition on Fµ (x, z) or suitable merit function; this is not needed in the stochastic setting. [sent-73, score-0.236]

31 Our stochastic approximation algorithm requires a slightly non-standard treatment because the target Fµ (x, z) moves as µ changes. [sent-74, score-0.302]

32 3 Background on interior-point methods We motivate and derive primal-dual interior-point methods starting from the logarithmic barrier method. [sent-77, score-0.323]

33 The logarithmic barrier approach for the constrained optimization problem (2) amounts to solving a sequence of unconstrained subproblems of the form m minimize fµ (x) ≡ f (x) − µ i=1 log(−ci (x)), (3) where µ > 0 is the barrier parameter, and m is the number of inequality constraints. [sent-82, score-0.817]

34 As µ becomes smaller, the barrier function fµ (x) acts more and more like the objective. [sent-83, score-0.284]

35 The philosophy of barrier methods diﬀers fundamentally from “exterior” penalty methods that penalize points violating the constraints [13, Chapter 17] because the logarithm in (3) prevents iterates from violating the constraints at all, hence the word “barrier”. [sent-84, score-0.672]

36 The central thrust of the barrier method is to progressively push µ to zero at a rate which allows the iterates to converge to the constrained optimum x⋆ . [sent-85, score-0.565]

37 Writing out a ﬁrst-order Taylor-series expansion to the optimality conditions ∇fµ (x) = 0 about a point x, the Newton step ∆x is the solution to the linear equations ∇2 fµ (x) ∆x = −∇fµ (x). [sent-86, score-0.13]

38 The barrier Hessian has long been known to be incredibly ill-conditioned—this fact becomes apparent by writing out ∇2 fµ (x) in full—but an analysis by Wright [25] shows that the ill-conditioning is not harmful under the right conditions. [sent-87, score-0.284]

39 The “right conditions” are that x be within a small distance1 from the central path or barrier trajectory, which is deﬁned to be the sequence of isolated minimizers x⋆ satisfying ∇fµ (x⋆ ) = 0 and c(x⋆ ) < 0. [sent-88, score-0.451]

40 The bad news: the barrier µ µ µ method is ineﬀectual at remaining on the barrier trajectory—it pushes iterates too close to the boundary where they are no longer well-behaved [7]. [sent-89, score-0.71]

41 Ordinarily, a convergence test is conducted for each value of µ, but this is not a plausible option for the stochastic setting. [sent-90, score-0.277]

42 Primal-dual methods form a Newton search direction for both the primal variables and the Lagrange multipliers. [sent-91, score-0.115]

43 Like classical barrier methods, they fail catastrophically outside the central path. [sent-92, score-0.355]

44 But their virtue is that they happen to be extremely good at remaining on the central path (even in the stochastic setting; see Sec. [sent-93, score-0.436]

45 (7) Because (2) is a convex optimization problem, we can derive a sensible update rule for the barrier parameter by guessing the distance between the primal and dual objectives [2]. [sent-102, score-0.424]

46 Since x must be close to the central path but far from the boundary, the favourable neighbourhood shrinks as µ nears 0. [sent-111, score-0.167]

47 4 Analysis of convergence First we establish conditions upon which the sequence of iterates generated by the algorithm converges almost surely to the solution (x⋆ , z ⋆ ) as the amount of data or iteration count goes to inﬁnity. [sent-112, score-0.272]

48 1 Asymptotic convergence A convergence proof from ﬁrst principles is beyond the scope of this paper; we build upon the martingale convergence proof of Spall and Cristion for non-stationary systems [21]. [sent-115, score-0.158]

49 They may be weakened by applying results from the stochastic approximation and optimization literature. [sent-117, score-0.401]

50 Unbiased observations: yk is a discrete-time martingale diﬀerence with respect to the true gradient ∇f (xk ); that is, E(yk | xk , history up to time k) = ∇f (xk ). [sent-119, score-0.454]

51 Step sizes: The maximum step sizes ak bounding ak (see Fig. [sent-121, score-0.301]

52 1) must approach ˆ ∞ ∞ zero (ˆk → 0 as k → ∞ and k=1 a2 < ∞) but not too quickly ( k=1 ak = ∞). [sent-122, score-0.126]

53 Bounded gradient estimates: for some ρ and for every k, E( yk ) < ρ. [sent-126, score-0.258]

54 These conditions allow us to directly apply standard theorems on constrained optimization for convex programming [2, 6, 7, 13]. [sent-135, score-0.201]

55 Then θ⋆ ≡ (x⋆ , z ⋆ ) is an isolated (locally unique within a δ-neighbourhood) solution to (2), and the iterates θk ≡ (xk , zk ) of the feasible interior-point stochastic approximation method (Fig. [sent-137, score-0.652]

56 2 Considerations regarding the central path The object of this section is to establish that computing the stochastic primal-dual search direction is numerically stable. [sent-141, score-0.526]

57 ) The concern is that noisy gradient measurements will lead to wildly perturbed search directions. [sent-143, score-0.293]

58 3, interior-point methods are surprisingly stable provided the iterates remain close to the central path, but the prospect of keeping close to the path seems particularly tenuous in the stochastic setting. [sent-145, score-0.513]

59 A key observation is that the central path is itself perturbed by the stochastic gradient estimates. [sent-146, score-0.582]

60 5 of [7], we show that the stochastic Newton step (6,7) stays on target. [sent-148, score-0.285]

61 We deﬁne the noisy central path as θ(µ, ε) = (x, z), where (x, z) is a solution to Fµ (x, z) = 0 with gradient estimate y ≡ ∇f (x) + ε. [sent-149, score-0.386]

62 One way to assess the quality of the Newton step is to compare it to the tangent line of the noisy central path at (µ, ε). [sent-151, score-0.29]

63 Since we know that Fµ (x, z) = 0, the Newton step (5) at (x, z) with perturbation µ⋆ and stochastic gradient estimate y ⋆ is the solution to H ZJ JT C ∆x ∆z =− y⋆ − y (µ⋆ − µ)1. [sent-154, score-0.47]

64 (10) In conclusion, if the tangent line (8) is a fairly reasonable approximation to the central path, then the stochastic Newton step (10) will make good progress toward θ(µ⋆ , ε⋆ ). [sent-156, score-0.462]

65 Having established that the stochastic gradient algorithm closely follows the noisy central path, the analysis of M. [sent-157, score-0.477]

66 Wright [26] directly applies, in which round-oﬀ error (ǫmachine ) is occasionally replaced by gradient noise (ε). [sent-159, score-0.136]

67 While this problem only involves simple bound constraints, we can use it to compare our method to existing approaches such as gradient projection. [sent-162, score-0.168]

68 We can treat the gradient of MSE as a sample expectation over responses of the form −xi (yi − xT β), so the on-line or stochastic update i β (new) = β + axi (yi − xT β), i (12) 2 improves the linear regression with only a single data point (a is the step size). [sent-176, score-0.472]

69 Since standard “batch” learning requires a full pass through the data for each gradient evaluation, the on-line update (12) may be the only viable option when faced with, for instance, a collection of 80 million images [16]. [sent-178, score-0.169]

70 2 The gradient descent direction can be a poor choice because it ignores the scaling of the problem. [sent-189, score-0.176]

71 Figure 2: (left) Performance of constrained stochastic gradient methods for diﬀerent step size sequences. [sent-191, score-0.489]

72 The trick is to separate the coeﬃcients into their positive (βpos ) and negative (βneg ) components following [3], thereby transforming the non-smooth, unconstrained optimization problem (11) into a smooth problem with convex, quadratic objective and bound constraints βpos , βneg ≥ 0. [sent-195, score-0.202]

73 The regularized delta rule (13) is then obtained from direct application of the primal-dual interior-point Newton search direction (6,7) with a stochastic gradient (see Eq. [sent-196, score-0.571]

74 1 Experiments We ran four small experiments to assess the reliability and shrinkage eﬀect of the interiorpoint stochastic gradient method for linear regression with L1 regularization; refer to Fig. [sent-199, score-0.565]

75 3 We also studied four alternatives to our method: 1) a subgradient method, 2) a smoothed, unconstrained approximation to (11), 3) a projected gradient method, and 4) the augmented Lagrangian approach described in [24]. [sent-202, score-0.326]

76 Each method was executed with a single pass on the data (100 iterations) with step sizes ak = 1/(k0 + k), where k0 = 50 by default. [sent-230, score-0.24]

77 The control variables of Experiments 1, 2 and 3 were, respectively, the step size parameter k0 , the inverse Gamma scale parameter ν, and the L1 penalty parameter λ. [sent-235, score-0.139]

78 2 shows the results of Experiments 1 and 2, with error n β exact − β on-line 1 averaged over the 20 data sets, in which β exact is the solution to (11), and β on-line is the estimate obtained after 100 iterations of the on-line or stochastic gradient method. [sent-239, score-0.485]

79 The stochastic interiorpoint method, however, always came closest to β exact and, for the range of values we tried, its solution was by far the least sensitive to the step size sequence and level of variance in the observations. [sent-241, score-0.439]

80 the stochastic interior-point method still best approximated the exact solution, and its performance did not degrade when λ was small. [sent-248, score-0.3]

81 After one pass through the data (middle)—equivalent to a single iteration of an exact solver—the interiorpoint stochastic gradient method shrank some of the data components, but didn’t quite discard irrelevant features altogether. [sent-255, score-0.542]

82 After 10 visits to the training data (right), the stochastic algorithm exhibited feature selection close to what we would normally expect from the Lasso (left). [sent-256, score-0.236]

83 2 Filtering spam Classifying email as spam or not is most faith- Figure 3: Performance of the methods fully modeled as an on-line learning problem in for various choices of the L penalty. [sent-258, score-0.921]

84 An eﬀective ﬁlter is one that minimizes misclassiﬁcation of incoming messages—throwing away a good email being considerably more deleterious than incorrectly placing a spam in the inbox. [sent-260, score-0.51]

85 Without any prior knowledge as to what spam looks like, any ﬁlter will be error-prone at initial stages of deployment. [sent-261, score-0.411]

86 We adapted the L1 -regularized delta rule to the spam ﬁltering problem by replacing the linear regression with a binary logistic regression [9]. [sent-263, score-0.66]

87 i i To our knowledge, no one has investigated this approach for on-line spam ﬁltering, though there is some work on logistic regression plus the Lasso for batch classiﬁcation in text corpora [8]. [sent-265, score-0.526]

88 We simulated the on-line spam ﬁltering task on the trec2005 corpus [4] containing emails from the legal investigations of Enron corporation. [sent-268, score-0.443]

89 3291 49499 Results for SpamBayes not spam 39393 spam 3 5515 47275 Results for Bogoﬁlter true not spam spam pred. [sent-282, score-1.644]

90 not spam 39382 spam 17 true not spam spam pred. [sent-283, score-1.644]

91 true not spam spam not spam 39389 spam 10 2803 49987 Results for Logistic + L1 Table 1: Contingency tables for on-line spam ﬁltering task on the trec2005 data set. [sent-285, score-2.055]

92 Following [5], we use contingency tables to present results of the on-line spam ﬁltering experiment (Table 1). [sent-287, score-0.446]

93 Our spam ﬁlter dominated SpamBayes on the trec2005 corpus, and performed comparably to Bogoﬁlter—one of the best spam ﬁlters to date [5]. [sent-291, score-0.822]

94 6 Conclusions Our experiments on a learning problem with noisy gradient measurements and bound constraints show that the interior-point stochastic approximation algorithm is a signiﬁcant improvement over other methods. [sent-297, score-0.569]

95 [5] , Online supervised spam ﬁlter evaluation, ACM Trans. [sent-319, score-0.411]

96 Clark, Stochastic approximation methods for constrained and unconstrained systems, Springer-Verlag, 1978. [sent-355, score-0.214]

97 Spall, Adaptive stochastic approximation by the simultaneous perturbation method, IEEE Transactions on Automatic Control, 45 (2000), pp. [sent-399, score-0.302]

98 Cristion, Model-free control of nonlinear stochastic systems with discretetime measurements, IEEE Transactions on Automatic Control, 43 (1998), pp. [sent-405, score-0.306]

99 Spall, Stochastic optimization with inequality constraints using simultaneous perturbations and penalty functions, in Proc. [sent-417, score-0.181]

100 Wright, Some properties of the Hessian of the logarithmic barrier function, Mathematical Programming, 67 (1994), pp. [sent-422, score-0.323]

similar papers computed by tfidf model

tfidf for this paper:

wordName wordTfidf (topN-words)

[('spam', 0.411), ('neg', 0.317), ('barrier', 0.284), ('pos', 0.237), ('stochastic', 0.236), ('xk', 0.161), ('zneg', 0.146), ('zpos', 0.146), ('newton', 0.137), ('gradient', 0.136), ('ak', 0.126), ('yk', 0.122), ('wright', 0.117), ('iterates', 0.11), ('zk', 0.099), ('email', 0.099), ('bogo', 0.098), ('path', 0.096), ('coe', 0.091), ('spall', 0.085), ('unconstrained', 0.08), ('ective', 0.078), ('mse', 0.076), ('ltering', 0.074), ('interiorpoint', 0.073), ('monro', 0.073), ('robbins', 0.073), ('spambayes', 0.073), ('di', 0.072), ('central', 0.071), ('delta', 0.07), ('constrained', 0.068), ('approximation', 0.066), ('lter', 0.063), ('optimization', 0.062), ('constraints', 0.06), ('feasible', 0.06), ('penalty', 0.059), ('jt', 0.059), ('hessian', 0.055), ('ho', 0.055), ('regression', 0.051), ('solution', 0.049), ('step', 0.049), ('cristion', 0.049), ('fiacco', 0.049), ('lots', 0.049), ('quali', 0.049), ('widrow', 0.049), ('regularization', 0.048), ('xt', 0.046), ('rule', 0.046), ('erent', 0.046), ('lagrange', 0.046), ('lasso', 0.046), ('subgradient', 0.044), ('lagrangian', 0.044), ('search', 0.043), ('interior', 0.043), ('mccormick', 0.043), ('nando', 0.043), ('perturbed', 0.043), ('convergence', 0.041), ('direction', 0.04), ('tangent', 0.04), ('establish', 0.04), ('logarithmic', 0.039), ('yi', 0.039), ('programming', 0.039), ('nonlinear', 0.039), ('measurements', 0.037), ('shrinkage', 0.037), ('weakened', 0.037), ('packages', 0.037), ('zij', 0.037), ('behind', 0.035), ('martingale', 0.035), ('centering', 0.035), ('contingency', 0.035), ('fundamentally', 0.035), ('noisy', 0.034), ('zj', 0.034), ('cients', 0.034), ('batch', 0.033), ('unreliable', 0.033), ('virtue', 0.033), ('pass', 0.033), ('exact', 0.032), ('primal', 0.032), ('method', 0.032), ('conditions', 0.032), ('corpus', 0.032), ('violating', 0.032), ('control', 0.031), ('squares', 0.031), ('logistic', 0.031), ('estimators', 0.031), ('mini', 0.03), ('tibshirani', 0.03), ('kkt', 0.03)]

similar papers list:

simIndex simValue paperId paperTitle

same-paper 1 1.0000008 25 nips-2008-An interior-point stochastic approximation method and an L1-regularized delta rule

Author: Peter Carbonetto, Mark Schmidt, Nando D. Freitas

2 0.11616081 214 nips-2008-Sparse Online Learning via Truncated Gradient

Author: John Langford, Lihong Li, Tong Zhang

Abstract: We propose a general method called truncated gradient to induce sparsity in the weights of online-learning algorithms with convex loss. This method has several essential properties. First, the degree of sparsity is continuous—a parameter controls the rate of sparsiﬁcation from no sparsiﬁcation to total sparsiﬁcation. Second, the approach is theoretically motivated, and an instance of it can be regarded as an online counterpart of the popular L1 -regularization method in the batch setting. We prove small rates of sparsiﬁcation result in only small additional regret with respect to typical online-learning guarantees. Finally, the approach works well empirically. We apply it to several datasets and ﬁnd for datasets with large numbers of features, substantial sparsity is discoverable. 1

3 0.11034346 165 nips-2008-On the Reliability of Clustering Stability in the Large Sample Regime

Author: Ohad Shamir, Naftali Tishby

Abstract: unkown-abstract

4 0.10328329 98 nips-2008-Hierarchical Semi-Markov Conditional Random Fields for Recursive Sequential Data

Author: Tran T. Truyen, Dinh Phung, Hung Bui, Svetha Venkatesh

Abstract: Inspired by the hierarchical hidden Markov models (HHMM), we present the hierarchical semi-Markov conditional random ﬁeld (HSCRF), a generalisation of embedded undirected Markov chains to model complex hierarchical, nested Markov processes. It is parameterised in a discriminative framework and has polynomial time algorithms for learning and inference. Importantly, we develop efﬁcient algorithms for learning and constrained inference in a partially-supervised setting, which is important issue in practice where labels can only be obtained sparsely. We demonstrate the HSCRF in two applications: (i) recognising human activities of daily living (ADLs) from indoor surveillance cameras, and (ii) noun-phrase chunking. We show that the HSCRF is capable of learning rich hierarchical models with reasonable accuracy in both fully and partially observed data cases. 1

5 0.09779904 21 nips-2008-An Homotopy Algorithm for the Lasso with Online Observations

Author: Pierre Garrigues, Laurent E. Ghaoui

Abstract: It has been shown that the problem of 1 -penalized least-square regression commonly referred to as the Lasso or Basis Pursuit DeNoising leads to solutions that are sparse and therefore achieves model selection. We propose in this paper RecLasso, an algorithm to solve the Lasso with online (sequential) observations. We introduce an optimization problem that allows us to compute an homotopy from the current solution to the solution after observing a new data point. We compare our method to Lars and Coordinate Descent, and present an application to compressive sensing with sequential observations. Our approach can easily be extended to compute an homotopy from the current solution to the solution that corresponds to removing a data point, which leads to an efﬁcient algorithm for leave-one-out cross-validation. We also propose an algorithm to automatically update the regularization parameter after observing a new data point. 1

6 0.086420253 155 nips-2008-Nonparametric regression and classification with joint sparsity constraints

7 0.083832212 173 nips-2008-Optimization on a Budget: A Reinforcement Learning Approach

8 0.083260812 145 nips-2008-Multi-stage Convex Relaxation for Learning with Sparse Regularization

9 0.080957592 202 nips-2008-Robust Regression and Lasso

10 0.078900404 179 nips-2008-Phase transitions for high-dimensional joint support recovery

11 0.078754716 24 nips-2008-An improved estimator of Variance Explained in the presence of noise

12 0.078052051 177 nips-2008-Particle Filter-based Policy Gradient in POMDPs

13 0.075342558 62 nips-2008-Differentiable Sparse Coding

14 0.073242232 206 nips-2008-Sequential effects: Superstition or rational behavior?

15 0.07256411 2 nips-2008-A Convex Upper Bound on the Log-Partition Function for Binary Distributions

16 0.066538639 195 nips-2008-Regularized Policy Iteration

17 0.066224977 1 nips-2008-A Convergent $O(n)$ Temporal-difference Algorithm for Off-policy Learning with Linear Function Approximation

18 0.065592952 135 nips-2008-Model Selection in Gaussian Graphical Models: High-Dimensional Consistency of \boldmath$\ell 1$-regularized MLE

19 0.065126568 175 nips-2008-PSDBoost: Matrix-Generation Linear Programming for Positive Semidefinite Matrices Learning

20 0.063356064 99 nips-2008-High-dimensional support union recovery in multivariate regression

similar papers computed by lsi model

lsi for this paper:

topicId topicWeight

[(0, -0.225), (1, 0.022), (2, -0.065), (3, 0.04), (4, 0.031), (5, 0.057), (6, -0.027), (7, 0.003), (8, -0.038), (9, 0.021), (10, -0.066), (11, -0.004), (12, -0.016), (13, 0.014), (14, -0.037), (15, -0.013), (16, -0.015), (17, -0.011), (18, 0.027), (19, 0.026), (20, 0.029), (21, 0.053), (22, -0.04), (23, 0.005), (24, -0.014), (25, 0.064), (26, 0.052), (27, -0.044), (28, -0.072), (29, -0.004), (30, -0.023), (31, -0.032), (32, -0.157), (33, -0.066), (34, -0.076), (35, -0.157), (36, -0.018), (37, 0.017), (38, 0.021), (39, 0.023), (40, -0.177), (41, -0.125), (42, 0.065), (43, -0.038), (44, -0.161), (45, -0.029), (46, 0.009), (47, -0.006), (48, -0.062), (49, 0.013)]

similar papers list:

simIndex simValue paperId paperTitle

same-paper 1 0.94513381 25 nips-2008-An interior-point stochastic approximation method and an L1-regularized delta rule

Author: Peter Carbonetto, Mark Schmidt, Nando D. Freitas

2 0.55698282 98 nips-2008-Hierarchical Semi-Markov Conditional Random Fields for Recursive Sequential Data

Author: Tran T. Truyen, Dinh Phung, Hung Bui, Svetha Venkatesh

3 0.53817022 149 nips-2008-Near-minimax recursive density estimation on the binary hypercube

Author: Maxim Raginsky, Svetlana Lazebnik, Rebecca Willett, Jorge Silva

Abstract: This paper describes a recursive estimation procedure for multivariate binary densities using orthogonal expansions. For d covariates, there are 2d basis coefﬁcients to estimate, which renders conventional approaches computationally prohibitive when d is large. However, for a wide class of densities that satisfy a certain sparsity condition, our estimator runs in probabilistic polynomial time and adapts to the unknown sparsity of the underlying density in two key ways: (1) it attains near-minimax mean-squared error, and (2) the computational complexity is lower for sparser densities. Our method also allows for ﬂexible control of the trade-off between mean-squared error and computational complexity.

4 0.53783721 165 nips-2008-On the Reliability of Clustering Stability in the Large Sample Regime

Author: Ohad Shamir, Naftali Tishby

Abstract: unkown-abstract

5 0.53578258 214 nips-2008-Sparse Online Learning via Truncated Gradient

Author: John Langford, Lihong Li, Tong Zhang

6 0.53281963 68 nips-2008-Efficient Direct Density Ratio Estimation for Non-stationarity Adaptation and Outlier Detection

7 0.52356255 175 nips-2008-PSDBoost: Matrix-Generation Linear Programming for Positive Semidefinite Matrices Learning

8 0.51867306 173 nips-2008-Optimization on a Budget: A Reinforcement Learning Approach

9 0.49968752 21 nips-2008-An Homotopy Algorithm for the Lasso with Online Observations

10 0.49679855 177 nips-2008-Particle Filter-based Policy Gradient in POMDPs

11 0.4916082 14 nips-2008-Adaptive Forward-Backward Greedy Algorithm for Sparse Learning with Linear Models

12 0.47786525 82 nips-2008-Fast Computation of Posterior Mode in Multi-Level Hierarchical Models

13 0.47465563 145 nips-2008-Multi-stage Convex Relaxation for Learning with Sparse Regularization

14 0.47004744 104 nips-2008-Improved Moves for Truncated Convex Models

15 0.46461186 54 nips-2008-Covariance Estimation for High Dimensional Data Vectors Using the Sparse Matrix Transform

16 0.46397296 24 nips-2008-An improved estimator of Variance Explained in the presence of noise

17 0.45417708 2 nips-2008-A Convex Upper Bound on the Log-Partition Function for Binary Distributions

18 0.45257807 185 nips-2008-Privacy-preserving logistic regression

19 0.44892135 20 nips-2008-An Extended Level Method for Efficient Multiple Kernel Learning

20 0.44729832 13 nips-2008-Adapting to a Market Shock: Optimal Sequential Market-Making

similar papers computed by lda model

lda for this paper:

topicId topicWeight

[(6, 0.073), (7, 0.068), (12, 0.035), (28, 0.145), (57, 0.029), (59, 0.019), (63, 0.451), (77, 0.033), (83, 0.053)]

similar papers list:

simIndex simValue paperId paperTitle

1 0.87642413 3 nips-2008-A Massively Parallel Digital Learning Processor

Author: Hans P. Graf, Srihari Cadambi, Venkata Jakkula, Murugan Sankaradass, Eric Cosatto, Srimat Chakradhar, Igor Dourdanovic

Abstract: We present a new, massively parallel architecture for accelerating machine learning algorithms, based on arrays of vector processing elements (VPEs) with variable-resolution arithmetic. Groups of VPEs operate in SIMD (single instruction multiple data) mode, and each group is connected to an independent memory bank. The memory bandwidth thus scales with the number of VPEs, while the main data flows are local, keeping power dissipation low. With 256 VPEs, implemented on two FPGAs (field programmable gate array) chips, we obtain a sustained speed of 19 GMACS (billion multiplyaccumulate per sec.) for SVM training, and 86 GMACS for SVM classification. This performance is more than an order of magnitude higher than that of any FPGA implementation reported so far. The speed on one FPGA is similar to the fastest speeds published on a Graphics Processor for the MNIST problem, despite a clock rate that is an order of magnitude lower. Tests with Convolutional Neural Networks show similar compute performances. This massively parallel architecture is particularly attractive for embedded applications, where low power dissipation is critical. 1 I n trod u cti on Machine learning demands higher and higher compute-performance, but serial processors are not improving that much anymore - at least not as quickly as they used to. Mainstream processor development is moving to multi-core systems, using shared memory technology to hide the parallel nature of the processors. But shared memory technology does not scale to hundreds or thousands of cores. In order to reach such levels of parallelization alternative approaches have to be developed. Massively parallel general-purpose computers had limited success so far, because of difficulties programming these machines, and they remain a niche market, mostly in highperformance computing. Yet processors specialized for certain application domains, such as graphics processors or routing processors 1, have been parallelized to several hundred cores and are successful mass products. They improve performance over general-purpose processors by focusing on a few key algorithmic elements, yet still maintain enough flexibility that they can be programmed for a variety of applications. We explore in this paper if a similar approach can lead to efficient machine learning processors. 1 e.g. Nvidia, Quadro FX 5600 graphics processor; Cisco, CRS-1 routing processor Several processors optimized for machine learning, in particular for neural networks, were developed during the 1980’s and 90’s. Examples are the Synapse-1 architecture [1], or the Connectionist Network Supercomputer, CNS1 [2]. Recently there has been less activity in this field, but some accelerators are sold today for specific applications, such as the Axeon [3] processor for power train control of cars. Beside digital processors a large number of analog circuits were built, emulating neural network structures. Extremely high performance with low power dissipation is achievable, see e.g. [4][5], but these networks have little flexibility. SVM implementations on FPGA have been demonstrated in recent years [6-8], yet reached only low compute-performances. All machine learning processors had only limited success so far, indicating how difficult it is to find a good combination of performance, flexibility, price and ease of use. An important consideration is that many applications of machine learning, such as video analysis, data mining, or personalization of services, show the most promise in embedded systems. Embedded learning requires high compute performance while dissipating little power, a combination that is difficult to achieve, and so far required application specific IC (ASIC). Our aim is to develop architectures that meet the requirements for embedded learning, but are programmable and therefore can be used in a wide range of applications. With the goal of analyzing different architectures we designed a development and testing environment where the parallel computation is mapped onto FPGA’s. Initially this system was intended only for experimentation, but its performance is so high that this platform is useful in its own right as accelerator for high-performance systems. While the experiments shown here emphasize high performance, the architecture has been designed from the start for low power dissipation. The main features for achieving this goal are: low-resolution arithmetic, keeping the main data flow local, low operating frequencies, and a modular design, so that unused parts can be powered down dynamically. All results shown here are from the test platform; migration to lowpower FPGA or chip designs are done in a later stage. 2 Al gori th ms - A ri th meti c - A rch i te ctu re For a substantial improvement over a general purpose processor, the algorithms, the arithmetic units, as well as the architecture have to be optimized simultaneously. This is not just an exercise in hardware design, but algorithms and their software implementations have to be developed concurrently. Most machine learning algorithms have not been developed with parallelization in mind. Therefore, we first need to find good parallel versions, identify their performance bottlenecks, and then extract common computational patterns that can be mapped into accelerator hardware. 2.1 Algorithms Characteristic for machine learning is that large amounts of data need to be processed, often with predictable data access patterns and no dependency between operations over large segments of the computation. This is why data-parallelization can often provide good accelerations on multi-core chips, clusters of machines, or even on loosely coupled networks of machines. Using MapReduce, speedups linear with the number of processors have been reported in [9] for several machine learning algorithms. Up to 16 cores were tested, and simulations indicate good scaling to more processors in some cases. Many algorithms, such as KNN, K-means clustering, LVQ, and Neural Networks can be reduced to forms where the computation is dominated by vector-matrix multiplications, which are easily parallelizable. For Convolutional Neural Networks (CNN) the data flow can be complex, yet the core of the computation is a convolution, an operation which has been studied extensively for parallel implementations. For Support Vector Machines (SVM), several parallel algorithms were described, but most saturate quickly for more than 16 processors. Scaling to larger numbers of processors has been demonstrated, applying MapReduce on a graphics processor with 128 cores [10]. Another implementation on a cluster of 48 dual-core machines (with 384 MMX units) [11] scales even super-linearly, and, according to simulations, scales to thousands of cores. Based on this analysis it is clear that vector-matrix and matrix-matrix multiplications for large vector dimensionalities and large numbers of vectors must be handled efficiently. Yet this alone is not sufficient since data access patterns vary greatly between algorithms. We analyze this here in more detail for SVM and CNN. These algorithms were chosen, because they are widely used for industrial applications and cover a broad range of computation, I/O, and memory requirements. The characteristics of the SVM training are summarized in Table 1. We use an approach similar to the one described in [11] to split different parts of the computation between a host CPU and the FPGA accelerator. For large dimensions d of the vectors the calculation of the columns of the kernel matrix dominates by far. This is needed to update the gradients, and in the present implementation, only this part is mapped onto the FPGA. If the dimensionality d is smaller than around 100, operations 2 and 5 can become bottlenecks and should also be mapped onto the accelerator. Challenging is that for each kernel computation a new data vector has to be loaded 4 into the processor, leading to very high I/O requirements. We consider here dimensions of 10 - 10 5 7 and numbers of training data of 10 - 10 , resulting easily in Gigabytes that need to be transferred to the processors at each iteration. 1 2 3 4 5 6 Operation Initialize all αx, Gx Do Find working set αi, αj Update αi, αj Get 2 columns of kernel matrix Update gradients Gx While not converged Computation 2n IO 2n Unit CPU I I I I I * 2n I*2 I * (2d+2dn) I*n CPU CPU FPGA CPU * * * * 2n 10 2nd n Table 1: Compute- and IO-requirements of each step for SVM training (SMO algorithm). n: number of training data; d: dimension of the vectors; G: gradients; α: support vector factors; I: number of iterations. The last column indicates whether the execution happens on the host CPU or the accelerator FPGA. It is assumed that the kernel computation requires a dot product between vectors (e.g. rbf, polynomial, tanh kernels). Neural network algorithms are essentially sequences of vector-matrix multiplications, but networks with special connectivity patterns, such as convolutional networks have very different IO characteristics than fully connected networks. Table 2 shows the computation and IO requirements for scanning several convolution kernels over one input plane. A full network requires multiple of these operations for one layer, with nonlinearities between layers. We map all operations onto the FPGA accelerator, since intermediate results are re-used right away. The most significant 2 difference to between the SVM and CNN is the Compute/IO ratio: SVM: ~ 1; CNN: ~ L*k > 100. Therefore the requirements for these two algorithms are very different, and handling both cases efficiently is quite a challenge for an architecture design. Operation Load L kernels For all input pixels Shift in new pixel Multiply kernels Shift out result 1 2 3 4 Computation IO 2 L* k n* m 2 n*m*L*k n*m Unit FPGA FPGA FPGA FPGA FPGA Table 2: Compute- and IO-requirements for CNN computation (forward pass), where l kernels of size k*k are scanned simultaneously over an input plane of size n*m. This is representative for implementations with kernel unrolling (kernel pixels processed in parallel). Internal shifts, computation of the non-linearity, and border effects not shown. 2.2 Arithmetic Hardware can be built much more compactly and runs with lower power dissipation, if it uses fixed-point instead of floating-point operations. Fortunately, many learning algorithms tolerate a low resolution in most of the computations. This has been investigated extensively for neural networks [12][13], but less so for other learning algorithms. Learning from data is inherently a noisy process, because we see only a sparse sampling of the true probability distributions. A different type of noise is introduced in gradient descent algorithms, when only a few training data are used at a time to move the optimization forward iteratively. This noise is particularly pronounced for stochastic gradient descent. There is no point in representing noisy variables with high resolution, and it is therefore a property inherent to many algorithms that low-resolution computation can be used. It is important, not to confuse this tolerance to low resolution with the resolution required to avoid numeric instabilities. Some of the computations have to be performed with a high resolution, in particular for variables that are updated incrementally. They maintain the state of the optimization and may change in very small steps. But usually by far the largest part of the computation can be executed at a low resolution. Key is that the hardware is flexible enough and can take advantage of reduced resolution while handling high resolution where necessary. Problem Adult Forest MNIST NORB Kernel: Float Obj. f. # SV 31,930.77 11,486 653,170.7 49,333 4,960.13 6,172 1,243.71 3,077 F-score 77.58 98.29 99.12 93.34 Kernel: 16 bit fixed point Obj. f. # SV F-score 31,930.1 11,490 77.63 652,758 49,299 98.28 4,959.64 6,166 99.11 1,244.76 3,154 93.26 F-sc. (4b in) NA NA 99.11 92.78 Table 3: Comparison of the results of SVM training when the kernels are represented with floating point numbers (32 or 64 bits) (left half) and with 16 bit fixed point (right half). The last column shows the results when the resolution of the training data is reduced from 8 bit to 4 bit. For NORB this reduces the accuracy; all other differences in accuracy are not significant. All are two class problems: Adult: n=32,562, d=122; Forest: n=522,000, d=54 (2 against the rest); MNIST: n=60,000, d=784 (odd–even); NORB: n=48,560, d=5,184. We developed a simulator that allows running the training algorithms with various resolutions in each of the variables. A few examples for SVM training are shown in Table 3. Reducing the resolution of the kernel values from double or float to 16 bit fixed point representations does not affect the accuracy for any of the problems. Therefore all the multiplications in the dot products for the kernel computation can be done in low resolutions (4–16 bit in the factors), but the accumulator needs sufficient resolution to avoid over/under flow (48 bit). Once the calculation of the kernel value is completed, it can be reduced to 16 bit. A low resolution of 16 bit is also tolerable for the α values, but a high resolution is required for the gradients (double). For Neural Networks, including CNN, several studies have confirmed that states and gradients can be kept at low resolutions (<16 bit), but the weights must be maintained at a high resolution (float) (see e.g. [12]). In our own evaluations 24 bits in the weights tend to be sufficient. Once the network is trained, for the classification low resolutions can be used for the weights as well (<16 bit). 2.3 A rc h i t e c t u re Figure 1: Left: Schematic of the architecture with the main data flows; on one FPGA 128 VPE are configured into four SIMD groups; L-S: Load-store units. Right: Picture of an FPGA board; in our experiments one or two of them are used, connected via PCI bus to a host CPU. Based on the analysis above, it is clear that the architecture must be optimized for processing massive amounts of data with relatively low precision. Most of the time, data access patterns are predictable and data are processed in blocks that can be stored contiguously. This type of computation is well suited for vector processing, and simple vector processing elements (VPE) with fixed-point arithmetic can handle the operations. Since typically large blocks of data are processed with the same operation, groups of VPE can work in SIMD (single instruction multiple data) mode. Algorithms must then be segmented to map the highvolume, low precision parts onto the vector accelerators and parts requiring high precision arithmetic onto the CPU. The most important design decision is the organization of the memory. Most memory accesses are done in large blocks, so that the data can be streamed, making complex caching unnecessary. This is fortunate, since the amounts of data to be loaded onto the processor are so large that conventional caching strategies would be overwhelmed anyway. Because the blocks tend to be large, a high data bandwidth is crucial, but latency for starting a block transfer is less critical. Therefore we can use regular DDR memories and still get high IO rates. This led to the design shown schematically in Figure 1, where independent memory banks are connected via separate IO ports for each group of 32 VPE. By connecting multiple of the units shown in Figure 1 to a CPU, this architecture scales to larger numbers of VPE. Parallel data IO and parallel memory access scale simultaneously with the number of parallel cores, and we therefore refer to this as the P3 (P-cube) architecture. Notice also that the main data flow is only local between a group of VPE and its own memory block. Avoiding movements of data over long distances is crucial for low power dissipation. How far this architecture can reasonably scale with one CPU depends on the algorithms, the amount of data and the vector dimensionality (see below). A few hundred VPE per CPU have provided good accelerations in all our tests, and much higher numbers are possible with multi-core CPUs and faster CPU-FPGA connections. 3 I mp l e men tati on of th e P 3 A rch i t ectu re This architecture fits surprisingly well onto some of the recent FPGA chips that are available with several hundred Digital Signal Processors (DSP) units and over 1,000 IO pins for data transfers. The boards used here contain each one Xilinx Virtex 5 LX330T-2 FPGA coupled to 4 independent DDR2 SDRAM with a total of 1GB, and 2 independent 4MB SSRAM memory banks (commercial board from AlphaData). One FPGA chip contains 192 DSP with a maximum speed of 550MHz, which corresponds to a theoretical compute-performance of 105.6 GMACS (18 bit and 25 bit operands). There is a total of 14 Mbit of on-chip memory, and the chip incorporates 960 pins for data IO. Due to routing overhead, not all DSP units can be used and the actual clock frequencies tend to be considerably lower than what is advertised for such chips (typically 230MHz or less for our designs). Nevertheless, we obtain high performances because we can use a large number of DSP units for executing the main computation. The main architecture features are: • Parallel processing (on one chip): 128 VPE (hardware DSP) are divided into 4 blocks of 32, each group controlled by one sequencer with a vector instruction set. • Custom Precision: Data are represented with 1 to 16 bit resolution. Higher resolutions are possible by operating multiple DSP as one processor. • Overlapping Computation and Communication: CPU-FPGA communication is overlapped with the FPGA computation. • Overlap Memory Operations with Computation: All loads and stores from the FPGA to off-chip memory are performed concurrently with computations. • High Off-chip Memory Bandwidth: 6 independent data ports, each 32 bits wide, access banked memories concurrently (12GB/s per chip). • • Streaming Data Flow, Simple Access Patterns: Load/store units are tailored for streaming input and output data, and for simple, bursty access patterns. Caching is done under application control with dual-port memory on chip. Load/store with (de)compression: For an increase of effective IO bandwidth the load/store units provide compression and decompression in hardware. Figure 2 shows the configuration of the VPEs for vector dot product computation used for SVM training and classification. For training, the main computation is the calculation of one column of the kernel matrix. One vector is pre-fetched and stored in on-chip memory. All other vectors are streamed in from off-chip memory banks 1-4. Since this is a regular and predictable access pattern, we can utilize burst-mode, achieving a throughput of close to one memory word per cycle. But the speed is nevertheless IO bound. When several vectors can be stored on-chip, as is the case for classification, then the speed becomes compute-bound. Figure 2: Architecture for vector dot-product computation. The left side shows a high-level schematic with the main data flow. The data are streamed from memory banks 1-4 to the VPE arrays, while memory banks 5 and 6, alternatively receive results or stream them back to the host. The right side shows how a group of VPE is pipelined to improve clock speed. The operation for SVM training on the FPGA corresponds to a vector-matrix multiplication and the one for classification to a matrix-matrix multiplication. Therefore the configuration of Figure 2 is useful for many other algorithms as well, where operations with large vectors and matrices are needed, such as Neural Networks. We implemented a specialized configuration for Convolutional Neural Networks, for more efficiency and lower power dissipation. The VPE are daisy-chained and operate as systolic array. In this way we can take advantage of the high computation to IO ratio (Table 2) to reduce the data transfers from memory. 4 E val u ati on s We evaluated SVM training and classification with the NORB and MNIST problems, the latter with up to 2 million training samples (data from [11]). Both are benchmarks with vectors of high dimensionality, representative for applications in image and video analysis. The computation is split between CPU and FPGA as indicated by Table 1. The DDR2 memory banks are clocked at 230MHz, providing double that rate for data transfers. The data may be compressed to save IO bandwidth. On the FPGA they are decompressed first and distributed to the VPE. In our case, a 32 bit word contains eight 4-bit vector components. Four 32 bit words are needed to feed all 32 VPEs of a group; therefore clocking the VPE faster than 115MHz does not improve performance. A VPE executes a multiplication plus add operation in one clock cycle, resulting in a theoretical maximum of 14.7 GMACS per chip. The sustained compute-rate is lower, about 9.4 GMACS, due to overhead (see Table 4). The computation on the host CPU overlaps with that on the FPGA, and has no effect on the speed in the experiments shown here. For the classification the VPE can be clocked higher, at 230 MHz. By using 4-bit operands we can execute 2 multiply-accumulates simultaneously on one DSP, resulting in speed that is more than four times higher and a sustained 43.0 GMACS limited by the number and speed of the VPE. Adding a second FPGA card doubles the speed, showing little saturation effects yet, but for more FPGA per CPU there will be saturation (see Fig. 3). The compute speed in GMACS obtained for NORB is almost identical. # 60k 2M Iterations 8,000 266,900 CPU time 754s -- speed 0.5 -- CPU+MMX time speed 240 s 1.57 531,534 s 1.58 CPU+FPGA time speed 40 s 9.42 88,589 s 9.48 CPU+2 FPGA time speed 21 s 17.9 48,723 s 17.2 Table 4: Training times and average compute speed for SVM training. Systems tested: CPU, Opteron, 2.2GHz; CPU using MMX; CPU with one FPGA; CPU with two FPGA boards. Results are shown for training sizes of 60k and 2M samples. Compute speed is in GMACS (just kernel computations). Training algorithm: SMO with second order working set selection. Parallelizations of SVM training have been reported recently for a GPU [10] and for a cluster [11], both using the MNIST data. In [10] different bounds for stopping were used than here and in [11]. Nevertheless, a comparison of the compute performance is possible, because based on the number of iterations we can compute the average GMACS for the kernel computations. As can be seen in Table 5 a single FPGA is similar in speed to a GPU with 128 stream processors, despite a clock rate that is about 5.5 times lower for I/O and 11 times lower for the VPE. The cluster with 384 MMX units is about 6 times faster than one FPGA with 128 VPE, but dissipates about two orders of magnitude more electric power. For the FPGA this calculation includes only the computation of the kernel values while the part on the CPU is neglected. This is justified for this study, because the rest of the calculations can be mapped on the FPGA as well and will increase the power dissipation only minimally. Number Clock Operand Power Average of cores speed type dissipation compute speed CPU (Opteron) 1 2.2 GHz float 40 W 0.5 GMACS GPU (from [10]) 128 1.35 GHz float 80 W 7.4 GMACS Cluster (from [11]) 384 1.6 GHz byte > 1 kW 54 GMACS FPGA 128 0.12 GHz 4 bit nibble 9W 9.4 GMACS Table 5: Comparison of performances for SVM training (MNIST data). GPU: Nvidia 8800 GTX. Cluster: 48 dual core CPU (Athlon), 384 MMX units. The GPU was training with 60k samples ([10], table 2, second order), the cluster trained with 2 million samples. Processor Figure 3: Acceleration of SVM training as a function of the number of VPE. MNIST n: 2,000,000, d=784; NORB: n=48,560, d=5,184. The points for 128 and 256 VPE are experimental, the higher ones are simulations. Curves MNIST, NORB: Multiple FPGA are attached to one CPU. Curve MNIST C: Each FPGA is attached to a separate host CPU. Scaling of the acceleration with the number of VPEs is shown in Figure 3. The reference speed is that of one FPGA attached to a CPU. The evaluation has been done experimentally for 128 and 256 VPEs, and beyond that with a simulator. The onset of saturation depends on the dimensionality of the vectors, but to a much lesser extent on the number of training vectors (up to the limit of the memory on the FPGA card). MNIST saturates for more than two FPGAs because then the CPU and FPGA computation times become comparable. For the larger vectors of NORB (d=5,184) this saturation starts to be noticeable for more than 4 FPGA. Alternatively, a system can be scaled by grouping multiple CPU, each with one attached FPGA accelerator. Then the scaling follows a linear or even super-linear acceleration (MNIST C) to several thousand VPE. If the CPUs are working in a cluster arrangement, the scaling is similar to the one described in [11]. For convolutional neural networks, the architecture of Figure 2 is modified to allow a block of VPE to operate as systolic array. In this way convolutions can be implemented with minimal data movements. In addition to the convolution, also sub-sampling and non-linear functions plus the logistics to handle multiple layers with arbitrary numbers of kernels in each layer are done on the FPGA. Four separate blocks of such convolvers are packed onto one FPGA, using 100 VPE. Clocked at 115MHz, this architecture provides a maximum of 11.5 GMACS. Including all the overhead the sustained speed is about 10 GMACS. 5 Con cl u s i on s By systematically exploiting characteristic properties of machine learning algorithms, we developed a new massively parallel processor architecture that is very efficient and can be scaled to thousands of processing elements. The implementation demonstrated here is more than an order of magnitude higher in performance than previous FPGA implementations of SVM or CNN. For the MNIST problem it is comparable to the fastest GPU implementations reported so far. These results underline the importance of flexibility over raw compute-speed for massively parallel systems. The flexibility of the FPGA allows more efficient routing and packing of the data and the use of computations with the lowest resolution an algorithm permits. The results of Table 5 indicate the potential of this architecture for low-power operation in embedded applications. R e f e re n c e s [1] Ramacher, et al. (1995) Synapse-1: A high-speed general purpose parallel neurocomputer system. In Proc. 9th Intl. Symposium on Parallel Processing (IPPS'95), pp. 774-781. [2] Asanovic, K., Beck, Feldman, J., Morgan, N. & Wawrzynek, J. (1994) A Supercomputer for Neural Computation, Proc. IEEE Intl. Joint Conference on Neural Networks, pp. 5-9, Orlando, Florida. [3] Neil, P., (2005) Combining hardware with a powerful automotive MCU for powertrain applications. In Industrial Embedded Resource Guide, p. 88. [4] Korekado, et al. (2003) A Convolutional Neural Network VLSI for Image Recognition Using Merged/Mixed Analog-Digital Architecture, in Proc. 7th KES 2003, Oxford, pp 169-176. [5] Murasaki, M., Arima, Y. & Shinohara, H. (1993) A 20 Tera-CPS Analog Neural Network Board. In Proc. Int. Joint Conf. Neural Networks, pp. 3027 – 3030. [6] Pedersen, R., Schoeberl, M. (2006), An Embedded Support Vector Machine, WISE 2006. [7] Dey, S., Kedia, M. Agarwal, N., Basu, A., Embedded Support Vector Machine: Architectural Enhancements and Evaluation, in Proc 20th Int. Conf. VLSI Design. [8] Anguita, D., Boni, A., Ridella, S., (2003) A Digital Architecture for Support Vector Machines: Theory, Algorithm, and FPGA Implementation, IEEE Trans. Neural Networks, 14/5, pp.993-1009. [9] Chu, C., Kim, S., Lin, Y., Yu, Y., Bradski, G., Ng, A. & Olukotun, K. (2007) Map-Reduce for Machine Learning on Multicore, Advances in Neural Information Processing Systems 19, MIT Press. [10] Catanzaro, B., Sundaram, N., & Keutzer, K. (2008) Fast Support Vector Machine Training and Classification on Graphics Processors, Proc. 25th Int. Conf. Machine Learning, pp 104-111. [11] Durdanovic, I., Cosatto, E. & Graf, H. (2007) Large Scale Parallel SVM Implementation. In L. Bottou, O. Chapelle, D. DeCoste, J. Weston (eds.), Large Scale Kernel Machines, pp. 105-138, MIT Press. [12] Simard, P & Graf, H. (1994) Backpropagation without Multiplication. In J. Cowan, G. Tesauro, J. Alspector, (eds.), Neural Information Processing Systems 6, pp. 232 – 239, Morgan Kaufmann. [13] Savich, A., Moussa, M., Areibi, S., (2007) The Impact of Arithmetic Representation on Implementing MLP-BP on FPGAs: A Study, IEEE Trans. Neural Networks, 18/1, pp. 240-252.

same-paper 2 0.86400104 25 nips-2008-An interior-point stochastic approximation method and an L1-regularized delta rule

Author: Peter Carbonetto, Mark Schmidt, Nando D. Freitas

3 0.85399729 212 nips-2008-Skill Characterization Based on Betweenness

Author: Ozgur Simsek, Andre S. Barreto

Abstract: We present a characterization of a useful class of skills based on a graphical representation of an agent’s interaction with its environment. Our characterization uses betweenness, a measure of centrality on graphs. It captures and generalizes (at least intuitively) the bottleneck concept, which has inspired many of the existing skill-discovery algorithms. Our characterization may be used directly to form a set of skills suitable for a given task. More importantly, it serves as a useful guide for developing incremental skill-discovery algorithms that do not rely on knowing or representing the interaction graph in its entirety. 1

4 0.78280091 78 nips-2008-Exact Convex Confidence-Weighted Learning

Author: Koby Crammer, Mark Dredze, Fernando Pereira

Abstract: Conﬁdence-weighted (CW) learning [6], an online learning method for linear classiﬁers, maintains a Gaussian distributions over weight vectors, with a covariance matrix that represents uncertainty about weights and correlations. Conﬁdence constraints ensure that a weight vector drawn from the hypothesis distribution correctly classiﬁes examples with a speciﬁed probability. Within this framework, we derive a new convex form of the constraint and analyze it in the mistake bound model. Empirical evaluation with both synthetic and text data shows our version of CW learning achieves lower cumulative and out-of-sample errors than commonly used ﬁrst-order and second-order online methods. 1

5 0.48759034 196 nips-2008-Relative Margin Machines

Author: Tony Jebara, Pannagadatta K. Shivaswamy

Abstract: In classiﬁcation problems, Support Vector Machines maximize the margin of separation between two classes. While the paradigm has been successful, the solution obtained by SVMs is dominated by the directions with large data spread and biased to separate the classes by cutting along large spread directions. This article proposes a novel formulation to overcome such sensitivity and maximizes the margin relative to the spread of the data. The proposed formulation can be eﬃciently solved and experiments on digit datasets show drastic performance improvements over SVMs. 1

6 0.48396495 75 nips-2008-Estimating vector fields using sparse basis field expansions

7 0.48353615 62 nips-2008-Differentiable Sparse Coding

8 0.48282626 210 nips-2008-Signal-to-Noise Ratio Analysis of Policy Gradient Algorithms

9 0.48278806 175 nips-2008-PSDBoost: Matrix-Generation Linear Programming for Positive Semidefinite Matrices Learning

10 0.48248553 133 nips-2008-Mind the Duality Gap: Logarithmic regret algorithms for online optimization

11 0.47856179 37 nips-2008-Biasing Approximate Dynamic Programming with a Lower Discount Factor

12 0.47339296 16 nips-2008-Adaptive Template Matching with Shift-Invariant Semi-NMF

13 0.46713936 202 nips-2008-Robust Regression and Lasso

14 0.46620277 141 nips-2008-Multi-Agent Filtering with Infinitely Nested Beliefs

15 0.46144009 2 nips-2008-A Convex Upper Bound on the Log-Partition Function for Binary Distributions

16 0.46055591 134 nips-2008-Mixed Membership Stochastic Blockmodels

17 0.45989156 21 nips-2008-An Homotopy Algorithm for the Lasso with Online Observations

18 0.45901307 228 nips-2008-Support Vector Machines with a Reject Option

19 0.4589881 194 nips-2008-Regularized Learning with Networks of Features

20 0.45799533 88 nips-2008-From Online to Batch Learning with Cutoff-Averaging