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117
Beyond the regret minimization barrier: an optimal algorithm for stochastic stronglyconvex optimization
 In Proceedings of the 24th Annual Conference on Learning Theory, volume 19 of JMLR Workshop and Conference Proceedings
, 2011
"... We give a novel algorithm for stochastic stronglyconvex optimization in the gradient oracle model which returns an O ( 1 T)approximate solution after T gradient updates. This rate of convergence is optimal in the gradient oracle model. This improves upon the previously log(T) known best rate of O( ..."
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Cited by 61 (3 self)
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We give a novel algorithm for stochastic stronglyconvex optimization in the gradient oracle model which returns an O ( 1 T)approximate solution after T gradient updates. This rate of convergence is optimal in the gradient oracle model. This improves upon the previously log(T) known best rate of O( T), which was obtained by applying an online stronglyconvex optimization algorithm with regret O(log(T)) to the batch setting. We complement this result by proving that any algorithm has expected regret of Ω(log(T)) in the online stochastic stronglyconvex optimization setting. This lower bound holds even in the fullinformation setting which reveals more information to the algorithm than just gradients. This shows that any onlinetobatch conversion is inherently suboptimal for stochastic stronglyconvex optimization. This is the first formal evidence that online convex optimization is strictly more difficult than batch stochastic convex optimization. 1
Privacy, Accuracy, and Consistency Too: A Holistic Solution to Contingency Table Release
, 2007
"... The contingency table is a work horse of official statistics, the format of reported data for the US Census, Bureau of Labor Statistics, and the Internal Revenue Service. In many settings such as these privacy is not only ethically mandated, but frequently legally as well. Consequently there is an e ..."
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Cited by 56 (6 self)
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The contingency table is a work horse of official statistics, the format of reported data for the US Census, Bureau of Labor Statistics, and the Internal Revenue Service. In many settings such as these privacy is not only ethically mandated, but frequently legally as well. Consequently there is an extensive and diverse literature dedicated to the problems of statistical disclosure control in contingency table release. However, all current techniques for reporting contingency tables fall short on at least one of privacy, accuracy, and consistency (among multiple released tables). We propose a solution that provides strong guarantees for all three desiderata simultaneously. Our approach can be viewed as a special case of a more general approach for producing synthetic data: Any privacypreserving mechanism for contingency table release begins with raw data and produces a (possibly inconsistent) privacypreserving set of marginals. From these tables alone – and hence without weakening privacy – we will find and output the “nearest ” consistent set of marginals. Interestingly, this set is no farther than the tables of the raw data, and consequently the additional error introduced by the imposition of consistency is no more than the error introduced by the privacy mechanism itself. The privacy mechanism of [20] gives the strongest known privacy guarantees, with very little error. Combined with the techniques of the current paper, we therefore obtain excellent privacy, accuracy, and consistency among the tables. Moreover, our techniques are surprisingly efficient. Our techniques apply equally well to the logical cousin of the contingency table, the OLAP cube.
A Fast Random Sampling Algorithm for Sparsifying Matrices
 IN APPROXRANDOM
, 2006
"... We describe a simple randomsampling based procedure for producing sparse matrix approximations. Our procedure and analysis are extremely simple: the analysis uses nothing more than the ChernoffHoeffding bounds. Despite the simplicity, the approximation is comparable and sometimes better than previ ..."
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Cited by 39 (1 self)
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We describe a simple randomsampling based procedure for producing sparse matrix approximations. Our procedure and analysis are extremely simple: the analysis uses nothing more than the ChernoffHoeffding bounds. Despite the simplicity, the approximation is comparable and sometimes better than previous work. Our algorithm computes the sparse matrix approximation in a single pass over the data. Further, most of the entries in the output matrix are quantized, and can be succinctly represented by a bit vector, thus leading to much savings in space.
Algorithms for Portfolio Management based on the Newton Method
 Portfolio Performance Evaluation, Investments, 4th edition, Irwin McGrawHill
, 1999
"... We experimentally study online investment algorithms first proposed by Agarwal and Hazan and extended by Hazan et al. which achieve almost the same wealth as the best constantrebalanced portfolio determined in hindsight. These algorithms are the first to combine optimal logarithmic regret bounds w ..."
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Cited by 33 (2 self)
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We experimentally study online investment algorithms first proposed by Agarwal and Hazan and extended by Hazan et al. which achieve almost the same wealth as the best constantrebalanced portfolio determined in hindsight. These algorithms are the first to combine optimal logarithmic regret bounds with efficient deterministic computability. They are based on the Newton method for offline optimization which, unlike previous approaches, exploits second order information. After analyzing the algorithm using the potential function introduced by Agarwal and Hazan, we present extensive experiments on actual financial data. These experiments confirm the theoretical advantage of our algorithms, which yield higher returns and run considerably faster than previous algorithms with optimal regret. Additionally, we perform financial analysis using meanvariance calculations and the Sharpe ratio. 1.
Extracting certainty from uncertainty: Regret bounded by variation in costs
 In COLT
, 2008
"... Prediction from expert advice is a fundamental problem in machine learning. A major pillar of the field is the existence of learning algorithms whose average loss approaches that of the best expert in hindsight (in other words, whose average regret approaches zero). Traditionally the regret of onlin ..."
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Cited by 38 (5 self)
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Prediction from expert advice is a fundamental problem in machine learning. A major pillar of the field is the existence of learning algorithms whose average loss approaches that of the best expert in hindsight (in other words, whose average regret approaches zero). Traditionally the regret of online algorithms was bounded in terms of the number of prediction rounds. CesaBianchi, Mansour and Stoltz [4] posed the question whether it is be possible to bound the regret of an online algorithm by the variation of the observed costs. In this paper we resolve this question, and prove such bounds in the fully adversarial setting, in two important online learning scenarios: prediction from expert advice, and online linear optimization. 1
Supplement: NonStochastic Bandit Slate Problems
"... Recall our special variant of Hedge: we are allowed to uses only distributions p(t) from some fixed convex subset P of the simplex of all distributions. The goal then is to minimize regret relative to an arbitrary distribution p ∈ P. Such a version of Hedge is given in Figure 1, and a statement of i ..."
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Cited by 13 (0 self)
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Recall our special variant of Hedge: we are allowed to uses only distributions p(t) from some fixed convex subset P of the simplex of all distributions. The goal then is to minimize regret relative to an arbitrary distribution p ∈ P. Such a version of Hedge is given in Figure 1, and a statement of its performance below. This algorithm is implicit in the work of [4, 6]. Algorithm MW(P) Initialization: An arbitrary probability distribution p(1) ∈ P on the experts, and some η> 0. For t = 1, 2,..., T: 1. Choose distribution p(t) over experts, and observe the cost vector ℓ(t). 2. Compute the probability vector ˆp(t + 1) using the following multiplicative update rule: for every expert i, ˆpi(t + 1) = pi(t) exp(−ηℓi(t))/Z(t) (1) where Z(t) = ∑ i pi(t) exp(−ηℓi(t)) is the normalization factor. 3. Set p(t + 1) to be the projection of ˆp(t + 1) on the set P using the RE as a distance function, i.e. p(t + 1) = arg minp∈P RE(p ‖ ˆp(t + 1)). Figure 1: The Multiplicative Weights Algorithm with Restricted Distributions Theorem 1.1. Assume that η> 0 is chosen so that for all t and i, ηℓi(t) ≥ −1. Then algorithm MW(P) generates distributions p(1),..., p(T) ∈ P, such that for any p ∈ P, T∑ T∑ ℓ(t) · p(t) − ℓ(t) · p ≤ η (ℓ(t)) 2 · p(t) + t=1 Here, (ℓ(t)) 2 is the vector that is the coordinatewise square of ℓ(t). t=1 RE(p ‖ p(1)) η Proof. We use the relative entropy between p and p(t), RE(p ‖ p(t)): = ∑ i pi ln(pi/pi(t)) as a “potential ” function. We have RE(p ‖ ˆp t+1) − RE(p ‖ p(t)) = ∑ pi(t) pi ln
Technion
"... A simple multiarmed bandit algorithm with optimal variationbounded regret ..."
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A simple multiarmed bandit algorithm with optimal variationbounded regret
Efficient Optimal Learning for Contextual Bandits
"... We address the problem of learning in an online setting where the learner repeatedly observes features x, selects among K actions, and receives reward r for the action taken. We provide the first efficient algorithm with an optimal regret. Our algorithm uses an oracle which returns an optimal policy ..."
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Cited by 26 (2 self)
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We address the problem of learning in an online setting where the learner repeatedly observes features x, selects among K actions, and receives reward r for the action taken. We provide the first efficient algorithm with an optimal regret. Our algorithm uses an oracle which returns an optimal policy given rewards for all actions for each x. The algorithm has running time polylog(N), where N is the number of policies that we compete with. This is exponentially faster than all previous algorithms that achieve optimal regret in this setting. Our formulation also enables us to create an algorithm with regret that is additive rather than multiplicative in feedback delay as in all previous work. 1.
Max Cut and the Smallest Eigenvalue
, 2008
"... We describe a new approximation algorithm for Max Cut. Our algorithm runs in Õ(n2) time, where n is the number of vertices, and achieves an approximation ratio of.50769. On instances in which an optimal solution cuts a 1 − ε fraction of edges, our algorithm finds a solution that cuts a 1 − 4ε 1/3 − ..."
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Cited by 3 (0 self)
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and Williamson [GW95], together with the SDP solver of Arora and Kale [AK07], give an approximation ratio of.878 in nearly linear time. While our algorithm is inferior in both running time and approximation ratio, it is the first algorithm to achieve an approximation better than 1/2 for Max Cut by any means
Results 1  10
of
117