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A Stochastic View of Optimal Regret through Minimax Duality
"... We study the regret of optimal strategies for online convex optimization games. Using von Neumann’s minimax theorem, we show that the optimal regret in this adversarial setting is closely related to the behavior of the empirical minimization algorithm in a stochastic process setting: it is equal to ..."
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Cited by 26 (10 self)
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We study the regret of optimal strategies for online convex optimization games. Using von Neumann’s minimax theorem, we show that the optimal regret in this adversarial setting is closely related to the behavior of the empirical minimization algorithm in a stochastic process setting: it is equal to the maximum, over joint distributions of the adversary’s action sequence, of the difference between a sum of minimal expected losses and the minimal empirical loss. We show that the optimal regret has a natural geometric interpretation, since it can be viewed as the gap in Jensen’s inequality for a concave functional—the minimizer over the player’s actions of expected loss—defined on a set of probability distributions. We use this expression to obtain upper and lower bounds on the regret of an optimal strategy for a variety of online learning problems. Our method provides upper bounds without the need to construct a learning algorithm; the lower bounds provide explicit optimal strategies for the adversary. 1
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 14 (0 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
The LastStep Minimax Algorithm
 Pages 279 290 of: Proc. 11th International Conference on Algorithmic Learning Theory
, 2000
"... We consider online density estimation with a parameterized density from an exponential family. In each trial t the learner predicts a parameter t . Then it receives an instance x t chosen by the adversary and incurs loss ln p(x t j t ) which is the negative loglikelihood of x t w.r.t. the predict ..."
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Cited by 8 (1 self)
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We consider online density estimation with a parameterized density from an exponential family. In each trial t the learner predicts a parameter t . Then it receives an instance x t chosen by the adversary and incurs loss ln p(x t j t ) which is the negative loglikelihood of x t w.r.t. the predicted density of the learner. The performance of the learner is measured by the regret dened as the total loss of the learner minus the total loss of the best parameter chosen oline. We develop an algorithm called the Laststep Minimax Algorithm that predicts with the minimax optimal parameter assuming that the current trial is the last one. For onedimensional exponential families, we give an explicit form of the prediction of the Laststep Minimax Algorithm and show that its regret is O(ln T ), where T is the number of trials. In particular, for Bernoulli density estimation the Laststep Minimax Algorithm is slightly better than the standard Laplace estimator. This work was done while...