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We introduce an efficient algorithm for the problem of online linear optimization in the bandit setting which achieves the optimal O * (√ T) regret. The setting is a natural generalization of the non-stochastic multi-armed bandit problem, and the existence of an efficient optimal algorithm has been posed as an open problem in a number of recent papers. We… (More)

Stochastic gradient descent (SGD) is a simple and popular method to solve stochastic optimization problems which arise in machine learning. For strongly convex problems, its convergence rate was known to be O(log(T)/T), by running SGD for T iterations and returning the average point. However, recent results showed that using a different algorithm, one can… (More)

This thesis studies two key properties of learning algorithms: their generalization ability and their stability with respect to perturbations. To analyze these properties, we focus on concentration inequalities and tools from empirical process theory. We obtain theoretical results and demonstrate their applications to machine learning. First, we show how… (More)

We present methods for online linear optimization that take advantage of benign (as opposed to worst-case) sequences. Specifically if the sequence encountered by the learner is described well by a known " predictable process " , the algorithms presented enjoy tighter bounds as compared to the typical worst case bounds. Additionally, the methods achieve the… (More)

We study the rates of growth of the regret in online convex optimization. First, we show that a simple extension of the algorithm of Hazan et al eliminates the need for a priori knowledge of the lower bound on the second derivatives of the observed functions. We then provide an algorithm, Adaptive Online Gradient Descent, which interpolates between the… (More)

These lecture notes contain material presented in the Statistical Learning Theory course at UC Berkeley, Spring'08. Various parts of these notes have been discovered together with 4 Bandit problems 29 5 Minimax Results and Lower Bounds 31 6 Variance Bounds 33 7 Stochastic Approximation 35 CONTENTS CHAPTER ONE INTRODUCTION The past two decades witnessed a… (More)

A number of learning problems can be cast as an Online Convex Game: on each round, a learner makes a prediction x from a convex set, the environment plays a loss function f , and the learner's long-term goal is to minimize regret. Algorithms have been proposed by Zinkevich, when f is assumed to be convex, and Hazan et al., when f is assumed to be strongly… (More)

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… (More)

We provide a principled way of proving Õ(√T) high-probability guarantees for partial-information (bandit) problems over arbitrary convex decision sets. First, we prove a regret guarantee for the full-information problem in terms of “local” norms, both for entropy and self-concordant barrier regularization, unifying these methods.… (More)

We show a principled way of deriving online learning algorithms from a minimax analysis. Various upper bounds on the minimax value, previously thought to be non-constructive, are shown to yield algorithms. This allows us to seamlessly recover known methods and to derive new ones, also capturing such " unorthodox " methods as Follow the Perturbed Leader and… (More)