We study the problem of repeated play in a zero-sum game in which the payoff matrix may change, in a possibly adversarial fashion, on each round; we call these Online Matrix Games. Finding the Nash Equilibrium (NE) of a two player zero-sum game is core to many problems in statistics, optimization, and economics, and for a fixed game matrix this can be easily reduced to solving a linear program. But when the payoff matrix evolves over time our goal is to find a sequential algorithm that can compete with, in a certain sense, the NE of the long-term-averaged payoff matrix. We design an algorithm with small NE regret--that is, we ensure that the long-term payoff of both players is close to minimax optimum in hindsight. Our algorithm achieves near-optimal dependence with respect to the number of rounds and depends poly-logarithmically on the number of available actions of the players. Additionally, we show that the naive reduction, where each player simply minimizes its own regret, fails to achieve the stated objective regardless of which algorithm is used. We also consider the so-called bandit setting, where the feedback is significantly limited, and we provide an algorithm with small NE regret using one-point estimates of each payoff matrix.
The reader familiar with Online Convex Optimization (OCO) may find it closely related to the OMG problem. In the OCO setting, a player is given a convex, closed, and bounded action set @math , and must repeatedly choose an action @math before the convex function @math is revealed. The player's goal is to obtain sublinear defined as @math . This problem is well studied and several algorithms such as Online Gradient Descent @cite_12 , Regularized Follow the Leader @cite_22 @cite_53 and Perturbed Follow the Leader @cite_6 achieve optimal individual regret bounds that scale as @math . The most natural (although incorrect) approach to attack the OMG problem is to equip each of the players with a sublinear individual regret algorithm. However, we will show in that if both players use an algorithm that guarantees sublinear individual regret, then it is impossible to achieve sublinear NE regret when the payoff matrices are chosen adversarially. In other words, the algorithms for the OCO setting cannot be directly applied to the OMG problem considered in this paper.
@cite_53 @cite_22 @cite_12 @cite_6
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We provide the first oracle efficient sublinear regret algorithms for adversarial versions of the contextual bandit problem. In this problem, the learner repeatedly makes an action on the basis of a context and receives reward for the chosen action, with the goal of achieving reward competitive with a large class of policies. We analyze two settings: i) in the transductive setting the learner knows the set of contexts a priori, ii) in the small separator setting, there exists a small set of contexts such that any two policies behave differently on one of the contexts in the set. Our algorithms fall into the Follow-The-Perturbed-Leader family (Kalai & Vempala, 2005) and achieve regret O(T3 4√K log(N)) in the transductive setting and O(T2 3d3 4K√log(N)) in the separator setting, where T is the number of rounds, K is the number of actions, N is the number of base-line policies, and d is the size of the separator. We actually solve the more general adversarial contextual semi-bandit linear optimization problem, whilst in the full information setting we address the even more general contextual combinatorial optimization. We provide several extensions and implications of our algorithms, such as switching regret and efficient learning with predictable sequences. This paper considers online convex optimization (OCO) with stochastic constraints, which generalizes Zinkevich's OCO over a known simple fixed set by introducing multiple stochastic functional constraints that are i.i.d. generated at each round and are disclosed to the decision maker only after the decision is made. This formulation arises naturally when decisions are restricted by stochastic environments or deterministic environments with noisy observations. It also includes many important problems as special cases, such as OCO with long term constraints, stochastic constrained convex optimization, and deterministic constrained convex optimization. To solve this problem, this paper proposes a new algorithm that achieves @math expected regret and constraint violations and @math high probability regret and constraint violations. Experiments on a real-world data center scheduling problem further verify the performance of the new algorithm. We consider the adversarial convex bandit problem and we build the first @math -time algorithm with @math -regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves @math -regret, and we show that a simple variant of this algorithm can be run in @math -time per step at the cost of an additional @math factor in the regret. These results improve upon the @math -regret and @math -time result of the first two authors, and the @math -regret and @math -time result of Hazan and Li. Furthermore we conjecture that another variant of the algorithm could achieve @math -regret, and moreover that this regret is unimprovable (the current best lower bound being @math and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order @math . We present a unified, black-box-style method for developing and analyzing online convex optimization (OCO) algorithms for full-information online learning in delayed-feedback environments. Our new, simplified analysis enables us to substantially improve upon previous work and to solve a number of open problems from the literature. Specifically, we develop and analyze asynchronous AdaGrad-style algorithms from the Follow-the-Regularized-Leader (FTRL) and Mirror-Descent family that, unlike previous works, can handle projections and adapt both to the gradients and the delays, without relying on either strong convexity or smoothness of the objective function, or data sparsity. Our unified framework builds on a natural reduction from delayed-feedback to standard (non-delayed) online learning. This reduction, together with recent unification results for OCO algorithms, allows us to analyze the regret of generic FTRL and Mirror-Descent algorithms in the delayed-feedback setting in a unified manner using standard proof techniques. In addition, the reduction is exact and can be used to obtain both upper and lower bounds on the regret in the delayed-feedback setting.