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.
Related to the OMG problem with bandit feedback is the seminal work of @cite_4 . They provide the first sublinear regret bound for Online Convex Optimization with bandit feedback, using a one-point estimate of the gradient. The one-point gradient estimate used in @cite_4 is similar to those independently proposed in @cite_35 and in @cite_44 . The regret bound provided in @cite_4 is @math , which is suboptimal. In @cite_53 , the authors give the first @math bound for the special case when the functions are linear. More recently, @cite_39 and @cite_16 designed the first efficient algorithms with @math regret for the general online convex optimization case; unfortunately, the dependence on the dimension @math in the regret rate is a very large polynomial. Our one-point matrix estimate is most closely related to the random estimator in @cite_8 for linear functions. It is possible to use the more sophisticated techniques from @cite_53 @cite_39 @cite_16 to improve our NE regret bound in section ; however, the result does not seem to be immediate and we leave this as future work.
@cite_35 @cite_4 @cite_8 @cite_53 @cite_39 @cite_44 @cite_16
2473549844 2120745256 2004001705 2301614296
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 . In the online linear optimization problem, a learner must choose, in each round, a decision from a set D ⊂ ℝn in order to minimize an (unknown and changing) linear cost function. We present sharp rates of convergence (with respect to additive regret) for both the full information setting (where the cost function is revealed at the end of each round) and the bandit setting (where only the scalar cost incurred is revealed). In particular, this paper is concerned with the price of bandit information, by which we mean the ratio of the best achievable regret in the bandit setting to that in the full-information setting. For the full information case, the upper bound on the regret is O*( √nT), where n is the ambient dimension and T is the time horizon. For the bandit case, we present an algorithm which achieves O*(n3 2 √T) regret — all previous (nontrivial) bounds here were O(poly(n)T2 3) or worse. It is striking that the convergence rate for the bandit setting is only a factor of n worse than in the full information case — in stark contrast to the K-arm bandit setting, where the gap in the dependence on K is exponential (√TK vs. √T log K). We also present lower bounds showing that this gap is at least √n, which we conjecture to be the correct order. The bandit algorithm we present can be implemented efficiently in special cases of particular interest, such as path planning and Markov Decision Problems. We study a general online convex optimization problem. We have a convex set S and an unknown sequence of cost functions c 1 , c 2 ,..., and in each period, we choose a feasible point x t in S, and learn the cost c t (x t ). If the function c t is also revealed after each period then, as Zinkevich shows in [25], gradient descent can be used on these functions to get regret bounds of O(√n). That is, after n rounds, the total cost incurred will be O(√n) more than the cost of the best single feasible decision chosen with the benefit of hindsight, min x Σ ct(x).We extend this to the "bandit" setting, where, in each period, only the cost c t (x t ) is revealed, and bound the expected regret as O(n3 4).Our approach uses a simple approximation of the gradient that is computed from evaluating c t at a single (random) point. We show that this biased estimate is sufficient to approximate gradient descent on the sequence of functions. In other words, it is possible to use gradient descent without seeing anything more than the value of the functions at a single point. The guarantees hold even in the most general case: online against an adaptive adversary.For the online linear optimization problem [15], algorithms with low regrets in the bandit setting have recently been given against oblivious [1] and adaptive adversaries [19]. In contrast to these algorithms, which distinguish between explicit explore and exploit periods, our algorithm can be interpreted as doing a small amount of exploration in each period. We consider the problem of online convex optimization against an arbitrary adversary with bandit feedback, known as bandit convex optimization. We give the first @math -regret algorithm for this setting based on a novel application of the ellipsoid method to online learning. This bound is known to be tight up to logarithmic factors. Our analysis introduces new tools in discrete convex geometry.