We study online optimization of smoothed piecewise constant functions over the domain [0, 1). This is motivated by the problem of adaptively picking parameters of learning algorithms as in the recently introduced framework by Gupta and Roughgarden (2016). Majority of the machine learning literature has focused on Lipschitz-continuous functions or functions with bounded gradients. This is with good reason—any learning algorithm suffers linear regret even against piecewise constant functions that are chosen adversarially, arguably the simplest of non-Lipschitz continuous functions. The smoothed setting we consider is inspired by the seminal work of Spielman and Teng (2004) and the recent work of Gupta and Roughgarden (2016)—in this setting, the sequence of functions may be chosen by an adversary, however, with some uncertainty in the location of discontinuities. We give algorithms that achieve sublinear regret in the full information and bandit settings.
In the multi-armed bandit problem, a gambler must decide which arm of non-identical slot machines to play in a sequence of trials so as to maximize his reward. This classical problem has received much attention because of the simple model it provides of the trade-off between exploration (trying out each arm to nd the best one) and exploitation (playing the arm believed to give the best payoff). Past solutions for the bandit problem have almost always relied on assumptions about the statistics of the slot machines. In this work, we make no statistical assumptions whatsoever about the nature of the process generating the payoffs of the slot machines. We give a solution to the bandit problem in which an adversary, rather than a well-behaved stochastic process, has complete control over the payoffs. In a sequence of plays, we prove that the per-round payoff of our algorithm approaches that of the best arm at the rate . We show by a matching lower bound that this is best possible. We also prove that our algorithm approaches the per-round payoff of any set of strategies at a similar rate: if the best strategy is chosen from a pool of strategies then our algorithm approaches the per-round payoff of the strategy at the rate . Finally, we apply our results to the problem of playing an unknown repeated matrix game. We show that our algorithm approaches the minimax payoff of the unknown game at the rate We consider the framework of stochastic multi-armed bandit problems and study the possibilities and limitations of strategies that perform an online exploration of the arms. The strategies are assessed in terms of their simple regret, a regret notion that captures the fact that exploration is only constrained by the number of available rounds (not necessarily known in advance), in contrast to the case when the cumulative regret is considered and when exploitation needs to be performed at the same time.We believe that this performance criterion is suited to situations when the cost of pulling an arm is expressed in terms of resources rather than rewards. We discuss the links between the simple and the cumulative regret. The main result is that the required exploration-exploitation trade-offs are qualitatively different, in view of a general lower bound on the simple regret in terms of the cumulative regret.