We consider a the general online convex optimization framework introduced by Zinkevich. In this setting, there is a sequence of convex functions. Each period, we must choose a signle point (from some feasible set) and pay a cost equal to the value of the next function on our chosen point. Zinkevich shows that, if the each function is revealed after the choice is made, then one can achieve vanishingly small regret relative the best single decision chosen in hindsight. We extend this to the bandit setting where we do not find out the entire functions but rather just their value at our chosen point. We show how to get vanishingly small regret in this setting. Our approach uses a simple approximation of the gradient that is computed from evaluating a function at a single (random) point. We show that this estimate is sufficient to mimic Zinkevich's gradient descent online analysis, with access to the gradient (only being able to evaluate the function at a single point).
A similar line of research has developed for the problem of online linear optimization @cite_1 @cite_10 @cite_9 . Here, one wants to solve the related but incomparable problem of optimizing a sequence of linear functions, over a possibly non-convex feasible set, modeling problems such as online shortest paths and online binary search trees (which are difficult to convexify). Kalai and Vempala @cite_1 show that, for such linear optimization problems in general, if the offline optimization problem is solvable efficiently, then regret can be bounded by @math also by an efficient online algorithm, in the full-information model. Awerbuch and Kleinberg @cite_10 generalize this to the bandit setting against an oblivious adversary (like ours). Blum and McMahan @cite_9 give a simpler algorithm that applies to adaptive adversaries, that may choose their functions @math depending on the previous points.
@cite_10 @cite_9 @cite_1
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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 address online linear optimization problems when the possible actions of the decision maker are represented by binary vectors. The regret of the decision maker is the difference between her realized loss and the minimal loss she would have achieved by picking, in hindsight, the best possible action. Our goal is to understand the magnitude of the best possible minimax regret. We study the problem under three different assumptions for the feedback the decision maker receives: full information, and the partial information models of the so-called “semi-bandit” and “bandit” problems. In the full information case we show that the standard exponentially weighted average forecaster is a provably suboptimal strategy. For the semi-bandit model, by combining the Mirror Descent algorithm and the INF Implicitely Normalized Forecaster strategy, we are able to prove the first optimal bounds. Finally, in the bandit case we discuss existing results in light of a new lower bound, and suggest a conjecture on the optimal regret in that case.