We study various discrete nonlinear combinatorial optimization problems in an online learning framework. In the first part, we address the question of whether there are negative results showing that getting a vanishing (or even vanishing approximate) regret is computational hard. We provide a general reduction showing that many (min-max) polynomial time solvable problems not only do not have a vanishing regret, but also no vanishing approximation @math -regret, for some @math (unless @math ). Then, we focus on a particular min-max problem, the min-max version of the vertex cover problem which is solvable in polynomial time in the offline case. The previous reduction proves that there is no @math -regret online algorithm, unless Unique Game is in @math ; we prove a matching upper bound providing an online algorithm based on the online gradient descent method. Then, we turn our attention to online learning algorithms that are based on an offline optimization oracle that, given a set of instances of the problem, is able to compute the optimum static solution. We show that for different nonlinear discrete optimization problems, it is strongly @math -hard to solve the offline optimization oracle, even for problems that can be solved in polynomial time in the static case (e.g. min-max vertex cover, min-max perfect matching, etc.). On the positive side, we present an online algorithm with vanishing regret that is based on the follow the perturbed leader algorithm for a generalized knapsack problem.
Online Learning, or Online Convex Optimization, is an active research domain. In this section, we only summarize works which are directly related to ours. We refer the reader to comprehensive books @cite_15 @cite_1 and references therein for a more complete overview. The first no-regret algorithm has been given by . Subsequently, and gave improved algorithms with regret @math where @math is the size of the action space. However, these algorithms have running-time @math which is exponential in the size of the input for many applications, in particular for combinatorial optimization problems. An intriguing question is whether there exists a no-regret online algorithm with running-time polynomial in @math . proved that no such algorithm exists in general settings without any assumption on the structure. Designing online polynomial-time algorithms with approximation and vanishing regret guarantees for combinatorial optimization problems is a major research agenda.
@cite_15 @cite_1
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We introduce the general and powerful scheme of predicting information re-use in optimization algorithms. This allows us to devise a computationally efficient algorithm for bandit convex optimization with new state-of-the-art guarantees for both Lipschitz loss functions and loss functions with Lipschitz gradients. This is the first algorithm admitting both a polynomial time complexity and a regret that is polynomial in the dimension of the action space that improves upon the original regret bound for Lipschitz loss functions, achieving a regret of @math . Our algorithm further improves upon the best existing polynomial-in-dimension bound (both computationally and in terms of regret) for loss functions with Lipschitz gradients, achieving a regret of @math . In this paper, we study the stochastic combinatorial multi-armed bandit (CMAB) framework that allows a general nonlinear reward function, whose expected value may not depend only on the means of the input random variables but possibly on the entire distributions of these variables. Our framework enables a much larger class of reward functions such as the @math function and nonlinear utility functions. Existing techniques relying on accurate estimations of the means of random variables, such as the upper confidence bound (UCB) technique, do not work directly on these functions. We propose a new algorithm called stochastically dominant confidence bound (SDCB), which estimates the distributions of underlying random variables and their stochastically dominant confidence bounds. We prove that SDCB can achieve @math distribution-dependent regret and @math distribution-independent regret, where @math is the time horizon. We apply our results to the @math -MAX problem and expected utility maximization problems. In particular, for @math -MAX, we provide the first polynomial-time approximation scheme (PTAS) for its offline problem, and give the first @math bound on the @math -approximation regret of its online problem, for any @math . A line of recent work provides welfare guarantees of simple combinatorial auction formats, such as selling m items via simultaneous second price auctions (SiSPAs) ( 2008, Bhawalkar and Roughgarden 2011, 2013). These guarantees hold even when the auctions are repeatedly executed and players use no-regret learning algorithms. Unfortunately, off-the-shelf no-regret algorithms for these auctions are computationally inefficient as the number of actions is exponential. We show that this obstacle is insurmountable: there are no polynomial-time no-regret algorithms for SiSPAs, unless RP @math NP, even when the bidders are unit-demand. Our lower bound raises the question of how good outcomes polynomially-bounded bidders may discover in such auctions. To answer this question, we propose a novel concept of learning in auctions, termed "no-envy learning." This notion is founded upon Walrasian equilibrium, and we show that it is both efficiently implementable and results in approximately optimal welfare, even when the bidders have fractionally subadditive (XOS) valuations (assuming demand oracles) or coverage valuations (without demand oracles). No-envy learning outcomes are a relaxation of no-regret outcomes, which maintain their approximate welfare optimality while endowing them with computational tractability. Our results extend to other auction formats that have been studied in the literature via the smoothness paradigm. Our results for XOS valuations are enabled by a novel Follow-The-Perturbed-Leader algorithm for settings where the number of experts is infinite, and the payoff function of the learner is non-linear. This algorithm has applications outside of auction settings, such as in security games. Our result for coverage valuations is based on a novel use of convex rounding schemes and a reduction to online convex optimization. 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 .