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.
Another direction is to design online learning algorithms using (offline polynomial-time) approximation algorithms as oracles. provided an algorithm which is inspired by Zinkevich's algorithm @cite_14 (gradient descent): at every step, the algorithm updates the current solution in the direction of the gradient and project back to the feasible set using an approximation algorithm. They showed that given an @math -approximation algorithm for a optimization problem, after @math prediction rounds (time steps) the online algorithm achieves an @math -regret bound of @math using @math calls to the approximation algorithm per round in average. Later on, gave an algorithm with @math -regret bound of @math using only @math calls to the approximation algorithm per round in average. These algorithms rely crucially on the linearity of the objective functions and it remains an interesting open question to design algorithms for online non-linear optimization problems.
@cite_14
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Generalized Linear Bandits (GLBs), a natural extension of the stochastic linear bandits, has been popular and successful in recent years. However, existing GLBs scale poorly with the number of rounds and the number of arms, limiting their utility in practice. This paper proposes new, scalable solutions to the GLB problem in two respects. First, unlike existing GLBs, whose per-time-step space and time complexity grow at least linearly with time @math , we propose a new algorithm that performs online computations to enjoy a constant space and time complexity. At its heart is a novel Generalized Linear extension of the Online-to-confidence-set Conversion (GLOC method) that takes online learning algorithm and turns it into a GLB algorithm. As a special case, we apply GLOC to the online Newton step algorithm, which results in a low-regret GLB algorithm with much lower time and memory complexity than prior work. Second, for the case where the number @math of arms is very large, we propose new algorithms in which each next arm is selected via an inner product search. Such methods can be implemented via hashing algorithms (i.e., "hash-amenable") and result in a time complexity sublinear in @math . While a Thompson sampling extension of GLOC is hash-amenable, its regret bound for @math -dimensional arm sets scales with @math , whereas GLOC's regret bound scales with @math . Towards closing this gap, we propose a new hash-amenable algorithm whose regret bound scales with @math . Finally, we propose a fast approximate hash-key computation (inner product) with a better accuracy than the state-of-the-art, which can be of independent interest. We conclude the paper with preliminary experimental resu