text: 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.
  @cite_15: Online learning is a well established learning paradigm which has both theoretical and practical appeals. The goal of online learning is to make a sequence of accurate predictions given knowledge of the correct answer to previous prediction tasks and possibly additional available information. Online learning has been studied in several research fields including game theory, information theory, and machine learning. It also became of great interest to practitioners due the recent emergence of large scale applications such as online advertisement placement and online web ranking. In this survey we provide a modern overview of online learning. Our goal is to give the reader a sense of some of the interesting ideas and in particular to underscore the centrality of convexity in deriving efficient online learning algorithms. We do not mean to be comprehensive but rather to give a high-level, rigorous yet easy to follow, survey.
  @cite_1: This monograph portrays optimization as a process. In many practical applications the environment is so complex that it is infeasible to lay out a comprehensive theoretical model and use classical algorithmic theory and mathematical optimization. It is necessary as well as beneficial to take a robust approach, by applying an optimization method that learns as one goes along, learning from experience as more aspects of the problem are observed. This view of optimization as a process has become prominent in varied fields and has led to some spectacular success in modeling and systems that are now part of our daily lives.
summary: 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.