Online $\mathrm{L}^{\natural}$-Convex Minimization
Ken Yokoyama, Shinji Ito, Tatsuya Matsuoka, Kei Kimura, Makoto Yokoo
TL;DR
The paper addresses online decision problems with nonlinear discrete objectives that fall outside the online submodular framework by introducing online $L^{\natural}$-convex minimization on integer lattices. It develops efficient full-information and bandit algorithms based on convex extensions (the Lovász extension and a piecewise-linear convex extension) and provides regret analyses, achieving $\mathbb{E}[R_T]=O(\hat{L}N\sqrt{dT})$ in the full information setting and $\mathbb{E}[R_T]=O(MdNT^{2/3})$ in the bandit setting, with a matching lower bound $\Omega(\hat{L}N\sqrt{dT})$ for the former. The framework is illustrated through applications to online spare-parts inventory and call-center shift scheduling, showing that these problems can be formulated as online $L^{\natural}$-convex minimization and solved with provable guarantees. Overall, the work extends online convex/ submodular optimization to discrete, lattice-based domains, enabling efficient handling of a broader class of nonlinear, combinatorial online problems with practical impact.
Abstract
An online decision-making problem is a learning problem in which a player repeatedly makes decisions in order to minimize the long-term loss. These problems that emerge in applications often have nonlinear combinatorial objective functions, and developing algorithms for such problems has attracted considerable attention. An existing general framework for dealing with such objective functions is the online submodular minimization. However, practical problems are often out of the scope of this framework, since the domain of a submodular function is limited to a subset of the unit hypercube. To manage this limitation of the existing framework, we in this paper introduce the online $\mathrm{L}^{\natural}$-convex minimization, where an $\mathrm{L}^{\natural}$-convex function generalizes a submodular function so that the domain is a subset of the integer lattice. We propose computationally efficient algorithms for the online $\mathrm{L}^{\natural}$-convex function minimization in two major settings: the full information and the bandit settings. We analyze the regrets of these algorithms and show in particular that our algorithm for the full information setting obtains a tight regret bound up to a constant factor. We also demonstrate several motivating examples that illustrate the usefulness of the online $\mathrm{L}^{\natural}$-convex minimization.