Fixed-Budget Real-Valued Combinatorial Pure Exploration of Multi-Armed Bandit

We study the real-valued combinatorial pure exploration of the multi-armed bandit in the fixed-budget setting. We first introduce the Combinatorial Successive Asign (CSA) algorithm, which is the first algorithm that can identify the best action even when the size of the action class is exponentially large with respect to the number of arms. We show that the upper bound of the probability of error of the CSA algorithm matches a lower bound up to a logarithmic factor in the exponent. Then, we introduce another algorithm named the Minimax Combinatorial Successive Accepts and Rejects (Minimax-CombSAR) algorithm for the case where the size of the action class is polynomial, and show that it is optimal, which matches a lower bound. Finally, we experimentally compare the algorithms with previous methods and show that our algorithm performs better.

Thompson Sampling for Real-Valued Combinatorial Pure Exploration of Multi-Armed Bandit

We study the real-valued combinatorial pure exploration of the multi-armed bandit (R-CPE-MAB) problem. In R-CPE-MAB, a player is given $d$ stochastic arms, and the reward of each arm $s\in\{1, \ldots, d\}$ follows an unknown distribution with mean $\mu_s$. In each time step, a player pulls a single arm and observes its reward. The player's goal is to identify the optimal \emph{action} $\boldsymbol{\pi}^{*} = \argmax_{\boldsymbol{\pi} \in \mathcal{A}} \boldsymbol{\mu}^{\top}\boldsymbol{\pi}$ from a finite-sized real-valued \emph{action set} $\mathcal{A}\subset \mathbb{R}^{d}$ with as few arm pulls as possible. Previous methods in the R-CPE-MAB assume that the size of the action set $\mathcal{A}$ is polynomial in $d$. We introduce an algorithm named the Generalized Thompson Sampling Explore (GenTS-Explore) algorithm, which is the first algorithm that can work even when the size of the action set is exponentially large in $d$. We also introduce a novel problem-dependent sample complexity lower bound of the R-CPE-MAB problem, and show that the GenTS-Explore algorithm achieves the optimal sample complexity up to a problem-dependent constant factor.

Combinatorial Pure Exploration of Multi-Armed Bandit with a Real Number Action Class

The combinatorial pure exploration (CPE) in the stochastic multi-armed bandit setting (MAB) is a well-studied online decision-making problem: A player wants to find the optimal \emph{action} $\boldsymbol{\pi}^*$ from \emph{action class} $\mathcal{A}$, which is a collection of subsets of arms with certain combinatorial structures. Though CPE can represent many combinatorial structures such as paths, matching, and spanning trees, most existing works focus only on binary action class $\mathcal{A}\subseteq\{0, 1\}^d$ for some positive integer $d$. This binary formulation excludes important problems such as the optimal transport, knapsack, and production planning problems. To overcome this limitation, we extend the binary formulation to real, $\mathcal{A}\subseteq\mathbb{R}^d$, and propose a new algorithm. The only assumption we make is that the number of actions in $\mathcal{A}$ is polynomial in $d$. We show an upper bound of the sample complexity for our algorithm and the action class-dependent lower bound for R-CPE-MAB, by introducing a quantity that characterizes the problem's difficulty, which is a generalization of the notion \emph{width} introduced in Chen et al.[2014].

Robust computation of optimal transport by $β$-potential regularization

Optimal transport (OT) has become a widely used tool in the machine learning field to measure the discrepancy between probability distributions. For instance, OT is a popular loss function that quantifies the discrepancy between an empirical distribution and a parametric model. Recently, an entropic penalty term and the celebrated Sinkhorn algorithm have been commonly used to approximate the original OT in a computationally efficient way. However, since the Sinkhorn algorithm runs a projection associated with the Kullback-Leibler divergence, it is often vulnerable to outliers. To overcome this problem, we propose regularizing OT with the \beta-potential term associated with the so-called $\beta$-divergence, which was developed in robust statistics. Our theoretical analysis reveals that the $\beta$-potential can prevent the mass from being transported to outliers. We experimentally demonstrate that the transport matrix computed with our algorithm helps estimate a probability distribution robustly even in the presence of outliers. In addition, our proposed method can successfully detect outliers from a contaminated dataset