Adversarial Bandits Robust to $S$-Switch Regret

We study the adversarial bandit problem under $S$ number of switching best arms for unknown $S$. For handling this problem, we adopt the master-base framework using the online mirror descent method (OMD). We first provide a master-base algorithm with basic OMD, achieving $\tilde{O}(S^{1/2}K^{1/3}T^{2/3})$. For improving the regret bound with respect to $T$, we propose to use adaptive learning rates for OMD to control variance of loss estimators, and achieve $\tilde{O}(\min\{\mathbb{E}[\sqrt{SKT\rho_T(h^\dagger)}],S\sqrt{KT}\})$, where $\rho_T(h^\dagger)$ is a variance term for loss estimators.

Rotting infinitely many-armed bandits

We consider the infinitely many-armed bandit problem with rotting rewards, where the mean reward of an arm decreases at each pull of the arm according to an arbitrary trend with maximum rotting rate $\varrho=o(1)$. We show that this learning problem has an $\Omega(\max\{\varrho^{1/3}T,\sqrt{T}\})$ worst-case regret lower bound where $T$ is the horizon time. We show that a matching upper bound $\tilde{O}(\max\{\varrho^{1/3}T,\sqrt{T}\})$, up to a poly-logarithmic factor, can be achieved by an algorithm that uses a UCB index for each arm and a threshold value to decide whether to continue pulling an arm or remove the arm from further consideration, when the algorithm knows the value of the maximum rotting rate $\varrho$. We also show that an $\tilde{O}(\max\{\varrho^{1/3}T,T^{3/4}\})$ regret upper bound can be achieved by an algorithm that does not know the value of $\varrho$, by using an adaptive UCB index along with an adaptive threshold value.

Scheduling Servers with Stochastic Bilinear Rewards

In this paper we study a multi-class, multi-server queueing system with stochastic rewards of job-server assignments following a bilinear model in feature vectors representing jobs and servers. Our goal is regret minimization against an oracle policy that has a complete information about system parameters. We propose a scheduling algorithm that uses a linear bandit algorithm along with dynamic allocation of jobs to servers. For the baseline setting, in which mean job service times are identical for all jobs, we show that our algorithm has a sub-linear regret, as well as a sub-linear bound on the mean queue length, in the horizon time. We further show that similar bounds hold under more general assumptions, allowing for non-identical mean job service times for different job classes and a time-varying set of server classes. We also show that better regret and mean queue length bounds can be guaranteed by an algorithm having access to traffic intensities of job classes. We present results of numerical experiments demonstrating how regret and mean queue length of our algorithms depend on various system parameters and compare their performance against a previously proposed algorithm using synthetic randomly generated data and a real-world cluster computing data trace.