Efficient Reinforcement Learning for Routing Jobs in Heterogeneous Queueing Systems

We consider the problem of efficiently routing jobs that arrive into a central queue to a system of heterogeneous servers. Unlike homogeneous systems, a threshold policy, that routes jobs to the slow server(s) when the queue length exceeds a certain threshold, is known to be optimal for the one-fast-one-slow two-server system. But an optimal policy for the multi-server system is unknown and non-trivial to find. While Reinforcement Learning (RL) has been recognized to have great potential for learning policies in such cases, our problem has an exponentially large state space size, rendering standard RL inefficient. In this work, we propose ACHQ, an efficient policy gradient based algorithm with a low dimensional soft threshold policy parameterization that leverages the underlying queueing structure. We provide stationary-point convergence guarantees for the general case and despite the low-dimensional parameterization prove that ACHQ converges to an approximate global optimum for the special case of two servers. Simulations demonstrate an improvement in expected response time of up to ~30% over the greedy policy that routes to the fastest available server.

Greedy $k$-Center from Noisy Distance Samples

We study a variant of the canonical $k$-center problem over a set of vertices in a metric space, where the underlying distances are apriori unknown. Instead, we can query an oracle which provides noisy/incomplete estimates of the distance between any pair of vertices. We consider two oracle models: Dimension Sampling where each query to the oracle returns the distance between a pair of points in one dimension; and Noisy Distance Sampling where the oracle returns the true distance corrupted by noise. We propose active algorithms, based on ideas such as UCB and Thompson sampling developed in the closely related Multi-Armed Bandit problem, which adaptively decide which queries to send to the oracle and are able to solve the $k$-center problem within an approximation ratio of two with high probability. We analytically characterize instance-dependent query complexity of our algorithms and also demonstrate significant improvements over naive implementations via numerical evaluations on two real-world datasets (Tiny ImageNet and UT Zappos50K).