Secure Best Arm Identification in the Presence of a Copycat

TL;DR

This paper introduces a secure algorithm for best arm identification in stochastic linear bandits that balances high identification accuracy with privacy against an observer, using coded arms without cryptography.

Contribution

It proposes a novel secure algorithm using coded arms that maintains high error exponents while keeping the best arm secret, without cryptographic methods.

Findings

Achieves an error exponent of  d with coded arms

Reveals almost no information about the best arm

Outperforms naive uniform play in security and efficiency

Abstract

Consider the problem of best arm identification with a security constraint. Specifically, assume a setup of stochastic linear bandits with $K$ arms of dimension $d$. In each arm pull, the player receives a reward that is the sum of the dot product of the arm with an unknown parameter vector and independent noise. The player's goal is to identify the best arm after $T$ arm pulls. Moreover, assume a copycat Chloe is observing the arm pulls. The player wishes to keep Chloe ignorant of the best arm. While a minimax--optimal algorithm identifies the best arm with an $\Omega\left(\frac{T}{\log(d)}\right)$ error exponent, it easily reveals its best-arm estimate to an outside observer, as the best arms are played more frequently. A naive secure algorithm that plays all arms equally results in an $\Omega\left(\frac{T}{d}\right)$ exponent. In this paper, we propose a secure algorithm that plays with \emph{coded arms}. The algorithm does not require any key or cryptographic primitives, yet achieves an $\Omega\left(\frac{T}{\log^2(d)}\right)$ exponent while revealing almost no information on the best arm.