The End of Optimism? An Asymptotic Analysis of Finite-Armed Linear Bandits
This reveals a fundamental limitation in standard sequential optimization tools used in areas like generalized linear bandits and reinforcement learning, indicating an incremental but critical theoretical insight.
The paper tackles the asymptotic instance-dependent regret in finite-armed linear bandits, showing that optimism-based algorithms and Thompson sampling cannot achieve the optimal rate and can be arbitrarily far from optimal in simple cases.
Stochastic linear bandits are a natural and simple generalisation of finite-armed bandits with numerous practical applications. Current approaches focus on generalising existing techniques for finite-armed bandits, notably the optimism principle and Thompson sampling. While prior work has mostly been in the worst-case setting, we analyse the asymptotic instance-dependent regret and show matching upper and lower bounds on what is achievable. Surprisingly, our results show that no algorithm based on optimism or Thompson sampling will ever achieve the optimal rate, and indeed, can be arbitrarily far from optimal, even in very simple cases. This is a disturbing result because these techniques are standard tools that are widely used for sequential optimisation. For example, for generalised linear bandits and reinforcement learning.