  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.
