We show that Thompson sampling has a Bayesian regret of at most $\tilde O(\sqrt{n})$ for $1$-dimensional bandit convex optimisation where $n$ is the time horizon and no assumptions are made on the loss function beyond convexity, boundedness and a mild Lipschitz assumption. For general high-dimensional problems we show that Thompson sampling can fail catastrophically. More positively, we show that Thompson sampling has Bayesian regret of $\tilde O(d^{2.5} \sqrt{n})$ for generalised linear bandits with an unknown convex monotone link function. Lastly, we prove that the standard information-theoretic machinery can never give a bound on the regret in the general case that improveson the best known bound of $\tilde O(d^{1.5} \sqrt{n})$.

We establish a lower bound on the eluder dimension in generalised linear model classes, showing that standard eluder dimension-based analysis cannot lead to first-order regret bounds. To address this, we introduce a localisation method for the eluder dimension; our analysis immediately recovers and improves on classic results for Bernoulli bandits, and allows for the first genuine first-order bounds for finite-horizon reinforcement learning tasks with bounded cumulative returns.