Bayes meets Bernstein at the Meta Level: an Analysis of Fast Rates in Meta-Learning with PAC-Bayes

Bernstein's condition is a key assumption that guarantees fast rates in machine learning. For example, the Gibbs algorithm with prior $\pi$ has an excess risk in $O(d_{\pi}/n)$, as opposed to the standard $O(\sqrt{d_{\pi}/n})$, where $n$ denotes the number of observations and $d_{\pi}$ is a complexity parameter which depends on the prior $\pi$. In this paper, we examine the Gibbs algorithm in the context of meta-learning, i.e., when learning the prior $\pi$ from $T$ tasks (with $n$ observations each) generated by a meta distribution. Our main result is that Bernstein's condition always holds at the meta level, regardless of its validity at the observation level. This implies that the additional cost to learn the Gibbs prior $\pi$, which will reduce the term $d_\pi$ across tasks, is in $O(1/T)$, instead of the expected $O(1/\sqrt{T})$. We further illustrate how this result improves on standard rates in three different settings: discrete priors, Gaussian priors and mixture of Gaussians priors.

The Survival Bandit Problem

We study the survival bandit problem, a variant of the multi-armed bandit problem introduced in an open problem by Perotto et al. (2019), with a constraint on the cumulative reward; at each time step, the agent receives a (possibly negative) reward and if the cumulative reward becomes lower than a prespecified threshold, the procedure stops, and this phenomenon is called ruin. This is the first paper studying a framework where the ruin might occur but not always. We first discuss that a sublinear regret is unachievable under a naive definition of the regret. Next, we provide tight lower bounds on the probability of ruin (as well as matching policies). Based on this lower bound, we define the survival regret as an objective to minimize and provide a policy achieving a sublinear survival regret (at least in the case of integral rewards) when the time horizon $T$ is known.