We consider online sequential decision problems where an agent must balance exploration and exploitation. We derive a set of Bayesian'optimistic' policies which, in the stochastic multi-armed bandit case, includes the Thompson sampling policy. We provide a new analysis showing that any algorithm producing policies in the optimistic set enjoys O ( AT) Bayesian regret for a problem with A actions after T rounds. We extend the regret analysis for optimistic policies to bilinear saddle-point problems which include zero-sum matrix games and constrained bandits as special cases. In this case we show that Thompson sampling can produce policies outside of the optimistic set and suffer linear regret in some instances. Finding a policy inside the optimistic set amounts to solving a convex optimization problem and we call the resulting algorithm'variational Bayesian optimistic sampling' (VBOS). The procedure works for any posteriors, i.e., it does not require the posterior to have any special properties, such as log-concavity, unimodality, or smoothness. The variational view of the problem has many useful properties, including the ability to tune the exploration-exploitation tradeoff, add regularization, incorporate constraints, and linearly parameterize the policy.

Machine learning (ML) models often exhibit unexpected failures once deployed in the "wild".

"shortcut learning": when a predictor relies on shortcuts, i.e., spurious correlations between the inputs

"shortcut learning": when a predictor relies on shortcuts, i.e., spurious correlations between the inputs

We also generalize our results to the settings of bounded size interventions and node-dependent interventional costs. For all the above settings, we provide the first known provable algorithms for efficiently computing (near)-optimal verifying sets on general graphs.

We also generalize our results to the settings of bounded size interventions and node-dependent interventional costs. For all the above settings, we provide the first known provable algorithms for efficiently computing (near)-optimal verifying sets on general graphs.

We do not foresee direct negative societal impact from the current work. Also, one may argue that our work is catalytic to enhancing the'disclosure risk' of individuals, i.e. an adversary might be able to make accurate Granted, no existing privacy frameworks can guard against this. We prove its ergodicity in Theorem S-3.1, which implies Theorem 3.3 . The model is such that the set { x: f ( x |) > 0 } does not depend on . The Metropolis-within-Gibbs sampler is aperiodic by construction, since some proposals can be rejected.