Different users of machine learning methods require different explanations, depending on their goals. To make machine learning accountable to society, one important goal is to get actionable options for recourse, which allow an affected user to change the decision f(x) of a machine learning system by making limited changes to its input x. We formalize this by providing a general definition of recourse sensitivity, which needs to be instantiated with a utility function that describes which changes to the decisions are relevant to the user. This definition applies to local attribution methods, which attribute an importance weight to each input feature. It is often argued that such local attributions should be robust, in the sense that a small change in the input x that is being explained, should not cause a large change in the feature weights. However, we prove formally that it is in general impossible for any single attribution method to be both recourse sensitive and robust at the same time. It follows that there must always exist counterexamples to at least one of these properties. We provide such counterexamples for several popular attribution methods, including LIME, SHAP, Integrated Gradients and SmoothGrad. Our results also cover counterfactual explanations, which may be viewed as attributions that describe a perturbation of x. We further discuss possible ways to work around our impossibility result, for instance by allowing the output to consist of sets with multiple attributions, and we provide sufficient conditions for specific classes of continuous functions to be recourse sensitive. Finally, we strengthen our impossibility result for the restricted case where users are only able to change a single attribute of x, by providing an exact characterization of the functions f to which impossibility applies.

Top two algorithms arose as an adaptation of Thompson sampling to best arm identification in multi-armed bandit models for parametric families of arms. They select the next arm to sample from by randomizing among two candidate arms, a leader and a challenger. Despite their good empirical performance, theoretical guarantees for fixed-confidence best arm identification have only been obtained when the arms are Gaussian with known variances. In this paper, we provide a general analysis of top-two methods, which identifies desirable properties of the leader, the challenger, and the (possibly non-parametric) distributions of the arms. As a result, we obtain theoretically supported top-two algorithms for best arm identification with bounded distributions. Our proof method demonstrates in particular that the sampling step used to select the leader inherited from Thompson sampling can be replaced by other choices, like selecting the empirical best arm.

We consider a stochastic bandit problem with a possibly infinite number of arms. We write $p^*$ for the proportion of optimal arms and $\Delta$ for the minimal mean-gap between optimal and sub-optimal arms. We characterize the optimal learning rates both in the cumulative regret setting, and in the best-arm identification setting in terms of the problem parameters $T$ (the budget), $p^*$ and $\Delta$. For the objective of minimizing the cumulative regret, we provide a lower bound of order $\Omega(\log(T)/(p^*\Delta))$ and a UCB-style algorithm with matching upper bound up to a factor of $\log(1/\Delta)$. Our algorithm needs $p^*$ to calibrate its parameters, and we prove that this knowledge is necessary, since adapting to $p^*$ in this setting is impossible. For best-arm identification we also provide a lower bound of order $\Omega(\exp(-cT\Delta^2p^*))$ on the probability of outputting a sub-optimal arm where $c>0$ is an absolute constant. We also provide an elimination algorithm with an upper bound matching the lower bound up to a factor of order $\log(T)$ in the exponential, and that does not need $p^*$ or $\Delta$ as parameter. Our results apply directly to the three related problems of competing against the $j$-th best arm, identifying an $\epsilon$ good arm, and finding an arm with mean larger than a quantile of a known order.

We study generalized Bayesian inference under misspecification, i.e. when the model is ‘wrong but useful’. Generalized Bayes equips the likelihood with a learning rate $\eta$. We show that for generalized linear models (GLMs), $\eta$-generalized Bayes concentrates around the best approximation of the truth within the model for specific $\eta eq 1$, even under severely misspecified noise, as long as the tails of the true distribution are exponential. We derive MCMC samplers for generalized Bayesian lasso and logistic regression and give examples of both simulated and real-world data in which generalized Bayes substantially outperforms standard Bayes.

We investigate and provide new insights on the sampling rule called Top-Two Thompson Sampling (TTTS). In particular, we justify its use for fixed-confidence best-arm identification. We further propose a variant of TTTS called Top-Two Transportation Cost (T3C), which disposes of the computational burden of TTTS. As our main contribution, we provide the first sample complexity analysis of TTTS and T3C when coupled with a very natural Bayesian stopping rule, for bandits with Gaussian rewards, solving one of the open questions raised by Russo (2016). We also provide new posterior convergence results for TTTS under two models that are commonly used in practice: bandits with Gaussian and Bernoulli rewards and conjugate priors.