A.1 Proof of Proposition 12 Proposition 1 The historical contrastive instance discrimination (HCID) can be modelled as a3 maximum likelihood problem optimized via Expectation Maximization.4 Maximum likelihood (ML) is a concept to describe the theoretic insights of clustering algorithms.6 PN n=1 Z(kn) = 1), and the last step of derivation13 employs Jensen's inequality [6, 7, 4]. Z(kn) log p(xq,kn; θE) (5) Expectation step focuses on estimating the posterior probability p(kn; xq,θE). We first gener-17 ate keys by a historical encoder: kt mn = Et m(xt), and xt Xtgt. Then, We calculate18 p(kn; xq,θE) = p(kt mn; xq,θE) = 1 (xq,kt mn), where 1 (xq,kt mn) = 1 if both belong to the19 positive pair; otherwise, 1 (xq,kt mn) = 0.20 Please note the notation "t m" shows that the k is encoded by a historical encoder.21

We address the consistency of a kernel ridge regression estimate of the conditional mean embedding (CME), which is an embedding of the conditional distribution of Y given X into a target reproducing kernel Hilbert space HY . The CME allows us to take conditional expectations of target RKHS functions, and has been employed in nonparametric causal and Bayesian inference. We address the misspecified setting, where the target CME is in the space of Hilbert-Schmidt operators acting from an input interpolation space between HX and L2, to HY . This space of operators is shown to be isomorphic to a newly defined vector-valued interpolation space. Using this isomorphism, we derive a novel and adaptive statistical learning rate for the empirical CME estimator under the misspecified setting. Our analysis reveals that our rates match the optimal O(logn/n) rates without assuming HY to be finite dimensional. We further establish a lower bound on the learning rate, which shows that the obtained upper bound is optimal.

Approximate Bayesian inference for neural networks is considered a robust alternative to standard training, often providing good performance on out-of-distribution data. However, Bayesian neural networks (BNNs) with high-fidelity approximate inference via full-batch Hamiltonian Monte Carlo achieve poor generalization under covariate shift, even underperforming classical estimation. We explain this surprising result, showing how a Bayesian model average can in fact be problematic under covariate shift, particularly in cases where linear dependencies in the input features cause a lack of posterior contraction. We additionally show why the same issue does not affect many approximate inference procedures, or classical maximum a-posteriori (MAP) training. Finally, we propose novel priors that improve the robustness of BNNs to many sources of covariate shift.

Out of the rich family of generalized linear bandits, perhaps the most well studied ones are logistic bandits that are used in problems with binary rewards: for instance, when the learner aims to maximize the profit over a user that can select one of two possible outcomes (e.g., 'click' vs'no-click'). Despite remarkable recent progress and improved algorithms for logistic bandits, existing works do not address practical situations where the number of outcomes that can be selected by the user is larger than two (e.g., 'click', 'show me later', 'never show again', 'no click'). In this paper, we study such an extension. We use multinomial logit (MNL) to model the probability of each one of K+1 2possible outcomes (+1 stands for the'not click' outcome): we assume that for a learner's action xt, the user selects one of K +1 2outcomes, say outcome i, with a MNL probabilistic model with corresponding unknown parameter θ i. Each outcome i is also associated with a revenue parameter ρi and the goal is to maximize the expected revenue. For this problem, we present MNL-UCB, an upper confidence bound (UCB)-based algorithm, that achieves regret O(dK T) with small dependency on problemdependent constants that can otherwise be arbitrarily large and lead to loose regret bounds. We present numerical simulations that corroborate our theoretical results.