We study the online learning with feedback graphs framework introduced by Mannor and Shamir (2011), in which the feedback received by the online learner is specified by a graph $G$ over the available actions. We develop an algorithm that simultaneously achieves regret bounds of the form: $O(\sqrt{\theta(G) T})$ with adversarial losses; $O(\theta(G)\mathrm{polylog}{T})$ with stochastic losses; and $O(\theta(G)\mathrm{polylog}{T} + \sqrt{\theta(G) C})$ with stochastic losses subject to $C$ adversarial corruptions. Here, $\theta(G)$ is the $clique~covering~number$ of the graph $G$. Our algorithm is an instantiation of Follow-the-Regularized-Leader with a novel regularization that can be seen as a product of a Tsallis entropy component (inspired by Zimmert and Seldin (2019)) and a Shannon entropy component (analyzed in the corrupted stochastic case by Amir et al. (2020)), thus subtly interpolating between the two forms of entropies. One of our key technical contributions is in establishing the convexity of this regularizer and controlling its inverse Hessian, despite its complex product structure.

We study the online learning with feedback graphs framework introduced by Mannor and Shamir (2011), in which the feedback received by the online learner is specified by a graph G over the available actions. We develop an algorithm that simultaneously achieves regret bounds of the form: O(\sqrt{\theta(G) T}) with adversarial losses; O(\theta(G)\mathrm{polylog}{T}) with stochastic losses; and O(\theta(G)\mathrm{polylog}{T} \sqrt{\theta(G) C}) with stochastic losses subject to C adversarial corruptions. Here, \theta(G) is the clique covering number of the graph G . Our algorithm is an instantiation of Follow-the-Regularized-Leader with a novel regularization that can be seen as a product of a Tsallis entropy component (inspired by Zimmert and Seldin (2019)) and a Shannon entropy component (analyzed in the corrupted stochastic case by Amir et al. (2020)), thus subtly interpolating between the two forms of entropies. One of our key technical contributions is in establishing the convexity of this regularizer and controlling its inverse Hessian, despite its complex product structure.

We develop unbiased implicit variational inference (UIVI), a method that expands the applicability of variational inference by defining an expressive variational family. UIVI considers an implicit variational distribution obtained in a hierarchical manner using a simple reparameterizable distribution whose variational parameters are defined by arbitrarily flexible deep neural networks. Unlike previous works, UIVI directly optimizes the evidence lower bound (ELBO) rather than an approximation to the ELBO. We demonstrate UIVI on several models, including Bayesian multinomial logistic regression and variational autoencoders, and show that UIVI achieves both tighter ELBO and better predictive performance than existing approaches at a similar computational cost.