In this paper, we study the conditional stochastic optimization (CSO) problem which covers a variety of applications including portfolio selection, reinforcement learning, robust learning, causal inference, etc. The sample-averaged gradient of the CSO objective is biased due to its nested structure, and therefore requires a high sample complexity for convergence. We introduce a general stochastic extrapolation technique that effectively reduces the bias. We show that for nonconvex smooth objectives, combining this extrapolation with variance reduction techniques can achieve a significantly better sample complexity than the existing bounds. Additionally, we develop new algorithms for the finite-sum variant of the CSO problem that also significantly improve upon existing results. Finally, we believe that our debiasing technique has the potential to be a useful tool for addressing similar challenges in other stochastic optimization problems.

Motion Compensation We compare our method to the traditional motion-compensated coding378 approach that forms the core of inter-picture coding in well established compression standards such379 as MPEG. Block matching is an essential component of these standards, allowing the compression of380 video content by up to three orders of magnitude with moderate loss of information. For each block381 in a frame, typical coders search for the most similar spatially displaced block in the previous frame382 (typically measured with MSE), and communicate the displacement coordinates to allow prediction383 of frame content by translating blocks of the (already transmitted) previous frame. We implemented384 a "diamond search" algorithm [29] operating on blocks of 8 8 pixels, with a maximal search385 distance of 8 pixels which balances accuracy of motion estimates and speed of estimation (the search386 step is computationally intensive). We use the estimated displacements to perform causal motion387 compensation (cMC), using displacement vectors estimated from the previous two observed frames388 (xt 1 and xt) to predict the next frame (xt+1) rather than the current one (as in MPEG).389

As trained intelligent systems become increasingly pervasive, multi-agent learning has emerged as a popular framework for studying complex interactions between autonomous agents. Yet, a formal understanding of how and when learners in heterogeneous environments benefit from sharing their respective experiences is still in its infancy. In this paper, we seek answers to these questions in the context of linear contextual bandits. We present a novel distributed learning algorithm based on the upper confidence bound (UCB) algorithm, which we refer to as H-LINUCB, wherein agents cooperatively minimize the group regret under the coordination of a central server. In the setting where the level of heterogeneity or dissimilarity across the environments is known to the agents, we show that H-LINUCB is provably optimal in regimes where the tasks are highly similar or highly dissimilar.

In this supplementary material, we provide more details regarding implementation details in Appendix B, more analysis of ERDA in Appendix C, full experimental results in Appendix D, and studies on parameters in Appendix E.