We study the sample complexity of learning neural networks, by providing new bounds on their Rademacher complexity assuming norm constraints on the parameter matrix of each layer. Compared to previous work, these complexity bounds have improved dependence on the network depth, and under some additional assumptions, are fully independent of the network size (both depth and width). These results are derived using some novel techniques, which may be of independent interest.

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Moulin [1980] characterizes the single-facility, deterministic strategy-proof mechanisms for social choice with single-peaked preferences as the set of generalized median rules. In contrast, we have only a limited understanding of multi-facility strategy-proof mechanisms, and recent work has shown negative worst case results for social cost. Our goal is to design strategy-proof, multi-facility mechanisms that minimize expected social cost. We first give a PAC learnability result for the class of multi-facility generalized median rules, and utilize neural networks to learn mechanisms from this class. Even in the absence of characterization results, we develop a computational procedure for learning almost strategy-proof mechanisms that are as good as or better than benchmarks from the literature, such as the best percentile and dictatorial rules.

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Partially Observable Markov Decision Processes (POMDPs) are a natural and general model in reinforcement learning that take into account the agent's uncertainty about its current state. In the literature on POMDPs, it is customary to assume access to a planning oracle that computes an optimal policy when the parameters are known, even though the problem is known to be computationally hard. Almost all existing planning algorithms either run in exponential time, lack provable performance guarantees, or require placing strong assumptions on the transition dynamics under every possible policy. In this work, we revisit the planning problem and ask: Are there natural and well-motivated assumptions that make planning easy?Our main result is a quasipolynomial-time algorithm for planning in (one-step) observable POMDPs. Specifically, we assume that well-separated distributions on states lead to wellseparated distributions on observations, and thus the observations are at least somewhat informative in each step. Crucially, this assumption places no restrictions on the transition dynamics of the POMDP; nevertheless, it implies that near-optimal policies admit quasi-succinct descriptions, which is not true in general (under standard hardness assumptions). Our analysis is based on new quantitative bounds for filter stability -i.e. the rate at which an optimal filter for the latent state forgets its initialization. Furthermore, we prove matching hardness for planning in observable POMDPs under the Exponential Time Hypothesis.