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Learning with noisy labels is a common challenge in supervised learning. Existing approaches often require practitioners to specify noise rates, i.e., a set of parameters controlling the severity of label noises in the problem, and the specifications are either assumed to be given or estimated using additional steps. In this work, we introduce a new family of loss functions that we name as peer loss functions, which enables learning from noisy labels and does not require a priori specification of the noise rates. Peer loss functions work within the standard empirical risk minimization (ERM) framework. We show that, under mild conditions, performing ERM with peer loss functions on the noisy data leads to the optimal or a near-optimal classifier as if performing ERM over the clean training data, which we do not have access to. We pair our results with an extensive set of experiments. Peer loss provides a way to simplify model development when facing potentially noisy training labels, and can be promoted as a robust candidate loss function in such situations. 

We prove quantitative convergence rates at which discrete Langevin-like processes converge to the invariant distribution of a related stochastic differential equation. We study the setup where the additive noise can be non-Gaussian and state-dependent and the potential function can be non-convex. We show that the key properties of these processes depend on the potential function and the second moment of the additive noise. We apply our theoretical findings to studying the convergence of Stochastic Gradient Descent (SGD) for non-convex problems and corroborate them with experiments using SGD to train deep neural networks on the CIFAR-10 dataset. 

Thompson sampling for multi-armed bandit problems is known to enjoy favorable performance in both theory and practice. However, its wider deployment is restricted due to a significant computational limitation: the need for samples from posterior distributions at every iteration. In practice, this limitation is alleviated by making use of approximate sampling methods, yet provably incorporating approximate samples into Thompson Sampling algorithms remains an open problem. In this work we address this by proposing two efficient Langevin MCMC algorithms tailored to Thompson sampling. The resulting approximate Thompson Sampling algorithms are efficiently implementable and provably achieve optimal instance-dependent regret for the Multi-Armed Bandit (MAB) problem. To prove these results we derive novel posterior concentration bounds and MCMC convergence rates for log-concave distributions which may be of independent interest.