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We consider the problem of estimating the discrete clustering structures under the sub-Gaussian mixture model. Our main results establish a hidden integrality property of a semidefinite programming (SDP) relaxation for this problem: while the optimal solution to the SDP is not integer-valued in general, its estimation error can be upper bounded by that of an idealized integer program. The error of the integer program, and hence that of the SDP, are further shown to decay exponentially in the signal-to-noise ratio. In addition, we show that the SDP relaxation is robust under the semirandom setting in which an adversary can modify the data generated from the mixture model. In particular, we generalize the hidden integrality property to the semirandom model and thereby show that SDP achieves the optimal error bound in this setting. These results together highlight the “global-to-local” mechanism that drives the performance of the SDP relaxation. To the best of our knowledge, our result is the first exponentially decaying error bound for convex relaxations of mixture models. A corollary of our results shows that in certain regimes, the SDP solutions are in fact integral and exact. More generally, our results establish sufficient conditions for the SDP to correctly recover the cluster memberships of [Formula: see text] fraction of the points for any [Formula: see text]. As a special case, we show that under the [Formula: see text]-dimensional stochastic ball model, SDP achieves nontrivial (sometimes exact) recovery when the center separation is as small as [Formula: see text], which improves upon previous exact recovery results that require constant separation. 

We consider reinforcement learning (RL) in episodic MDPs with adversarial full-information reward feedback and unknown fixed transition kernels. We propose two model-free policy optimization algorithms, POWER and POWER++, and establish guarantees for their dynamic regret. Compared with the classical notion of static regret, dynamic regret is a stronger notion as it explicitly accounts for the non-stationarity of environments. The dynamic regret attained by the proposed algorithms interpolates between different regimes of non-stationarity, and moreover satisfies a notion of adaptive (near-)optimality, in the sense that it matches the (near-)optimal static regret under slow-changing environments. The dynamic regret bound features two components, one arising from exploration, which deals with the uncertainty of transition kernels, and the other arising from adaptation, which deals with non-stationary environments. Specifically, we show that POWER++ improves over POWER on the second component of the dynamic regret by actively adapting to non-stationarity through prediction. To the best of our knowledge, our work is the first dynamic regret analysis of model-free RL algorithms in non-stationary environments.