Parallel implementations of stochastic gradient descent (SGD) have received significant research attention, thanks to its excellent scalability properties.

In this supplementary material, we provide "full versions" of Sections 2-4 from the main submission, Fact 2.5 (Uniform bound on entries of Gaussian vector) . For g N (0, Id), g/ nullg null is identical in distribution to v . We can expand the expectation and apply Fact B.2 to get We will also need the following stability result for affine linear thresholds. Putting all of these ingredients together, we can now complete the proof of the main Lemma B.1 of By Lemma B.6 applied to the projection of f to the two-dimensional This notion is motivated by Lemma 4.4 in Section C.1 where we study the critical points We first collect some elementary consequences of closeness. Suppose < 2/ 2. If ( v In the rest of the paper we will take to be small, so Lemma 3.3 will always apply.

We consider the problem of spherical Gaussian Mixture models with $k \geq 3$ components when the components are well separated. A fundamental previous result established that separation of $\Omega(\sqrt{\log k})$ is necessary and sufficient for identifiability of the parameters with polynomial sample complexity (Regev and Vijayaraghavan, 2017). We show that $\tilde{O}(kd/\epsilon^2)$ samples suffice, closing the gap from polynomial to linear, and thus giving the first sample-optimal upper bound for the parameter estimation of well-separated Gaussian mixtures (up to logarithmic factors). We accomplish this by proving a new result for the Expectation-Maximization (EM) algorithm: we show that EM converges locally, under separation $\Omega(\sqrt{\log k})$. The previous best-known guarantee required $\Omega(\sqrt{k})$ separation (Yan, et al., 2017). Unlike prior work, our results do not assume or use prior knowledge of the (potentially different) mixing weights or variances of the Gaussian components. Furthermore, our results show that the finite-sample error of EM does not depend on non-universal quantities such as pairwise distances between means of Gaussian components.

This paper is concerned with the squared F(robenius)-norm regularized factorization form for noisy low-rank matrix recovery problems. Under a suitable assumption on the restricted condition number of the Hessian matrix of the loss function, we establish an error bound to the true matrix for those local minima whose ranks are not more than the rank of the true matrix. Then, for the least squares loss function, we achieve the KL property of exponent 1/2 for the F-norm regularized factorization function over its global minimum set under a restricted strong convexity assumption. These theoretical findings are also confirmed by applying an accelerated alternating minimization method to the F-norm regularized factorization problem.