We present a unified framework to analyze the global convergence of Langevin dynamics based algorithms for nonconvex finite-sum optimization with $n$ component functions. At the core of our analysis is a direct analysis of the ergodicity of the numerical approximations to Langevin dynamics, which leads to faster convergence rates. Specifically, we show that gradient Langevin dynamics (GLD) and stochastic gradient Langevin dynamics (SGLD) converge to the almost minimizer within $\tilde O\big(nd/(\lambda\epsilon) \big)$ and $\tilde O\big(d^7/(\lambda^5\epsilon^5) \big)$ stochastic gradient evaluations respectively, where $d$ is the problem dimension, and $\lambda$ is the spectral gap of the Markov chain generated by GLD. Both of the results improve upon the best known gradient complexity results. Furthermore, for the first time we prove the global convergence guarantee for variance reduced stochastic gradient Langevin dynamics (VR-SGLD) to the almost minimizer after $\tilde O\big(\sqrt{n}d^5/(\lambda^4\epsilon^{5/2})\big)$ stochastic gradient evaluations, which outperforms the gradient complexities of GLD and SGLD in a wide regime. Our theoretical analyses shed some light on using Langevin dynamics based algorithms for nonconvex optimization with provable guarantees.

We consider the robust phase retrieval problem of recovering the unknown signal from the magnitude-only measurements, where the measurements can be contaminated by both sparse arbitrary corruption and bounded random noise. We propose a new nonconvex algorithm for robust phase retrieval, namely Robust Wirtinger Flow to jointly estimate the unknown signal and the sparse corruption. We show that our proposed algorithm is guaranteed to converge linearly to the unknown true signal up to a minimax optimal statistical precision in such a challenging setting. Compared with existing robust phase retrieval methods, we achieve an optimal sample complexity of $O(n)$ in both noisy and noise-free settings. Thorough experiments on both synthetic and real datasets corroborate our theory.

We propose a nonconvex estimator for joint multivariate regression and precision matrix estimation in the high dimensional regime, under sparsity constraints. A gradient descent algorithm with hard thresholding is developed to solve the nonconvex estimator, and it attains a linear rate of convergence to the true regression coefficients and precision matrix simultaneously, up to the statistical error. Compared with existing methods along this line of research, which have little theoretical guarantee, the proposed algorithm not only is computationally much more efficient with provable convergence guarantee, but also attains the optimal finite sample statistical rate up to a logarithmic factor. Thorough experiments on both synthetic and real datasets back up our theory.