The alternating gradient descent (AGD) is a simple but popular algorithm which has been applied to problems in optimization, machine learning, data ming, and signal processing, etc. The algorithm updates two blocks of variables in an alternating manner, in which a gradient step is taken on one block, while keeping the remaining block fixed. When the objective function is nonconvex, it is well-known the AGD converges to the first-order stationary solution with a global sublinear rate. In this paper, we show that a variant of AGD-type algorithms will not be trapped by "bad" stationary solutions such as saddle points and local maximum points. In particular, we consider a smooth unconstrained optimization problem, and propose a perturbed AGD (PA-GD) which converges (with high probability) to the set of second-order stationary solutions (SS2) with a global sublinear rate. To the best of our knowledge, this is the first alternating type algorithm which takes @math iterations to achieve SS2 with high probability [where polylog @math is polynomial of the logarithm of dimension @math of the problem].
There has been a line of work on stochastic gradient descent algorithms, where properly scaled Gaussian noise is added to the iterates of the gradient at each time [also known as stochastic gradient Langevin dynamics, (SGLD)]. Some theoretical works have pointed out that SGLD not only converges to the local minimum points asymptotically but also may escape from local minima @cite_30 @cite_16 . Unfortunately, these algorithms require a large number of iterations with @math steps to achieve the optimal point. There are fruitful results that show some carefully designed algorithms can escape from strict saddle point efficiently, such as negative-curvature-originated-from noise (Neon) @cite_20 , Neon2 @cite_32 , Neon @math @cite_15 and gradient descent with one-step escaping (GOSE) @cite_3 . The Neon-type of algorithms utilizes the stochastic first-order updates to find the negative curvature direction, and GOSE just needs one negative curvature descent step with calculation of eigenvectors when the iterates of the algorithm are near the saddle point for saving the computational burden.
@cite_30 @cite_32 @cite_3 @cite_15 @cite_16 @cite_20
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We study the Stochastic Gradient Langevin Dynamics (SGLD) algorithm for non-convex optimization. The algorithm performs stochastic gradient descent, where in each step it injects appropriately scaled Gaussian noise to the update. We analyze the algorithm's hitting time to an arbitrary subset of the parameter space. Two results follow from our general theory: First, we prove that for empirical risk minimization, if the empirical risk is point-wise close to the (smooth) population risk, then the algorithm achieves an approximate local minimum of the population risk in polynomial time, escaping suboptimal local minima that only exist in the empirical risk. Second, we show that SGLD improves on one of the best known learnability results for learning linear classifiers under the zero-one loss. Stochastic Gradient Langevin Dynamics (SGLD) is a popular variant of Stochastic Gradient Descent, where properly scaled isotropic Gaussian noise is added to an unbiased estimate of the gradient at each iteration. This modest change allows SGLD to escape local minima and suffices to guarantee asymptotic convergence to global minimizers for sufficiently regular non-convex objectives (Gelfand and Mitter, 1991). The present work provides a nonasymptotic analysis in the context of non-convex learning problems, giving finite-time guarantees for SGLD to find approximate minimizers of both empirical and population risks. As in the asymptotic setting, our analysis relates the discrete-time SGLD Markov chain to a continuous-time diffusion process. A new tool that drives the results is the use of weighted transportation cost inequalities to quantify the rate of convergence of SGLD to a stationary distribution in the Euclidean @math -Wasserstein distance. We present a unified framework to analyze the global convergence of Langevin dynamics based algorithms for nonconvex finite-sum optimization with @math 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 @math and @math stochastic gradient evaluations respectively, where @math is the problem dimension, and @math 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 @math 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. Negative curvature descent (NCD) method has been utilized to design deterministic or stochastic algorithms for non-convex optimization aiming at finding second-order stationary points or local minima. In existing studies, NCD needs to approximate the smallest eigen-value of the Hessian matrix with a sufficient precision (e.g., ϵ2≪1) in order to achieve a sufficiently accurate second-order stationary solution (i.e., λmin(∇2f( ))≥−ϵ2). One issue with this approach is that the target precision ϵ2 is usually set to be very small in order to find a high quality solution, which increases the complexity for computing a negative curvature. To address this issue, we propose an adaptive NCD to allow for an adaptive error dependent on the current gradient's magnitude in approximating the smallest eigen-value of the Hessian, and to encourage competition between a noisy NCD step and gradient descent step. We consider the applications of the proposed adaptive NCD for both deterministic and stochastic non-convex optimization, and demonstrate that it can help reduce the the overall complexity in computing the negative curvatures during the course of optimization without sacrificing the iteration complexity.