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. We propose a reduction for non-convex optimization that can (1) turn a stationary-point finding algorithm into a local-minimum finding one, and (2) replace the Hessian-vector product computations with only gradient computations. It works both in the stochastic and the deterministic settings, without hurting the algorithm's performance. As applications, our reduction turns Natasha2 into a first-order method without hurting its performance. It also converts SGD, GD, SCSG, and SVRG into local-minimum finding algorithms outperforming some best known results. We propose a family of nonconvex optimization algorithms that are able to save gradient and negative curvature computations to a large extent, and are guaranteed to find an approximate local minimum with improved runtime complexity. At the core of our algorithms is the division of the entire domain of the objective function into small and large gradient regions: our algorithms only perform gradient descent based procedure in the large gradient region, and only perform negative curvature descent in the small gradient region. Our novel analysis shows that the proposed algorithms can escape the small gradient region in only one negative curvature descent step whenever they enter it, and thus they only need to perform at most @math negative curvature direction computations, where @math is the number of times the algorithms enter small gradient regions. For both deterministic and stochastic settings, we show that the proposed algorithms can potentially beat the state-of-the-art local minima finding algorithms. For the finite-sum setting, our algorithm can also outperform the best algorithm in a certain regime. Accelerated gradient (AG) methods are breakthroughs in convex optimization, improving the convergence rate of the gradient descent method for optimization with smooth functions. However, the analysis of AG methods for non-convex optimization is still limited. It remains an open question whether AG methods from convex optimization can accelerate the convergence of the gradient descent method for finding local minimum of non-convex optimization problems. This paper provides an affirmative answer to this question. In particular, we analyze two renowned variants of AG methods (namely Polyak's Heavy Ball method and Nesterov's Accelerated Gradient method) for extracting the negative curvature from random noise, which is central to escaping from saddle points. By leveraging the proposed AG methods for extracting the negative curvature, we present a new AG algorithm with double loops for non-convex optimization this is in contrast to a single-loop AG algorithm proposed in a recent manuscript , which directly analyzed the Nesterov's AG method for non-convex optimization and appeared online on November 29, 2017. However, we emphasize that our work is an independent work, which is inspired by our earlier work and is based on a different novel analysis. , which converges to second-order stationary point @math such that @math and @math with @math iteration complexity, improving that of gradient descent method by a factor of @math and matching the best iteration complexity of second-order Hessian-free methods for non-convex optimization. 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. Two classes of methods have been proposed for escaping from saddle points with one using the second-order information carried by the Hessian and the other adding the noise into the first-order information. The existing analysis for algorithms using noise in the first-order information is quite involved and hides the essence of added noise, which hinder further improvements of these algorithms. In this paper, we present a novel perspective of noise-adding technique, i.e., adding the noise into the first-order information can help extract the negative curvature from the Hessian matrix, and provide a formal reasoning of this perspective by analyzing a simple first-order procedure. More importantly, the proposed procedure enables one to design purely first-order stochastic algorithms for escaping from non-degenerate saddle points with a much better time complexity (almost linear time in terms of the problem's dimensionality). In particular, we develop a first-order stochastic algorithm based on our new technique and an existing algorithm that only converges to a first-order stationary point to enjoy a time complexity of @math for finding a nearly second-order stationary point @math such that @math and @math (in high probability), where @math denotes the objective function and @math is the dimensionality of the problem. To the best of our knowledge, this is the best theoretical result of first-order algorithms for stochastic non-convex optimization, which is even competitive with if not better than existing stochastic algorithms hinging on the second-order information.