In this paper, we propose a new adaptive stochastic gradient Langevin dynamics (ASGLD) algorithmic framework and its two specialized versions, namely adaptive stochastic gradient (ASG) and adaptive gradient Langevin dynamics(AGLD), for non-convex optimization problems. All proposed algorithms can escape from saddle points with at most @math iterations, which is nearly dimension-free. Further, we show that ASGLD and ASG converge to a local minimum with at most @math iterations. Also, ASGLD with full gradients or ASGLD with a slowly linearly increasing batch size converge to a local minimum with iterations bounded by @math , which outperforms existing first-order methods.
In this section, we provide an overview on saddle point escape for non-convex learning, which has attracted a lot of attention in the machine learning community recently. Generally speaking, the most straightforward way of escaping from a saddle point is to examine the second-order Hessian matrix and identify an eigen-direction associated with a negative eigenvalue as an escape direction. Not surprisingly, most early attempts in this area are based on second-order curvature information. For example, @cite_18 used a cubic regularization with Hessian information to obtain dimension-free @math time complexity to escape saddle points. The same time complexity was obtained by @cite_21 by using a second-order trust region algorithm. Later, escape time complexity based on second-order approach was sharpened to @math by @cite_19 , where only Hessian-vector product information is needed (hence an @math penalty is incurred). @cite_3 also proposed a cubic regularization method based on @cite_10 to achieve the same @math escape time.
@cite_18 @cite_21 @cite_3 @cite_19 @cite_10
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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. A central challenge to using first-order methods for optimizing nonconvex problems is the presence of saddle points. First-order methods often get stuck at saddle points, greatly deteriorating their performance. Typically, to escape from saddles one has to use second-order methods. However, most works on second-order methods rely extensively on expensive Hessian-based computations, making them impractical in large-scale settings. To tackle this challenge, we introduce a generic framework that minimizes Hessian based computations while at the same time provably converging to second-order critical points. Our framework carefully alternates between a first-order and a second-order subroutine, using the latter only close to saddle points, and yields convergence results competitive to the state-of-the-art. Empirical results suggest that our strategy also enjoys a good practical performance. (This is a theory paper) In this paper, we consider first-order methods for solving stochastic non-convex optimization problems. The key building block of the proposed algorithms is first-order procedures to extract negative curvature from the Hessian matrix through a principled sequence starting from noise, which are referred to NEgative-curvature-Originated-from-Noise or NEON and are of independent interest. Based on this building block, we design purely first-order stochastic algorithms for escaping from non-degenerate saddle points with a much better time complexity (almost linear time in the problem's dimensionality). In particular, we develop a general framework of first-order stochastic algorithms with a second-order convergence guarantee based on our new technique and existing algorithms that may only converge to a first-order stationary point. For finding a nearly second-order stationary point such that ‖∇F( )‖≤ϵ and ∇2F( )≥− √ ϵ I (in high probability), the best time complexity of the presented algorithms is ˜ O (d ϵ3.5), where F(⋅) denotes the objective function and d is the dimensionality of the problem. To the best of our knowledge, this is the first theoretical result of first-order stochastic algorithms with an almost linear time in terms of problem's dimensionality for finding second-order stationary points, which is even competitive with existing stochastic algorithms hinging on the second-order information. A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for these local methods to find the global minimum is the proliferation of local minima with much higher error than the global minimum. Here we argue, based on results from statistical physics, random matrix theory, neural network theory, and empirical evidence, that a deeper and more profound difficulty originates from the proliferation of saddle points, not local minima, especially in high dimensional problems of practical interest. Such saddle points are surrounded by high error plateaus that can dramatically slow down learning, and give the illusory impression of the existence of a local minimum. Motivated by these arguments, we propose a new approach to second-order optimization, the saddle-free Newton method, that can rapidly escape high dimensional saddle points, unlike gradient descent and quasi-Newton methods. We apply this algorithm to deep or recurrent neural network training, and provide numerical evidence for its superior optimization performance.