Loss functions with a large number of saddle points are one of the main obstacles to training many modern machine learning models. Gradient descent (GD) is a fundamental algorithm for machine learning and converges to a saddle point for certain initial data. We call the region formed by these initial values the "attraction region." For quadratic functions, GD converges to a saddle point if the initial data is in a subspace of up to n-1 dimensions. In this paper, we prove that a small modification of the recently proposed Laplacian smoothing gradient descent (LSGD) [Osher, et al, arXiv:1806.06317] contributes to avoiding saddle points without sacrificing the convergence rate of GD. In particular, we show that the dimension of the LSGD's attraction region is at most floor((n-1) 2) for a class of quadratic functions which is significantly smaller than GD's (n-1)-dimensional attraction region.
@cite_24 showed that GD converges to a local minimizer almost surely if the initial point is randomly chosen. The proof is based on the stable manifold theorem and hence no upper bound of the number of steps is provided. How to escape from saddle points efficiently has been a core problem in non-convex optimization. There has been much recent work on developing algorithms which provably avoid saddle points. The existing algorithms use either the perturbed noisy gradient or partially or fully the Hessian information.
@cite_24
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Although gradient descent (GD) almost always escapes saddle points asymptotically [, 2016], this paper shows that even with fairly natural random initialization schemes and non-pathological functions, GD can be significantly slowed down by saddle points, taking exponential time to escape. On the other hand, gradient descent with perturbations [, 2015, , 2017] is not slowed down by saddle points—it can find an approximate local minimizer in polynomial time. This result implies that GD is inherently slower than perturbed GD, and justifies the importance of adding perturbations for efficient non-convex optimization. While our focus is theoretical, we also present experiments that illustrate our theoretical findings. A commonly used heuristic in non-convex optimization is Normalized Gradient Descent (NGD) - a variant of gradient descent in which only the direction of the gradient is taken into account and its magnitude ignored. We analyze this heuristic and show that with carefully chosen parameters and noise injection, this method can provably evade saddle points. We establish the convergence of NGD to a local minimum, and demonstrate rates which improve upon the fastest known first order algorithm due to Ge e al. (2015). The effectiveness of our method is demonstrated via an application to the problem of online tensor decomposition; a task for which saddle point evasion is known to result in convergence to global minima. Nesterov's accelerated gradient descent (AGD), an instance of the general family of "momentum methods", provably achieves faster convergence rate than gradient descent (GD) in the convex setting. However, whether these methods are superior to GD in the nonconvex setting remains open. This paper studies a simple variant of AGD, and shows that it escapes saddle points and finds a second-order stationary point in @math iterations, faster than the @math iterations required by GD. To the best of our knowledge, this is the first Hessian-free algorithm to find a second-order stationary point faster than GD, and also the first single-loop algorithm with a faster rate than GD even in the setting of finding a first-order stationary point. Our analysis is based on two key ideas: (1) the use of a simple Hamiltonian function, inspired by a continuous-time perspective, which AGD monotonically decreases per step even for nonconvex functions, and (2) a novel framework called improve or localize, which is useful for tracking the long-term behavior of gradient-based optimization algorithms. We believe that these techniques may deepen our understanding of both acceleration algorithms and nonconvex optimization. Author(s): Anandkumar, A; Ge, R | Abstract: Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have degenerate saddle points such that the first and second order derivatives cannot distinguish them with local optima. In this paper we use higher order derivatives to escape these saddle points: we design the first efficient algorithm guaranteed to converge to a third order local optimum (while existing techniques are at most second order). We also show that it is NP-hard to extend this further to finding fourth order local optima.