Although gradient descent (GD) almost always escapes saddle points asymptotically , 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 [Ge et al., 2015 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.
Du et al. REF showed that PGD or noisy GD can escape saddle point faster.
Gradient Descent Can Take Exponential Time to Escape Saddle Points
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ArXiv
Computer Science Mathematics