An Alternative View: When Does SGD Escape Local Minima?¶
Authors: Robert Kleinberg, Yuanzhi Li, Yang Yuan
Published: 2018 (Conference Paper)
Source: International Conference on Machine Learning
Algorithm: SGD
arXiv: 1802.06175
Summary¶
Provides a theoretical explanation for why SGD escapes sharp local minima, reframing SGD as optimizing a smoothed (convolved) version of the loss. The one-point convexity condition identified is broad enough to encompass neural network loss surfaces, bridging the gap between theory and practical SGD success.
Abstract¶
Stochastic gradient descent (SGD) is widely used in machine learning. Although being commonly viewed as a fast but not accurate version of gradient descent (GD), it always finds better solutions than GD for modern neural networks. In order to understand this phenomenon, we take an alternative view that SGD is working on the convolved (thus smoothed) version of the loss function. We show that, even if the function f has many bad local minima or saddle points, as long as for every point x, the weighted average of the gradients of its neighborhoods is one point convex with respect to the desired solution x*, SGD will get close to, and then stay around x* with constant probability. More specifically, SGD will not get stuck at "sharp" local minima with small diameters, as long as the neighborhoods of these regions contain enough gradient information. The neighborhood size is controlled by step size and gradient noise. Our result identifies a set of functions that SGD provably works, which is much larger than the set of convex functions. Empirically, we observe that the loss surface of neural networks enjoys nice one point convexity properties locally, therefore our theorem helps explain why SGD works so well for neural networks.
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Tags¶
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Stochastic gradient descent
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Local minima
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Non-convex optimization
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Neural networks
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Optimization theory