Passed & Spurious: Descent Algorithms and Local Minima in Spiked MatrixTensor Models
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Proceedings of the 36th International Conference on Machine Learning, PMLR 97:43334342, 2019.
Abstract
In this work we analyse quantitatively the interplay between the loss landscape and performance of descent algorithms in a prototypical inference problem, the spiked matrixtensor model. We study a loss function that is the negative loglikelihood of the model. We analyse the number of local minima at a fixed distance from the signal/spike with the KacRice formula, and locate trivialization of the landscape at large signaltonoise ratios. We evaluate analytically the performance of a gradient flow algorithm using integrodifferential PDEs as developed in physics of disordered systems for the Langevin dynamics. We analyze the performance of an approximate message passing algorithm estimating the maximum likelihood configuration via its state evolution. We conclude by comparing the above results: while we observe a drastic slow down of the gradient flow dynamics even in the region where the landscape is trivial, both the analyzed algorithms are shown to perform well even in the part of the region of parameters where spurious local minima are present.
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