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Register a new userWho is Afraid of Big Bad Minima? Analysis of Gradient-Flow in a Spiked Matrix-Tensor Model
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Gradient-based algorithms are effective for many machine learning tasks, but despite ample recent effort and some progress, it often remains unclear why they work in practice in optimising high-dimensional non-convex functions and why they find good minima instead of being trapped in spurious ones. Here we present a quantitative theory explaining this behaviour in a spiked matrix-tensor model. Our framework is based on the Kac-Rice analysis of stationary points and a closed-form analysis of gradient-flow originating from statistical physics. We show that there is a well defined region of parameters where the gradient-flow algorithm finds a good global minimum despite the presence of exponentially many spurious local minima. We show that this is achieved by surfing on saddles that have strong negative direction towards the global minima, a phenomenon that is connected to a BBP-type threshold in the Hessian describing the critical points of the landscapes.
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In this work we analyse quantitatively the interplay between the loss landscape and performance of descent algorithms in a prototypical inference problem, the spiked matrix-tensor model. We study a loss function that is the negative log-likelihood of the model. We analyse the number of local minima at a fixed distance from the signal/spike with the Kac-Rice formula, and locate trivialization of the landscape at large signal-to-noise ratios. We evaluate in a closed form the performance of a gradient flow algorithm using integro-differential 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.
Despite the widespread use of gradient-based algorithms for optimizing high-dimensional non-convex functions, understanding their ability of finding good minima instead of being trapped in spurious ones remains to a large extent an open problem. Here we focus on gradient flow dynamics for phase retrieval from random measurements. When the ratio of the number of measurements over the input dimension is small the dynamics remains trapped in spurious minima with large basins of attraction. We find analytically that above a critical ratio those critical points become unstable developing a negative direction toward the signal. By numerical experiments we show that in this regime the gradient flow algorithm is not trapped; it drifts away from the spurious critical points along the unstable direction and succeeds in finding the global minimum. Using tools from statistical physics we characterize this phenomenon, which is related to a BBP-type transition in the Hessian of the spurious minima.
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We analyze the connection between minimizers with good generalizing properties and high local entropy regions of a threshold-linear classifier in Gaussian mixtures with the mean squared error loss function. We show that there exist configurations that achieve the Bayes-optimal generalization error, even in the case of unbalanced clusters. We explore analytically the error-counting loss landscape in the vicinity of a Bayes-optimal solution, and show that the closer we get to such configurations, the higher the local entropy, implying that the Bayes-optimal solution lays inside a wide flat region. We also consider the algorithmically relevant case of targeting wide flat minima of the (differentiable) mean squared error loss. Our analytical and numerical results show not only that in the balanced case the dependence on the norm of the weights is mild, but also, in the unbalanced case, that the performances can be improved.
Three widely used scenaria for defining cuts on transverse energies of jets in ep collisions are discussed. All of them are shown to suffer from the same sort of unphysical behaviour when the cut regions are subject to additional constraints. This feature is inherent in the very way dijet cross sections are defined and cannot be avoided. In particular, the symmetric cut scenario is shown to be equally suitable for the comparison with NLO QCD calculations as the asymmetric or sum-like ones.
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