Algorithms and Barriers in the Symmetric Binary Perceptron Model

We study the random binary symmetric perceptron problem, focusing on the behavior of rare high-margin solutions. While most solutions are isolated, we demonstrate that these rare solutions are part of clusters of extensive entropy, heuristically corresponding to non-trivial fixed points of an approximate message-passing algorithm. We enumerate these clusters via a local entropy, defined as a Franz-Parisi potential, which we rigorously evaluate using the first and second moment methods in the limit of a small constraint density $\alpha$ (corresponding to vanishing margin $\kappa$) under a certain assumption on the concentration of the entropy. This examination unveils several intriguing phenomena:i) We demonstrate that these clusters have an entropic barrier in the sense that the entropy as a function of the distance from the reference high-margin solution is non-monotone when $\knew \le 1.429 \sqrt{-\alpha/\log{\alpha}}$, while it is monotone otherwise, and that they have an energetic barrier in the sense that there are no solutions at an intermediate distance from the reference solution when $\knew \le 1.239 \sqrt{-\alpha/ \log{\alpha}}$. The critical scaling of the margin $\kappa$ in $\sqrt{-\alpha/\log\alpha}$ corresponds to the one obtained from the earlier work of Gamarnik et al. for the overlap-gap property, a phenomenon known to present a barrier to certain efficient algorithms. ii) We establish using the replica method that the complexity (the logarithm of the number of clusters of such solutions) versus entropy (the logarithm of the number of solutions in the clusters) curves are partly non-concave and correspond to very large values of the Parisi parameter, with the equilibrium being reached when the Parisi parameter diverges.

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Algorithms and Barriers in the Symmetric Binary Perceptron Model

Gamarnik

Kızıldağ

Perkins

et al. 2022

2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)

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For many computational problems involving randomness, intricate geometric features of the solution space have been used to rigorously rule out powerful classes of algorithms. This is often accomplished through the lens of the multi Overlap Gap Property (m-OGP), a rigorous barrier against algorithms exhibiting input stability. In this paper, we focus on the algorithmic tractability of two models: (i) discrepancy minimization, and (ii) the symmetric binary perceptron (SBP), a random constraint satisfaction problem as well as a toy model of a single-layer neural network.Our first focus is on the limits of online algorithms. By establishing and leveraging a novel geometrical barrier, we obtain sharp hardness guarantees against online algorithms for both the SBP and discrepancy minimization. Our results match the best known algorithmic guarantees, up to constant factors. Our second focus is on efficiently finding a constant discrepancy solution, given a random matrix M ∈ R M ×n . In a smooth setting, where the entries of M are i.i.d. standard normal, we establish the presence of m-OGP for n = Θ(M log M ). Consequently, we rule out the class of stable algorithms at this value. These results give the first rigorous evidence towards a conjecture of Altschuler and Niles-Weed [ANW21, Conjecture 1].Our methods use the intricate geometry of the solution space to prove tight hardness results for online algorithms. The barrier we establish is a novel variant of the m-OGP. Furthermore, it regards m-tuples of solutions with respect to correlated instances, with growing values of m, m = ω(1). Importantly, our results rule out online algorithms succeeding even with an exponentially small probability.

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Algorithms and Barriers in the Symmetric Binary Perceptron Model

Cited by 10 publications

References 98 publications

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Algorithms and Barriers in the Symmetric Binary Perceptron Model

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