The problem of optimizing over random structures emerges in many areas of science and engineering, ranging from statistical physics to machine learning and artificial intelligence. For many such structures, finding optimal solutions by means of fast algorithms is not known and often is believed not to be possible. At the same time, the formal hardness of these problems in the form of the complexity-theoretic NP-hardness is lacking. A new approach for algorithmic intractability in random structures is described in this article, which is based on the topological disconnectivity property of the set of pairwise distances of near-optimal solutions, called the Overlap Gap Property. The article demonstrates how this property 1) emerges in most models known to exhibit an apparent algorithmic hardness; 2) is consistent with the hardness/tractability phase transition for many models analyzed to the day; and, importantly, 3) allows to mathematically rigorously rule out a large class of algorithms as potential contenders, specifically the algorithms that exhibit the input stability (insensitivity).

The overlap gap property: A topological barrier to optimizing over random structures

We consider the symmetric binary perceptron model, a simple model of neural networks that has gathered significant attention in the statistical physics, information theory and probability theory communities, with recent connections made to the performance of learning algorithms in Baldassi et al. '15. 
We establish that the partition function of this model, normalized by its expected value, converges to a lognormal distribution. As a consequence, this allows us to establish several conjectures for this model: (i) it proves the contiguity conjecture of Aubin et al. '19 between the planted and unplanted models in the satisfiable regime; (ii) it establishes the sharp threshold conjecture; (iii) it proves the frozen 1-RSB conjecture in the symmetric case, conjectured first by Krauth-Mezard '89 in the asymmetric case. 
In a recent concurrent work of Perkins-Xu [PX21], the last two conjectures were also established by proving that the partition function concentrates on an exponential scale. This left open the contiguity conjecture and the lognormal limit characterization, which are established here. In particular, our proof technique relies on a dense counter-part of the small graph conditioning method, which was developed for sparse models in the celebrated work of Robinson and Wormald.

Proof of the Contiguity Conjecture and Lognormal Limit for the Symmetric Perceptron

We consider the symmetric binary perceptron model, a simple model of neural networks that has gathered significant attention in the statistical physics, information theory and probability theory communities, with recent connections made to the performance of learning algorithms in Baldassi et al. '15. We establish that the partition function of this model, normalized by its expected value, converges to a log-normal distribution. As a consequence, this allows us to establish several conjectures for this model: (i) it proves the contiguity conjecture of Aubin et al. '19 between the planted and unplanted models in the satisfiable regime; (ii) it establishes the sharp threshold conjecture; (iii) it proves the frozen 1-RSB conjecture in the symmetric case, conjectured first by Krauth-Mézard '89 in the asymmetric case. In a recent work of Perkins-Xu '21, the last two conjectures were also established by proving that the partition function concentrates on an exponential scale, under an analytical assumption on a real-valued function. This left open the contiguity conjecture and the lognor-mal limit characterization, which are established here unconditionally, with the analytical assumption verified. In particular, our proof technique relies on a dense counter-part of the small graph conditioning method, which was developed for sparse models in the celebrated work of Robinson and Wormald. 

/pdf/proof-of-the-contiguity-conjecture-and-lognormal-limit-for-2kzrne42.pdf

The binary (or Ising) perceptron is a toy model of a single-layer neural network and can be viewed as a random constraint satisfaction problem with a high degree of connectivity. The model and its symmetric variant, the symmetric binary perceptron (SBP), have been studied widely in statistical physics, mathematics, and machine learning.The SBP exhibits a dramatic statistical-to-computational gap: the densities at which known efficient algorithms find solutions are far below the threshold for the existence of solutions. Furthermore, the SBP exhibits a striking structural property: at all positive constraint densities almost all of its solutions are ‘totally frozen’ singletons separated by large Hamming distance [1], [2]. This suggests that finding a solution to the SBP may be computationally intractable. At the same time, however, the SBP does admit polynomial-time search algorithms at low enough densities. A conjectural explanation for this conundrum was put forth in [3]: efficient algorithms succeed in the face of freezing by finding exponentially rare clusters of large size. However, it was discovered recently that such rare large clusters exist at all subcritical densities, even at those well above the limits of known efficient algorithms [4]. Thus the driver of the statistical-to-computational gap exhibited by this model remains a mystery. In this paper, we conduct a different landscape analysis to explain the statistical-to-computational gap exhibited by this problem. We show that at high enough densities the SBP exhibits the multi Overlap Gap Property (m-OGP), an intricate geometrical property known to be a rigorous barrier for large classes of algorithms. Our analysis shows that the m-OGP threshold (a) is well below the satisfiability threshold; and (b) matches the best known algorithmic threshold up to logarithmic factors as $m\rightarrow\infty$. We then prove that the m-OGP rules out the class of stable algorithms for the SBP above this threshold. We conjecture that the $m\rightarrow\infty$ limit of the m-OGP threshold marks the algorithmic threshold for the problem. Furthermore, we investigate the stability of known efficient algorithms for perceptron models and show that the Kim-Roche algorithm [5], devised for the asymmetric binary perceptron, is stable in the sense we consider.

Algorithms and Barriers in the Symmetric Binary Perceptron Model

In this review article we discuss connections between the physics of disordered systems, phase transitions in inference problems, and computational hardness. We introduce two models representing the behavior of glassy systems, the spiked tensor model and the generalized linear model. We discuss the random (non-planted) versions of these problems as prototypical optimization problems, as well as the planted versions (with a hidden solution) as prototypical problems in statistical inference and learning. Based on ideas from physics, many of these problems have transitions where they are believed to jump from easy (solvable in polynomial time) to hard (requiring exponential time). We discuss several emerging ideas in theoretical computer science and statistics that provide rigorous evidence for hardness by proving that large classes of algorithms fail in the conjectured hard regime. This includes the overlap gap property, a particular mathematization of clustering or dynamical symmetry-breaking, which can be used to show that many algorithms that are local or robust to changes in their input fail. We also discuss the sum-of-squares hierarchy, which places bounds on proofs or algorithms that use low-degree polynomials such as standard spectral methods and semidefinite relaxations, including the Sherrington–Kirkpatrick model. Throughout the manuscript we present connections to the physics of disordered systems and associated replica symmetry breaking properties.

Disordered systems insights on computational hardness

We prove, under an assumption on the critical points of a real-valued function, that the symmetric Ising perceptron exhibits the `frozen 1-RSB' structure conjectured by Krauth and Mezard in the physics literature; that is, typical solutions of the model lie in clusters of vanishing entropy density. Moreover, we prove this in a very strong form conjectured by Huang, Wong, and Kabashima: a typical solution of the model is isolated with high probability and the Hamming distance to all other solutions is linear in the dimension. The frozen 1-RSB scenario is part of a recent and intriguing explanation of the performance of learning algorithms by Baldassi, Ingrosso, Lucibello, Saglietti, and Zecchina. We prove this structural result by comparing the symmetric Ising perceptron model to a planted model and proving a comparison result between the two models. Our main technical tool towards this comparison is an inductive argument for the concentration of the logarithm of number of solutions in the model.

https://dl.acm.org/doi/pdf/10.1145/3406325.3451119

Changji Xu

Papers

Frozen 1-RSB structure of the symmetric Ising perceptron

Frozen $1$-RSB structure of the symmetric Ising perceptron