Tight Lipschitz Hardness for Optimizing Mean Field Spin Glasses

Abstract

We study the problem of algorithmically optimizing the Hamiltonian H N of a spherical or Ising mixed p-spin glass. The maximum asymptotic value OPT of H N /N is characterized by a variational principle known as the Parisi formula, proved first by Talagrand and in more generality by Panchenko. Recently developed approximate message passing algorithms efficiently optimize H N /N up to a value ALG given by an extended Parisi formula, which minimizes over a larger space of functional order parameters. These two objectives are equal for spin glasses exhibiting a no overlap gap property. However, ALG < OPT can also occur, and no efficient algorithm producing an objective value exceeding ALG is known.We prove that for mixed even p-spin models, no algorithm satisfying an overlap concentration property can produce an objective larger than ALG with non-negligible probability. This property holds for all algorithms with suitably Lipschitz dependence on the disorder coefficients of H N . It encompasses natural formulations of gradient descent, approximate message passing, and Langevin dynamics run for bounded time and in particular includes the algorithms achieving ALG mentioned above. To prove this result, we substantially generalize the overlap gap property framework introduced by Gamarnik and Sudan to arbitrary ultrametric forbidden structures of solutions. Contents

Algorithms and Barriers in the Symmetric Binary Perceptron Model

“…We believe it might be possible to remove this factor by considering a more intricate overlap pattern, e.g. similar to those considered in [Wei20,BH21,HS21].…”

“…More recently, by leveraging the ensemble m−OGP; Bresler and Huang [BH21] established nearly tight low-degree hardness results for the random k−SAT problem: they show that low-degree polynomials fail to return a satisfying assignment when the clause density is only a constant factor off by the computational threshold. In yet another work, Huang and Sellke [HS21] construct a very intricate forbidden structure consisting of an ultrametric tree of solutions, which they refer to as the branching OGP. By leveraging this branching OGP, they rule out overlap concentrated algorithms 4 at the algorithmic threshold for the problem of optimizing mixed, even p−spin model Hamiltonian.…”

“…See Conjecture 3.7 for details. This conjecture is backed up by the evidence that for many random computational problems including random k−SAT [BH21], independent sets in sparse random graphs [RV17,Wei20], and mixed even p−spin model [HS21], the m−OGP matches or nearly matches the best known algorithmic threshold.…”

“…Recall from our prior discussion that for many random computational problems, the m−OGP threshold coincides (or nearly coincides) with conjectured algorithmic threshold. Examples include the problem of finding the largest independent set in random sparse graphs [RV17,Wei20], NAE-k-SAT [GS17b], random k−SAT [BH21], mixed even p−spin model [HS21], and so on. In light of the preceding discussion, this is also the case for the SBP model: the limit of known algorithms is at Θ(κ 2 ), whereas, as we establish in Theorem 2.4, the ensemble m−OGP holds for densities Ω κ 2 log 2 1 κ in the regime κ → 0.…”

Performance and limitations of the QAOA at constant levels on large sparse hypergraphs and spin glass models

“…Although it was previously known that local quantum algorithms such as the QAOA are limited in the low circuit-depth regime where they do not see the whole graph, our result shows for the first time that significant barriers remain even when the whole graph is seen. One natural path forward is to compare the energy achieved by the constant-p QAOA to that by the AMP algorithm [Mon19,AMS21b], which holds the current record on the q-spin models among polynomial-time classical algorithms [HS21]. It would be very interesting to see whether the QAOA can achieve a better energy than AMP.…”