We prove the existence of a shattered phase within the replica-symmetric phase of the pure spherical p-spin models for p sufficiently large. In this phase, we construct a decomposition of the sphere into well-separated small clusters, each of which has exponentially small Gibbs mass, yet which together carry all but an exponentially small fraction of the Gibbs mass. We achieve this via quantitative estimates on the derivative of the Franz-Parisi potential, which measures the Gibbs mass profile around a typical sample. Corollaries on dynamics are derived, in particular we show the two-times correlation function of stationary Langevin dynamics must have an exponentially long plateau. We further show that shattering implies disorder chaos for the Gibbs measure in the optimal transport sense; this is known to imply failure of sampling algorithms which are stable under perturbation in the same metric.

The convex body chasing problem, introduced by Friedman and Linial [FL93], is a competitive analysis problem on any normed vector space. In convex body chasing, for each timestep t ∈ N, a convex body K t ⊆ R d is given as a request, and the player picks a point x t ∈ K t . The player aims to ensure that the total distance moved T −1 t=0 ||x t − x t+1 || is within a bounded ratio of the smallest possible offline solution.In this work, we consider the nested version of the problem, in which the sequence (K t ) must be decreasing. For Euclidean spaces, we consider a memoryless algorithm which moves to the so-called Steiner point, and show that in an appropriate sense it is exactly optimal among memoryless algorithms. For general finite dimensional normed spaces, we combine the Steiner point and our recent algorithm in [ABC + 18] to obtain a new algorithm which is nearly optimal for all ℓ p d spaces with p ≥ 1, closing a polynomial gap. * This work was done while M. Sellke and Y. Li were at Microsoft Research.

Mean-field spin glasses are families of random energy functions (Hamiltonians) on highdimensional product spaces. In this paper we consider the case of Ising mixed p-spin models, namely Hamiltonians H N : Σ N → R on the Hamming hypercube Σ N = {±1} N , which are defined by the property that {H N (σ)} σ∈ΣN is a centered Gaussian process with covariance E{H N (σ 1 )H N (σ 2 )} depending only on the scalar product σ 1 , σ 2 .The asymptotic value of the optimum max σ∈ΣN H N (σ) was characterized in terms of a variational principle known as the Parisi formula, first proved by Talagrand and, in a more general setting, by Panchenko. The structure of superlevel sets is extremely rich and has been studied by a number of authors. Here we ask whether a near optimal configuration σ can be computed in polynomial time.We develop a message passing algorithm whose complexity per-iteration is of the same order as the complexity of evaluating the gradient of H N , and characterize the typical energy value it achieves. When the p-spin model H N satisfies a certain no-overlap gap assumption, for any ε > 0, the algorithm outputs σ ∈ Σ N such that H N (σ) ≥ (1 − ε) max σ ′ H N (σ ′ ), with high probability. The number of iterations is bounded in N and depends uniquely on ε. More generally, regardless of whether the no-overlap gap assumption holds, the energy achieved is given by an extended variational principle, which generalizes the Parisi formula.

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

In the chasing convex bodies problem, an online player receives a request sequence of N convex sets K 1 , . . . , K N contained in a normed space R d . The player starts at x 0 ∈ R d , and after observing each K n picks a new point x n ∈ K n . At each step the player pays a movement cost of ||x n − x n−1 ||. The player aims to maintain a constant competitive ratio against the minimum cost possible in hindsight, i.e. knowing all requests in advance. The existence of a finite competitive ratio for convex body chasing was first conjectured in 1991 by Friedman and Linial in [FL93]. This conjecture was recently resolved in [BLLS18] which proved an exponential 2 O(d) upper bound on the competitive ratio.In this paper, we drastically improve the exponential upper bound. We give an algorithm achieving competitive ratio d for arbitrary normed spaces, which is exactly tight for ℓ ∞ . In Euclidean space, our algorithm achieves nearly optimal competitive ratio O( √ d log N ), compared to a lower bound of √ d. Our approach extends the recent work [BKL + 18] which chases nested convex bodies using the classical Steiner point of a convex body. We define the functional Steiner point of a convex function and apply it to the work function to obtain our algorithm.