Mark Sellke
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View article: Geometry Meets Incentives: Sample-Efficient Incentivized Exploration with Linear Contexts
Geometry Meets Incentives: Sample-Efficient Incentivized Exploration with Linear Contexts Open
In the incentivized exploration model, a principal aims to explore and learn over time by interacting with a sequence of self-interested agents. It has been recently understood that the main challenge in designing incentive-compatible algo…
View article: On the Discontinuous Breaking of Replica Symmetry and Shattering in Mean-Field Spin Glasses
On the Discontinuous Breaking of Replica Symmetry and Shattering in Mean-Field Spin Glasses Open
We show that in mean-field spin glasses, a discontinuous breaking of replica symmetry at the critical inverse temperature $β_c$ implies the existence of an intermediate shattered phase. This confirms a prediction from physics regarding the…
View article: Strong Low Degree Hardness for the Number Partitioning Problem
Strong Low Degree Hardness for the Number Partitioning Problem Open
In the number partitioning problem (NPP) one aims to partition a given set of $N$ real numbers into two subsets with approximately equal sum. The NPP is a well-studied optimization problem and is famous for possessing a statistical-to-comp…
View article: Tight Low Degree Hardness for Optimizing Pure Spherical Spin Glasses
Tight Low Degree Hardness for Optimizing Pure Spherical Spin Glasses Open
We prove constant degree polynomial algorithms cannot optimize pure spherical $p$-spin Hamiltonians beyond the algorithmic threshold $\mathsf{ALG}(p)=2\sqrt{\frac{p-1}{p}}$. The proof goes by transforming any hypothetical such algorithm in…
View article: Enhanced binding for a quantum particle coupled to scalar quantized field
Enhanced binding for a quantum particle coupled to scalar quantized field Open
Enhanced binding of a quantum particle coupled to a quantized field means that the Hamiltonian of the particle alone does not have a bound state, while the particle-field Hamiltonian does. For the Pauli--Fierz model, this is usually shown …
View article: On Marginal Stability in Low Temperature Spherical Spin Glasses
On Marginal Stability in Low Temperature Spherical Spin Glasses Open
We show marginal stability of near-ground states in spherical spin glasses is equivalent to full replica symmetry breaking at zero temperature near overlap $1$. This connection has long been implicit in the physics literature, which also l…
View article: Improved Lower Bound for Frankl’s Union-Closed Sets Conjecture
Improved Lower Bound for Frankl’s Union-Closed Sets Conjecture Open
We verify an explicit inequality conjectured in [Gilmer, 2022, arXiv:2211.09055], thus proving that for any nonempty union-closed family $\mathcal{F} \subseteq 2^{[n]}$, some $i\in [n]$ is contained in at least a $\frac{3-\sqrt{5}}{2} \app…
View article: Free Energy Universality of Spherical Spin Glasses
Free Energy Universality of Spherical Spin Glasses Open
We prove the free energy and ground state energy of spherical spin glasses are universal under the minimal moment assumptions. Previously such universality was known only for Ising spin glasses and random symmetric matrices, the latter bei…
View article: Localization of Random Surfaces with Monotone Potentials and an FKG-Gaussian Correlation Inequality
Localization of Random Surfaces with Monotone Potentials and an FKG-Gaussian Correlation Inequality Open
The seminal 1975 work of Brascamp-Lieb-Lebowitz initiated the rigorous study of Ginzberg-Landau random surface models. It was conjectured therein that fluctuations are localized on $\mathbb Z^d$ when $d\geq 3$ for very general potentials, …
View article: Optimization Algorithms for Multi-species Spherical Spin Glasses
Optimization Algorithms for Multi-species Spherical Spin Glasses Open
This paper develops approximate message passing algorithms to optimize multi-species spherical spin glasses. We first show how to efficiently achieve the algorithmic threshold energy identified in our companion work (Huang and Sellke in ar…
View article: No Free Prune: Information-Theoretic Barriers to Pruning at Initialization
No Free Prune: Information-Theoretic Barriers to Pruning at Initialization Open
The existence of "lottery tickets" arXiv:1803.03635 at or near initialization raises the tantalizing question of whether large models are necessary in deep learning, or whether sparse networks can be quickly identified and trained without …
View article: Optimizing mean field spin glasses with external field
Optimizing mean field spin glasses with external field Open
We consider the Hamiltonians of mean-field spin glasses, which are certain\nrandom functions $H_N$ defined on high-dimensional cubes or spheres in $\\mathbb\nR^N$. The asymptotic maximum values of these functions were famously obtained\nby…
View article: Mean square displacement of Brownian paths perturbed by bounded pair potentials
Mean square displacement of Brownian paths perturbed by bounded pair potentials Open
We study Brownian paths perturbed by semibounded pair potentials and prove upper bounds on the mean square displacement. As a technical tool we derive infinite dimensional versions of key inequalities that were first used in [Sellke; arXiv…
View article: A Constructive Proof of the Spherical Parisi Formula
A Constructive Proof of the Spherical Parisi Formula Open
The Parisi formula for the free energy is among the crown jewels in the theory of spin glasses. We present a simpler proof of the lower bound in the case of the spherical mean-field model. Our method follows the TAP approach developed rece…
View article: Sampling from Mean-Field Gibbs Measures via Diffusion Processes
Sampling from Mean-Field Gibbs Measures via Diffusion Processes Open
We consider Ising mixed $p$-spin glasses at high-temperature and without external field, and study the problem of sampling from the Gibbs distribution $μ$ in polynomial time. We develop a new sampling algorithm with complexity of the same …
View article: Strong Topological Trivialization of Multi-Species Spherical Spin Glasses
Strong Topological Trivialization of Multi-Species Spherical Spin Glasses Open
We study the landscapes of multi-species spherical spin glasses. Our results determine the phase boundary for annealed trivialization of the number of critical points, and establish its equivalence with a quenched strong topological trivia…
View article: Optimization Algorithms for Multi-Species Spherical Spin Glasses
Optimization Algorithms for Multi-Species Spherical Spin Glasses Open
This paper develops approximate message passing algorithms to optimize multi-species spherical spin glasses. We first show how to efficiently achieve the algorithmic threshold energy identified in our companion work, thus confirming that t…
View article: Shattering in Pure Spherical Spin Glasses
Shattering in Pure Spherical Spin Glasses Open
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, …
View article: Incentivizing Exploration with Linear Contexts and Combinatorial Actions
Incentivizing Exploration with Linear Contexts and Combinatorial Actions Open
We advance the study of incentivized bandit exploration, in which arm choices are viewed as recommendations and are required to be Bayesian incentive compatible. Recent work has shown under certain independence assumptions that after colle…
View article: On Size-Independent Sample Complexity of ReLU Networks
On Size-Independent Sample Complexity of ReLU Networks Open
We study the sample complexity of learning ReLU neural networks from the point of view of generalization. Given norm constraints on the weight matrices, a common approach is to estimate the Rademacher complexity of the associated function …
View article: Asymptotically Optimal Pure Exploration for Infinite-Armed Bandits
Asymptotically Optimal Pure Exploration for Infinite-Armed Bandits Open
We study pure exploration with infinitely many bandit arms generated i.i.d. from an unknown distribution. Our goal is to efficiently select a single high quality arm whose average reward is, with probability $1-δ$, within $\varepsilon$ of …
View article: The Threshold Energy of Low Temperature Langevin Dynamics for Pure Spherical Spin Glasses
The Threshold Energy of Low Temperature Langevin Dynamics for Pure Spherical Spin Glasses Open
We study the Langevin dynamics for spherical $p$-spin models, focusing on the short time regime described by the Cugliandolo-Kurchan equations. Confirming a prediction of [Cugliandolo-Kurchan, Phys. Rev. Lett. 1993], we show the asymptotic…
View article: Local algorithms for maximum cut and minimum bisection on locally treelike regular graphs of large degree
Local algorithms for maximum cut and minimum bisection on locally treelike regular graphs of large degree Open
Given a graph of degree over vertices, we consider the problem of computing a near maximum cut or a near minimum bisection in polynomial time. For graphs of girth , we develop a local message passing algorithm whose complexity is , and tha…
View article: Tight Space Lower Bound for Pseudo-Deterministic Approximate Counting
Tight Space Lower Bound for Pseudo-Deterministic Approximate Counting Open
We investigate one of the most basic problems in streaming algorithms: approximating the number of elements in the stream. In 1978, Morris famously gave a randomized algorithm achieving a constant-factor approximation error for streams of …
View article: Algorithmic Threshold for Multi-Species Spherical Spin Glasses
Algorithmic Threshold for Multi-Species Spherical Spin Glasses Open
We study efficient optimization of the Hamiltonians of multi-species spherical spin glasses. Our results characterize the maximum value attained by algorithms that are suitably Lipschitz with respect to the disorder through a variational p…
View article: A Universal Law of Robustness via Isoperimetry
A Universal Law of Robustness via Isoperimetry Open
Classically, data interpolation with a parametrized model class is possible as long as the number of parameters is larger than the number of equations to be satisfied. A puzzling phenomenon in deep learning is that models are trained with …
View article: Almost Quartic Lower Bound for the Fröhlich Polaron's Effective Mass via Gaussian Domination
Almost Quartic Lower Bound for the Fröhlich Polaron's Effective Mass via Gaussian Domination Open
We prove the Fröhlich polaron has effective mass at least $\frac{α^4}{(\log α)^6}$ when the coupling strength $α$ is large. This nearly matches the quartic growth rate $C_*α^4$ predicted by Landau and Pekar in 1948 and complements a recent…