arXiv subjects

Mark Sellke

Publications and source records attributed to Mark Sellke.

At least 37 records · Page 2Linked to original sources

arXiv2025

Strong Low Degree Hardness for the Number Partitioning Problem

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-computational gap: when the $N$ numbers to be partitioned are i.i.d. standard gaussian, the optimal discrepancy is $2^{-Θ(N)}$ with high probability, but the best known polynomial-time algorithms only find solutions with a discrepancy of $2^{-Θ(\log^2 N)}$. This gap is a common feature in optimization problems over random combinatorial structures, and indicates the need for a study that goes beyond worst-case analysis. We provide evidence of a nearly tight algorithmic barrier for the number partitioning problem. Namely we consider the family of low coordinate degree algorithms (with randomized rounding into the Boolean cube), and show that degree $D$ algorithms fail to solve the NPP to accuracy beyond $2^{-\widetilde O(D)}$. According to the low degree heuristic, this suggests that simple brute-force search algorithms are nearly unimprovable, given any allotted runtime between polynomial and exponential in $N$. Our proof combines the isolation of solutions in the landscape with a conditional form of the overlap gap property: given a good solution to an NPP instance, slightly noising the NPP instance typically leaves no good solutions near the original one. In fact our analysis applies whenever the $N$ numbers to be partitioned are independent with uniformly bounded density.

math.ST↗

arXiv2025

Strong Topological Trivialization of Multi-Species Spherical Spin Glasses

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 trivialization property. Namely in the "trivial" regime, the number of critical points is constant, all are well-conditioned, and all approximate critical points are close to a true critical point. As a consequence, we deduce that Langevin dynamics at sufficiently low temperature has logarithmic mixing time. Our approach begins with the Kac--Rice formula. We characterize the annealed trivialization phase by explicitly solving a suitable multi-dimensional variational problem, obtained by simplifying certain asymptotic determinant formulas from (Ben Arous--Bourgade--McKenna 2023, McKenna 2024). To obtain more precise quenched results, we develop general purpose techniques to avoid sub-exponential correction factors and show non-existence of approximate critical points. Many of the results are new even in the 1-species case.

math.PR↗

arXiv2025

Enhanced binding for a quantum particle coupled to scalar quantized field

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 via the binding condition, which works less well in the case of a linear coupling to a scalar field. In particular, the case of a single particle linearly coupled to a scalar field has been open so far. Using a method relying on functional integrals and the Gaussian correlation inequality, we obtain enhanced binding for this case. From a statistical mechanics point of view, our result describes a localization phase transition (in the strength of the pair potential) for a Brownian motion subject to an external and an attractive pair potential.

math-ph↗

arXiv2025

Tight Low Degree Hardness for Optimizing Pure Spherical Spin Glasses

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 into a Lipschitz one, for which hardness was shown previously by the author and B. Huang.

math.PR↗

arXiv2025

Nonparametric MLE for Gaussian Location Mixtures: Certified Computation and Generic Behavior

We study the nonparametric maximum likelihood estimator $\widehatπ$ for Gaussian location mixtures in one dimension. It has been known since (Lindsay, 1983) that given an $n$-point dataset, this estimator always returns a mixture with at most $n$ components, and more recently (Wu-Polyanskiy, 2020) gave a sharp $O(\log n)$ bound for subgaussian data. In this work we study computational aspects of $\widehatπ$. We provide an algorithm which for small enough $\varepsilon>0$ computes an $\varepsilon$-approximation of $\widehatπ$ in Wasserstein distance in time $K+Cnk^2\log\log(1/\varepsilon)$. Here $K$ is data-dependent but independent of $\varepsilon$, while $C$ is an absolute constant and $k=|supp(\widehatπ)|\leq n$ is the number of atoms in $\widehatπ$. We also certifiably compute the exact value of $|supp(\widehatπ)|$ in finite time. These guarantees hold almost surely whenever the dataset $(x_1,\dots,x_n)\in [-cn^{1/4},cn^{1/4}]$ consists of independent points from a probability distribution with a density (relative to Lebesgue measure). We also show the distribution of $\widehatπ$ conditioned to be $k$-atomic admits a density on the associated $2k-1$ dimensional parameter space for all $k\leq \sqrt{n}/3$, and almost sure locally linear convergence of the EM algorithm. One key tool is a classical Fourier analytic estimate for non-degenerate curves.

math.ST↗

arXiv2025

Shattering in Pure Spherical Spin Glasses

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.

math.PR↗

arXiv2024

Tight Space Lower Bound for Pseudo-Deterministic Approximate Counting

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 length at most N in space $O(\log \log N)$. We investigate the pseudo-deterministic complexity of the problem and prove a tight $Ω(\log N)$ lower bound, thus resolving a problem of Goldwasser-Grossman-Mohanty-Woodruff.

cs.DS↗

arXiv2024

Incentivizing Exploration with Linear Contexts and Combinatorial Actions

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 collecting enough initial samples, the popular Thompson sampling algorithm becomes incentive compatible. We give an analog of this result for linear bandits, where the independence of the prior is replaced by a natural convexity condition. This opens up the possibility of efficient and regret-optimal incentivized exploration in high-dimensional action spaces. In the semibandit model, we also improve the sample complexity for the pre-Thompson sampling phase of initial data collection.

cs.GT↗

arXiv2024

Free Energy Universality of Spherical Spin Glasses

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 being a celebrated result of Bai--Yin. Our methods extend to $\ell^q$ balls for $q>2$, thus resolving a conjecture of Chen-Sen, and to tensor PCA.

math.PR↗

arXiv2024

No Free Prune: Information-Theoretic Barriers to Pruning at Initialization

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 ever training the dense models that contain them. However, efforts to find these sparse subnetworks without training the dense model ("pruning at initialization") have been broadly unsuccessful arXiv:2009.08576. We put forward a theoretical explanation for this, based on the model's effective parameter count, $p_\text{eff}$, given by the sum of the number of non-zero weights in the final network and the mutual information between the sparsity mask and the data. We show the Law of Robustness of arXiv:2105.12806 extends to sparse networks with the usual parameter count replaced by $p_\text{eff}$, meaning a sparse neural network which robustly interpolates noisy data requires a heavily data-dependent mask. We posit that pruning during and after training outputs masks with higher mutual information than those produced by pruning at initialization. Thus two networks may have the same sparsities, but differ in effective parameter count based on how they were trained. This suggests that pruning near initialization may be infeasible and explains why lottery tickets exist, but cannot be found fast (i.e. without training the full network). Experiments on neural networks confirm that information gained during training may indeed affect model capacity.

stat.ML↗

arXiv2024

Improved Lower Bound for Frankl's Union-Closed Sets Conjecture

We verify an explicit inequality conjectured recently by Gilmer, thus proving that for any nonempty union-closed family $F \subseteq 2^{[n]}$, some $i\in [n]$ is contained in at least a $\frac{3-\sqrt{5}}{2} \approx 0.38$ fraction of the sets in $F$. One case, an explicit one-variable inequality, is checked by computer calculation.

math.CO↗

arXiv2024

A Constructive Proof of the Spherical Parisi Formula

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 recently in e.g. (Subag, 2018): we obtain an ultrametric tree of pure states, each with approximately the same free energy as the entire model, which are hierarchically arranged in accordance with the Parisi ansatz. We construct this tree ``layer by layer'' given the minimizer to Parisi's variational problem. On overlap intervals with full RSB, the tree is built by an optimization algorithm due to Subag. On overlap intervals with finite RSB, the tree is constructed by a new truncated second moment argument; a similar argument also characterizes the free energy of the resulting pure states. Notably we do not use the Aizenman--Sims--Starr scheme, and require interpolation bounds only up to the 1RSB level. Our methods also yield results for large deviations of the ground state, including the entire upper tail rate function for all 1RSB models without external field.

math.PR↗

arXiv2024

The Threshold Energy of Low Temperature Langevin Dynamics for Pure Spherical Spin Glasses

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 energy achieved is exactly $E_{\infty}(p)=2\sqrt{\frac{p-1}{p}}$ in the low temperature limit. The upper bound uses hardness results for Lipschitz optimization algorithms and applies for all temperatures. For the lower bound, we prove the dynamics reaches and stays above the lowest energy of any approximate local maximum. In fact the latter behavior holds for any Hamiltonian obeying natural smoothness estimates, even with disorder-dependent initialization and on exponential time-scales.

cond-mat.dis-nn↗

arXiv2024

Localization of Random Surfaces with Monotone Potentials and an FKG-Gaussian Correlation Inequality

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, matching the behavior of the Gaussian free field. We confirm this behavior for all even potentials $U:\mathbb R\to\mathbb R$ satisfying $U'(x)\geq \min(\varepsilon x,\frac{1+\varepsilon}{x})$ on $x\in \mathbb R^+$. Given correspondingly stronger growth conditions on $U$, we show power or stretched exponential tail bounds on all transient graphs, which determine the maximum field value up to constants in many cases. Further extensions include non-wired boundary conditions and iterated Laplacian analogs such as the membrane model. Our main tool is an FKG-based generalization of the Gaussian correlation inequality, which is used to dominate the finite-volume Gibbs measures by mixtures of centered Gaussian fields.

math.PR↗

arXiv2024

Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localization

We consider the Sherrington-Kirkpatrick model of spin glasses at high-temperature and no external field, and study the problem of sampling from the Gibbs distribution $μ$ in polynomial time. We prove that, for any inverse temperature $β<1/2$, there exists an algorithm with complexity $O(n^2)$ that samples from a distribution $μ^{alg}$ which is close in normalized Wasserstein distance to $μ$. Namely, there exists a coupling of $μ$ and $μ^{alg}$ such that if $(x,x^{alg})\in\{-1,+1\}^n\times \{-1,+1\}^n$ is a pair drawn from this coupling, then $n^{-1}\mathbb E\{||x-x^{alg}||_2^2\}=o_n(1)$. The best previous results, by Bauerschmidt and Bodineau and by Eldan, Koehler, and Zeitouni, implied efficient algorithms to approximately sample (under a stronger metric) for $β<1/4$. We complement this result with a negative one, by introducing a suitable "stability" property for sampling algorithms, which is verified by many standard techniques. We prove that no stable algorithm can approximately sample for $β>1$, even under the normalized Wasserstein metric. Our sampling method is based on an algorithmic implementation of stochastic localization, which progressively tilts the measure $μ$ towards a single configuration, together with an approximate message passing algorithm that is used to approximate the mean of the tilted measure.

math.PR↗

arXiv2024

On Size-Independent Sample Complexity of ReLU Networks

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 class. Previously Golowich-Rakhlin-Shamir (2020) obtained a bound independent of the network size (scaling with a product of Frobenius norms) except for a factor of the square-root depth. We give a refinement which often has no explicit depth-dependence at all.

cs.LG↗

arXiv2024

Optimization Algorithms for Multi-Species Spherical Spin Glasses

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 the Lipschitz hardness result proved therein is tight. Next we give two generalized algorithms which produce multiple outputs and show all of them are approximate critical points. Namely, in an $r$-species model we construct $2^r$ approximate critical points when the external field is stronger than a "topological trivialization" phase boundary, and exponentially many such points in the complementary regime. We also compute the local behavior of the Hamiltonian around each. These extensions are relevant for another companion work on topological trivialization of the landscape.

math.PR↗

arXiv2023

Optimizing Mean Field Spin Glasses with External Field

We consider the Hamiltonians of mean-field spin glasses, which are certain random functions $H_N$ defined on high-dimensional cubes or spheres in $\mathbb R^N$. The asymptotic maximum values of these functions were famously obtained by Talagrand and later by Panchenko and by Chen. The landscape of approximate maxima of $H_N$ is described by various forms of replica symmetry breaking exhibiting a broad range of possible behaviors. We study the problem of efficiently computing an approximate maximizer of $H_N$. We give a two-phase message pasing algorithm to approximately maximize $H_N$ when a no overlap-gap condition holds. This generalizes several recent works by allowing a non-trivial external field. For even Ising spin glasses with constant external field, our algorithm succeeds exactly when existing methods fail to rule out approximate maximization for a wide class of algorithms. Moreover we give a branching variant of our algorithm which constructs a full ultrametric tree of approximate maxima.

cond-mat.dis-nn↗