arXiv subjects

Matthew X. Burns

Publications and source records attributed to Matthew X. Burns.

7 recordsLinked to original sources

arXiv2026

Log-Averaged Mirror Prox for Fast, Large-Scale Optimal Transport in Linear Space

We propose Log-Averaged Mirror Prox (LAMP), a linear-space primal-dual method for large-scale optimal transport. LAMP implements primal mirror prox updates by tracking an averaged dual sequence, reducing storage complexity from ${O}(nm)$ to $O(n+m)$ while preserving dense, GPU-friendly reductions. Consequently, LAMP preserves the last-iterate $\widetilde{O}( nm\varepsilon^{-1})$ arithmetic complexity of conservatively parameterized primal-dual mirror prox. We further analyze LAMP as a direct optimal transport solver in a more performant parameter regime, providing a last-iterate sub-optimality certificate dependent on infeasibility and an explicit $O(1/t)$ term. Moreover, we give a computable sufficient condition for best-iterate convergence to a saddle-point. Numerical experiments with an optimized CUDA implementation show that LAMP outperforms first-order baselines in several high-accuracy (entropic) optimal transport problems. LAMP is further shown to scale up to problems with $n=m=2^{18}$ marginal supports, which were previously beyond the reach of primal-dual first-order methods.

math.OC↗

arXiv2026

Accuracy Certificates for Convex Optimization at Accelerated Rates via Primal-Dual Averaging

Many works in convex optimization provide rates for achieving a small primal gap. However, this quantity is typically unavailable in practice. In this work, we show that solving a regularized surrogate with algorithms based on simple primal-dual averaging provides non-asymptotic convergence guarantees for a \textit{computable} optimality certificate. We first analyze primal and dual methods based on one average, namely modified dual averaging and generalized conditional gradient, and establish $\tilde{O}(\varepsilon^{-1})$ certificate complexities. Motivated by asymmetries in the one-average case, we analyze a self-dual, two-average method that preserves symmetry while losing certificate guarantees. To recover certificate convergence, we propose a three-average method that achieves an accelerated $\tilde{O}(\varepsilon^{-1/2})$ certificate complexity. Furthermore, we prove primal-dual algorithm correspondences for the one, two, and three-average cases. In particular, the primal three-average accelerated method mirrors the well-known gradient extrapolation method in the dual. By interpreting our results through the lens of zero-sum matrix games and Fisher markets, we further connect primal-dual averaging methods to game theory and market dynamics.

math.OC↗

arXiv2026

General Oscillator-Based Ising Machine Models with Phase-Amplitude Dynamics and Polynomial Interactions

We present an oscillator model with both phase and amplitude dynamics for oscillator-based Ising machines (OIMs). The model targets combinatorial optimization problems with polynomial cost functions of arbitrary order and addresses fundamental limitations of previous OIM models through a mathematically rigorous formulation with a well-defined energy function and corresponding dynamics. The model demonstrates monotonic energy decrease and reliable convergence to low-energy states. Empirical evaluations on 3-SAT problems show significant performance improvements over existing phase-amplitude models. Additionally, we propose a flexible, generalizable framework for designing higher-order oscillator interactions, from which we derive a practical method for oscillator binarization without compromising performance. This work strengthens both the theoretical foundation and practical applicability of oscillator-based Ising machines for complex optimization problems.

cs.ET↗

arXiv2026

Limitations in Parallel Ising Machine Networks: Theory and Practice

Analog Ising machines (IMs) occupy an increasingly prominent area of computer architecture research, offering high-quality and low latency/energy solutions to intractable computing tasks. However, IMs have a fixed capacity, with little to no utility in out-of-capacity problems. Previous works have proposed parallel, multi-IM architectures to circumvent this limitation. In this work we theoretically and numerically investigate tradeoffs in parallel IM networks to guide researchers in this burgeoning field. We propose formal models of parallel IM excution models, then provide theoretical guarantees for probabilistic convergence. Numerical experiments illustrate our findings and provide empirical insight into high and low synchronization frequency regimes. We also provide practical heuristics for parameter/model selection, informed by our theoretical and numerical findings.

cs.ET↗

arXiv2026

Improved Analysis of Restarted Accelerated Gradient and Augmented Lagrangian Methods via Inexact Proximal Point Frameworks

This paper studies a class of double-loop (inner-outer) algorithms for convex composite optimization. For unconstrained problems, we develop a restarted accelerated composite gradient method that attains the optimal first-order complexity in both the convex and strongly convex settings. For linearly constrained problems, we introduce inexact augmented Lagrangian methods, including a basic method and an outer-accelerated variant, and establish near-optimal first-order complexity for both methods. The established complexity bounds follow from a unified analysis based on new inexact proximal point frameworks that accommodate relative and absolute inexactness, acceleration, and strongly convex objectives. Numerical experiments on LASSO and linearly constrained quadratic programs demonstrate the practical efficiency of the proposed methods.

math.OC↗

arXiv2025

Provable Accuracy Bounds for Hybrid Dynamical Optimization and Sampling

Analog dynamical accelerators (DXs) are a growing sub-field in computer architecture research, offering order-of-magnitude gains in power efficiency and latency over traditional digital methods in several machine learning, optimization, and sampling tasks. However, limited-capacity accelerators require hybrid analog/digital algorithms to solve real-world problems, commonly using large-neighborhood local search (LNLS) frameworks. Unlike fully digital algorithms, hybrid LNLS has no non-asymptotic convergence guarantees and no principled hyperparameter selection schemes, particularly limiting cross-device training and inference. In this work, we provide non-asymptotic convergence guarantees for hybrid LNLS by reducing to block Langevin Diffusion (BLD) algorithms. Adapting tools from classical sampling theory, we prove exponential KL-divergence convergence for randomized and cyclic block selection strategies using ideal DXs. With finite device variation, we provide explicit bounds on the 2-Wasserstein bias in terms of step duration, noise strength, and function parameters. Our BLD model provides a key link between established theory and novel computing platforms, and our theoretical results provide a closed-form expression linking device variation, algorithm hyperparameters, and performance.

cs.LG↗

arXiv2024

GALIC: Hybrid Multi-Qubitwise Pauli Grouping for Quantum Computing Measurement

Observable estimation is a core primitive in NISQ-era algorithms targeting quantum chemistry applications. To reduce the state preparation overhead required for accurate estimation, recent works have proposed various simultaneous measurement schemes to lower estimator variance. Two primary grouping schemes have been proposed: fully commutativity (FC) and qubit-wise commutativity (QWC), with no compelling means of interpolation. In this work we propose a generalized framework for designing and analyzing context-aware hybrid FC/QWC commutativity relations. We use our framework to propose a noise-and-connectivity aware grouping strategy: Generalized backend-Aware pauLI Commutation (GALIC). We demonstrate how GALIC interpolates between FC and QWC, maintaining estimator accuracy in Hamiltonian estimation while lowering variance by an average of 20\% compared to QWC. We also explore the design space of near-term quantum devices using the GALIC framework, specifically comparing device noise levels and connectivity. We find that error suppression has a more than $10\times$ larger impact on device-aware estimator variance than qubit connectivity with even larger correlation differences in estimator biases.

quant-ph↗