Classical and Quantum Speedups for Non-Convex Optimization via Energy Conserving Descent

arXiv stat.ML / 4/15/2026

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Key Points

The paper provides the first analytical study of the Energy Conserving Descent (ECD) method for global optimization of non-convex problems, focusing on a one-dimensional setting.
It formalizes both a stochastic ECD dynamics with energy-preserving noise (sECD) and a quantum analog based on a Hamiltonian formulation (qECD), establishing a route to a quantum algorithm via Hamiltonian simulation.
For positive double-well objectives, the authors derive expected hitting times from local to global minima and prove exponential speedups of sECD and qECD over gradient descent baselines.
For non-convex objectives with tall barriers, qECD is shown to provide an additional speedup over sECD, suggesting stronger quantum advantage in harder landscapes.

Abstract

The Energy Conserving Descent (ECD) algorithm was recently proposed (De Luca & Silverstein, 2022) as a global non-convex optimization method. Unlike gradient descent, appropriately configured ECD dynamics escape strict local minima and converge to a global minimum, making it appealing for machine learning optimization. We present the first analytical study of ECD, focusing on the one-dimensional setting for this first installment. We formalize a stochastic ECD dynamics (sECD) with energy-preserving noise, as well as a quantum analog of the ECD Hamiltonian (qECD), providing the foundation for a quantum algorithm through Hamiltonian simulation. For positive double-well objectives, we compute the expected hitting time from a local to the global minimum. We prove that both sECD and qECD yield exponential speedup over respective gradient descent baselines--stochastic gradient descent and its quantization. For objectives with tall barriers, qECD achieves a further speedup over sECD.