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

2026-04-14

AI Summary

This work addresses the challenge of non-convex optimization, where algorithms often become trapped in local minima and struggle to converge efficiently to the global optimum. To overcome this, the authors propose the Energy-Conserving Descent (ECD) method and provide the first theoretical analysis of its properties. They further introduce a stochastic variant, sECD, by incorporating energy-preserving noise, and develop a quantum version, qECD, leveraging quantum Hamiltonian simulation. Through modeling via stochastic differential equations and first-hitting-time analysis on a one-dimensional double-well objective function, both sECD and qECD achieve exponential acceleration over classical gradient descent. Notably, in high-barrier regimes, qECD substantially outperforms sECD, demonstrating a clear quantum advantage.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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.

Problem

• non-convex optimization
• global minimum
• local minima
• energy barriers
• optimization speedup

Innovation

• Energy Conserving Descent
• non-convex optimization
• quantum speedup
• stochastic dynamics
• Hamiltonian simulation