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

TL;DR

This paper analyzes Energy Conserving Descent (ECD), demonstrating exponential speedups over gradient descent in non-convex optimization, and introduces quantum analogs for further acceleration.

Contribution

First analytical study of ECD in one dimension, introducing stochastic and quantum versions with proven exponential speedups over classical methods.

Findings

Both sECD and qECD outperform gradient descent in expected hitting times.

qECD achieves additional speedup over sECD for tall barrier problems.

The paper provides the foundation for quantum algorithms based on ECD Hamiltonian simulation.

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.