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

arXiv: https://arxiv.org/abs/2604.13022 · HTML: https://mlsanity.com/paper/2604.13022/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 16 references that are in the map.
- First posted on arXiv 14 Apr 2026.

## Builds on (most cited first, 6 of 6)

- [An Alternative View: When Does SGD Escape Local Minima?](https://mlsanity.com/paper/1802.06175/) (2018)
- [On Learning Rates and Schr"odinger Operators](https://mlsanity.com/paper/2004.06977/) (2020)
- [Power-law escape rate of SGD](https://mlsanity.com/paper/2105.09557/) (2021)
- [Born-Infeld (BI) for AI: Energy-Conserving Descent (ECD) for Optimization](https://mlsanity.com/paper/2201.11137/) (2022)
- [On Quantum Speedups for Nonconvex Optimization via Quantum Tunneling Walks](https://mlsanity.com/paper/2209.14501/) (2022)
- [Improving Energy Conserving Descent for Machine Learning: Theory and Practice](https://mlsanity.com/paper/2306.00352/) (2023)

## Abstract (arXiv)

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 in a tractable model where the barrier-crossing mechanism can be computed explicitly. For one-dimensional double-well objectives in the under-guessing regime, we compute the expected dynamical hitting times from a local minimum to the global minimum. We prove that both sECD and qECD exhibit exponential improvements in continuous hitting time relative to their respective gradient-based baselines, stochastic gradient descent (SGD) and quantum tunneling walk (QTW). For objectives with tall barriers, qECD admits a further hitting time improvement over sECD. Mechanistically, ECD sidesteps the exponential cost associated with rare-escape events of SGD from local minima by moving from dissipative to energy-conserving dynamics.