# Global Convergence of Third-Order Langevin Dynamics for Non-Convex Optimization via Simulated Annealing

arXiv: https://arxiv.org/abs/2609.28611 · HTML: https://mlsanity.com/paper/2609.28611/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 58 references that are in the map.
- First posted on arXiv 23 Sep 2026.

## Builds on (most cited first, 2 of 2)

- [Fast and Efficient Parallel Sampling Using Higher Order Langevin Dynamics](https://mlsanity.com/paper/2510.18242/) (2025)
- [High-Order Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2508.17545/) (2025)

## Abstract (arXiv)

We study global convergence guarantees of third-order Langevin dynamics for non-convex optimization via simulated annealing with fixed friction and decreasing noise. An explicit three-block distorted entropy transfers dissipation from the noisy auxiliary variable to the full state. Under dissipativity, regularity, and low-temperature functional-inequality assumptions, logarithmic cooling drives the objective values to the global minimum in probability at the barrier-controlled kinetic rate. For the exact-force-integral and midpoint three-stage discretizations, polynomially decreasing steps preserve this rate on the physical time scale. The cubic local endpoint estimate gives a less restrictive sufficient step-size condition than the available frozen-force kinetic result. A comparison with the one-gradient UBU integrator shows how its centered stochastic local error leads, under the same strong-coupling analysis, to a smaller sufficient iteration exponent. Numerical experiments are conducted to illustrate our theory. For a double well objective, third-order Langevin terminal-success point estimates are higher than UBU at both a common horizon and an equal gradient budget. For a high-dimensional nonconvex neural-network objective using synthetic data, independently tuned UBU and third-order Langevin schemes both outperform overdamped Langevin dynamics; the third-order Langevin point estimate is higher. For the same neural-network objective on real data, we show the same point-estimate ordering for best-basin probability and post-quench test accuracy. Numerical code and associated experiment results are publicly available at https://github.com/gagawjbytw/simulated-annealing-third-order-langevin.