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About this paper

Fan Chen, Sinho Chewi, Jianfeng Lu, Matthew S Zhang

Proximal BPS samples smooth, strongly log-concave targets using reflected auxiliary Gaussian offsets and local half-turn particle moves with gradient-correcting bounces, preserving dependence between updates. Given an order-2 Rényi warm start, its expected gradient-query complexity for total-variation error ε is Õ(√κ d^(1/4)) when the initial divergence is O(1), improving dimension dependence over the authors’ BPS bound while retaining its √κ dependence; the guarantee excludes warm-start construction.

Key insights

• A key insight is that deterministically reflecting an auxiliary variable in an augmented space can effectively transfer momentum, enabling sustained progress across successive localized problems.
• The 'conditional half-turn process', with its fixed duration of 'pi', allows the particle's harmonic motion to approximately reverse initial momentum, acting as a form of kinetic acceleration.
• The rigorous convergence analysis is achieved through a modified L^2 hypocoercivity framework, adapted for a discrete-time Markov chain, which quantifies how reflection and half-turn dynamics lead to one-step decay.

Summary by alphaXiv.