Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyond - PowerPoint PPT Presentation

Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyond Xuechen Li 1,2 Denny Wu 1,2 Lester Mackey 3 Murat A. Erdogdu 1,2 1 University of Toronto 2 Vector Institute 3 Microsoft Research

The Problem and Our Work Given smooth potential f : R d → R , sample from given density p ( x ) ∝ exp( − f ( x )) . • We study both strongly convex and non-convex potentials. • Many papers study individual algorithms [1, 2, 3, 4, 5]. However, there has yet to be a unifying theoretical framework. • We provide a theorem that gives the convergence rate of sampling algorithms obtained by discretizing an exponentially contracting diffusion based on local properties of the numerical method. • A direct extension is we obtain faster converging algorithms with the class of stochastic Runge-Kutta (SRK) methods.

Exponential W 2 -Contraction of Diffusions Diffusion X t has exponential W 2 -contraction if two instances X t , x , X t , y initiated respectively from x and y satisfy W 2 ( X t , x , X t , y ) ≤ e − α t � x − y � 2 , for all x , y ∈ R d , t ≥ 0 . Informal: The marginals of the continuous-time diffusion become the same very quickly regardless of the initial state. Example: When f is strongly convex, the Langevin diffusion characterized by the SDE √ d X t = −∇ f ( X t ) d t + 2 d B t has exponential W 2 -contraction.

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Uniform Orders of Local Deviation Recall local deviation D ( k ) = X ( k ) − ˜ X k . A numerical scheme has h h uniform mean-square and mean orders of ( p 1 , p 2 ) if for all k ∈ N E (1) � � D ( k ) �� h � 2 ≤ λ 1 h 2 p 1 , � = E 2 |F t k − 1 (1) E k �� � E (2) D ( k ) � 2 ≤ λ 2 h 2 p 2 , � �� = E � E h |F t k − 1 (2) k 2 for constants λ 1 and λ 2 independent of h . Remark: Bounds like (1) appeared explicitly in previous works (see e.g. [1]). To the best of our knowledge, (2) did not appear explicitly in previous works.

A General Theorem Theorem (Informal) Diffusion has a stationary distribution p ( x ) ∝ exp( − f ( x )) and exhibits exponential W 2 -contraction. Acting on this diffusion, a numerical discretization with uniform mean-square and mean 2 has rate ˜ orders of ( p 1 , p 2 ) for p 2 ≥ p 1 + 1 O ( ǫ − 1 / ( p 1 − 1 / 2) ) in W 2 . Remark 1: Connects the numerical SDE and sampling literatures: Take any classical SDE discretization method, instantly know the convergence rate when it’s used for sampling! Remark 2: Can also be used for discretizing the underdamped Langenvin diffusion! Check out our examples in the paper.

Convergence Rates for EM and SRK Result Diffusion Smoothness Unif. Orders Rate ˜ O ( d ǫ − 2 ) Langevin (1 . 0 , 1 . 5) EM (Durmus et al.) 1st ˜ O ( d ǫ − 1 ) Langevin (1 . 5 , 2 . 0) EM (Ex. 1) 1st & 2nd ˜ O ( d ǫ − 2 / 3 ) Langevin (2 . 0 , 2 . 5) SRK-LD (This work) 1st-3rd ˜ O ( d ǫ − 2 ) General (1 . 0 , 1 . 5) EM (Ex. 2) 1st ˜ O ( d 3 / 4 m 2 ǫ − 1 ) General (1 . 5 , 2 . 0) SRK-ID (This work) 1st Table: Convergence rates in W 2 , i.e. number of iterations required to reach ǫ accuracy to the target in W 2 . Top three for strongly convex f ; bottom two for non-convex f that admits uniformly dissipative diffusion. EM = Euler-Maruyama SRK = Stochastic Runge-Kutta

Thanks to you and my coauthors: Denny Wu Lester Mackey Murat A. Erdogdu

Our poster: East Exhibition Hall B + C #162 [1] Xiang Cheng, Niladri S Chatterji, Peter L Bartlett, and Michael I Jordan. Underdamped Langevin MCMC: A non-asymptotic analysis. [2] Arnak S Dalalyan. Theoretical guarantees for approximate sampling from smooth and log-concave densities. [3] Alain Durmus, Eric Moulines, et al. Nonasymptotic convergence analysis for the unadjusted langevin algorithm. [4] Yin Tat Lee, Zhao Song, and Santosh S Vempala. Algorithmic theory of odes and sampling from well-conditioned logconcave densities. [5] Santosh S Vempala and Andre Wibisono. Rapid convergence of the unadjusted langevin algorithm: Log-sobolev suffices.