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Sharp convergence rates for Langevin dynamics in the nonconvex setting
Xiang Cheng, Niladri S. Chatterji, Yasin Abbasi-Yadkori, Peter L. Bartlett, Michael I. Jordan
TL;DR
The paper studies nonconvex sampling under structural assumptions beyond log-concavity, addressing limited nonasymptotic understanding of dimension dependence. It analyzes overdamped and underdamped Langevin algorithms and shows polynomial dependence on dimension and accuracy, with exponential dependence on LR^2; underdamped dynamics improve runtime analogously to the strongly convex setting.
Problem
The paper addresses limited nonasymptotic understanding of how sampling convergence rates depend on dimension, especially for useful nonconvex settings.
Method
The paper analyzes overdamped and underdamped Langevin MCMC for distributions satisfying structural assumptions more general than log-concavity.
Results
Sampling is possible in time polynomial in dimension d and target accuracy ε, while convergence depends exponentially on LR^2.
Takeaways & Limitations
Underdamped dynamics improve runtime over the first-order method, mirroring the strongly convex case, when the problem is not too badly nonconvex.
Takeaways & Limitations
The guarantees retain exponential dependence on LR^2, and the paper identifies alleviating this dependence as an open direction.
Abstract
from arXiv · showhide
We study the problem of sampling from a distribution $p^*(x) \propto \exp\left(-U(x)\right)$, where the function $U$ is $L$-smooth everywhere and $m$-strongly convex outside a ball of radius $R$, but potentially nonconvex inside this ball. We study both overdamped and underdamped Langevin MCMC and establish upper bounds on the number of steps required to obtain a sample from a distribution that is within $ε$ of $p^*$ in $1$-Wasserstein distance. For the first-order method (overdamped Langevin MCMC), the iteration complexity is $\tilde{\mathcal{O}}\left(e^{cLR^2}d/ε^2\right)$, where $d$ is the dimension of the underlying space. For the second-order method (underdamped Langevin MCMC), the iteration complexity is $\tilde{\mathcal{O}}\left(e^{cLR^2}\sqrt{d}/ε\right)$ for an explicit positive constant $c$. Surprisingly, the iteration complexity for both these algorithms is only polynomial in the dimension $d$ and the target accuracy $ε$. It is exponential, however, in the problem parameter $LR^2$, which is a measure of non-log-concavity of the target distribution.
1. Introduction
The paper studies gradient-based Langevin samplers for potentials that are smooth everywhere and strongly convex outside a ball, while allowing nonconvexity inside. It establishes polynomial dependence on dimension and accuracy, with exponential dependence on LR^2, and shows a faster underdamped rate.
- The target potential is L-smooth everywhere and m-strongly convex outside a ball of radius R, while remaining potentially nonconvex inside.
- The work addresses convergence rates for sampling algorithms, emphasizing explicit dependence on dimension in nonconvex settings.
- Overdamped Langevin MCMC discretizes a first-order stochastic differential equation driven by gradients of the potential and Brownian motion.
- Underdamped Langevin diffusion provides a second-order alternative that can be discretized into an MCMC algorithm and has been observed to converge faster than first-order methods.
- Both algorithms achieve sampling time polynomial in dimension d and target accuracy ε, while convergence depends exponentially on LR^2, a measure of nonconvexity.
- The overdamped method requires approximately e O(e^cLR^2 d/ε^2) steps, whereas underdamped Langevin MCMC achieves a quadratic speed-up in d and ε.
2. Notation, definitions and assumptions
The paper assumes a continuously differentiable potential with Lipschitz gradients, a stationary point at zero, and strong convexity outside a ball. It defines couplings between probability measures and notes that its strong-convexity condition is stronger than related assumptions.
- Assumptions on U: The potential U is continuously differentiable on R^d with globally Lipschitz-continuous gradients.There exists L > 0 controlling the gradient differences.
- Assumptions on U: U has a stationary point at zero, imposed without loss of generality by shifting coordinates.The paper states that a stationary point can be found and used as the origin.
- Assumptions on U: U is strongly convex whenever two points are more than R apart.The condition uses constants m,R > 0 and applies for all x,y with ||x − y||_2 > R.
- Assumptions on U: The condition number is defined as κ := L/m.
- Assumptions on U: The strong-convexity assumption applies to all y, unlike related one-point conditions that fix y = 0.The paper compares this requirement with assumptions in several recent works.
- Coupling and Wasserstein distance: A coupling is a joint distribution whose marginals are the specified probability measures.The paper represents such couplings through transference plans in Γ(μ,ν).
3. Overdamped Langevin diffusion
The overdamped Langevin diffusion is discretized into an implementable MCMC algorithm whose distribution converges to p*(y) ∝ e^−U(y). Its iteration complexity is exponential in LR^2 but polynomial in dimension and target accuracy.
- Overdamped Langevin diffusion: The overdamped Langevin diffusion is an SDE used to construct the sampling method.The paper introduces its SDE before defining the discretized process.
- Overdamped Langevin diffusion: The invariant distribution of the continuous SDE is p*(y) ∝ e^−U(y), so its y-marginal is the target distribution.
- Overdamped Langevin diffusion: The discretized process uses step size δ, with floor function ⌊·⌋ determining the time discretization.
- Overdamped Langevin diffusion: Algorithm 1 implements the discretized SDE, and x_i^δ has the same distribution as the SDE solution at time iδ.The theorem establishes the convergence rate for this algorithm.
- Convergence guarantee: For constant LR^2, overdamped MCMC requires e^O(d/ε^2) iterations, matching the strongly log-concave rate.The general rate is described as exponential in LR^2 and scales with d/ε^2.
- Convergence guarantee: The proof combines Eberle’s continuous-time convergence argument with Durmus and Moulines’s discretization bound.
4. Underdamped Langevin diffusion
The section presents underdamped Langevin diffusion, its discretization as Algorithm 2, and a coupling-based convergence analysis in 1-Wasserstein distance. Under the stated structural assumptions, the resulting iteration complexity is exponentially dependent on LR^2 but improves quadratically over overdamped Langevin MCMC in dimension and accuracy.
- Algorithm and diffusion: Underdamped Langevin diffusion is a second-order stochastic process whose discretization yields the implementable underdamped Langevin MCMC algorithm.The algorithm uses step size δ and Gaussian updates based on the previous position and velocity iterates.
- Algorithm and diffusion: Algorithm 2’s iterates have the same distribution as the corresponding discretized SDE at matching times.This distributional equivalence connects the implementable algorithm to the continuous-time analysis.
- Convergence guarantee: For constant LR^2, underdamped Langevin MCMC requires e O(d/ε) iterations, a quadratic improvement over overdamped Langevin MCMC.The general rate remains exponentially dependent on LR^2, while the stated comparison concerns dimension and target accuracy.
- Coupling analysis: The underdamped convergence proof couples a discretized process with a stationary continuous-time process using synchronous and reflection coupling.The coupling switches between shared Brownian noise and reflected noise according to the processes’ state.
- Coupling analysis: The Lyapunov function contracts exponentially, with synchronous coupling exploiting strong convexity when the coupled state is sufficiently separated.The contraction is established with probability one and transfers to the Lyapunov function used to control 1-Wasserstein distance.
5. Discussion
The discussion emphasizes polynomial sampling guarantees under a structural assumption broader than log-concavity and identifies underdamped dynamics as a source of improved runtime. It also outlines open directions for reducing the exponential dependence on LR^2 and weakening assumptions or enriching the algorithms.
- Contributions: The paper studies sampling from distributions under a structural assumption more general than log-concavity, with runtime polynomial in dimension and accuracy.The discussion frames this as a tractable non-log-concave sampling guarantee.
- Contributions: Underdamped dynamics improve runtime in this setting, mirroring the speedup observed in the strongly convex case.The discussion presents this as a demonstrated consequence of using second-order dynamics.
- Open directions: Future work includes structural assumptions that could alleviate the exponential dependence on LR^2 and guarantees under weaker conditions such as weak convexity outside a ball.The authors also suggest algorithms using more than a gradient oracle or higher-order discretizations.
Appendix B. Two Small Constants
Appendix B explains the small smoothing and coupling constants used to regularize the analysis. The construction smooths the norm near the origin and discretizes coupling updates while preserving the iteration-complexity bounds.
- Smoothing constant β: β is chosen as a small inverse-polynomial quantity and controls the smoothing scale for approximating the Euclidean norm.The exact polynomial depends on problem parameters such as L, 1/m, d, and R.
- Smoothing constant β: Because ℓ is strongly convex near the origin, the coupling dynamics are designed to have no noise in that region.This avoids complications caused by the smoothing-induced curvature inside the β/2 ball.
- Coupling constant ν: ν discretizes switches between synchronous and reflection coupling at deterministic finite intervals to ensure existence and regularity of the coupling solution.The resulting discretization error σ_t is shown to be o(ν^2).
- Coupling constant ν: ν is distinct from the algorithmic step size δ, and unlike ν, δ cannot be made arbitrarily small without additional computation costs.The proof treats ν as an inverse-polynomial auxiliary constant that does not alter the iteration-complexity bound.
- Smoothing constant β: The smoothed norm ℓ is close to the Euclidean norm, equals it outside a β/2 neighborhood, and has continuous second derivatives everywhere.This regularization makes derivatives well-defined while changing the norm by at most a controlled β-dependent error.
Appendix C. Proofs for overdamped Langevin Monte Carlo
The appendix constructs the overdamped coupling and Lyapunov function used to establish convergence to the invariant distribution. Smooth norm approximations and carefully chosen coupled dynamics support the contraction argument and the equivalence of constructed processes to the target SDE.
- Lyapunov analysis: A Lyapunov function built from a distance function f is used to show convergence of the coupled distributions in 1-Wasserstein distance.The construction combines the smoothed distance with auxiliary stochastic processes.
- Coupling construction: The proof begins by constructing a coupling between two overdamped Langevin SDEs and verifying that one coupled process has the invariant marginal distribution.The modified dynamics remain distributionally equivalent to the original SDE.
- Coupling construction: The coupling uses a state-dependent rule that combines synchronous and reflection behavior according to the distance between the processes.The switching function is designed to be Lipschitz and gradient-Lipschitz.
C.2 Proof of Theorem 2
The proof combines Itô-calculus estimates, Lyapunov contraction, and a final Wasserstein-distance argument to establish the iteration bound for Theorem 2.
- Proof strategy: The proof proceeds in three steps: analyze f(ℓ(z_t)), establish contraction of L_t, then convert that contraction into iteration complexity.The final step also yields a bound in 1-Wasserstein distance.
- Step 1: The analysis bounds the evolution of f(ℓ(z_t)) using properties of f, q, γ_t, and the discretization error Δ_t.The bounds split the dynamics across regions determined by ||z_t||_2 and use smoothness and strong-convexity assumptions.
- Step 2: The Lyapunov function L_t contracts at a sufficiently fast rate after applying the derived differential inequalities and Grönwall’s inequality.The argument takes expectations over the Brownian motion and controls the initial Lyapunov value.
- Step 3: The number of steps is obtained by setting t = nδ in the contraction inequality and translating the resulting bound on E[f(ℓ(z_nδ))] into E[||z_nδ||_2].This translation implies a 1-Wasserstein-distance guarantee under the theorem’s assumptions on ε.
Appendix D. Proofs for Underadmped Langevin Monte Carlo
The underdamped Langevin proof establishes contraction of a Lyapunov function by combining case-based dynamics, derivative estimates, and Grönwall’s inequality.
- Lyapunov contraction: The proof shows that L_t contracts over successive intervals, including cases where the switching variable changes between interval endpoints.The transition from μ_{k−1}=0 to μ_k=1 is the technically difficult case because L_t can jump, but the jump is shown to be negative.
- From contraction to complexity: The proof combines contraction with bounds on discretization terms to obtain the theorem’s iteration-complexity guarantee.The resulting Lyapunov contraction is linked to convergence in 1-Wasserstein distance.
- Differential estimates: Itô’s Lemma and derivative calculations control the evolution of the coupling quantities and establish the main differential inequality.The proof separately bounds several terms and uses properties of f, q, γ_t, and the state regions.
- Contraction bound: The main inequality is d(μ_k · (f(r_s) − ξ_s) − σ_s − φ_s) ≤ −C_m(μ_k · (f(r_s) − ξ_s) − (σ_s + φ_s)) + 5β ds.Grönwall’s Lemma converts this differential bound into the claimed contraction result.
D.3 Main results for synchronous coupling
This section proves contraction for synchronous coupling over synchronization periods and uses that contraction to control the underdamped dynamics across switching cases.
- Synchronization contraction: exp(−C_m T_sync) is the contraction factor for f(r_s) over a synchronization period, with probability one.The result is weaker than contraction at every time but sufficient for the proof.
D.4 Discretization Error Bound
The discretization analysis bounds the errors introduced by gradient evaluation and discrete switching, then combines these bounds with Lyapunov contraction to prove the final complexity result.
- Error decomposition: The discretization errors are separated into ξ_t, σ_t, and φ_t, arising respectively from gradient evaluation, discrete switching updates, and a martingale term.The analysis proves E[φ_t] = 0 and bounds the other two terms.
- Gradient discretization: The gradient error ξ_t arises because the underdamped update uses ∇U(x_k) instead of the continuously varying ∇U(x_t).Its bound relies on moment estimates.
- Switching discretization: The switching error σ_t depends on the discrete interval ν and can be made arbitrarily small by reducing ν.The analysis derives polynomial bounds in L, 1/m, d, and R for sufficiently small ν.
- Final complexity: Combining the discretization bounds with recursive Lyapunov contraction yields the final number-of-steps bound.The proof selects ν and β so their contributions are dominated by the remaining terms before deriving the step count.
F.1 Proof of Lemma 32
The proof controls the evolution of a Lyapunov function by bounding its component terms and applying Itô’s and Grönwall’s lemmas. Recursive inequalities then establish the desired bound and complete the lemma.
- The Lyapunov function’s time evolution is analyzed using the stochastic dynamics and Itô’s lemma.
- The proof bounds three terms separately, using inequalities such as Cauchy–Schwarz, Young’s, Jensen’s, and the assumptions on U.
- Applying Grönwall’s lemma over successive intervals and summing the resulting geometric terms yields a bound for all positive integers k.
- Auxiliary lemmas provide bounds for the Lyapunov function and related quantities when δ ≤ 1/1000.
- Taking expectations removes the Brownian-motion term, and the invariant distribution property is used in the final estimate.
Appendix G. Existence of Coupling
The appendix establishes existence and regularity of the coupled stochastic processes by verifying smoothness, Lipschitz, and filtration conditions. It also derives Gaussian transition laws for the underdamped diffusion and verifies Brownian-motion structure through quadratic variation.
- The existence proof proceeds inductively across intervals [kν, (k + 1)ν], using the algebraic equivalence of the dynamics on each interval.
- Smoothness of ∇U and Lipschitz and boundedness properties of the diffusion coefficients verify the conditions required for a stochastic differential-equation solution.
- The coupled process (x_t, y_t, u_t, v_t) is unique and continuous in time with probability one and adapted to the generated filtration.
- The transformed noise process is shown to be Brownian by verifying its continuous-martingale property and quadratic variation through Lévy’s characterization.
- Conditioned on (x_kδ, u_kδ), the underdamped diffusion has a Gaussian transition distribution whose means and covariances are computed explicitly.