Source-linked AI summary
User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient
Arnak S. Dalalyan, Avetik G. Karagulyan
TL;DR
The paper asks how to sample from smooth, strongly log-concave densities when Langevin methods are discretized and gradient evaluations may be inaccurate. It analyzes first- and second-order LMC variants and establishes Wasserstein-2 error guarantees, including improved varying-step results and efficient second-order sampling. The results also identify a limitation for applying the inaccurate-gradient theorem directly to SGLD.
Problem
The paper addresses nonasymptotic error guarantees for Langevin sampling from smooth, strongly log-concave densities, including when accurate log-density gradients are unavailable.
Method
The authors analyze discretizations of Langevin diffusion using first-order LMC with constant or varying step sizes, inaccurate-gradient models, and second-order methods leveraging Hessian information.
Results
The paper establishes Wasserstein-2 guarantees that improve or extend prior LMC results, including a varying-step advantage and nearly LMCO-level guarantees for LMCO’.
Takeaways & Limitations
Lipschitz Hessians permit larger step sizes and improved convergence rates, while second-order LMC methods can improve first-order LMC in ill-conditioned settings.
Takeaways & Limitations
Applied directly to SGLD, Theorem 4 is of limited interest because its stochastic-gradient error term can dominate the other terms.
Abstract
from arXiv · showhide
In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We provide nonasymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings.
1. Introduction
The paper develops user-friendly, nonasymptotic Wasserstein guarantees for Langevin Monte Carlo sampling from smooth, strongly log-concave targets. It improves constant- and varying-step results, extends analysis to inaccurate gradients, and introduces a computationally efficient second-order method.
- Problem and framework: The paper studies LMC for sampling from smooth, strongly log-concave target densities and measures approximation error using the Wasserstein-2 distance.The authors motivate W2 as useful for statistical and machine-learning applications because it directly bounds first- and second-order moment accuracy.
- Problem and framework: The proposed guarantees are nonasymptotic, easy to check, and designed to guide iteration-count and step-size choices without advance knowledge of the target precision.Such bounds also support stopping rules and computational-complexity analysis.
- First-order methods: The paper simplifies constant-step LMC and LMCO guarantees with improved constants and proposes a varying-step LMC that removes a logarithmic iteration factor.The varying-step result is stated as avoiding a logarithmic factor in the iterations needed to reach precision ϵ.
- Inaccurate gradients: The analysis extends LMC guarantees to deterministic and stochastic gradient errors, including settings without unbiasedness or independence assumptions.Theorem 4 applies under the base assumptions, while Theorem 5 additionally requires a smooth Hessian; the resulting bound is tighter than the earlier result it extends.
- Second-order methods: The new second-order method LMCO’ has per-iteration cost comparable to LMC and nearly the guarantees of LMCO when the Hessian is Lipschitz continuous.Its implementation avoids matrix exponentials and matrix inversions, requiring matrix-vector multiplications instead.
- Illustration: For ϵ = 0.001, the varying-step iteration bound averages 4.6 times fewer iterations than one comparison bound and almost 3 times fewer than the constant-step bound.The varying-step advantage becomes more important as the target precision becomes smaller.
2. Guarantees in the Wasserstein distance with accurate gradient
The paper gives Wasserstein-2 guarantees for LMC targeting smooth, strongly log-concave densities, including constant- and varying-step schemes. A varying-step choice is independent of target precision and removes logarithmic iteration factors.
- Setup: The LMC Markov chain is the Euler discretization of Langevin diffusion, whose invariant density is the target π.The analysis studies the Wasserstein-2 distance between the iterate distribution ν_k and π.
- Constant-step LMC: Under the theorem’s conditions, choosing h and K appropriately ensures W2(νK, π) ≤ ε.These conditions provide a nonasymptotic stopping guarantee for the LMC output.
- Varying-step LMC: The varying-step LMC improves the constant-step bound by a logarithmic factor in p/ε2.The paper motivates this scheme as an explicit step-size selection that clarifies how varying steps improve convergence.
- Varying-step LMC: The varying step-size is independent of ε, and K = K1 + (27M2/2m3)(p/ε2) iterations suffice for precision ε.K1 depends neither on dimension p nor on precision ε under the stated result.
- Mixture extension: The mixture LMC extends Langevin-based sampling to mixtures whose conditional component potentials satisfy the strong log-concavity assumption.The result follows directly from the varying-step guarantee.
3. Guarantees for the inaccurate gradient version
The paper extends LMC guarantees to inaccurate gradient evaluations, covering deterministic bias and stochastic noise under relaxed dependence assumptions. It also shows how these guarantees apply to subsampling and clarifies limitations for SGLD.
- Model and assumptions: The noisy LMC uses gradient evaluations with deterministic and stochastic inaccuracies, formalized through parameters δ and σ.Theorem 4 analyzes this setting under Condition N.
- Model and assumptions: Theorem 4 requires independence of each Langevin update from past gradient noise, but does not require the noise vectors themselves to be independent.This relaxes the independence assumption used in earlier work.
- Theoretical guarantee: Theorem 4 extends earlier guarantees to biased gradient noise and improves the small-σ dependence from O(σ) to O(σ2).The sharper bound follows from a recursive inequality and removes the need for unbiasedness.
- Applications: For uniformly subsampled gradients, the subsampling contribution has the same order as the second error term, so precision does not deteriorate significantly while computation is reduced.This conclusion applies under suitable assumptions with δ = 0 and constant σ2.
- Applications: The theorem accommodates deterministic approximation bias, such as quadrature error, and stochastic variance, such as Monte Carlo error.The two error types correspond to δ and σ, respectively.
- Limitations: For SGLD, the theorem is of limited interest because stochastic-gradient noise can dominate the other error terms.In the example discussed, the final term is of order (nhp)1/2 and remains larger after using s subsamples.
4. Guarantees under additional smoothness
Under Lipschitz continuity of the Hessian, the paper derives guarantees for inaccurate-gradient LMC and analyzes two Hessian-based discretizations. The second-order methods have comparable theoretical bounds, while the modified version has lower per-iteration cost.
- Inaccurate-gradient LMC: Lipschitz continuity of the Hessian permits improved convergence guarantees for the inaccurate-gradient LMC.Theorem 5 assumes conditions F and N and uses a step-size h ≤ 2/(m+M).
- Inaccurate-gradient LMC: Theorem 5 separates the effects of step-size h, iteration count K, deterministic tolerance δ, and stochastic precision σ.For Wasserstein error of order ϵ, the discussion recommends σ of order √ϵ and δ of order mϵ/√p.
- Inaccurate-gradient LMC: With accurate gradients, Theorem 5 recovers the constant-step-size result of [15, Theorem 8] with optimized constants.The comparison is illustrated through the optimal iteration counts plotted in Figure 2.
- Second-order methods: The LMCO and LMCO’ algorithms use the Hessian in addition to the gradient to exploit its Lipschitz continuity.LMCO uses Hessian-based matrix exponentials, whereas LMCO’ approximates those exponentials with linear functions.
- Second-order methods: The second-order guarantees are established for exact gradient and Hessian evaluations, with step-size h ≤ m/M2.The analysis covers both LMCO and LMCO’.
- Second-order methods: LMCO and LMCO’ have similar theoretical guarantees, but LMCO’ has significantly lower per-iteration cost because it avoids matrix exponentials and matrix inversions.LMCO’ only needs the simpler operations described in the paper.
5. Relation with optimization
The paper frames LMC as a sampling analogue of gradient descent and connects second-order sampling methods to Newton’s method. It also shows that variable step-sizes can improve LMC convergence, while the LMCO’ guarantees do not directly recover standard Newton guarantees.
- LMC is presented as the sampling analogue of gradient descent, extending optimization convergence results to sampling.
- Sampling to precision ε requires O((p/ε2) log(p/ε)) LMC iterations, compared with O(log(1/ε)) steps for approximate optimization.
- As τ → 0, the LMC sequence for fτ converges to the gradient-descent sequence because πτ concentrates at the minimizer.
- LMCO is the sampling counterpart of Newton’s method: its stochastic perturbation vanishes in the τ → 0 limit, yielding Newton updates.
- Theorem 6 does not recover standard Newton guarantees because it analyzes LMCO for small step-sizes, whereas Newton corresponds to a fixed-step-size limit.
6. Conclusion
The paper develops user-friendly finite-sample guarantees for Langevin sampling with fixed or variable step-sizes, inaccurate gradients, and second-order information. It also relates sampling to optimization and identifies several open directions, including lower bounds and nonsmooth sampling.
- The results provide finite-sample guarantees for strongly log-concave sampling using fixed-step LMC and extend them to deterministic and random gradient errors.
- With a Lipschitz continuous second derivative, larger step-sizes yield improved convergence rates.
- Optimization algorithms can arise as limiting cases of sampling algorithms, including gradient descent from LMC and Newton’s method from LMCO.
- LMCO’ improves LMC’s dependence on the scale while keeping each iteration computationally cheaper than LMCO.
- Accurate gradient evaluations combined with a suitable variable step-size provide a logarithmic improvement in LMC convergence.
- Future work includes lower bounds, user-friendly posterior-mean guarantees, and sampling from nonsmooth densities.
7. Proofs
The proofs couple discrete LMC-type chains with stationary Langevin diffusions and derive recursive Wasserstein-error bounds. Auxiliary contraction, remainder, stochastic-process, and recursion lemmas then control first- and second-order discretization errors.
- The central proof strategy bounds W2(νk+1, π) through a recursive inequality involving W2(νk, π).
- A stationary Langevin diffusion provides the coupling reference, with W2 error represented through the L2 distance between the chain iterate and diffusion state.
- For noisy gradients, the recursion incorporates deterministic and stochastic remainder terms, with the noiseless case recovered by setting σ = δ = 0.
- Strong convexity and small step-sizes make the gradient update contractive, supplying the multiplicative error-reduction factor.
- Unrolling the recursion yields a Wasserstein bound combining initial error, contraction, and discretization error, including the term αM(h3p)1/2.
- Gronwall-type control, integration by parts, conditional-expectation identities, and auxiliary lemmas bound the stochastic remainder terms.
- The second-order proofs construct continuous-time comparison processes whose one-step distributions match LMCO or LMCO’ iterates, then control their difference from Langevin diffusion.