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Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization

Pan Xu, Jinghui Chen, Difan Zou, Quanquan Gu

arXiv:1707.06618v3stat.MLcs.LGmath.OC

TL;DR

The paper studies how to obtain global convergence guarantees for Langevin-based algorithms on nonconvex finite-sum optimization, beyond methods that may reach only stationary points. It directly analyzes the ergodicity of discrete-time algorithms and proves convergence guarantees for GLD, SGLD, and SVRG-LD, including a regime where SVRG-LD outperforms the other two.

  • Problem

    Existing nonconvex optimization algorithms are guaranteed only to converge to stationary points, motivating the question of efficiently reaching a global minimum.

  • Method

    The paper uses a unified framework that directly analyzes the ergodicity of discrete-time Langevin-based Markov chains rather than continuous-time diffusion.

  • Results

    GLD, SGLD, and SVRG-LD globally converge to an almost minimizer with the stated stochastic-gradient complexities, with SVRG-LD outperforming GLD and SGLD when 1/ε^3 ≤ n ≤ 1/ε^5.

  • Takeaways & Limitations

    The results provide provable global-convergence guarantees for Langevin dynamics based algorithms in nonconvex finite-sum optimization.

  • Takeaways & Limitations

    The paper leaves open whether SVRG-LD is universally better than GLD for nonconvex optimization.

Abstract

from arXiv · show

We present a unified framework to analyze the global convergence of Langevin dynamics based algorithms for nonconvex finite-sum optimization with $n$ component functions. At the core of our analysis is a direct analysis of the ergodicity of the numerical approximations to Langevin dynamics, which leads to faster convergence rates. Specifically, we show that gradient Langevin dynamics (GLD) and stochastic gradient Langevin dynamics (SGLD) converge to the almost minimizer within $\tilde O\big(nd/(λε) \big)$ and $\tilde O\big(d^7/(λ^5ε^5) \big)$ stochastic gradient evaluations respectively, where $d$ is the problem dimension, and $λ$ is the spectral gap of the Markov chain generated by GLD. Both results improve upon the best known gradient complexity results (Raginsky et al., 2017). Furthermore, for the first time we prove the global convergence guarantee for variance reduced stochastic gradient Langevin dynamics (SVRG-LD) to the almost minimizer within $\tilde O\big(\sqrt{n}d^5/(λ^4ε^{5/2})\big)$ stochastic gradient evaluations, which outperforms the gradient complexities of GLD and SGLD in a wide regime. Our theoretical analyses shed some light on using Langevin dynamics based algorithms for nonconvex optimization with provable guarantees.

1 Introduction

The paper addresses the open problem of globally minimizing nonconvex finite-sum objectives, where existing first-order methods generally guarantee only convergence to stationary points. It develops a unified, direct discrete-time analysis of Langevin algorithms and establishes global convergence guarantees for GLD, SGLD, and SVRG-LD.

  • Existing nonconvex optimization methods are guaranteed only to reach stationary points, which may be local minima, local maxima, or saddle points.
  • Langevin dynamics use sampling distributions concentrated around the global minimum to motivate optimization algorithms for nonconvex objectives.
  • The framework directly analyzes the ergodicity of discrete-time numerical approximations rather than continuous-time diffusion, avoiding discretization error.
  • Our Contributions: GLD globally converges to an almost minimizer within eO(nd/(λε)) stochastic gradient evaluations, improving the complexity implied by prior analysis.

2 Review of Langevin Dynamics Based Algorithms

The section reviews GLD, SGLD, and SVRG-LD as Langevin-based algorithms for nonconvex optimization, distinguishing full-gradient, stochastic-gradient, and variance-reduced updates.

  • Gradient Langevin Dynamics: GLD discretizes Langevin diffusion with Euler–Maruyama, adding Gaussian noise to the gradient-descent update.The noise magnitude is controlled by the inverse temperature parameter β, while η is the step size.
  • Stochastic Gradient Langevin Dynamics: SGLD replaces the full gradient with an unbiased stochastic-gradient estimator computed from a random minibatch.This reduces the cost of each iteration when n is large, but stochastic-gradient variance can slow convergence.
  • Stochastic Variance Reduced Gradient Langevin Dynamics: SVRG-LD uses a semi-stochastic gradient and periodically updated snapshots to reduce stochastic-gradient variance.Its update samples a subset of component functions and uses a snapshot every L iterations.
  • Scope: All three algorithms were originally proposed for posterior sampling or Bayesian learning but can also be applied to nonconvex optimization.The paper reviews their optimization use as Langevin dynamics based algorithms.

3 Main Theory

The paper analyzes global convergence of GLD, SGLD, and SVRG-LD for nonconvex finite-sum optimization under smoothness and dissipativity assumptions. Its direct analysis of discrete-time Markov-chain ergodicity yields convergence guarantees to an almost minimizer and gradient-complexity comparisons across the algorithms.

  • Optimization framework: Under smoothness and dissipativity assumptions, the paper bounds optimization error by separating the algorithm-dependent iterate-to-stationarity gap from the stationary-distribution gap to the global minimum.The stationary distribution is a Gibbs distribution concentrated around the global minimizer, while the iterate-to-stationarity term is controlled by algorithmic convergence.
  • GLD: GLD’s first error term vanishes exponentially through Markov-chain ergodicity, with the rate controlled by the discrete-time spectral gap λ.The spectral gap λ is stated to be of order e^-e O(d).
  • GLD: GLD achieves ε-precision convergence to the almost minimizer after the iteration complexity stated in Corollary 3.4, improving on the earlier continuous-time spectral-gap analysis.The earlier result cited in the paper gives O(d/(ε^4λ*) log^5(1/ε)) iterations, whereas the paper’s comparison emphasizes the improved GLD analysis.
  • SGLD: SGLD achieves E[F_n(Y_K)] − E[F_n(X_π)] ≤ ε within K = O(d/(λε) log(1/ε)) stochastic gradient evaluations.The analysis notes that larger mini-batches can reduce stochastic-gradient variance and offset a divergent logarithmic term.
  • SVRG-LD: SVRG-LD receives the first global-convergence and gradient-complexity guarantee for a nonconvex finite-sum optimization problem.The paper distinguishes this result from prior SVRG-LD work that established sampling error convergence without implying nonconvex optimization convergence.
  • Gradient-complexity comparison: When n ≤ 1/ε^5, SVRG-LD has better gradient complexity than SGLD, and when n ≥ 1/ε^3, it is better than both GLD and SGLD.The comparison uses total stochastic-gradient evaluations because full-gradient evaluations can be expensive in large-scale machine-learning problems.

4 Proof Sketch of the Main Results

The proof decomposes optimization error into algorithmic convergence, discretization, and stationary-distribution terms, then directly analyzes the discrete Markov chain's ergodicity. This framework yields the main GLD, SGLD, and SVRG-LD guarantees by combining corresponding bounds.

  • Roadmap of the Proof: The analysis decomposes optimization error into the gap between the iterate and stationary GLD, the gap between invariant measures, and the gap between Langevin diffusion and the global minimum.The first term is algorithm-dependent, while the latter terms concern discretization and model error.
  • 4.1 Roadmap of the Proof: Directly analyzing geometric convergence of the discretized Markov chain bypasses the slow approximation and spectral-gap terms used in prior SGLD analyses.The paper contrasts its red-arrow analysis of X_k to stationarity with prior blue-arrow analysis of X_k to X(t).
  • Bounding I1: Geometric Ergodicity of GLD: Under the stated assumptions, GLD has a unique invariant measure, with geometric convergence controlled by a contraction parameter depending on inverse temperature.The exact computation of this contraction parameter requires additional nontrivial effort outside the paper's scope.
  • Bounding I2: Convergence to Stationary Distribution: The invariant-measure discrepancy depends on the numerical step size, inverse temperature, and an upper bound from the Poisson-equation solution.This term compares the stationary distributions of GLD and Langevin diffusion.
  • Proof of the Main Theorems: Combining the GLD bounds with algorithm-specific comparisons yields the SGLD and SVRG-LD optimization-error guarantees.The SGLD and SVRG-LD proofs add bounds comparing their outputs with GLD outputs.

5 Conclusions and Future Work

The conclusion presents the paper's analysis framework and supporting stochastic-process equations, including Fokker–Planck and Backward Kolmogorov equations. These tools describe stationary distributions and the evolution of expectations for Langevin dynamics.

  • Conclusions and Future Work: The paper presents a new framework for non-asymptotic convergence analysis of Langevin dynamics based algorithms for nonconvex finite-sum optimization.
  • Supporting Equations: The Fokker–Planck equation describes the evolution of the probability density associated with a stochastic differential equation.It is used to determine whether the SDE has a stationary distribution.
  • Supporting Equations: The stationary distribution associated with the Langevin SDE is the Gibbs measure.
  • Supporting Equations: The Backward Kolmogorov equation describes the evolution of E[g(X(t))|X(0) = x] for a smooth test function g.A Taylor expansion of this quantity is introduced afterward.

B Proof of Corollaries

The corollary proofs select step sizes, iteration counts, batch sizes, and inner-loop lengths to obtain iteration and gradient-complexity bounds for GLD, SGLD, and SVRG-LD.

  • GLD Corollary: For GLD, choosing η = O(ϵ) and Θ = O(d^2/ρ^(d/2)) yields the stated iteration complexity.
  • SGLD Corollary: For SGLD, setting η = O(ϵ) and K = O(d/(λϵ) log(1/ϵ)) controls the iterate error at ϵ precision.
  • SVRG-LD Corollary: For SVRG-LD, the proof requires η = O(ϵ) and Kη = O(1/λ log(3Θ/ϵ)) before combining the resulting constraints.
  • SVRG-LD Corollary: SVRG-LD's gradient complexity counts B stochastic-gradient evaluations per iteration plus K/L full-gradient calculations.
  • SVRG-LD Corollary: Optimizing the batch and inner-loop parameters gives B = √nϵ^-3/2 and L = √nϵ^3/2, yielding the optimal SVRG-LD gradient complexity.

C.1 Proof of Lemma 4.1

The proof establishes geometric ergodicity of GLD by combining dissipativity, quadratic Lyapunov control, minorization, and invariant-measure arguments for the numerical Markov chain.

  • Ergodicity Setup: Earlier work shows that Euler–Maruyama discretization can lose convergence properties even when the underlying Langevin diffusion converges exponentially.This motivates a direct ergodicity analysis of the numerical approximation.
  • Geometric Ergodicity: The proof invokes a minorization condition on a compact set to establish the needed Markov-chain convergence properties.For Langevin diffusion, this condition holds under the dissipative and smooth assumptions.
  • Geometric Ergodicity: Under the dissipative and smoothness assumptions, the GLD numerical approximation has a unique invariant measure μ.
  • Geometric Ergodicity: An essentially quadratic Lyapunov function bounds the objective and supports the Markov-chain ergodicity argument.The proof uses V(x) to control function growth under the stated assumptions.
  • Geometric Ergodicity: The contraction rate is represented by ρ_β ∈ (0, 1), which depends on the inverse temperature, while κ depends on smoothness, dissipativity, and dimension parameters.
  • Invariant-Measure Comparison: The GLD proof then uses stationarity and invariant-measure comparisons to bound the expected objective and complete the optimization-error analysis.

C.3 Proof of Lemma 4.4

The proof bounds the expected function-value gap between SGLD and GLD by comparing their iterate distributions through continuous-time Markov interpolations and divergence inequalities.

  • Moment bounds: Uniform second-moment and exponential second-moment bounds control the Wasserstein and function-value terms appearing in the comparison.The moment bounds depend on the dissipativity, smoothness, dimension, inverse temperature, and gradient-growth parameters.
  • Markov-process construction: A continuous-time Markov process bridges Langevin diffusion and the GLD discretization, while a second Markov process represents SGLD with matching one-time marginals.The stochastic-gradient interpolation itself is not Markov, so the proof introduces a Markov process with the same one-time marginals.
  • Divergence control: The K-L divergence between GLD and SGLD iterate distributions is bounded using Girsanov’s formula and the data-processing inequality.The proof first relates path-process divergences to iterate distributions, then controls the resulting terms through the constructed processes.
  • Expected gap: The expected function-value gap between SGLD and GLD is bounded by combining distributional distance estimates with moment bounds.The final bound follows by combining intermediate inequalities and assuming Kη > 1 and C1 ≥ C2.

C.4 Proof of Lemma 4.5

The proof of Lemma 4.5 compares GLD and SVRG-LD through continuous-time Markov constructions, divergence bounds, and control of the variance-reduced gradient terms.

  • Expected gap: The function-value gap between GLD and SVRG-LD is upper-bounded by applying distributional comparison inequalities to their terminal iterates.The proof begins by applying Lemma C.8 and then bounds the associated Wasserstein and K-L terms.
  • Markov-process construction: A continuous-time interpolation represents SVRG-LD, and an auxiliary Markov process preserves its one-time marginals despite the semi-stochastic gradient randomness.The interpolation is not Markov, motivating the auxiliary process used in the divergence analysis.
  • Divergence control: Girsanov’s formula and the data-processing inequality reduce the terminal-distribution comparison to bounds on the continuous-time processes.The resulting expressions exploit Gaussian Brownian increments and the stepwise structure of the drift terms.
  • Variance-reduction terms: The SVRG-LD comparison additionally bounds inner-loop gradient-error sums using Lemma C.10, Lemma D.1, and iterate L2 bounds.The resulting estimate incorporates the mini-batch size and inner-loop length before being substituted into the final comparison.

D.3 Proof of Lemma C.4

The proof of Lemma C.4 uses the Poisson equation for Langevin dynamics to relate time-averaged GLD values to the Gibbs expectation and then controls the discretization remainder.

  • Poisson-equation representation: The Poisson equation is chosen with g = Fn to express the difference between the GLD time average and the Gibbs expectation.The generator equation provides the representation used for the subsequent averaging argument.
  • Remainder control: Bounded expectations of the Poisson solution and its derivatives control the remainder terms in the discretization analysis.The proof invokes dissipativity and smoothness assumptions, together with polynomial-growth bounds, and simplifies them through a constant Cψ.
  • Iteration averaging: The proof averages the resulting relation over iterations and uses equality in distribution between GLD iterates and the continuous-time Markov interpolation.This distributional equivalence justifies replacing discrete-process expectations with interpolation expectations in the derivation.
  • Inverse-temperature extension: The argument first treats β = 1 and then extends the conclusion to arbitrary inverse temperature by a scaling argument.The proof explicitly states that the same scaling technique applies to general β.

D.5 Proof of Lemma C.6

The proof of Lemma C.6 establishes uniform L2 bounds for GLD and SVRG-LD iterates by deriving recursive moment inequalities and closing them under small-step-size conditions.

  • GLD bound: The GLD iterates satisfy a recursive second-moment inequality with contraction coefficient 1 − 2ηm + 2η^2M^2.The recursion also contains additive terms involving b, G, dimension d, and inverse temperature β.
  • GLD bound: Iterating the GLD recursion yields a uniform second-moment bound for all iterations when the coefficient lies in the required range.The proof handles separately the cases where the coefficient is nonpositive or lies between zero and one.
  • SVRG-LD bound: SVRG-LD obtains the same type of uniform L2 control by induction, using unbiased semi-stochastic gradients and bounds on the variance-reduction error.The base case uses Z0 = 0, while the induction step combines Lemma C.10 with gradient-growth estimates.
  • Exponential bound: The exponential L2 bound requires sufficiently small η and β ≥ max{2/(m − M^2η), 4η}.Under these conditions, the proof bounds the exponential moment recursively and obtains the stated control.

E.1 Proof of Lemma D.1

The proof begins by applying Assumption 3.2, then uses properties of the minimizer x* of F_n and Assumption 3.1.

  • The proof first invokes Assumption 3.2.
  • x* is identified as the minimizer of F_n.
  • The minimizer satisfies ∇F_n(x*) = 0.
  • The proof bounds the minimizer norm by ∥x*∥2 ≤ b/m.
  • The argument next invokes Assumption 3.1.
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