Global Convergence of Third-Order Langevin Dynamics for Non-Convex Optimization via Simulated Annealing

1 Introduction

• Third-order Langevin dynamics extends simulated-annealing global-minimization analysis to a setting previously studied for sampling, with guarantees there limited to strongly log-concave targets.The authors describe this non-convex global-minimization setting as novel for the dynamics.Source: §1 · p. 3
• Cooling noise in the third-order chain makes direct dissipation degenerate because noise acts only on the last component.As its amplitude is cooled, the coefficient of that direct dissipation vanishes.Source: §1 · p. 4
• An explicit three-block Fisher-information matrix transfers dissipation from Z through V to X while friction remains fixed and noise decreases.The approach combines the fixed-friction annealing strategy of He, Tan, and Wu with the third-order diffusion and three-stage scheme of Mou et al..Source: §1 · p. 4
• The continuous-time rate follows from a three-block dissipation certificate, while local coupling and related controls yield convergence results for exact-force and midpoint schemes.The contribution summary also identifies a uniform restart estimate for discrete comparison.Source: §1 · p. 4
• In synthetic and real-data neural-network examples, both kinetic schemes outperform overdamped Langevin, and third-order Langevin has higher best-basin point estimates than UBU.The real-data example also reports the highest post-quench test-accuracy point estimate for third-order Langevin.Source: §1 · pp. 4–5
• The cubic local endpoint estimate permits a less restrictive sufficient step-size threshold than the available kinetic frozen-force result.For polynomial grids, the third-order midpoint exponent is strictly larger than the other exponents compared; these are proved sufficient conditions, not lower bounds or an intrinsic integrator ordering.Source: §1 · p. 5

2 Model Setup and Assumptions

• At frozen temperature, the generator has the Gibbs law as an invariant distribution.The transport preserves the energy, while the final friction/noise pair preserves the Gaussian auxiliary-variable factor.Source: §2 · p. 6
• The objective is C4 with bounded Hessian, dissipative growth, and third and fourth derivatives of at most polynomial growth; temperature is C1 and non-increasing.The assumptions further specify an asymptotic logarithmic cooling scale and derivative control.Source: §2 · p. 6
• The continuous-time initial law must be absolutely continuous, have finite second moment, and have finite relative Fisher information with respect to the initial Gibbs law.For discrete schemes, the additional assumptions require bounded third derivatives and a finite fourth moment, while allowing singular initialization.Source: §2 · pp. 6–7
• The low-temperature Gibbs law is assumed to satisfy a logarithmic Sobolev inequality with barrier parameter D, which landscape hypotheses can identify with critical depth.The setup assumes D < E, where E is the cooling constant.Source: §2 · pp. 6–7

3.1 Dissipation and initial-density approximation

• The distorted entropy removes the γε factor from KL dissipation and transfers dissipation from z through v to x, with M positive definite.The coefficient choices provide sufficient conditions for coercivity uniformly over Hessians with operator norm at most K; they are not optimized.Source: §3.1 · p. 8
• Lemma 3.2 extends integrated entropy identities to initial laws with finite relative Fisher information and finite second moment by approximation with regular densities.Under Assumption 2.1, the entropy remains finite at every finite time, and the approximation supports comparison estimates for general initial and restart laws.Source: §3.1 · p. 9

3.2 Annealing estimate

• Under Assumptions 2.1–2.3, Theorem 3.4 gives a continuous-time annealing estimate with the kinetic barrier-controlled exponent [HTW24].The remark identifies D with the critical depth E* and says this exponent matches underdamped and overdamped annealing; the quantitative distinction here is at the discretization level.Source: §3.2 · p. 15
• For every r0 < 1 − D/E, the entropy differential inequality yields polynomial decay at exponent r0, which a Gibbs-tail bound converts to the stated annealing estimate.The bound follows by absorbing the cooling contribution into frozen decay and applying a supersolution comparison.Source: §3.2 · pp. 18–19
• After any restart time s ≥ t∗, the comparison holds uniformly for laws with bounded initial second moment and finite entropy, independently of their initial Fisher information.The constants depend on the moment bound and model parameters, not on the restarting time or initial Fisher information.Source: §3.2 · p. 19
• The diffusion has rank d in a 3d-dimensional state space, and cooling drives its nonzero eigenvalues to zero; the entropy weight A0/ε compensates for the ε factor in entropy dissipation.The temperature-dependent estimates retain the dependence needed for the discrete comparison.Source: §3.2 · p. 19

4.1 Discretization schemes and convergence result

• When a ≥ (1 + ϑ)/4, both schemes retain the continuous-time exponent, with iteration exponent 4rc(δ)/(5 + ϑ).Theorem 4.1 gives convergence when a > ϑ/2; the grid satisfies sk ≍ k^(1/(1+a)).Source: §4.1 · p. 21
• The third-order midpoint scheme improves both discrete sufficient iteration exponents, achieving 4rc(δ)/(5 + ϑ), versus UBU’s 3rc(δ)/(4 + ϑ) and frozen-force kinetic annealing’s 2rc(δ)/3.For convergence, Theorem 4.1 requires a > ϑ/2; UBU’s rate-preserving sufficient condition is a ≥ (1 + ϑ)/3.Source: §4.1 · p. 22

4.2 Local comparison and smoothing

• Both numerical schemes have uniformly bounded state moments through order four, and their laws at positive grid points have smooth positive densities with a quantitative entropy bound.The Gaussian one-step covariance supplies the grid-point regularity used to initialize the distorted entropy.Source: §4.2 · pp. 23, 26
• A finite controlled coupling yields one-unit smoothing for the exact inhomogeneous transition: Lip(Pt,T f) ≤ e^C Lip(f), uniformly in t, T, and initial states.The lemma assumes ∥∇3U∥∞ < ∞.Source: §4.2 · p. 28
• A degree-at-most-five path shift preserves the endpoint, enabling a local change-of-measure comparison that extends to arbitrary increments and longer times.For sufficiently small increments, the construction uses Girsanov’s theorem and Pinsker’s inequality; the resulting estimate supports smoothing for bounded measurable tests.Source: §4.2 · pp. 28–30

4.3 Proof of Theorem 4.1

• Telescoping numerical and exact transitions in the terminal window controls their endpoint discrepancy while retaining temperature variation within each step.The window has length T − τ_m ≍ T^β and step sizes h_j ≍ T^-a; the proof uses a uniform restart estimate and bounded third moments.Source: §4.3 · pp. 30–31
• The proof concludes the bound (63), with a > ϑ/2 ensuring that every relevant polynomial exponent is positive.It restarts the annealed process in a late-time window, applies dissipation there, and combines the resulting entropy control with a Gibbs-tail estimate.Source: §4.3 · pp. 30–32

5 Numerical Illustration

• The experiments compare OLD, ULD, and third-order Langevin, with kinetic parameters selected on validation paths and held fixed for independent evaluation.UBU is used for ULD to align gradient evaluations and force quadrature with the midpoint third-order scheme.Source: §5 · p. 32

• Third-order Langevin has higher reported point estimates than UBU across the double-well and both neural-network experiments.On the real-data objective, its best-basin estimate is 0.074 higher and its post-quench test-accuracy estimate is 0.843 versus 0.834.Source: §5.1 · p. 34, §5.2 · p. 36, §5.3 · p. 37

5.1 Annealing with the theory-driven cooling schedule

• At the common horizon, terminal success is 0.421 for third-order Langevin versus 0.402 for UBU; at equal gradient budget, the estimates are 0.415 versus 0.405.At the common horizon, the terminal-success difference is 0.0198 with a 95% interval of approximately [−0.002, 0.041]; at equal budget, it is 0.0098 with standard error 0.0110.Source: §5.1 · p. 34

5.2 A neural-network example with synthetic data

• On the synthetic neural-network objective, UBU and third-order Langevin improve best-basin probability over OLD by 0.145 and 0.174, respectively.The third-order estimate exceeds UBU by 0.030, with a 95% interval of [−0.019, 0.079].Source: §5.2 · p. 36

5.3 A neural-network example with real data

• On the UCI Sonar objective, third-order Langevin's best-basin estimate is 0.074 higher than UBU, and its post-quench test-accuracy estimate is 0.843 versus 0.834.With the initial step halved, third-order Langevin remains highest on both reported point estimates.Source: §5.3 · p. 37

6 Conclusion

• Logarithmic cooling gives convergence in probability at the barrier-controlled kinetic rate, while cubic local endpoint estimates yield less restrictive sufficient step-size conditions for the exact-force and midpoint schemes.The comparison also gives a better sufficient iteration exponent than the existing kinetic frozen-force result and the UBU bound under the same strong-coupling method.Source: §6 · p. 37
• Third-order Langevin has higher terminal-success point estimates than UBU in the double-well experiments.The comparison holds at both a common horizon and an equal gradient budget.Source: §6 · p. 37
• On the neural-network objective, both kinetic methods outperform overdamped Langevin in best-basin recovery, and third-order Langevin has the higher point estimate on synthetic data and the highest post-quench test-accuracy point estimate on real data.The best-basin ordering on real data is the same, and the qualitative conclusions persist under step refinement.Source: §6 · p. 38

A Underdamped UBU Comparison

• The appendix provides the UBU result summarized in Table 1 and used for the comparison in Remark 4.3.Source: A Underdamped UBU Comparison · p. 42

A.1 Scheme and convergence result

• UBU is the underdamped comparator, defined for ULD with friction parameter γu > 0.Source: §A.1 · p. 42
• A frozen-temperature UBU step uses the exact Gaussian flow of the force-free part of ULD, whose mean is (x + F_r v, q_r v), followed by the scheme’s stated updates.Here q_r = e^-γu r and F_r = (1 - q_r)/γu.Source: §A.1 · p. 42
• UBU is a one-gradient symmetric splitting with independent Brownian increments in its two half steps, and its local-error analysis retains a centered stochastic term of order h^(5/2).Source: §A.1 · p. 42
• Under Assumptions 2.1, 2.3, and 2.4, sufficiently small c_h, and a finite fourth moment for the initial numerical law, the theorem applies when a > 2ϑ/3.Source: §A.1 · pp. 42–43
• Taking a ≥ (1 + ϑ)/3 retains the continuous-time exponent, with iteration exponent r_u(δ, a)/(1 + a) up to an arbitrarily small loss.The step scale satisfies s_k ≍ k^(1/(1+a)).Source: §A.1 · p. 43

A.2 Proof of the UBU comparison

• The UBU one-step analysis separates a conditionally centered stochastic velocity error from the remaining error and establishes local bounds with uniform moment control.The proof first freezes the temperature, then unfreezes it; the resulting estimates yield a discrete Lyapunov recursion for sufficiently small steps.Source: §A.2 · pp. 45, 47
• The final Gaussian convolution yields a smooth, positive density for the law.Source: §A.2 · p. 48
• For restarts with T − s ≍ T^β, s ≍ T, and β > ϑ, the ULD estimate gives restart decay, while semigroup smoothing controls bounded measurable tests by their Lipschitz constants.These estimates are applied at the start of the comparison window; telescoping the UBU and exact ULD kernels then uses the local-error lemma.Source: §A.2 · pp. 48, 50
• The estimates establish (91), and grid asymptotics yield the iteration statement.The preceding bound includes a stretched-exponential term that is smaller than any fixed inverse power of T.Source: §A.2 · p. 51
• The third-order scheme’s stochastic force-integral error is O(√ε h^7/2), below its O(h^3) deterministic endpoint error, while UBU has an O(√ε h^5/2) term.Under the present strong-coupling argument, the difference follows from how noise reaches the force evaluation; a weak-error or martingale analysis using UBU error centering could sharpen its bound.Source: §A.2 · p. 51