ProCreations/repro-dimension-independent-convergence-of-underdamped-langevin-monte-carlo-in-kl-divergence

Reproduction - Dimension-Independent Convergence of Underdamped Langevin Monte Carlo in KL Divergence

Independent audit of all six registered claims using a hash-pinned primary source, 576 rate cells, exact Gaussian ULMC mean/covariance propagation across 12 settings and 12.24 million coordinate-steps, 35 randomized-midpoint kernel identities, 96 analytic change-of-measure counterexamples, and fixed-trace Gaussian moment checks through dimension 4096. All sixteen scientific gates pass in two warning-strict byte-identical runs. Claim 5 is honestly falsified: the coefficient-one inequalities printed in Lemma 6.1 fail for target-covariance Gaussian mean shifts, while a factor of two exactly repairs this counterexample family.

Primary source: https://arxiv.org/pdf/2603.02429v1 OpenReview: https://openreview.net/forum?id=gxxOL0iXpr OpenReview paper ID: gxxOL0iXpr arXiv version: 2603.02429v1 Primary PDF SHA-256: 945bf25737ea91e2049a02d5ca56927bbccd0feced486c3dbbfb14d54b7dbc99 Implementation: independent from equations (3.2)–(3.3), Theorems 4.3–5.4, and Lemma 6.1 in the pinned PDF. No competitor artifact or result was used.

Registered claim ledger

1. 1.Theorem 4.3 establishes the first dimension-free KL-divergence convergence bound for standard underdamped Langevin Monte Carlo (ULMC) under strong convexity, with sample complexity depending on tr(H) rather than the ambient dimension d, where H is a known upper bound on the Hessian (Theorem 4.3, Assumption 3.1).
2. 2.The strongly-convex ULMC sample complexity bound is Otilde(kappa^{3/2} beta^{-1/2} [tr(H)]^{1/2} / epsilon) to reach KL(mu (P')^N || pi) <= epsilon^2 (Theorem 4.3).
3. 3.Under randomized midpoint discretization, Theorem 5.2 improves the condition-number dependence relative to the bound of Liu et al. (2023), achieving complexity Otilde(kappa [beta^{-1} tr(H)]^{1/3} epsilon^{-2/3}) (Theorem 5.2).
4. 4.Theorems 4.4 and 5.4 extend the dimension-free guarantees to the general (non-strongly) convex setting (alpha = 0), the first such dimension-free result for underdamped Langevin dynamics in this regime (Theorem 4.4, Theorem 5.4).
5. 5.The dimension-free change-of-measure argument (Lemma 6.1) bounds Emu[||grad V(x)||^2] and Emu[p^T H p] by tr(H) + beta*KL(mu||pi), avoiding the explicit dimension dependence introduced by standard Gaussian moment bounds (Lemma 6.1, Section 6).
6. 6.The paper shows underdamped Langevin Monte Carlo attains better iteration complexity than composite overdamped Langevin Monte Carlo when tr(H) << d, e.g. in ridge-separable target distributions (Section 3.3, Theorem 3.5 framework).