Error estimates for tamed Euler and Randomized Euler schemes for SDEs with locally Lipschitz drift with applications to non-logconcave sampling and optimization

Angelos Ntousis; Iosif Lytras

arxiv: 2605.24937 · v1 · pith:Q46DFTVHnew · submitted 2026-05-24 · 🧮 math.PR · cs.NA· math.NA· math.OC· stat.CO· stat.ML

Error estimates for tamed Euler and Randomized Euler schemes for SDEs with locally Lipschitz drift with applications to non-logconcave sampling and optimization

Iosif Lytras , Angelos Ntousis This is my paper

Pith reviewed 2026-06-29 23:52 UTC · model grok-4.3

classification 🧮 math.PR cs.NAmath.NAmath.OCstat.COstat.ML

keywords tamed Euler schemeslocally Lipschitz driftSDE discretizationnon-logconcave samplinglogarithmic Sobolev inequalityLangevin Monte Carloerror estimatesnon-convex optimization

0 comments

The pith

Tamed Euler schemes deliver near-optimal sampling complexities for distributions with superlinear drift under logarithmic Sobolev inequalities.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops local error frameworks for tamed and randomized Euler discretizations of SDEs whose drift is locally Lipschitz with superlinear growth. These frameworks produce non-asymptotic bounds on the discretization error in KL divergence for the KL-accelerated tamed unadjusted Langevin algorithm and in total variation for the new tamed randomized midpoint scheme. Under the assumption that the invariant measure satisfies a logarithmic Sobolev inequality, the bounds translate into iteration complexities of order tilde O of epsilon to the minus one half for the first scheme and tilde O of epsilon to the minus one for the second. The same results also supply non-asymptotic guarantees for non-convex optimization problems. Readers care because these rates are near-optimal and the schemes remain stable where classical Euler-Maruyama fails.

Core claim

The paper claims that two new local-error frameworks, built on the shifted-composition method, give finite-time non-asymptotic error estimates for kTULA in KL divergence and for tRLMC in total variation that hold for arbitrary locally Lipschitz drifts; specializing to targets obeying a logarithmic Sobolev inequality produces the near-optimal iteration complexities in sampling and corresponding bounds for optimization.

What carries the argument

The shifted-composition approach applied to tamed schemes, which controls the local truncation error despite the absence of global Lipschitz continuity.

If this is right

kTULA achieves tilde O of epsilon to the minus one half complexity in KL, TV, and Wasserstein metrics for sampling.
tRLMC achieves tilde O of epsilon to the minus one complexity in total variation and Wasserstein for the first time under super-linear growth.
Non-asymptotic excess risk bounds follow for non-convex optimization via both schemes.
The frameworks apply to general SDE approximation beyond the sampling setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar taming techniques could be combined with other acceleration methods to improve the complexity further.
The results suggest that randomized midpoint schemes may be preferable when total variation guarantees are required.
Explicit dependence on the log-Sobolev constant could guide parameter tuning in applications.

Load-bearing premise

The target distribution obeys a logarithmic Sobolev inequality, allowing conversion of local discretization errors into global iteration counts.

What would settle it

Running the schemes on a low-dimensional test SDE with known log-Sobolev constant and superlinear drift, then checking whether the measured error after the predicted number of steps stays below the stated bound.

read the original abstract

In this paper, we study the numerical discretization of stochastic differential equations with locally Lipschitz, super-linearly growing drift, and the resulting implications for sampling from non-log-concave distributions satisfying a logarithmic Sobolev inequality. In this regime, the classical Euler--Maruyama scheme underlying the unadjusted Langevin algorithm (ULA) is known to be unstable. We analyze the KL-accelerated tamed unadjusted Langevin algorithm (kTULA) and introduce a new tamed randomized midpoint scheme, termed tRLMC. Building on the shifted-composition approach of \cite{chewi2024local}, we develop two new local-error frameworks that yield finite-time, non-asymptotic error estimates against the underlying SDE -- in KL divergence for kTULA, and in total variation for tRLMC -- valid for general locally Lipschitz drift. Specializing these frameworks to the sampling problem under a logarithmic Sobolev inequality, we obtain a near-optimal $\widetilde{O}(\varepsilon^{-1/2})$ iteration complexity for kTULA in KL divergence, with corresponding guarantees in total variation and Wasserstein distance. We further establish, for the first time, a non-asymptotic guarantee in total variation for a tamed randomized Langevin scheme under super-linear drift growth, together with the corresponding Wasserstein-distance bound, both with $\widetilde{O}(\varepsilon^{-1})$ complexity for tRLMC. As a consequence, both schemes yield non-asymptotic bounds for a non-convex excess-risk optimization problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper delivers the first non-asymptotic TV bounds for a tamed randomized Langevin scheme under super-linear drift and near-optimal rates for accelerated tamed ULA in KL.

read the letter

The main new pieces are the tRLMC scheme itself and the two local-error frameworks that extend the shifted-composition idea to give finite-time KL and TV estimates for general locally Lipschitz super-linear drifts. Those frameworks are then specialized under an LSI to produce the stated iteration complexities: roughly O(ε^{-1/2}) for kTULA in KL and O(ε^{-1}) for tRLMC in TV. The TV guarantee for the randomized tamed scheme appears to be new.

The work is solid on the parts that matter for sampling applications. It correctly identifies that standard Euler-Maruyama is unstable in this regime and shows how taming plus randomization buys stability while still recovering usable rates once an LSI is assumed. The argument structure is consistent: the general error bounds do not rely on the LSI, and the LSI is only invoked later to turn local errors into global iteration counts.

The LSI is a real limitation for how non-convex the target can be, but the paper is explicit about it. The tilde-O notation hides logarithmic factors, which is common but worth checking in the proofs. No circularity is visible from the abstract or the stress-test description.

This is for people working on non-convex sampling and on numerical methods for SDEs with bad growth. It is worth sending to a serious referee because the claimed first TV result and the reusable error frameworks are concrete enough to evaluate.

Referee Report

0 major / 3 minor

Summary. The manuscript develops two local-error frameworks for discretizations of SDEs with locally Lipschitz super-linear drift, using a shifted-composition argument. For the KL-accelerated tamed ULA (kTULA) it obtains non-asymptotic KL error bounds against the continuous process; for the new tamed randomized midpoint scheme (tRLMC) it obtains total-variation bounds. Specializing under a logarithmic Sobolev inequality on the target yields iteration complexities ilde O(ε^{-1/2}) for kTULA in KL (with TV and W_2 corollaries) and ilde O(ε^{-1}) for tRLMC in TV (with W_2 corollary). The same bounds are applied to a non-convex excess-risk optimization problem.

Significance. If the local-error frameworks are valid, the work supplies the first non-asymptotic TV guarantee for a tamed randomized Langevin scheme under super-linear growth and near-optimal sampling rates for kTULA. The explicit dependence on the LSI constant and the extension of shifted composition to these schemes are concrete contributions to the analysis of non-log-concave sampling.

minor comments (3)

§3.2, Definition 3.3: the precise statement of the 'shifted composition' operator should be restated verbatim from Chewi et al. (2024) so that the subsequent error propagation lemmas are self-contained.
Theorem 4.5 (tRLMC TV bound): the dependence on the step-size restriction h ≤ c / (L^2 + M^2) is stated only in the proof; moving the explicit constant into the theorem statement would improve readability.
Table 1: the row for 'tRLMC (this work)' lists TV complexity ilde O(ε^{-1}); it would be helpful to add a footnote clarifying that this is the first such guarantee under super-linear growth.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment, the recognition of the local-error frameworks and complexity results, and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; new frameworks independent of inputs

full rationale

The paper cites an external reference (chewi2024local) for the shifted-composition approach and then develops two new local-error frameworks for KL and TV that apply to general locally Lipschitz super-linear drift. These frameworks are specialized under an explicit LSI assumption to obtain the iteration complexities. No derivation step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central non-asymptotic guarantees are presented as novel analysis rather than re-labeling of inputs. The LSI is stated as an additional ingredient required only for the sampling rates, not smuggled via citation. This is a standard self-contained derivation with external support.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the logarithmic Sobolev inequality for the target measure and the local Lipschitz plus superlinear growth condition on the drift; both are standard domain assumptions rather than new postulates.

axioms (2)

domain assumption The target probability measure satisfies a logarithmic Sobolev inequality.
Invoked to specialize the general KL and TV error bounds to concrete iteration complexities for sampling.
domain assumption The drift coefficient is locally Lipschitz and exhibits super-linear growth.
Defines the regime in which classical Euler-Maruyama is unstable and the tamed schemes are analyzed.

pith-pipeline@v0.9.1-grok · 5827 in / 1276 out tokens · 25359 ms · 2026-06-29T23:52:30.838396+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

36 extracted references · 11 canonical work pages · 1 internal anchor

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Diffusions hypercontractives

Dominique Bakry and Michel ´Emery. Diffusions hypercontractives. InS ´eminaire de Probabilit´es XIX 1983/84: Proceedings, pages 177–206. Springer, 2006

1983

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Geometric ergodicity of modified euler schemes for sdes with super-linearity.arXiv preprint arXiv:2412.19377, 2024

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work page arXiv 2024

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work page arXiv 2025

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Non-asymptotic estimates for tusla algorithm for non-convex learning with applications to neural networks with relu activation function, 2023

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work page arXiv 2023

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Taming neural networks with tusla: Nonconvex learning via adaptive stochastic gradient langevin algorithms.SIAM Journal on Mathematics of Data Science, 5(2):323–345, 2023

Attila Lovas, Iosif Lytras, Mikl ´os R´asonyi, and Sotirios Sabanis. Taming neural networks with tusla: Nonconvex learning via adaptive stochastic gradient langevin algorithms.SIAM Journal on Mathematics of Data Science, 5(2):323–345, 2023

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Lytras and S

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Contractive kinetic langevin samplers beyond global lipschitz continuity.arXiv preprint arXiv: 2509.12031, 2025

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work page arXiv 2025

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Lytras and P

Iosif Lytras and Panayotis Mertikopoulos. Tamed langevin sampling under weaker conditions.arXiv preprint arXiv:2405.17693, 2024

work page arXiv 2024

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ktula: A langevin sampling algorithm with improved kl bounds under super-linear log-gradients.arXiv preprint arXiv:2506.04878, 2025

Iosif Lytras, Sotirios Sabanis, and Ying Zhang. ktula: A langevin sampling algorithm with improved kl bounds under super-linear log-gradients.arXiv preprint arXiv:2506.04878, 2025

work page arXiv 2025

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Non-asymptotic bounds for sampling algorithms without log-concavity.arXiv preprint arXiv:1808.07105, 2018

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work page arXiv 2018

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Towards a complete analysis of langevin monte carlo: Beyond poincar´e inequality

Alireza Mousavi-Hosseini, Tyler K Farghly, Ye He, Krishna Balasubramanian, and Murat A Erdogdu. Towards a complete analysis of langevin monte carlo: Beyond poincar´e inequality. InThe Thirty Sixth Annual Conference on Learning Theory, pages 1–35. PMLR, 2023

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Non-asymptotic convergence bounds for modified tamed unadjusted langevin algorithm in non-convex setting, 2024

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work page arXiv 2024

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Non-convex learning via stochastic gradient langevin dynamics: a nonasymptotic analysis

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