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arxiv: 2110.12922 · v2 · submitted 2021-10-25 · 🧮 math.PR · cs.LG· stat.ML

On quantitative Laplace-type convergence results for some exponential probability measures, with two applications

Valentin De Bortoli , Agn\`es Desolneux This is my paper

Pith reviewed 2026-05-24 13:03 UTC · model grok-4.3

classification 🧮 math.PR cs.LGstat.ML
keywords Laplace methodexponential measuresWasserstein distancenorm-like potentialsgeneralized Jacobiancoarea formulamaximum entropy modelsstochastic gradient Langevin dynamics
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The pith

For norm-like potentials, exponential measures converge quantitatively to their zero-temperature limits in Wasserstein-1 distance when a generalized Jacobian is invertible.

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

The paper establishes quantitative bounds on how fast measures with densities proportional to exp(-U(x)/ε) approach their limiting distributions as ε tends to zero. These bounds hold in the 1-Wasserstein metric for potentials U that are norm-like, provided a generalized Jacobian remains invertible at the relevant points. The argument replaces the classical twice-differentiability requirement with geometric measure theory, specifically the coarea formula. The results are applied to maximum-entropy models and to the low-temperature behavior of stochastic gradient Langevin dynamics on non-convex landscapes.

Core claim

For norm-like potentials U, if a generalized Jacobian is invertible at the points of interest, then the measures π_ε converge to the limiting measure π_0 supported on the minimizers of U, and the convergence is quantitative in the Wasserstein distance of order 1.

What carries the argument

Invertibility condition on the generalized Jacobian of U, which permits application of the coarea formula to obtain explicit Wasserstein-1 bounds without classical Hessian assumptions.

If this is right

  • The same quantitative bounds relate microcanonical and macrocanonical distributions in maximum-entropy models.
  • The iterates of stochastic gradient Langevin dynamics converge in Wasserstein-1 to the set of global minimizers at low temperature for non-convex objectives.
  • The convergence rates hold without requiring the potential to be twice differentiable everywhere.
  • The coarea-formula approach yields explicit constants that depend on the geometry of the level sets of U.

Where Pith is reading between the lines

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

  • The same technique might be tested on other sampling schemes whose stationary measures are of Laplace type.
  • It remains open whether analogous bounds can be obtained in Wasserstein distance of order p greater than 1 under the same Jacobian condition.
  • The results suggest that invertibility of a first-order object can substitute for second-order non-degeneracy in several concentration problems.

Load-bearing premise

The potential U must be norm-like and its generalized Jacobian must be invertible at the relevant points.

What would settle it

A concrete norm-like potential U for which the generalized Jacobian is invertible at all required points yet the Wasserstein-1 distance between π_ε and π_0 fails to approach zero at the rate claimed as ε tends to zero.

Figures

Figures reproduced from arXiv: 2110.12922 by Agn\`es Desolneux, Valentin De Bortoli.

Figure 1
Figure 1. Figure 1: Left: graph of the polynomial x 7→ P(x), that has 4 zeros. Right: verifying the scaling relation of Proposition 6 by plotting ε 7→ 2πε[P 2 ], which is equivalent to ε as ε goes to 0. Two-dimensional ellipse In this second example, we consider the function F : R 2 → R given for any x = (x1, x2) ∈ R 2 by F(x) = a1x 2 1 + a2x 2 2 − 1 with a1, a2 > 0. For ε > 0, we define πε and π Ψ ε whose densities w.r.t the… view at source ↗
Figure 2
Figure 2. Figure 2: Distributions πε and π Ψ ε (first line), and histogram of their samples from them (second line). This experiment shows that the limit distribution of π Ψ ε as ε goes to 0 is the uniform microcanonical model given by the uniform distribution on the zeros of P. (dπ Ψ ε /dλ)(x) = JF(x) exp[−kF(x)k 2/ε]/ R R2 JF(˜x) exp[−kF(˜x)k 2/ε]d˜x , To sample from πε and π Ψ ε , we use two Markov chains given by the Unad… view at source ↗
Figure 3
Figure 3. Figure 3: Left: histogram of angles of the samples [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left: the function x 7→ u(x, 0) with u given in (13). Right: the function x 7→ u(x, 0.5). The global minimizers are given by the dotted red lines. 3.2.3 The importance of the thermodynamic barrier To conclude this section, we investigate the role of the thermodynamic barrier in order to establish quantitative parametric Laplace-type results. This quantity should not be confused with the concept of kinetic … view at source ↗
Figure 5
Figure 5. Figure 5: Difference between the thermodynamic barrier (blue) and the kinetic barrier (red). 4 Proofs In this section, we gather the proofs of the previous sections. In Section 4.1 we prove Theorem 3. Then, in Section 4.2 we provide the proofs of the results of Section 3.1. Finally, the proofs of the results of Section 3.2 are given in Section 4.3. 4.1 Proof of Theorem 3 In this section, we prove Theorem 3. We recal… view at source ↗
read the original abstract

Laplace-type results characterize the limit of sequence of measures $(\pi_\varepsilon)_{\varepsilon >0}$ with density w.r.t the Lebesgue measure $(\mathrm{d} \pi_\varepsilon / \mathrm{d} \mathrm{Leb})(x) \propto \exp[-U(x)/\varepsilon]$ when the temperature $\varepsilon>0$ converges to $0$. If a limiting distribution $\pi_0$ exists, it concentrates on the minimizers of the potential $U$. Classical results require the invertibility of the Hessian of $U$ in order to establish such asymptotics. In this work, we study the particular case of norm-like potentials $U$ and establish quantitative bounds between $\pi_\varepsilon$ and $\pi_0$ w.r.t. the Wasserstein distance of order $1$ under an invertibility condition of a generalized Jacobian. One key element of our proof is the use of geometric measure theory tools such as the coarea formula. We apply our results to the study of maximum entropy models (microcanonical/macrocanonical distributions) and to the convergence of the iterates of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm at low temperatures for non-convex minimization.

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.

Referee Report

0 major / 3 minor

Summary. The paper establishes quantitative bounds in the 1-Wasserstein distance between the family of measures π_ε with Lebesgue density proportional to exp(−U(x)/ε) and the limiting measure π_0 supported on the minimizers of U, for the special case of norm-like potentials U. The main result requires an invertibility condition on a generalized Jacobian and is proved using the coarea formula from geometric measure theory. Two applications are developed: one to the equivalence of microcanonical and macrocanonical maximum-entropy distributions, and one to the low-temperature convergence of the iterates of stochastic gradient Langevin dynamics for non-convex objectives.

Significance. If the stated W1 bounds hold under the given hypotheses, the work supplies explicit quantitative rates for Laplace-type concentration in a non-smooth regime where the classical Hessian-invertibility assumption fails. The reliance on the coarea formula to handle the geometry of the level sets is a technically sound extension of existing GMT-based arguments. The two applications illustrate that the abstract condition can be verified in concrete statistical and algorithmic settings, which increases the result’s utility for both theoretical probability and optimization practice.

minor comments (3)
  1. The precise definition of the generalized Jacobian and the points at which its invertibility is required should be stated in a single, self-contained paragraph early in the main-result section so that the hypotheses can be checked directly in the applications.
  2. In the SGLD application, the passage from the continuous-time Langevin diffusion to the discrete iterates at low temperature would benefit from an explicit reference to the step-size and discretization-error controls that are used to transfer the W1 bound.
  3. A short remark comparing the obtained W1 rate with the classical smooth-case rate (when the Hessian is invertible) would help readers gauge the price paid for allowing non-smooth norm-like potentials.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript, the assessment of its significance, and the recommendation for minor revision. We appreciate the recognition of the technical approach using the coarea formula and the utility of the applications.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives quantitative W1 bounds for Laplace-type measures with norm-like potentials U under an explicit invertibility assumption on a generalized Jacobian. The proof invokes the coarea formula and other tools from geometric measure theory as independent external results. Applications proceed by direct verification of the stated hypotheses in each case. No equations reduce by construction to fitted parameters, self-definitions, or unverified self-citations; the central claim retains independent mathematical content grounded in standard analysis without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, invented entities, or ad-hoc axioms are described beyond standard assumptions of geometric measure theory and probability.

axioms (2)
  • standard math Coarea formula applies to the level sets of the potential U
    Invoked as key element of the proof for quantitative bounds.
  • domain assumption Generalized Jacobian invertibility condition holds at minimizers
    Stated as the condition under which the main result holds.

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    > 0, there existsm > 0 such that for anyk ∈ {1, . . . , N} and v ∈ Rd, v⊤H(xk 0, xk 0)v ≥ m∥v∥2 with H(x, y) = DF(x)⊤DF(y) for any x, y ∈ Rd. For anyk ∈ {1, . . . , N}, there existsWk ⊂ Vk such that for anyx, y ∈ Wk we have∥H(x, y)−H(xk 0, xk 0)∥2 ≤ m/2. Therefore we have for anyx, y ∈ Wk ∥F (x) − F(y)∥2 = ∫ 1 0 ∫ 1 0 ⟨DF(x + t(y − x))(y − x), DF(x + s(y ...

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