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arxiv: 2301.09428 · v2 · submitted 2023-01-23 · 💻 cs.LG · cond-mat.dis-nn· cond-mat.stat-mech

Explaining the effects of non-convergent sampling in the training of Energy-Based Models

Elisabeth Agoritsas , Giovanni Catania , Aur\'elien Decelle , Beatriz Seoane This is my paper

Pith reviewed 2026-05-24 10:16 UTC · model grok-4.3

classification 💻 cs.LG cond-mat.dis-nncond-mat.stat-mech
keywords Energy-based modelsnon-convergent samplingMarkov chainsdynamical processempirical statisticsdiffusion modelsBoltzmann machine
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The pith

EBMs trained with non-persistent short Markov chain runs reproduce empirical data statistics through a precise dynamical process rather than equilibrium convergence.

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

The paper shows analytically that Energy-Based Models trained by estimating gradients with short, non-convergent Markov chains starting from random points can exactly match a set of empirical statistics from the data. This match happens through the specific dynamics created by the incomplete sampling, not by converging to the model's equilibrium distribution. A reader would care because the result explains why short-run sampling strategies produce high-quality samples efficiently in practice. It also supplies the analytical basis for treating EBMs as diffusion-style models.

Core claim

EBMs trained with non-persistent short runs to estimate the gradient can perfectly reproduce a set of empirical statistics of the data, not at the level of the equilibrium measure, but through a precise dynamical process. The authors derive this for generic EBMs, work it out explicitly in two solvable models, and verify the predictions numerically on a ConvNet EBM and a Boltzmann machine.

What carries the argument

Non-persistent short Markov chain runs that begin from random initial conditions and induce a dynamical process whose statistics are encoded exactly into the trained parameters.

If this is right

  • EBMs become usable as diffusion models because the dynamical encoding replaces the need for equilibrium sampling.
  • Short runs from random starts constitute an efficient, high-quality sampling method whose effect is now explained from first principles.
  • The effect can be computed in closed form for solvable models, giving exact predictions for how parameters shift under non-convergent training.
  • Numerical checks on neural-network EBMs confirm that the dynamical matching holds beyond the solvable cases.

Where Pith is reading between the lines

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

  • The same dynamical-matching principle could be applied to other approximate-sampling generative models to reduce the cost of training.
  • Training objectives might be redesigned to optimize the dynamical statistics directly instead of an equilibrium loss.
  • The link to diffusion models raises the question of whether EBMs inherit convergence or mixing guarantees known for diffusion processes.

Load-bearing premise

The short runs must begin from random initial conditions so that the incomplete sampling dynamics alone, without equilibrium convergence, exactly capture the empirical statistics.

What would settle it

Train an EBM with the described short runs, then compare the statistics of samples produced by the same short-run procedure against the statistics obtained from long equilibrated chains on the same trained model; mismatch in the equilibrated case would support the claim.

Figures

Figures reproduced from arXiv: 2301.09428 by Aur\'elien Decelle, Beatriz Seoane, Elisabeth Agoritsas, Giovanni Catania.

Figure 1
Figure 1. Figure 1: Left: Evolution of J (k) α (t) for different values of k (different colors) and initial conditions for J (k) α (0). Right: Convergence for a large training time t, where the higher k, the faster the convergence; dash-dotted lines are the numerical integration of Eq. (18), full lines show the exponential fit J (k) α (t) − J (k),∞ α ∼A exp(−t/τ ) with τ given by Eq. (20). Inset: the asymptotic values J (k) α… view at source ↗
Figure 2
Figure 2. Figure 2: Left: Numerical resolution of the learning dynamics in presence of two modes. The dash-lines represent the resolution using a convergent dynamics k → ∞, while the plain ones correspond to k= 1. Right: (inset) evolution of J (k) α (t) for k= 1 at various stages of the learning. (Main) The error on the correlation function, E (2) = ( [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Generation results obtained with a ConvNet EBM trained on CIFAR-10 using k = 150 Langevin MCMC sampling steps from random initial conditions. Left: We use the Frechet Inception Distance Score (FID) to evaluate the generation quality as a function of the sampling time k ′ . The different colors correspond to different training epochs. We see again that the best score is achieved at k ′ = k (corresponding to… view at source ↗
Figure 4
Figure 4. Figure 4: Left: Error over the covariance matrices E (2) = P i<j (⟨xixj ⟩k′ ,p0 −⟨xixj ⟩pD ) 2 / N 2  between the training set and the data generated at different learning ages vs the generation time k ′ , for a BM trained with data sampled from a 2D ferromagnetic Ising model with N = 72 at k= 5. Similarly to [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Error over the eigenvalues of the covariance matrix generated after k ′ steps of MCMC vs the dataset covariance matrix. The setting is the same of [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
read the original abstract

In this paper, we quantify the impact of using non-convergent Markov chains to train Energy-Based models (EBMs). In particular, we show analytically that EBMs trained with non-persistent short runs to estimate the gradient can perfectly reproduce a set of empirical statistics of the data, not at the level of the equilibrium measure, but through a precise dynamical process. Our results provide a first-principles explanation for the observations of recent works proposing the strategy of using short runs starting from random initial conditions as an efficient way to generate high-quality samples in EBMs, and lay the groundwork for using EBMs as diffusion models. After explaining this effect in generic EBMs, we analyze two solvable models in which the effect of the non-convergent sampling in the trained parameters can be described in detail. Finally, we test these predictions numerically on a ConvNet EBM and a Boltzmann machine.

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

2 major / 2 minor

Summary. The manuscript claims that EBMs trained with non-persistent short Markov chain runs (starting from random initial conditions) to estimate gradients can exactly reproduce a chosen set of empirical statistics of the data via the finite-step dynamics of those runs, rather than by matching the equilibrium measure. An analytical derivation is presented for generic EBMs, followed by exact solutions for two solvable models that make the effect explicit, and numerical validation on a ConvNet EBM and a Boltzmann machine. The work positions this as an explanation for the success of short-run sampling strategies and as groundwork for EBMs as diffusion models.

Significance. If the central analytical claim holds under its assumptions, the result supplies a first-principles account of why short non-convergent chains succeed in EBM training and training-for-sampling, together with concrete solvable cases and numerical checks. These elements constitute a genuine contribution that could inform both theory and practice in energy-based and diffusion-style models.

major comments (2)
  1. [generic EBMs section] Generic EBMs section: the exact analytical result for generic EBMs is stated to follow from the training dynamics of short runs, yet the derivation inherits the assumption that each chain begins from random, data-independent initial conditions without additional justification that this holds when the initial distribution is data-dependent or when chain length varies; this assumption is load-bearing for the generality claim.
  2. [solvable models sections] Solvable models sections: the exact solutions are presented for two models, but the manuscript does not report an error analysis or sensitivity study with respect to finite chain length or initialization variance; without this, it is unclear whether the claimed exact reproduction remains robust outside the idealized limits used in the derivations.
minor comments (2)
  1. The abstract should briefly specify which empirical statistics are exactly reproduced by the dynamical process.
  2. [numerical experiments section] Numerical experiments section: figure captions and text should state the precise chain lengths, initialization distributions, and number of gradient steps used in the ConvNet and Boltzmann machine experiments to support reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments and the positive assessment of the significance of our work. We address the major comments below and will revise the manuscript accordingly to improve clarity and robustness.

read point-by-point responses

  1. Referee: [generic EBMs section] Generic EBMs section: the exact analytical result for generic EBMs is stated to follow from the training dynamics of short runs, yet the derivation inherits the assumption that each chain begins from random, data-independent initial conditions without additional justification that this holds when the initial distribution is data-dependent or when chain length varies; this assumption is load-bearing for the generality claim.

    Authors: The derivation in the generic EBMs section is developed specifically under the assumption of short runs initialized from random, data-independent initial conditions, which is the setting used in the non-persistent short-run training strategies that the paper seeks to explain. This assumption is stated in the manuscript and is central to the result, as different initializations would lead to different dynamics. We do not claim the result holds for data-dependent initial distributions. To address the concern, we will revise the text to more explicitly state the scope of the assumption and provide a brief justification for focusing on data-independent random initials, namely that this matches the practical strategy whose success we aim to account for. Regarding chain length variation, the result holds for any fixed finite length under the stated conditions. revision: partial

  2. Referee: [solvable models sections] Solvable models sections: the exact solutions are presented for two models, but the manuscript does not report an error analysis or sensitivity study with respect to finite chain length or initialization variance; without this, it is unclear whether the claimed exact reproduction remains robust outside the idealized limits used in the derivations.

    Authors: We acknowledge that while the solvable models allow for exact derivations in specific cases, an analysis of robustness to variations in chain length and initialization would be beneficial. In the revised version, we will add a sensitivity study section or subsection, including both analytical considerations where feasible and numerical experiments to quantify the error or deviation as chain length and initialization variance change. This will help demonstrate the robustness of the effect beyond the idealized limits. revision: yes

Circularity Check

0 steps flagged

No circularity: analytical derivation from short-run Markov dynamics is self-contained

full rationale

The paper derives its central claim analytically from the explicit form of the gradient estimator using finite-length Markov chains initialized at random (data-independent) noise. No step reduces a prediction to a fitted parameter by construction, invokes a self-citation as the sole justification for a uniqueness claim, or renames an input statistic as an output. The assumption on initial conditions is stated explicitly in the abstract and generic-EBM section rather than smuggled in; the subsequent solvable-model sections simply instantiate the same derivation. The result therefore stands or falls on the correctness of the dynamical calculation itself, not on any definitional loop.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, invented entities, or ad-hoc axioms; the derivation rests on standard properties of Markov chains and gradient estimation in EBMs.

axioms (1)
  • standard math Markov chain Monte Carlo sampling reaches a stationary distribution under standard conditions
    Invoked when contrasting convergent vs. non-convergent short runs

pith-pipeline@v0.9.0 · 5705 in / 1205 out tokens · 29586 ms · 2026-05-24T10:16:40.223082+00:00 · methodology

discussion (0)

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Reference graph

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    write newline

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