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REVIEW 4 major objections 5 minor 46 references

Joint Learning of Energy-based Models and their Partition Function

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper's central claim is that a probabilistic energy-based model and its log-partition can be learned jointly by an unbiased, MCMC-free stochastic gradient scheme that provably recovers maximum likelihood in function space and…

desk verdict A clean min-min surrogate for EBM MLE with a solid theoretical core, but the experimental claims rest on a shared neural tau outside the proven guarantees. read the letter →

arxiv 2501.18528 v3 pith:7ZGWKIH6 submitted 2025-01-30 cs.LG stat.ML

Michael E. Sander , Vincent Roulet , Tianlin Liu , Mathieu Blondel This is my paper · ORCID
classification cs.LGstat.ML
keywords energy-basedmodelslog-partitionfunctionmaximumlikelihoodestimationFenchel-Younglossessparsemaxstructuredpredictionmultilabelclassificationlabelranking
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes a way to train probabilistic energy-based models over combinatorially large discrete output spaces, such as all subsets or all permutations of labels, without Markov chain Monte Carlo. The idea is to learn the energy function and its log-partition simultaneously: a second network outputs the normalizing constant for each input, and both are optimized by a doubly stochastic gradient descent that only needs samples from a fixed reference distribution. In the space of continuous functions, the authors prove that minimizing this joint objective is exactly equivalent to maximum likelihood, with the learned log-partition converging to the true log-sum-exp. The same machinery extends to the wider family of Fenchel-Young losses, yielding the first tractable way to train with the sparsemax loss in these large discrete spaces. Experiments on multilabel classification and label ranking show the learned log-partition also generalizes to unseen inputs.

What carries the argument

The load-bearing object is the min-min objective in equations (12)-(13), in which the log-partition is promoted from a quantity to be computed into an optimization variable $\tau\in\mathcal{F}(\mathcal{X})$. The derivation rests on a convex-conjugate identity: $\Omega^*_1(h)=\min_{\tau\in\mathbb{R}}\tau+\Omega^*_+(h-\tau)$, which replaces the intractable normalization over $\mathcal{Y}$ with a one-dimensional minimization against the conjugate of the regularizer. In the MLE case this becomes $\mathcal{L}_{g,\tau}(x)=\tau(x)+\mathbb{E}_{y'\sim q(\cdot|x)}[\exp(g(x,y')-\tau(x))-1]$, giving the doubly stochastic estimator: sample $(x,y)$ from the data, sample $y'$ from the reference measure $q$, and update both networks with unbiased gradients. The second network for $\tau$ is what allows the log-partition to be evaluated on unseen inputs.

What would settle it

On a dataset whose true log-partition is computable in closed form, train with a capacity-limited $\tau$ network and send the number of reference samples $B'\to\infty$; if at the optimizer the held-out gap $\tau_v(x)-\mathrm{LSE}_{g_w}(x)$ does not shrink toward zero, or if the achieved logistic loss stays strictly above the MLE optimum, then the parameter-space objective has not reproduced the function-space equivalence.

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Extended reading notes

Core claim

On continuous functions, the paper's central result is that the MLE objective $\mathcal{L}_{\mathrm{MLE}}(g)$ and the joint min-min objective $\mathcal{L}_{\mathrm{MLE}}(g,\tau)$ share the same minimum over $g$, with the optimal $\tau^\star(x)$ equal to the log-sum-exp $\mathrm{LSE}_{g^\star}(x)$ and the implied conditional distribution equal to $q(y|x)\exp(g^\star(x,y)-\tau^\star(x))$. The new loss treats $\tau$ as a Lagrange multiplier for the normalization constraint $\sum_{y\in\mathcal{Y}} p_g(y|x)=1$ and reads $\mathcal{L}_{g,\tau}(x)=\tau(x)+\sum_{y'\in\mathcal{Y}} q(y'|x)(\exp(g(x,y')-\tau(x))-1)$. Since this is an expectation against the reference measure $q$, gradients are unbiased whenever one can sample from $q$, removing the need for MCMC. In the finite-sum setting, giving each training example a free per-example value $v_i$ recovers exact MLE even for models nonlinear in the parameters, and with linear couplings the objective is jointly convex in $(w,v)$. The same min-min equivalence is proved for the wider family of Fenchel-Young losses obtained by replacing the KL term with an $f$-divergence, which includes the sparsemax loss.

Load-bearing premise

The practical version assumes that a neural-network log-partition trained in parameter space lands close to the function-space optimum, because the exact maximum-likelihood equivalence is proved only for arbitrary continuous functions or for free per-example values, not for the neural-network parameterization used in the experiments.

Editorial extensions

If this is right

  • Probabilistic EBMs over sets and permutations become trainable by plain SGD with unbiased gradients, bypassing MCMC samplers and the #P-complete partition sum that makes exact MLE intractable.
  • The learned log-partition network yields a normalization-constant estimate on unseen inputs, which earlier methods that compute or treat the partition as a scalar did not provide.
  • The method gives the first tractable way to optimize the sparsemax loss on general combinatorially large output spaces, where prior approaches required a $k$-best oracle that may not exist.
  • For linear energy models with per-example log-partition values, the training objective is jointly convex, so SGD converges at the standard rates for convex objectives.
  • Using fewer reference samples acts as a regularizer and can improve test scores over exact MLE, an effect the paper documents on several datasets.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves implicit that the learned $\tau$ network could double as a proposal distribution for importance sampling or rejection sampling from $p_g(\cdot|x)$, turning the learned log-partition into a sampler accelerator; this would be a direct test of whether the approximation quality transfers to generative uses.
  • The observed regularization effect of small $B'$ suggests treating the reference-sample count as a bias-variance dial rather than an accuracy knob; a systematic study could map how the generalization gain varies with dataset size and output-space cardinality.
  • Because the stochastic gradient is unbiased only when $\tau$ equals the true log-partition, monitoring the gap $\tau_v(x)-\mathrm{LSE}_{g_w}(x)$ on a validation set during training could serve as a diagnostic for when the min-min objective silently stops approximating MLE.
  • The Fenchel-Young generalization hints that other $f$-divergence regularizers beyond chi-square, such as Tsallis entropies, could be trained by the same doubly stochastic scheme whenever their conjugate is easy to evaluate.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a method for learning conditional energy-based models p(y|x) ∝ q(y|x) exp(g(x,y)) over combinatorially large discrete spaces by jointly minimizing an energy function g and a log-partition function τ. The key objective is the min-min problem (12)-(13), which is optimized with a doubly stochastic SGD scheme (Algorithm 1) that samples outputs from a reference distribution q and never runs MCMC. The paper proves that in the space of continuous functions the min-min objective equals the MLE objective with τ*(x) = LSE_{g*}(x) (Proposition 1), that in the finite-sum setting with free per-example values τv(xi)=vi the empirical MLE is recovered exactly (Proposition 2), and that the construction extends to Fenchel-Young losses based on f-divergences, including sparsemax (Proposition 4). Experiments on multilabel classification and label ranking report improvements over MCMC and min-max baselines and show visual agreement between a learned neural τ and the closed-form log-partition on a unary model.

Significance. The variational proofs in Appendices C.2-C.5 are careful and the core identity is correct: minimizing (13) over τ yields the first-order condition that pins τ to the log-sum-exp, and the doubly stochastic estimator is unbiased under sampling from q. The finite-sum result with per-example τ is exact even for nonlinear g_w, and joint convexity for linear g with per-example τ gives concrete convergence rates. The sparsemax extension is a genuine contribution. The caveat is that the advertised 'MLE without MCMC' claim is proven only in function space or for per-example τ, while several headline experiments and the generalization claim about the learned log-partition use a shared neural-network τ that falls outside both guarantees. The paper explicitly acknowledges the approximation ('thanks to the universality of neural networks... should be close') but provides no bound and no quantitative measurement of the gap in the settings where τ is needed. If this gap can be closed, or the claims appropriately scoped, the paper would be a solid contribution; in its current form the central claim overreaches the theorem coverage.

major comments (4)
  1. [Section 3.1 (Propositions 1 and 2) and Section 4.2 (Table 2)] Proposition 1 is a function-space statement, and Proposition 2 covers only the finite-sum setting with free per-example values τv(xi)=vi. The label-ranking experiments in Table 2 and the generalization experiment in Figure 2 use a shared MLP (or ResNet/ICNN) τ network, which is covered by neither theorem. The statement in Section 3.1 that 'thanks to the universality of neural networks, minimization in the space of parameters should be close to minimization in the space of continuous functions' is an unquantified heuristic. This gap is load-bearing: in the objective (12)-(13), whenever τv(x) differs from LSE_gw(x), the gradient with respect to g is no longer the gradient of the MLE objective, so SGD can reduce the surrogate by exploiting τ's misspecification rather than by improving the energy model. To support the central claim, I would ask for either (a) a restriction of the headline claims to per-example τ, which is already exact for finite-sum MLE and is what Table 1 uses, with neural τ presented as an amortized heuristic validated explicitly; or (b) a quantitative analysis of the surrogate gap, e.g., a bound on the excess risk in terms of sup_x |τv(x) - LSE_g(x)|, together with a measurement of this quantity on held-out data and, for label ranking, a comparison against the per-example τ control.
  2. [Section 4.1 (Figure 2)] The claim that the learned log-partition 'generalizes to unseen data points' is supported only by a visual comparison on the unary model, where the true log-partition has the closed form (17). This testbed is the one setting where τ is actually unnecessary, since LSE_g is tractable. The more relevant settings are the pairwise multilabel model and the permutation models, where LSE is intractable, and no quantitative accuracy measure is reported there. I recommend reporting numerical errors (e.g., mean/median absolute or relative error against a closed-form LSE where available, or against a high-accuracy estimate on small instances) and extending the check, even if only on small k, to the pairwise and permutation settings.
  3. [Tables 1-2 and Figures 1-3] All experimental results appear to be single runs: the tables report point estimates with no standard deviations or confidence intervals, and the learning curves are single traces. Given that hyperparameters are selected on validation sets and the reported differences between methods are often a few points, it is difficult to assess whether the improvements are significant. Reporting means and standard deviations over several seeds, at least for the main tables, is necessary to support the empirical claims.
  4. [Section 3.2 (joint convexity paragraph)] The convergence-rate statements (O(1/√t) and O(1/t)) are proved in Appendix C.6 only for linear g_w with per-example τv(xi)=vi. The sentence 'we have obtained convergence rates for learning EBMs in arbitrary combinatorial spaces' should be qualified to this jointly convex setting; it does not cover the neural-network parameterization of g or τ used in most experiments. This is not an error in the proof, but the presentation currently invites a broader reading than the theorem supports.
minor comments (5)
  1. [Throughout] There are several typos: 'Bolzmann' in Section 1, 'mutiplier' in Section 3.1 and in the contributions list, and inconsistent capitalization of 'Resnet' vs 'ResNet' across Tables 1-2 and the appendix.
  2. [Appendix C.5] The final displayed equation writes π_{g*,q}(y|x) = q(y|x)(f_+^*)'(g*(x,y)), omitting the '- τ*(x)' that appears in the statement of Proposition 4 in the main text. The appendix should match the proposition.
  3. [Table 1 caption] The caption says 'with constant τ model, that is τv(xi) := vi', but v_i varies per example; calling this a 'constant' model is misleading. I suggest 'per-example τ' or 'free per-example values' instead.
  4. [Figure 1 caption] The caption states that the loss and gradient in plots (a) and (b) are computed using (15) even for the proposed method; this is informative but should also be stated in the main text near the discussion of convergence to exact MLE, since the plotted curves are not the training objective actually optimized.
  5. [Section 3.4] The claim of obtaining 'the first tractable method for optimizing the sparsemax loss in combinatorially-large spaces' is plausible, but the comparison with prior k-best-oracle methods (Pillutla et al., 2018) is brief; a short discussion of the oracle cost and why sampling from q avoids it would help readers assess the claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the min-min equivalence is a genuine variational identity, and the learned log-partition is benchmarked against an external closed form.

full rationale

The central equivalence claims (Propositions 1, 2, and 4) are genuine variational identities, not definitions of the target in terms of the input. The objective L_{g,tau}(x) = tau(x) + E_{y'~q}[exp(g(x,y') - tau(x)) - 1] is constructed so that the inner minimization over tau recovers LSE_g(x) by first-order conditions, and the proofs in Appendix C derive this from convex-conjugate calculus (Lemmas 1 and 2) rather than assuming the conclusion. This is a standard variational representation of the log-sum-exp, and the paper proves the equality for the continuous-function and finite-sum settings. The learned tau is not fitted to the closed-form LSE and then reported as a prediction: in Figure 2, tau is trained jointly with g on training data and evaluated on held-out inputs against the analytic softplus LSE of the unary model, which is an external benchmark internal to the paper. The Fenchel-Young/sparsemax extension follows from the same duality machinery and is an original derivation, not a renaming of a known result. Self-citations (e.g., Blondel et al. 2020a/2022, Sander et al. 2023) are used for background, baselines, or standard optimization tools and are not load-bearing; no 'uniqueness theorem' from the authors is invoked to forbid alternatives or force the proposed choice. The parameterization gap between the function-space guarantee and the neural-network experiments — the unquantified appeal to 'universality of neural networks' — is a correctness/scope concern about whether approximate MLE is achieved in practice, not a circularity, since the paper explicitly states that in parameter space the approach 'only performs approximate MLE' and does not claim the parameter-space optimum equals the function-space optimum.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The derivation itself is nearly parameter-free: one free scalar per input in the finite-sum theorem and no fitted constants in the MLE equivalence. The practical method, however, carries tuning knobs (B', tau architecture, learning rate and regularization) and one ad hoc premise: a neural tau approximates the log-partition well enough for the parameter-space objective to stay close to MLE. The reference measure q is an input to the method, not derived; uniform q in experiments is a domain assumption. No invented entities are introduced: tau is a standard Lagrange multiplier for the normalization constraint.

free parameters (3)
  • B' (number of reference samples y' per input) = ranging from 1 to 100000, tuned per experiment
    Algorithm 1 draws B' samples from q per input; Figures 1 and 3 show the estimator's noise depends on B', and the authors report a regularization effect on test f1, so B' is a result-affecting tuning knob, not a fixed constant.
  • learning rate and weight decay = tuned per dataset on a validation set
    The authors state hyperparameters are optimized on a validation set before refitting on train plus validation; standard practice, but part of the empirical claim.
  • tau network architecture = constant, MLP, ICNN, or ResNet
    Tables 4, 5, and 7 show scores vary with the tau family, and the generalization claim of the learned log-partition depends on this choice; architecture selection is part of the method's practical specification.
assumptions (4)
  • standard math Variational representation of log-sum-exp and Fenchel biconjugacy (Lemmas 1 and 2)
    The entire min-min objective is built on conjugate calculus, namely Omega*_1(h) = min_tau [tau + Omega*_+(h - tau)], cited to Boyd 2004 and used in Propositions 1, 2, and 4.
  • domain assumption Reference measure q(y|x) is fixed, strictly positive, and sampleable for every x
    The doubly stochastic estimator (14) requires sampling from q; the proof of Proposition 4 assumes q > 0, and the experiments use uniform q, so the method's viability depends on choosing a sampleable and reasonable reference measure.
  • domain assumption Continuity of g and q, and f strictly convex differentiable with (0,+infinity) contained in dom f'
    Propositions 1 and 4 state these regularity conditions to make the pointwise minimizer tau*(x) continuous and the min-expectation interchange valid, as shown in Appendix C.5, Steps 1 to 3.
  • ad hoc to paper Neural-network parameterization of tau is close enough to the function-space solution
    Section 3.1 asserts that 'thanks to the universality of neural networks, minimization in the space of parameters should be close to minimization in the space of continuous functions'; no bound is given, and the headline experiments depend on this gap being small.

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Cite this review

Pith. "Pith review of Joint Learning of Energy-based Models and their Partition Function." pith.science (2026). https://pith.science/paper/7ZGWKIH6

@misc{pith2026250118528,
  author       = {Pith},
  title        = {Pith review of: Joint Learning of Energy-based Models and their Partition Function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ZGWKIH6}},
  note         = {Machine review of arXiv:2501.18528}
}
read the original abstract

Energy-based models (EBMs) offer a flexible framework for parameterizing probability distributions using neural networks. However, learning EBMs by exact maximum likelihood estimation (MLE) is generally intractable, due to the need to compute the partition function (normalization constant). In this paper, we propose a novel formulation for approximately learning probabilistic EBMs in combinatorially-large discrete spaces, such as sets or permutations. Our key idea is to jointly learn both an energy model and its log-partition, both parameterized as a neural network. Our approach not only provides a novel tractable objective criterion to learn EBMs by stochastic gradient descent (without relying on MCMC), but also a novel means to estimate the log-partition function on unseen data points. On the theoretical side, we show that our approach recovers the optimal MLE solution when optimizing in the space of continuous functions. Furthermore, we show that our approach naturally extends to the broader family of Fenchel-Young losses, allowing us to obtain the first tractable method for optimizing the sparsemax loss in combinatorially-large spaces. We demonstrate our approach on multilabel classification and label ranking.

Figures

Figures reproduced from arXiv: 2501.18528 by the authors.

Figure 1
Figure 1. Convergence of the proposed approach as a function of the number of iterations, when varying the number of prior samples y ′ drawn. To be able to compute the exact MLE objective (15), we use the unary multilabel model (Section 4.1) on the cal500 dataset (174 classes and therefore 2 174 possible configurations) as the test bed. Note that the loss and gradient in plots (a) and (b) are computed using (15) even for our … view at source ↗
Figure 2
Figure 2. Generalization ability of the learned log-partition function for multilabel classification. As a testbed to compare the learned log-partition (dark purple), we use the unary model (see Section 4.1 for details), which enjoys a closed-form expression for the exact log-partition (light purple). We pick randomly 100 test samples (x-axis) on 5 multilabel classification datasets, after training models with hyper-parameter… view at source ↗
Figure 3
Figure 3. Convergence of the proposed approach as a function of the number of prior samples y ′ drawn. To be able to compute the exact MLE objective (15), we use the unary multilabel model (Section 4.1) as the test bed. Note that the loss and gradient in the left and center columns are computed using (15) even for our method. We make two key observations: i) our approach converges to exact MLE as predicted by our theory, ii) … view at source ↗

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.