Title: A Difference-of-Convex Functions Approach to Energy-Based Iterative Reasoning
Venue: NeurIPS 2025
Accepted: yes
Keywords: Energy-based models

Iterative reasoning

Difference-of-convex function optimization
Abstract: While energy-based models have recently proven to be a powerful framework for learning to reason with neural networks, their practical application is still limited by computational cost. That is, existing methods for energy-based iterative reasoning suffer from computational bottlenecks by relying on expensive optimization routines during training and especially during inference. Furthermore, these routines may not always converge to minimal energy states, potentially leading to suboptimal reasoning. To address these limitations, we propose a novel and efficient algorithm for energy-based iterative reasoning based on a difference-of-convex (DC) functions approach. Our algorithm achieves a significant speedup compared to prior methods while offering theoretical convergence guarantees ensuring locally minimal energy states. In addition, we achieve state-of-the-art or superior performance on continuous reasoning tasks, as demonstrated by our experiments on multiple benchmark datasets from continuous algorithmic reasoning. As such, our method offers a leap in computational efficiency, enabling faster inference with theoretical guarantees, and hence unlocking the potential of energy-based models for iterative reasoning applications.


Peer reviews:
comments: SUMMARY:
This paper proposes an approach to energy-based reasoning where a difference of convex functions is used to represent the energy function. The paper provides theoretical results illustrating how this approach can approximate arbitrarily complex functions, while also ensuring that optimization converges to valid minima. Finally, the paper illustrates the applicability of the method on several numerical tasks, with an interesting energy landscape recovered in token reasoning. Overall, this paper has a nice blend of both emperical practice as well as theoretical analysis

STRENGTHS AND WEAKNESSES:
Strengths:

- The paper is clear and well motivated
- The paper provides theoretical justification on their approach, with the approach
having a provable minimum that it can converge to.
- The paper illustrates a set of promising results on real-world settings

Weaknesses 

- Most of the setting shown in the paper are a little toy, but I don't think this really 
detracts from the paper, as I think it sufficient to illustrate the promise of the method

QUESTIONS:
- Would the method presented in the paper also work with non single layer MLP networks?
- How robust are the energy minima recovered from this approach? Do they have less local
minima than prior work on energy based reasoning?
- Could the authors provide a bit more clarification on why the subgradient in Equation 8 of the paper
is the subgradient of Equation 4? I derived a slightly different form when going through the computation.

LIMITATIONS:
The paper adequetly discusses the limitations of the approach.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-29

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a novel approach for iterative reasoning that formulates the problem of reasoning as energy minimization, which, with appropriate neural network parameterization, is transformed into a difference of convex functions. The authors provide theoretical convergence guarantees as well as expressivity results for the parameterization used. Finally, the DCA routine is scaled to perform batched optimization. The experiments are performed on matrix-based tasks (completion, inversion, decomposition, etc) and energy landscape optimization in the embedding space of language models to show the efficacy of the proposed method compared to previous approaches for energy-based iterative reasoning.\n\nSTRENGTHS AND WEAKNESSES:\n**Quality**: The authors provide theoretical convergence guarantees for DCAReasoner as well as expressivity results for the parameterization used. The experiments demonstrate the improvement in efficiency compared to previous energy-based reasoning approaches. However, the choice of tasks used in the experiments and the lack of comparison with other kinds of baselines (see question 1 below) fall short in providing a comprehensive picture.\n\n**Significance**: While the general idea of energy-based iterative reasoning is quite promising. The choice of experiments and baselines could be expanded to make the paper more impactful.\n\n**Clarity**: The paper is well written and easy to follow. \n\n**Originality**: The use of DC optimization for energy-based iterative reasoning and its proof of convergence both incorporate original ideas.\n\nQUESTIONS:\n1. What is the motivation behind selecting the task in section 5.3? The paper can be made significantly more impactful if a more realistic task, ideally in discrete space (for example, Sudoku or Zebra Puzzles), is chosen to demonstrate the effectiveness of the proposed method. In that setting, it would be nice to compare the performance of DCAReasoner with iterative feed-forward approaches like discrete diffusion and search-based auto-regressive methods (for example, [1,2])\n\n2. As I understand from the description, the DistilBERT was fine-tuned on the symptom-to-diagnosis dataset (please correct me if I'm misinterpreting here). If so, what was the accuracy of the DistilBERT classifier?\n\n# References\n[1] [[2409.10502] Causal Language Modeling Can Elicit Search and Reasoning Capabilities on Logic Puzzles](https://arxiv.org/abs/2409.10502)\n\n[2] [[2502.06768] Train for the Worst, Plan for the Best: Understanding Token Ordering in Masked Diffusions](https://arxiv.org/abs/2502.06768)\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces DCAReasoner, an energy-based iterative reasoning algorithm leveraging a Difference-of-Convex decomposition. It provides finite-step convergence guarantees and universal function approximation, and achieves 3–27× inference speedups on tasks including polynomial root finding, matrix inversion, and quadratic programming, while matching or exceeding state-of-the-art accuracy.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n\n* Theoretical guarantees. Finite-step convergence to a local minimum and universal approximation results under mild sign-constraints ensure both reliability and expressivity.\n* End-to-end trainability. By deriving closed-form updates for each DC subproblem, the entire K-step inference can be unrolled into a differentiable computation graph, enabling direct gradient-based optimization of the energy model parameters.\n* Fast training. Closed-form DC updates replace inner gradient-descent loops, reducing both per-epoch compute and memory overhead—training runs substantially faster than methods that nest gradient-descent in the forward pass.\n* Significant inference speedups. Achieves 3–27× faster inference on polynomial root finding, matrix inversion, and QP benchmarks while matching or exceeding prior accuracy.\n* Broad applicability. Demonstrated across continuous algorithmic reasoning, high-dimensional embedding-space classification, and convex/non-convex energy landscapes.\n\n## Weakness \n* Generalization Beyond Training Distribution: Because the energy function $E_\\theta(x;y) $ is learned to minimize MSE on the training pairs $(y_i,x_i)$, it may overfit to those specific problem instances. It’s unclear how well it generalizes to out‐of‐distribution y (e.g.\\ different matrix condition numbers or QP formulations).\n* High-Dimensional Scalability: Solving the DC subproblem still requires inverting a Hessian-like operator or solving a fairly large convex system at each step. While tractable up to a few hundred dimensions, it may become prohibitive in, say, $\\geq10^4$-dimensional settings without further approximations.\n* Robustness to Noise and Model Mis-specification: There is no evaluation of how DCAReasoner handles noisy or approximate y (e.g.\\ measurement error in the input matrix) or errors in the DC decomposition. Such perturbations could cause the DCA iterates to converge to spurious local minima.\n\nQUESTIONS:\n* The acronym “DCA” appears before it is explicitly defined. Could you please clarify and provide its full form at first use?\n* What criteria guided your choice of the number of DCA iterations K in each experiment? How sensitive are your results—both in terms of accuracy and convergence speed—to different values of K and to the initialization x_0?\n* The paper does not fully specify the hyperparameters and configurations used for the baseline methods. Could you include detailed settings for all baselines to ensure reproducibility and a fair comparison?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors proposed a new algorithm named DCAReasoner for energy-based continuous iterative reasoning. It is build upon a tailored class of energy functions for which derived theoretical approximation guarantees. In addition, the authors presented theoretical convergence guarantees for the inherent DCA routine. That is, the authors showed that it converges to local minima in finitely many steps independent of the starting point. Empirically, the authors have proven that it yields improved performance and inference times.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n1. The authors introduce a tailored form of energy functions and present a DCA routine for powering novel energy learning algorithm.\n\n2. The authors derive theoretical convergence guarantees of our DCA routine to local solutions.\n\n3. The authors show that our DCA routine converges in finitely many steps and, hence, offers a clear termination criterion.\n\n4. Under additional assumptions, the authors show how the energy learning algorithm can be scaled for batch optimization and present theoretical approximation guarantees for our form of energy function.\n\nWeaknesses\n\n1. Restrictive Assumption Limiting Expressiveness\nThe paper's core practical contribution relies on Assumption 1 (requiring all αᵢ ≤ 0), which significantly constrains the model's expressiveness. While the authors provide theoretical approximation guarantees for \"convexly-regular\" functions, this is a somewhat artificial class they define. The restriction to non-positive weights in the energy function may limit the model's ability to learn complex reasoning patterns that require both positive and negative activations. The authors acknowledge this could affect universal approximation properties but don't thoroughly investigate the practical implications of this limitation.\n\n2. Limited Experimental Scope and Generalization. The experimental evaluation is quite narrow:\n\n\t•\tSmall-scale problems: All continuous reasoning tasks involve only 20×20 matrices, which doesn't demonstrate scalability to realistic problem sizes\n\n\t•\t Artificial benchmarks: The tasks (matrix inversion, completion, etc.) are relatively simple mathematical operations that may not reflect the complexity of real-world reasoning scenarios\n\n\t•\tToken embedding experiment is preliminary: The medical diagnosis task, while interesting, is quite simple (22 classes) and doesn't convincingly demonstrate the method's potential for complex language reasoning applications\n\nQUESTIONS:\nsee Weaknesses\n\nLIMITATIONS:\nYES", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-05"}