Energy-Based Reasoners: Your Recurrent Reasoner Should Be an Energy-Based Model

William Xingxu Chen; Alexi Gladstone; Zeyu Michael Li; Heng Ji; Yilun Du

cs.LGNov 6, 2026

Energy-Based Reasoners: Your Recurrent Reasoner Should Be an Energy-Based Model

Authors: William Xingxu Chen, Alexi Gladstone, Zeyu Michael Li, Heng Ji, Yilun Du

Organizations: Duke University · UIUC · Harvard University

Abstract

Looping is a powerful inductive bias for reasoning. By repeatedly iterating a recurrent block, looped models can scale their effective depth and enable multi-step reasoning in latent space. However, existing looped models are not inherently guided by an explicit measure of progress, making it difficult to determine when to stop looping and how to improve upon the current state. To enable this, we introduce Energy-Based Reasoners (EBRs), which learn an energy function over latent reasoning states and formulate looped updates as gradient descent on this energy. The energy scores latent states, while its gradient guides refinement and provides a signal for stopping. To train EBRs effectively, we further introduce exploration-based training to discover additional modes and to better shape the energy landscape over a broader set of states. Across four algorithmic reasoning benchmarks, EBRs match or outperform prior recurrent reasoning models, achieving 99.9% accuracy on Sudoku-Extreme while using 345 $\times$ less inference compute than the next-best model and maintaining comparable training compute. On all multi-solution benchmarks, EBRs achieve state-of-the-art coverage of distinct valid solutions while maintaining near-100% single-sample accuracy. These results suggest that energy-based models are a promising approach for strong reasoning performance.

Figures & tables

Figure 1: Reasoning as energy minimization. Energy-Based Reasoners (EBRs) solve a puzzle $x$ by gradient descent on a learned energy $E_{\theta}(z;x)$ over latent reasoning states $z$ . The surface shows this energy for one Sudoku puzzle on the plane through three states of a descent trajectory.

Figure 2

Figure 2: Segment-wise training. Each segment consists of

n

descent steps. The first

n-H

steps are computed without retaining a backpropagation graph (gray), while gradients propagate through the final

H

steps (blue), whose states are supervised by

\mathcal{L}

( eq. 3 ). After each segment, the loss is propagated and converged trajectories stop; otherwise, the detached endpoint initializes the next segment.

Figure 3: State-based XM.

K

candidate trajectories are rolled out without gradients, and the trajectory–target pair with the lowest prediction loss is selected. The selected trajectory is replayed, with gradients propagated through its final

H

steps ( fig. 2 ).

Method	# Params	Sudoku-Extreme (↑)	Maze-Hard (↑)
Single sample (%)
TRM [ 12 ]	5–7M	87.4	85.3
EqR [ 18 ]	5–7M	93.0	–
FPRM [ 17 ]	7M	94.2	87.0
GRAM [ 19 ]	10M	86.6 $\pm$ 2.2	–
Looped Flows [ 26 ]	5–7M	97.9 $\pm$ 0.4	86.7 $\pm$ 1.1

Table 1: Accuracy on Sudoku-Extreme and Maze-Hard. EBRs achieve strong single-sample accuracy on both benchmarks while using the fewest parameters. For multiple samples, the number of samples, N, is shown in parentheses. EBRs select the lowest-energy solution, while baseline results are taken from their papers and follow their respective evaluation protocols.

Figure 4: Test-time scaling on Sudoku-Extreme. Exact accuracy as we increase (a) the number of iterations with one sample and (b) the number of samples. EBRs select the lowest-energy sample. GRAM and TRM are from Baek et al. [19] . EqR is evaluated from its released checkpoint.

Figure 5: Training and inference efficiency on Sudoku-Extreme. (a) Single-sample exact accuracy at 16 iterations versus training compute. (b) Exact accuracy versus inference FLOPs per puzzle. We compare EBRs with and without convergence-based stopping against GRAM, EqR and TRM, using

N{=}8

samples for EBRs,

N{=}20

and

50

for GRAM,

N{=}128

for EqR, and

N{=}1

for TRM. GRAM and TRM are from Baek et al. [19] , and EqR is its reported accuracy [ 18 ] . Convergence-based stopping allows EBRs to maintain high accuracy with

10.8\times

less inference computation compared to EBRs with fixed iterations.

Figure 6: Correct solutions converge to low-energy stationary states. Energy (a), tangent gradient norm (b), and relative stepwise residual (c) throughout descent on Sudoku-Extreme, grouped by whether the final prediction is correct. Solved trajectories reach lower energies with small gradients and residuals, while incorrect trajectories continue to exhibit larger gradients and residuals.

Figure 7: Energy-based selection closely approaches oracle accuracy. Sudoku-Extreme exact accuracy when selecting one of

N

independently generated samples using the lowest energy, lowest gradient norm, or convergence criterion. Oracle uses test label.

Table 10

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

	TRM	EBR
Block output	the next latent state	a scalar energy ( eq. 7 )
State update	replaced by the block output	a step along $-\nabla_{z}E_{\theta}$
Latent state	hierarchical, $z_{H}$ and $z_{L}$	a single $z$
Initial state	one fixed vector for every input	fresh draw from a normal
Halting	learned Q head	residual of the descent ( eq. 4 )

Appendix

Table A.1 : Key mechanism differences between TRM and EBRs.

			N-Queens		Graph Coloring
	Sudoku	Maze	$8{\times}8$	$10{\times}10$	8-vertex	10-vertex
Model
Hidden size	256	256	512	512	512	512
Parameters	1.8M	1.8M	7.1M	7.1M	7.1M	7.1M
Per-cell energy	linear	softplus	linear	linear	linear	linear
Descent

Appendix

Table A.2 : Per-task settings. Settings shared by all tasks are given in the text.

			N-Queens		Graph Coloring
	Sudoku	Maze	$8{\times}8$	$10{\times}10$	8-vertex	10-vertex
Training steps	60k	50k	30k	45k	70k	70k
Wall time (hours)	3	3	4	8	5	8

Appendix

Table A.3 : Training time of one seed on a single NVIDIA H200 GPU, including training evaluations.

	Training steps		Total EFLOPs		$N{=}1$ result
Task	EBR	GRAM	EBR	GRAM	EBR	GRAM
Sudoku-Extreme	60k	65.1k	1.92	2.26	99.41 $\pm$ 0.12	86.6 $\pm$ 2.2
Maze-Hard	50k	–	1.16	–	87.37 $\pm$ 1.1	–
N-Queens $8{\times}8$	30k	47.0k	2.45	1.08	100 $\pm$ 0	99.5 $\pm$ 0.3
N-Queens $10{\times}10$	45k	140.8k	5.82	4.74	99.51 $\pm$ 0.45	89.7 $\pm$ 2.7
Graph Coloring, 8-vertex	70k	45.5k	2.46	0.67	0 $\pm$ 0	8.3 $\pm$ 2.5

Appendix

Table B.1 : Training compute and single-sample results of EBRs and GRAM. Total training FLOPs of the full run and the

N{=}1

result, as in tables 1 and 3 . GRAM reports no Maze-Hard results.

$\tau$	Avg. iterations	Pass@1	Pass@8	TFLOPs / puzzle	Saving
No stopping	64	99.41 $\pm$ 0.12	99.92 $\pm$ 0.05	6.60	1.0 $\times$
0.1	7.6 $\pm$ 1.4	99.41 $\pm$ 0.12	99.92 $\pm$ 0.05	0.82	8.0 $\times$
0.2	5.6 $\pm$ 0.6	99.41 $\pm$ 0.12	99.92 $\pm$ 0.05	0.61	10.8 $\times$
0.5	3.6 $\pm$ 0.1	99.42 $\pm$ 0.12	99.86 $\pm$ 0.07	0.39	17.1 $\times$
1	3.5 $\pm$ 0.1	99.41 $\pm$ 0.12	99.88 $\pm$ 0.07	0.38	17.4 $\times$
2	3.5 $\pm$ 0.1	98.81 $\pm$ 0.17	99.89 $\pm$ 0.06	0.38	17.6 $\times$

Appendix

Table C.2 : Effect of the stopping threshold

\tau

on Sudoku-Extreme. Iterations are averaged over samples, with a cap of 64. Pass@8 keeps the lowest-energy of eight samples. TFLOPs are per puzzle for eight samples and include the cost of the gradient check, and the saving is relative to running all 64 iterations.

Figure D.1 : Different shortest paths for the same Maze-Hard instance. (a) The benchmark’s labeled path and (b) an alternative both reach the goal (G) from the start (S) in 111 steps. (c) Path-specific edges are colored, and shared edges are gray. Exact match accepts only (a) , whereas shortest-path accuracy accepts both.

Inference	Labeled-path exact match	Any shortest path
EBR, Pass@1	87.37 $\pm$ 1.10	94.07 $\pm$ 0.91
EBR, Pass@5	88.27 $\pm$ 0.32	94.77 $\pm$ 0.59

Appendix

Table D.1 : Exact-match and shortest-path accuracy (%) on Maze-Hard.

Figure E.1 : Reverse SXM with partial solution sets. We restrict reverse SXM to at most

k

valid targets per training example and measure their accuracy and coverage.

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Energy-guided Recursive Model

Yifei Zhao, Ying Tang

Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China · School of Physics, University of Electronic Science and Technology of China, Chengdu 611731, China

Language Modeling Recursive Language Models Energy-Based Models Inference-Time Search Algorithmic Reasoning Inference-Time Guidance+1

Open paper AskPDF arXiv

Recursive models show promise on reasoning and language tasks, yet their test-time scaling lacks a principled criterion for selecting trajectories or determining recurrent depth. We introduce Energy-guided Recursive Model (ERM), which uses Hopfield-type memories of valid local and global structures to assign intrinsic energies to candidate trajectories. These energies guide candidate selection and suggest an effective range of recurrent depths, implying that deeper recurrence does not necessarily improve reasoning accuracy. They also enable sampling methods such as parallel tempering to improve exploration. For reasoning tasks, ERM achieves optimal solutions on Sudoku (

98.97\%

), Pencil Puzzle Bench (PPBench,

88.04\%

) and Maze (

99.30\%

), reaching the best accuracy in recursive modeling. On language modeling, ERM reduces RedPajama-V2 perplexity by

1.74\%

with marginal inference overhead. The results support energy guidance as a practical framework for improving test-time scaling in recursive models.

Abstract

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