What Limits Recursive Reasoning Models: Optimization, Architecture and Test-Time Scaling

Yuliana Shakhvalieva; Dmitrii Kharchev; Viacheslav Bezrukov; Inessa Fedorova; Dmitry Bocharov; Ivan Oseledets; Valerii Ternovskii

cs.LGSep 30, 2026

What Limits Recursive Reasoning Models: Optimization, Architecture and Test-Time Scaling

Authors: Yuliana Shakhvalieva, Dmitrii Kharchev, Viacheslav Bezrukov, Inessa Fedorova, Dmitry Bocharov, Ivan Oseledets, Valerii Ternovskii

Organizations: RND NLP, DAIMLD, Russian Federation

Abstract

Recursive reasoning models apply a small shared Transformer block many times to refine a latent state. This gives them large effective depth with few parameters and makes them strong on algorithmic tasks. Such compact solvers are natural candidates for tools that an LLM can call on narrow algorithmic subproblems. However, existing models such as HRM, TRM and URM differ in architecture, gradient propagation and training procedure simultaneously. This makes it hard to tell what drives their performance, and their optimization is still poorly understood and often unstable. In this work we address both of these gaps. First, we study these questions under a unified experimental pipeline spanning six algorithmic domains. Individual controlled ablations are performed on representative domains, while the resulting recipe is evaluated across the full suite. The study reveals a surprisingly simple recipe for stable and generalizable recursive reasoning: an intermediate gradient horizon, large physical batches and controlled updates of the recurrent state. An explicit hierarchical architecture is not needed. Second, we combine these findings into a stable 13.6M-parameter model that achieves the strongest overall performance among the evaluated recursive baselines, with particularly large gains on out-of-distribution generalization. It raises Arithmetic OOD accuracy to 71.2%, from 36.2% for the strongest baseline, while reaching 98.41% on Sudoku and 59.5% pass@2 on ARC-AGI-1. Our results show that, within the recursive architectures studied here, performance depends strongly on how recurrence is optimized and stabilized. More broadly, it shows how AI systems can be improved by optimizing their components one at a time.

Figures & tables

Figure 1: Recursive reasoning performance is determined by both recurrent computation and its optimization. We systematically study the factors that govern recursive models: gradient propagation through the recurrent trajectory, recurrent-state stabilization, architectural choices, and test-time computation. Our final 13.6M-parameter model combines the most effective components and achieves strong performance across algorithmic domains while improving out-of-distribution generalization.

Figure 2: Evaluation domains used in our study. Sudoku, Maze, and ARC-AGI follow established recursive-reasoning benchmarks, while Game of Life and Arithmetic provide controlled out-of-distribution evaluation.

Figure 3: Overview of the final recursive reasoning architecture. A shared Transformer block repeatedly refines latent recurrent states. Controlled state updates and truncated gradient propagation stabilize the recursive trajectory.

Setup	HRM	TRM	URM	Ours
Parameters	27M	7M	7M	13.6M
Modules	2 ( $H+L$ )	1 shared	1 shared	1 shared
Layers	$4+4$	2	2*	4
Hidden size	512	512	512	512
Cycles $(H,L)$	$(2,2)$	$(3,6)$	$(3,6)$	$(4,2)$
Gradient path	Last $H$ + last $L$	Last $H$ + all $L$	Last $H$ + all $L$	$(K_{L},K_{H})=(2,2)$

Table 1: Comparison of recursive reasoning recipes.

Metric	Dense	Qwen3-30B-A3B	HRM	TRM	URM	Ours
Arithmetic ID	78.50	0.10	92.00	98.80	95.60	96.34
Arithmetic OOD	7.49	0.00	13.80	36.20	28.50	71.16
Game of Life ID	8.17	0.00	5.00	7.11	55.42	66.08
Game of Life OOD	7.54	0.00	0.30	7.14	55.26	65.80
Maze	8.00	0.20	71.10	80.00	80.20	84.70
Sudoku	18.30	0.00	50.00	87.40	77.60	98.41

Table 2: Exact accuracy on the evaluation domains, in percent, from EMA weights. Arithmetic is split into the in-distribution and out-of-distribution target buckets of Section 3 ; Game of Life is reported on its in-range set and as the mean over its out-of-distribution sets. ARC-AGI is evaluated with pass@1 and pass@2. HRM, TRM, URM, and the dense control are retrained in our codebase; general-purpose models are evaluated from released checkpoints without fine-tuning. Best values are in bold.

Figure 4: Exact accuracy on Arithmetic as a function of the differentiated gradient horizon. Forward recursion is fixed at $L_{\mathrm{cycles}}=2$ and $H_{\mathrm{cycles}}=4$ ; only $(K_{L},K_{H})$ varies. Left: in-distribution accuracy. Right: out-of-distribution accuracy.

Figure 5: Effect of physical batch size versus gradient accumulation at a fixed effective batch size of 8192 on Game of Life. Configurations correspond to physical batch sizes of 8192, 4096, and 2048 with 1, 2, and 4 accumulation steps, respectively. Left: ID and OOD accuracy. Right: mean pre-clipping gradient norm.

Configuration	GoL ID	GoL OOD	Sudoku
Base	$61.95\pm 3.32$	$63.87\pm 2.90$	$97.29\pm 1.06$
+ gate	$\underline{64.24\pm 2.59}$	$\mathbf{65.89\pm 2.77}$	$98.41\pm 0.25$
+ grad clip	$62.41\pm 1.56$	$63.87\pm 1.54$	$\mathbf{98.50\pm 0.20}$
+ dropout	$61.82\pm 0.34$	$62.09\pm 0.41$	$98.36\pm 0.25$
+ relative noise (final)	$\mathbf{66.08\pm 0.11}$	$\underline{65.80\pm 0.13}$	$\underline{98.41\pm 0.21}$

Table 3: Cumulative stabilization ablation on Game of Life and Sudoku. Post-cycle normalization and EMA are fixed in all configurations. Values report mean

\pm

standard deviation over three seeds. Best values are in bold and second-best values are underlined.

Panel of Figure 6 — Figure 6: Architectural and test-time scaling ablations. (a) Effect of sharing parameters between recurrent timescales. (b) Effect of additional recurrent computation at inference time.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Domain	Input structure	Primary capability	Controlled OOD
Sudoku	$9\times 9$ grid	Constraint satisfaction	–
Maze	$30\times 30$ grid	Search and planning	–
Game of Life	Variable-size binary grid	Iterative dynamics	Patterns, horizon
Arithmetic	RPN expression	Symbolic composition / search	Operands, value range
ARC-AGI-1	Grid transformations	Abstract rule induction	–
ARC-AGI-2	Grid transformations	Abstract rule induction	–

Appendix

Table 4: Summary of the six evaluation domains. Game of Life and Arithmetic provide explicit controlled distribution shifts; the remaining domains are retained primarily for comparability with prior recursive reasoning models.

Domain	$d_{\mathrm{core}}$	$d_{H}=d_{L}$	Conv.	$\eta$	$z_{L}$ -gate
Arithmetic	.025	.010	no	.005	yes
Sudoku	.010	.000	no	.005	yes
Game of Life	.100	.000	no	.005	yes
Maze	.010	.000	no	.005	yes
ARC-AGI-1	.025	.025	yes	.003	yes
ARC-AGI-2	.025	.025	yes	.005	yes

Appendix

Table 5: Domain-specific recurrent regularization.

d_{\mathrm{core}}

denotes dropout in the embedding, attention, residual, and feed-forward paths, while

d_{H}

and

d_{L}

denote dropout applied to the recurrent states

z_{H}

and

z_{L}

\eta

is the relative state and embedding noise scale. “Conv.” indicates convolutional input mixing. Controlled

z_{L}

updates use

\tau=0.7

Domain	Physical batch	Epochs	Eval. interval	Peak LR	Min. LR ratio	$\beta_{2}$	Weight decay	Puzzle LR
Arithmetic	4096	2000	50	$5{\times}10^{-4}$	.01	.95	1.0	.005
Sudoku	4096	2000	50	$1{\times}10^{-4}$	1.0	.95	0.1	.010
Game of Life	8192	2000	50	$1{\times}10^{-4}$	.10	.95	0.1	.010
Maze	1024	54000	500	$1{\times}10^{-4}$	.01	.995	0.1	.010
ARC-AGI-1	768	300000	10000	$1{\times}10^{-4}$	1.0	.95	0.1	.010
ARC-AGI-2	768	300000	10000	$1{\times}10^{-4}$	1.0	.95	0.1	.010

Appendix

Table 6: Domain-specific optimization configurations. All runs use 2,000 learning-rate warmup steps,

\beta_{1}=0.9

, global gradient clipping at norm 1, and EMA weights. Puzzle embeddings use a separate learning rate (“Puzzle LR”) and the same weight decay as the remaining parameters.

Model	Layers $(H,L)$	Cycles $(H,L)$	Relative compute
Dense Transformer	$8,-$	–	$1$
HRM	$4,4$	$(2,2)$	$3T$
TRM	$2,2$	$(3,6)$	$5.25T$
URM	$2,2$	$(3,6)$	$5.25T$
Ours	$4,4$	$(4,2)$	$6T$

Appendix

Table 7: Inference cost of one answer relative to a single forward pass of the non-recurrent dense control.

T

denotes the number of ACT steps.

Figure 7: Effect of the gradient-path length on Arithmetic. Top: final EMA best_full exact accuracy as a function of the number of high-level cycles retained in the backward graph, for one or two retained low-level cycles. Bottom: OOD accuracy over training. The selected $(2,2)$ model obtains the highest final OOD score.

$g_{L}$	$g_{H}$	Final ID	Final OOD	Best OOD	Best update
1	1	97.35	13.87	19.67	87,920
1	2	99.54	7.93	20.91	87,920
1	3	99.60	8.80	17.88	428,610
1	4	99.31	65.53	68.54	428,610
2	1	98.12	23.71	28.61	351,680
2	2	96.34	71.16	71.16	428,610

Appendix

Table 8: EMA best_full exact accuracy under gradient truncation. Accuracies are percentages. Final values are measured at 439,600 updates. “Best OOD” is the maximum over the 40 recorded checkpoints and uses the OOD split for checkpoint selection.

Figure 8: Single-seed Game of Life training trajectory. Left: EMA best_full exact accuracy on ID and five OOD splits. Right: the unweighted OOD mean for online and EMA parameters; shading spans the minimum and maximum EMA OOD split. Accuracy peaks at epoch 1150 and declines before the final checkpoint.

Split	Online final	Online peak	EMA final	EMA peak	EMA token acc.	Best forced step
ID	63.70	65.99	64.01	65.98	75.81	63.75 (4)
OOD $+1$ step	63.71	65.73	64.04	65.94	75.83	63.79 (4)
OOD $+2$ steps	63.65	65.73	63.94	65.83	75.92	63.67 (3)
OOD $+3$ steps	63.02	65.22	63.35	65.37	75.45	63.03 (3)
OOD $+10$ steps	63.46	65.94	64.04	65.87	75.92	63.76 (4)
OOD unseen patterns	63.12	65.34	63.46	65.36	75.69	63.12 (5)

Appendix

Table 9: Final and peak validation performance. Values are percentages. Exact columns use the logged best_full exact-accuracy series; token accuracy is the EMA accuracy series at the final checkpoint. All EMA peaks occur at 211,140 updates (epoch 1150). “Best forced step” reports the highest single-step exact score at the final checkpoint, followed by its logged ACT step in parentheses.

Figure 9: Final and peak EMA best_full quality and recurrent-depth profile. Left: final and peak best_full exact accuracy for every evaluation split. Right: EMA single-step exact accuracy at the final complete checkpoint. Most of the single-step gain occurs between forced ACT steps 1 and 2; all six curves then remain in a narrow band through step 24.

Figure 10: Exact validation accuracy during joint four-domain training. Online and EMA parameters are evaluated at six checkpoints. Arithmetic improves throughout training, but Game of Life remains below 0.1% exact accuracy and neither Maze nor Sudoku produces a single exact solution. The panels use different vertical scales.

Domain	Final token acc.	Best exact acc.	Final exact acc.
Arithmetic	63.97	28.22	28.22
Game of Life	51.68	0.09	0.06
Maze	87.50	0.00	0.00
Sudoku	20.21	0.00	0.00

Appendix

Table 10: EMA validation metrics for joint training. Values are percentages. “Best” is selected over the six checkpoints; final metrics are measured at 194,562 updates. Token accuracy can be high even when the full structured output is always incorrect.

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Recursive models create computational depth through parameter reuse, offering a parameter-efficient alternative to increasing model size. However, most recursive models repeatedly apply one learned transformation or a prescribed sequence of transformations, restricting computation to a fixed layer order. We introduce the Random Recursive Model (RRM), which maintains a pool of

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T

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