Steering Recurrent Reasoners at Inference Time with Readout Feedback

Shunsuke Kamiya; Masanori Koyama; Seongcheol Jeong; Fumiya Uchiyama; Kenji Kubo; Kohei Hayashi; Masahiro Suzuki; Yutaka Matsuo

cs.LGAug 25, 2026

Steering Recurrent Reasoners at Inference Time with Readout Feedback

Authors: Shunsuke Kamiya, Masanori Koyama, Seongcheol Jeong, Fumiya Uchiyama, Kenji Kubo, Kohei Hayashi, Masahiro Suzuki, Yutaka Matsuo

Organizations: Graduate School of Engineering, The University of Tokyo

Abstract

Recurrent models, which repeatedly update latent states with shared computation blocks, have emerged as powerful architectures for solving complex reasoning tasks. Existing inference-time methods scale computation by running more steps or sampling more trajectories, but ignore information revealed within each trajectory. Here we show that recurrent models can be improved at inference time by using their own readout probabilities to steer latent dynamics without retraining. We introduce Readout Feedback (RoFB), a test-time intervention that converts intermediate predictions into token-wise pairwise coupling forces injected into the latent dynamics. Across three recurrent models (AKOrN, ItrSA++, TRM) on Sudoku and Maze, RoFB yields clear gains in four of six model-task pairs and a small gain in one. The four positive pairs achieve performance unattainable by merely running more steps or selecting from multiple trajectories, at comparable or lower computational cost. These results suggest that closed-loop steering of latent dynamics can serve as a complementary inference-time control mechanism for recurrent reasoning models.

Figures & tables

Figure 1: Conventional inference vs. our method. (a) Conventional inference without feedback for a recurrent model. (b) Inference with RoFB. RoFB injects a feedback coupling into the latent dynamics based on the distance between readout probabilities, leaving all model weights unchanged.

Model	Hierarchy	Latent Space	Initial Value	Input Injection
AKOrN	single-level	$(\mathbb{S}^{m-1})^{D/m}$	Random, unlearned	Every step
ItrSA++	two-level	$\mathbb{S}^{D-1}(\sqrt{D})$	Random, unlearned	Every $L$ steps
TRM	two-level	$\mathbb{S}^{D-1}(\sqrt{D})$	Fixed, learned	Every step

Table 1: Architectural differences of the models used in this work. The table summarizes the key architectural features of the three recurrent models (AKOrN, ItrSA++, and TRM) analyzed in this paper, including hierarchy, latent geometry, initialization, and input injection.

Figure 2: AKOrN inference dynamics on a Sudoku puzzle. (Top left) Total cell entropy, $\sum_{i}H(\mathbf{p}^{(i)})$ , over 256 inference steps. The shaded region indicates steps at which the predicted board is correct. (Top right) Predicted boards at $t=50$ and $t=200$ . Green and red indicate correct and incorrect predictions on blank cells, respectively. (Bottom) Token-wise cosine similarity matrices and spherical multidimensional scaling (MDS) visualizations at $t=50$ (left pair) and $t=200$ (right pair). The matrices include all tokens, grouped by predicted digit, with blank-cell tokens preceding given-cell tokens. The MDS plots show blank-cell tokens only, colored by predicted digit.

Figure 3: Compute-accuracy comparison of baseline vs. RoFB on model-task pairs. Color encodes

K_{\text{vote}}\in\{1,4,16,64\}

( Baseline blue / RoFB red , shade darkens with

K_{\text{vote}}

from 1 / 1 to 64 / 64 ). Shape encodes the inference time steps

T

(

\blacktriangle,\blacksquare

\bullet

for

T=64,128,256

;

N_{\text{block}}\in\{8,12,16\}

for TRM). Each AKOrN and ItrSA++ point is the mean over 8 inference seeds; TRM on Maze uses a compressed y-axis due to baseline saturation. The remaining

K_{\text{vote}}

values and all numerical values are in Appendix G .

Appendix figures & tables31 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: AKOrN inference dynamics on a Maze puzzle. (Top left) Total cell entropy, $\sum_{i}H(\mathbf{p}^{(i)})$ , over 256 inference steps, shown on a logarithmic vertical axis. The shaded region indicates steps at which the predicted board is correct. (Top right) Predicted boards at $t=50$ and $t=200$ . Green and red indicate correct and incorrect predictions on passage cells relative to the reference path, respectively. Darker shades indicate cells predicted to lie on the path. (Bottom) Token-wise cosine similarity matrices and spherical multidimensional scaling (MDS) visualizations at $t=50$ (left pair) and $t=200$ (right pair). The matrices include all tokens, grouped by predicted class in the order non-path, path, wall, start, and goal. The MDS plots show passage-cell tokens only, excluding the start and goal, colored by predicted class.

Figure 5: Cosine gap by solution correctness. Cells are grouped by terminal predicted labels. White dots show medians, thick bars the interquartile range, and thin bars the observed range; violin widths are normalized separately.

Figure 6: Silhouette score by solution correctness. Grouping and plot conventions follow Figure 5 . Higher scores indicate stronger separation of latent tokens according to their predicted classes.

Figure 7: Maximum normalized cell entropy by solution correctness. Lower values indicate higher confidence. Entropy is displayed on logarithmic axes with panel-specific ranges; densities are estimated in log space. Other plot conventions follow Figure 5 .

Figure 8: Clustering and confidence in terminal readout states. Each point is one board; green circles denote correct outputs and red crosses incorrect outputs. Entropy axes are linear for Sudoku and logarithmic for Maze, with panel-specific ranges. Dashed lines mark $s=0.500$ and $h_{\max}=0.100$ as visual reference levels, not optimized decision boundaries.

Task	$\lambda$	$\alpha$	$t_{\min}$	$\tau$	$N_{\text{val}}$	$n_{\text{trial}}$	$n_{\text{chunks}}$
Sudoku	$[0.01,\,2.0]$	$[0.01,\,0.50]$	$\{0,8,\dots,128\}$	$[0.005,\,2.0]$	500	30	4
Maze	$[0.005,\,0.5]$	$[0.01,\,0.50]$	$\{0,8,\dots,128\}$	$[0.005,\,2.0]$	100	30	2

Appendix

Table 2: Search space and Optuna configuration of the four-axis RoFB hyperparameter sweep.

\lambda

and

\tau

are sampled log-uniformly;

\alpha

uniformly;

t_{\min}

as integers in steps of 8 (Kuramoto updates for AKOrN, self-attention updates for ItrSA++, and

L

-level updates for TRM). The Successive Halving Pruner uses

\eta=3

\min_{\text{resource}}=1

for both tasks.

Task $\times$ Model	$\lambda$	$\alpha$	$t_{\min}$	$\tau$	best val acc	metric
AKOrN $\times$ Sudoku	1.949	0.281	16	1.552	100.0	board accuracy
AKOrN $\times$ Maze	0.394	0.195	128	0.141	75.0	shortest path accuracy
ItrSA++ $\times$ Sudoku	0.028	0.012	64	0.461	70.6	board accuracy
ItrSA++ $\times$ Maze	0.165	0.327	104	0.028	85.0	shortest path accuracy
TRM $\times$ Sudoku	0.829	0.182	88	1.462	74.8	board accuracy
TRM $\times$ Maze	0.167	0.216	88	0.939	91.0	shortest path accuracy

Appendix

Table 3: Selected RoFB hyperparameters and validation accuracy. Aggregated validation accuracies are reported in %; entries are point estimates.

Model-task pair	$T_{\max}$	$T_{\text{sweep}}$	$r_{\text{step}}$	$t_{\min}$	$r_{\text{tot}}^{(T_{\max})}$	$N_{\text{val}}$	$N_{\text{test}}$	$\widetilde{\mathcal{C}}_{\text{sweep}}$
AKOrN $\times$ Sudoku	256	256	$0.018$	16	$1.017$	500	20,000	$0.381$
AKOrN $\times$ Maze	256	256	$0.195$	128	$1.098$	100	1,000	$1.646$
ItrSA++ $\times$ Sudoku	256	256	$0.052$	64	$1.039$	500	20,000	$0.390$
ItrSA++ $\times$ Maze	256	256	$0.160$	104	$1.095$	100	1,000	$1.643$
TRM $\times$ Sudoku	16	288	$0.012$	88	$1.008$	500	20,000	$6.806$
TRM $\times$ Maze	16	192	$0.092$	88	$1.050$	100	1,000	$18.879$

Appendix

Table 4: Per-task-pair parameters for the RoFB compute model of Eq. ( 25 ).

T_{\max}

is each model’s standard evaluation horizon (Kuramoto step for AKOrN, self-attention step for ItrSA++, outer block for TRM);

T_{\text{sweep}}

is the same horizon expressed in the unit used internally for the

t_{\min}

search (TRM: one outer block

=H_{\mathrm{cycles}}\cdot L_{\mathrm{cycles}}

L

-level steps

=18

on Sudoku and

12

on Maze, hence

16\times 18=288

and

16\times 12=192

r_{\text{step}}

is the FLOP fraction of the batched matrix multiplication ( torch.bmm ) of shape

(B,N,N)\times(B,N,D)

relative to one baseline forward step.

r_{\text{tot}}^{(T_{\max})}

r_{\text{tot}}(T_{\max})

evaluated at the cell’s

t_{\min}

\widetilde{\mathcal{C}}_{\text{sweep}}

is the amortized sweep cost of Eq. ( 25 ), computed with

n_{\text{trial}}{=}30

\eta_{\text{ASHA}}{=}0.5

, and the

(N_{\text{val}},N_{\text{test}})

shown. The

\widetilde{\mathcal{C}}_{\text{sweep}}

column is plotted only on the appendix figures (Figs. 9 , 11 ); the main figure (Fig. 3 ) shows only the per-inference forward overhead.

Figure 9: Same operating points as Fig. 12 , with the per-cell hyperparameter-sweep cost (

\widetilde{\mathcal{C}}_{\text{sweep}}

in Table 4 ) added to the RoFB

x

-coordinate. Sweep cost shifts the RoFB scatter rigidly to the right by a constant per cell; the Pareto improvement is preserved across all six cells.

Figure 10: Same operating points as Fig. 12 , with the

x

-axis replaced by measured wall-clock time on a single NVIDIA GH200 (forward only, no sweep cost; wall-clock measured in the seed-42 run, accuracies as in Fig. 12 ). Within-cell baseline-vs-RoFB comparisons are dtype-matched (fp32 for AKOrN/ItrSA++; bf16 for TRM); cross-cell wall comparisons should be read together with this dtype split.

Figure 11: Same as Fig. 10 but with the hyperparameter-sweep wall added to the RoFB

x

-coordinate (RoFB only).

Figure 12: Complete compute-accuracy grid with all

K_{\text{vote}}\in\{1,2,4,8,16,32,64\}

. As in Fig. 3 , each AKOrN and ItrSA++ point is the mean accuracy over 8 inference seeds (42–49) with the same checkpoint, RoFB hyperparameters and test boards; the seed-to-seed SD is at most

0.7

pp at every point, so error bars are omitted. TRM has a deterministic initial state and is a single run. The FLOPs axis is analytic and does not depend on the seed. Rows: Sudoku board accuracy, Maze shortest-path accuracy. Encoding as in Fig. 3 .

Model	$K_{\text{vote}}$	$T_{\text{eval}}{=}64$ ( $N_{\text{block}}{=}8$ )			$T_{\text{eval}}{=}128$ ( $N_{\text{block}}{=}12$ )			$T_{\text{eval}}{=}256$ ( $N_{\text{block}}{=}16$ )
		Base	RoFB	$\Delta$	Base	RoFB	$\Delta$	Base	RoFB	$\Delta$
AKOrN	1	63.3	59.8	$-3.6$	74.4	73.9	$-0.5$	79.1	81.7	$+2.6$
	2	70.6	69.7	$-0.9$	79.2	81.8	$+2.7$	82.4	87.5	$+5.0$
	4	76.3	78.3	$+2.0$	82.9	87.9	$+5.0$	85.1	91.5	$+6.4$
	8	81.2	85.1	$+3.9$	85.9	91.8	$+5.9$	87.5	94.2	$+6.7$
	16	85.3	90.4	$+5.1$	88.7	94.8	$+6.1$	89.6	96.2	$+6.6$

Appendix

Table 5: Full operating-point grid on Sudoku (board-exact accuracy).

Model	$K_{\text{vote}}$	$T_{\text{eval}}{=}64$ ( $N_{\text{block}}{=}8$ )			$T_{\text{eval}}{=}128$ ( $N_{\text{block}}{=}12$ )			$T_{\text{eval}}{=}256$ ( $N_{\text{block}}{=}16$ )
		Base	RoFB	$\Delta$	Base	RoFB	$\Delta$	Base	RoFB	$\Delta$
AKOrN	1	69.8	69.8	$+0.0$	72.2	72.2	$+0.0$	72.6	75.8	$+3.2$
	2	71.6	71.6	$+0.0$	73.4	73.4	$+0.0$	73.7	76.5	$+2.8$
	4	73.1	73.1	$+0.0$	74.0	74.0	$+0.0$	74.1	77.2	$+3.1$
	8	73.8	73.8	$+0.0$	74.0	74.0	$+0.0$	73.9	76.9	$+3.0$
	16	73.8	73.8	$+0.0$	74.3	74.3	$+0.0$	74.3	77.2	$+2.9$

Appendix

Table 6: Full operating-point grid on Maze (shortest-path accuracy).

Model on Task	$T_{\text{eval}}$	Baseline	RoFB	$\Delta$	95% CI of $\Delta$	$\Delta$ min – max	Seeds with $\Delta>0$
AKOrN on Sudoku	64	76.5 $\pm$ 0.2	78.2 $\pm$ 0.1	$+1.71$ $\pm$ 0.23	–	$+1.34$ – $+1.98$	8/8
	128	82.9 $\pm$ 0.1	87.5 $\pm$ 0.1	$+4.61$ $\pm$ 0.17	–	$+4.41$ – $+4.96$	8/8
	256	85.1 $\pm$ 0.1	91.5 $\pm$ 0.1	$+6.38$ $\pm$ 0.19	$[6.05,\,6.70]$	$+6.18$ – $+6.66$	8/8
AKOrN on Maze	64	72.4 $\pm$ 0.5	72.4 $\pm$ 0.5	$+0.00$ $\pm$ 0.00	–	$+0.00$ – $+0.00$	0/8
	128	73.8 $\pm$ 0.5	73.8 $\pm$ 0.5	$+0.00$ $\pm$ 0.00	–	$+0.00$ – $+0.00$	0/8
	256	74.2 $\pm$ 0.4	77.0 $\pm$ 0.3	$+2.83$ $\pm$ 0.25	$[1.49,\,4.21]$	$+2.50$ – $+3.20$	8/8

Appendix

Table 7: Paired gains and inference-seed robustness at

K_{\text{vote}}{=}4

Parameter	Five validation values relative to the selected value	Test values
Feedback strength $\lambda$	$\{0.5,0.75,1,1.25,1.5\}\lambda^{*}$	$\{0.5,1,1.5\}\lambda^{*}$
Start time $t_{\min}$	$t_{\min}^{*}+\{-2,-1,0,1,2\}d$	$t_{\min}^{*}+\{-2,0,2\}d$
Entropy threshold $\alpha$	$\{0.5,0.75,1,1.25,1.5\}\alpha^{*}$	$\{0.5,1,1.5\}\alpha^{*}$
Gate temperature $\tau$	$\{0.25,0.5,1,2,4\}\tau^{*}$	$\{0.25,1,4\}\tau^{*}$

Appendix

Table 8: Hyperparameter ranges for validation and test.

Figure 13: Test-set gains when each hyperparameter is varied separately. Rows identify the model and task; columns identify the varied parameter. The vertical axis shows accuracy gains in percentage points, on a common scale across all panels and using the baselines described above. Bars show pointwise 95% confidence intervals; diamonds and dotted lines mark the reference setting and its gain.

Sudoku: all instances
Geometry measure	AKOrN	ItrSA++	TRM
Change in $\Delta_{\cos}$	$+0.0412$ $[0.0360,\,0.0464]$	$+0.0086$ $[0.0075,\,0.0097]$	$+0.1580$ $[0.1532,\,0.1628]$
Change in silhouette	$+0.0508$ $[0.0361,\,0.0654]$	$+0.0185$ $[0.0124,\,0.0248]$	$+0.1487$ $[0.1411,\,0.1565]$
Maze: all instances
Change in $\Delta_{\cos}$	$+0.1805$ $[0.1692,\,0.1921]$	$+0.1171$ $[0.1124,\,0.1220]$	$+0.0803$ $[0.0797,\,0.0808]$
Change in silhouette	$+0.0410$ $[0.0381,\,0.0439]$	$+0.1000$ $[0.0964,\,0.1038]$	$+0.0325$ $[0.0323,\,0.0327]$

Appendix

Table 9: RoFB increases terminal inter-class separation across all instances. Values are dimensionless mean differences (RoFB minus the no-force reference). Brackets show 95% paired-bootstrap confidence intervals (CIs) over instances.

Sudoku: incorrect under both conditions
Geometry measure	AKOrN	ItrSA++	TRM
Number of instances	271	566	477
Change in $\Delta_{\cos}$	$+0.0082$ $[0.0070,\,0.0094]$	$+0.0026$ $[0.0016,\,0.0035]$	$+0.0191$ $[0.0161,\,0.0221]$
Change in silhouette	$+0.0095$ $[0.0031,\,0.0159]$	$+0.0055$ $[0.0025,\,0.0085]$	$+0.0270$ $[0.0235,\,0.0307]$
Maze: incorrect under both conditions
Number of instances	209	140	72

Appendix

Table 10: RoFB increases separation among instances incorrect under both conditions.

Sudoku
Intervention	AKOrN	ItrSA++	TRM
Removal $-$ magnitude-matched RoFB	$-2.300$ $[-3.400,\,-1.200]$	$-2.700$ $[-3.550,\,-1.900]$	$-3.250$ $[-4.400,\,-2.100]$
Magnitude-restored removal $-$ RoFB	$-18.250$ $[-20.000,\,-16.500]$	$-7.150$ $[-8.300,\,-6.050]$	$-6.950$ $[-8.300,\,-5.550]$
Maze
Removal $-$ magnitude-matched RoFB	$-8.889$ $[-10.889,\,-7.000]$	$-11.000$ $[-13.111,\,-9.000]$	$+0.111$ $[0.000,\,0.333]$
Magnitude-restored removal $-$ RoFB	$-12.222$ $[-14.444,\,-10.111]$	$-22.444$ $[-25.222,\,-19.778]$	$-0.111$ $[-0.333,\,0.000]$

Appendix

Table 11: Accuracy differences after removing the separation-related component.

Alternative	Method-specific parameter searched jointly with $\beta$
Margin repulsion	Distance margin $m\in\{0.5,1,1.5\}$
Probability-weighted contrastive loss	Contrastive temperature $\kappa\in\{0.05,0.1,0.2\}$ ; separate from the fixed gate temperature $\tau$
Within-class variance plus centroid margin	Centroid-distance margin $m\in\{0.5,1,1.5\}$ ; both loss coefficients remain one
K-means centroid attraction	Cluster count $G\in\{8,9,10\}$ on Sudoku or $\{1,2,3\}$ on Maze

Appendix

Table 12: Validation search: three force multipliers crossed with three values of the method-specific parameter.

G

counts state clusters, not inference candidates. Candidate counts remain four for AKOrN and ItrSA++ and one for TRM.

Figure 14: Absolute accuracy differences from standard RoFB (%; paired-bootstrap 95% confidence intervals). Columns: AKOrN, ItrSA++, TRM; rows: Sudoku, Maze. The dashed line is standard RoFB with its native entropy gate and saved strength. The alternatives retain that gate and adapter and use the validation-selected pre-gate direction scaling above. Intervals use 10,000 paired bootstrap resamples and are unadjusted; stars indicate exact McNemar

p<0.05

after Holm correction across the prespecified 24 comparisons (four alternatives across six model–task pairs).

Sudoku
Intervention	AKOrN	ItrSA++	TRM
Original RoFB ( $h(d)=-d$ )	$91.52$ $[91.12,\,91.89]$	$77.00$ $[73.11,\,80.47]$	$74.17$ $[73.55,\,74.77]$
Attractive ( $h(d)=1-d$ )	$72.35$ $[71.72,\,72.96]$	$75.40$ $[71.44,\,78.97]$	$56.88$ $[56.19,\,57.56]$
Mixed ( $h(d)=1-2d$ )	$90.82$ $[90.41,\,91.21]$	$77.20$ $[73.32,\,80.66]$	$73.33$ $[72.71,\,73.94]$
Const ( $h(d)=-1$ )	$91.62$ $[91.23,\,92.00]$	$77.00$ $[73.11,\,80.47]$	$74.72$ $[74.11,\,75.32]$
Maze

Appendix

Table 13: Coupling ablations at the standard horizon.

(a) Candidate-selected accuracy (%)
Intervention	AKOrN	ItrSA++	TRM
Sudoku
Original RoFB	93.2	77.0	71.0
Warmup $\equiv 1$	93.8	78.0	72.0
Entropy gate $\equiv 1$	89.0 †	77.0	69.4
Both $\equiv 1$	82.0	76.2	63.0

Appendix

Table 14: Gate-factor ablations.

Sudoku
Condition / difference	AKOrN	ItrSA++	TRM
No-force reference	$0.0982$	$0.118$	$0.00825$
RoFB	$0.0774$	$0.108$	$0.00591$
RoFB $-$ reference	$-0.0209$ $[-0.0318,\,-0.00998]$	$-0.00908$ $[-0.0135,\,-0.00467]$	$-0.00234$ $[-0.00329,\,-0.00142]$
Maze
No-force reference	$0.00293$	$0.00183$	$1.48\times 10^{-6}$

Appendix

Table 15: Mean normalized readout entropy with and without RoFB.

Sudoku
Intervention	AKOrN	ItrSA++	TRM
Fixed-scale entropy-gradient intervention	$+0.700$ $[0.000,\,1.400]$	$+0.050$ $[-0.650,\,0.750]$	$+0.450$ $[-0.350,\,1.250]$
Magnitude-matched entropy-gradient intervention	$-0.100$ $[-1.250,\,1.100]$	$-1.200$ $[-2.150,\,-0.250]$	$-1.200$ $[-2.400,\,-0.050]$
Online RoFB	$+7.150$ $[5.800,\,8.500]$	$+0.800$ $[-0.050,\,1.650]$	$+5.600$ $[4.250,\,7.000]$
Maze
Fixed-scale entropy-gradient intervention	$-0.111$ $[-0.333,\,0.000]$	$-0.111$ $[-0.333,\,0.000]$	$+0.000$ $[0.000,\,0.000]$

Appendix

Table 16: Accuracy gains over paired no-feedback baselines.

Architecture	Natural frequencies	Gain for each training seed	Mean $\pm$ SD	Positive seeds
Modified	Off	$+0.715/-4.170/+0.070$	$-1.128\pm 2.654$	2/3
Modified	On	$+4.280/+4.245/+3.280$	$+3.935\pm 0.568$	3/3
Original (reproduced)	Off	$-0.067/-0.022/-0.272$	$-0.120\pm 0.133$	0/3
Original (reproduced)	On	$+1.511/+0.233/+0.744$	$+0.830\pm 0.643$	3/3

Appendix

Table 17: Effect of natural frequencies on RoFB gains in Sudoku.

	pass@1		pass@5		pass@100
Aug.	Baseline	RoFB	Baseline	RoFB	Baseline	RoFB
1	35.333	35.500	37.833	37.500	37.833	37.500
3	38.833	39.000	43.500	43.167	43.500	43.167
7	39.333	39.000	46.667	46.667	47.333	47.000
20	41.000	40.667	49.500	49.500	51.333	51.333
100	41.667	41.667	50.167	50.167	55.000	55.000

Appendix

Table 18: ARC-AGI-1 results on 300 evaluation puzzles with validation-selected RoFB. Scores are percentages.

Architecture
	Sudoku Extreme	Maze Hard
oscillator dim $m$	4	4
embed width $D$	512	512
# blocks $L$	1	1
# heads	8	8
$\gamma$ (step size)	1.0	1.0

Appendix

Table 19: AKOrN: full architecture and training hyperparameters.

Architecture
	Sudoku Extreme	Maze Hard
embed_dim	384	384
num_heads	12	12
num_layers	1	1
num_iter ( $=n_{\mathrm{block}}^{\mathrm{train}}$ )	16	32
num_rep_attn ( $R$ )	4	4

Appendix

Table 20: ItrSA++: full architecture and training hyperparameters. We use the affine-free variant (

\mathtt{rmsnorm\_affine}{=}\mathrm{False}

) as the canonical baseline. Inference horizons follow the convention

T=R\cdot n_{\mathrm{block}}

with

R{=}\mathrm{num\_rep\_attn}{=}4

Architecture
	Sudoku-Extreme	Maze-30x30-Hard
$H_{\mathrm{cycles}}$	3	3
$L_{\mathrm{cycles}}$	6	4
$H_{\mathrm{layers}}$	0 (shared with $L$ )	0 (shared with $L$ )
$L_{\mathrm{layers}}$	2	2
hidden_size	512	512

Appendix

Table 21: TRM: full architecture and training hyperparameters.

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Sep 29, 2026

Scheduling Recursive Reasoning in Looped Transformers

Recurrent reasoning models have attracted growing attention for scaling test-time computation, typically by iteratively refining latent states with shared parameters. However, these models apply each learned update with a fixed unit scale, which can be conservative when updates make persistent progress and overly aggressive when they fluctuate, limiting the benefit of additional loops. To understand how the scale should vary along the trajectory, we first analyze the sensitivity of terminal loss to recurrent update scale. We show that its temporal average admits an exact decomposition into persistent-progress and centered-fluctuation contributions. Based on this, we introduce the Trajectory Adaptive Progress-Fluctuation Scheduler (TAPS), which tracks their balance across recurrent updates and adapts the step size online. Theoretically, we establish sufficient conditions under which TAPS reduces expected terminal loss and reaches a target quality in fewer recurrent loops. Empirically, we show that TAPS improves terminal accuracy across structured reasoning tasks without retraining. By further incorporating the progress-fluctuation principle into training, TAPS yields additional accuracy gains with up to 1.56 times wall-clock speedup at matched baseline accuracy. The broad applicability of TAPS is supported by its effectiveness across diverse recurrent architectures and inference strategies. Together, these results establish update scale as complementary control axis of recurrent inference alongside architecture and depth.

Boyuan Wang, Chengyao Yu, Jiaxi Ren +3

Southern University of Science and Technology · The Chinese University of Hong Kong, Shenzhen · Shenzhen Loop Area Institute

Adaptive Inference Recurrent Transformers Inference-Time Scaling Learning Rate Scheduling

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