Neural network models with latent recurrent processing, where identical layers are recursively applied to the latent state, have gained attention as promising models for performing reasoning tasks. A strength of such models is that they enable test-time scaling, where the models can enhance their performance in the test phase without additional training. Models such as the Hierarchical Reasoning Model (HRM) and Artificial Kuramoto Oscillatory Neurons (AKOrN) can facilitate deeper reasoning by increasing the number of recurrent steps, thereby enabling the completion of challenging tasks, including Sudoku, Maze solving, and AGI benchmarks. In this work, we introduce confidence-based voting (C-voting), a test-time scaling strategy designed for recurrent models with multiple latent candidate trajectories. Initializing the latent state with multiple candidates using random variables, C-voting selects the one maximizing the average of top-1 probabilities of the predictions, reflecting the model's confidence. Additionally, it yields 4.9% higher accuracy on Sudoku-hard than the energy-based voting strategy, which is specific to models with explicit energy functions. An essential advantage of C-voting is its applicability: it can be applied to recurrent models without requiring an explicit energy function. Finally, we introduce a simple attention-based recurrent model with randomized initial values named ItrSA++, and demonstrate that when combined with C-voting, it outperforms HRM on Sudoku-extreme (95.2% vs. 55.0%) and Maze (78.6% vs. 74.5%) tasks.
Figures & tables
Figure 1: (a): The overview of confidence-based voting (C-voting). The candidates of the latent state are initialized by random variables, and the recurrent model updates the latent states from t=0 to t=T using input x . Calculating probabilities from the final states, C-voting selects the most confident prediction. (b): The performance comparisons of ItrSA++ with C-voting to other state-of-the-art models. The proposed method outperforms in Sudoku-hard, Sudoku-extreme, and Maze-hard tasks.
Figure 2: A performance comparison between C-voting and E-voting in AKOrN.
Figure 3: Scaling analysis of ItrSA++ with C-voting.
Figure 4: Board accuracy for Sudoku-extreme task of HRM with C-voting.
Figure 5: Visualization of calibration of ItrSA++.
Figure 6: Distributions of confidence of ItrSA++ at t=32 .
Figure 7: Time step dependency of confidence.
Appendix figures & tables8 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 8: Test-time scaling in ItrSA++
Figure 9: Board accuracy of a transformer with C-voting.
Parameter
Value
Optimizer
AdamW
β for Adam
(0.9,0.95)
Weight decay
0.01
Gradient clipping threshold
1.0
Learning rate
5×10−4
Batch size
64
Appendix
Table 1: Hyperparameters for ItrSA++.
Parameter
Value
Optimizer
Adam-atan2 ( Everett et al., 2024 )
β for Adam
(0.9,0.95)
Weight decay
1.0
Gradient clipping threshold
1.0
Learning rate
1×10−4
Warm-up steps
2000
Appendix
Table 2: Hyperparameters for modified HRM.
Parameter
Value
Optimizer
Adam
β for Adam
(0.9,0.999)
Gradient clipping threshold
1.0
Learning rate
5×10−4
Batch size
100
EMA decay rate
0.995
Appendix
Table 3: Hyperparameters for AKOrN.
Variant
Board acc. in test
w/o GT
0.0%
w/o EMA
0.0%
Cross attn. to linear
59.2%
SWiGLU to FFN
62.0%
ItrSA++(Proposed)
63.2%
Appendix
Table 4: Ablation study for ItrSA++
Figure 10: Comparison of voting effects across different metrics.
Figure 11: Random seed dependency of board accuracy.