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C-Voting: Confidence-Based Test-Time Voting without Explicit Energy Functions
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Proposes C-voting, a test-time voting strategy for recurrent models with random latent initializations that selects the candidate with highest average top-1 probability, and a lightweight attention-based recurrent model ItrSA++ that with C-voting beats HRM on Sudoku-extreme and Maze-hard.
Problem
Test-time voting over random latent initializations in recurrent reasoning models usually relies on an explicit energy function (E-voting), which many models such as HRM and recurrent transformers lack; the paper seeks a model-agnostic alternative and a compact recurrent architecture that matches SO
Approach
Select among K randomly initialized latent trajectories by the model's confidence, defined as the mean over blank positions of the top-1 softmax probability of the final readout, instead of by energy. It needs no energy function and works with any recurrent model whose readout produces class logits. A new small attention-based recurrent model, ItrSA++, is trained with random initialization and pai
Key results
- C-voting with 4096 samples on AKOrN reaches 94.4±0.1% board accuracy on Sudoku-hard, 4.9 points above reported E-voting (89.5±2.5%).
- ItrSA++ with C-voting reaches 95.2% on Sudoku-extreme versus HRM's 55.0%.
- ItrSA++ with C-voting reaches 78.6% on Maze-hard versus HRM's 74.5%, with about 3M parameters versus HRM's 27M.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace E-voting with C-voting (4096 samples) | AKOrN · Sudoku-hard | board accuracy: +4.9% (94.4% vs 89.5%) | Helped |
| Add cross-attention mixing of random init with input (vs linear layer) (ablation) | ItrSA++ · Sudoku-extreme | board accuracy: 63.2% vs 59.2% | Helped |
| Replace SwiGLU with FFN (ablation) | ItrSA++ · Sudoku-extreme | board accuracy: 62.0% vs 63.2% | Hurt |
| Remove exponential moving average of parameters (ablation) | ItrSA++ · Sudoku-extreme | board accuracy: 63.2% to 0.0% | Hurt |
| Remove gradient truncation during training (ablation) | ItrSA++ · Sudoku-extreme | board accuracy: 63.2% to 0.0% | Hurt |
| Use top-1 probability vs negative entropy / log-probability as voting metric (ablation) | ItrSA++ C-voting · Sudoku-extreme and Maze-hard | board accuracy: almost no difference (Sudoku-extreme); ~0.1% (Maze-hard) | No clear effect |
| Apply C-voting with increasing samples to HRM (randomized init, ACT exploration 1) | HRM · Sudoku-extreme | board accuracy: limited gains | Mixed |
| Apply C-voting to ItrSA++ (no explicit energy) vs HRM | ItrSA++ + C-voting · Maze-hard | board accuracy: 78.6% vs 74.5% for HRM | Helped |
| Apply C-voting to ItrSA++ vs original HRM | ItrSA++ + C-voting · Sudoku-extreme | board accuracy: 95.2% vs 55.0% | Helped |
| Apply C-voting to a standard transformer with linear input-latent mixing | Transformer · Sudoku-hard | board accuracy: limited improvement | No clear effect |
Surprising findings
- AKOrN fails to learn Maze-hard, obtaining 0% test board accuracy.
- Applying C-voting to HRM yields only limited gains because HRM relies on a fixed initial state, which randomization disrupts.
Limitations
- C-voting gains on Maze-hard are weak, since the model is often confidently wrong regardless of initialization.
- HRM gains are limited because its fixed-initialization design conflicts with randomized sampling.
- Average top-1 confidence is only a proxy for board accuracy, and the reason it beats energy-based selection is not clear.
Experimental setup
Datasets: Sudoku Sudoku-hard Sudoku-extreme Maze-hard
Benchmarks: Sudoku-hard Sudoku-extreme Maze-hard
Metrics: board accuracy
Baselines: HRM AKOrN with E-voting AKOrN Transformer
Builds on: AKOrN Hierarchical Reasoning Model Perceiver Energy-based voting (E-voting)
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