CUSUM-Shaped Inference-Time Monitoring and Targeted Re-Decoding for Quantized Small Language Model Reasoning

Low-bit quantization makes small reasoning models inexpensive to deploy but can degrade their chains of thought. This motivates decoder-side monitors that intervene when generation becomes unreliable. We show that a natural candidate, the centered token log-probability increment $\log p(w_t)+H_t$, is the wrong observable for this purpose. Under the model's own sampling law it is a mean-zero martingale by construction, so it measures sampling self-consistency rather than trajectory health and is nearly silent during confident repetition, where both $\log p(w_t)$ and entropy are close to zero. We introduce a training-free decoding controller that combines (i) a degeneration-aware alarm score fusing token uncertainty with explicit verbatim repetition and (ii) a calibrated e-process-inspired sequential detector. The raw product process is Ville-valid under a conditional-mean null, while the deployed CUSUM-floored statistic is treated as an empirical change detector because the score is history-dependent and autocorrelated. On GSM8K with DeepSeek-R1-Distill-Qwen-1.5B in FP16 and INT4, calibration turns a monitor that fires on 93--95% of generations into a selective detector of failing traces ($φ\approx 0.3$, precision $\approx 0.6$ against a 0.38 base rate). In this pilot, the controller reduces measured verbatim-degeneration signals and yields a positive but statistically inconclusive INT4 accuracy change from 63% to 69% (paired McNemar $p=0.18$, $n=100$), at a 28% token-budget cost. We also find that non-termination, rather than looping, is the dominant failure mode on GSM8K. The main contribution is methodological: an explanation of why centered token log-probability is inadequate for decoder monitoring and a calibrated, cautiously evaluated replacement.

Post-training quantization (PTQ) is widely used to deploy large language models efficiently, but its effect on reasoning models is not well understood. Across math, coding, and science QA, we find that aggressive PTQ reduces accuracy while increasing chain-of-thought (CoT) length. Surprisingly, we show that in up to 52% of the quantized models' failures, models reach the right answer in intermediate reasoning steps but do not output it as a final answer. To understand why quantization leads to this increase in overthinking errors, we measure the token-level KL divergence between quantized and full-precision output distributions. Positions with high KL divergence correlate strongly with high next-token entropy, and at these positions quantized models disproportionately sample overthinking markers such as "wait", "but", and "alternatively". We show that simply introducing a training-free logit penalty on a curated set of overthinking markers can reduce CoT length by 12--23% while preserving or improving accuracy across 5 models (1.5B-32B parameters), 3 quantization methods, and 5 benchmarks, yielding a favorable Pareto frontier of accuracy against reasoning cost compared to penalizing other token sets. Overthinking errors produced by quantized models are particularly reduced by up to 58%.

Large Reasoning Models (LRMs) rely on long reasoning traces, making inference expensive. While low-bit quantization reduces per-token decoding cost, we show that aggressive 2-bit inference can fail to deliver end-to-end speedup because instability in the generation process inflates total token count. Instead of merely lowering answer accuracy, 2-bit quantization often produces much longer traces with repetitive loops, budget exhaustion, delayed commitment, and unclosed reasoning segments. We analyze full reasoning traces of Qwen3 reasoning models across mathematical and commonsense benchmarks and show that accuracy degradation is tightly linked to these process-level failures. To address them, we introduce two lightweight controls: FP16 planning, which gives the 2-bit model a short high-precision outline, and loop rescue, which detects repetitive traces and either commits to an earlier answer or falls back to FP16. On MATH-500, loop rescue improves Qwen3-8B accuracy from 17.2% to 74.2%, while planning plus loop rescue improves Qwen3-32B from 65.0% to 87.2%. Overall, our results show that extreme low-bit reasoning becomes practical when its failures are treated as controllable generation pathologies: with lightweight detection and selective FP16 support, 2-bit inference can recover accuracy while preserving real end-to-end speed. Our code is available at: https://github.com/brain-lab-research/quantized-reasoning.