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Thinking by Subtraction: Confidence-Driven Contrastive Decoding for LLM Reasoning

Lexiang Tang, Weihao Gao, Bingchen Zhao, Lu Ma, Qiao jin, Bang Yang, Yuexian Zou

TL;DR

This work proposes Thinking by Subtraction, a confidence-driven contrastive decoding approach that improves reasoning reliability through targeted token-level intervention that significantly improves accuracy across mathematical reasoning benchmarks while substantially reducing output length.

Abstract

Recent work on test-time scaling for large language model (LLM) reasoning typically assumes that allocating more inference-time computation uniformly improves correctness. However, prior studies show that reasoning uncertainty is highly localized: a small subset of low-confidence tokens disproportionately contributes to reasoning errors and unnecessary output expansion. Motivated by this observation, we propose Thinking by Subtraction, a confidence-driven contrastive decoding approach that improves reasoning reliability through targeted token-level intervention. Our method, Confidence-Driven Contrastive Decoding, detects low-confidence tokens during decoding and intervenes selectively at these positions. It constructs a contrastive reference by replacing high-confidence tokens with minimal placeholders, and refines predictions by subtracting this reference distribution at low-confidence locations. Experiments show that CCD significantly improves accuracy across mathematical reasoning benchmarks while substantially reducing output length, with minimal KV-cache overhead. As a training-free method, CCD enhances reasoning reliability through targeted low-confidence intervention without computational redundancy. Our code will be made available at: https://github.com/bolo-web/CCD.

Thinking by Subtraction: Confidence-Driven Contrastive Decoding for LLM Reasoning

TL;DR

This work proposes Thinking by Subtraction, a confidence-driven contrastive decoding approach that improves reasoning reliability through targeted token-level intervention that significantly improves accuracy across mathematical reasoning benchmarks while substantially reducing output length.

Abstract

Recent work on test-time scaling for large language model (LLM) reasoning typically assumes that allocating more inference-time computation uniformly improves correctness. However, prior studies show that reasoning uncertainty is highly localized: a small subset of low-confidence tokens disproportionately contributes to reasoning errors and unnecessary output expansion. Motivated by this observation, we propose Thinking by Subtraction, a confidence-driven contrastive decoding approach that improves reasoning reliability through targeted token-level intervention. Our method, Confidence-Driven Contrastive Decoding, detects low-confidence tokens during decoding and intervenes selectively at these positions. It constructs a contrastive reference by replacing high-confidence tokens with minimal placeholders, and refines predictions by subtracting this reference distribution at low-confidence locations. Experiments show that CCD significantly improves accuracy across mathematical reasoning benchmarks while substantially reducing output length, with minimal KV-cache overhead. As a training-free method, CCD enhances reasoning reliability through targeted low-confidence intervention without computational redundancy. Our code will be made available at: https://github.com/bolo-web/CCD.
Paper Structure (39 sections, 15 equations, 10 figures, 4 tables, 1 algorithm)

This paper contains 39 sections, 15 equations, 10 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Overview
  4. Preliminaries: Confidence-Based Uncertainty and Contrastive Decoding
  5. Confidence-Driven Contrastive Decoding
  6. Contrastive Distribution Construction.
  7. Confidence-Driven Logit Subtraction.
  8. Theoretical Analysis
  9. From Confidence Correction to Contrastive Distribution Construction.
  10. High-Confidence Tokens as Semantic Anchors.
  11. Masking Anchors Increases Uncertainty.
  12. Dual KV Cache Maintenance
  13. Algorithms
  14. Empirical and Statistical Evaluation
  15. Experimental Setup
  16. ...and 24 more sections

Figures (10)

  • Figure 1: Trajectory-level relationship between token confidence and final answer correctness. Across 2,880 reasoning trajectories generated by Qwen3-8B on AIME24.
  • Figure 2: Overview of Confidence-Driven Contrastive Decoding (CCD). The decoding process consists of four key components: (1) online estimation of token-level confidence; (2) confidence-driven token selection that partitions chain-of-thought tokens into low-confidence (LC) and high-confidence (HC) sets; (3) contrastive decoding triggered at LC-CoT tokens to refine uncertain predictions; and (4) dual key--value (KV) cache maintenance to support selective intervention without disrupting standard autoregressive decoding.
  • Figure 3: Ablation on different replacement intervals
  • Figure 4: Ablation on using different special token.
  • Figure 6: Token-level confidence improvement at low-confidence decoding positions.
  • ...and 5 more figures