# Beyond Log Likelihood: Probability-Based Objectives for Supervised Fine-Tuning across the Model Capability Continuum

arXiv: https://arxiv.org/abs/2510.00526 · HTML: https://mlsanity.com/paper/2510.00526/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 63 references that are in the map.
- First posted on arXiv 1 Oct 2025.

## Builds on (most cited first, 30 of 45)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2204.05862/) (2022)
- [Scaling Instruction-Finetuned Language Models](https://mlsanity.com/paper/2210.11416/) (2022)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation](https://mlsanity.com/paper/1606.04797/) (2016)
- [Can a Suit of Armor Conduct Electricity? A New Dataset for Open Book Question Answering](https://mlsanity.com/paper/1809.02789/) (2018)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [MMLU-Pro: A More Robust and Challenging Multi-Task Language Understanding Benchmark](https://mlsanity.com/paper/2406.01574/) (2024)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [Is Your Code Generated by ChatGPT Really Correct? Rigorous Evaluation of Large Language Models for Code Generation](https://mlsanity.com/paper/2305.01210/) (2023)
- [What Disease does this Patient Have? A Large-scale Open Domain Question Answering Dataset from Medical Exams](https://mlsanity.com/paper/2009.13081/) (2020)
- [PubMedQA: A Dataset for Biomedical Research Question Answering](https://mlsanity.com/paper/1909.06146/) (2019)
- [WizardLM: Empowering large pre-trained language models to follow complex instructions](https://mlsanity.com/paper/2304.12244/) (2023)
- [Length-Controlled AlpacaEval: A Simple Way to Debias Automatic Evaluators](https://mlsanity.com/paper/2404.04475/) (2024)
- [Cross-Task Generalization via Natural Language Crowdsourcing Instructions](https://mlsanity.com/paper/2104.08773/) (2021)
- [SFT Memorizes, RL Generalizes: A Comparative Study of Foundation Model Post-training](https://mlsanity.com/paper/2501.17161/) (2025)
- [Instruction Tuning for Large Language Models: A Survey](https://mlsanity.com/paper/2308.10792/) (2023)
- [MedMCQA : A Large-scale Multi-Subject Multi-Choice Dataset for Medical domain Question Answering](https://mlsanity.com/paper/2203.14371/) (2022)
- [Understanding the Effects of RLHF on LLM Generalisation and Diversity](https://mlsanity.com/paper/2310.06452/) (2023)
- [Fine-Tuning Pretrained Language Models: Weight Initializations, Data Orders, and Early Stopping](https://mlsanity.com/paper/2002.06305/) (2020)
- [Magpie: Alignment Data Synthesis from Scratch by Prompting Aligned LLMs with Nothing](https://mlsanity.com/paper/2406.08464/) (2024)
- [Magicoder: Empowering Code Generation with OSS-Instruct](https://mlsanity.com/paper/2312.02120/) (2023)
- [MedXpertQA: Benchmarking Expert-Level Medical Reasoning and Understanding](https://mlsanity.com/paper/2501.18362/) (2025)
- [Better Fine-Tuning by Reducing Representational Collapse](https://mlsanity.com/paper/2008.03156/) (2020)
- [REASONING GYM: Reasoning Environments for Reinforcement Learning with Verifiable Rewards](https://mlsanity.com/paper/2505.24760/) (2025)

## Built on by (most cited first, 18 of 18)

- [ProFit: Leveraging High-Value Signals in SFT via Probability-Guided Token Selection](https://mlsanity.com/paper/2601.09195/) (2026)
- [Rethinking Generalization in Reasoning SFT: A Conditional Analysis on Optimization, Data, and Model Capability](https://mlsanity.com/paper/2604.06628/) (2026)
- [Good SFT Optimizes for SFT, Better SFT Prepares for Reinforcement Learning](https://mlsanity.com/paper/2602.01058/) (2026)
- [Probability-Entropy Calibration: An Elastic Indicator for Adaptive Fine-tuning](https://mlsanity.com/paper/2602.01745/) (2026)
- [Large Language Model Post-Training: A Unified View of Off-Policy and On-Policy Learning](https://mlsanity.com/paper/2604.07941/) (2026)
- [InfoSFT: Learn More and Forget Less with Information-Aware Token Weighting](https://mlsanity.com/paper/2605.14967/) (2026)
- [RIFT: Repurposing Negative Samples via Reward-Informed Fine-Tuning](https://mlsanity.com/paper/2601.09253/) (2026)
- [Gradients Must Earn Their Influence: Unifying SFT with Generalized Entropic Objectives](https://mlsanity.com/paper/2602.11424/) (2026)
- [LATMiX: Learnable Affine Transformations for Microscaling Quantization of LLMs](https://mlsanity.com/paper/2602.17681/) (2026)
- [Graph homophily booster: Reimagining the role of discrete features in heterophilic graph learning](https://mlsanity.com/paper/2602.07256/) (2026)
- [ReMix: Reinforcement routing for mixtures of LoRAs in LLM finetuning](https://mlsanity.com/paper/2603.10160/) (2026)
- [From Tokens to Regions: CUDA-Sensitive Instruction Tuning for GPU Kernel Generation](https://mlsanity.com/paper/2606.16231/) (2026)
- [Data Difficulty and the Generalization--Extrapolation Tradeoff in LLM Fine-Tuning](https://mlsanity.com/paper/2605.12906/) (2026)
- [A Unifying Lens on Supervised Fine-Tuning Through Target Distribution Design](https://mlsanity.com/paper/2606.11189/) (2026)
- [Strategically Diverse Sampling for Self-Training](https://mlsanity.com/paper/2609.31571/) (2026)
- [Alpha Diffusion Language Models: Factorization Alone Is Not the Problem](https://mlsanity.com/paper/2609.38066/) (2026)
- [Facts Don't Speak Louder than Words: The Behavioral Essence of Long-CoT Distillation](https://mlsanity.com/paper/2610.17607/) (2026)
- [Geometric Calibration of Supervised Fine-Tuning](https://mlsanity.com/paper/2610.18634/) (2026)

## Abstract (arXiv)

Supervised fine-tuning (SFT) is the standard approach for post-training large language models (LLMs), yet it often shows limited generalization. We trace this limitation to its default training objective: negative log likelihood (NLL). While NLL is classically optimal when training from scratch, post-training operates in a different paradigm and could violate its optimality assumptions, where models already encode task-relevant priors and supervision can be long and noisy. In this work, we systematically study various probability-based objectives and characterize when and why different objectives succeed or fail under varying conditions. Through comprehensive experiments and extensive ablation studies across 8 model backbones, 27 benchmarks, and 7 domains, we uncover a critical dimension that governs objective behavior: the model-capability continuum. Near the model-strong end, prior-leaning objectives that downweight low-probability tokens (e.g., -p, -p10, thresholded variants) consistently outperform NLL; toward the model-weak end, NLL dominates; in between, no single objective prevails. Our theoretical analysis further elucidates how objectives trade places across the continuum, providing a principled foundation for adapting objectives to model capability. The code is available at https://github.com/GaotangLi/Beyond-Log-Likelihood.