# Effective Does Not Mean Useful: Conditional Functional Substitutability for Redundancy and Scaling in Transformers

arXiv: https://arxiv.org/abs/2609.39259 · HTML: https://mlsanity.com/paper/2609.39259/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 29 of its 34 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 29 of 29)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Emerging Properties in Self-Supervised Vision Transformers](https://mlsanity.com/paper/2104.14294/) (2021)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Training data-efficient image transformers & distillation through attention](https://mlsanity.com/paper/2012.12877/) (2020)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Analyzing Multi-Head Self-Attention: Specialized Heads Do the Heavy Lifting, the Rest Can Be Pruned](https://mlsanity.com/paper/1905.09418/) (2019)
- [Are Sixteen Heads Really Better than One?](https://mlsanity.com/paper/1905.10650/) (2019)
- [LLM-Pruner: On the Structural Pruning of Large Language Models](https://mlsanity.com/paper/2305.11627/) (2023)
- [How to train your ViT? Data, Augmentation, and Regularization in Vision Transformers](https://mlsanity.com/paper/2106.10270/) (2021)
- [ShortGPT: Layers in Large Language Models are More Redundant Than You Expect](https://mlsanity.com/paper/2403.03853/) (2024)
- [Explaining Neural Scaling Laws](https://mlsanity.com/paper/2102.06701/) (2021)
- [Towards Best Practices of Activation Patching in Language Models: Metrics and Methods](https://mlsanity.com/paper/2309.16042/) (2023)
- [Mixture-of-Depths: Dynamically allocating compute in transformer-based language models](https://mlsanity.com/paper/2404.02258/) (2024)
- [AdaViT: Adaptive Vision Transformers for Efficient Image Recognition](https://mlsanity.com/paper/2111.15668/) (2021)
- [Revisiting Model Stitching to Compare Neural Representations](https://mlsanity.com/paper/2106.07682/) (2021)
- [The Hydra Effect: Emergent Self-repair in Language Model Computations](https://mlsanity.com/paper/2307.15771/) (2023)
- [The Quantization Model of Neural Scaling](https://mlsanity.com/paper/2303.13506/) (2023)
- [Well-Read Students Learn Better: On the Importance of Pre-training Compact Models](https://mlsanity.com/paper/1908.08962/) (2019)
- [A Dynamical Model of Neural Scaling Laws](https://mlsanity.com/paper/2402.01092/) (2024)
- [Explorations of Self-Repair in Language Models](https://mlsanity.com/paper/2402.15390/) (2024)
- [Leveraging redundancy in attention with Reuse Transformers](https://mlsanity.com/paper/2110.06821/) (2021)
- [Superposition Yields Robust Neural Scaling](https://mlsanity.com/paper/2505.10465/) (2025)
- [Understanding Scaling Laws with Statistical and Approximation Theory for Transformer Neural Networks on Intrinsically Low-dimensional Data](https://mlsanity.com/paper/2411.06646/) (2024)
- [A Mixture of h-1 Heads is Better than h Heads](https://mlsanity.com/paper/2005.06537/) (2020)
- [Conditional Co-Ablation: Recovering Self-Repair Backups in Transformer Circuits](https://mlsanity.com/paper/2607.01940/) (2026)
- [SmartTrim: Adaptive Tokens and Attention Pruning for Efficient Vision-Language Models](https://mlsanity.com/paper/2305.15033/) (2023)

## Abstract (arXiv)

Modern neural networks scale predictably, yet the mechanisms behind these regularities remain unclear. Neural redundancy is typically characterized by component importance or representational similarity, both indirect proxies. We view redundancy as an input-conditioned, dynamic relation: intermediate computational states are functionally redundant when they induce similar downstream responses. We introduce Conditional Functional Substitutability (CFS) to directly characterize such functional substitution. CFS exposes functional relations and reduction potential missed by conventional importance- and similarity-based measures. Across modalities and Transformer families, CFS reveals systematic functional reorganization with scale. Controlled scaling further shows that performance gains need not track growth in substitutability, while fixed-capacity models with more independent functional structure perform better, providing a functional account of diminishing returns. Predicted CFS further enables dynamic computation with a better performance--computation trade-off than importance-based component selection, suggesting new directions for redundancy-aware computation and more efficient model scaling.