# Affinity Is Not Enough: Recovering the Free Energy Principle in Mixture-of-Experts

arXiv: https://arxiv.org/abs/2605.00604 · HTML: https://mlsanity.com/paper/2605.00604/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Expert routing strategies, one of 12 directions in Mixture-of-experts.
- Builds on 26 of its 35 references that are in the map.
- First posted on arXiv 1 May 2026.

## Placed in

- Mixture-of-experts / [Expert routing strategies](https://mlsanity.com/moe/expert-routing-strategies/)

## Builds on (most cited first, 26 of 26)

- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity](https://mlsanity.com/paper/2101.03961/) (2021)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context](https://mlsanity.com/paper/1901.02860/) (2019)
- [Mixtral of Experts](https://mlsanity.com/paper/2401.04088/) (2024)
- [RWKV: Reinventing RNNs for the Transformer Era](https://mlsanity.com/paper/2305.13048/) (2023)
- [Surrogate Gradient Learning in Spiking Neural Networks](https://mlsanity.com/paper/1901.09948/) (2019)
- [Mixture-of-Experts with Expert Choice Routing](https://mlsanity.com/paper/2202.09368/) (2022)
- [Retentive Network: A Successor to Transformer for Large Language Models](https://mlsanity.com/paper/2307.08621/) (2023)
- [ST-MoE: Designing Stable and Transferable Sparse Expert Models](https://mlsanity.com/paper/2202.08906/) (2022)
- [Better & Faster Large Language Models via Multi-token Prediction](https://mlsanity.com/paper/2404.19737/) (2024)
- [Spikformer: When Spiking Neural Network Meets Transformer](https://mlsanity.com/paper/2209.15425/) (2022)
- [Training Spiking Neural Networks Using Lessons From Deep Learning](https://mlsanity.com/paper/2109.12894/) (2021)
- [Predictive Coding Approximates Backprop along Arbitrary Computation Graphs](https://mlsanity.com/paper/2006.04182/) (2020)
- [SpikeGPT: Generative Pre-trained Language Model with Spiking Neural Networks](https://mlsanity.com/paper/2302.13939/) (2023)
- [Deep active inference agents using Monte-Carlo methods](https://mlsanity.com/paper/2006.04176/) (2020)
- [Predictive Coding: Towards a Future of Deep Learning beyond Backpropagation?](https://mlsanity.com/paper/2202.09467/) (2022)
- [Layerwise Recurrent Router for Mixture-of-Experts](https://mlsanity.com/paper/2408.06793/) (2024)
- [Path integrals, particular kinds, and strange things](https://mlsanity.com/paper/2210.12761/) (2022)
- [Graph Mixture of Experts and Memory-augmented Routers for Multivariate Time Series Anomaly Detection](https://mlsanity.com/paper/2412.19108/) (2024)
- [ExpertFlow: Efficient Mixture-of-Experts Inference via Predictive Expert Caching and Token Scheduling](https://mlsanity.com/paper/2410.17954/) (2024)
- [Predictive Coding beyond Gaussian Distributions](https://mlsanity.com/paper/2211.03481/) (2022)
- [MoxE: Mixture of xLSTM Experts with Entropy-Aware Routing for Efficient Language Modeling](https://mlsanity.com/paper/2505.01459/) (2025)
- [Variational Routing: A Scalable Bayesian Framework for Calibrated Mixture-of-Experts Transformers](https://mlsanity.com/paper/2603.09453/) (2026)

## Abstract (arXiv)

Sparse MoE routing fails at domain transitions, where the current token belongs to one distribution and the next to another. In a controlled experiment (4 experts, 5 seeds), standard affinity routing assigns only 0.006 +/- 0.001 probability to the correct expert at the transition. Three lightweight gate modifications raise this to 0.748 +/- 0.002 (124x), cutting experts needed for 99% coverage from infeasible to a small constant: temporal memory (beta), a per-expert LIF membrane potential accumulating routing context across tokens; precision-weighted gating (Pi), a per-expert inverse variance of recent prediction error, yielding 31x contrast between reliable and unreliable experts; and anticipatory routing, a next-state predictor conditioned on the beta-accumulated hidden state. The mechanisms draw from Friston's Free Energy Principle and use LIF dynamics from spiking neural networks. An ablation across all 2^3 subsets reveals a super-additive beta x Ant interaction: anticipation alone gives nothing (+0.000 +/- 0.001); beta alone gives modest gain (+0.295 +/- 0.013); combined they close 75% of the oracle gap (+0.741 +/- 0.002, exceeding the sum by +0.446 +/- 0.014). This is structural: a stateless predictor cannot detect approaching transitions because pre-transition tokens are distributionally identical to within-domain tokens. In a character-level MoE LM (5 seeds), beta-routing reduces transition-step BPC from 6.56 +/- 0.01 (Standard) to 4.01 +/- 0.15 (beta-MoE); the beta + Ant gate places 0.86 +/- 0.02 probability on the correct domain expert before that domain appears in input, vs 0.42 +/- 0.12 for Standard MoE. Reference implementations (~200 lines each): https://github.com/russellwmy/affinity-is-not-enough