# Stagewise Reinforcement Learning and the Geometry of the Regret Landscape

arXiv: https://arxiv.org/abs/2601.07524 · HTML: https://mlsanity.com/paper/2601.07524/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 43 of its 77 references that are in the map.
- First posted on arXiv 12 Jan 2026.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 30 of 43)

- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [IMPALA: Scalable Distributed Deep-RL with Importance Weighted Actor-Learner Architectures](https://mlsanity.com/paper/1802.01561/) (2018)
- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [Reinforcement Learning and Control as Probabilistic Inference: Tutorial and Review](https://mlsanity.com/paper/1805.00909/) (2018)
- [The Pitfalls of Simplicity Bias in Neural Networks](https://mlsanity.com/paper/2006.07710/) (2020)
- [Contextual Decision Processes with Low Bellman Rank are PAC-Learnable](https://mlsanity.com/paper/1610.09512/) (2016)
- [Emergent Complexity and Zero-shot Transfer via Unsupervised Environment Design](https://mlsanity.com/paper/2012.02096/) (2020)
- [The Statistical Complexity of Interactive Decision Making](https://mlsanity.com/paper/2112.13487/) (2021)
- [Deep learning generalizes because the parameter-function map is biased towards simple functions](https://mlsanity.com/paper/1805.08522/) (2018)
- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [Goal Misgeneralization: Why Correct Specifications Aren't Enough For Correct Goals](https://mlsanity.com/paper/2210.01790/) (2022)
- [Sudden Drops in the Loss: Syntax Acquisition, Phase Transitions, and Simplicity Bias in MLMs](https://mlsanity.com/paper/2309.07311/) (2023)
- [Goal Misgeneralization in Deep Reinforcement Learning](https://mlsanity.com/paper/2105.14111/) (2021)
- [Acquisition of Chess Knowledge in AlphaZero](https://mlsanity.com/paper/2111.09259/) (2021)
- [Preconditioned Stochastic Gradient Langevin Dynamics for Deep Neural Networks](https://mlsanity.com/paper/1512.07666/) (2015)
- [Loose lips sink ships: Mitigating Length Bias in Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2310.05199/) (2023)
- [The Transient Nature of Emergent In-Context Learning in Transformers](https://mlsanity.com/paper/2311.08360/) (2023)
- [SimBa: Simplicity Bias for Scaling Up Parameters in Deep Reinforcement Learning](https://mlsanity.com/paper/2410.09754/) (2024)
- [Consistency and fluctuations for stochastic gradient Langevin dynamics](https://mlsanity.com/paper/1409.0578/) (2014)
- [The promises and pitfalls of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1811.10072/) (2018)
- [The Local Learning Coefficient: A Singularity-Aware Complexity Measure](https://mlsanity.com/paper/2308.12108/) (2023)
- [Deep Learning is Singular, and That's Good](https://mlsanity.com/paper/2010.11560/) (2020)
- [From Lists to Emojis: How Format Bias Affects Model Alignment](https://mlsanity.com/paper/2409.11704/) (2024)
- [Loss Landscape Degeneracy and Stagewise Development in Transformers](https://mlsanity.com/paper/2402.02364/) (2024)
- [Differentiation and Specialization of Attention Heads via the Refined Local Learning Coefficient](https://mlsanity.com/paper/2410.02984/) (2024)
- [Why neural networks find simple solutions: the many regularizers of geometric complexity](https://mlsanity.com/paper/2209.13083/) (2022)
- [Dynamical versus Bayesian Phase Transitions in a Toy Model of Superposition](https://mlsanity.com/paper/2310.06301/) (2023)
- [Simplicity Bias in 1-Hidden Layer Neural Networks](https://mlsanity.com/paper/2302.00457/) (2023)
- [Understanding and Controlling a Maze-Solving Policy Network](https://mlsanity.com/paper/2310.08043/) (2023)

## Built on by (most cited first, 4 of 4)

- [Patterning: The Dual of Interpretability](https://mlsanity.com/paper/2601.13548/) (2026)
- [Interpreting Reinforcement Learning Agents with Susceptibilities](https://mlsanity.com/paper/2605.08007/) (2026)
- [On Distinguishing Capability Elicitation from Capability Creation in Post-Training: A Free-Energy Perspective](https://mlsanity.com/paper/2605.08368/) (2026)
- [Structure and Scale in Simplicial Sequence Modelling](https://mlsanity.com/paper/2606.01302/) (2026)

## Abstract (arXiv)

Singular learning theory characterizes Bayesian learning as an evolving tradeoff between accuracy and complexity, with transitions between qualitatively different solutions as sample size increases. We extend this theory to reinforcement learning, proving that the concentration of a generalized posterior over policies is governed by the local learning coefficient (LLC), an invariant of the geometry of the regret function. This theory predicts that deep reinforcement learning with SGD should proceed from simple policies with high regret to complex policies with low regret. We verify this prediction empirically in a gridworld environment exhibiting stagewise policy development: phase transitions over training manifest as "opposing staircases" where regret decreases sharply while the LLC increases.