# A Bellman Optimality Equation for Plasticity

arXiv: https://arxiv.org/abs/2609.10776 · HTML: https://mlsanity.com/paper/2609.10776/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Deep RL algorithms and representations, one of 15 directions in Deep reinforcement learning methods.
- Builds on 6 of its 9 references that are in the map.
- First posted on arXiv 9 Sep 2026.

## Placed in

- Deep reinforcement learning methods / [Deep RL algorithms and representations](https://mlsanity.com/deeprl/deep-rl-algorithms-and-representations/)

## Builds on (most cited first, 6 of 6)

- [Understanding plasticity in neural networks](https://mlsanity.com/paper/2303.01486/) (2023)
- [Variational Information Maximisation for Intrinsically Motivated Reinforcement Learning](https://mlsanity.com/paper/1509.08731/) (2015)
- [Continual Backprop: Stochastic Gradient Descent with Persistent Randomness](https://mlsanity.com/paper/2108.06325/) (2021)
- [A Unified Bellman Optimality Principle Combining Reward Maximization and Empowerment](https://mlsanity.com/paper/1907.12392/) (2019)
- [Plasticity as the Mirror of Empowerment](https://mlsanity.com/paper/2505.10361/) (2025)
- [A Unified Bellman Equation for Causal Information and Value in Markov Decision Processes](https://mlsanity.com/paper/1703.01585/) (2017)

## Abstract (arXiv)

In continual reinforcement learning, carefully managing the stability-plasticity tradeoff remains a core challenge. Recent work by Abel et al. (2025) formalized this dilemma by defining plasticity as the generalized directed information from an agent's observations to its actions, and empowerment as the generalized directed information from its actions to its observations. This formulation successfully reframes the traditional stability-plasticity tradeoff as an empowerment-plasticity tradeoff. However, while extensive literature exists on optimizing for empowerment, there is currently no research addressing the optimization of plasticity under this new definition. This paper presents preliminary work toward optimizing plasticity within Markov decision processes. We show that there exists a Bellman optimality equation for optimizing plasticity similar to previous work for empowerment.