# A Unified Bellman Optimality Principle Combining Reward Maximization and Empowerment

arXiv: https://arxiv.org/abs/1907.12392 · HTML: https://mlsanity.com/paper/1907.12392/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Jul 2019.

## Built on by (most cited first, 15 of 15)

- [Which Mutual-Information Representation Learning Objectives are Sufficient for Control?](https://mlsanity.com/paper/2106.07278/) (2021)
- [INFOrmation Prioritization through EmPOWERment in Visual Model-Based RL](https://mlsanity.com/paper/2204.08585/) (2022)
- [Policy Information Capacity: Information-Theoretic Measure for Task Complexity in Deep Reinforcement Learning](https://mlsanity.com/paper/2103.12726/) (2021)
- [Backprop-Free Reinforcement Learning with Active Neural Generative Coding](https://mlsanity.com/paper/2107.07046/) (2021)
- [Complex behavior from intrinsic motivation to occupy action-state path space](https://mlsanity.com/paper/2205.10316/) (2022)
- [Active Predicting Coding: Brain-Inspired Reinforcement Learning for Sparse Reward Robotic Control Problems](https://mlsanity.com/paper/2209.09174/) (2022)
- [Towards Empowerment Gain through Causal Structure Learning in Model-Based RL](https://mlsanity.com/paper/2502.10077/) (2025)
- [Embodied Co-Design for Rapidly Evolving Agents: Taxonomy, Frontiers, and Challenges](https://mlsanity.com/paper/2512.04770/) (2025)
- [Bellman: A Toolbox for Model-Based Reinforcement Learning in TensorFlow](https://mlsanity.com/paper/2103.14407/) (2021)
- [Information-Theoretic Policy Pre-Training with Empowerment](https://mlsanity.com/paper/2510.05996/) (2025)
- [Unifying Goal-Conditioned RL and Unsupervised Skill Learning via Control-Maximization](https://mlsanity.com/paper/2605.06145/) (2026)
- [Variational Inference for Model-Free and Model-Based Reinforcement Learning](https://mlsanity.com/paper/2209.01693/) (2022)
- [Generative Intrinsic Optimization: Intrinsic Control with Model Learning](https://mlsanity.com/paper/2310.08100/) (2023)
- [Understanding Action Effects through Instrumental Empowerment in Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2508.15652/) (2025)
- [A Bellman Optimality Equation for Plasticity](https://mlsanity.com/paper/2609.10776/) (2026)

## Abstract (arXiv)

Empowerment is an information-theoretic method that can be used to intrinsically motivate learning agents. It attempts to maximize an agent's control over the environment by encouraging visiting states with a large number of reachable next states. Empowered learning has been shown to lead to complex behaviors, without requiring an explicit reward signal. In this paper, we investigate the use of empowerment in the presence of an extrinsic reward signal. We hypothesize that empowerment can guide reinforcement learning (RL) agents to find good early behavioral solutions by encouraging highly empowered states. We propose a unified Bellman optimality principle for empowered reward maximization. Our empowered reward maximization approach generalizes both Bellman's optimality principle as well as recent information-theoretical extensions to it. We prove uniqueness of the empowered values and show convergence to the optimal solution. We then apply this idea to develop off-policy actor-critic RL algorithms which we validate in high-dimensional continuous robotics domains (MuJoCo). Our methods demonstrate improved initial and competitive final performance compared to model-free state-of-the-art techniques.