# Online Markov decision processes with Kullback-Leibler control cost

arXiv: https://arxiv.org/abs/1401.3198 · HTML: https://mlsanity.com/paper/1401.3198/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jan 2014.

## Built on by (most cited first, 7 of 7)

- [On the Properties of Kullback-Leibler Divergence Between Multivariate Gaussian Distributions](https://mlsanity.com/paper/2102.05485/) (2021)
- [Blending Data-Driven Priors in Dynamic Games](https://mlsanity.com/paper/2402.14174/) (2024)
- [An Energy-Based Mechanism for Compositional Behavior](https://mlsanity.com/paper/2512.04745/) (2025)
- [On Convex Data-Driven Inverse Optimal Control for Nonlinear, Non-stationary and Stochastic Systems](https://mlsanity.com/paper/2306.13928/) (2023)
- [Distributed Algorithms for Linearly-Solvable Optimal Control in Networked Multi-Agent Systems](https://mlsanity.com/paper/2102.09104/) (2021)
- [Discrete fully probabilistic design: towards a control pipeline for the synthesis of policies from examples](https://mlsanity.com/paper/2112.11210/) (2021)
- [Optimal Stability of KL Divergence under Gaussian Perturbations](https://mlsanity.com/paper/2604.11026/) (2026)

## Abstract (arXiv)

This paper considers an online (real-time) control problem that involves an agent performing a discrete-time random walk over a finite state space. The agent's action at each time step is to specify the probability distribution for the next state given the current state. Following the set-up of Todorov, the state-action cost at each time step is a sum of a state cost and a control cost given by the Kullback-Leibler (KL) divergence between the agent's next-state distribution and that determined by some fixed passive dynamics. The online aspect of the problem is due to the fact that the state cost functions are generated by a dynamic environment, and the agent learns the current state cost only after selecting an action. An explicit construction of a computationally efficient strategy with small regret (i.e., expected difference between its actual total cost and the smallest cost attainable using noncausal knowledge of the state costs) under mild regularity conditions is presented, along with a demonstration of the performance of the proposed strategy on a simulated target tracking problem. A number of new results on Markov decision processes with KL control cost are also obtained.