[Paper Review] Universal Reinforcement Learning

Vivek F. Farias, Ciamac C. Moallemi+2 more|ArXiv.org|Jul 20, 2007

Computability, Logic, AI Algorithms15 references4 citations

TL;DR

This paper proposes the active LZ algorithm, a universal reinforcement learning method inspired by Lempel-Ziv data compression, for optimal control in unknown environments with finite memory dependence. It guarantees asymptotic average cost optimality when future outcomes depend on a bounded history of actions and observations, using context-tree modeling and adaptive exploration to converge to the optimal policy without prior knowledge of the system's structure or memory length.

ABSTRACT

We consider an agent interacting with an unmodeled environment. At each time, the agent makes an observation, takes an action, and incurs a cost. Its actions can influence future observations and costs. The goal is to minimize the long-term average cost. We propose a novel algorithm, known as the active LZ algorithm, for optimal control based on ideas from the Lempel-Ziv scheme for universal data compression and prediction. We establish that, under the active LZ algorithm, if there exists an integer $K$ such that the future is conditionally independent of the past given a window of $K$ consecutive actions and observations, then the average cost converges to the optimum. Experimental results involving the game of Rock-Paper-Scissors illustrate merits of the algorithm.

Motivation & Objective

• To develop a reinforcement learning algorithm that achieves optimal long-term average cost in environments with unknown, finite-memory dependencies.
• To eliminate the need for prior knowledge of the system's transition kernel or memory length K.
• To combine principles from universal data compression (Lempel-Ziv) and dynamic programming for model-free optimal control.
• To establish theoretical guarantees on convergence to the optimal average cost under weak assumptions.
• To enable learning in partially observed or non-Markovian environments with bounded historical dependence.

Proposed method

• The active LZ algorithm uses a Lempel-Ziv-inspired context tree to model the conditional distribution of future observations given past actions and observations.
• It maintains estimates of transition probabilities using a growing tree structure that adapts to observed sequences of actions and observations.
• The algorithm performs adaptive exploration by balancing exploitation of estimated optimal actions with exploration of less-visited contexts in the tree.
• It employs a discounted value iteration framework with a time-varying discount factor α approaching 1, ensuring long-term optimality.
• A doubling-trick scheme is used to achieve average cost optimality without prior knowledge of α, by successively increasing the time horizon and adjusting exploration rates.
• The method leverages the Borel-Cantelli lemma and concentration bounds to show that suboptimal actions are taken only finitely often with high probability.

Experimental results

Research questions

• RQ1 Can a reinforcement learning algorithm achieve asymptotic average cost optimality without prior knowledge of the system's transition kernel or memory length?
• RQ2 How can universal data compression principles be adapted to solve optimal control problems in unknown environments?
• RQ3 What conditions on the environment’s memory structure ensure convergence to the optimal policy?
• RQ4 Can adaptive exploration strategies based on context trees guarantee long-term optimality in reinforcement learning?
• RQ5 What is the rate of convergence of the fraction of suboptimal actions to zero under the proposed algorithm?

Key findings

• The active LZ algorithm guarantees asymptotic average cost optimality when the environment’s future is conditionally independent of the past given a finite window of K consecutive actions and observations.
• The fraction of time the algorithm selects suboptimal actions converges to zero at a rate of O((1/log T)^c) for some c < 1, indicating a slow but guaranteed convergence.
• The algorithm achieves optimality without prior knowledge of the discount factor α or the memory length K, using a doubling-trick scheme with βk = Ω(1/log log k).
• Theoretical analysis shows that, with high probability, the greedy action is suboptimal for at most O(2^k * (log log 2^k)^3 / log 2^k) time steps in the k-th epoch, ensuring long-term optimality.
• The method is robust to unknown system structure and can be applied to problems such as Rock-Paper-Scissors with finite-memory opponents and joint source-channel coding with fixed decoders.
• The framework suggests potential improvements by replacing the LZ-based context tree with the context-tree weighting method, which may yield faster convergence rates.