Bayesian Reinforcement Learning with Gaussian Processes Huanren Zhang Electrical and Computer Engineering Purdue University

Outline Introduction to Reinforcement Learning (RL) Markov Decision Processes (MDPs) Traditional RL Solution Methods Gaussian Processes (GPs) Gaussian Process Temporal Difference (GPTD) Experiment Conclusion

Reinforcement Learning (RL) An agent interacts with the environment and learns how to map situations to actions in order to maximize the reward. Involves sequences of decision Almost all Artificial Intelligence (AI) problems can be formulated as RL problems

Reinforcement Learning (RL) Evaluative feedback (reward or reinforcement)  Indicates how good the action taken is, but not whether it is correct or wrong. Balance between exploration and exploitation  Exploitation— make most of the current information that has already got  Exploration — explore the unknown states that may cause higher return in the long run. Online learning

Markov Decision Processes (MDPs) RL problems can be formulated as Markov Decision Processes (MDPs) An MDP is a tuple  State space: S  Action space: A  Reward function  State transition function represents the probability of making a transition from state s to state s’ taking action a By a model, we mean the reward function and state-transition function.

Maze world problem

Traditional RL Solution Methods Dynamic Programming (DP) Monte Carlo (MC) Methods Temporal Difference (TD) Methods  All the methods are based on estimate of value function under certain policy  The value of a state is the total amount of reward an agent can expect to accumulate starting from that state

Maze world problem The value for the states that are near the goal should be greater than those far away from the goal.

Temporal Difference (TD) Methods Learn directly from experience Bootstrap: update estimate based on other learned estimate Do Not need a model Updating rule:  δ t : the temporal difference  α t : time dependent learning rate  γ: discounting rate

TD Method (With Optimistic Policy Iteration)

Policy Learned by TD method (After 100 trails)

Gaussian Processes (GPs) A Bayesian approach: provides full posterior over values, not just point estimates Forces us to make our assumptions explicit Non-parametric – priors are placed and inference is performed directly in function space (kernels) Domain knowledge intuitively coded in priors

Gaussian Processes (GPs) “An indexed set of jointly Gaussian random variables" The index set X may be just about any set. For a Gaussian Process F, Kernel function k(x,x’) is symmetric positive definite (Mercer kernel).

Conditioning Theorem

GP regression

GPTD Methods Generative model for the reward of the trajectory s 1,, s 2,…,s t, In compact form

GPTD Methods Conditioning theorem where

Can we use the uncertainty information to help balance exploitation and exploration? New value function (improved GPTD): Parameter c balances the importance between exploitation and exploration. Information theory – higher uncertainty means more information. Visiting states with higher uncertainty gives higher information gain – another kind of value for a state

Experiment Gaussian kernel is used:  ||s-s’|| represents the Euclidean distance between two states s and s’ : adjacent state will have similar value function in maze problem. Use Optimistic Policy Iteration (OPI) to determine policy  Take actions that lead to the highest expected returned based on current value estimate.

GPTD Method

Policy Learned by GPTD

GPTD for Multi-goal Maze

Policy Learned by GPTD

Improved GPTD

Policy Learned by Improved GPTD

Conclusion Gaussian process gives how certainty the estimate is along with the estimate. GPTD will give much better results than traditional RL methods The main contribution of the project is the proposal of one way to utilize the uncertainty to balance exploration and exploitation in RL; and the experiment shows its effectiveness

Reference Bishop, C. M Pattern Recognition and Machine Learning. Secaucus, NJ, USA: Springer-Verlag New York, Inc. Engel, Y.; Mannor, S.; and Meir, R Bayesian meets bellman: The gaussian process approach to temporal difference learning. International Conference on Machine Learning. Engel, Y.; Mannor, S.; and Meir, R Reinforcement learning with gaussian process. International Conference on Machine Learning. Engel, Y Algorithms and Representations for Reinforcement Learning. Ph.D. Dissertation, The Hebrew University of Jerusalem, Israel. Puterman, M. L Markov Decision Process: Discrete Stochastic Dynamic Programming. Wiley-Interscience, New York, NY. Russell, S. J., and Norvig, P Artificial Intelligence: A Modern Approach (2nd Edition). Prentice Hall. Sutton, R. S., and Barto, A. G Reinforcement Learning: An Introduction. The MIT Press, Cambridge, MA.

Questions? Bayesian Reinforcement Learning with Gaussian Processes