Reinforcement Learning Learning algorithms Yishay Mansour TelAviv University

Reinforcement Learning: Learning algorithms Yishay Mansour Tel-Aviv University

Outline • Last week – Goal of Reinforcement Learning – Mathematical Model (MDP) – Planning • Value iteration • Policy iteration • This week: Learning Algorithms – Model based – Model Free

Planning - Basic Problems. Given a complete MDP model. Policy evaluation - Given a policy p, estimate its return. Optimal control - Find an optimal policy p* (maximizes the return from any start state).

Planning - Value Functions Vp(s) The expected value starting at state s following p. Qp(s, a) The expected value starting at state s with action a and then following p. V*(s) and Q*(s, a) are define using an optimal policy p*. V*(s) = maxp Vp(s)

Algorithms - optimal control CLAIM: A policy p is optimal if and only if at each state s: Vp(s) = MAXa {Qp(s, a)} (Bellman Eq. ) The greedy policy with respect to Qp(s, a) is p(s) = argmaxa{Qp(s, a) }

MDP - computing optimal policy 1. Linear Programming 2. Value Iteration method. 3. Policy Iteration method.

Planning versus Learning Tightly coupled in Reinforcement Learning Goal: maximize return while learning.

Example - Elevator Control Learning (alone): Model the arrival model well. Planning (alone) : Given arrival model build schedule Real objective: Construct a schedule while updating model

Learning Algorithms Given access only to actions perform: 1. policy evaluation. 2. control - find optimal policy. Two approaches: 1. Model based (Dynamic Programming). 2. Model free (Q-Learning).

Learning - Model Based Estimate the model from the observation. (Both transition probability and rewards. ) Use the estimated model as the true model, and find optimal policy. If we have a “good” estimated model, we should have a “good” estimation.

Learning - Model Based: off policy • Let the policy run for a “long” time. – what is “long” ? ! • Build an “observed model”: – Transition probabilities – Rewards • Use the “observed model” to estimate value of the policy.

Learning - Model Based off-policy algorithm • Observe a trajectory generated by a policy π – off-policy: not need to control • For every s, s’ ε S and a ε A: – obsδ(s, a, s’)= #(s, a, s’) / #(s, a, *) – obs. R(s, a) = average(r | s, a) • Find an optimal policy for (S, a, obsδ, obs. R)

Learning - Model Based • Claim: if the model is “accurate” we will find a near-optimal policy. • Hidden assumption: – Each (s, a, *) is sampled many times. – This is the “responsibility” of the policy π • Simple question: How many samples we need for each (s, a, *)

Learning - Model Based sample size Sample size (optimal policy): Naive: O(|S| log (|S| |A|) ) samples per (s, a) (approximates each transition d(s, a, s’) well. ) Better: O(log (|S| |A|) ) samples per (s, a) (Sufficient to approximate optimal policy. ) [KS, NIPS’ 98]

Learning - Model Based: on policy • The learner has control over the action. – The immediate goal is to lean a model • As before: – Build an “observed model”: • Transition probabilities and Rewards – Use the “observed model” to estimate value of the policy. • Accelerating the learning: – How to reach “new” places ? !

Learning - Model Based: on policy Well sampled nodes Relatively unknown nodes

Learning: Policy improvement • Assume that we can perform: – Given a policy p, – Compute V and Q functions of p • Can run policy improvement: – p = Greedy (Q) • Process converges if estimates are accurate.

Learning: Monte Carlo Methods • Assume we can run in episodes – Terminating MDP – Discounted return • Simplest: sample the return of state s: – Wait to reach state s, – Compute the return from s, – Average all the returns.

Learning: Monte Carlo Methods • First visit: – For each state in the episode, – Compute the return from first occurrence – Average the returns • Every visit: – Might be biased! • Computing optimal policy: – Run policy iteration.

Learning - Model Free Policy evaluation: TD(0) An online view: At state st we performed action at, received reward rt and moved to state st+1. Our “estimation error” is Errt =rt+g. V(st+1)-V(st), The update: Vt +1(st) = Vt(st ) + a Errt Note that for the correct value function we have: E[r+g. V(s’)-V(s)] =E[Errt]=0

Learning - Model Free Optimal Control: off-policy Learn online the Q function. Qt+1 (st , at ) = Qt (st , at )+ a [ rt+g Vt (st+1) - Qt (st , at )] OFF POLICY: Q-Learning Any underlying policy selects actions. Assumes every state-action performed infinitely often Learning rate dependency. Convergence in the limit: GUARANTEED [DW, JJS, S, TS]

Learning - Model Free Optimal Control: on-policy Learn online the Q function. Qt+1 (st , at ) = Qt (st , at )+ a [ rt+g Qt (st+1, at+1) - Qt (st , at )] ON-Policy: SARSA at+1 the e-greedy policy for Qt. The policy selects the action! Need to balance exploration and exploitation. Convergence in the limit: GUARANTEED [DW, JJS, S, TS]

Learning - Model Free Policy evaluation: TD( ) Again: At state st we performed action at, received reward rt and moved to state st+1. Our “estimation error” Errt =rt+g. V(st+1)-V(st), Update every state s: Vt +1(s) = Vt(s ) + a Errt e(s) Update of e(s) : When visiting s: incremented by 1: e(s) = e(s)+1 For all other s’: decremented by g factor: e(s’) = g e(s’)

Summary Markov Decision Process: Mathematical Model. Planning Algorithms. Learning Algorithms: Model Based Monte Carlo TD(0) Q-Learning SARSA TD( )