1CS 188: Artificial IntelligenceFall 2010Lecture 11: Reinforcement Learning9/30/2010Dan Klein – UC BerkeleyMany slides over the course adapted from either Stuart Russell or Andrew Moore1Reinforcement Learning Reinforcement learning: Still assume an MDP: A set of states s ∈ S A set of actions (per state) A A model T(s,a,s’) A reward function R(s,a,s’) Still looking for a policy π(s) New twist: don’t know T or R I.e. don’t know which states are good or what the actions do Must actually try actions and states out to learn[DEMO]22Passive Learning Simplified task You don’t know the transitions T(s,a,s’) You don’t know the rewards R(s,a,s’) You are given a policy π(s) Goal: learn the state values … what policy evaluation did In this case: Learner “along for the ride” No choice about what actions to take Just execute the policy and learn from experience We’ll get to the active case soon This is NOT offline planning! You actually take actions in the world and see what happens…3Example: Direct Evaluation Episodes:xy(1,1) up -1(1,2) up -1(1,2) up -1(1,3) right -1(2,3) right -1(3,3) right -1(3,2) up -1(3,3) right -1(4,3) exit +100(done)(1,1) up -1(1,2) up -1(1,3) right -1(2,3) right -1(3,3) right -1(3,2) up -1(4,2) exit -100(done)V(2,3) ~ (96 + -103) / 2 = -3.5V(3,3) ~ (99 + 97 + -102) / 3 = 31.3γ = 1, R = -1+100-1004[DEMO – Optimal Policy]3Recap: Model-Based Policy Evaluation Simplified Bellman updates to calculate V for a fixed policy: New V is expected one-step-look-ahead using current V Unfortunately, need T and R5π(s)ss, π(s)s, π(s),s’s’Model-Based Learning Idea: Learn the model empirically through experience Solve for values as if the learned model were correct Simple empirical model learning Count outcomes for each s,a Normalize to give estimate of T(s,a,s’) Discover R(s,a,s’) when we experience (s,a,s’) Solving the MDP with the learned model Iterative policy evaluation, for example6π(s)ss, π(s)s, π(s),s’s’4Example: Model-Based Learning Episodes:xyT(<3,3>, right, <4,3>) = 1 / 3T(<2,3>, right, <3,3>) = 2 / 2+100-100γ = 1(1,1) up -1(1,2) up -1(1,2) up -1(1,3) right -1(2,3) right -1(3,3) right -1(3,2) up -1(3,3) right -1(4,3) exit +100(done)(1,1) up -1(1,2) up -1(1,3) right -1(2,3) right -1(3,3) right -1(3,2) up -1(4,2) exit -100 (done)7Model-Free Learning Want to compute an expectation weighted by P(x): Model-based: estimate P(x) from samples, compute expectation Model-free: estimate expectation directly from samples Why does this work? Because samples appear with the right frequencies!85Sample-Based Policy Evaluation? Who needs T and R? Approximate the expectation with samples (drawn from T!)9π(s)ss, π(s)s1’s2’s3’s, π(s),s’s’Almost! But we only actually make progress when we move to i+1.Temporal-Difference Learning Big idea: learn from every experience! Update V(s) each time we experience (s,a,s’,r) Likely s’ will contribute updates more often Temporal difference learning Policy still fixed! Move values toward value of whatever successor occurs: running average!10π(s)ss, π(s)s’Sample of V(s):Update to V(s):Same update:6Exponential Moving Average Exponential moving average Makes recent samples more important Forgets about the past (distant past values were wrong anyway) Easy to compute from the running average Decreasing learning rate can give converging averages11Example: TD Policy EvaluationTake γ = 1, α = 0.5(1,1) up -1(1,2) up -1(1,2) up -1(1,3) right -1(2,3) right -1(3,3) right -1(3,2) up -1(3,3) right -1(4,3) exit +100(done)(1,1) up -1(1,2) up -1(1,3) right -1(2,3) right -1(3,3) right -1(3,2) up -1(4,2) exit -100(done)12[DEMO – Grid V’s]7Problems with TD Value Learning TD value leaning is a model-free way to do policy evaluation However, if we want to turn values into a (new) policy, we’re sunk: Idea: learn Q-values directly Makes action selection model-free too!ass, as,a,s’s’13Active Learning Full reinforcement learning You don’t know the transitions T(s,a,s’) You don’t know the rewards R(s,a,s’) You can choose any actions you like Goal: learn the optimal policy … what value iteration did! In this case: Learner makes choices! Fundamental tradeoff: exploration vs. exploitation This is NOT offline planning! You actually take actions in the world and find out what happens…148Detour: Q-Value Iteration Value iteration: find successive approx optimal values Start with V0*(s) = 0, which we know is right (why?) Given Vi*, calculate the values for all states for depth i+1: But Q-values are more useful! Start with Q0*(s,a) = 0, which we know is right (why?) Given Qi*, calculate the q-values for all q-states for depth i+1:15Q-Learning Q-Learning: sample-based Q-value iteration Learn Q*(s,a) values Receive a sample (s,a,s’,r) Consider your old estimate: Consider your new sample estimate: Incorporate the new estimate into a running average:[DEMO – Grid Q’s]169Q-Learning Properties Amazing result: Q-learning converges to optimal policy If you explore enough If you make the learning rate small enough … but not decrease it too quickly! Basically doesn’t matter how you select actions (!) Neat property: off-policy learning learn optimal policy without following it (some caveats)S ES E[DEMO – Grid Q’s]17Exploration / Exploitation Several schemes for forcing exploration Simplest: random actions (ε greedy) Every time step, flip a coin With probability ε, act randomly With probability 1-ε, act according to current policy Problems with random actions? You do explore the space, but keep thrashing around once learning is done One solution: lower ε over time Another solution: exploration functions1810Exploration Functions When to explore Random actions: explore a fixed amount Better idea: explore areas whose badness is not (yet) established Exploration function Takes a value estimate and a count, and returns an optimistic utility, e.g. (exact form not important)19[DEMO – Auto Grid Q’s]Q-Learning Q-learning produces tables of q-values:[DEMO – Crawler
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