1CS 188: Artificial IntelligenceFall 2009Lecture 12: Reinforcement Learning II10/6/2009Dan 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]42The Story So Far: MDPs and RL If we know the MDP Compute V*, Q*, π* exactly Evaluate a fixed policy π If we don’t know the MDP We can estimate the MDP then solve We can estimate V for a fixed policy π We can estimate Q*(s,a) for the optimal policy while executing an exploration policy5 Model-based DPs Value and policy Iteration Policy evaluation Model-based RL Model-free RL: Value learning Q-learningThings we know how to do:Techniques:Model-Free Learning Model-free (temporal difference) learning Experience world through episodes Update estimates each transition Over time, updates will mimic Bellman updates6ass, as’Q-Value Iteration (model-based, requires known MDP)Q-Learning (model-free, requires only experienced transitions)r3Q-Learning We’d like to do Q-value updates to each Q-state: But can’t compute this update without knowing T, R Instead, compute average as we go Receive a sample transition (s,a,r,s’) This sample suggests But we want to average over results from (s,a) (Why?) So keep a running average[DEMO – Grid Q’s]7Q-Learning Properties Will converge to optimal policy If you explore enough (i.e. visit each q-state many times) If you make the learning rate small enough Basically doesn’t matter how you select actions (!) Off-policy learning: learns optimal q-values, not the values of the policy you are followingS ES E[DEMO – Grid Q’s]84Exploration / 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 Regret: expected gap between rewards during learning and rewards from optimal action Q-learning with random actions will converge to optimal values, but possibly very slowly, and will get low rewards on the way Results will be optimal but regret will be large How to make regret small?9Exploration Functions When to explore Random actions: explore a fixed amount Better ideas: explore areas whose badness is not (yet) established, explore less over time One way: exploration function Takes a value estimate and a count, and returns an optimistic utility, e.g. (exact form not important)10[DEMO – Crawler]5Q-Learning Q-learning produces tables of q-values:[DEMO – Crawler Q’s]11Q-Learning In realistic situations, we cannot possibly learn about every single state! Too many states to visit them all in training Too many states to hold the q-tables in memory Instead, we want to generalize: Learn about some small number of training states from experience Generalize that experience to new, similar states This is a fundamental idea in machine learning, and we’ll see it over and over again126Example: Pacman Let’s say we discover through experience that this state is bad: In naïve q learning, we know nothing about this state or its q states: Or even this one!13[DEMO – RL Pacman]Feature-Based Representations Solution: describe a state using a vector of features (properties) Features are functions from states to real numbers (often 0/1) that capture important properties of the state Example features: Distance to closest ghost Distance to closest dot Number of ghosts 1 / (dist to dot)2 Is Pacman in a tunnel? (0/1) …… etc. Is it the exact state on this slide? Can also describe a q-state (s, a) with features (e.g. action moves closer to food)147Linear Feature Functions Using a feature representation, we can write a q function (or value function) for any state using a few weights: Advantage: our experience is summed up in a few powerful numbers Disadvantage: states may share features but actually be very different in value!15Function Approximation Q-learning with linear q-functions: Intuitive interpretation: Adjust weights of active features E.g. if something unexpectedly bad happens, disprefer all states with that state’s features Formal justification: online least squares16Exact Q’sApproximate Q’s8Example: Q-Pacman17[DEMO – RL Pacman]0 2002040010203040010203020222426Linear RegressionPrediction Prediction189Ordinary Least Squares (OLS)0 200Error or “residual”PredictionObservation19Minimizing ErrorApproximate q update explained:20Imagine we had only one point x with features f(x):“target” “prediction”100 2 4 6 8 10 12 14 16 18 20-15-10-5051015202530Degree 15 polynomialOverfittingPolicy Search2211Policy Search Problem: often the feature-based policies that work well aren’t the ones that approximate V / Q best E.g. your value functions from project 2 were probably horrible estimates of future rewards, but they still produced good decisions We’ll see this distinction between modeling and prediction again later in the course Solution: learn the policy that maximizes rewards rather than the value that predicts rewards This is the idea behind policy search, such as what controlled the upside-down helicopter23Policy Search Simplest policy search: Start with an initial linear value function or q-function Nudge each feature weight up and down and see if your policy is better than before Problems: How do we tell the policy got better? Need to run many sample episodes! If there are a lot of features, this can be impractical2412Policy Search* Advanced policy search: Write a stochastic (soft) policy: Turns out you can efficiently approximate the derivative of the returns with respect to the parameters w (details in the book, optional material) Take uphill steps, recalculate derivatives, etc.25Take a Deep Breath… We’re done with search and planning! Next, we’ll look at how to reason with probabilities Diagnosis
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