Reinforcement Learning Known Unknown Assumed Current state Available actions Experienced rewards Transition model Reward structure Markov transitions Fixed reward for (s,a,s’) Problem: Find values for fixed policy  (policy evaluation) Model-based learning: Learn the model, solve for values Model-free learning: Solve for values directly (by sampling)

The Story So Far: MDPs and RL Things we know how to do: Techniques: If we know the MDP Compute V*, Q*, * exactly Evaluate a fixed policy  If we don’t know the MDP We can estimate the MDP then solve We can estimate V for a fixed policy  We can estimate Q*(s,a) for the optimal policy while executing an exploration policy Model-based DPs Value and policy Iteration Policy evaluation Model-based RL Model-free RL: Value learning Q-learning

Recap: Optimal Utilities The utility of a state s: V*(s) = expected utility starting in s and acting optimally The utility of a q-state (s,a): Q*(s,a) = expected utility starting in s, taking action a and thereafter acting optimally The optimal policy: *(s) = optimal action from state s s is a state s a (s, a) is a q-state s, a (s,a,s’) is a transition s,a,s’ s’

Model-Free Learning Model-free (temporal difference) learning s Experience world through episodes Update estimates each transition Over time, updates will mimic Bellman updates a s, a r s’ Q-Value Iteration (model-based, requires known MDP) Q-Learning (model-free, requires only experienced transitions)

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! s (s) s, (s) s’ Sample of V(s): Update to V(s): Same update:

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 averages

Example: TD Policy Evaluation (1,1) up -1 (1,2) up -1 (1,3) right -1 (2,3) right -1 (3,3) right -1 (3,2) up -1 (4,3) 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) Take  = 1,  = 0.5

Problems 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! a s s, a s,a,s’ s’

Active Learning Full reinforcement learning In this case: 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…

Model-Based Active Learning In general, want to learn the optimal policy, not evaluate a fixed policy Idea: adaptive dynamic programming Learn an initial model of the environment: Solve for the optimal policy for this model (value or policy iteration) Refine model through experience and repeat Crucial: we have to make sure we actually learn about all of the model

Example: Greedy ADP Imagine we find the lower path to the good exit first Some states will never be visited following this policy from (1,1) We’ll keep re-using this policy because following it never collects the regions of the model we need to learn the optimal policy ? ?

What Went Wrong? Problem with following optimal policy for current model: Never learn about better regions of the space if current policy neglects them Fundamental tradeoff: exploration vs. exploitation Exploration: must take actions with suboptimal estimates to discover new rewards and increase eventual utility Exploitation: once the true optimal policy is learned, exploration reduces utility Systems must explore in the beginning and exploit in the limit ? ?

Detour: 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:

Q-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

Q-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 following S E S E

Exploration / Exploitation Several schemes for forcing exploration Simplest: random actions ( greedy) Every time step, flip a coin With probability , act randomly With probability 1-, 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?

Exploration 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)

Q-Learning Q-learning produces tables of q-values:

Q-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 again

Example: 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!

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)

Linear 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!

Function 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 squares Exact Q’s Approximate Q’s

Example: Q-Pacman

Linear Regression Prediction Prediction 10 20 30 40 22 24 26 40 20 20 Figure 1: scatter(1:20,10+(1:20)+2*randn(1,20),'k','filled'); a=axis; a(3)=0; axis(a); Prediction Prediction

Ordinary Least Squares (OLS) Error or “residual” Observation Prediction Figure 1: scatter(1:20,10+(1:20)+2*randn(1,20),'k','filled'); a=axis; a(3)=0; axis(a); 20

Minimizing Error Imagine we had only one point x with features f(x): Approximate q update explained: “target” “prediction”

Overfitting Degree 15 polynomial 30 25 20 15 10 5 -5 -10 -15 2 4 6 8 -5 -10 -15 2 4 6 8 10 12 14 16 18 20

Policy Search

Policy 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 helicopter

Policy Search Simplest policy search: Problems: 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 impractical

Policy 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.

Take a Deep Breath… We’re done with search and planning! Next, we’ll look at how to reason with probabilities Diagnosis Tracking objects Speech recognition Robot mapping … lots more! Last part of course: machine learning