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##### Reinforcement Learning

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**Reinforcement Learning**• Control learning • Control polices that choose optimal actions • Q learning • Convergence Chapter 13 Reinforcement Learning**Control Learning**Consider learning to choose actions, e.g., • Robot learning to dock on battery charger • Learning to choose actions to optimize factory output • Learning to play Backgammon Note several problem characteristics • Delayed reward • Opportunity for active exploration • Possibility that state only partially observable • Possible need to learn multiple tasks with same sensors/effectors Chapter 13 Reinforcement Learning**One Example: TD-Gammon**Tesauro, 1995 Learn to play Backgammon Immediate reward • +100 if win • -100 if lose • 0 for all other states Trained by playing 1.5 million games against itself Now approximately equal to best human player Chapter 13 Reinforcement Learning**a0**a1 a2 s0 s1 s2 r1 r2 r0 Reinforcement Learning Problem Environment action state reward Agent ... Goal: learn to choose actions that maximize r0 + r1 + 2r2 + …, where 0 < 1 Chapter 13 Reinforcement Learning**Markov Decision Process**Assume • finite set of states S • set of actions A • at each discrete time, agent observes state st S and choose action at A • then receives immediate reward rt • and state changes to st+1 • Markov assumption: st+1 = (st, at) and rt = r(st, at) • i.e., rt and st+1 depend only on current state and action • functions and r may be nondeterministic • functions and r no necessarily known to agent Chapter 13 Reinforcement Learning**Agent’s Learning Task**Execute action in environment, observe results, and • learn action policy : SA that maximizes E[rt + rt+1 + 2rt+2 + …] from any starting state in S • here 0 < 1 is the discount factor for future rewards Note something new: • target function is : SA • but we have no training examples of form <s,a> • training examples are of form <<s,a>,r> Chapter 13 Reinforcement Learning**Value Function**To begin, consider deterministic worlds … For each possible policy the agent might adopt, we can define an evaluation function over states where rt,rt+1,… are generated by following policy starting at state s Restated, the task is to learn the optimal policy * Chapter 13 Reinforcement Learning**0**0 0 81 G 100 G 100 0 0 0 72 90 81 0 0 0 0 100 81 90 72 81 100 0 0 81 90 r(s,a) (immediate reward) values Q(s,a) values G G 90 100 0 81 90 100 One optimal policy V*(s) values Chapter 13 Reinforcement Learning**What to Learn**We might try to have agent learn the evaluation function V* (which we write as V*) We could then do a lookahead search to choose best action from any state s because A problem: • This works well if agent knows a : SAS, and r :SA • But when it doesn’t, we can’t choose actions this way Chapter 13 Reinforcement Learning**Q Function**Define new function very similar to V* If agent learns Q, it can choose optimal action even without knowing d! Q is the evaluation function the agent will learn Chapter 13 Reinforcement Learning**Training Rule to Learn Q**Note Q and V* closely related: Which allows us to write Q recursively as Let denote learner’s current approximation to Q. Consider training rule where s' is the state resulting from applying action a in state s Chapter 13 Reinforcement Learning**Q Learning for Deterministic Worlds**For each s,a initialize table entry Observe current state s Do forever: • Select an action a and execute it • Receive immediate reward r • Observe the new state s' • Update the table entry for as follows: • ss' Chapter 13 Reinforcement Learning**63**63 R R 100 100 90 72 81 81 aright next state: s2 initial state: s1 Updating Chapter 13 Reinforcement Learning**Convergence**converges to Q. Consider case of deterministic world where each <s,a> visited infinitely often. Proof: define a full interval to be an interval during which each <s,a> is visited. During each full interval the largest error in table is reduced by factor of Let be table after n updates, and n be the maximum error in ; that is Chapter 13 Reinforcement Learning**Convergence (cont)**For any table entry updated on iteration n+1, the error in the revised estimate is Chapter 13 Reinforcement Learning**Nondeterministic Case**What if reward and next state are non-deterministic? We redefine V,Q by taking expected values Chapter 13 Reinforcement Learning**Nondeterministic Case**Q learning generalizes to nondeterministic worlds Alter training rule to where Can still prove converge of to Q [Watkins and Dayan, 1992] Chapter 13 Reinforcement Learning**Temporal Difference Learning**Q learning: reduce discrepancy between successive Q estimates One step time difference: Why not two steps? Or n? Blend all of these: Chapter 13 Reinforcement Learning**Temporal Difference Learning**Equivalent expression: TD() algorithm uses above training rule • Sometimes converges faster than Q learning • converges for learning V* for any 0 1 (Dayan, 1992) • Tesauro’s TD-Gammon uses this algorithm Chapter 13 Reinforcement Learning**Subtleties and Ongoing Research**• Replace table with neural network or other generalizer • Handle case where state only partially observable • Design optimal exploration strategies • Extend to continuous action, state • Learn and use d : SAS, d approximation to • Relationship to dynamic programming Chapter 13 Reinforcement Learning**RL Summary**• Reinforcement learning (RL) • control learning • delayed reward • possible that the state is only partially observable • possible that the relationship between states/actions unknown • Temporal Difference Learning • learn discrepancies between successive estimates • used in TD-Gammon • V(s) - state value function • needs known reward/state transition functions Chapter 13 Reinforcement Learning**RL Summary**• Q(s,a) - state/action value function • related to V • does not need reward/state trans functions • training rule • related to dynamic programming • measure actual reward received for action and future value using current Q function • deterministic - replace existing estimate • nondeterministic - move table estimate towards measure estimate • convergence - can be shown in both cases Chapter 13 Reinforcement Learning